Volume 57, Issue 1

In this article, we deal with the following p p -fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity: M ( [ u ] s , A p ) ( − Δ ) p , A s u + V ( x ) ∣ u ∣ p − 2 u = λ ∫ R N ∣ u ∣ p μ , s * ∣ x − y ∣ μ d y ∣ u ∣ p μ , s * − 2 u + k ∣ u ∣ q − 2 u , x ∈ R N , M({\left[u]}_{s,A}^{p}){\left(-\Delta )}_{p,A}^{s}u+V\left(x){| u| }^{p-2}u=\lambda \left(\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{{| u| }^{{p}_{\mu ,s}^{* }}}{{| x-y| }^{\mu }}{\rm{d}}y\right){| u| }^{{p}_{\mu ,s}^{* }-2}u+k{| u| }^{q-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N}, where 0 < s < 1 < p 0\lt s\lt 1\lt p , p s < N ps\lt N , p < q < 2 p s , μ * p\lt q\lt 2{p}_{s,\mu }^{* } , 0 < μ < N 0\lt \mu \lt N , λ \lambda , and k k are some positive parameters, p s , μ * = p N − p μ 2 N − p s {p}_{s,\mu }^{* }=\frac{pN-p\frac{\mu }{2}}{N-ps} is the critical exponent with respect to the Hardy-Littlewood-Sobolev inequality, and functions V V and M M satisfy the suitable conditions. By proving the compactness results using the fractional version of concentration compactness principle, we establish the existence of nontrivial solutions to this problem.

Jleli and Samet in [ On a new generalization of metric spaces , J. Fixed Point Theory Appl. 20 (2018), 128 (20 pages)] introduced the notion of ℱ -metric space as a generalization of traditional metric space and proved Banach contraction principle in the setting of this generalized metric space. The objective of this article is to use ℱ -metric space and establish some common fixed point theorems for ( β \beta - ψ \psi )-contractions. Our results expand, generalize, and consolidate several known results in the literature. As applications of the main result, the solution for fuzzy initial-value problems in the background of a generalized Hukuhara derivative was discussed.

Riesz-Caputo fractional derivative refers to a fractional derivative that reflects both the past and the future memory effects. This study gives sufficient conditions for the existence of solutions for a coupled system of fractional order hybrid differential equations involving the Riesz-Caputo fractional derivative. For this motive, the results are obtained via classical results due to Dhage.

This article introduces new nonparametric statistical methods for prediction in case of data containing right-censored observations and left-censored observations simultaneously. The methods can be considered as new versions of Hill’s A ( n ) {A}_{\left(n)} assumption for double-censored data. Two bounds are derived to predict the survival function for one future observation X n + 1 {X}_{n+1} based on each version, and these bounds are compared through two examples. Two interesting features are provided based on the proposed methods. The first one is the detailed graphical presentation of the effects of right and left censoring. The second feature is that the lower and upper survival functions can be derived.

Generally, fractional partial integro-differential equations (FPIDEs) play a vital role in modeling various complex phenomena. Because of the several applications of FPIDEs in applied sciences, mathematicians have taken a keen interest in developing and utilizing the various techniques for its solutions. In this context, the exact and analytical solutions are not very easy to investigate the solution of FPIDEs. In this article, a novel analytical approach that is known as the Laplace adomian decomposition method is implemented to calculate the solutions of FPIDEs. We obtain the approximate solution of the nonlinear FPIDEs. The results are discussed using graphs and tables. The graphs and tables have shown the greater accuracy of the suggested method compared to the extended cubic-B splice method. The accuracy of the suggested method is higher at all fractional orders of the derivatives. A sufficient degree of accuracy is achieved with fewer calculations with a simple procedure. The presented method requires no parametrization or discretization and, therefore, can be extended for the solutions of other nonlinear FPIDEs and their systems.

In this short article, we prove the existence of projected solutions to a class of quasi-variational hemivariational inequalities with non-self-constrained mapping, which generalizes the results of Allevi et al. ( Quasi-variational problems with non-self map on Banach spaces: Existence and applications , Nonlinear Anal. Real World Appl. 67 (2022), 103641, DOI: https://doi.org/10.1016/j.nonrwa.2022.103641.)

Spectral collocation method, named linear barycentric rational interpolation collocation method (LBRICM), for convection-diffusion (C-D) equation with constant coefficient is considered. We change the discrete linear equations into the matrix equation. Different from the classical methods to solve the C-D equation, we solve the C-D equation with the time variable and space variable obtained at the same time. Furthermore, the convergence rate of the C-D equation by LBRICM is proved. Numerical examples are presented to test our analysis.

In this study, based on two new local fractional integral operators involving generalized Mittag-Leffler kernel, Hermite-Hadamard inequality about these two integral operators for generalized h h -preinvex functions is obtained. Subsequently, an integral identity related to these two local fractional integral operators is constructed to obtain some new Ostrowski-type local fractional integral inequalities for generalized h h -preinvex functions. Finally, we propose three examples to illustrate the partial results and applications. Meanwhile, we also propose two midpoint-type inequalities involving generalized moments of continuous random variables to show the application of the results.

In this article, Sturm-Liouville problem with one boundary condition including an eigenparameter is considered, and the asymptotic expansion of its eigenparameter is calculated. The problem also has a symmetric single-well potential, which is an important function in quantum mechanics.

In this article, we analyze a rate of attraction of poles of an approximated function to poles of incomplete multipoint Padé approximants and use it to derive a sharp bound on the geometric rate of convergence of multipoint Hermite-Padé approximants to a vector of approximated functions in the Montessus de Ballore theorem when a table of interpolation points is Newtonian.

We calculate the essential norm of bounded diagonal infinite matrices acting on Köthe sequence spaces. As a consequence of our result, we obtain a recent criteria for the compactness of multiplication operator acting on Köthe sequence spaces.

In this study, we address a Cauchy problem within the context of the one-dimensional Timoshenko system, incorporating a distributed delay term. The heat conduction aspect is described by the Lord-Shulman theory. Our demonstration establishes that the dissipation resulting from the coupling of the Timoshenko system with Lord-Shulman’s heat conduction is sufficiently robust to stabilize the system, albeit with a gradual decay rate. To support our findings, we convert the system into a first-order form and, utilizing the energy method in Fourier space, and derive point wise estimates for the Fourier transform of the solution. These estimates, in turn, provide evidence for the slow decay of the solution.

This study is devoted to designing two hybrid computational algorithms to find approximate solutions for a class of singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters. In our approaches, the time discretization is first performed by the well-known Rothe method and Taylor series procedures, which reduce the underlying model problem into a sequence of boundary value problems (BVPs). Hence, a matrix collocation technique based on novel shifted Delannoy functions (SDFs) is employed to solve each BVP at each time step. We show that our proposed hybrid approximate techniques are uniformly convergent in order O ( Δ τ s + M − 1 2 ) {\mathcal{O}}\left(\Delta {\tau }^{s}+{M}^{-\tfrac{1}{2}}) for s = 1 , 2 s=1,2 , where Δ τ \Delta \tau is the time step and M M is the number of SDFs used in the approximation. Numerical simulations are performed to clarify the good alignment between numerical and theoretical findings. The computational results are more accurate as compared with those of existing numerical values in the literature.

In this article, we study the split variational inclusion and fixed point problems using Bregman weak relatively nonexpansive mappings in the p p -uniformly convex smooth Banach spaces. We introduce an inertial shrinking projection self-adaptive iterative scheme for the problem and prove a strong convergence theorem for the sequences generated by our iterative scheme under some mild conditions in real p p -uniformly convex smooth Banach spaces. The algorithm is designed to select its step size self-adaptively and does not require the prior estimate of the norm of the bounded linear operator. Finally, we provide some numerical examples to illustrate the performance of our proposed scheme and compare it with other methods in the literature.

A T T -symmetric univalent function is a complex valued function that is conformally mapping the unit disk onto itself and satisfies the symmetry condition ϕ [ T ] ( ζ ) = [ ϕ ( ζ T ) ] 1 ∕ T {\phi }^{\left[T]}\left(\zeta )={\left[\phi \left({\zeta }^{T})]}^{1/T} for all ζ \zeta in the unit disk. In other words, it is a complex function that preserves the unit disk’s shape and orientation and is symmetric about the unit circle. They are used in the study of geometric function theory and the theory of univalent functions. In recent effort, we extend the class of fractional anomalous diffusion equations in a symmetric complex domain. we aim to present the analytic univalent solution for such a class using special functions technique. Our analysis and comparative findings are further supported by the geometric simulations for the univalent solution such as the convexity and starlikeness of the diffusion. As a consequence of illustration of a list of conditions yielding the univalent solutions (normalize analytic function in the open unit disk), the normalization of diffusion shape is achieved.

This study investigates the existence of solutions for implicit fractional differential equations with fractional-order integral boundary conditions. We create the required conditions to ensure unique solution and Ulam-Hyers-Rassias stability. We also give examples to highlight the major findings.

In this study, we define new class of holomorphic functions associated with tangent function. Furthermore, we examine the differential subordination implementation results related to Janowski and tangent functions. Also, we investigate some extreme point theorem and partial sums results, necessary and sufficient conditions, convex combination, closure theorem, growth and distortion bounds, and radii of close-to-starlikeness and starlikeness for this newly defined functions class of holomorphic functions.

In this article, we consider the following Choquard equation with upper critical exponent: − Δ u = μ f ( x ) ∣ u ∣ p − 2 u + g ( x ) ( I α * ( g ∣ u ∣ 2 α * ) ) ∣ u ∣ 2 α * − 2 u , x ∈ Ω , -\Delta u=\mu f\left(x){| u| }^{p-2}u+g\left(x)({I}_{\alpha }* \left(g{| u| }^{{2}_{\alpha }^{* }})){| u| }^{{2}_{\alpha }^{* }-2}u,\hspace{1.0em}x\in \Omega , where μ > 0 \mu \gt 0 is a parameter, N > 4 N\gt 4 , 0 < α < N 0\lt \alpha \lt N , I α {I}_{\alpha } is the Riesz potential, N N − 2 < p < 2 \frac{N}{N-2}\lt p\lt 2 , Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} is a bounded domain with smooth boundary, and f f and g g are continuous functions. For μ \mu small enough, using variational methods, we establish the relationship between the number of solutions and the profile of potential g g .

This study deals with the one-parameter family { D q } q ∈ [ 0 , 1 ] {\left\{{D}_{q}\right\}}_{q\in \left[0,1]} of Bernstein-type operators introduced by Gupta and called the limit q q -Durrmeyer operators. The continuity of this family with respect to the parameter q q is examined in two most important topologies of the operator theory, namely, the strong and uniform operator topologies. It is proved that { D q } q ∈ [ 0 , 1 ] {\left\{{D}_{q}\right\}}_{q\in \left[0,1]} is continuous in the strong operator topology for all q ∈ [ 0 , 1 ] q\in \left[0,1] . When it comes to the uniform operator topology, the continuity is preserved solely at q = 0 q=0 and fails at all q ∈ ( 0 , 1 ] . q\in \left(0,1]. In addition, a few estimates for the distance between two limit q q -Durrmeyer operators have been derived in the operator norm on C [ 0 , 1 ] C\left[0,1] .

This article is devoted to exploring the solutions of several systems of the first-order partial differential difference equations (PDDEs) with product type u ( z + c ) [ α 1 u ( z ) + β 1 u z 1 + γ 1 u z 2 + α 2 v ( z ) + β 2 v z 1 + γ 2 v z 2 ] = 1 , v ( z + c ) [ α 1 v ( z ) + β 1 v z 1 + γ 1 v z 2 + α 2 u ( z ) + β 2 u z 1 + γ 2 u z 2 ] = 1 , \left\{\begin{array}{l}u\left(z+c){[}{\alpha }_{1}u\left(z)+{\beta }_{1}{u}_{{z}_{1}}+{\gamma }_{1}{u}_{{z}_{2}}+{\alpha }_{2}v\left(z)+{\beta }_{2}{v}_{{z}_{1}}+{\gamma }_{2}{v}_{{z}_{2}}]=1,\\ v\left(z+c){[}{\alpha }_{1}v\left(z)+{\beta }_{1}{v}_{{z}_{1}}+{\gamma }_{1}{v}_{{z}_{2}}+{\alpha }_{2}u\left(z)+{\beta }_{2}{u}_{{z}_{1}}+{\gamma }_{2}{u}_{{z}_{2}}]=1,\end{array}\right. where c = ( c 1 , c 2 ) ∈ C 2 c=\left({c}_{1},{c}_{2})\in {{\mathbb{C}}}^{2} , α j , β j , γ j ∈ C , j = 1 , 2 {\alpha }_{j},{\beta }_{j},{\gamma }_{j}\in {\mathbb{C}},\hspace{0.33em}j=1,2 . Our theorems about the forms of the transcendental solutions for these systems of PDDEs are some improvements and generalization of the previous results given by Xu, Cao and Liu. Moreover, we give some examples to explain that the forms of solutions of our theorems are precise to some extent.

The aim of this study is to obtain several inequalities involving the Berezin number and the Berezin norm for various combinations of operators acting on a reproducing kernel Hilbert space. First, we present some bounds regarding the Berezin number associated with W * Q + W * Q ′ {W}^{* }Q+{W}^{* }Q^{\prime} , where W W , Q Q , and Q ′ Q^{\prime} are three bounded linear operators. Next, several Berezin norm and Berezin number inequalities for the sum of n n operators are established.

An osculating curve is a type of curve in space that holds significance in the study of differential geometry. In this article, we investigate certain geometric invariants of osculating curves on smooth and regularly immersed surfaces under conformal transformations in Euclidean space R 3 {{\mathbb{R}}}^{3} . The primary objective of this article is to explore conditions sufficient for the conformal invariance of the osculating curve under both conformal transformations and isometries. We also compute the tangential and normal components of the osculating curves, demonstrating that they remain invariant under the isometry of the surfaces in R 3 {{\mathbb{R}}}^{3} .

Inequalities are essential in pure and applied mathematics. In particular, Opial’s inequality and its generalizations have been playing an important role in the study of the existence and uniqueness of initial and boundary value problems. In this work, some new Opial-type inequalities are given and applied to generalized Riemann-Liouville-type integral operators.

In this work, we present a sophisticated operating algorithm, the reproducing kernel Hilbert space method, to investigate the approximate numerical solutions for a specific class of fractional Begley-Torvik equations (FBTE) equipped with fractional integral boundary condition. Such fractional integral boundary condition allows us to understand the non-local behavior of FBTE along with the given domain. The algorithm methodology depends on creating an orthonormal basis based on reproducing kernel function that satisfies the constraint boundary conditions so that the solution is finally formulated in the form of a uniformly convergent series in ϖ 3 [ a , b ] {\varpi }_{3}\left[a,b] . From a numerical point of view, some illustrative examples are provided to determine the appropriateness of algorithm design and the effect of using non-classical boundary conditions on the behavior of solutions approach.

In this article, we establish some truncated second main theorems for holomorphic curves into projective spaces with some special hypersurfaces and give some applications. In addition, the defect relation, the algebraically degenerate conditions and uniqueness theorem for holomorphic curves with some special divisors may be improved.

The main goal of this article is to study the existence and uniqueness of periodic solutions for the implicit problem with nonlinear fractional differential equation involving the Caputo tempered fractional derivative. The proofs are based upon the coincidence degree theory of Mawhin. To show the efficiency of the stated result, two illustrative examples will be demonstrated.

The purpose this study is to present and investigate the q q -Beta-Baskakov-Szasz-Stancu operator. The operators are accompanied by Voronovskaja-type consequences, which include both an exact approximation order and a quantitative assessment, specifically within compact disks.

In this article, we investigate the existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions. Using the theory of resolvent operators, some fixed point theorems, and an estimation technique of Kuratowski measure of noncompactness, we first establish some existence results of mild solutions for the proposed equation. Subsequently, we show by applying a newly established lemma that these solutions have regularity property under some conditions. Finally, as a sample of application, the obtained results are applied to a class of non-autonomous nonlocal partial integrodifferential equations.

Let R R be a unital associative ring. Our motivation is to prove that left derivations in column finite matrix rings over R R are equal to zero and demonstrate that a left derivation d : T → T d:{\mathcal{T}}\to {\mathcal{T}} in the infinite upper triangular matrix ring T {\mathcal{T}} is determined by left derivations d j {d}_{j} in R ( j = 1 , 2 , … ) R\left(j=1,2,\ldots ) satisfying d ( ( a i j ) ) = ( b i j ) d\left(\left({a}_{ij}))=\left({b}_{ij}) for any ( a i j ) ∈ T \left({a}_{ij})\in {\mathcal{T}} , where b i j = d j ( a 11 ) , i = 1 , 0 , i ≠ 1 . {b}_{ij}=\left\{\begin{array}{ll}{d}_{j}\left({a}_{11}),& i=1,\\ 0,& i\ne 1.\end{array}\right. The similar results about Jordan left derivations are also obtained when R R is 2-torsion free.

We consider the following nonlinear nonlocal elliptic problem: − a + b ∫ R 3 ∣ ∇ ψ ∣ 2 d x Δ ψ + λ ψ = ∫ R 3 G ( ψ ( y ) ) ∣ x − y ∣ α d y G ′ ( ψ ) , x ∈ R 3 , -\left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}{| \nabla \psi | }^{2}{\rm{d}}x\right)\Delta \psi +\lambda \psi =\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}\frac{G\left(\psi (y))}{{| x-y| }^{\alpha }}{\rm{d}}y\right)G^{\prime} \left(\psi ),\hspace{1em}x\in {{\mathbb{R}}}^{3}, where a , b > 0 a,b\gt 0 are constants, λ > 0 \lambda \gt 0 is a parameter, α ∈ ( 0 , 3 ) \alpha \in \left(0,3) , and G ∈ C 1 ( R , R ) G\in {{\mathcal{C}}}^{1}\left({\mathbb{R}},{\mathbb{R}}) . By using variational methods, we establish the existence of least energy solutions for the above equation under conditions on the nonlinearity G G we believe to be almost necessary. Some qualitative properties of the least energy solutions are also obtained

In this article, we use the notion of lacunary statistical convergence of order ( α , β ) \left(\alpha ,\beta ) to introduce new sequence spaces by lacunary sequence, invariant means defined by Musielak-Orlicz function ℳ = ( ℵ k ) {\mathcal{ {\mathcal M} }}=\left({\aleph }_{k}) . We also examine some topological properties and prove inclusion relations between newly constructed sequence spaces.

In this manuscript, we prove new extensions of Nashine, Wardowski, Feng-Liu, and Ćirić-type contractive inequalities using orbitally lower semi-continuous functions in an orbitally complete b b -metric space. We accomplish new multivalued common fixed point results for two families of dominated set-valued mappings in an ordered complete orbitally b b -metric space. Some new definitions and illustrative examples are given to validate our new results. To show the novelty of our results, applications are given to obtain the solution of nonlinear integral and fractional differential equations. Our results expand the hypothetical consequences of Nashine et al. ( Feng–Liu-type fixed point result in orbital b-metric spaces and application to fractal integral equation , Nonlinear Anal. Model. Control. 26 (2021), no. 3, 522–533) and Rasham et al. ( Common fixed point results for new Ciric-type rational multivalued-contraction with an application , J. Fixed Point Theory Appl. 20 (2018), no. 1, Paper No. 45).

Global in time weak solution to a regularized periodic three-dimensional Boussinesq system is proved to exist in energy spaces. This solution depends continuously on the initial data. In particular, it is unique. The main novelty is the global in time aspect of this solution. The proofs use the coupling between the temperature and the velocity of the fluid, energy methods, and compactness argument.

In this article, a functional boundary value problem involving mixed fractional derivatives with p ( x ) p\left(x) -Laplacian operator is investigated. Based on the fixed point theorems and Mawhin’s coincidence theory’s extension theory, some existence theorems are obtained in the case of non-resonance and the case of resonance. Some examples are supplied to verify our main results.

This study first establishes several maximum and minimum principles involving the nonlocal Monge-Ampère operator and the multi-term time-space fractional Caputo-Fabrizio derivative. Based on the maximum principle established above, on the one hand, we show that a family of multi-term time-space fractional parabolic Monge-Ampère equations has at most one solution; on the other hand, we establish some comparison principles of linear and nonlinear multi-term time-space fractional parabolic Monge-Ampère equations.

In this article, we define three-dimensional q -Riordan arrays and q -Riordan representations for these arrays. Also, we give four cases of infinite multiplication three-dimensional matrices of these arrays. As applications, we obtain three-dimensional q q -Pascal-like matrix and its inverse matrix by Heine’s binomial formula, using combinatorial identities. Finally, we consider the generalization of three-dimensional q q -Pascal-like matrix and give some identities involving q q -binomial coefficients.

The normal curve is a space curve that plays an important role in the field of differential geometry. This research focuses on analyzing the properties of normal curves on smooth immersed surfaces, considering their invariance under isometric transformations. The primary contribution of this article is to explore the requirements for the image of a normal curve that preserves its invariance under isometric transformations. In this article, we investigate the invariant condition for the component of the position vector of the normal curves under isometry and compute the expression for the normal and geodesic curvature of such curves. Moreover, it has been investigated that the geodesic curvature and Christoffel symbols remain unchanged under the isometry of surfaces in R 3 {{\mathbb{R}}}^{3} .

In this article, we apply the Fourier transform to prove the Hyers-Ulam and Hyers-Ulam-Rassias stability for the first- and second-order nonlinear differential equations with initial conditions. Additionally, we extend the results to investigate the Mittag-Leffler-Hyers-Ulam and Mittag-Leffler-Hyers-Ulam-Rassias stability of these differential equations using the proposed method.

In this article, we investigate a class of measure differential inclusions of evolution type involving non-autonomous operator with nonlocal condition defined on the half-line. By fixed point theorem, we first obtain some sufficient conditions to ensure the solution set is nonempty, compact, and R δ {R}_{\delta } -set on compact interval. Subsequently, by means of the inverse limit method, we generalize the results on compact interval to noncompact interval. Finally, an example is given to demonstrate the effectiveness of obtained results.

In this article, we study the Daugavet property and the diametral diameter two properties (DD2Ps) in complex Banach spaces. The characterizations for both Daugavet and Δ \Delta -points are revisited in the context of complex Banach spaces. We also provide relationships between some variants of alternative convexity and smoothness, nonsquareness, and the Daugavet property. As a consequence, every strongly locally uniformly alternatively convex or smooth (sluacs) Banach space does not contain Δ \Delta -points from the fact that such spaces are locally uniformly nonsquare. We also study the convex diametral local diameter two property and the polynomial Daugavet property in the vector-valued function space A ( K , X ) A\left(K,X) . From an explicit computation of the polynomial Daugavetian index of A ( K , X ) A\left(K,X) , we show that the space A ( K , X ) A\left(K,X) has the polynomial Daugavet property if and only if either the base algebra A A or the range space X X has the polynomial Daugavet property. Consequently, we obtain that the polynomial Daugavet property, Daugavet property, DD2Ps, and property ( D {\mathcal{D}} ) are equivalent for infinite-dimensional uniform algebras.

In this study, using convolution theorem of the Laplace transforms, a monotonicity rule for the ratio of two Laplace transforms, Bernstein’s theorem for completely monotonic functions, and other analytic techniques, the authors verify decreasing property of a ratio between three derivatives of a function involving trigamma function and find the necessary and sufficient conditions for a function defined by three derivatives of a function involving trigamma function to be completely monotonic. These results confirm previous guesses posed by Qi and generalize the corresponding known conclusions.

In this article, we study the unicity of meromorphic functions concerning small functions and derivatives-differences. The results obtained in this article extend and improve some results of Chen et al. [ Uniqueness problems on difference operators of meromorphic functions ] and Chen and Huang [ Uniqueness of meromorphic functions concerning their derivatives and shifts with partially shared values ].

The study of special functions has become an enthralling area in mathematics because of its properties and wide range of applications that are relevant into other fields of knowledge. Developing topics in special functions involves the investigation of Apostol-type polynomials encompassing the combinations, extensions, and generalizations of some classical polynomials such as Bernoulli, Euler, Genocchi, and tangent polynomials. One particular type of these polynomials is the Apostol-Frobenius-Euler polynomials of order a a denoted by H n α ( z ; u ; λ ) {H}_{n}^{\alpha }\left(z;\hspace{0.33em}u;\hspace{0.33em}\lambda ) . Using the saddle point method, Corcino et al. obtained approximations for the higher-order tangent polynomials. They also established a new method to derive its approximations with enlarged region of validity. In this article, it is found that these methods are applicable to the higher-order Apostol-Frobenius-Euler polynomials. Consequently, approximations of higher-order Apostol-Frobenius-Euler polynomials in terms of the hyperbolic functions are obtained for large values of the parameter n n , and its uniform approximations with enlarged region of validity are also derived. Moreover, approximations of the generalized Apostol-type Frobenius-Euler polynomials of order α \alpha with parameters a , b , a,b, and c c are obtained by applying the same methods. Graphs are provided to show the accuracy of the exact values of these polynomials and their corresponding approximations for some specific values of the parameters.

In this article, we study the Hyers-Ulam stability of Davison functional equation f ( x y ) + f ( x + y ) = f ( x y + x ) + f ( y ) f\left(xy)+f\left(x+y)=f\left(xy+x)+f(y) on some unbounded restricted domains. Using the obtained results, we study an interesting asymptotic behavior of Davison functions. We also investigate the Hyers-Ulam stability of Davison functional equation and its generalized form given by f ( x y ) + g ( x + y ) = h ( x y + x ) + k ( y ) , f\left(xy)+g\left(x+y)=h\left(xy+x)+k(y), for x , y ∈ R ⩾ 0 = { t ∈ R : t ⩾ 0 } x,y\in {{\mathbb{R}}}^{\geqslant 0}=\left\{t\in {\mathbb{R}}:t\geqslant 0\right\} .

Developing a model of fractional differential systems and studying the existence and stability of a solution is considebly one of the most important topics in the field of analysis. Therefore, this manuscript was dedicated to deriving a new type of fractional system that arises from the combination of three sequential fractional derivatives with fractional pantograph equations. Also, the fixed-point technique was used to evaluate the existence and uniqueness of solutions to the supposed hybrid model. Furthermore, stability results for the intended system in the sense of the Mittag-Leffler-Ulam have been investigated. Ultimately, an illustrative example has been highlighted in order to reinforce the theoretical results and suggest applications for this article.

This article is devoted to constructing sequences of integral operators with the same Voronovskaja formula as the generalized Baskakov operators, but having different behavior in other respects. For them, we investigate the eigenstructure, the inverses, and the corresponding Voronovskaja type formulas. A general result of Voronovskaja type for composition of operators is given and applied to the new operators. The asymptotic behavior of differences between the operators is investigated, and as an application, we obtain a formula involving Euler’s gamma function.

Fractal interpolation has been conventionally treated as a method to construct a univariate continuous function interpolating a given finite data set with the distinguishing property that the graph of the interpolating function is the attractor of a suitable iterated function system. On the one hand, attempts have been made to extend the univariate fractal interpolation from a finite data set to a countably infinite set. On the other hand, fractal interpolation in higher dimensions, particularly the theory of fractal interpolation surfaces (FISs), has received increasing attention for more than a quarter century. This article targets a two-fold extension of the notion of fractal interpolation by providing a general framework to construct FISs for a prescribed set consisting of countably infinite data on a rectangular grid. By using this as a crucial tool, we obtain a parameterized family of bivariate fractal functions simultaneously interpolating and approximating a prescribed bivariate continuous function. Some elementary properties of the associated nonlinear (not necessarily linear) fractal operators are established, thereby allowing the interaction of the notion of fractal interpolation with the theory of nonlinear operators.

The aim of this work is to study the global existence in time of solutions for the tridiagonal system of reaction-diffusion by order m m . Our techniques of proof are based on compact semigroup methods and some L 1 {L}^{1} -estimates. We show that global solutions exist. Our investigation can be applied for a wide class of nonlinear terms of reaction.

In this work, we investigate the online MapReduce processing problem on m m uniform parallel machines, aiming at minimizing the makespan. Each job consists of two sets of tasks, namely, the map tasks and the reduce tasks. A job’s map tasks can be arbitrarily split and processed on different machines simultaneously, while its reduce tasks can only be processed after all its map tasks have been completed. We assume that the reduce tasks are preemptive, but cannot be processed on different machines in parallel. We provide a new lower bound for this problem and present an online algorithm with a competitive ratio of 2 − 1 m 2-\frac{1}{m} ( m m is the number of machines) when the speeds of the machines are 1.

In this study, the basic theory of the sequential Henstock-Kurzweil delta integral on time scales will be discussed. First, we give the notion and the elementary properties of this integral; then we show the equivalence of the Henstock-Kurzweil delta integral and the sequential Henstock-Kurzweil delta integral on time scales. In addition, we consider the Cauchy criterion and the Fundamental Theorems of Calculus. Finally, we prove Henstock’s lemma and give some convergence theorems. As an application, we consider the existence theorem of a kind of functional dynamic equations.

In this study, we introduce the notion of α \alpha -convex sequences which is a generalization of the convexity concept. For this class of sequences, we establish a discrete version of Fejér inequality without imposing any symmetry condition. In our proof, we use a new approach based on the choice of an appropriate sequence, which is the unique solution to a certain second-order difference equation. Moreover, we obtain a refinement of the standard (right) Fejér inequality for convex sequences.

We deal with the existence of three distinct solutions for a poly-Laplacian system with a parameter on finite graphs and a ( p , q ) \left(p,q) -Laplacian system with a parameter on locally finite graphs. The main tool is an abstract critical point theorem in [G. Bonanno and S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition , Appl. Anal. 89 (2010), no. 1, 1–10]. A key point in this study is that we overcome the difficulty to prove that the Gâteaux derivative of the variational functional for poly-Laplacian operator admits a continuous inverse, which is caused by the special definition of the poly-Laplacian operator on graph and mutual coupling of two variables in system.

We introduce a novel framework for embedding anisotropic variable exponent Sobolev spaces into spaces of anisotropic variable exponent Hölder-continuous functions within rectangular domains. We establish a foundational approach to extend the concept of Hölder continuity to anisotropic settings with variable exponents, providing deeper insight into the regularity of functions across different directions. Our results not only broaden the understanding of anisotropic function spaces but also open new avenues for applications in mathematical and applied sciences.

If T 1 {{\mathbb{T}}}_{1} and T 2 {{\mathbb{T}}}_{2} are commuting d d -tuples of Hilbert space operators in B ( ℋ ) d B{\left({\mathcal{ {\mathcal H} }})}^{d} such that ( T 1 * ⊗ I + I ⊗ T 2 * , T 1 ⊗ I + I ⊗ T 2 ) \left({{\mathbb{T}}}_{1}^{* }\otimes I+I\otimes {{\mathbb{T}}}_{2}^{* },{{\mathbb{T}}}_{1}\otimes I+I\otimes {{\mathbb{T}}}_{2}) is strictly m m -isometric (resp., m m -symmetric) for some positive integer m m , then there exist a scalar d d -tuple λ \lambda and positive integers m i {m}_{i} , 1 ≤ i ≤ 2 1\le i\le 2 , such that m = m 1 + 2 m 2 − 2 m={m}_{1}+2{m}_{2}-2 , ( T 1 * + λ ¯ , T 1 + λ ) \left({{\mathbb{T}}}_{1}^{* }+\overline{\lambda },{{\mathbb{T}}}_{1}+\lambda ) is m 1 {m}_{1} isometric, and T 2 − λ {{\mathbb{T}}}_{2}-\lambda is m 2 {m}_{2} -nilpotent (resp., m = m 1 + m 2 − 1 m={m}_{1}+{m}_{2}-1 , ( T 1 * + λ ¯ , T 1 + λ ) \left({{\mathbb{T}}}_{1}^{* }+\overline{\lambda },{{\mathbb{T}}}_{1}+\lambda ) is m 1 {m}_{1} -symmetric and ( T 2 * − λ ¯ , T 2 − λ ) \left({{\mathbb{T}}}_{2}^{* }-\overline{\lambda },{{\mathbb{T}}}_{2}-\lambda ) is m 2 {m}_{2} symmetric).

This study is devoted to exploring the existence and the precise form of finite-order transcendental entire solutions of second-order trinomial partial differential-difference equations L ( f ) 2 + 2 h L ( f ) f ( z 1 + c 1 , z 2 + c 2 ) + f ( z 1 + c 1 , z 2 + c 2 ) 2 = e g ( z 1 , z 2 ) L{(f)}^{2}+2hL(f)f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})+f{\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})}^{2}={e}^{g\left({z}_{1},{z}_{2})} and L ˜ ( f ) 2 + 2 h L ˜ ( f ) ( f ( z 1 + c 1 , z 2 + c 2 ) − f ( z 1 , z 2 ) ) + ( f ( z 1 + c 1 , z 2 + c 2 ) − f ( z 1 , z 2 ) ) 2 = e g ( z 1 , z 2 ) , \tilde{L}{(f)}^{2}+2h\tilde{L}(f)(f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})-f\left({z}_{1},{z}_{2}))+{(f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})-f\left({z}_{1},{z}_{2}))}^{2}={e}^{g\left({z}_{1},{z}_{2})}, where L ( f ) L(f) and L ˜ ( f ) \tilde{L}(f) are defined in (2.1) and (2.2), respectively, and g ( z ) g\left(z) is a polynomial in C 2 {{\mathbb{C}}}^{2} . Our results are the extensions of some of the previous results of Liu et al. Also, we exhibit a series of examples to explain that the forms of transcendental entire solutions of finite-order in our results are precise.

In this article, we examine two problems: a fractional Sturm-Liouville boundary value problem on a compact star graph and a fractional Sturm-Liouville transmission problem on a compact metric graph, where the orders α i {\alpha }_{i} of the fractional derivatives on the i th edge lie in ( 0 , 1 ) (0,1) . Our main objective is to introduce quantum graph Hamiltonians incorporating fractional-order derivatives. To this end, we construct a fractional Sturm-Liouville operator on a compact star graph. We impose boundary conditions that reduce to well-known Neumann-Kirchhoff conditions and separated conditions at the central vertex and pendant vertices, respectively, when α i → 1 {\alpha }_{i}\to 1 . We show that the corresponding operator is self-adjoint. Moreover, we investigate a discontinuous boundary value problem involving a fractional Sturm-Liouville operator on a compact metric graph containing a common edge between the central vertices of two star graphs. We construct a new Hilbert space to show that the operator corresponding to this fractional-order transmission problem is self-adjoint. Furthermore, we explain the relations between the self-adjointness of the corresponding operator in the new Hilbert space and in the classical L 2 {L}^{2} space.

In this article, we consider a problem of exact controllability in the processes described by a nonlinear damped thermoviscoelastic plate. First, we prove the global well-posedness result for the nonlinear functions that are continuous with respect to time and globally Lipschitz with respect to space variable. Next, we perform a spectral analysis of the linear and uncontrolled problem. Then, we prove that the corresponding solutions decay exponentially to zero at a rate determined explicitly by the physical constants. Finally, we prove the exact controllability of the linear and the nonlinear problems by proving that the corresponding controllability mappings are surjective.

In this article, we present two new algorithms referred to as the improved modified gradient-based iterative (IMGI) algorithm and its relaxed version (IMRGI) for solving the complex conjugate and transpose (CCT) Sylvester matrix equations, which often arise from control theory, system theory, and so forth. Compared with the gradient-based iterative (GI) (A.-G. Wu, L.-L. Lv, and G.-R. Duan, Iterative algorithms for solving a class of complex conjugate and transpose matrix equations , Appl. Math. Comput. 217 (2011), 8343–8353) and the relaxed GI (RGI) (W.-L. Wang, C.-Q. Song, and S.-P. Ji, Iterative solution to a class of complex matrix equations and its application in time-varying linear system , J. Appl. Math. Comput. 67 (2021), 317–341) algorithms, the proposed ones can make full use of the latest information and need less computations, which leads to higher computational efficiency. With the real representation of a complex matrix as a tool, we establish sufficient and necessary conditions for the convergence of the IMGI and the IMRGI algorithms. Finally, some numerical examples are given to illustrate the effectiveness and advantages of the proposed algorithms.

Let f f be a function defined on the real line, and T f {T}_{f} be the corresponding superposition operator which maps h h to T f ( h ) {T}_{f}\left(h) , i.e., T f ( h ) = f ∘ h {T}_{f}\left(h)=f\circ h . In this article, the sufficient and necessary conditions such that T f {T}_{f} maps periodic Hölder-Lipschitz spaces H p α {H}_{p}^{\alpha } into itself with 0 < α < 1 p 0\lt \alpha \lt \frac{1}{p} and 1 p < α < 1 \frac{1}{p}\lt \alpha \lt 1 , where α \alpha is the smoothness index, are shown. Our result in the case 0 < α < 1 p 0\lt \alpha \lt \frac{1}{p} may be the first result about the superposition operator problems of smooth function space containing unbounded functions.

In this study, we introduce the λ \lambda -analogue of Lah numbers and λ \lambda -analogue of r r -Lah numbers in the view of degenerate version, respectively. We investigate their properties including recurrence relation and several identities of λ \lambda -analogue of Lah numbers arising from degenerate differential operators. Using these new identities, we study the normal ordering of degenerate integral power of the number operator in terms of boson operators, which is represented by means of λ \lambda -analogue of Lah numbers and λ \lambda -analogue of r r -Lah numbers, respectively.

This article is concerned with the following Kirchhoff equation: − a + b ∫ R 3 ∣ ∇ u ∣ 2 d x Δ u = g ( u ) + h ( x ) in R 3 , -\left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}{| \nabla u| }^{2}{\rm{d}}x\right)\Delta u=g\left(u)+h\left(x)\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{3}, where a a and b b are positive constants and h ≠ 0 h\ne 0 . Under the Berestycki-Lions type conditions on g g , we prove that the equation has at least two positive solutions by using variational methods. Furthermore, we obtain the existence of ground state solutions.

The purpose of this article is to discuss about the so-called semi-greedy bases in p p -Banach spaces. Specifically, we will review existing results that characterize these bases in terms of almost-greedy bases, and, also, we analyze quantitatively the behavior of certain constants. As new results, by avoiding the use of certain classical results in p p -convexity, we aim to quantitatively improve specific bounds for bi-monotone 1-semi-greedy bases.

Krasnoselskii’s iteration is a classical and important method for approximating the fixed point of an operator that satisfies certain conditions. Many authors have used this approach to obtain several famous fixed point theorems for different types of operators. It is well known that Kirk’s iteration can be seen as a generalization of Krasnoselskii’s iteration, in which the iterates are generated by a certain generalized averaged mapping. This approximation method is of great practical significance because the iterative formula contains more information related to the operator in question. The purpose of this study is to define weak ( α n , β i ) \left({\alpha }_{n},{\beta }_{i}) -convex orbital Lipschitz operators. These concepts not only extend the previously introduced Popescu-type convex orbital ( λ , β ) \left(\lambda ,\beta ) -Lipschitz operators in Fixed-point results for convex orbital operators , (Demonstr. Math. 56 (2023), 20220184), but also encompass many classical contractive operators. Popescu also proved a fixed point result for his proposed operator using the graphic contraction principle and obtained an approximation of the fixed point with Krasnoselskii’s iterates. To extend Popescu’s main results from Krasnoselskii’s iterative scheme to Kirk’s iterative scheme, several fixed point theorems are established, in which an appropriate Kirk’s iterative algorithm can be used to approximate the fixed point of a k k -fold averaged mapping associated with our presented convex orbital Lipschitz operators. These results not only generalize, but also complement the existing results documented in the previous literature.

This article is focused on deriving the approximate model for surface wave propagation on an elastic isotropic half-plane under the effects of the rotation and magnetic field along with the prescribed vertical and tangential face loads. The method of study depends on the slow time perturbation of the prevalent demonstration for the Rayleigh wave eigen solutions through harmonic functions. A perturbed pseudo-hyperbolic equation on the interface of the media is subsequently derived, governing the propagation of the surface wave. The established asymptotic formulation is tested by comparison with the exact secular equation. In the absence of the magnetic field, the specific value of Poisson’s ratio, ν = 0.25 \nu =0.25 , is highlighted, where the rotational effect vanishes at the leading order.

In this article, we study the following nonlinear k k -Hessian system with singular weights S k 1 k ( σ ( D 2 u 1 ) ) = λ b ( ∣ x ∣ ) f ( − u 1 , − u 2 ) , in Ω , S k 1 k ( σ ( D 2 u 2 ) ) = λ h ( ∣ x ∣ ) g ( − u 1 , − u 2 ) , in Ω , u 1 = u 2 = 0 , on ∂ Ω , \left\{\begin{array}{ll}{S}_{k}^{\frac{1}{k}}(\sigma ({D}^{2}{u}_{1}))=\lambda b\left(| x| )f\left(-{u}_{1},-{u}_{2}),& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ {S}_{k}^{\frac{1}{k}}(\sigma ({D}^{2}{u}_{2}))=\lambda h\left(| x| )g\left(-{u}_{1},-{u}_{2}),& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ {u}_{1}={u}_{2}=0,& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where λ > 0 \lambda \gt 0 , 1 ≤ k ≤ N 1\le k\le N is an integer, Ω \Omega stands for the open unit ball in R N {{\mathbb{R}}}^{N} , and S k ( σ ( D 2 u ) ) {S}_{k}(\sigma ({D}^{2}u)) is the k k -Hessian operator of u u . By using the fixed point index theory, we prove the existence and nonexistence of negative k k -convex radial solutions. Furthermore, we establish the multiplicity result of negative k k -convex radial solutions based on a priori estimate achieved. More precisely, there exists a constant λ ∗ > 0 {\lambda }^{\ast }\gt 0 such that the system admits at least two negative k k -convex radial solutions for λ ∈ ( 0 , λ ∗ ) \lambda \in \left(0,{\lambda }^{\ast }) .

In this study, we introduce the following additive functional equation: g ( λ u + v + 2 y ) = λ g ( u ) + g ( v ) + 2 g ( y ) g\left(\lambda u+v+2y)=\lambda g\left(u)+g\left(v)+2g(y) for all λ ∈ C \lambda \in {\mathbb{C}} , all unitary elements u , v u,v in a unital Poisson C * {C}^{* } -algebra P P , and all y ∈ P y\in P . Using the direct method and the fixed point method, we prove the Hyers-Ulam stability of the aforementioned additive functional equation in unital Poisson C * {C}^{* } -algebras. Furthermore, we apply to study Poisson C * {C}^{* } -algebra homomorphisms and Poisson C * {C}^{* } -algebra derivations in unital Poisson C * {C}^{* } -algebras.

This article aims to provide a modified Noor iterative scheme to approximate the fixed points of generalized nonexpansive mappings with property ( E E ) called MN-iteration. We establish the strong and weak convergence results in a uniformly convex Banach space. Additionally, numerical experiments of the iterative technique are demonstrated using a signal recovery application in a compressed sensing situation. Ultimately, an illustrative analysis regarding Noor, SP-, and MN-iteration procedures is obtained via polysomnographic techniques. The images obtained are called polynomiographs. Polynomiographs have importance for both the art and science aspects. The obtained graphs describe the pattern of complex polynomials and also the convergence properties of the iterative method. They can also be used to increase the functionality of the existing polynomiography software.

The shrinking of support in non-linear parabolic p p -Laplacian equations with a positive initial condition u 0 {u}_{0} that decayed as ∣ x ∣ → ∞ | x| \to \infty was explored in the Cauchy problem. Proofs were provided for establishing exact local estimates for the boundary of the support of the solutions.

The problem of common solutions for nonlinear equations has significant theoretical and practical value. In this article, we first introduce a new concept of a pair of ( α , Θ ) \left(\alpha ,\Theta ) -type contractions, and then, we present some common fixed point results for the contractions in complete metric spaces. Finally, our results are applied to consider the existence, uniqueness and approximation of common solutions for two classes of nonlinear fractional differential equations.

In this article, a family of delay differential equations with pseudo compact almost automorphic coefficients is considered. By introducing a concept of Bi-pseudo compact almost automorphic functions and establishing the properties of these functions, and using Halanay’s inequality and Banach fixed point theorem, some results on the existence, uniqueness and global exponential stability of pseudo compact automorphic solutions of the equations are obtained. Our results extend some recent works. Moreover, an example is given to illustrate the validity of our results.

The primary focus of this work lies in the exploration of the limiting dynamics governing fractional stochastic discrete wave equations with nonlinear noise. First, we establish the well-posedness of solutions to these stochastic equations and subsequently demonstrate the existence of periodic measures for the considered equations.

The present study is related to the existence and the asymptotic behavior of the solution of a nonlinear elliptic Steklov problem imposed on a nanostructure depending on the thickness parameter ε \varepsilon (nano-scale), distributed on the boundary of the domain when the parameter ε \varepsilon goes to 0, under some appropriate conditions on the data involved in the problem. We use epi-convergence method in order to establish the limit behavior by characterizing the weak limits of the energies for the solutions. An intermediate step in the proof provides a homogenization result for the considered structure.

In this article, based on the real representation and Kronecker product, Cramer’s rule for a class of coupled Sylvester commutative quaternion matrix equations is studied and its expression is obtained. The proposed algorithm is very simple and convenient because it only involves real operations. Some numerical examples are provided to illustrate the feasibility of the proposed algorithm.

In the present work, we establish a quantitative estimate for the perturbed sampling Kantorovich operators in Orlicz spaces, in terms of the modulus of smoothness, defined by means of its modular functional. From the obtained result, we also deduce the qualitative order of approximation, by considering functions in suitable Lipschitz classes. This allows us to apply the above results in certain Orlicz spaces of particular interest, such as the interpolation spaces, the exponential spaces and the L p {L}^{p} -spaces, 1 ≤ p < + ∞ 1\le p\lt +\infty . In particular, in the latter case, we also provide an estimate established using a direct proof based on certain properties of the L p {L}^{p} -modulus of smoothness, which are not valid in the general case of Orlicz spaces. The possibility of using a direct approach allows us to improve the estimate that can be deduced as a consequence of the one achieved in Orlicz spaces. In the final part of the article, we furnish some estimates and the corresponding qualitative order of approximation in the space of uniformly continuous and bounded functions.

The purpose of this work is to study a mathematical model that describes a contact between a deformable body and a rigid foundation. A linear viscoelastic Kelvin-Voigt constitutive law with time-fractional derivatives describes the material’s behavior. The contact is modeled with Signorini’s condition coupled with Coulomb’s friction law. We derive a variational formulation of the model, and we prove the existence of a weak solution using the theory of monotone operators and Caputo derivative and the Rothe method. We also introduce the penalized problem and prove its solvability using the Galerkin method. Furthermore, we study the convergence of its solution to the solution of the original problem as the penalization parameter tends to zero.

In this study, subordination, superordination, and sandwich theorems are established for a class of p -valent analytic functions involving a generalized integral operator that has as a special case p -valent Sălăgean integral operator. Relevant connections of the new results with several well-known ones are given as a conclusion for this investigation.

This article presents a higher-order circular intuitionistic fuzzy time series forecasting method for predicting the stock change index, which is shown to be an improvement over traditional time series forecasting methods. The method is based on the principles of circular intuitionistic fuzzy set theory. It uses both positive and negative membership values and a circular radius to handle uncertainty and imprecision in the data. The circularity of the time series is also taken into consideration, leading to more accurate and robust forecasts. The higher-order forecasting capability of this method provides more comprehensive predictions compared to previous methods. One of the key challenges we face when using the amount featured as a case study in our article to project the future value of ratings is the influence of the stock market index. Through rigorous experiments and comparison with traditional time series forecasting methods, the results of the study demonstrate that the proposed higher-order circular intuitionistic fuzzy time series forecasting method is a superior approach for predicting the stock change index.

Binary relations (BIRs) have many applications in computer science, graph theory, and rough set theory. This study discusses the combination of BIRs, fuzzy substructures of quantale, and rough fuzzy sets. Approximation of fuzzy subsets of quantale with the help of BIRs is introduced. In quantale, compatible and complete relations in terms of aftersets and foresets with the help of BIRs are defined. Furthermore, we use compatible and complete relations to approximate fuzzy substructures of quantale, and these approximations are interpreted by aftersets and foresets. This concept generalizes the concept of rough fuzzy quantale. Finally, using BIRs, quantale homomorphism is used to build a relationship between the approximations of fuzzy substructures of quantale and the approximations of their homomorphic images.

In the era of big data, efficient classification of rapidly growing data volumes is a critical challenge. Traditional algorithms often fall short in handling the scale and complexity of big data, leading to inefficiencies in classification accuracy and processing times. This study aims to address these limitations by introducing a novel approach to algorithm selection, which is essential for advancing big data classification methods. We developed an advanced classification algorithm that integrates a fuzzy multi-criteria decision-making (MCDM) model, specifically tailored for big data environments. This integration involves leveraging the analytical strengths of MCDM, particularly the analytic hierarchy process, to systematically evaluate and select the most suitable classification algorithms. Our method uniquely combines the precision of fuzzy logic with the comprehensive evaluative capabilities of MCDM, setting it apart from conventional approaches. The proposed model is meticulously designed to assess key performance indicators such as accuracy, true rate, and processing efficiency in various big data scenarios. Our findings reveal that the proposed model significantly enhances classification accuracy and processing efficiency compared to traditional algorithms. The model demonstrated a marked improvement in true rates and overall classification performance, showcasing its effectiveness in handling large-scale data challenges. These results underline the model’s potential as a pragmatic solution for big data classification, offering substantial improvements over existing methodologies. The study contributes a groundbreaking perspective to the field of big data classification, addressing critical gaps in current practices. By combining fuzzy logic with MCDM, the proposed model offers a more nuanced and effective approach to algorithm selection, catering to the intricate demands of big data environments. This research not only enhances the understanding of classification behaviors in big data but also paves the way for future advancements in data mining technologies. Its implications extend beyond theoretical value, providing practical tools for practitioners and researchers in the realm of big data analytics.

Wearable sensors (WS) play a vital role in health assistance to improve the patient monitoring process. However, the existing data collection process faces difficulties in error corrections, rehabilitation, and training validations. Therefore, the data analysis requires additional effort to reduce the overall problems in sports rehabilitation. The existing research difficulties are overcome by applying the proposed spatial data correlation with a support vector machine (SDC-SVM). The algorithm uses the hyperplane function that recognizes sportsperson activities and improves overall activity recognition efficiency. The sensor data are analyzed according to the input margin, and the classification process is performed. In addition, feature correlation and input size are considered to maximize the overall classification procedure of WS data correlation using the size and margin of the input and previously stored data. In both the differentiation and classification instances, the spatiotemporal features of data are extracted and analyzed using support vectors. The proposed SDC-SVM method can improve recognition accuracy, F 1 score, and computing time for the varying WS inputs, classifications, and subjects.

In a dynamic market marked by disruptions like pandemics and recessions, organizations face significant challenges in efficiently managing logistics processes and activities. The primary objective of this article is to propose an integrated four-phase model for assessing the efficiency of retail distribution warehouses based on principal component analysis-data envelopment analysis-improved fuzzy step-wise weight assessment ratio analysis-combined compromise solution (PCA-DEA-IMF SWARA-CoCoSo). The model provides a synergistic effect of all positive sides of the considered methods. PCA-DEA methods are used to reduce the number of variables and to identify efficient warehouses. IMF SWARA is applied to determine criteria weights, while the CoCoSo method is employed in the last phase for ranking efficient warehouses. The model incorporates 18 inputs and 3 outputs, derived from both literature and real-world systems. The proposed model identifies the most efficient warehouses, which can serve as benchmarks for improving the performance of less efficient ones. After implementing PCA-DEA, only seven warehouses were identified as efficient. Subsequently, fixed and variable costs are identified as the two most important criteria. Results of the considered case study indicate that warehouse A4 emerges as the best one, whereas A6 is the least preferred warehouse. This research offers valuable insights and practical implications for organizations operating in dynamic markets, assisting them in achieving operational excellence and improving their supply chain performance.

Dedekind sums and their generalizations are defined in terms of Bernoulli functions and their generalizations. As a new generalization of the Dedekind sums, the degenerate poly-Dedekind sums, which are obtained from the Dedekind sums by replacing Bernoulli functions by degenerate poly-Bernoulli functions of arbitrary indices are introduced in this article and are shown to satisfy a reciprocity relation.

The main goal of the present study is to introduce an operational collocation scheme based on sixth-kind Chebyshev polynomials (SCPs) to solve a category of optimal control problems involving a variable-order dynamical system (VODS). To achieve this goal, the collocation method based on SCPs, the pseudo-operational matrix for the fractional integral operator, and the dual operational matrix are adopted. More precisely, an algebraic equation is obtained instead of the objective function and a system of algebraic equation is derived instead of the VODS. The constrained equations obtained from joining the objective function to the VODS are ultimately optimized using the method of the Lagrange multipliers. Detailed convergence analysis of the suggested method is given as well. Four illustrative examples along with several tables and figures are formally provided to support the efficiency and preciseness of the numerical scheme.

Hail, an intense convective catastrophic weather, is seriously hazardous to people’s lives and properties. This article proposes a multi-step cyclone hail weather recognition model, called long short-term memory (LSTM)-C3D, based on radar images, integrating attention mechanism and network voting optimization characteristics to achieve intelligent recognition and accurate classification of hailstorm weather based on long short-term memory networks. Based on radar echo data in the strong-echo region, LSTM-C3D can selectively fuse the long short-term time feature information of hail meteorological images and effectively focus on the significant features to achieve intelligent recognition of hail disaster weather. The meteorological scans of 11 Doppler weather radars deployed in various regions of the Hunan Province of China are used as the specific experimental and application objects for extensive validation and comparison experiments. The results show that the proposed method can realize the automatic extraction of radar reflectivity image features, and the accuracy of hail identification in the strong-echo region reaches 91.3%. It can also effectively realize the prediction of convective storm movement trends, laying the theoretical foundation for reducing the misjudgment of extreme disaster weather.

In this article, we study the application of NetworkX, a Python library for dealing with traffic networks, to the problem of signal optimization at a single intersection. We use the shortest-path algorithms such as Bellman-Ford (Dynamic Programming), A star (A*), and Dijkstra’s algorithm to compute an optimal solution to the problem. We consider both undersaturated and oversaturated traffic conditions. The results show that we find optimal results with short Central Processor Unit (CPU) time using all the applied algorithms, where Dijkstra’s algorithm slightly outperformed others. Moreover, we show that bee colony optimization can find the optimal solution for all tested problems with different degrees of computational complexity for less CPU time, which is a new contribution to knowledge in this field.

In this research article, a control approach for port-Hamiltonian PH systems based in a neural network (NN) quaternion-based control strategy is presented. First, the dynamics is converted by the implementation of a Poisson bracket in order to facilitate the mathematical model in order to obtain a feasible formulation for the controller design based on quaternion NNs. In this study, two controllers for this kind of of system are presented: the first one consists in the controller design for a PH system about its equilibrium points taking into consideration the position and momentum. This mean is achieved by dividing the quaternion neural controller into scalar and vectorial parts to facilitate the controller derivation by selecting a Lyapunov functional. The second control strategy consists in designing the trajectory tracking controller, in which a reference moment is considered in order to drive this variable to the final desired position according to a reference variable; again, a Lyapunov functional is implemented to obtain the desired control law. It is important to mention that both controllers take into advantage that the energy consideration and that the representation of many physical systems could be implemented in quaternions. Besides the angular velocity, trajectory tracking of a three-phase induction motor is presented as a third numerical experiment. Two numerical experiments are presented to validate the theoretical results evinced in this study. Finally, a discussion and conclusion section is provided.

Ontology serves as a structured knowledge representation that models domain-specific concepts, properties, and relationships. Ontology matching (OM) aims to identify similar entities across distinct ontologies, which is essential for enabling communication between them. At the heart of OM lies the similarity feature (SF), which measures the likeness of entities from different perspectives. Due to the intricate nature of entity diversity, no single SF can be universally effective in heterogeneous scenarios, which underscores the urgency to construct an SF with high discriminative power. However, the intricate interactions among SFs make the selection and combination of SFs an open challenge. To address this issue, this work proposes a novel kernel principle component analysis and evolutionary algorithm (EA) to automatically construct SF for OM. First, a two-stage framework is designed to optimize SF selection and combination, ensuring holistic SF construction. Second, a cosine similarity-driven kPCA is presented to capture intricate SF relationships, offering precise SF selection. Finally, to bolster the practical application of EA in the SF combination, a novel evaluation metric is developed to automatically guide the algorithm toward more reliable ontology alignments. In the experiment, our method is compared with the state-of-the-art OM methods in the Benchmark and Conference datasets provided by the ontology alignment evaluation initiative. The experimental results show its effectiveness in producing high-quality ontology alignments across various matching tasks, significantly outperforming the state-of-the-art matching methods.

Water is a vital resource essential to the survival and development of all creatures. With the rapid growth of industry and agriculture, people face a severe threat of ecological destruction and environmental pollution while living earthly lives. Water pollution, in particular, harms people’s health the most. As a result, water supply security has become a top priority. As a critical point in water supply safety, monitoring water quality effectively and forecasting sudden water contamination on time has become a research hotspot worldwide. With the rapid development and wide applications of artificial intelligence and computer vision technologies, biological activity identification-based intelligent water quality monitoring methods have drawn widespread attention. They were taking fish activities as the water-quality indicator has gained extensive attention by introducing advanced computer vision and artificial intelligence technologies with low cost and ease of carrying. This article comprehensively reviews recent progress in the research and applications of machine vision-based intelligent water quality monitoring and early warning techniques based on fish activity behavior recognition. In detail, it addresses water quality-oriented fish detection and tracking, activity recognition, and abnormal behavior recognition-based intelligent water quality monitoring. It analyzes and compares the performance and their favorite application conditions. Finally, it summarizes and discusses the difficulties and hotspots of water quality monitoring based on the fish’s abnormal behavior recognition and their future development trends.

The Guzheng, an ancient and widely cherished musical instrument in China, serves as a significant cultural heritage with its enchanting melodies. The advent of artificial intelligence offers a novel avenue for the automatic recognition of guzheng music. This article introduces a pitch detection and recognition approach leveraging an enhanced capsule network. By integrating relative spectrum-aware linear prediction and Mel-scale frequency cepstral coefficients into novel features and feeding them into an optimized capsule network, the method achieves precise pitch recognition from audio inputs. Evaluation on a custom dataset indicates a high accuracy in identifying distinct pitches across the guzheng’s 21 strings, with an average recognition rate of 98.15%. Furthermore, to assess the algorithm’s resilience to interference, comparative experiments against three other network models were conducted in various noise conditions. Our approach outperformed all others, maintaining over 96% accuracy even in noisy environments, demonstrating superior anti-interference capabilities.

The global phenomenon known as cyberbullying is a form of modern harassment that cannot be entirely stopped but can be avoided. Most current solutions to the cyberbullying problem have relied on tools and methods to identify online bullying. However, end users do not have free access to these tools. The goal of this study is to create a model to combat cyberbullying on social media sites based on users’ appearance. In this article, we present a cyberbullying detection system constructed using the Word2Vec word-embedding method and a deep learning convolutional neural network combined with bidirectional long short-term memory (CNN-BiLSTM), as well as the XLM-Roberta transformer, to develop a model for cyberbullying detection. We carried out two experiments based on binary (hate speech or non-hate speech bullying comments) and multiclass (religion, age, gender, ethnicity, and non-bullying tweets) datasets collected from Kaggle online discussions and Twitter. To evaluate the model’s performance, we used standard measurement metrics, such as precision, recall, F 1-score, and accuracy. Through a comparison of the results, it is noted that the XLM-Roberta model outperformed the CNN-BiLSTM model, resulting in 84% accuracy using the Kaggle online discussion dataset and 94% accuracy using the Twitter dataset.

In the field of civil engineering education, accurately evaluating the effectiveness of budget courses is crucial. However, traditional methods of evaluation tend to be cumbersome and subjective. In recent years, machine learning technology has demonstrated immense potential in educational evaluation. Nevertheless, in practical application, the machine learning-based evaluation model for civil engineering budget courses faces the predicament of inadequate evaluation accuracy. To solve this problem, the squirrel search algorithm technology was used to establish support vector machine parameters and create optimization algorithms. The performance of the proposed optimization algorithm was tested, and the results showed that the accuracy of the proposed algorithm was 0.927, which was better than similar prediction algorithms. Then, the empirical analysis of the proposed civil engineering budget course evaluation model showed that student satisfaction and student examination scores had increased to 92 and 94 points, respectively. The above results reveal that the proposed optimization algorithm and course evaluation model have good performance. Therefore, the implementation of the proposed curriculum evaluation method can significantly improve the learning efficiency of students and the teaching quality of civil engineering budgeting methods courses.

The present article deals with new fixed point theorems by means of G G -strongly contractive maps. The research findings are demonstrated in a spherically complete ultrametric space with a graph for single-valued mappings. The special cases of the results that extend the current ones are offered, along with some examples that illustrate our results. Besides, an application utilized in dynamic programming that endorses the acquired observations is also provided.

In this study, we give a modification of Mellin convolution-type operators. In this way, we obtain the rate of convergence with the modulus of the continuity of the m m th-order Mellin derivative of function f f , but without the derivative of the operator. Then, we express the Taylor formula including Mellin derivatives with integral remainder. Later, a Voronovskaya-type theorem is proved. In the last part, we state order of approximation of the modified operators, and the obtained results are restated for the Mellin-Gauss-Weierstrass operator.

In this work, we consider a special nondegenerate equation with two weights. We investigate multiplicity result of this biharmonic equation. Mainly, our purpose is to obtain this result using an alternative Ricceri’s theorem. Moreover, we give some compact embeddings in variable exponent Sobolev spaces with second order to prove the main idea.

In this article, we construct a new sequence of positive linear operators H n : B [ 0 , 1 ] → C [ 0 , 1 ] {H}_{n}:B{[}0,1]\to C{[}0,1] using the hypergeometric distribution of probability theory and the rational values of f at the equally spaced control points k ∕ n k/n ( k = 0 , 1 , … , n ) \left(k=0,1,\ldots ,n) of the unit interval [0,1]. Moreover, we obtain some approximation properties of these operators. It is important to note that hypergeometric distribution has a special interest in probability theory because of its natural behaviour. Namely, unlike all other discrete distributions, the previous steps in the hypergeometric distribution affect the next steps. In other discrete distributions, the process starts from the beginning at each stage, whereas in the hypergeometric distribution, the previous steps determine the structure of the next steps, since the previous steps are not replaced.

In this study, we deal with Kantorovich-type generalization of the Brass-Stancu operators. For the sequence of these operators, we study L p {L}^{p} -convergence and give some upper estimates for the L p {L}^{p} -norm of the approximation error via first-order averaged modulus of smoothness and the first-order K K -functional. Moreover, we show that the Kantorovich generalization of each Brass-Stancu operator satisfies variation detracting property and is bounded with respect to the norm of the space of functions of bounded variation on [ 0 , 1 ] \left[0,1] . Finally, we present graphical and numerical examples to compare the convergence of these operators to given functions under different parameters.

In this study, we will express the 2-crossed module of groups from a higher-dimensional categorical perspective. According to simplicial homotopy theory, a 2-crossed module is the Moore complex of a 2-truncated simplicial group. Therefore, the 2-crossed module is an algebraic homotopy model for the homotopy 3-types. Tricategories are a three-dimensional generalization of the bicategory concept. Any tricategory is triequivalent to the Gray category, where Gray is a category enriched over the monoidal category 2Cat equipped with the Gray tensor product. Briefly, a Gray category is a semi-strict 3-category for homotopy 3-types. Naturally, the tricategory perspective is used in homotopy theory. The 2-crossed module is associated with the concept of the Gray category. The aim of this study is to obtain a single object tricategory from any 2-crossed module of groups.

In this article, we present a one-parameter fractional multiplicative integral identity and use it to derive a set of inequalities for multiplicatively s s -convex mappings. These inequalities include new discoveries and improvements upon some well-known results. Finally, we provide an illustrative example with graphical representations, along with some applications to special means of real numbers within the domain of multiplicative calculus.

In this article, we first study the inverse source problem for parabolic with memory term. We show that our problem is ill-posed in the sense of Hadamard. Then, we construct the convergence result when the parameter tends to zero. We also investigate the regularized solution using the Fourier truncation method. The error estimate between the regularized solution and the exact solution is obtained.

For μ ∈ C 1 ( I ) \mu \in {C}^{1}\left(I) , μ > 0 \mu \gt 0 , and λ ∈ C ( I ) \lambda \in C\left(I) , where I I is an open interval of R {\mathbb{R}} , we consider the set of functions f ∈ C 2 ( I ) f\in {C}^{2}\left(I) satisfying the second-order differential inequality d d t μ d f d t + λ f ≥ 0 \frac{{\rm{d}}}{{\rm{d}}t}\left(\phantom{\rule[-0.75em]{}{0ex}},\mu \frac{{\rm{d}}f}{{\rm{d}}t}\right)+\lambda f\ge 0 in I I . The considered set includes several classes of generalized convex functions from the literature. In particular, if μ ≡ 1 \mu \equiv 1 and λ = k 2 \lambda ={k}^{2} , k > 0 k\gt 0 , we obtain the class of trigonometrically k k -convex functions, while if μ ≡ 1 \mu \equiv 1 and λ = − k 2 \lambda =-{k}^{2} , k > 0 k\gt 0 , we obtain the class of hyperbolic k k -convex functions. In this article, we establish a Fejér-type inequality for the introduced set of functions without any symmetry condition imposed on the weight function and discuss some special cases of weight functions. Moreover, we provide characterizations of the classes of trigonometrically and hyperbolic k k -convex functions.

In this study, some mappings related to the Fejér-type inequalities for G A GA -convex functions are defined over the interval [ 0 , 1 ] {[}0,1] . Some Fejér-type inequalities for G A GA -convex functions are proved using these mappings. Properties of these mappings are considered and consequently we obtain refinements of some known results.

This article is mainly concerned to link the Hermite-Hadamard and the Jensen-Mercer inequalities by using majorization theory and fractional calculus. We derive the Hermite-Hadamard-Jensen-Mercer-type inequalities in conticrete form, which serve as both discrete and continuous inequalities at the same time, for majorized tuples in the framework of the famous Atangana-Baleanu fractional operators. Also, the main inequalities encompass the previously established inequalities as special cases. Using generalized Mercer’s inequality, we also investigate the weighted forms of our major inequalities for certain monotonic tuples. Furthermore, the derivation of new integral identities enables us to construct bounds for the discrepancy of the terms concerning the main results. These bounds are constructed by incorporating the convexity of ∣ f ′ ∣ | f^{\prime} | and ∣ f ′ ∣ q ( q > 1 ) {| f^{\prime} | }^{q}\hspace{0.33em}\left(q\gt 1) and making use of power mean and Hölder inequalities along with the established identities.

In this article, we will demonstrate some Hardy’s inequalities by utilizing Hölder inequality, integration by parts, and chain rule of the conformable fractional calculus. When α = 1 \alpha =1 , we can obtain some of classical Copson’s and Hardy’s inequalities. In the end, we will obtain an application of Hardy’s inequality utilizing the conformable fractional calculus.

In this article, we propose a novel integral transform coined as quaternion quadratic phase S-transform (Q-QPST), which is an extension of the quadratic phase S-transform and study the uncertainty principles associated with the Q-QPST. The Q-QPST possesses some desirable characteristics that are absent in conventional time-frequency transforms, especially for dealing with the time-varying quaternion-valued signals. First, we propose the definition of Q-QPST and then we explore some mathematical properties of the of quaternion Q-QPST, including the linearity, modulation, shift, orthogonality relation, and reconstruction formula. Second, we derive the associated Heisenberg’s uncertainty inequality and the corresponding logarithmic version for Q-QPST. Finally, an illustrative example and some potential applications of the Q-QPST are introduced.

This manuscript is devoted to constructing a novel iterative scheme and reckoning of fixed points for generalized contraction mappings in hyperbolic spaces. Also, we establish Δ \Delta and strong convergence results by the considered iteration under the class of mappings satisfying condition (E). Moreover, some qualitative results of the suggested iteration, like weak w 2 {w}^{2} -stability and data dependence results, are discussed. Furthermore, to test the efficiency and effectiveness of the proposed iteration, practical experiments are given. To support the theoretical results, illustrative examples are presented. Finally, our results improve and generalize several classical results in the literature of fixed point iterations.

In this paper, we introduce a new class of nonlinear mappings and compare it to other classes of nonlinear mappings that have appeared in the literature. We establish various existence and convergence theorems for this class of mappings under different assumptions in Banach spaces, particularly Banach spaces with a normal structure. In addition, we provide examples to substantiate the findings presented in this study. We prove the existence of a common fixed point for a family of commuting α \alpha -partially nonexpansive self-mappings. Furthermore, we extend the results reported by Suzuki ( Fixed point theorems and convergence theorems for some generalized nonexpansive mappings , J. Math. Anal. Appl. 340 (2008), no. 2, 1088–1095), Llorens-Fuster ( Partially nonexpansive mappings , Adv. Theory Nonlinear Anal. Appl. 6 (2022), no. 4, 565–573), and Dhompongsa et al. ( Edelstein’s method and fixed point theorems for some generalized nonexpansive mappings , J. Math. Anal. Appl. 350 (2009), no. 1, 12–17). Finally, we present an open problem concerning the existence of fixed points for α \alpha -partially nonexpansive mappings in the context of uniformly nonsquare Banach spaces.

In this article, we study a generalized Yosida variational inclusion problem involving multi-valued operator with XOR operation. It is shown that the generalized Yosida variational inclusion problem involving multi-valued operator with XOR operation is equivalent to a fixed point equation. We have proved that the generalized Yosida approximation operator is Lipschitz continuous. Finally, we prove an existence and convergence result for our problem.

This article presents the notions of extended b-gauge space ( U , Q φ ; Ω ) \left(U,{Q}_{\varphi ;\Omega }) and extended J φ ; Ω {{\mathcal{J}}}_{\varphi ;\Omega } - families of generalized extended pseudo-b-distances on U U . Furthermore, we look at these extended J φ ; Ω {{\mathcal{J}}}_{\varphi ;\Omega } - families on U U and define the extended J φ ; Ω {{\mathcal{J}}}_{\varphi ;\Omega } -sequential completeness. We also look into some fixed and periodic point theorems for set-valued mappings in the new space with a graph that does not meet the completeness condition of the space. Additionally, this article includes some examples to explain the corresponding results and highlights some important consequences of our obtained results.

It is of strong theoretical significance and application prospects to explore three-block nonconvex optimization with nonseparable structure, which are often modeled for many problems in machine learning, statistics, and image and signal processing. In this article, by combining the Bregman distance and Peaceman-Rachford splitting method, we propose a novel three-block Bregman Peaceman-Rachford splitting method (3-BPRSM). Under a general assumption, global convergence is presented via optimality conditions. Furthermore, we prove strong convergence when the augmented Lagrange function satisfies Kurdyka-Łojasiewicz property. In addition, if the association function possessing the Kurdyka-Łojasiewicz property exhibits a distinctive structure, then linear and sublinear convergence rate of 3-BPRSM can be guaranteed. Finally, a preliminary numerical experiment demonstrates the effectiveness.

This study is concerned with topological structure of the solution sets to evolution inclusions of neutral type involving measures on compact intervals. By using Górniewicz-Lassonde fixed-point theorem, the existence of solutions and the compactness of solution sets for neutral measure differential inclusions are obtained. Second, based on the R δ {R}_{\delta } -structure equivalence theorem, by constructing a continuous function that can make the solution set homotopic at a single point, the R δ {R}_{\delta } -type structure of the solution sets of this kind of differential inclusion is obtained.

In this study, we generalize fuzzy metric-like, non-Archimedean fuzzy metric-like, and all the variants of fuzzy metric spaces. We propose the idea of fuzzy metric-unlike and non-Archimedean fuzzy metric-unlike, respectively. We also propose the idea of ( α , F ) \left(\alpha ,F) -Geraghty-type generalized F F -contraction mappings utilizing fuzzy metric-unlike and non-Archimedean fuzzy metric-unlike spaces. We investigate the presence of unique fixed points using the recently introduced contraction mappings. In order to complement our study, we consider an application to dynamic market equilibrium.

The objective of this study is to determine the criteria under which the infinite system of integral equations in three variables has a solution in the Banach tempering sequence space c 0 β {c}_{0}^{\beta } and ℓ 1 β {\ell }_{1}^{\beta } , utilizing the Meir-Keeler condensing operators. Our research builds upon the findings of Malik and Jalal. Furthermore, we provide illustrative examples to demonstrate the implications of our established conditions.

In this work, we consider a class of fuzzy fractional delay integro-differential equations with the generalized Caputo-type Atangana-Baleanu (ABC) fractional derivative. By using the monotone iterative method, we not only obtain the existence and uniqueness of the solution for the given problem with the initial condition but also give the monotone iteration sequence converging to the unique solution of the problem. Furthermore, we also give the continuous dependence of the unique solution on initial value. Finally, an example is presented to illustrate the main results obtained. The results presented in this study are new and open a new avenue of research for fuzzy fractional delay integro-differential equations with the generalized ABC fractional derivative.

By the Klein-Gordon potential, we call a convolution-type integral with a kernel, which is the fundamental solution of the Klein-Gordon equation and also a solution of the Cauchy problem to the same equation. An interesting question having several important applications (in general) is what boundary condition can be imposed on the Klein-Gordon potential on the boundary of a given domain so that the Klein-Gordon equation with initial conditions complemented by this “transparent” boundary condition would have a unique solution within that domain still given by the Klein-Gordon potential. It amounts to finding the trace of the Klein-Gordon potential to the boundary of the given domain. In this article, we analyze this question and construct a novel initial boundary-value problem for the Klein-Gordon equation in characteristic coordinates.

In 2008, Cai and Jiu showed that the Cauchy problem of the Navier-Stokes equations, with damping α ∣ u ∣ β − 1 u \alpha {| u| }^{\beta -1}u for α > 0 \alpha \gt 0 and β ≥ 1 \beta \ge 1 has global weak solutions in L 2 ( R 3 ) {L}^{2}\left({{\mathbb{R}}}^{3}) . In this article, we study the uniqueness and the continuity in L 2 ( R 3 ) {L}^{2}\left({{\mathbb{R}}}^{3}) of this global weak solution. We also prove the large time decay for this global solution for β ≥ 10 3 \beta \ge \frac{10}{3} .

In this article, we study the global existence, uniqueness, and continuity for the solution of incompressible convective Brinkman-Forchheimer on the whole space R 3 {{\mathbb{R}}}^{3} when 4 μ β ≥ 1 4\mu \beta \ge 1 . Additionally, we give an asymptotic type of convergence of the global solution.

The present study introduces the Haar wavelet method, which utilizes collocation points to approximate solutions to the Emden-Fowler Pantograph delay differential equations (PDDEs) of general order. This semi-analytic method requires the transformation of the original differential equation into a system of nonlinear differential equations, which is then solved to determine the Haar coefficients. The method’s application to fourth-, fifth-, and sixth-order PDDEs is discussed, along with an examination of convergence that involves the determination of an upper bound and the formulation of the rate of convergence for the method. Numerical simulations and error tables are presented to demonstrate the effectiveness and precision of this approach. The error tables clearly illustrate that the method’s accuracy improves progressively with increasing resolution.

In this article, we focus on the global regularity of n -dimensional liquid crystal equations with fractional dissipation terms ( − Δ ) α u {\left(-\Delta )}^{\alpha }u and ( − Δ ) β d {\left(-\Delta )}^{\beta }d . We show that the equations have a unique global smooth solution if α ≥ 1 2 + n 4 \alpha \ge \frac{1}{2}+\frac{n}{4} and β ≥ 1 2 + n 4 \beta \ge \frac{1}{2}+\frac{n}{4} .

We consider the boundary value problem generated by a system of Dirac equations with polynomials of spectral parameter in the boundary condition. We investigate the continuity of the scattering function and provide Levinson-type formula, which shows that the increment of the scattering function’s logarithm is related to the number of eigenvalues of the boundary value problem.

In previous work, Fayssal considered a thermoelastic laminated beam with structural damping, where the heat conduction is given by the classical Fourier’s law and acting on both the rotation angle and the transverse displacements established an exponential stability result for the considered problem in case of equal wave speeds and a polynomial stability for the opposite case. This article deals with a laminated beam system along with structural damping, past history, and the presence of both temperatures and microtemperature effects. Employing the semigroup approach, we establish the existence and uniqueness of the solution. With the help of convenient assumptions on the kernel, we demonstrate a general decay result for the solution of the considered system, with no constraints regarding the speeds of wave propagations. The result obtained is new and substantially improves earlier results in the literature.

In this article, we investigate the behavior of weak solutions for the three-dimensional Navier-Stokes-Voigt-Brinkman-Forchheimer fluid model with memory and Tresca friction law within a thin domain. We analyze the asymptotic behavior as one dimension of the fluid domain approaches zero. We derive the limit problem and obtain the specific Reynolds equation, while also establishing the uniqueness of the limit velocity and pressure distributions.

We prove the nonexistence of global solutions for the following wave equations with structural damping and nonlinear memory source term u t t + ( − Δ ) α 2 u + ( − Δ ) β 2 u t = ∫ 0 t ( t − s ) δ − 1 ∣ u ( s ) ∣ p d s {u}_{tt}+{\left(-\Delta )}^{\tfrac{\alpha }{2}}u+{\left(-\Delta )}^{\tfrac{\beta }{2}}{u}_{t}=\underset{0}{\overset{t}{\int }}{\left(t-s)}^{\delta -1}{| u\left(s)| }^{p}{\rm{d}}s and u t t + ( − Δ ) α 2 u + ( − Δ ) β 2 u t = ∫ 0 t ( t − s ) δ − 1 ∣ u s ( s ) ∣ p d s , {u}_{tt}+{\left(-\Delta )}^{\tfrac{\alpha }{2}}u+{\left(-\Delta )}^{\tfrac{\beta }{2}}{u}_{t}=\underset{0}{\overset{t}{\int }}{\left(t-s)}^{\delta -1}{| {u}_{s}\left(s)| }^{p}{\rm{d}}s, posed in ( x , t ) ∈ R N × [ 0 , ∞ ) \left(x,t)\in {{\mathbb{R}}}^{N}\times \left[0,\infty ) , where u = u ( x , t ) u=u\left(x,t) is the real-valued unknown function, p > 1 p\gt 1 , α , β ∈ ( 0 , 2 ) \alpha ,\beta \in \left(0,2) , δ ∈ ( 0 , 1 ) \delta \in \left(0,1) , by using the test function method under suitable sign assumptions on the initial data. Furthermore, we give an upper bound estimate of the life span of solutions.

In this article, we study the inviscid limit of the solution to the Cauchy problem of a one-dimensional viscous conservation law, where the second-order term is nonlinear. Under the assumption that the inviscid equation admits a piecewise smooth solution with two noninteracting entropy shocks, we prove that the solution of the viscous equation converges uniformly to the piecewise smooth inviscid solution away from the shocks, even the strength of shocks is not small.

This study deals with the limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays and multiplicative noise on unbounded thin domains. We first prove the existence and uniqueness of pullback tempered random attractors for the systems and then establish the upper semicontinuity of these attractors when the thin domains collapse onto R {\mathbb{R}} .

We present and analyze two kinds of nonconforming finite element methods for three-field Biot’s consolidation model in poroelasticity. We employ the Crouzeix-Raviart element for one of the displacement component and conforming linear element for the remaining component, the lowest order Raviart-Thomas element (or the first-order Brezzi-Douglas-Marini element) for the fluid flux, and the piecewise constant for the pressure. We provide the corresponding analysis, including the well-posedness and a priori error estimates, for the fully discrete scheme coupled with the backward Euler finite difference for the time discretization. Such scheme ensures that the discrete Korn’s inequality is satisfied without adding any stabilization terms. In particular, it is free of poroelasticity locking. Numerical results are presented to compare the accuracy and locking-free performance of the two introduced schemes.