Sahand Communications in Mathematical Analysis

Abstract In this article, we first define hyperstructures known as Krasner hypermodules. Then, the concept of topological Krasner hypermodules is explored, examining their fundamental properties
and the notion of continuous mappings that exist between such topological hyperstructures. Next, the concept of Hausdorff topology is introduced and its relation to Krasner hypermodules is examined. The relationship between locally compact Krasner hypermodules and the role of open neighborhoods in their topological structure is then analyzed. Several theorems are presented and proven to clarify these relationships. By applying relative topology to subhypermodules, their associated properties are analyzed. In other words, the aim is to use specific topologies to identify the various substructural features of this type of hypermodule. Additionally, the quotient topology induced by the $\theta^*$-relation on the Krasner hypermodule is investigated to understand how this relation affects the topological structure of the hypermodule. Finally, it is shown that the topological Krasner hypermodule induced by $\tau_\theta$​, the finest and strongest topology on it, ultimately forms a module.

Abstract In this paper, we introduce a generalized normed space, which we refer to as a $G$-normed space. We define the concepts of a $G$-continuous and a $G$-bounded linear operator on this space and explore some related properties. Given the central role of the Banach-Steinhaus Theorem in the theory of normed spaces, particularly in the study of bounded linear operators, we conclude by formulating and proving a version of the Banach-Steinhaus Theorem for $G$-normed spaces.

Abstract This paper introduces an inertial shrinking projection algorithm for approximating fixed points of relatively nonexpansive mappings in uniformly convex and smooth Banach spaces. By incorporating inertial terms, the method improves convergence speed and stability compared to classical projection techniques. The analysis relies on geometric properties such as the Kadec-Klee condition and the continuity of the duality mapping to ensure strong convergence. The proposed algorithm generalizes several existing iterative schemes and operates under mild assumptions. Numerical results in both finite-dimensional and function spaces confirm its practical effectiveness. 

Abstract Fractional operators and integral inequalities have become a focal point of research due to their applications in mathematics, physics, engineering, and applied sciences. This paper introduces a new identity for the Caputo-Fabrizio fractional integral operators. Employing the Peano kernel method, we derive Simpson's type inequalities for $\left( s,m\right) $-convex functions through twice-differentiable functions, accompanied by graphical illustrations to analyze their behavior. Several new corollaries are established, with insightful remarks enhancing their interpretation.
Additionally, applications to special means, $q$-digamma functions, modified Bessel function, Simpson's formula, matrix inequality, and midpoint formula are explored, underscoring the utility and adaptability of these results across various mathematical and applied domains.

Abstract In this article, we introduce a new class of convex functions called $\alpha$-inverse cosine convex functions ($\alpha$-ICCF), which extends the traditional classes. We analyze various algebraic and geometric properties by illustrating the graphs of several significant $\alpha$-ICCF via visual representations. Utilizing this novel class, we derive the Hermite-Hadamard (HH) inequality and certain refinements for functions whose first derivative in absolute value is $\alpha$-ICCF. The primary tools employed in deriving the main results include Hölder's inequality, Hölder-Iscan inequality and power-mean integral inequality. Our findings demonstrate that the approximations obtained using Hölder-Iscan and the improved power-mean integral inequality are superior to those derived from other methods. In particular, when $\alpha=1$, the derived results will coincide with those of classical ICCF. This innovative concept of $\alpha$-inverse cosine convexity opens new avenues for research, encouraging further exploration of such convexity classes.

Abstract Taking into account the recent view of fixed point theory as expressed by Jalali and Samet [On Banach's fixed point theorem in perturbed metric spaces, J. Appl. Anal. Comput.  14 (2) (2025), 992--1001], we first introduce a perturbed metric space equipped with a graph and then present new concepts and notions related to this space. Next, we prove some fixed point theorems related to this new space. Several consequences and an example are also presented to demonstrate the effectiveness of the main results. Following the idea of this article, one can continue this new way to obtain fixed points of the mappings that do not satisfy classic contractions in such spaces endowed with a graph or a partial order.

Abstract This study introduces the sequence spaces $c(S^\star)$ and $c_0(S^\star)$, defined as the domain of the matrix $S^\star$ constructed from a hybrid structure involving Schr\"{o}der and Catalan numbers. A comprehensive investigation is conducted into the fundamental topological and structural properties of these sequence spaces, such as completeness and their relationships and embedding within classical sequence spaces. Furthermore, the dual space structures corresponding to these newly defined spaces are thoroughly characterized. In the final sections, various classes of matrix transformations and compact linear operators acting on these sequence spaces are examined, emphasizing their significance in functional analysis.

Abstract Boundary and initial value problems,  including nonlinear difference equations and nonlinear differential equations, are the mathematical models of many physics and engineering problems and natural phenomena. Usually, due to the lack of a solid theory for solving these types of equations, these equations are solved by using numerical and approximate methods. In this paper, first some elementary and basic definitions and concepts of discrete and continuous multiplicative calculus are given.  Next we apply some ideas and methods to obtain invariant functions with respect to their associated derivative. These invariant functions are used to solve several types of nonlinear difference and differential equations that have appeared in natural sciences and physical problems. 
After that, these methods are expanded for solving nonlinear difference and differential equations through discrete and continuous multiplicative differential equations. 
Finally, some applications of multiplicative forms of differential equations are given which  simplify numerical methods for solving nonlinear biological problems and exponential approximations for nonlinear functions.

Abstract In this study, the $q$-calculus is employed to introduce a novel generalization of the Mittag-Leffler function. In the following, we investigate a novel $q$-exponential function with five parameters, resulting in the generalized $q$-Mittag-Leffler function.  Some $q$-integral representations and fractional $q$-derivative (Caputo  and Hilfer) for this $q$-Mittag-Leffler type function are derived. Moreover, we obtain the solutions to the $q$-fractional kinetic equations includes this function by applying $q$-Laplace and $q$-Sumudu transforms defined using fractional $q$-calculus operators of the Riemann-Liouville (R-L) type. A few particular scenarios to illustrate the use of our primary finding. Further, we state some significant and special cases of our main results. Finally, we present the obtained solutions in the form of numerical graphs using MATLAB 16.

Abstract This paper investigates the application of the Quasilinearization Method (QLM) for approximating non-linear delay differential equations (DDEs), which are prevalent in fields such as control systems and population dynamics. QLM effectively transforms these complex non-linear problems into a system of linear equations, a key advantage for computational efficiency. Our work provides two main contributions: a rigorous mathematical proof demonstrating the quadratic convergence of the proposed technique and numerical examples that illustrate its practical applicability and reliability. We apply QLM to DDEs with various non-linear forms, including quadratic and exponential types and with fixed, discrete delays. The results confirm that the method is highly accurate, computationally efficient and easy to implement, making it  a valuable tool for future research.

Abstract The concept of convexity represents one of the fundamental notions of mathematical analysis and optimization. Various extensions of the concept of convexity have contributed to a wide range of applications, including the study of integral inequalities, approximation theory and other areas of applied mathematics. In this paper, we start from exponentially convex functions defined with respect to $s$ and introduce a new class of $m$-exponentially convex functions with respect to $s$. Furthermore, the basic algebraic properties of this class are analyzed. In the second part, special attention is devoted to the application and the meaning of the extension of the Hermite–Hadamard inequality, where a more general framework is provided compared to the existing ones. The obtained results, in addition to their theoretical significance, also point to the potential for applications in data analysis, optimization and further generalizations.

Abstract This paper introduces and explores the concept of statistical derivatives within the framework of deferred N\"{o}rlund summability, complemented by illustrative examples. Leveraging this approach, we establish a new Korovkin-type theorem for a specific class of algebraic test functions, namely $1$, $x$ and $x^{2}$, within the Banach space $\mathfrak{C}[0,1]$. Our findings serve as a significant generalization of several classical and statistical Korovkin-type results in approximation theory. Furthermore, we examine the rate of convergence associated with statistical derivatives under deferred N\"{o}rlund summability, providing insights into the effectiveness of this summability method. To validate our theoretical results, we present numerical examples alongside graphical visualizations created using MATLAB, offering a clearer perspective on the convergence behavior of the proposed operators.

Abstract This paper studies the scalability of tensor products of several frames, which may come from different Hilbert spaces. Although the scalability of single frames and tensor products of two frames has been explored, the case with more than two frames has not been studied and solved yet. We establish sufficient
and necessary conditions under which such tensor products are scalable, and we describe these conditions using operator theory, spectral analysis, and numerical methods. To clarify the issue, we provide examples and counterexamples. Finally, we briefly mention how this can be applied in signal processing and sparse representation.