Band 56, Heft 1

The main objective of this study is to define a new class of bipolar soft (BS) separation axioms known as BS T ˜ ˜ i {\widetilde{\widetilde{T}}}_{i} -space ( i = 0 , 1 , 2 , 3 , 4 ) \left(i=0,1,2,3,4) . This type is defined in terms of ordinary points. We prove that BS T ˜ ˜ i {\widetilde{\widetilde{T}}}_{i} -space implies BS T ˜ ˜ i − 1 {\widetilde{\widetilde{T}}}_{i-1} -space for i = 1 , 2 i=1,2 ; however, the opposite is incorrect, as demonstrated by an example. For i = 0 , 1 , 2 , 3 , 4 i=0,1,2,3,4 , we investigate that every BS T ˜ ˜ i {\widetilde{\widetilde{T}}}_{i} -space is soft T ˜ i {\widetilde{T}}_{i} -space; and we set up a condition in which the reverse is true. Moreover, we point out that a BS subspace of a BS T ˜ ˜ i {\widetilde{\widetilde{T}}}_{i} -space is a BS T ˜ ˜ i {\widetilde{\widetilde{T}}}_{i} -space for i = 0 , 1 , 2 , 3 i=0,1,2,3 .

In this article, we investigate a class of analytic functions defined on the unit open disc U = { z : ∣ z ∣ < 1 } {\mathcal{U}}=\left\{z:| z| \lt 1\right\} , such that for every f ∈ P α ( β , γ ) f\in {{\mathcal{P}}}_{\alpha }\left(\beta ,\gamma ) , α > 0 \alpha \gt 0 , 0 ≤ β ≤ 1 0\le \beta \le 1 , 0 < γ ≤ 1 0\lt \gamma \le 1 , and ∣ z ∣ < 1 | z| \lt 1 , the inequality Re f ′ ( z ) + 1 − γ α γ z f ″ ( z ) − β 1 − β > 0 {\rm{Re}}\left\{\frac{f^{\prime} \left(z)+\frac{1-\gamma }{\alpha \gamma }z{f}^{^{\prime\prime} }\left(z)-\beta }{1-\beta }\right\}\gt 0 holds. We find conditions on the numbers α , β \alpha ,\beta , and γ \gamma such that P α ( β , γ ) ⊆ S P ( λ ) {{\mathcal{P}}}_{\alpha }\left(\beta ,\gamma )\subseteq SP\left(\lambda ) , for λ ∈ ( − π 2 , π 2 ) \lambda \in \left(-\frac{\pi }{2},\frac{\pi }{2}) , where S P ( λ ) SP\left(\lambda ) denotes the set of all λ \lambda -spirallike functions. We also make use of Ruscheweyh’s duality theory to derive conditions on the numbers α , β , γ \alpha ,\beta ,\gamma and the real-valued function φ \varphi so that the integral operator V φ ( f ) {V}_{\varphi }(f) maps the set P α ( β , γ ) {{\mathcal{P}}}_{\alpha }\left(\beta ,\gamma ) into the set S P ( λ ) SP\left(\lambda ) , provided φ \varphi is non-negative normalized function ( ∫ 0 1 φ ( t ) d t = 1 ) \left({\int }_{0}^{1}\varphi \left(t){\rm{d}}t=1) and V φ ( f ) ( z ) = ∫ 0 1 φ ( t ) f ( t z ) t d t . {V}_{\varphi }(f)\left(z)=\underset{0}{\overset{1}{\int }}\varphi \left(t)\frac{f\left(tz)}{t}{\rm{d}}t.

In this article, some new generalizations of Darbo’s fixed-point theorem are given and the solvability of an infinite system of weighted fractional integral equations of a function with respect to another function is studied. Also, with the help of a proper example, we illustrate our findings.

We investigate the existence and nonexistence of nonnegative radial solutions to exterior problems of the form Δ H m u ( q ) + λ ψ ( q ) K ( r ( q ) ) f ( r 2 − Q ( q ) , u ( q ) ) = 0 {\Delta }_{{{\mathbb{H}}}^{m}}u\left(q)+\lambda \psi \left(q)K\left(r\left(q))f\left({r}^{2-Q}\left(q),u\left(q))=0 in B 1 c {B}_{1}^{c} , under the Dirichlet boundary conditions u = 0 u=0 on ∂ B 1 \partial {B}_{1} and lim r ( q ) → ∞ u ( q ) = 0 {\mathrm{lim}}_{r\left(q)\to \infty }u\left(q)=0 . Here, λ ≥ 0 \lambda \ge 0 is a parameter, Δ H m {\Delta }_{{{\mathbb{H}}}^{m}} is the Kohn Laplacian on the Heisenberg group H m = R 2 m + 1 {{\mathbb{H}}}^{m}={{\mathbb{R}}}^{2m+1} , m > 1 m\gt 1 , Q = 2 m + 2 Q=2m+2 , B 1 {B}_{1} is the unit ball in H m {{\mathbb{H}}}^{m} , B 1 c {B}_{1}^{c} is the complement of B 1 {B}_{1} , and ψ ( q ) = ∣ z ∣ 2 r 2 ( q ) \psi \left(q)=\frac{| z{| }^{2}}{{r}^{2}\left(q)} . Namely, under certain conditions on K K and f f , we show that there exists a critical parameter λ ∗ ∈ ( 0 , ∞ ] {\lambda }^{\ast }\in \left(0,\infty ] in the following sense. If 0 ≤ λ < λ ∗ 0\le \lambda \lt {\lambda }^{\ast } , the above problem admits a unique nonnegative radial solution u λ {u}_{\lambda } ; if λ ∗ < ∞ {\lambda }^{\ast }\lt \infty and λ ≥ λ ∗ \lambda \ge {\lambda }^{\ast } , the problem admits no nonnegative radial solution. When 0 ≤ λ < λ ∗ 0\le \lambda \lt {\lambda }^{\ast } , a numerical algorithm that converges to u λ {u}_{\lambda } is provided and the continuity of u λ {u}_{\lambda } with respect to λ \lambda , as well as the behavior of u λ {u}_{\lambda } as λ → λ ∗ − \lambda \to {{\lambda }^{\ast }}^{-} , are studied. Moreover, sufficient conditions on the the behavior of f ( t , s ) f\left(t,s) as s → ∞ s\to \infty are obtained, for which λ ∗ = ∞ {\lambda }^{\ast }=\infty or λ ∗ < ∞ {\lambda }^{\ast }\lt \infty . Our approach is based on partial ordering methods and fixed point theory in cones.

The question of whether there is a true isometry approximating the ε \varepsilon -isometry defined in the bounded subset of the n n -dimensional Euclidean space has long been considered an interesting question. In 1982, Fickett published the first article on this topic, and in early 2000, Alestalo et al. and Väisälä improved Fickett’s result significantly. Recently, the second author of this article published a paper improving the previous results. The main purpose of this article is to significantly improve all of the aforementioned results by applying a basic and intuitive method.

We study the uniqueness, the continuity in L 2 {L}^{2} , and the large time decay for the Leray solutions of the 3D incompressible Navier-Stokes equations with the nonlinear exponential damping term a ( e b ∣ u ∣ 2 − 1 ) u a\left({e}^{b| u{| }^{{\bf{2}}}}-1)u , ( a , b > 0 a,b\gt 0 ).

The Laplace transform method is applied in this article to study the semi-Hyers-Ulam-Rassias stability of a Volterra integro-differential equation of order n, with convolution-type kernel. This kind of stability extends the original Hyers-Ulam stability whose study originated in 1940. A general integral equation is formulated first, and then some particular cases (polynomial function and exponential function) for the function from the kernel are considered.

In this article, we define the following: Let N 0 {{\mathbb{N}}}_{0} be the set of all nonnegative integers and D = ( d i ) i ∈ N 0 D={\left({d}_{i})}_{i\in {{\mathbb{N}}}_{0}} a family of additive mappings of a ∗ \ast -ring R R such that d 0 = i d R {d}_{0}=i{d}_{R} . D D is called a Jordan ( α , β ) \left(\alpha ,\beta ) - higher ∗ \ast - derivation (resp. a Jordan triple ( α , β ) \left(\alpha ,\beta ) - higher ∗ \ast - derivation ) of R R if d n ( a 2 ) = ∑ i + j = n d i ( β j ( a ) ) d j ( α i ( a ∗ i ) ) {d}_{n}\left({a}^{2})={\sum }_{i+j=n}{d}_{i}\left({\beta }^{j}\left(a)){d}_{j}\left({\alpha }^{i}\left({a}^{{\ast }^{i}})) (resp. d n ( a b a ) = ∑ i + j + k = n d i ( β j + k ( a ) ) d j ( β k ( α i ( b ∗ i ) ) ) d k ( α i + j ( a ∗ i + j ) ) {d}_{n}\left(aba)={\sum }_{i+j+k=n}{d}_{i}\left({\beta }^{j+k}\left(a)){d}_{j}\left({\beta }^{k}\left({\alpha }^{i}\left({b}^{{\ast }^{i}}))){d}_{k}\left({\alpha }^{i+j}\left({a}^{{\ast }^{i+j}})) ) for all a , b ∈ R a,b\in R and each n ∈ N 0 n\in {{\mathbb{N}}}_{0} . We show that the two notions of Jordan ( α , β ) \left(\alpha ,\beta ) -higher ∗ \ast -derivation and Jordan triple ( α , β ) \left(\alpha ,\beta ) -higher ∗ \ast -derivation on a 6-torsion free semiprime ∗ \ast -ring are equivalent.

The Kirchhoff model is derived from the vibration problem of stretchable strings. This article focuses on the long-time dynamics of a class of higher-order coupled Kirchhoff systems with nonlinear strong damping. The existence and uniqueness of the solutions of these equations in different spaces are proved by prior estimation and the Faedo-Galerkin method. Subsequently, the family of global attractors of these problems is proved using the compactness theorem. In this article, we systematically propose the definition and proof process of the family of global attractors and enrich the related conclusions of higher-order coupled Kirchhoff models. The conclusions lay a theoretical foundation for future practical applications.

In this article, an approximation of the image of the closed ball of the space L p {L}_{p} ( p > 1 p\gt 1 ) centered at the origin with radius r r under Hilbert-Schmidt integral operator F ( ⋅ ) : L p → L q F\left(\cdot ):{L}_{p}\to {L}_{q} , 1 p + 1 q = 1 \frac{1}{p}+\frac{1}{q}=1 is considered. An error evaluation for the given approximation is obtained.

The aim of this article is to introduce α \alpha - ψ \psi -proximal contraction in the setting of ℱ -metric space and prove the existence of best proximity points for these contractions. As applications of our main results, we obtain coupled best proximity points on ℱ -metric space equipped with an arbitrary binary relation.

In this manuscript, we first generate topological structures by subset neighborhoods and ideals and apply to establish some generalized rough-set models. Then, we present other types of generalized rough-set models directly defined by the concepts of subset neighborhoods and ideals. We explore the main characterizations of the proposed approximation spaces and compare them in terms of approximation operators and accuracy measures. The obtained results and given examples show that the second type of the proposed approximation spaces is better than the first one in cases of u u and ⟨ u ⟩ \langle u\rangle , whereas the relationships between the rest of the six cases are posted as an open question. Moreover, we demonstrate the advantages of the current models to decrease the upper approximation and increase the lower approximation compared to the existing approaches in published literature. Algorithms and a flow chart are given to illustrate how the exact and rough sets are determined for each approach. Finally, we analyze the information system of dengue fever to confirm the efficiency of our approaches to maximize the value of accuracy and shrink the boundary regions.

In this research work, we proposed a Haar wavelet collocation method (HWCM) for the numerical solution of first- and second-order nonlinear hyperbolic equations. The time derivative in the governing equations is approximated by a finite difference. The nonlinear hyperbolic equation is converted into its full algebraic form once the space derivatives are replaced by the finite Haar series. Convergence analysis is performed both in space and time, where the computational results follow the theoretical statements of convergence. Many test problems with different nonlinear terms are presented to verify the accuracy, capability, and convergence of the proposed method for the first- and second-order nonlinear hyperbolic equations.

In this article, a numerical solution for the Rosenau-Kawahara equation is considered. A new conservative numerical approximation scheme is presented to solve the initial boundary value problem of the Rosenau-Kawahara equation, which preserves the original conservative properties. The proposed scheme is based on the finite difference method. The existence of the numerical solutions for the scheme has been shown by Browder fixed point theorem. The priori bound and error estimates, as well as the conservation of discrete mass and discrete energy for the finite difference solutions, are discussed. The discrepancies of discrete mass and energy are computed and shown by the curves of these quantities over time. Unconditional stability, second-order convergence, and uniqueness of the scheme are proved based on the discrete energy method. Numerical examples are given to show the effectiveness of the proposed scheme and confirm the theoretical analysis.

In this article, we study the Fekete-Szegö functional associated with a new class of analytic functions related to the class of bounded turning by using the principle of quasi-subordination. We derived the coefficient estimates including the classical Fekete-Szegö inequality for functions belonging to this class. We also improved some existing results.

Local fractional integral inequalities of Hermite-Hadamard type involving local fractional integral operators with Mittag-Leffler kernel have been previously studied for generalized convexities and preinvexities. In this article, we analyze Hermite-Hadamard-type local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) \left({\tilde{h}}_{1},{\tilde{h}}_{2}) -preinvex function comprising local fractional integral operators and Mittag-Leffler kernel. In addition, two examples are discussed to ensure that the derived consequences are correct. As an application, we construct an inequality to establish central moments of a random variable.

The main aim of this article is to present some novel geometric properties for three distinct normalizations of the generalized k k -Bessel functions, such as the radii of uniform convexity and of α \alpha -convexity. In addition, we show that the radii of α \alpha -convexity remain in between the radii of starlikeness and convexity, in the case when α ∈ [ 0 , 1 ] , \alpha \in {[}0,1], and they are decreasing with respect to the parameter α . \alpha . The key tools in the proof of our main results are infinite product representations for normalized k k -Bessel functions and some properties of real zeros of these functions.

In this article, weak and strong convergence theorems of the M-iteration method for 𝒢-nonexpansive mapping in a uniformly convex Banach space with a directed graph were established. Moreover, weak convergence theorem without making use of Opial’s condition is proved. The rate of convergence between the M-iteration and some other iteration processes in the literature was also compared. Specifically, our main result shows that the M-iteration converges faster than the Noor and SP iterations. Finally, the numerical examples to compare convergence behavior of the M-iteration with the three-step Noor iteration and the SP-iteration were given. As application, some numerical experiments in real-world problems were provided, focused on image deblurring and signal recovering problems.

Let S γ , A , B ∗ ( D ) {S}_{\gamma ,A,B}^{\ast }\left({\mathbb{D}}) be the usual class of g g -starlike functions of complex order γ \gamma in the unit disk D = { ζ ∈ C : ∣ ζ ∣ < 1 } {\mathbb{D}}=\left\{\zeta \in {\mathbb{C}}:| \zeta | \lt 1\right\} , where g ( ζ ) = ( 1 + A ζ ) ∕ ( 1 + B ζ ) g\left(\zeta )=\left(1+A\zeta )/\left(1+B\zeta ) , with γ ∈ C \ { 0 } , − 1 ≤ A < B ≤ 1 , ζ ∈ D \gamma \left\in {\mathbb{C}}\backslash \left\{0\right\}\right,-1\le A\lt B\le 1,\zeta \in {\mathbb{D}} . First, we obtain the bounds of all the coefficients of homogeneous expansions for the functions f ∈ S γ , A , B ∗ ( D ) f\in {S}_{\gamma ,A,B}^{\ast }\left({\mathbb{D}}) when ζ = 0 \zeta =0 is a zero of order k + 1 k+1 of f ( ζ ) − ζ f\left(\zeta )-\zeta . Second, we generalize this result to several complex variables by considering the corresponding biholomorphic mappings defined in a bounded complete Reinhardt domain. These main theorems unify and extend many known results.

Let G G be a group and R R be a G G -graded commutative ring with nonzero unity 1. In this article, we introduce the concept of graded weakly 1-absorbing primary ideals which is a generalization of graded 1-absorbing primary ideal. A proper graded ideal P P of R R is said to be a graded weakly 1-absorbing primary ideal of R R if whenever nonunit elements x , y , z ∈ h ( R ) x,y,z\in h\left(R) such that 0 ≠ x y z ∈ P 0\ne xyz\in P , then x y ∈ P xy\in P or z n ∈ P {z}^{n}\in P , for some n ∈ N n\in {\mathbb{N}} . Several properties of graded weakly 1-absorbing primary ideals are investigated.

The paradigm of choice practice represents the psychological theory of learning in the development of moral judgment. It is concerned with evaluating the implications of several choices and selecting one of them to implement. The goal of this work is to provide a generic functional equation to observe the behavior of animals in such circumstances. Our suggested functional equation can be employed to describe several well-known psychology and learning theories. The fixed point theorem proposed by Banach is utilized to show that the solution of a given functional problem exists and is unique. In addition, the stability of the given functional equation’s solution is discussed in terms of the Hyers-Ulam and Hyers-Ulam-Rassias results. Furthermore, two examples are provided to highlight the relevance of the significant outcomes in the context of the literature.

In this article, a definition of a b ( α n , β n ) {b}_{\left({\alpha }_{n},{\beta }_{n})} -best approximations of b ( α n , β n ) {b}_{\left({\alpha }_{n},{\beta }_{n})} -hypermetric spaces over Banach algebras is given. Our objective is to prove the concept of extension of fixed point theorems in n n -ary orthogonal b ( α n , β n ) {b}_{\left({\alpha }_{n},{\beta }_{n})} -hypermetric spaces over Banach algebras.

In this article, one standard and four nonstandard finite difference methods are used to solve a cross-diffusion malignant invasion model. The model consists of a system of nonlinear coupled partial differential equations (PDEs) subject to specified initial and boundary conditions, and no exact solution is known for this problem. It is difficult to obtain theoretically the stability region of the classical finite difference scheme to solve the set of nonlinear coupled PDEs, this is one of the challenges of this class of method in this work. Three nonstandard methods abbreviated as NSFD1, NSFD2, and NSFD3 are considered from the study of Chapwanya et al., and these methods have been constructed by the use of a more general function replacing the denominator of the discrete derivative and nonlocal approximations of nonlocal terms. It is shown that NSFD1, which preserves positivity when used to solve classical reaction-diffusion equations, does not inherit this property when used for the cross-diffusion system of PDEs. NSFD2 and NSFD3 are obtained by appropriate modifications of NSFD1. NSFD2 is positivity-preserving when the functional relationship [ ψ ( h ) ] 2 = 2 ϕ ( k ) {\left[\psi \left(h)]}^{2}=2\phi \left(k) holds, while NSFD3 is unconditionally dynamically consistent with respect to positivity. First, we show that NSFD2 and NSFD3 are not consistent methods. Second, we tried to modify NSFD2 in order to make it consistent but we were not successful. Third, we extend NSFD3 so that it becomes consistent and still preserves positivity. We denote the extended version of NSFD3 as NSFD5. Finally, we compute the numerical rate of convergence in time for NSFD5 and show that it is close to the theoretical value. NSFD5 is consistent under certain conditions on the step sizes and is unconditionally positivity-preserving.

In this article, with the help of an integral representation of the Dirichlet lambda function, by means of a monotonicity rule for the ratio of two integrals with a parameter, and by virtue of complete monotonicity and another property of an elementary function involving the exponential function, the authors find increasing property and logarithmic convexity of two functions containing the gamma function and the Dirichlet lambda function.

As a crucial medium of information dissemination, text holds a pivotal role in a multitude of applications. However, text detection in complex and unstructured environments presents significant challenges, such as the presence of cluttered backgrounds, variations in appearance, and uneven lighting conditions. To address this issue, this study proposes a text detection framework that leverages multistage edge detection and contextual information. This framework deviates from traditional approaches by incorporating four primary processing steps, including text visual saliency region detection to accentuate the text regions and diminish background interference, multistage edge detection to enhance the conventional stroke width transform results, a texture-based and connected components-based integration to accurately distinguish text from the background, and a context fusion step to recover missing text regions and improve the recall of text detection. The proposed method was evaluated on two widely used benchmark datasets, i.e., the international conference on document analysis and recognition (ICDAR) 2005 dataset and the ICDAR 2011 dataset, and the results indicate the advancedness of the method.

In this article, a class of cyclic (noncyclic) operators are defined on Banach spaces via concept of measure of noncompactness using some abstract functions. The best proximity point (pair) results are manifested for the said operators. The obtained main results are applied to demonstrate the existence of optimum solutions of a system of fractional differential equations involving ( k , ψ ) \left(k,\psi ) -Hilfer fractional derivatives.

The purpose of this article is to describe the properties of the pair of solutions of several systems of Fermat-type partial differential difference equations. Our theorems exhibit the forms of finite order transcendental entire solutions for these systems, which are some extensions and improvement of the previous theorems given by Xu, Cao, Liu, etc. Furthermore, we give a series of examples to show that the existence conditions and the forms of transcendental entire solutions with finite order of such systems are precise.

In this article, we study the regularity criteria of the weak solutions to the Boussinesq equations involving the horizontal component of velocity or the horizontal derivatives of the two components of velocity in anisotropic Lorentz spaces. This result reveals that the velocity field plays a dominant role in regularity theory of the Boussinesq equations.

The aim of this study is to investigate the boundedness, essential norm, and compactness of generalized Stević-Sharma operator from the minimal Möbius invariant space into Bloch-type space.

In this article, we are interested to study the elliptic equation under the Caputo derivative. We obtain several regularity results for the mild solution based on various assumptions of the input data. In addition, we derive the lower bound of the mild solution in the appropriate space. The main tool of the analysis estimation for the mild solution is based on the bound of the Mittag-Leffler functions, combined with analysis in Hilbert scales space. Moreover, we provide a regularized solution for our problem using the Fourier truncation method. We also obtain the error estimate between the regularized solution and the mild solution. Our current article seems to be the first study to deal with elliptic equations with Caputo derivatives on the unbounded domain.

In this article, we introduce a novel block preconditioner for block two-by-two linear equations by expanding the dimension of the coefficient matrix. Theoretical results on the eigenvalues distribution of the preconditioned matrix are obtained, and a feasible implementation is discussed. Some numerical examples, including the solution of the Navier-Stokes equations, are presented to support the theoretical findings and demonstrate the preconditioner’s efficiency.

This article presents the general solution f : G → V f:{\mathcal{G}}\to {\mathcal{V}} of the following functional equation: f ( x ) − 4 f ( x + y ) + 6 f ( x + 2 y ) − 4 f ( x + 3 y ) + f ( x + 4 y ) = 0 , x , y ∈ G , f\left(x)-4f\left(x+y)+6f\left(x+2y)-4f\left(x+3y)+f\left(x+4y)=0,\hspace{1.0em}x,y\in {\mathcal{G}}, where ( G , + ) \left({\mathcal{G}},+) is an abelian group and V {\mathcal{V}} is a linear space. We also investigate its Hyers-Ulam stability on some restricted domains. We apply the obtained results to present some asymptotic behaviors of this functional equation in the framework of normed spaces. Finally, we provide some characterizations of inner product spaces associated with the mentioned functional equation.

In this article, we are concerned with the following critical nonlocal equation with variable exponents: ( − Δ ) p ( x , y ) s u = λ f ( x , u ) + ∣ u ∣ q ( x ) − 2 u in Ω , u = 0 in R N \ Ω , \left\{\begin{array}{ll}{\left(-\Delta )}_{p\left(x,y)}^{s}u=\lambda f\left(x,u)+{| u| }^{q\left(x)-2}u& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=0& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}\backslash \Omega \right,\end{array}\right. where Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} is a bounded domain with Lipschitz boundary, N ≥ 2 N\ge 2 , p ∈ C ( Ω × Ω ) p\in C(\Omega \times \Omega ) is symmetric, f : C ( Ω × R ) → R f:C\left(\Omega \times {\mathbb{R}})\to {\mathbb{R}} is a continuous function, and λ \lambda is a real positive parameter. We also assume that { x ∈ R N : q ( x ) = p s ∗ ( x ) } ≠ ∅ \left\{x\in {{\mathbb{R}}}^{N}:q\left(x)={p}_{s}^{\ast }\left(x)\right\}\ne \varnothing , and p s ∗ ( x ) = N p ˜ ( x ) ⁄ ( N − s p ˜ ( x ) ) {p}_{s}^{\ast }\left(x)=N\tilde{p}\left(x)/\left(N-s\tilde{p}\left(x)) is the critical Sobolev exponent for variable exponents. We prove the existence of non-trivial solutions in the case of low perturbations ( λ \lambda small enough) by using the mountain pass theorem, the concentration-compactness principles for fractional Sobolev spaces with variable exponents, and the Moser iteration method. The features of this article are the following: (1) the function f f does not satisfy the usual Ambrosetti-Rabinowitz condition and (2) this article contains the presence of critical terms, which can be viewed as a partial extension of the previous results concerning the the existence of solutions to this problem in the case of s = 1 s=1 and subcritical case.

First, we introduce the concepts of equicontinuity, expansivity , ergodic shadowing property, and chain transitivity in uniform space. Second, we study the dynamical properties of equicontinuity, expansivity , ergodic shadowing property, and chain transitivity in the hyperspace of uniform space. Let ( X , μ ) \left(X,\mu ) be a uniform space, ( C ( X ) , C μ ) \left(C\left(X),{C}^{\mu }) be a hyperspace of ( X , μ ) \left(X,\mu ) , and f : X → X f:X\to X be uniformly continuous. By using the relationship between original space and hyperspace, we obtain the following results: (a) the map f f is equicontinous if and only if the induced map C f {C}^{f} is equicontinous; (b) if the induced map C f {C}^{f} is expansive, then the map f f is expansive; (c) if the induced map C f {C}^{f} has ergodic shadowing property, then the map f f has ergodic shadowing property; (d) if the induced map C f {C}^{f} is chain transitive, then the map f f is chain transitive. In addition, we also study the topological conjugate invariance of ( G , h ) \left(G,h) -shadowing property in metric G G - space and prove that the map S S has ( G , h ) \left(G,h) -shadowing property if and only if the map T T has ( G , h ) \left(G,h) -shadowing property. These results generalize the conclusions of equicontinuity, expansivity, ergodic shadowing property, and chain transitivity in hyperspace.

A proper graded ideal P P of a commutative graded ring R R is called graded weakly 1-absorbing primary if whenever x , y , z x,y,z are nonunit homogeneous elements of R R with 0 ≠ x y z ∈ P 0\ne xyz\in P , then either x y ∈ P xy\in P or z z is in the graded radical of P P . In this article, we explore more results on graded weakly 1-absorbing primary ideals.

The detection problem of a finite number of source points acting on a steady incompressible fluid flow from overdetermined boundary data was studied. The approach used in this study deals with the topological sensitivity technique. An asymptotic analysis of a prescribed cost function with respect to the domain perturbation was developed. Some numerical results to illustrate the efficiency and robustness of the developed source point detection algorithm were presented.

The main goal of this study was to look into some new integral transformations that are associated with a generalized k k -Bessel function. Integral formulas for the generalized k k -Bessel function have been established using the Laplace transform, Euler transform, Whittaker transform, and k k -transforms. The results presented here have the potential to be helpful, and some special cases of corollaries are explicitly demonstrated.

In this article, we introduce a new family of real hypersurfaces in the complex hyperbolic quadric Q n ∗ = S O 2 , n o ∕ S O 2 S O n {{Q}^{n}}^{\ast }=S{O}_{2,n}^{o}/S{O}_{2}S{O}_{n} , namely, the ruled real hypersurfaces foliated by complex hypersurfaces. Berndt described an example of such a real hypersurface in Q n ∗ {{Q}^{n}}^{\ast } as a homogeneous real hypersurface generated by a A {\mathfrak{A}} -principal horocycle in a real form R H n {\mathbb{R}}{H}^{n} . So, in this article, we compute a detailed expression of the shape operator for ruled real hypersurfaces in Q n ∗ {{Q}^{n}}^{\ast } and investigate their characterizations in terms of the shape operator and the integrable distribution C = { X ∈ T M ∣ X ⊥ ξ } {\mathcal{C}}=\left\{X\in TM| X\perp \xi \right\} . Then, by using these observations, we give two kinds of classifications of real hypersurfaces in Q n ∗ {{Q}^{n}}^{\ast } satisfying η \eta -parallelism under either η \eta -commutativity of the shape operator or integrability of the distribution C {\mathcal{C}} . Moreover, we prove that the unit normal vector field of a real hypersurface with η \eta -parallel shape operator in Q n ∗ {{Q}^{n}}^{\ast } is A {\mathfrak{A}} -principal. On the other hand, it is known that all contact real hypersurfaces in Q n ∗ {{Q}^{n}}^{\ast } have a A {\mathfrak{A}} -principal normal vector field. Motivated by these results, we give a characterization of contact real hypersurfaces in Q n ∗ {{Q}^{n}}^{\ast } in terms of η \eta -parallel shape operator.

In this work, we study new asymptotic properties of positive solutions of the even-order neutral differential equation with the noncanonical operator. The new properties are iterative, which means they can be used several times. We also use these properties to obtain new criteria for oscillation of the studied equation.

In this study, we obtained the stability of the multi-variable bi-Jensen-type functional equation: n 2 f x 1 + ⋯ + x n n , y 1 + ⋯ + y n n = ∑ i = 1 n ∑ j = 1 n f ( x i , y j ) . {n}^{2}f\left(\frac{{x}_{1}+\cdots +{x}_{n}}{n},\frac{{y}_{1}+\cdots +{y}_{n}}{n}\right)=\mathop{\sum }\limits_{i=1}^{n}\mathop{\sum }\limits_{j=1}^{n}f\left({x}_{i},{y}_{j}).

In this article, we introduce and study some strong convergence theorems for a mixed-type SP-iteration for three asymptotically nonexpansive self-mappings and three asymptotically nonexpansive nonself-mappings in uniformly convex hyperbolic spaces. In addition to that, we provide an illustrative example. The findings here expand and improve upon some of the relevant conclusions found in the published literature.

Let n n and m m be two positive integers, and the second-order Fermat-type functional equation f n + ( f ″ ) m ≡ 1 {f}^{n}+{({f}^{^{\prime\prime} })}^{m}\equiv 1 does not have a nonconstant meromorphic solution in the complex plane, except ( n , m ) ∈ { ( 1 , 1 ) , ( 1 , 2 ) , ( 1 , 3 ) , ( 2 , 1 ) , ( 3 , 1 ) } \left(n,m)\in \left\{\left(1,1),\left(1,2),\left(1,3),\left(2,1),\left(3,1)\right\} . The research gives a ready-to-use scheme to study certain Fermat-type functional differential equations in the complex plane by using the Nevanlinna theory, the complex method, and the Weierstrass factorization theorem.

Recently, integral transforms are a powerful tool used in many areas of mathematics, physics, engineering, and other fields and disciplines. This article is devoted to the study of one important integral transform, which is called the modified degenerate Laplace transform (MDLT). The fundamental formulas and properties of the MDLT are obtained. Furthermore, as an application of the acquired MDLT, we solved a simple differential equation and fractional-order kinetic equations. The outcomes covered here are general in nature and easily reducible to new and known outcomes.

In this article, we investigate the existence and the precise form of finite-order transcendental entire solutions of some system of Fermat-type quadratic binomial and trinomial shift equations in C n {{\mathbb{C}}}^{n} . Our results are the generalizations of the results of [H. Y. Xu, S. Y. Liu, and Q. P. Li, Entire solutions for several systems of nonlinear difference and partial differential-difference equations of Fermat-type , J. Math. Anal. Appl. 483 (2020), 123641, 1–22, DOI: https://doi.org/10.1016/j.jmaa.2019.123641.] and [H. Y. Xu and Y. Y. Jiang, Results on entire and meromorphic solutions for several systems of quadratic trinomial functional equations with two complex variables , RACSAM 116 (2022), 8, DOI: https://doi.org/10.1007/s13398-021-01154-9.] to a large extent. Most interestingly, as a consequence of our main result, we have shown that the system of quadratic trinomial shift equation has no solution when it reduces to a system of quadratic trinomial difference equation. In addition, some examples relevant to the content of the article have been exhibited.

Being informed that one of our articles is cited in the paper mentioned in the title, we downloaded it, and we were surprised to see that, practically, all the results from our paper were reproduced in Section 3 of Meghea and Stamin’s article. Having in view the title of the article, one is tempted to think that the remarks mentioned in the paper are original and there are examples given as to where and how (at least) some of the reviewed results are effectively applied. Unfortunately, a closer look shows that most of those remarks in Section 3 are, in fact, extracted from our article, and it is not shown how a specific result is used in a certain application. So, our aim in the present note is to discuss the content of Section 3 of Meghea and Stamin’s paper, emphasizing their Remark 8, in which it is asserted that the proof of Lemma 7 in our article is “full of errors.”

In this article, we considered the pseudo-parabolic equation with Caputo-Fabrizio fractional derivative. This equation has many applications in different fields, such as science, technology, and so on. In this article, we gave the formula of mild solution, which is represented in the form of Fourier series by some operators . In the linear case, we investigated the continuity of the mild solution with respect to the fractional order. For the nonlinear case, we investigated the existence and uniqueness of a global solution. The main proof technique is based on the Banach fixed point theorem combined with some Sobolev embeddings. For more detailed, we obtained two other interesting results: the continuity of mild solution with respect to the derivative order and the convergence of solution as the coefficient k approaches to zero.

The aim of this article is to introduce a new type of operator similar to those of A. Petruşel and G. Petruşel type ( Fixed point results for decreasing convex orbital operators , J. Fixed Point Theory Appl. 23 (2021), no. 35) and prove some fixed-point theorems which generalize and complement several results in the theory of nonlinear operators.

We introduced a new general class of Prešić-type operators, by enriching the known class of Prešić contractions. We established conditions under which enriched Prešić operators possess a unique fixed point, proving the convergence of two different iterative methods to the fixed point. We also gave a data dependence result that was finally applied in proving the global asymptotic stability of the equilibrium of a certain k -th order difference equation.

In this article, we discuss equilibrium problems and their resolvents on complete geodesic spaces. In particular, we consider asymptotic behavior and continuity of resolvents with positive parameter in a complete geodesic space whose curvature is bounded above. Furthermore, we apply these results to resolvents of convex functions.

In this article, we solve the system of additive functional equations: 2 f ( x + y ) − g ( x ) = g ( y ) , g ( x + y ) − 2 f ( y − x ) = 4 f ( x ) \left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}2f\left(x+y)-g\left(x)=g(y),\\ g\left(x+y)-2f(y-x)=4f\left(x)\end{array}\right. and prove the Hyers-Ulam stability of the system of additive functional equations in complex Banach spaces. Furthermore, we prove the Hyers-Ulam stability of f f -hom-ders in Banach algebras.

In this work, we present a new proximal gradient algorithm based on Tseng’s extragradient method and an inertial technique to solve the convex minimization problem in real Hilbert spaces. Using the stepsize rules, the selection of the Lipschitz constant of the gradient of functions is avoided. We then prove the weak convergence theorem and present the numerical experiments for image recovery. The comparative results show that the proposed algorithm has better efficiency than other methods.

In this article, we discuss the existence of a unique solution to a ψ \psi -Hilfer fractional differential equation involving the p p -Laplacian operator subject to nonlocal ψ \psi -Riemann-Liouville fractional integral boundary conditions. Banach’s fixed point theorem is the main tool of our study. Examples are given for illustrating the obtained results.

Cauchy problems with fractal-fractional differential operators with a power law, exponential decay, and the generalized Mittag-Leffler kernels are considered in this work. We start with deriving some important inequalities, and then by using the linear growth and Lipchitz conditions, we derive the conditions under which these equations admit unique solutions. A numerical scheme was suggested for each case to derive a numerical solution to the equation. Some examples of fractal-fractional differential equations were presented, and their exact solutions were obtained and compared with the used numerical scheme. A nonlinear case was considered and solved, and numerical solutions were presented graphically for different values of fractional orders and fractal dimensions.

This article introduces a type of dominating property, partially inherited from L. Chen’s, and proves an existence and uniqueness theorem concerning common best proximity points. A certain kind of boundary value problem involving the so-called Caputo derivative can be formulated so that our result applies.

In this article, we introduce and study a new class of hybrid fractional q q -integro-difference equations involving Riemann-Liouville q q -derivatives, supplemented with nonlocal boundary conditions containing Riemann-Liouville q q -integrals of different orders. The existence of a unique solution to the given problem is shown by applying Banach’s fixed point theorem. We also present the existing criteria for solutions to the problem at hand by applying Krasnoselskii’s fixed point theorem and Leray-Schauder’s nonlinear alternative. Illustrative examples are given to demonstrate the application of the obtained results. Some new results follow as special cases of this work.

In this article, we present a structure result concerning fuzzy fractals generated by an orbital fuzzy iterated function system ( ( X , d ) , ( f i ) i ∈ I , ( ρ i ) i ∈ I ) \left(\left(X,d),{({f}_{i})}_{i\in I},{\left({\rho }_{i})}_{i\in I}) . Our result involves the following two main ingredients: (a) the fuzzy fractal associated with the canonical iterated fuzzy function system ( ( I N , d Λ ) , ( τ i ) i ∈ I , ( ρ i ) i ∈ I ) \left(\left({I}^{{\mathbb{N}}},{d}_{\Lambda }),{\left({\tau }_{i})}_{i\in I},{\left({\rho }_{i})}_{i\in I}) , where d Λ {d}_{\Lambda } is Baire’s metric on the code space I N {I}^{{\mathbb{N}}} and τ i : I N → I N {\tau }_{i}:{I}^{{\mathbb{N}}}\to {I}^{{\mathbb{N}}} is given by τ i ( ( ω 1 , ω 2 , … ) ) ≔ ( i , ω 1 , ω 2 , … ) {\tau }_{i}\left(\left({\omega }_{1},{\omega }_{2},\ldots )):= \left(i,{\omega }_{1},{\omega }_{2},\ldots ) for every ( ω 1 , ω 2 , … ) ∈ I N \left({\omega }_{1},{\omega }_{2},\ldots )\in {I}^{{\mathbb{N}}} and every i ∈ I i\in I ; (b) the canonical projections of certain iterated function systems associated with the fuzzy fractal under consideration.

We prove that if a self-affine arc γ ∈ R 2 \gamma \in {{\mathbb{R}}}^{2} does not satisfy weak separation condition, then it is a segment of a parabola or a straight line. If a self-affine arc γ \gamma is not a segment of a parabola or a line, then it is a component of the attractor of a Jordan multizipper with the same set of generators.

In this article, a new system of Hadamard-type hybrid fractional differential inclusions equipped with Dirichlet boundary conditions was constructed. By virtue of a fixed-point theorem due to B. C. Dhage, ( Existence results for neutral functional differential inclusions in Banach algebras , Nonlinear Anal. 64 (2006), no. 6, 1290–1306, doi: https://doi.org/10.1016/j.na.2005.06.036), the existence results of solutions for the considered problem are derived in a new norm space for multivalued maps. A numerical example is provided to illustrate our main results.

In this article, using critical point theory and variational methods, we investigate the existence of at least three solutions for a class of double eigenvalue discrete anisotropic Kirchhoff-type problems. An example is presented to demonstrate the applicability of our main theoretical findings.

In this article, we introduce and study the notion of split generalized equilibrium problem with multiple output sets (SGEPMOS). We propose a new iterative method that employs viscosity approximation technique for approximating the common solution of the SGEPMOS and common fixed point problem for an infinite family of multivalued demicontractive mappings in real Hilbert spaces. Under mild conditions, we prove a strong convergence theorem for the proposed method. Our method uses self-adaptive step size that does not require prior knowledge of the operator norm. The results presented in this article unify, complement, and extend several existing recent results in the literature.

The study presented in this article involves q -calculus connected to fractional calculus applied in the univalent functions theory. Riemann-Liouville fractional integral of q -hypergeometric function is defined here, and investigations are conducted using the theories of differential subordination and superordination. Theorems and corollaries containing new subordination and superordination results are proved for which best dominants and best subordinants are given, respectively. As an application of the results obtained by the means of the two theories, the statement of a sandwich-type theorem concludes the study.

The aim of this article is to investigate the neutrosophic Nörlund ℐ -statistically convergent sequence space. We present some neutrosophic normed spaces (NNSs) in Nörlund convergent spaces. In addition, we also examine various topological and algebraic properties of these convergent sequence spaces. Theorems are proved in light of the NNS theory approach. Results are obtained via different perspectives and new examples are produced to justify the counterparts and show the existence of the introduced notions. The results established in this research work supply an exhaustive foundation in NNS and make a significant contribution to the theoretical development of NNS in the literature. The original aspect of this study is the first wholly up-to-date and thorough examination of the features and implementation of neutrosophic Nörlund ℐ -statistically convergent sequences in NNS, based upon the standard definition.

The goal of this study is to analyse the eigenvalues and weak eigenfunctions of a new type of multi-interval Sturm-Liouville problem (MISLP) which differs from the standard Sturm-Liouville problems (SLPs) in that the Strum-Liouville equation is defined on a finite number of non-intersecting subintervals and the boundary conditions are set not only at the endpoints but also at finite number internal points of interaction. For the self-adjoint treatment of the considered MISLP, we introduced some self-adjoint linear operators in such a way that the considered multi-interval SLPs can be interpreted as operator-pencil equation. First, we defined a concept of weak solutions (eigenfunctions) for MISLPs with interface conditions at the common ends of the subintervals. Then, we found some important properties of eigenvalues and corresponding weak eigenfunctions. In particular, we proved that the spectrum is discrete and the system of weak eigenfunctions forms a Riesz basis in appropriate Hilbert space.

The harmonic numbers and generalized harmonic numbers appear frequently in many diverse areas such as combinatorial problems, many expressions involving special functions in analytic number theory, and analysis of algorithms. The aim of this article is to derive some identities involving generalized harmonic numbers and generalized harmonic functions from the beta functions F n ( x ) = B ( x + 1 , n + 1 ) , ( n = 0 , 1 , 2 , … ) {F}_{n}\left(x)=B\left(x+1,n+1),\left(n=0,1,2,\ldots ) using elementary methods. For instance, we show that the Hurwitz zeta function ζ ( x + 1 , r ) \zeta \left(x+1,r) and r ! r\&#x0021; are expressed in terms of those numbers and functions, for every r = 2 , 3 , 4 , 5 r=2,3,4,5 .

In the 1970s, Rota began to build completely rigid foundations for the theory of umbral calculus based on relatively modern ideas of linear functions and linear operators. Since then, umbral calculus has been used in the study of special functions in various fields. In this article, we derive some new and interesting identities related to degenerate derangement polynomials and some special polynomials by using λ \lambda -Sheffer sequences and λ \lambda -umbral calculus, which are defined by Kim-Kim ( Degenerate Sheffer sequences and λ \lambda - Sheffer sequences , J. Math. Anal. Appl. 493 (2021), 124521, 21pp).

It is known that the Stirling numbers of the second kind are related to normal ordering in the Weyl algebra, while the unsigned Stirling numbers of the first kind are related to normal ordering in the shift algebra. Recently, Kim-Kim introduced a λ \lambda -analogue of the unsigned Stirling numbers of the first kind and that of the r r -Stirling numbers of the first kind. In this article, we introduce a λ \lambda -analogue of the shift algebra (called λ \lambda -shift algebra) and investigate normal ordering in the λ \lambda -shift algebra. From the normal ordering in the λ \lambda -shift algebra, we derive some identities about the λ \lambda -analogue of the unsigned Stirling numbers of the first kind.

Nonlocal operators with different kernels were used here to obtain more general harmonic oscillator models. Power law, exponential decay, and the generalized Mittag-Leffler kernels with Delta-Dirac property have been utilized in this process. The aim of this study was to introduce into the damped harmonic oscillator model nonlocalities associated with these mentioned kernels and see the effect of each one of them when computing the Bode diagram obtained from the Laplace and the Sumudu transform. For each case, we applied both the Laplace and the Sumudu transform to obtain a solution in a complex space. For each case, we obtained the Bode diagram and the phase diagram for different values of fractional orders. We presented a detailed analysis of uniqueness and an exact solution and used numerical approximation to obtain a numerical solution.

In this article, we consider compositions of positive integers with 2 s and 3 s . We see that these compositions lead us to results that involve Padovan numbers, and we give some tiling models of these compositions. Moreover, we examine some tiling models of the compositions related to the Padovan polynomials and prove some identities using the tiling model’s method. Next, we obtain various identities of the compositions of positive integers with 2 s and 3 s related to the Padovan numbers. The number of palindromic compositions of this type is determined, and some numerical arithmetic functions are defined. Finally, we provide a table that compares all of the results obtained from compositions of positive integers with 2 s and 3 s .

In this article, we investigate the relationships between the instantaneous invariants of a one-parameter spatial movement and the local invariants of the axodes. Specifically, we provide new proofs for the Euler-Savary and Disteli formulas using the E. Study map in spatial kinematics, showcasing its elegance and efficiency. In addition, we introduce two line congruences and thoroughly analyze their spatial equivalence. Our findings contribute to a deeper understanding of the interplay between spatial movements and axodes, with potential applications in fields such as robotics and mechanical engineering.

Like q q -calculus, Hahn calculus (or q , ω q,\omega -calculus) is constructed by defining a difference derivative operator and an integral operator. The q , ω q,\omega -analogs of the integral representations of the Laplace transform and related special functions, such as gamma and beta, are proposed in this article. Then, some basic properties similar to classical and q q -analogs are investigated. Finally, a few examples are given to solve q , ω q,\omega -initial value problems via the newly introduced q , ω q,\omega -Laplace transform.

Discrete analogues of the sine integral and complementary exponential integral functions are investigated. Hypergeometric representation, power series, and Laplace transforms are derived for each. The difficulties in extending these definitions to other common trigonometric integral functions are discussed.

In this article, we the study generalized family of positive linear operators based on two parameters, which are a broad family of many well-known linear positive operators, e.g., Baskakov-Durrmeyer, Baskakov-Szász, Szász-Beta, Lupaş-Beta, Lupaş-Szász, genuine Bernstein-Durrmeyer, and Pǎltǎnea. We first find direct estimates in terms of the second-order modulus of continuity, then we find an estimate of error in the Ditzian-Totik modulus of smoothness. Then we discuss the rate of approximation for functions in the Lipschitz class. Furthermore, we study the pointwise Grüss-Voronovskaja-type result and also establish the Grüss-Voronovskaja-type asymptotic formula in quantitative form.

The aim of this article is to construct univariate Bernstein-type operators ( ℬ m x G ) ( x , z ) \left({{\mathcal{ {\mathcal B} }}}_{m}^{x}G)\left(x,z) and ( ℬ n z G ) ( x , z ) , \left({{\mathcal{ {\mathcal B} }}}_{n}^{z}G)\left(x,z), their products ( P m n G ) ( x , z ) \left({{\mathcal{P}}}_{mn}G)\left(x,z) , ( Q n m G ) ( x , z ) \left({{\mathcal{Q}}}_{nm}G)\left(x,z) , and their Boolean sums ( S m n G ) ( x , z ) \left({{\mathcal{S}}}_{mn}G)\left(x,z) , ( T n m G ) ( x , z ) \left({{\mathcal{T}}}_{nm}G)\left(x,z) on elliptic region, which interpolate the given real valued function G G defined on elliptic region on its boundary. The bound of the remainders of each approximation formula of corresponding operators are computed with the help of Peano’s theorem and modulus of continuity, and the rate of convergence for functions of Lipschitz class is computed.

The primary goal of this research is to investigate the approximate numerical solution of variational inequalities using quasimonotone operators in infinite-dimensional real Hilbert spaces. In this study, the sequence obtained by the proposed iterative technique for solving quasimonotone variational inequalities converges strongly toward a solution due to the viscosity-type iterative scheme. Furthermore, a new technique is proposed that uses an inertial mechanism to obtain strong convergence iteratively without the requirement for a hybrid version. The fundamental benefit of the suggested iterative strategy is that it substitutes a monotone and non-monotone step size rule based on mapping (operator) information for its Lipschitz constant or another line search method. This article also provides a numerical example to demonstrate how each method works.

In the current manuscript, we combine the q -fractional integral operator and q -fractional derivative to investigate a coupled hybrid fractional q -differential systems with sequential fractional q -derivatives. The existence and uniqueness of solutions for the proposed system are established by means of Leray-Schauder’s alternative and the Banach contraction principle. Furthermore, the Ulam-Hyers and Ulam-Hyers-Rassias stability results are discussed. Finally, two illustrative examples are given to highlight the theoretical findings.

The fundamental objective of this article is to investigate about the boundary value problem with the uses of a generalized conformable fractional derivative introduced by Zarikaya et al. ( On generalized the conformable calculus , TWMS J. App. Eng. Math. 9 (2019), no. 4, 792–799, http://jaem.isikun.edu.tr/web/images/articles/vol.9.no.4/11.pdf). In the development of the this article, by using classical methods of fractional calculus, we find a definition of the generalized fractional Wronskian according to the fractional differential operator defined by Zarikaya, a fractional version of the Sturm-Picone theorem, and in addition, the stability criterion given by the Hyers-Ulam theorem is studied with the use of the aforementioned fractional derivatives.

In this study, we have investigated the oscillatory properties of the following fractional difference equation: ∇ α + 1 χ ( κ ) ⋅ ∇ α χ ( κ ) − p ( κ ) г ( ∇ α χ ( κ ) ) + q ( κ ) G ∑ μ = κ − α + 1 ∞ ( μ − κ − 1 ) ( − α ) χ ( μ ) = 0 , {\nabla }^{\alpha +1}\chi \left(\kappa )\cdot {\nabla }^{\alpha }\chi \left(\kappa )-p\left(\kappa )&#x0433;\left({\nabla }^{\alpha }\chi \left(\kappa ))+q\left(\kappa ){\mathcal{G}}\left(\mathop{\sum }\limits_{\mu =\kappa -\alpha +1}^{\infty }{\left(\mu -\kappa -1)}^{\left(-\alpha )}\chi \left(\mu )\right)=0, where κ ∈ N 0 \kappa \in {{\mathbb{N}}}_{0} , ∇ α {\nabla }^{\alpha } denotes the Liouville fractional difference operator of order α ∈ ( 0 , 1 ) \alpha \in \left(0,1) , p p , and q q are nonnegative sequences, and г &#x0433; and G {\mathcal{G}} are real valued continuous functions, all of which satisfy certain assumptions. Using the generalized Riccati transformation technique, mathematical inequalities, and comparison results, we have found a number of new oscillation results. A few examples have been built up in this context to illustrate the main findings. The conclusion of this study is regarded as an expansion of continuous time to discrete time in fractional contexts.

In this article, we have established some new bounds of Fejér-type Hermite-Hadamard inequality for k k -fractional integrals involving r r -times differentiable preinvex functions. It is noteworthy that in the past, there was no weighted version of the left and right sides of the Hermite-Hadamard inequality for k k -fractional integrals for generalized convex functions available in the literature.

In this article, the behavior of hidden bifurcation in a two-dimensional (2D) scroll via saturated function series controlled by the coefficient harmonic linearization method is presented. A saturated function series approach for chaos generation. The systematic saturated function series methodicalness improved here can make multi-scroll and grid scroll chaotic attractors from a 3D linear autonomous system with a plain saturated function series supervisor. We have used a hidden bifurcation method in grid scroll., where the method of hidden bifurcation presented by Menacer, et al. in (2016) for Chua multi-scroll attractors. This additional parameter, which is absent from the initial problem, is perfectly adapted to unfold the structure of the multispiral chaotic attractor. The novelty of this article is twofold: first, the saturated function series model for hidden bifurcation in a 2 – D scroll; and second, the control of hidden bifurcation behavior by the value of the harmonic coefficient k 3 {k}_{3} .

The classification of the phase portraits is one of the classical and difficult problems in the qualitative theory of polynomial differential systems in R 2 {{\mathbb{R}}}^{2} , particularly for quadratic systems. Even with the hundreds of studies on the topology of real planar quadratic vector fields, fully characterizing their phase portraits is still a difficult problem. This paper is devoted to classifying the phase portraits of two polynomial vector fields with two usual invariant algebraic curves, by investigating the geometric solutions within the Poincaré disc. One can notice that these systems yield 26 topologically different phase portraits.

In this article, a queueing inventory system with finite sources of demands, retrial demands, service time, lead time, ( s , S ) \left(s,S) replenishment policy, and demands search from the orbit was studied. When the lead time is exponentially distributed (resp. lead time is generally distributed), generalized stochastic Petri net (GSPN) (resp. Markov regenerative stochastic Petri net [MRSPN]) is proposed for this inventory system. The quantitative analysis of this stochastic Petri net model was obtained by continuous time Markov chain for the GSPN model (resp. the supplementary variable method for the MRSPN model). The probability distributions are obtained, witch allowed us to compute performance measures and the expected cost rate of the studied system.

The aim of this article is to study the known susceptible-infectious (SI) epidemic model using fractional order reaction-diffusion fractional partial differential equations [FPDEs] in order to better describe the dynamics of a reaction-diffusion SI with a nonlinear incidence rate describing the infection dynamics of the HIV/AIDS virus. We initially examined the nonnegativity, global existence, and boundedness for solutions of the proposed system. After determining that the proposed model has two steady states, we derived sufficient conditions for the global and local asymptotic stability of the equilibrium of the proposed system and their relationship to basic reproduction in the case of fractional ordinary differential equations and FPDEs by analyzing the eigenvalues and using the appropriately chosen Lyapunov function. Finally, we used numerical examples to illustrate our theoretical results.

In this article, we explain how a recent Lyapunov theorem on stability plays a role in the study of the strong stabilizability problem in Hilbert spaces. We explore a degenerate controlled system and investigate the properties of a feedback control to stabilize such system in depth. The spectral theory of an appropriate pencil operator is used to generate robustness constraints for a stabilizing control.

The Hermite-Hadamard inequality is regarded as one of the most favorable inequalities from the research point of view. Currently, mathematicians are working on extending, improving, and generalizing this inequality. This article presents conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in weighted and unweighted forms by using the idea of majorization and convexity together with generalized conformable fractional integral operators. They not only represent continuous and discrete inequalities in compact form but also produce generalized inequalities connecting various fractional operators such as Hadamard, Katugampola, Riemann-Liouville, conformable, and Rieman integrals into one single form. Also, two new integral identities have been investigated pertaining a differentiable function and three tuples. By using these identities and assuming ∣ f ′ ∣ | f^{\prime} | and ∣ f ′ ∣ q ( q > 1 ) | f^{\prime} {| }^{q}\hspace{0.33em}\left(q\gt 1) as convex, we deduce bounds concerning the discrepancy of the terms of the main inequalities.

The nonlinear fractional-order cubic-quintic-heptic Duffing problem will be solved through a new numerical approximation technique. The suggested method is based on the Pell-Lucas polynomials’ operational matrix in the fractional and integer orders. The studied problem will be transformed into a nonlinear system of algebraic equations. The numerical expansion containing unknown coefficients will be obtained numerically via applying Newton’s iteration method to the claimed system. Convergence analysis and error estimates for the introduced process will be discussed. Numerical applications will be given to illustrate the applicability and accuracy of the proposed method.

In this article, the stochastic fractional Davey-Stewartson equations (SFDSEs) that result from multiplicative Brownian motion in the Stratonovich sense are discussed. We use two different approaches, namely the Riccati-Bernoulli sub-ordinary differential equations and sine-cosine methods, to obtain novel elliptic, hyperbolic, trigonometric, and rational stochastic solutions. Due to the significance of the Davey-Stewartson equations in the theory of turbulence for plasma waves, the discovered solutions are useful in explaining a number of fascinating physical phenomena. Moreover, we illustrate how the fractional derivative and Brownian motion affect the exact solutions of the SFDSEs using MATLAB tools to plot our solutions and display a number of three-dimensional graphs. We demonstrate how the multiplicative Brownian motion stabilizes the SFDSE solutions at around zero.

Fractional Hahn boundary value problems are significant tools to describe mathematical and physical phenomena depending on non-differentiable functions. In this work, we develop certain aspects of the theory of fractional Hahn boundary value problems involving fractional Hahn derivatives of the Caputo type. First, we construct the Green function for an α th \alpha {\rm{th}} -order fractional boundary value problem, with 1 < α < 2 1\lt \alpha \lt 2 , and discuss some important properties of the Green function. The solutions to the proposed problems are obtained in terms of the Green function. The uniqueness of the solutions is proved by various fixed point theorems. The Banach’s contraction mapping theorem, the Schauder’s theorem, and the Browder’s theorem are used.

The contribution of fractional calculus in the development of different areas of research is well known. This article presents investigations involving fractional calculus in the study of analytic functions. Riemann-Liouville fractional integral is known for its extensive applications in geometric function theory. New contributions were previously obtained by applying the Riemann-Liouville fractional integral to the convolution product of multiplier transformation and Ruscheweyh derivative. For the study presented in this article, the resulting operator is used following the line of research that concerns the study of certain new subclasses of analytic functions using fractional operators. Riemann-Liouville fractional integral of the convolution product of multiplier transformation and Ruscheweyh derivative is applied here for introducing a new class of analytic functions. Investigations regarding this newly introduced class concern the usual aspects considered by researchers in geometric function theory targeting the conditions that a function must meet to be part of this class and the properties that characterize the functions that fulfil these conditions. Theorems and corollaries regarding neighborhoods and their inclusion relation involving the newly defined class are stated, closure and distortion theorems are proved, and coefficient estimates are obtained involving the functions belonging to this class. Geometrical properties such as radii of convexity, starlikeness, and close-to-convexity are also obtained for this new class of functions.

Cross-border e-commerce platform (CBECP) plays a very important role in the development of a cross-border e-commerce (CBEC). How to select the best CBECP scientifically and reasonably is a very critical multi-attribute group decision-making (MAGDM) issue. With the uncertainty of people’s cognition of the objective world, the decision-making process is full of a lot of fuzzy information. In view of the great advantages of probabilistic dual hesitation fuzzy set (FS) in expressing decision-making information, and in combination with the very extensive use of the Dice similarity measure (DSM), a new MAGDM method is proposed for the optimal CBECP selection (CBECPS) under the probabilistic dual hesitation fuzzy (PDHF) environment. First, on the basis of reviewing a large number of documents on the CBECPS for CBEC, the evaluation index system for the CBECPS is constructed; second, several new DSMs are proposed in the PDHF environment; third, based on the two newly proposed probabilistic dual hesitant weighted generalized Dice similarity measures, two novel MAGDM methods are provided for CBECPS, which are used for CBECPS; finally, the two established MAGDM techniques are compared with the existing decision-making methods, and the parameter analysis is carried out to illustrate the effectiveness and superiority of the two established MAGDM techniques. The two established techniques can not only be used for CBECPS of CBEC, but also be extended to similar related research.

The use of multi-criteria decision-making (MCDM) methods to select the most appropriate one from a range of alternatives considering multiple criteria is a suitable methodology for making informed decisions. When constructing a decision or objective matrix (DOM) for MCDM procedure, either crisp numerical values or fuzzy linguistic terms can be used. A review of relevant literature indicates that decision experts often prefer to give linguistic terms (instead of crisp numerical values) based on their domain knowledge, to establish a fuzzy DOM. However, previous research articles have not adequately studied the selection between fuzzy and crisp DOM in MCDM, especially under the context of assessing the financial performance (FP) of listed firms – a notably complex decision-making problem. As such, the primary motivation of this study is to bridge this research gap through comparative analyses of fuzzy and crisp DOM in MCDM. Along this path, and in order to handle fuzzy DOM, this work also proposes two new fuzzy MCDM methods: fuzzy preference ranking on the basis of ideal-average distance (PROBID) and fuzzy sPROBID (simpler PROBID), extending the applicability of the original crisp PROBID and sPROBID methods. Moreover, for the first time in the literature, this work compares the FP rankings obtained using fuzzy MCDM methods with an objective benchmark we have identified, i.e., the real-life stock return (SR)-based ranking. The case study of ranking the FP of 32 listed firms demonstrates that the fuzzy MCDM methods produce higher correlation results with the SR-based ranking. The results also suggest that the proposed fuzzy sPROBID method with triangular fuzzy DOM performs the best for assessing the FP of firms in terms of Spearman’s rank correlation coefficient with the SR-based ranking. Overall, the contributions of this work are three-fold: first, it proposes two new fuzzy MCDM methods (i.e., fuzzy PROBID and fuzzy sPROBID); second, it advances the application of fuzzy MCDM methods in assessing and ranking the FP of listed firms to make rational investment decisions in the financial market; third, it studies the selection between fuzzy and crisp DOM through comparisons with an objective benchmark.

Due to the absence of a negative of three membership functions, there are drawbacks to the existing definition of a picture fuzzy graph (PFG). In that definition of bipolar picture fuzzy graph (BPFG), membership function, neutral membership function, nonmembership function, negative of membership function, negative of neutral membership function, and negative of nonmembership function are involved. A BPFG is the extension of PFG. In this manuscript, we present some properties of symmetric difference, and rejection of BPFG.