Volume 58, Issue 1

In this study, we introduce a kind of Stancu variant of the complex Bernstein-Durrmeyer-type operators in movable compact disks. Their approximation properties for analytic functions in the movable compact disks are investigated.

The circular Pythagorean fuzzy set is an expansion of the circular intuitionistic fuzzy set (CIFS), in which each component is represented by a circle. Nevertheless, even though CIFS improves the intuitionistic fuzzy set representation, it is still restricted to the inflexible intuitionistic fuzzy interpretation triangle (IFIT) space, where the square sum of membership and nonmembership in a circular Pythagorean fuzzy environment and the sum of membership and nonmembership in a circular intuitionistic fuzzy environment cannot exceed one. To overcome this restriction, we provide a fresh extension of the CIFS called the circular n,m -rung orthopair fuzzy set (C n,m -ROFS), which allows the IFIT region to be expanded or contracted while maintaining the features of CIFS. Consequently, decision makers can assess items over a wider and more flexible range when using a C n,m -ROFS, allowing for the making of more delicate decisions. In addition, we define several basic algebraic and arithmetic operations on C n,m -ROFS, such as intersection, union, multiplication, addition, and scalar multiplication, and we discuss their key characteristics together with some of the known relations over C n,m -ROFS. In addition, we present and study the new circular n,m -rung orthopair fuzzy weighted average/geometric aggregation operators and their properties. Further, a strategy for resolving multicriteria decision-making problems in a C n,m -ROF environment is provided. The suggested strategy is tested on two situations: the best teacher selection problem and the best school selection problem. To confirm and illustrate the efficacy of the suggested methodology, a comparative analysis with the intuitionistic fuzzy weighted average, intuitionistic fuzzy weighted geometric, q-rung orthopair fuzzy weighted averaging, q-rung orthopair fuzzy geometric averaging, circular PFWA max {{\rm{PFWA}}}_{{\rm{\max }}} , and circular PFWA min {{\rm{PFWA}}}_{{\rm{\min }}} operators approaches is also carried out. Ultimately, in the final section, there are discussions and ideas for future research.

This article studies the Triebel-Lizorkin-type spaces built on grand Morrey spaces on Euclidean spaces. We establish a number of characterizations on the grand Triebel-Lizorkin-Morrey spaces such as the Peetre maximal function characterizations, the Lusin area function characterizations, and the g λ ∗ {g}_{\lambda }^{\ast } function characterizations. The boundedness of the Fourier multipliers on the grand Triebel-Lizorkin-Morrey spaces is also obtained.

In this article, we determine coefficient estimates and study the Fekete-Szegö problem for classes Λ n ( b ) {\Lambda }_{n}\left(b) and Λ n c ( b ) {\Lambda }_{n}^{c}\left(b) , where b b is a nonzero complex order. In addition, we determine estimates for logarithmic coefficients of functions within these classes. Moreover, we extend our study to include estimates for logarithmic coefficients of functions in the classes F n ( b ) {{\mathscr{F}}}_{n}\left(b) and S n c ( b ) {{\mathscr{S}}}_{n}^{c}\left(b) , studied by Kanas and Darwish.

Narayana numbers are well-known and have many applications in the field of combinatorics. The Narayana numbers form a triangular array, where the sum of the n th row is the n th Catalan number. Here we normalize the Narayana numbers by dividing each entry in the n th row by the n th Catalan number. Now each row of these normalized Narayana numbers defines a discrete probability distribution. We investigate two new properties of these normalized Narayana numbers: instead of summing along the rows, we derive the limit of the sums along the columns and the limit of the sums along the short diagonals.

It has been noted that lower type of an entire function completely ignores the value of lower order. The question arises for entire functions of irregular growth that what happens when we replace order by an arbitrary nonzero finite number. Here in this article our aim is to solve this problem by defining new lower ( p , q ) \left(p,q) -type by using a ( p , q ) \left(p,q) -scale, ( p ≥ q ≥ 1 ) \left(p\ge q\ge 1) for an entire function. Moreover, a relationship has been established between lower ( p , q ) \left(p,q) -type of entire function solutions of linear homogeneous partial differential equation of second order with coefficients occurring in series expansion and Laguerre polynomial approximation errors in sup norm.

This article presents two types of the new convolutions for the Hartley integral transform associated with the Hermite functions, gives rise to the identification of some commutative and non-commutative Banach algebras, and to the Young inequalities which, in a certain sense, can be seen as the exceptional Young inequalities. Moreover, the solvability of a class of the integral equations is investigated, and their solutions in a series form can be obtained by using the constructed convolutions.

In this article, we first prove some auxiliary results in the form of lemmas using an improved Schwarz lemma at the boundary recently proved by Mercer. Furthermore, we establish some new inequalities for rational functions on the unit disk in the complex plane with prescribed poles and restricted zeros. It is of interest to know that in the bounds of theorems we prove, four extremal coefficients of the numerator polynomial are incorporated rather than usually two coefficients in the existing literature. Moreover, our results strengthen some recent results concerning the inequalities for rational functions and, in turn, produce refinements of some polynomial inequalities as particular cases.

In this article, we introduce new sequence spaces defined via an Orlicz function within the framework of a 2-normed space and incorporating the Lucas difference matrix and its associated matrix domain. The study provides a detailed examination of the topological and geometric properties of these spaces, exploring their structural characteristics and underlying relationships. Furthermore, inclusion relations among the defined sequence spaces are established, offering a deeper understanding of their interconnections and generalizations within the broader context of sequence space theory. This work contributes to the advancement of functional analysis by extending the theoretical framework of sequence spaces in 2-normed structures, laying the foundation for further research and applications in this domain.

The demand for advanced predictive tools has surged in the intricate landscape of global financial markets. Traditional predictive tools based on crisp models offer foundational insights, while the evolving complexities in global financial markets necessitate more nuanced analytical techniques. This research delves deep into Bayesian networks (FBN) as a potential tool for financial risk prediction (FRP). Integrating the probabilistic reasoning of Bayesian Networks with the uncertainty-handling capabilities of fuzzy logic, FBNs present a promising avenue for capturing the multifaceted dynamics of financial data. A comprehensive methodology was employed, encompassing data collection, data preprocessing, and transformation. The FBN model’s construction was rooted in established methodologies, emphasizing feature selection, parameter estimation, and a systematic validation process. The model’s empirical robustness was ensured through rigorous validation and testing mechanisms. The results found that the FBN accuracy achieved a mean absolute error (MAE) of 9.78 and a root mean square error (RMSE) of 11.64, when compared to traditional models such as linear regression, which had MAE and RMSE values of 15.70 and 18.39, respectively. The obtained results illuminate the FBN’s standout performance in FRP. The FBN excels in capturing the underlying intricacies of financial data, offering unparalleled predictive accuracy. Its predictions are closer to actual average value but exhibit fewer large deviations, making it an invaluable tool in the financial analytics arsenal demonstrably outpacing traditional crisp models.

We introduce two new classes of single-valued contractions of polynomial type defined on a metric space. For the first one, called the class of polynomial contractions, we establish two fixed point theorems. Namely, we first consider the case when the mapping is continuous. Next we weaken the continuity condition. In particular, we recover Banach’s fixed point theorem. The second class, called the class of almost polynomial contractions, includes the class of almost contractions introduced by Berinde [ Approximating fixed points of weak contractions using the Picard iteration , Nonlinear Anal. Forum 9 (2004), no. 1, 43–53]. A fixed point theorem is established for almost polynomial contractions. The obtained result generalizes that derived by Berinde in the above reference. Several examples showing that our generalizations are significant are provided.

The aim of this article is to study a semi-functional partial linear regression model (SFPLR) for spatial data with responses missing at random (MAR). The estimators are constructed using the kernel method, and some asymptotic properties, such as the probability convergence rates of the nonparametric component and the asymptotic distribution of the parametric and nonparametric components, are established under certain conditions. Next, the performance and superiority of these estimators are presented and examined through a study on simulated data, comparing our semi-functional partially linear model with the MAR estimator to the semi-functional partially linear model with the full-case estimator, and the functional nonparametric regression model estimator with MAR. The results indicate that the proposed estimators outperform traditional estimators as the amount of randomly missing data increases. Additionally, a study is conducted on real data regarding the modeling of pollution levels using our model, incorporating covariates such as average daily temperature as a functional variable, alongside maximum daily mixing height, total daily precipitation, and daily primary aerosol emission rates as explanatory variables.

In this study, we investigate systems known as nonlinear fractional delay differential (nLFDD) systems, characterized by finite state delays and fractional orders within the range of 0 < η ≤ 1 < ζ ≤ 2 0\lt \eta \le 1\lt \zeta \le 2 , situated infinite-dimensional settings. We utilize the controllability Gramian matrix to establish both necessary and sufficient conditions for the controllability of linear fractional delay differential systems that fall within the order range of 0 < η ≤ 1 < ζ ≤ 2 0\lt \eta \le 1\lt \zeta \le 2 . Moreover, the Schauder fixed point theorem is employed to delineate the sufficient conditions required for the controllability of nLFDD systems, which are defined by finite state delays and fractional orders in the specified range. To substantiate the theoretical constructs put forth, we provide two illustrative examples.

In this work, several bilateral inequalities for K K - g g -frames in subspaces are established, drawing support from two kinds of operators induced, respectively, by the K K - g g -frame itself and the K K -dual pair, which, compared with previous ones concerning this topic, possess novel structures. It is indicated that new types of inequalities for some other generalized frames can be naturally presented following our approaches.

Padé approximations constructed from orthogonal and Faber polynomials on some compact set E E serve as tools to detect poles of an approximated function around the set E E . The goal of this article is to study the rate of such pole detection using indicators introduced by Gonchar ( Poles of rows of the Padé table and meromorphic continuation of functions , Sb. Math. 43 (1981), 527–546). Particularly, we compute the values of these indicators corresponding to our extensions of Padé approximation for the poles of the approximated function within the domain of its meromorphy. Our computations extend the indicator formulas found in the article by Gonchar.

In this article, we are interested in the existence of nontrivial solutions for the following nonhomogeneous Choquard equation involving the p p -biharmonic operator: M ∫ Ω ∣ Δ u ∣ p d x Δ p 2 u − Δ p u = λ ( ∣ x ∣ − μ ⁎ ∣ u ∣ q ) ∣ u ∣ q − 2 u + ∣ u ∣ p * − 2 u + f , in Ω , u = Δ u = 0 , on ∂ Ω , \left\{\begin{array}{l}M\left(\mathop{\displaystyle \int }\limits_{\Omega }{| \Delta u| }^{p}{\rm{d}}x\right){\Delta }_{p}^{2}u-{\Delta }_{p}u=\lambda \left({| x| }^{-\mu }\ast {| u| }^{q}){| u| }^{q-2}u+{| u| }^{{p}^{* }-2}u+f,\hspace{1em}{\rm{in}}\hspace{0.33em}\Omega ,\\ u=\Delta u=0,\hspace{1em}{\rm{on}}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} , N ≥ 3 N\ge 3 is a smooth bounded domain, 1 < p < N 2 1\lt p\lt \frac{N}{2} , 0 < μ < N 0\lt \mu \lt N , p < 2 q < p * p\lt 2q\lt {p}^{* } , p * = N p N − 2 p {p}^{* }=\frac{Np}{N-2p} denotes the Sobolev conjugate of p p . M M is a nondecreasing and continuous function, and λ > 0 \lambda \gt 0 is a parameter. Δ p 2 u ≔ Δ ( ∣ Δ u ∣ p − 2 Δ u ) {\Delta }_{p}^{2}u:= \Delta \left({| \Delta u| }^{p-2}\Delta u) is the operator of fourth order called the p-biharmonic operator, Δ p u ≔ div ( ∣ ∇ u ∣ p − 2 ∇ u ) {\Delta }_{p}u:= \hspace{0.1em}\text{div}\hspace{0.1em}\left({| \nabla u| }^{p-2}\nabla u) is the p p -Laplacian operator. f ≥ 0 f\ge 0 , f ∈ L p p − 1 ( Ω ) f\in {L}^{\tfrac{p}{p-1}}\left(\Omega ) , and ∣ f ∣ p p − 1 {| f| }_{\tfrac{p}{p-1}} is sufficiently small. Using the concentration-compactness principle together with the mountain pass theorem, we obtain the existence of nontrivial solutions for the aforementioned problem in both nondegenerate and degenerate cases.

Shape-preserving approximation is a significant approximation method that has many application areas, such as computer-based geometric design, image processing, geodesy, chemistry, and robotics. Due to this important fact, the aim of this work is to study the shape-preserving properties of a new class of Kantorovich-type operators, which are constructed via Bézier bases based on shape parameter λ ∈ [ − 1 , 1 ] \lambda \in {[}-1,1] . In pursuance of this goal, we commence our calculations by rewriting these operators as a summation of the classical Kantorovich-type operators and a summation of integral values of first-order divided differences of function f f over the interval [0, 1]. Then, adopting this representation, we establish the shape-preserving properties of these operators, such as linearity, positivity, and specifically monotonicity and convexity-preserving properties in connection with function f f on the interval [0, 1]. Our results reveal that while operators preserve monotonicity entirely on the interval [0, 1] for all λ ∈ [ − 1 , 1 ] \lambda \in {[}-1,1] , they fail to preserve convexity for some λ \lambda on the same interval. We endorse this claim by counterexamples and give an altered result on convexity preservation for a specific class of functions when λ ∈ [ 0 , 1 ] \lambda \in {[}0,1] .

In this article, we present several Geršhgorin-type theorems for Z 1 {Z}_{1} -eigenvalues of tensors, which improve the results provided by Wang et al. ( Some upper bounds on Z t {Z}_{t} - eigenvalues of tensors , Appl. Math. Comput. 329 (2018), 266–277). Bounds for the largest Z 1 {Z}_{1} -eigenvalue of nonnegative tensors are given. Furthermore, based on the Geršhgorin-type theorems, we introduce some sufficient criteria for the positivity of even order tensors.

The main purpose of this work is to study the ideal topology defined by the minimal and maximal ideals on a topological space. Also, we define and investigate the concepts of ideal quotient and annihilator of any subfamily of 2 X {2}^{X} , where 2 X {2}^{X} is the power set of X X . We obtain some of their fundamental properties. In addition, several relationships among the above notions have been discussed. Moreover, we define a new topology on an ideal topological space, called sharp topology, via the sharp operator defined in this study, which turns out to be finer than the original topology. Furthermore, a decomposition of open sets (in the original topology) has been obtained. Finally, we conclude our work with some interesting applications.

This work investigates the existence of singular limit solutions for nonlinear elliptic systems. Our main approach focuses on using the nonlinear domain decomposition method to establish a new Liouville-type result.

In this article, we investigate the following Schrödinger equation: − Δ u = h ( x ) g ( u ) + λ u in R N , ∫ R N ∣ u ∣ 2 d x = a u ∈ H 1 ( R N ) , \left\{\begin{array}{ll}-\Delta u=h\left(x)g\left(u)+\lambda u\hspace{1.0em}& \hspace{-0.2em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}\hspace{0.1em}\text{d}\hspace{0.1em}x=a\hspace{1.0em}& u\in {H}^{1}\left({{\mathbb{R}}}^{N}),\end{array}\right. where N ≥ 3 N\ge 3 , a > 0 a\gt 0 , and λ ∈ R \lambda \in {\mathbb{R}} arises as a Lagrange multiplier. Assuming that the sublinear nonlinearity f ∈ C ( R N × R , R ) f\in C\left({{\mathbb{R}}}^{N}\times {\mathbb{R}},{\mathbb{R}}) is locally defined for ∣ u ∣ | u| small; by using generalized minimax approach, we prove the existence of infinitely many normalized solutions { ( u k , λ k ) } k ∈ N {\{\left({u}_{k},{\lambda }_{k})\}}_{k\in {\mathbb{N}}} for the problem, where λ k < 0 {\lambda }_{k}\lt 0 and ∣ u k ( x ) ∣ → 0 | {u}_{k}\left(x)| \to 0 as ∣ x ∣ → ∞ | x| \to \infty . Furthermore, as k → ∞ k\to \infty , we show that ‖ ∇ u k ‖ L 2 ( R N ) → 0 {\Vert \nabla {u}_{k}\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{N})}\to 0 , ‖ u k ‖ L ∞ ( R N ) → 0 {\Vert {u}_{k}\Vert }_{{L}^{\infty }\left({{\mathbb{R}}}^{N})}\to 0 , and λ k → 0 {\lambda }_{k}\to 0 .

This work addresses the issue of financial risk analysis by introducing a novel expected shortfall (ES) regression model, which employs expectile regression to define the shortfall threshold in financial risk management. We develop a nonparametric estimator for this model and provide mathematical support by proving both pointwise and uniform complete convergence of the estimator. These asymptotic results are derived under traditional assumptions and include precise convergence rates, emphasizing the impact of the regressor’s dimensionality on the estimation approach. A key feature of our contribution is the straightforward implementation of the estimator, demonstrated through applications on both simulated and real data. Our findings indicate that the new ES-expectile model is more effective than the standard model based on quantile regression, offering improved relevance in financial risk management.

This article investigates new analytical wave solutions within the beta ( β \beta ) fractional framework (F κ \kappa IIAE and F κ \kappa IIBE) of the Kuralay II equations, which are significant in the field of nonlinear optics. To achieve this, we employ the improved Kudryashov method, the R method, and the Sardar sub-equation method. The study successfully derives a variety of soliton solutions, including dark, bright, singular, and others, some of which are illustrated through 2D and 3D graphics. For the first time, this research graphically demonstrates the impact of the beta fractional derivative on these solutions. The findings provide valuable insights that may aid in the advancement of future models. The methodologies applied here are not only effective and straightforward to implement but also robust for addressing other fractional, nonlinear partial differential equations.

In recent years, research into the multiplicity of solutions to the p p -Laplace operator problem has attracted attention, and several important results have been investigated and others still remain open. Problems involving a critical point are indeed interesting and relevant, especially challenging. Motivated by such questions, in this article, we are interested, through a critical point theorem, to investigate the existence of at least three distinct weak solutions for a generalized p p -Laplacian problem with parameters under appropriate hypotheses, applicable in physics, for instance, in fluid mechanics, and in Newtonian fluids. In this sense, as a direct consequence of the main result, we finish the work with two other results of weak solutions.

This article is devoted to describing the entire solutions of several systems of the first-order nonlinear partial differential difference equations (PDDEs). Using the Nevanlinna theory and the Hadamard factorization theory of meromorphic functions, we establish some interesting results to reveal the existence and the forms of the finite-order transcendental entire solutions of several systems of the first-order nonlinear PDDEs: f ( z 1 + c 1 , z 2 + c 2 ) ( a 1 g z 1 + a 2 g z 2 ) = m 1 , g ( z 1 + c 1 , z 2 + c 2 ) ( a 3 f z 1 + a 4 f z 2 ) = m 2 , f ( z 1 + c 1 , z 2 + c 2 ) ( a 1 f z 1 + a 2 g z 1 ) = m 1 , g ( z 1 + c 1 , z 2 + c 2 ) ( a 3 f z 2 + a 4 g z 2 ) = m 2 , \begin{array}{l}\left\{\begin{array}{l}f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})({a}_{1}{g}_{{z}_{1}}+{a}_{2}{g}_{{z}_{2}})={m}_{1},\\ g\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})({a}_{3}{f}_{{z}_{1}}+{a}_{4}{f}_{{z}_{2}})={m}_{2},\end{array}\right.\\ \left\{\begin{array}{l}f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})({a}_{1}{f}_{{z}_{1}}+{a}_{2}{g}_{{z}_{1}})={m}_{1},\\ g\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})({a}_{3}{f}_{{z}_{2}}+{a}_{4}{g}_{{z}_{2}})={m}_{2},\end{array}\right.\end{array} and f ( z 1 + c 1 , z 2 + c 2 ) ( a 1 f z 2 + a 2 g z 1 ) = m 1 , g ( z 1 + c 1 , z 2 + c 2 ) ( a 3 f z 1 + a 4 g z 2 ) = m 2 , \left\{\begin{array}{l}f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})({a}_{1}{f}_{{z}_{2}}+{a}_{2}{g}_{{z}_{1}})={m}_{1},\\ g\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})({a}_{3}{f}_{{z}_{1}}+{a}_{4}{g}_{{z}_{2}})={m}_{2},\end{array}\right. where a 1 , a 2 , a 3 , a 4 , c 1 , c 2 ∈ C {a}_{1},{a}_{2},{a}_{3},{a}_{4},{c}_{1},{c}_{2}\in {\mathbb{C}} , m 1 , m 2 ∈ C − { 0 } {m}_{1},{m}_{2}\in {\mathbb{C}}-\left\{0\right\} . Moreover, some examples are given to explain that there are significant differences in the forms of solutions from some previous systems of functional equations.

Given a square matrix A A , we are able to construct numerous equalities involving reasonable mixed operations of A A and its conjugate transpose A ∗ {A}^{\ast } , Moore-Penrose inverse A † {A}^{\dagger } and group inverse A # {A}^{\#} . Such kind of equalities can be generally represented in the equation form f ( A , A ∗ , A † , A # ) = 0 f\left(A,{A}^{\ast },{A}^{\dagger },{A}^{\#})=0 . In this article, the author constructs a series of simple or complicated matrix equalities composed of A A , A ∗ {A}^{\ast } , A † {A}^{\dagger } , A # {A}^{\#} and their algebraic operations, as well as established various explicit formulas for calculating the ranks of these matrix expressions. Many applications of these matrix rank equalities are presented, including a broad range of necessary and sufficient conditions for a square matrix to be range-Hermitian and Hermitian/skew-Hermitian, respectively.

For a given Banach algebra ℳ {\mathcal{ {\mathcal M} }} and a continuous endomorphism ω \omega on ℳ {\mathcal{ {\mathcal M} }} , we define ℳ {\mathcal{ {\mathcal M} }} to be ω ℒ {\omega }_{{\mathcal{ {\mathcal L} }}} -biprojective and ω ¯ \overline{\omega } -contractible. We then explore the relationship between them. Additionally, we show that l 1 ( N ∨ ) {l}^{1}\left({{\mathbb{N}}}_{\vee }) is a ω ℒ {\omega }_{{\mathcal{ {\mathcal L} }}} -biprojective Banach algebra. Finally, we examine the concept of ω \omega -pseudo amenability and ω \omega -approximate biprojectivity in Banach algebras. We demonstrate that for every unital Banach algebra ℳ {\mathcal{ {\mathcal M} }} , ω \omega -approximate biprojectivity and ω \omega -pseudo contractibility coincide.

A challenging part of studying geometric function theory is figuring out the sharp boundaries for coefficient-related problems that crop up in the Taylor-Maclaurin series of univalent functions. The objective of this article is to define the families of Sakaguchi-type starlike functions with respect to symmetric points based on q q -operator and to investigate the precise boundaries for a range of issues, including the first three initial coefficient estimates, Fekete-Szegö type and the Zalcman inequalities by subordinating to the function of the three leaves. Additionally, we discussed initial coefficients and Fekete-Szegö type inequalities for functions of the form ℱ − 1 {{\mathcal{ {\mathcal F} }}}^{-1} and z ℱ ( z ) \frac{z}{{\mathcal{ {\mathcal F} }}\left(z)} and 1 2 log ℱ ( z ) z \frac{1}{2}\log \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{{\mathcal{ {\mathcal F} }}(z)}{z}\right) linked with the function of the three leaves.

In this article, we have introduced general transformation to solving the general quadratic equations. It is of interest to know about the existence and form of the solutions of general quadratic functional equations. By utilizing Nevanlinna theory for several complex variables and the transformation, we study the existence and form of the solutions to the general quadratic partial differential or partial differential-difference equations of the form a f 2 + 2 α f g + b g 2 + 2 β f + 2 γ g + C = 0 a{f}^{2}+2\alpha fg+b{g}^{2}+2\beta f+2\gamma g+C=0 in C 2 {{\mathbb{C}}}^{2} . Consequently, we obtain certain corollaries of the main results of this article concerning trinomial equations which generalize many results. In addition, some examples relevant to the content of the article has been exhibited.

The generalized trigonometric functions occur as an eigenfunction of the Dirichlet problem for the one-dimensional p p -Laplacian. In this study, the authors investigate some weighted power mean inequalities for the p p -generalized trigonometric functions with certain conditions on the parameters p p , λ \lambda .

This article concerns about optimality conditions for boundary-value problems related to differential inclusions (DFIs) of higher orders. We intend to attain optimality conditions when a general Lagrange functional takes place in the cost function. Moreover, it is intended that these conditions are applicable to the non-convex case as well. The notion of locally adjoint mapping for both convex and non-convex functions is used via Hamiltonian functions and arg-max sets of set-valued functions to obtain results. The presented main problem turns into a problem in the calculus of variations with some simplifications. It is noteworthy to see that the famous Euler-Poisson equation arises in this case. Furthermore, a higher-order semilinear optimal control problem is considered as an application, and its sufficient conditions, including Weierstrass-Pontryagin maximum principle, are derived. Then, the dual problems for the presented primal problems are established and their duality theorems are proved. Finally, the third-order polyhedral DFI with duality relations is considered.

This article comprises the study of a new class of starlike functions, which is defined by q q -fractional operator and subordination to the q q -analog of sine function. This q q -sine function is defined by using the q q -exponential function and has the symmetric geometric structure. Certain coefficient problems are discussed for the newly defined class S q sin ∗ {S}_{q\sin }^{\ast } , which include four coefficient bounds and the upper bounds of second and third Hankel determinants. Kruskal’s inequality for this class is also a part of this study.

Let FPT, MNC, HVI, SEPDE, SMHVI, PGCDD, and NLBC represent the fixed point theorem, measure of noncompactness, hemivariational inequality, system of nonlinear evolutionary partial differential equations, system of mixed hemivariational inequalities, partial generalized Clarke directional derivative, and nonlocal boundary condition in Banach spaces, respectively. We present and examine a complicated system acquired by the mixture of the SEPDE and SMHVI with PGCDDs when the constraint set is not necessary to be of boundedness and the issue is driven by NLBCs, that is termed as an system of partial differential hemivariational inequalities (SPDHVI) with PGCDDs. Also, it is shown that the set of solutions of SMHVI with PGCDDs implicated in the SPDHVI is the nonempty bounded set of both closedness and convexity. Besides, the measurable and upper semicontinuous natures for multivalued operator U : [ 0 , T ] × V → C b v ( U ) {\mathcal{U}}:\left[0,T]\times V\to Cbv\left(U) (see (3.20)) get validated. Finally, certain existence outcomes of SPDHVI get established with the help of the properties of PGCDD, the FPT of condensing multivalued mappings, and the technique of MNC.

In this work, using the concepts of q q -calculus, we first define the q q -Jung-Kim-Srivastava and q q -Bernardi integral operators for multivalent functions. Then, we use these operators to establish the generalized integral operator ℬ q , p − m − λ f ( z ) {{\mathcal{ {\mathcal B} }}}_{q,p}^{-m-\lambda }f\left(z) for multivalent functions. By using the newly defined operator ℬ q , p − m − λ f ( z ) {{\mathcal{ {\mathcal B} }}}_{q,p}^{-m-\lambda }f\left(z) , we define a new class of multivalent analytic functions and diskuss various interesting features of the functions belonging to this class. Certain interesting examples of our developments are also considered. Furthermore, we explore the applications of the Carathéodory lemma in the form of specific results.

Suppose that a weakly singular linear regression model M {\mathscr{M}} and its two competing restricted models M 1 {{\mathscr{M}}}_{1} and M 2 {{\mathscr{M}}}_{2} are given. We first compare the superiorities of the best linear unbiased estimators (BLUEs) under M {\mathscr{M}} and M 1 {{\mathscr{M}}}_{1} related to the covariance matrix criterion and derive some necessary and sufficient conditions. Based on these conditions and the covariance matrix criterion, we then compare the superiorities of the BLUEs under M 1 {{\mathscr{M}}}_{1} and M 2 {{\mathscr{M}}}_{2} and obtain some results. We finally apply these results to measure the influence of an observation on the BLUE.

This article presents a characterization of non-singular measurable transformation, denoted as T T , mapping from Ω \Omega to itself, along with bicomplex-valued BC {\mathbb{BC}} -measurable function u u defined on Ω \Omega , which induces a weighted composition operator. The study then proceeds to fully identify their D {\mathbb{D}} -compactness and D {\mathbb{D}} -closedness within the range of bicomplex Lorentz spaces denoted as L p , q BC ( Ω , M , ϑ ) {L}_{p,q}^{{\mathbb{BC}}}(\Omega ,{\mathfrak{M}},{\vartheta }) , where ( Ω , M , ϑ ) (\Omega ,{\mathfrak{M}},{\vartheta }) represents a σ \sigma -finite complete BC {\mathbb{BC}} -measure space, ϑ = ϑ 1 e 1 + ϑ 2 e 2 {\vartheta }={{\vartheta }}_{1}{e}_{1}+{{\vartheta }}_{2}{e}_{2} is a BC {\mathbb{BC}} -measure, and the parameters satisfy 1 < p ≤ ∞ 1\lt p\le \infty , 1 ≤ q ≤ ∞ 1\le q\le \infty .

This study investigates the conditions under which various curves maintain their characteristics when mapped between two regular surfaces in four-dimensional Euclidean space E 4 {{\mathbb{E}}}^{4} . We particularly focus on rectifying, osculating, and normal curves, examining their behavior under isometric, conformal, and homothetic conformal maps. By analyzing their behavior, this study reveals conditions under which these curves remain invariant when mapped between two surfaces. The findings contribute to a deeper understanding of geometric transformations in four-dimensional spaces.

We study the boundedness of commutators of the Hardy-Littlewood maximal function and the sharp maximal function on weighted Morrey spaces when the symbols of the commutators belong to weighted Lipschitz spaces (weighted Morrey-Campanato spaces). Some new characterizations for weighted Lipschitz functions are obtained in terms of the boundedness of the commutators.

The objective of this study is to examine a class of positive linear operators, defined in terms of the b ψ , k λ , μ {b}_{\psi ,k}^{\lambda ,\mu } basis and to analyze their approximation properties. Direct estimates for the ( λ , μ ) \left(\lambda ,\mu ) -Durrmeyer-Stancu-type operators are obtained using the first modulus of continuity and in a certain Lipschitz-type space. Approximation properties of these operators in Lebesgue spaces are also given. Finally, illustrative graphics are provided to support the results and to compare the rate of convergence.

We present a novel generalized Mittag-Leffler-type function of arbitrary order in this study. We look into its fundamental characteristics, such as differential formulas, recurrence relations, integral representations, the Euler, Laplace, Mellin, Whittaker, and Mellin-Barnes transforms. Additionally, we express it using the H-function, generalized hypergeometric function, and Fox-Wright function. Further, for this generalized Mittag-Leffler-type function of arbitrary order, we create fractional integral and differential operators. Also, we derive several intriguing special instances of our fundamental results.

In this study, we introduce the Kantorovich-type modified Szàsz-Charlier operators and examine their approximation properties within the framework of fractional modeling and control theory. These operators are defined using the Korovkin-type theorem, and their local approximation properties are analyzed in detail. Additionally, the approximation rate of these operators is estimated using the modulus of continuity and functions in the Lipschitz class. To gain a deeper understanding of their approximation capabilities, we calculate their central moments. Furthermore, the convergence rates are determined based on the modulus of continuity, and a Voronovskaja-type asymptotic formula is provided for these operators. Our study also investigates the approximation speed in weighted spaces, which is crucial for addressing complex dynamical systems in science and engineering. The approximation speed in weighted spaces is evaluated using the weighted modulus of continuity and the Peetre- K function. Finally, illustrative graphics generated using Maple are presented to visualize the convergence of operators to specific functions.

The article gives the corresponding split quaternions in Clifford analysis and the split Fourier transform (FT). Also, we investigate some properties of the split FT and apply to generalizations of the quaternion FT.

Inspired by some recent work on the quantum detection problem by discrete frames and continuous frames, in this article, we examine the quantum detection problem with G-frames. By employing operator theory and G-frames theory, we give some equivalent descriptions for injective g-frames and classifications of G-frames for solving injectivity problem. Also, we provide an approach to obtain an injective G-frame by using an injective frame. Moreover, we show that the injectivity of a G-frame is preserved under a linear isomorphism. In addition, injective G-frames exhibit instability in infinite-dimensional settings. Some examples are also provided to demonstrate the existence of injective G-frames.

We give an operator characterization of disjointly weakly compact sets and show that disjointly weakly compact sets coincide with reciprocal Dunford-Pettis sets. We compare disjointly weakly compact sets with almost Grothendieck sets, relatively compact sets, and weak reciprocal Dunford-Pettis sets. Consequently, we obtain new characterizations of the weak Grothendieck property, Schur property, and Banach lattices whose dual have an order continuous norm.

We aim to develop a q q -analog of recently introduced Motzkin sequence spaces by Erdem et al. [ Motzkin sequence spaces and Motzkin core , Numer. Funct. Anal. Optim. 45 (2024), no. 4–6, 283–303] by using q q -Motzkin numbers and introduce sequence spaces c ( M ( q ) ) c\left({\mathfrak{M}}\left(q)) and c 0 ( M ( q ) ) . {c}_{0}\left({\mathfrak{M}}\left(q)). We investigate some topological properties, compute bases, and obtain their duals. For X ∈ { c ( M ( q ) ) , c 0 ( M ( q ) ) } X\in \{c\left({\mathfrak{M}}\left(q)),{c}_{0}\left({\mathfrak{M}}\left(q))\} and Y ∈ { ℓ ∞ , c , c 0 , ℓ 1 } , Y\in \{{\ell }_{\infty },c,{c}_{0},{\ell }_{1}\}, some results pertaining to characterization of matrix class ( X , Y ) \left(X,Y) is given. We devote the final section to obtain necessary and sufficient conditions for a matrix operator to be compact on the space c 0 ( M ( q ) ) {c}_{0}\left({\mathfrak{M}}\left(q)) via Hausdorff measure of noncompactness.

This article investigates the existence and multiplicity of nontrivial solutions for Schrödinger-Bopp-Podolsky systems with critical nonlinearity in R 3 {{\mathbb{R}}}^{3} . Under appropriate assumptions about the potential and nonlinear terms, the existence and multiplicity of solutions are obtained by using the concentration-compactness principle and the symmetric mountain pass theorem. To some extent, we generalize the previous results.

We investigate the conditions under which a diagonal solution exists for a linear matrix inequality that arises when studying the asymptotic stability of differential-difference systems with both discrete and distributed delays. Our primary contribution is the characterization of the existence of positive definite diagonal matrices that ensure the asymptotic stability of these systems through Lyapunov-Krasovskii functionals. We establish several necessary and sufficient conditions for the existence of such positive definite diagonal solutions and extend recent results related to positive systems with difference-differential equations.

We describe the discriminant of deformations of simple quasi m-boundary equivalence classes for m ≥ 2 m\ge 2 . All quasi simple m m -boundary classes are right equivalent to Arnold’s singularities (ADE). Consequently, their respective discriminants are cylinders over the standard ones, together with some new descriptions involving either submanifolds or generalized Whitney umbrella. Our main results are as follows: (1) Theorem 3.5 on the bifurcation diagram and caustic of quasi A k {{\mathbb{A}}}_{k} singularities. (2) Theorem 3.9 on the bifurcation diagram and caustic of quasi D 4 + {{\mathbb{D}}}_{4}^{+} singularity. (3) Theorem 3.10 on the bifurcation diagram and caustic of quasi D 4 − {{\mathbb{D}}}_{4}^{-} singularity. (4) Theorem 3.11 on the bifurcation diagram and caustic of quasi H p , k {{\mathbb{H}}}_{p,k} singularities. The subsequent section provides a concise overview of the primary results.

Consider A {\mathcal{A}} and ℬ {\mathcal{ {\mathcal B} }} to be nonassociative unital algebras. Under the assumption that either one of them has finite dimensions or that both are finite dimensions, a generalized derivation is an additive map ℱ : A → A {\mathcal{ {\mathcal F} }}:{\mathcal{A}}\to {\mathcal{A}} associated with a derivation d {\mathcal{d}} of A {\mathcal{A}} if ℱ ( u v ) = ℱ ( u ) v + u d ( v ) {\mathcal{ {\mathcal F} }}\left(uv)={\mathcal{ {\mathcal F} }}\left(u)v+u{\mathcal{d}}\left(v) for all u , v ∈ A u,v\in {\mathcal{A}} . The objective of this study is to characterize and elucidate the structure of a generalized derivation on the tensor product of nonassociative algebras. Specifically, we prove that if ℱ {\mathcal{ {\mathcal F} }} is a generalized derivation of A ⊗ ℬ {\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }} associated with a derivation d {\mathcal{d}} of A ⊗ ℬ {\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }} , then ℱ = ℒ u + d {\mathcal{ {\mathcal F} }}={{\mathcal{ {\mathcal L} }}}_{u}+{\mathcal{d}} , where ℒ u {{\mathcal{ {\mathcal L} }}}_{u} is a left multiplication by u u and u u belongs to the left nucleus of A ⊗ ℬ {\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }} (i.e., ℒ u r = u r {{\mathcal{ {\mathcal L} }}}_{u}r=ur for all r ∈ A ⊗ ℬ r\in {\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }} and u ∈ N l ( A ⊗ ℬ ) u\in {N}_{l}\left({\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }}) ). Moreover, every generalized derivation of A ⊗ ℬ {\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }} can be represented as the sum of the derivations of the three categories: (i) w + a d u w+a{\mathcal{d}}u , where u , w ∈ N ( A ⊗ ℬ ) u,w\in N\left({\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }}) , (ii) ℒ z ⊗ f {{\mathcal{ {\mathcal L} }}}_{z}\otimes f , where f f is a derivation of ℬ {\mathcal{ {\mathcal B} }} and z ∈ Z ( A ) z\in Z\left({\mathcal{A}}) (the center of A {\mathcal{A}} ), and (iii) g ⊗ ℒ w g\otimes {{\mathcal{ {\mathcal L} }}}_{w} , where g g is a derivation of A {\mathcal{A}} and w ∈ Z ( ℬ ) w\in Z\left({\mathcal{ {\mathcal B} }}) .

This article investigates the existence and optimal controls for a class of Hilfer fractional evolution equations of order in ( 0 , 1 ) \left(0,1) with type of [ 0 , 1 0,1 ]. The first part focuses on analyzing the existence and uniqueness of the mild solution using the fixed point theorem and semigroup theory. In the second part, an optimal control problem that admits an optimal pair is considered. Furthermore, the necessary optimality conditions for an optimal control problem governed by semilinear fractional diffusion equations are presented.

The coexistence of a traveling pulse and a periodic traveling wave was established in a convecting shallow water model when taking a nonlinear buoyancy term u u x u{u}_{x} . In this brief communication, we show that the mechanical balance underlying this coexistence is disrupted by a stronger nonlinear dissipation ( u 2 u x ) x {\left({u}^{2}{u}_{x})}_{x} , which arises from an enhanced buoyancy term u 2 u x {u}^{2}{u}_{x} . Consequently, the convecting shallow water model exhibits either a unique periodic wave or a unique solitary wave, each within a fixed range of wave speeds. Furthermore, we show that the wave speed is monotonic with respect to the wave amplitude and is smaller than that observed in the model with the buoyancy term u u x u{u}_{x} . A numerical study is performed to verify the theoretical study.

This article is concerned with the existence of nontrivial solutions to critical growth Kohn-Laplace equations with jumping nonlinearities. Or, more specifically, we consider the following Kohn-Laplace problem: − Δ H u = b u + − a u − + ∣ u ∣ Q ∗ − 2 u , in Ω , u = 0 , on ∂ Ω , \left\{\begin{array}{ll}-{\Delta }_{H}u=b{u}^{+}-a{u}^{-}+{| u| }^{{Q}^{\ast }-2}u,\hspace{1.0em}& {\rm{in}}\hspace{0.33em}\Omega ,\\ u=0,\hspace{1.0em}& {\rm{on}}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Δ H {\Delta }_{H} is the Kohn-Laplacian on the Heisenberg group H n {{\mathbb{H}}}^{n} with n > 1 n\gt 1 , Q ∗ = 2 + 2 n {Q}^{\ast }=2+\frac{2}{n} is the critical Sobolev exponent for Δ H {\Delta }_{H} , Ω ⊂ H n \Omega \subset {{\mathbb{H}}}^{n} denotes a smooth bounded domain, both a a and b b are greater than zero, and u ± = max { ± u , 0 } {u}^{\pm }=\max \left\{\pm u,0\right\} . When the pair ( a , b ) \left(a,b) belongs to some designated regions of R 2 {{\mathbb{R}}}^{2} , the existence of nontrivial solutions is proved for the aforementioned Kohn-Laplace equation. The proof is based on two abstract existence results recently obtained by Perera and Sportelli in [Theorems 4.1–4.2].

Monotonicity, oscillation, and asymptotic behavior of solutions of nonlinear fractional differential equations are investigated. Fractional differential equations are classified according to their oscillation properties, and a comparison between the classical differential operator and the Riemann-Liouville operator is made.

We investigate the existence and multiplicity of solutions for a class of the generalized coupled system involving poly-Laplacian and the parameter λ \lambda on finite graphs. By using the Mountain pass lemma together with the cut-off technique, we obtain that system has at least a nontrivial weak solution ( u λ , v λ ) \left({u}_{\lambda },{v}_{\lambda }) for every large parameter λ \lambda when the nonlinear term F ( x , u , v ) F\left(x,u,v) satisfies superlinear growth conditions only in a neighborhood of origin point (0, 0). We also obtain a concrete form for the lower bound of λ \lambda and the trend of ( u λ , v λ ) \left({u}_{\lambda },{v}_{\lambda }) with the change of λ \lambda . Moreover, by using a revised Clark’s theorem together with cut-off technique, we obtain that system has a sequence of solutions tending to 0 for every λ > 0 \lambda \gt 0 when the nonlinear term F ( x , u , v ) F\left(x,u,v) satisfies sublinear growth conditions only in a neighborhood of origin point (0, 0).

In this work, we derive a discrete-time chemostat model from a continuous-time population model using the forward Euler method. We first determine the model’s fixed points and analyze their local stability. By applying the central manifold theorem and bifurcation theory, we establish the existence of flip and Neimark-Sacker bifurcations, considering the step size as the bifurcation parameter. Theoretical findings are validated through numerical simulations, which also uncover novel dynamic behaviors.

The Duhamel principle is a mathematical principle that allows us to solve linear partial differential equations. This system is generalized by the concept of the k k -symbol fractional calculus. We demonstrate that in specific functional domains, the suggested system produces a global solution. Convergence toward the stable point is investigated. Some special cases are illustrated including the rotated Koebe function, which is used for creating the optimal solutions, where these functions are convex univalent in the open unit disk.

The Peyrard-Bishop DNA model is investigated in this study. Two most reliable and efficient analytical techniques, the Jacobi elliptic function method, and the tanh \tanh - coth \coth method, are being employed for finding new and novel soliton solutions of the suggested model. The Painlevé-test (P-test) is applied on the proposed DNA model to test the integrability. Few extracted results are graphically examined for studying the dynamical behavior.

In this work, we prove some fixed point results in generalized Banach spaces ( G B S {\mathfrak{G}}{\mathfrak{B}}{\mathfrak{S}} s) in the sense of Perov using the vector degree of nondensifiability tools. The given result generalizes Darbo’s and Krasnoselskii’s theorems, which are connected with the vector measure of noncompactness. An existence result for coupled Caputo fractional delay differential equations in the G B S C ( [ − τ , T ] , R ) × C ( [ − τ , T ] , R ) {\mathfrak{G}}{\mathfrak{B}}{\mathfrak{S}}\hspace{0.33em}{\mathcal{C}}\left(\left[-\tau ,T],{\mathbb{R}})\times {\mathcal{C}}\left(\left[-\tau ,T],{\mathbb{R}}) is given to show the significance and the applicability of our theoretical results.

The focal point of this article is to study a functional equation of sum form involving four unknown mappings. We primarily concentrate on exploring all possible solutions of the equation. An attempt is made to delve into the intricacies of obtaining these solutions without assuming any regularity condition on the mappings. Our study further aims to examine the stability of these general solutions, thus shedding light on the behaviour of sum form functional equations under perturbations. The article concludes by reflecting upon the relevance of the general solutions in information theory and diversity index.

In this study, we establish new generalizations and results for Simpson, midpoint, and trapezoid-type integral inequalities within the framework of multiplicative calculus. We begin by proving a new identity for multiplicatively differentiable functions. Using this identity, we then obtain a new Simpson-type inequality for multiplicatively s s -convex functions. Additionally, we derive novel integral inequalities related to the right and left sides of the Hermite-Hadamard inequality for multiplicatively s s -convex functions. We also demonstrate that some of the results obtained here improve upon known results, while others are generalizations. Finally, we present some applications to special means.

Reverse Pachpatte-type inequalities are concave generalizations of the well-known Bennett-Leindler-type inequalities. We establish reverse nabla Pachpatte-type dynamic inequalities taking account of concavity. It is the first time that converses of Pachpatte-type inequalities are obtained in the nabla time scale calculus as well as for its special cases such as continuous and discrete cases and for the dual results obtained in the delta time scale calculus. Moreover, some of our results extend the related ones when concavity has been removed.

In this work, we investigate the stabilization problem for coupled biharmonic Schrödinger equations with fractional internal damping. First, we establish that the system is well-posed using the semigroup theory of linear operators. Second, we demonstrate that the system is strongly stable. Finally, we prove a polynomial decay rate using multiplier techniques combined with the frequency domain method, and we show the lack of exponential decay of the energy in a specific case.

In this study, we investigate a nonlinear heat equation incorporating both a viscoelastic term and a reaction-diffusion term that depends on space-time variables. Initially, we establish the local Hadamard well-posedness results using the standard Faedo-Galerkin method. Subsequently, we demonstrate that the solution exhibits finite-time blowup for initial energy values that are both negative and nonnegative. Finally, we establish the global existence of the solution and provide general decay estimates for the energy functions with small initial energy, utilizing Martinez’s inequality.

The aim of this article is to establish some new extensions and variants of Jensen’s discrete and Simic-type inequalities for HA-convex and uniformly HA-convex functions. We introduce uniformly HA-convex functions, which are the generalized class of HA-convex (harmonic-arithmetic)-convex functions and provide some examples too. As an application point of view for the new Jensen’s bounds, we present some new improved bounds for Shannon’s entropy. Finally, some analysis is given by graphical illustration to prove the improvements of bounds.

This study focuses on differential equations incorporating generalized fractional derivatives of the Caputo type. The concept of Ulam-type stability (US) is analyzed in the context of both initial value problems and boundary value problems (BVPs) for the fractional differential equations under investigation. Particular attention is given to addressing certain misconceptions that arise when applying US to BVPs. To mitigate these issues, we propose incorporating a parameter into the boundary conditions as a potential solution. The dependency of the solution on this parameter is established, and a method is outlined for selecting the parameter appropriately. This approach ensures that the solution of the fractional equation is strongly influenced by the arbitrarily chosen solution of the associated inequality. The theoretical findings are further clarified through illustrative examples.

In this article, we incorporate multivariate Taylor polynomials into the definition of the Bernstein operators to get a faster approximation to multivariate functions by these combined operators. We also give various numerical simulations including graphical illustrations and error estimations. Our results improve not only the linear approximation by classical Bernstein polynomials but also the nonlinear approximation obtained by max-product operations.

In this article, we introduce a new family of ( λ , ψ ) \left(\lambda ,\psi ) -Bernstein-Kantorovich operators which depends on a parameter λ \lambda , derived from the basis functions of Bézier curves and an integrable function ψ \psi . In this approach, all moments and central moments of the new operators can be obtained in terms of two numbers M 1 , ψ {M}_{1,\psi } and M 2 , ψ {M}_{2,\psi } , which are the integrals of ψ \psi and ψ 2 {\psi }^{2} , respectively. For operators L n ( f ; x ) {L}_{n}\left(f;\hspace{0.33em}x) with L n ( 1 ; x ) = 1 {L}_{n}\left(1;\hspace{0.33em}x)=1 , the order of approximation to a function f f by L n ( f ; x ) {L}_{n}\left(f;\hspace{0.33em}x) is more controlled by the term L n ( ( t − x ) 2 ; x ) {L}_{n}\left({\left(t-x)}^{2};\hspace{0.33em}x) . For our operators K n , λ , ψ ( f ; x ) {K}_{n,\lambda ,\psi }\left(f;\hspace{0.33em}x) , the second central moment K n , λ , ψ ( ( t − x ) 2 ; x ) {K}_{n,\lambda ,\psi }\left({\left(t-x)}^{2};\hspace{0.33em}x) depend on M 1 , ψ {M}_{1,\psi } and M 2 , ψ {M}_{2,\psi } . This means that in our new approach, it is possible to search for a function ψ \psi with different values of M 1 , ψ {M}_{1,\psi } and M 2 , ψ {M}_{2,\psi } to make K n , λ , ψ ( ( t − x ) 2 ; x ) {K}_{n,\lambda ,\psi }\left({\left(t-x)}^{2};\hspace{0.33em}x) smaller. Using this new approach, we show that there exists a function ψ \psi such that the order of approximation to a function f f by our new ( λ , ψ ) \left(\lambda ,\psi ) -Bernstein-Kantorovich operators is better than the classical λ \lambda -Bernstein-Kantorovich operators on the interval [0, 1]. Moreover, we obtain some direct and local approximation properties of new operators. We also show that our operators preserve monotonicity properties. Furthermore, we illustrate the approximation results of our operators graphically and numerically.

This article is dedicated to present various concepts on α \alpha -time scale, including power series, Taylor series, binomial series, exponential function, gamma function, and Bessel functions of the first kind. We introduce the α \alpha -exponential function as a series, examine its absolute and uniform convergence, and establish its additive identity by employing the α \alpha -Gauss binomial formula. Furthermore, we define the α \alpha -gamma function and prove α \alpha -analogue of the Bohr-Mollerup theorem. Specifically, we demonstrate that the α \alpha -gamma function is the unique logarithmically convex solution of f ( s + 1 ) = ϕ ( s ) f ( s ) f\left(s+1)=\phi \left(s)f\left(s) , f ( 1 ) = 1 f\left(1)=1 , where ϕ ( s ) \phi \left(s) refers to the α \alpha -number. In addition, we present Euler’s infinite product form and asymptotic behavior of α \alpha -gamma function. As an application, we propose α \alpha -analogue of the cylindrical diffusion equation, from which α \alpha -Bessel and modified α \alpha -Bessel equations are derived. We explore the solutions of the α \alpha -cylindrical diffusion equation using the separation of variables technique, revealing analogues of the Bessel and modified Bessel functions of order zero of the first kind. Finally, we illustrate the graphs of the α \alpha -analogues of exponential and gamma functions and investigate their reductions to discrete and ordinary counterparts.

As a Bloch variant of ( q , p ) \left(q,p) -mixing linear operators, we introduce the notion of ( q , p ) \left(q,p) -mixing Bloch maps. We prove Pietsch’s domination theorem and Maurey’s splitting theorem in this Bloch context, following the corresponding results from the linear case.

In this study, we utilize the Gram-Schmidt orthogonalization method to construct a new set of orthogonal polynomials called OB n ( x , q ) {{\rm{OB}}}_{n}(x,q) from the q -Bernoulli polynomials. We demonstrate the relationship between polynomials OB n ( x , q ) {{\rm{OB}}}_{n}(x,q) and the little q -Legendre polynomials, and derive a generalized formula for OB n ( x , q ) {{\rm{OB}}}_{n}(x,q) by leveraging the little q -Legendre polynomials. Furthermore, we present some properties of polynomials OB n ( x , q ) {{\rm{OB}}}_{n}(x,q) . Finally, we introduce a hybrid of block-pulse function and orthogonal polynomials OB n ( x , q ) {{\rm{OB}}}_{n}(x,q) and examine various properties of these polynomials.

Bede et al. (B. Bede, L. Coroianu, and S. G. Gal, Approximation and shape preserving properties of the nonlinear Meyer-König and Zeller operator of max-product kind , Numer. Funct. Anal. Optim. 31 (2010), no. 3, 232–253) defined the max-product Meyer-König and Zeller operator. They examined the approximation and shape preserving properties of this operator, and they found the order of approximation to be y ( 1 − y ) m \frac{\sqrt{y}(1-y)}{\sqrt{m}} by the modulus of continuity and claimed that this order of approximation could only be improved in certain subclasses of the functions. In contrast to this claim, we demonstrate that we can obtain a better order of approximation without reducing the function class (by the classical modulus of continuity). We find the degree of approximation to be ( 1 − y ) y 1 α m 1 − 1 α \frac{(1-y){y}^{\tfrac{1}{\alpha }}}{{m}^{1-\tfrac{1}{\alpha }}} , α = 2 , 3 , … \alpha =2,3,\ldots . Since 1 − 1 α 1-\frac{1}{\alpha } tends to 1 for enough big α \alpha , we improve this degree of approximation.

This study investigates the renewal reward process under the assumption that the random variables describing the discrete interference of chance follow a generalized gamma distribution. A moment-based approximation method is employed to derive novel results for the renewal function, enabling an approximation of the ergodic distribution of the process. Furthermore, the limiting distribution of the ergodic distribution is also derived. The theoretical findings are illustrated through a specific example in which the demand random variable η 1 {\eta }_{1} is represented by a third-order Erlang distribution with parameter θ = 1 \theta =1 .

In this article, we express a numerical form of the convergence using the suitable modulus of smoothness for linear compositions of the Mellin convolution operators. Later, with the help of a different modulus of smoothness than before, called the logarithmic weighted modulus of smoothness in the weighted Mellin-Lebesgue spaces, a rate of convergence is obtained, and then the global smoothness preservation property is also expressed through the logarithmic weighted modulus of smoothness in the weighted Mellin-Lebesgue spaces. It is noteworthy that two different moduli of smoothness are used in this article. One is given on the weighted space of continuous functions, the other on the weighted Mellin-Lebesgue spaces comprising all Lebesgue measurable functions. Moreover, the striking results in this article are that the stated quantitative-type theorems are expressed in terms of the modulus of smoothness of the m m th order Mellin derivative of the given function. In the last section, the results obtained are given for classical Mellin-Gauss-Weierstrass operators.

This article introduces two-variable polynomials constructed with generalized Laguerre polynomials. The study focuses on deriving generating function relationships, establishing closed-form summation formulas, and proving a symmetry identity for the proposed polynomials. In addition, a theorem concerning bilinear and bilateral generating functions is rigorously established. The connections between the newly defined polynomials and other well-known polynomial families are also explored, highlighting their potential to unify and extend existing mathematical frameworks. These findings provide a comprehensive perspective on the theoretical significance and applicability of the proposed polynomials in advanced mathematical research.

In this article, we consider the initial value problem for a system of Klein-Gordon equations in anti-de Sitter spacetime. We assume that the mass matrix is diagonalizable with positive eigenvalues. Under the assumption for the small initial data, we prove the global existence of solutions to the initial value problem. The existence result is derived based on the assumption of power-type nonlinearity.

This study examines Durrmeyer-type exponential sampling series to obtain a quantitative estimate by using the concept of the logarithmic modulus of smoothness defined with the help of a suitable modular functional on Mellin-Orlicz spaces. Also, we want to underline that the spaces on which we study are reduced versions of Mellin-Orlicz spaces. Additionally, we obtain a further estimate for a particular case of Mellin-Lebesgue space, i.e., X 0 p , {X}_{0}^{p}, by utilizing an appropriate logarithmic modulus of smoothness that is different from the previous one. Also, we obtain direct estimates on the order of approximation for two different types of logarithmic Lipschitz classes by using previously mentioned results in involved sections.

This article introduces a novel concept of convergence, referred to as ℐ {\mathcal{ {\mathcal I} }} - α β \alpha \beta -statistical relative uniform convergence, for double sequences of functions. This notion, which is proposed for the first time in this article, is explored in depth, leading to the establishment of a Korovkin-type approximation theorem within this framework. The illustrative example demonstrates that the proposed convergence is indeed stronger than previously known forms. Additionally, this article investigates the rate of ℐ 2 {{\mathcal{ {\mathcal I} }}}_{2} - α β \alpha \beta -statistical relative uniform convergence, providing explicit computations to support the findings. The results contribute to the understanding of ideal statistical convergence and open up new perspectives for approximation theory.

In this article, we provide a note on rigidity results for mean field-type equations with multiple exponential terms, including sinh-Gordon and Tzitzeica equations, on closed two-dimensional manifolds with positive Ricci curvature bounds. It is based on two nonlinear transformations and a geometric identity developed by Dolbeault et al.

Our primary objective is to study the solvability of two kinds of fractional Kirchhoff double-phase problem involving logarithmic nonlinearity in R N {{\mathbb{R}}}^{N} via the variational approach. To address the modulation coefficient in fractional double-phase operators, a new functional framework and the corresponding fractional Musielak-Sobolev space are provided. Among others, the completeness, reflexivity, and uniformly convexity of the space are established, and the continuity, boundedness, and ( S + ) \left({S}_{+}) -property of the fractional double-phase operator are given.

This study innovates a novel technique of the nonlinear fractional Langevin equation of Hilfer-Hadamard type, incorporating an initial condition. The research demonstrates that this problem can be reformulated as an integral equation featuring a Mittag-Leffler function within the kernel. Through rigorous analysis, we establish the existence and uniqueness of solutions for this problem without imposing a contractive assumption. Furthermore, the study extends these findings to encompass two specific types of fractional differential equations characterized by two fractional derivatives and a variable coefficient. Theoretical conclusions are supported by the provision of two illustrative examples.

This study examines some dynamical aspects of the nonlinear fractional Chen-Lee-Liu (CLL) model. This model has applications in the evaluation of optical fiber communication systems. To study the CLL model we transformed it to a Hamiltonian system. Using the qualitative theory of planar integrable systems, we investigated the bifurcation of the resulting system and plotted its phase portraits. The system was integrated over the possible intervals of real wave propagation to obtain new exact solutions for the model. Graphical representations of some solutions are provided. We also analyzed the influence of fractional order and group velocity dispersion on two solution types. These effects are expressed in terms of amplitude and width of the wave solution. By introducing a periodic external influence into the model, we explored quasi-periodic and chaotic behaviors through 2D and 3D phase portraits and time series analyses. Sensitivity analysis was employed to investigate chaotic patterns. Numerical simulations demonstrated that variations in frequency and amplitude significantly impact the system’s dynamical characteristics.