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In this paper, an attempt has been made to establish the sufficient conditions for the existence and multiplicity of positive solutions by using the fixed point index theory and the Avery–Peterson fixed point theorem respectively for the following class of nonlinear singular fractional differential equation D 0 + β ⁢ ( ϕ p ⁢ ( D 0 + α ⁢ u ⁢ ( t ) ) ) + f ⁢ ( t , u ⁢ ( t ) , … , u ( n - 2 ) ⁢ ( t ) ) = 0 , t ∈ ( 0 , 1 ) , D_{0^{+}}^{\beta}(\phi_{p}(D_{0^{+}}^{\alpha}u(t)))+f(t,u(t),\dots,u^{(n-2)}(t% ))=0,\quad t\in(0,1), with the boundary conditions u ( k ) ⁢ ( 0 ) = 0 , 0 ≤ k ≤ n - 2 , D 0 + α ⁢ u ⁢ ( 0 ) = 0 , D 0 + α - 1 ⁢ u ⁢ ( 1 ) = ∫ 0 1 D 0 + α - 1 ⁢ u ⁢ ( t ) ⁢ 푑 A ⁢ ( t ) , u^{(k)}(0)=0,\quad 0\leq k\leq n-2,\qquad D_{0^{+}}^{\alpha}u(0)=0,\qquad D_{0% ^{+}}^{\alpha-1}u(1)=\int_{0}^{1}D_{0^{+}}^{\alpha-1}u(t)\,dA(t), where 0 < β ≤ 1 {0<\beta\leq 1} , n - 1 < α ≤ n {n-1<\alpha\leq n} , n ≥ 3 {n\geq 3} , A : [ 0 , 1 ] → ℝ {A:[0,1]\to\mathbb{R}} is a function of bounded variation, ϕ p ⁢ ( s ) = | s | p - 2 ⁢ s {\phi_{p}(s)=|s|^{p-2}s} for p > 1 {p>1} and ϕ q ⁢ ( s ) {\phi_{q}(s)} is the inverse of ϕ p ⁢ ( s ) {\phi_{p}(s)} , where p , q {p,q} satisfies the relation 1 p + 1 q = 1 {\frac{1}{p}+\frac{1}{q}=1} , f may have singularity at t = 0 {t=0} . [ABSTRACT FROM AUTHOR]

The main objective of this paper is to investigate exponential attractors for a nonlocal delayed reaction-diffusion equation on an unbounded domain. We first obtain the existence of a globally attractive absorbing set for the dynamical system generated by the equation under the assumption that the nonlinear term is bounded. Then, we construct exponential attractors of the equation directly in its natural phase space, i.e., a Banach space with explicit fractal dimension by combining squeezing properties of the system as well as a covering lemma of finite dimensional subspaces of a Banach space. Our result generalizes the methods established in Hilbert spaces and weighted spaces, and the fractal dimension of the obtained exponential attractor does not depend on the entropy number but only depends on some inner characteristic of the studied equation. [ABSTRACT FROM AUTHOR]

Given Banach space operators A, B, let δ A , B denote the generalised derivation δ (X) = (L A - R B) (X) = A X - X B and ▵ A , B the length two elementary operator ▵ A , B (X) = (I - L A R B) (X) = X - A X B . This note considers the structure of m-symmetric operators δ ▵ A 1 , B 1 , ▵ A 2 , B 2 m (I) = (L ▵ A 1 , B 1 - R ▵ A 2 , B 2 ) m (I) = 0 . It is seen that there exist scalars λ i ∈ σ a (B 1) , 1 ≤ i ≤ 2 , such that δ λ 1 A 1 , λ 2 A 2 m (I) = 0 . Translated to Hilbert space operators A and B this implies that if δ ▵ A ∗ , B ∗ , ▵ A , B m (I) = 0 , then there exists λ ¯ ∈ σ a (B ∗) such that δ (λ A) ∗ , λ A m (I) = 0 = δ λ ¯ B , λ B ∗ m (I) . We prove that the operator δ ▵ A ∗ , B ∗ , ▵ A , B m is compact if and only if (i) there exists a real number α and finite sequnces (i) { a j } j = 1 n ⊆ σ (A) , { b j } j = 1 n ⊆ σ (B) such that a j b j = 1 - α , 1 ≤ j ≤ n ; (ii) decompositions ⊕ j = 1 n H j and ⊕ j = 1 n H J of H such that ⊕ j = 1 n (A - a j I) | H j and ⊕ j = 1 n (B - b j I) | H j are nilpotent. If δ ▵ A ∗ , B ∗ , ▵ A , B m (I) = 0 implies δ ▵ A ∗ , B ∗ , ▵ A , B (I) = 0 , then A and B satisfy a (Putnam-Fuglede type) commutativity theorem; conversely, a sufficient condition for δ ▵ A ∗ , B ∗ , ▵ A , B m (I) = 0 to imply δ ▵ A ∗ , B ∗ , ▵ A , B (I) = 0 is that λ A and λ ¯ B satisfy the commutativity property for scalars lambda ¯ ∈ σ a (B ∗) . An analogous result is seen to hold for the operators ▵ δ A ∗ , B ∗ , δ A , B m and ▵ δ A ∗ , B ∗ , δ A , B m (I) . Perturbation by commuting nilpotents is considered. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, the split variational inequality problem with multiple output sets is considered in a more general framework of reflexive Banach spaces. We introduce a relaxed Tseng extragradient method for approximating the solution of this problem when the cost operators are pseudomonotone and non-Lipschitz. Moreover, our proposed algorithm employs the inertial technique and a new self-adaptive step size to guarantee high rate of convergence. Strong convergence of the algorithm is proved in this case without the need for the sequential weak continuity condition. Finally, some numerical experiments are given to show the applicability of our proposed algorithm. [ABSTRACT FROM AUTHOR]

The local analysis of convergence for two competing methods of order seven or eight is developed to solve Banach space‐valued equations. Previous studies have used the eighth or ninth derivative of the operator involved, which do not appear on the methods, to show the convergence of these methods on the finite‐dimensional Euclidean space. In addition, no computable error distances or isolation of the solution results are provided in this study. These problems limit the applicability of this method to solving equations with operators that are at least nine times differentiable. In the current study, only conditions on the first derivative, appearing in these methods, are employed to show the convergence of these methods. Moreover, computable error bounds depend on the distance in the world as well as the isolation of the solution. Results are provided based on generalized continuity conditions on the first derivative. Furthermore, the more interesting semi‐local analysis of convergence not given before these methods is presented using a majorizing sequence. Finally, a great deal of impressive numerical results has been shown on real‐world problems. [ABSTRACT FROM AUTHOR]

In the present work, we construct a sixth order derivative free family without memory for solving nonlinear operator equations in Banach spaces. We further modify this to a ninth order family with memory without any additional functional evaluation. Local convergence analysis of both these methods have been studied using assumptions only on the first derivative. Numerical computations validate the theoretical results and show the superiority of our methods over the existing ones. Basins of attraction have also been presented to see the dynamical behaviour of the proposed methods. [ABSTRACT FROM AUTHOR]

Let Ω be a nonempty closed convex subset of a real Hilbert space H . Let ℑ be a nonspreading mapping from Ω into itself. Define two sequences { ψ n } n = 1 ∞ and { ϕ n } n = 1 ∞ as follows: { ψ n + 1 = π n ψ n + (1 − π n) ℑ ψ n , ϕ n = 1 n ∑ n t = 1 ψ t , for n ∈ N , where 0 ≤ π n ≤ 1 , and π n → 0 . In 2010, Kurokawa and Takahashi established weak and strong convergence theorems of the sequences developed from the above Baillion-type iteration method (Nonlinear Anal. 73:1562–1568, 2010). In this paper, we prove weak and strong convergence theorems for a new class of (η , β) -enriched strictly pseudononspreading ((η , β) -ESPN) maps, more general than that studied by Kurokawa and W. Takahashi in the setup of real Hilbert spaces. Further, by means of a robust auxiliary map incorporated in our theorems, the strong convergence of the sequence generated by Halpern-type iterative algorithm is proved thereby resolving in the affirmative the open problem raised by Kurokawa and Takahashi in their concluding remark for the case in which the map ℑ is averaged. Some nontrivial examples are given, and the results obtained extend, improve, and generalize several well-known results in the current literature. [ABSTRACT FROM AUTHOR]

In this work, we give three kinds of criteria for weak sets in Orlicz–Bochner sequence spaces l(Φ)(X)$l_{(\Phi)}(X)$ without constraints, conditions posited in each criterion are necessary and sufficient. As an application, we give criteria for weak sets in Orlicz sequence spaces. Well‐known conclusions are exhibited once more, such as Schur's theorem, Banach–Alaoglu's theorem, and the boundedly compact principle of finite dimension space. The results obtained show that the weak compactness may not be extrapolated straightforwardly from X$X$ to l(Φ)(X)$l_{(\Phi)}(X)$, for example, l∞(X)$l_{\infty }(X)$. [ABSTRACT FROM AUTHOR]

This research is concerned with the existence and uniqueness of solutions for a coupled system of Ψ –Riemann–Liouville fractional differential equations. To achieve this objective, we establish a set of necessary conditions by formulating the problem as an integral equation and utilizing well-known fixed-point theorems. By employing these mathematical tools, we demonstrate the existence and uniqueness of solutions for the proposed system. Additionally, to illustrate the practical implications of our findings, we provide several examples that showcase the main results obtained in this study. [ABSTRACT FROM AUTHOR]

This article introduces a multistep method for developing sequences that solve Banach space-valued equations. It provides error estimates, a radius of convergence, and uniqueness results. Our approach improves the applicability of the recommended method and addresses challenges in applied science. The theoretical advancements are supported by comprehensive computational results, demonstrating the practical applicability and robustness of the earlier method. We ensure more reliable and precise solutions to Banach space-valued equations by providing computable error estimates and a clear radius of convergence for the considered method. We conclude that our work significantly improves the practical utility of multistep methods, offering a rigorous and computable approach to solving complex equations in Banach spaces, with strong theoretical and computational results. [ABSTRACT FROM AUTHOR]

This paper studies the dynamic of a chain of infinitely many kinematic points guaranteeing the uniform convergence in the ℓ∞$$ {\ell}&#x0005E;{\infty } $$ sense. A sufficient condition and some necessary conditions for infinite many kinematic points rendezvousing are given in terms of the spectrum and the invariant subspaces of the dynamic operator. The necessary and sufficient condition for finitely many kinematic points rendezvousing is characterized by the eigenvalues and the invariant subspaces of the dynamic operator. The estimate on the rate of the rendezvous is derived based on the spectrum of the dynamic operator. [ABSTRACT FROM AUTHOR]

We extend the well-known Gelfand–Phillips property for Banach spaces to locally convex spaces, defining a locally convex space E to be b-Gelfand–Phillips if every limited set in E, which is bounded in the strong topology β (E , E ′) on E, is precompact in β (E , E ′). Several characterizations of b-Gelfand–Phillips spaces are given. The problem of preservation of the b-Gelfand–Phillips property by standard operations over locally convex spaces is considered. Also we explore the b-Gelfand–Phillips property in spaces C(X) of continuous functions on a Tychonoff space X. If τ and T are two locally convex topologies on C(X) such that T p ⊆ τ ⊆ T ⊆ T k , where T p is the topology of pointwise convergence and T k is the compact-open topology on C(X), then the b-Gelfand–Phillips property of the function space (C (X) , τ) implies the b-Gelfand–Phillips property of (C (X) , T). If additionally X is metrizable, then the function space (C (X) , T) is b-Gelfand–Phillips. [ABSTRACT FROM AUTHOR]

In our research, we broaden the scope of Fourier–Stieltjes transforms to encompass locally compact groups, denoted as G. We achieve this extension by leveraging the induced representation from a closed subgroup K. From this, we deduce the Fourier transform f ^ of a Haar-integrable function f defined on G. Specifically, we express f ^ as the Fourier–Stieltjes transform μ ^ of the measure μ = f λ , where λ denotes the Haar measure of G. Our work is significant because when applied to Lie groups with compact subgroups K, our Fourier–Stieltjes transform m ^ exhibits more nuanced characteristics compared to the traditionally defined one via the Gel’fand transform, which is standard in the context of Lie groups. We rigorously substantiate this observation. One of the principal challenges we confront is the construction of the “trigonometric functions”, which serve as the foundation for building the Fourier transform. [ABSTRACT FROM AUTHOR]

The local analysis of convergence for Newton's method has been extensively studied by numerous researchers under a plethora of sufficient conditions. However, the complexity of extending the convergence domain requires very general conditions such as the ones depending on the majorant principle in order to include as large classes of operators as possible. In the present article, such an analysis is developed under the weak majorant condition. The new results extend earlier ones using similar information. Finally, the numerical examples complement the theory. [ABSTRACT FROM AUTHOR]

In this paper, we studied the AA-iterative algorithm for finding fixed points of the class of nonlinear generalized (α, β)-nonexpansive mappings. First, we proved weak convergence and then proved several strong convergence results of the scheme in a ground setting of uniformly convex Banach spaces. We gave a few numerical examples of generalized (α, β)-nonexpansive mappings to illustrate the major outcomes. One example was constructed over a subset of a real line while the other one was on the two dimensional space with a taxicab norm. We considered both these examples in our numerical computations to show that our iterative algorithm was more effective in the rate of convergence corresponding to other fixed point algorithms of the literature. Some 2D and 3D graphs were obtained that supported graphically our results and claims. As applications of our major results, we solved a class of fractional differential equations, 2D Voltera differential equation, and a convex minimization problem. Our findings improved and extended the corresponding results of the current literature [ABSTRACT FROM AUTHOR]

The computational study of fixed-point problems in distance spaces is an active and important research area. The purpose of this paper is to construct a new iterative scheme in the setting of Banach space for approximating solutions of fixed-point problems. We first prove the strong convergence of the scheme for a general class of contractions under some appropriate assumptions on the domain and a parameter involved in our scheme. We then study the qualitative aspects of our scheme, such as the stability and order of convergence for the scheme. Some nonlinear problems are then considered and solved numerically by our new iterative scheme. The numerical simulations and graphical visualizations prove the high accuracy and stability of the new fixed-point scheme. Eventually, we solve a 2D nonlinear Volterra Integral Equation (VIE) via the application of our main outcome. Our results improve many related results in fixed-point iteration theory. [ABSTRACT FROM AUTHOR]

The stability problem arises when a functional equation is replaced by an inequality. When the equation admits a unique solution, we say that the equation is stable. In this paper, an Arunkumar-Agilan type additive functional equation is considered. By applying the direct method introduced by Hyers and the method of Gavruta, we established the Hyers-Ulam-Rassias stability condition for such an additive functional equation.

This article is dedicated to establishing the existence and uniqueness of solutions for the following problem: Dαx(t)=F(t,x(t))x(0)=x0,\left\{\begin{array}{l}{D}^{\alpha }x\left(t)=F\left(t,x\left(t))\hspace{1.0em}\\ x\left(0)={x}_{0},\hspace{1.0em}\end{array}\right. where x0{x}_{0} is the singular generalized function and F satisfies L∞{L}^{\infty } logarithmic type, Dα{D}^{\alpha } is the Caputo derivative of order m−1

Abstract Let Ω be a nonempty closed convex subset of a real Hilbert space H $\mathfrak{H}$ . Let ℑ be a nonspreading mapping from Ω into itself. Define two sequences { ψ n } n = 1 ∞ $\{\psi _{{n}}\}_{n=1}^{\infty}$ and { ϕ n } n = 1 ∞ $\{\phi _{{n}}\}_{n=1}^{\infty}$ as follows: { ψ n + 1 = π n ψ n + ( 1 − π n ) ℑ ψ n , ϕ n = 1 n ∑ n t = 1 ψ t , $$\begin{aligned} \textstyle\begin{cases} \psi _{n+1}=\pi _{n}\psi _{{n}}+(1-\pi _{n})\Im \psi _{{n}}, \\ \phi _{{n}}=\dfrac{1}{n}\underset{t=1}{\overset{n}{\sum}}\psi _{t}, \end{cases}\displaystyle \end{aligned}$$ for n ∈ N $n\in \mathit{N}$ , where 0 ≤ π n ≤ 1 $0\leq \pi _{n}\leq 1$ , and π n → 0 $\pi _{n} \rightarrow 0$ . In 2010, Kurokawa and Takahashi established weak and strong convergence theorems of the sequences developed from the above Baillion-type iteration method (Nonlinear Anal. 73:1562–1568, 2010). In this paper, we prove weak and strong convergence theorems for a new class of ( η , β ) $(\eta ,\beta )$ -enriched strictly pseudononspreading ( ( η , β ) $(\eta ,\beta )$ -ESPN) maps, more general than that studied by Kurokawa and W. Takahashi in the setup of real Hilbert spaces. Further, by means of a robust auxiliary map incorporated in our theorems, the strong convergence of the sequence generated by Halpern-type iterative algorithm is proved thereby resolving in the affirmative the open problem raised by Kurokawa and Takahashi in their concluding remark for the case in which the map ℑ is averaged. Some nontrivial examples are given, and the results obtained extend, improve, and generalize several well-known results in the current literature.

In this paper, we show that a sufficient and necessary condition for that the Hoeffding concentration inequality of Banach space-valued (B -valued, for simplicity) random variables holds is that the Banach space in question admits the Rademacher type p for some 1 ≤ p ≤ 2 ; which is equivalent to that the Bernstein concentration inequality holds for such B -valued random variables. This is done by applying a refined McDiarmid inequality (stated and proved in this paper) via the entropy method. This result is also applied to the density property of induced subgraphs of random Cayley graphs generated by a finite abelian group. [ABSTRACT FROM AUTHOR]

The aim of this paper is to solve second order nonlinear boundary value problems (BVPs) approximately. To do so, we utilize Thakur et al. iterative scheme (Filomat 3(10):2711- 2720, 2016) involving a weak contraction mapping in an arbitrary Banach space. Firstly, we study the convergence behaviour and the two numerical examples to show the better rate of convergence. Further, we apply this method to find the approximate solution of nonlinear boundary value problems (BVPs) and two illustrative numerical examples for BVPs are provided. Thus, several corresponding results in the literature are generalised and improved. [ABSTRACT FROM AUTHOR]

In this paper, we introduce three new inertial subgradient extragradient methods for solving variational inequalities involving quasi-monotone operators in the setting of 2-uniformly convex and uniformly smooth Banach spaces. We dispense with the well-known requirement of the stepsizes of the subgradient extragradient method on the prior knowledge of the Lipschitz constant of the cost function in our proposed algorithms. Furthermore, we give many numerical examples to test the robustness of our proposed algorithms and compare their performance with several algorithms in the literature. In addition, we use our proposed algorithms in the restoration process of some degraded images and compare the quality of the restored images using our proposed algorithms and some recent algorithms in the literature. Finally, from the results of the numerical simulations, our proposed algorithms are competitive and promising. [ABSTRACT FROM AUTHOR]

In this paper, we studied the AA-iterative algorithm for finding fixed points of the class of nonlinear generalized $ (\alpha, \beta) $-nonexpansive mappings. First, we proved weak convergence and then proved several strong convergence results of the scheme in a ground setting of uniformly convex Banach spaces. We gave a few numerical examples of generalized $ (\alpha, \beta) $-nonexpansive mappings to illustrate the major outcomes. One example was constructed over a subset of a real line while the other one was on the two dimensional space with a taxicab norm. We considered both these examples in our numerical computations to show that our iterative algorithm was more effective in the rate of convergence corresponding to other fixed point algorithms of the literature. Some 2D and 3D graphs were obtained that supported graphically our results and claims. As applications of our major results, we solved a class of fractional differential equations, 2D Voltera differential equation, and a convex minimization problem. Our findings improved and extended the corresponding results of the current literature.

Abstract In this article, we study the structure of a multiple variable mapping. Indeed, we reduce the system of several mixed additive-quadratic equations defining a multivariable mapping to obtain a single functional equation, say, the multimixed additive-quadratic equation. We also show that such mappings under some conditions can be multi-additive, multi-quadratic and multi-additive-quadratic. Moreover, we establish the Hyers–Ulam stability of the multimixed additive-quadratic equation, using the so-called direct (Hyers) method. Additionally, we present a concrete example (the numerical approximation) regarding the stability of some two variable mappings into real numbers. Applying some characterization results, we indicate two examples for the case that a multimixed additive-quadratic mapping (in the special cases) cannot be stable.

Analyzing the covariance structure of data is a fundamental task of statistics. While this task is simple for low‐dimensional observations, it becomes challenging for more intricate objects, such as multi‐variate functions. Here, the covariance can be so complex that just saving a non‐parametric estimate is impractical and structural assumptions are necessary to tame the model. One popular assumption for space‐time data is separability of the covariance into purely spatial and temporal factors. In this article, we present a new test for separability in the context of dependent functional time series. While most of the related work studies functional data in a Hilbert space of square integrable functions, we model the observations as objects in the space of continuous functions equipped with the supremum norm. We argue that this (mathematically challenging) setup enhances interpretability for users and is more in line with practical preprocessing. Our test statistic measures the maximal deviation between the estimated covariance kernel and a separable approximation. Critical values are obtained by a non‐standard multiplier bootstrap for dependent data. We prove the statistical validity of our approach and demonstrate its practicability in a simulation study and a data example. [ABSTRACT FROM AUTHOR]

Let X$X$ and Y$Y$ be complex Banach spaces with BX$B_X$ denoting the open unit ball of X$X$. This paper studies various aspects of the holomorphic Lipschitz spaceHL0(BX,Y)$\mathcal {H}L_0(B_X,Y)$, endowed with the Lipschitz norm. This space consists of the functions in the intersection of the sets Lip0(BX,Y)$\operatorname{Lip}_0(B_X,Y)$ of Lipschitz mappings and H∞(BX,Y)$\mathcal {H}^\infty (B_X,Y)$ of bounded holomorphic mappings, from BX$B_X$ to Y$Y$. Thanks to the Dixmier–Ng theorem, HL0(BX,C)$\mathcal {H}L_0(B_X, \mathbb {C})$ is indeed a dual space, whose predual G0(BX)$\mathcal {G}_0(B_X)$ shares linearization properties with both the Lipschitz‐free space and Dineen–Mujica predual of H∞(BX)$\mathcal {H}^\infty (B_X)$. We explore the similarities and differences between these spaces, and combine techniques to study the properties of the space of holomorphic Lipschitz functions. In particular, we get that G0(BX)$\mathcal {G}_0(B_X)$ contains a 1‐complemented subspace isometric to X$X$ and that G0(X)$\mathcal {G}_0(X)$ has the (metric) approximation property whenever X$X$ has it. We also analyze when G0(BX)$\mathcal {G}_0(B_X)$ is a subspace of G0(BY)$\mathcal {G}_0(B_Y)$, and we obtain an analog of Godefroy's characterization of functionals with a unique norm preserving extension in the holomorphic Lipschitz context. [ABSTRACT FROM AUTHOR]

In this manuscript, we initiate a large class of enriched (d , Z) - Z -contractions defined on Banach spaces and prove the existence and uniqueness of the fixed point of these contractions. We also provide an example to support our results and give an existence condition for the uniqueness of the solution to the integral equation. The results provided in the manuscript extend, generalize, and modify the existence results. Our research introduces novel fixed-point results under various contractive conditions. Furthermore, we discuss the iterated function system associated with enriched (d , Z) - Z -contractions in Banach spaces and define the enriched Z -Hutchinson operator. A result regarding the convergence of Krasnoselskii's iteration method and the uniqueness of the attractor via enriched (d , Z) - Z -contractions is also established. Our discoveries not only confirm but also significantly build upon and broaden several established findings in the current body of literature. [ABSTRACT FROM AUTHOR]

In this article, we study the structure of a multiple variable mapping. Indeed, we reduce the system of several mixed additive-quadratic equations defining a multivariable mapping to obtain a single functional equation, say, the multimixed additive-quadratic equation. We also show that such mappings under some conditions can be multi-additive, multi-quadratic and multi-additive-quadratic. Moreover, we establish the Hyers–Ulam stability of the multimixed additive-quadratic equation, using the so-called direct (Hyers) method. Additionally, we present a concrete example (the numerical approximation) regarding the stability of some two variable mappings into real numbers. Applying some characterization results, we indicate two examples for the case that a multimixed additive-quadratic mapping (in the special cases) cannot be stable. [ABSTRACT FROM AUTHOR]

The aim of this note is threefold: first, to present a few relevant facts about the way in which the technique of enriching contractive mappings was introduced; secondly, to expose the main contributions in the area of enriched mappings established by the authors and their collaborators by using this technique; and third, to survey some related developments in the very recent literature which were authored by other researchers. [ABSTRACT FROM AUTHOR]

This work addresses the qualitative analysis of a novel non-linear coupled system of fractional differential problems (FDPs) using Caputo and Liouville–Riemann fractional derivatives. Fractional calculus has demonstrated significant applicability across various fields, including financial systems, optimal control, epidemiological models, chaotic systems, and engineering. The proposed model builds on existing research by formulating a non-linear coupled fractional boundary value problem with mixed boundary conditions. The primary advantages of our method include its ability to capture the dynamics of complex systems more accurately and its flexibility in handling different types of fractional derivatives. The model's solution was derived using advanced mathematical techniques, and the results confirmed the existence and uniqueness of the solutions. This approach not only generalizes classical differential equation methods but also offers a robust framework for modeling real-world phenomena governed by fractional dynamics. The study concludes with the validation of the theoretical findings through illustrative examples, highlighting the method's efficacy and potential for further applications. [ABSTRACT FROM AUTHOR]

This paper introduces a novel F fixed-point iteration method that leverages Green's function for solving the nonlinear Troesch problem in Banach spaces, which are symmetric spaces. The Troesch problem, characterized by its challenging boundary conditions and nonlinear nature, is significant in various physical and engineering applications. The proposed method integrates fixed-point theory with Green's function techniques to develop an iteration process that ensures convergence, stability, and accuracy. The numerical experiments demonstrate the method's efficiency and robustness, highlighting its potential for broader applications in solving nonlinear differential equations in Banach spaces. [ABSTRACT FROM AUTHOR]

Over the past two decades, significant advancements have been made in understanding the stability according to Hyers–Ulam involving different functional equations (FEs). This study investigates the generalized stability of norm-based (norm-additive) FEs within the framework of arbitrary (noncommutative) groups and p-uniformly convex spaces. Specifically, we analyze two key functional equations, ∥ η (g h) ∥ = ∥ η (g) + η (h) ∥ and ∥ η (g h − 1) ∥ = ∥ η (g) − η (h) ∥ for every g , h ∈ G , where (G , ·) denotes an arbitrary group and B is considered to be a p-uniformly convex space. The surjectivity of the function η : G → B is a critical assumption in our analysis. Drawing upon the foundational works of L. Cheng and M. Sarfraz, this paper applies the large perturbation method tailored for p-uniformly convex spaces, where p ≥ 1 . This study extends previous research by offering a deeper exploration of the conditions under which these functional equations demonstrate Hyers–Ulam stability. In this study, the additive functional equation demonstrates a fundamental form of symmetry, where the order of operands does not affect the results. This symmetry under permutation of arguments is crucial for the analysis of stability. In the context of norm-additive FEs, this stability criterion investigates how small changes in the inputs of a functional equation affect the outputs, especially when the function is expected to follow an additive form. [ABSTRACT FROM AUTHOR]

In this paper, we study, in an infinite dimensional Banach space X, the existence and uniqueness of mild solutions to the Cauchy problem for the semilinear differential evolution equations with nonlocal conditions where HDα,β;ψ is the ψ-Hilfer operator, for 0 < α < 1, 0 < β < 1, and A(t): X → X is a continuous operator for each t ∈ [0, T]. The function f: [0, T] × X → X is not necessarily lipschitzian. We first derive a variation of constants formula. Then we use theWessinger and the Krasnoselskii's fixed point theorems to achieve our results. These results generalize a recent work by K. Balachandran et al. We provide an example to illustrate our abstract results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and discuss the upper, lower, upper modified and lower modified (n, p) -th von Neumann-Jordan constant for p ≥ 1, which is a further generalization for von Neumann-Jordan constant in Banach spaces. We give some relationships between these constants and characterize uniformly non-ln¹ Banach spaces by upper and upper modified (n, p) - th von Neumann-Jordan constant. Moreover, the exact value of this constant is calculated for the spaces lp, Lp and L∞, - l1. [ABSTRACT FROM AUTHOR]

In this paper, a fractional food chain system consisting of a Holling type II functional response was studied in view of a fractional derivative operator. The considered fractional derivative operator provided nonsingular as well as a nonlocal kernel which was significantly better than other derivative operators. Fractional order modeling of a model was also useful to model the behavior of real systems and in the investigation of dynamical systems. This model depicted the relationship among four types of species: prey, susceptible intermediate predators (IP), infected intermediate predators, and apex predators. One of the significant aspects of this model was the inclusion of Michaelis-Menten type or Holling type II functional response to represent the predator-prey link. A functional response depicted the rate at which the normal predator consumed the prey. The qualitative property and assumptions of the model were discussed in detail. The present work discussed the dynamics and analytical behavior of the food chain model in the context of fractional modeling. This study also examined the existence and uniqueness related analysis of solutions to the food chain system. In addition, the Ulam-Hyers stability approach was also discussed for the model. Moreover, the present work examined the numerical approach for the solution and simulation for the model with the help of graphical presentations. [ABSTRACT FROM AUTHOR]

In this research article, we study the coupled system of nonlinear Langevin fractional q-difference equations associated with two different fractional orders in Banach Space. The existence, uniqueness, and stability in the sense of Ulam are established for the proposed system. Our approach is based on the technique of measure of noncompactness combined with Mönch fixed point theorem, the implementation Banach contraction principle fixed point theorem, and the employment of Urs’s stability approach. Two examples illustrating the effectiveness of the theoretical results are presented. [ABSTRACT FROM AUTHOR]

In this paper, we have established a Hyers-Ulam-Rassias stability result of a pexider type functional equation. The work is in the framework of Banach spaces. The result is established using direct method as well as a fixed point theorem approach followed by some corollaries. Apart from its main objective of obtaining the stability result, the present paper is also a demonstration of the comparative study of the results in Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the local and semilocal convergence of an iterative method for solving nonlinear systems of equations. We first establish the conditions under which these methods converge locally to the solution. Then, we extend our analysis to examine the semilocal convergence of these methods, considering their behavior when starting from initial guesses that are not necessarily close to the solution. Iterative approaches for solving nonlinear systems of equations must take into account the radius of convergence, computable upper error bounds, and the uniqueness of solutions. These points have not been addressed in earlier studies. Moreover, we provide numerical examples to demonstrate the theoretical findings and compare the performance of these methods under different circumstances. Finally, we conclude that our examination offers a significant understanding of the convergence characteristics of previous iterative techniques for solving nonlinear equation systems. [ABSTRACT FROM AUTHOR]

We explore an oscillator with nonlinear functions and equilibrium lines that displays chaos. The equilibrium stability and complexity of the oscillator have been analysed and investigated. The presence of multiple equilibrium lines sets it apart from previously reported oscillators. The synchronization of the oscillator is considered as an application for secure communications. An observer is designed by considering a transmitted signal as a state, in other words, by injecting a linear function satisfying Lipschitz's condition to the proposed oscillator. Moreover, the adaptive control of the new oscillator is obtained. [ABSTRACT FROM AUTHOR]

In this paper, an inertial subgradient extragradient algorithm is proposed to solve the pseudomonotone variational inequality problems in Banach space. This iterative scheme employs a new line-search rule. Strong convergence theorems for the proposed algorithms are established under the assumptions that the operators are non-Lipschitz continuous. Furthermore, several numerical experiments are given to show that our method has better convergence performance than the known ones in the literatures. [ABSTRACT FROM AUTHOR]

The main goal of this research article is to investigate the stability of generalized norm-additive functional equations. This study demonstrates that these equations are Hyers-Ulam stable for surjective functions from an arbitrary group G to a real Banach space B using the large perturbation method. Furthermore, hyperstability results are investigated for a generalized Cauchy equation. [ABSTRACT FROM AUTHOR]

We develop a new Kantorovich-like convergence analysis of Newton-type methods to solve nonsingular and singular nonlinear equations in Banach spaces. The outer or generalized inverses are exchanged by a finite sum of linear operators making the implementation of these methods easier than in earlier studies. The analysis uses relaxed generalized continuity of the derivatives of operators involved required to control the derivative and on real majorizing sequences. The same approach can also be implemented on other iterative methods with inverses. The examples complement the theory by verifying the convergence conditions and demonstrating the performance of the methods. [ABSTRACT FROM AUTHOR]

This work is devoted to establish the existence of multiple positive radial solutions to the following equations associated with certain boundary conditions. The Six Functionals Fixed Point Theorem serves as the basis for the technique. To illustrate the key findings, an application is given. [ABSTRACT FROM AUTHOR]

We study numerical ranges of antilinear operators on both Hilbert and Banach spaces. We prove that the numerical range of an antilinear operator on at least two-dimensional space is always a disc, and so a convex set. This improves existing results. We also study other properties known for numerical ranges of linear operators and discuss similarities and differences in the antilinear setting. [ABSTRACT FROM AUTHOR]

We study abstract sufficient criteria for cost-uniform open-loop stabilizability of linear control systems in a Banach space with a bounded control operator, which build up and generalize a sufficient condition for null-controllability in Banach spaces given by an uncertainty principle and a dissipation estimate. For stabilizability these estimates are only needed for a single spectral parameter and, in particular, their constants do not depend on the growth rate w.r.t. this parameter. Our result unifies and generalizes earlier results obtained in the context of Hilbert spaces. As an application we consider fractional powers of elliptic differential operators with constant coefficients in L p (R d) for p ∈ [ 1 , ∞) and thick control sets. [ABSTRACT FROM AUTHOR]

This article focuses on the problem of integral boundary value for Riemann-Liouville derivatives equipped with (p; q)-difference calculus in Banach space. To provide further clarification, our focus lies in establishing the existence of a solution to our problem using the measure of noncompactness (m.n.) and the Mönch's fixed point theorem. Our investigation in the Banach space encompasses two nonlinear terms with two distinct orders of derivatives. Our paper concludes with an illustrative example and conclusion. [ABSTRACT FROM AUTHOR]

This paper study was designed to establish solutions for mixed functional fractional integral equations that involve the Riemann-Liouville fractional operator and the Erd'elyi-Kober fractional operator to describe biological population dynamics in Banach space. The results rely on the measure of non-compactness and theoretical concepts from fractional calculus. Darbo's fixed-point theorem for Banach spaces has been utilized. Moreover, the solvability of a specific non-linear integral equation that models the spread of infectious diseases with a seasonally varying periodic contraction rate has been explored by using the Banach contraction principle. Finally, two numerical examples demonstrate the practical application of these findings in the realm of fractional integral equation theory. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to revisit previous works of the author with Helffer and Sjöstrand (arXiv:1001.4171v1. 2010; Int Equ Op Theory 93(3):36, 2021) on the stability of semigroups which were proved in the Hilbert case by considering the Banach case at the light of a paper by Latushkin and Yurov (Discrete Contin Dyn Syst 33:5203–5216, 2013). [ABSTRACT FROM AUTHOR]

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