Alternative Of Leray-Schauder Research Articles

Existence, Uniqueness, and Stability of a Nonlinear Tripled Fractional Order Differential System

This manuscript investigates the existence, uniqueness, and different forms of Ulam stability for a system of three coupled differential equations involving the Riemann–Liouville (RL) fractional operator. The Leray–Schauder alternative is employed to confirm the existence of solutions, while the Banach contraction principle is used to establish their uniqueness. Stability conditions are derived utilizing classical nonlinear functional analysis techniques. Theoretical findings are illustrated with an example. The proposed system generalizes third-order ordinary differential equations (ODEs) with different boundary conditions (BCs).

Study of Quantum Difference Coupled Impulsive System with Respect to Another Function

In this paper, we study a quantum difference coupled impulsive system with respect to another function. Some quantum derivative and integral asymmetric graphs with respect to another function are shown to illustrate the behavior of parameters. Existence and uniqueness results are established via Banach contraction mapping principle and Leray–Schauder alternative. Examples illustrating the obtained results are also included. Our results are new and significantly contribute to the literature to this new subject on quantum calculus on finite intervals with respect to another function.

On a (k;chi)-Hilfer fractional system with coupled nonlocal boundary conditions including various fractional derivatives and Riemann–Stieltjes integrals

In the present research, we investigate the existence and uniqueness of solutions for a system of (k; χ)-Hilfer fractional differential equations, subject to coupled nonlocal boundary conditions, which contain various fractional derivatives and Riemann–Stieltjes integrals. The uniqueness result relies on the Banach contraction mapping principle, while the existence results depend on the Leray–Schauder alternative and Krasnosel’skiĭ fixed point theorem. Examples are also constructed to illustrate the obtained results.

Study on a Nonlocal Fractional Coupled System Involving (k,ψ)-Hilfer Derivatives and (k,ψ)-Riemann–Liouville Integral Operators

This paper deals with a nonlocal fractional coupled system of (k,ψ)-Hilfer fractional differential equations, which involve, in boundary conditions, (k,ψ)-Hilfer fractional derivatives and (k,ψ)-Riemann–Liouville fractional integrals. The existence and uniqueness of solutions are established for the considered coupled system by using standard tools from fixed point theory. More precisely, Banach and Krasnosel’skiĭ’s fixed-point theorems are used, along with Leray–Schauder alternative. The obtained results are illustrated by constructed numerical examples.

Solvability and Mittag–Leffler–Ulam stability for fractional Duffing problem with three sequential fractional derivatives

In this paper, we discuss the existence, uniqueness, and the Mittag–Leffler–Ulam stability of solutions for fractional Duffing equations involving three fractional derivatives. Uniqueness result for solution of the underlying Duffing problem is given with the help of Banach's fixed point theorem, where the existence result is computed using Leray–Schauder's alternative. Also, the Mittag–Leffler–Ulam stability results are computed by employing generalized singular Gronwall's inequality. An illustrative example is also given.

Mild solution in the α-norm for some partial integrodifferential equations involving a nonlocal condition

Abstract In this work, we investigate the existence of mild solutions in the α \alpha -norm for a class of nonlocal integrodifferential equations. We employ the theory of resolvent operators introduced by R. Grimmer. The Leray-Schauder alternative is the principal working tool for our analysis.

Existence and uniqueness of blow-up solution to a fully fractional thermostat model

In this article, we deal with a fully fractional thermostat model in the settings of the Riemann–Liouville fractional derivatives. The equivalences between the fractional differential equations and the corresponding Volterra–Fredholm integral equations are rigorously derived. By choosing an appropriate weighted Banach space of continuous functions, we employ two standard fixed-point theorems, Leray–Schauder alternative and Banach contraction principle, to establish the existence of blow-up solutions — the unbounded mathematical solutions in the operational interval. Furthermore, an implicit numerical scheme based on the right product rectangle rule is presented, which provides the numerical approximation of the obtained solution. Some examples are provided to validate our theoretical findings, along with numerical simulations of the solutions.

Existence and H-U stability of a tripled system of sequential fractional differential equations with multipoint boundary conditions

In this paper, we introduce a new coupled system of sequential fractional differential equations with coupled boundary conditions. We establish existence and uniqueness results using the Leray–Schauder alternative and Banach contraction principle. We examine the stability of the solutions involved in the Hyers–Ulam type. As an application, we present a few examples to illustrate the main results.

Coupled system of three sequential Caputo fractional differential equations: Existence and stability analysis

Recently, many studies on fractional coupled systems involving different sequential fractional derivatives have appeared during the past several years. The paper is dealing with a coupled system of three sequential Caputo fractional differential equations, and the designed system absorbs none of the commutativity and the semigroup properties. The Banach contraction principle is used for proving the existence and uniqueness results. We prove the existence of at least one is obtained by using the Leray–Schauder alternative. The Ulam–Hyers–Rassias stability of the considered system is defined and discussed. An illustrative example is also presented.

Systems of Sequential ψ1-Hilfer and ψ2-Caputo Fractional Differential Equations with Fractional Integro-Differential Nonlocal Boundary Conditions

In this paper, we introduce and study a new class of coupled and uncoupled systems, consisting of mixed-type ψ1-Hilfer and ψ2-Caputo fractional differential equations supplemented with asymmetric and symmetric integro-differential nonlocal boundary conditions (systems (2) and (13), respectively). As far as we know, this combination of ψ1-Hilfer and ψ2-Caputo fractional derivatives in coupled systems is new in the literature. The uniqueness result is achieved via the Banach contraction mapping principle, while the existence result is established by applying the Leray–Schauder alternative. Numerical examples illustrating the obtained results are also presented.

On a System of Sequential Caputo Fractional Differential Equations with Nonlocal Boundary Conditions

We obtain existence and uniqueness results for the solutions of a system of Caputo fractional differential equations which contain sequential derivatives, integral terms, and two positive parameters, supplemented with general coupled Riemann–Stieltjes integral boundary conditions. The proofs of our results are based on the Banach fixed point theorem and the Leray–Schauder alternative.

Existence and uniqueness for a coupled system of fractional equations involving Riemann-Liouville and Caputo derivatives with coupled Riemann-Stieltjes integro-multipoint boundary conditions

<abstract><p>Recently, coupled systems of fractional differential equations play a central role in the modelling of many systems in e.g., financial economics, ecology, and many more. This study investigates the existence and uniqueness of solutions for a nonlinear coupled system of fractional differential equations involving Riemann-Liouville and Caputo derivatives with coupled Riemann-Stieltjes integro-multipoint boundary conditions. The main tools are known fixed point theorems, namely, Leray-Schauder alternative, Banach fixed point theorem, and the Krasnoselskii fixed point theorem. The new system, which can be considered as a generalized version of many previous fascinating systems, is where the article's novelty lies. Examples are presented to illustrate the results. In this way, we generalize several earlier results.</p></abstract>

On a nonlinear coupled Caputo-type fractional differential system with coupled closed boundary conditions

<abstract><p>We introduce a novel notion of coupled closed boundary conditions and investigate a nonlinear system of Caputo fractional differential equations equipped with these conditions. The existence result for the given problem is proved via the Leray-Schauder alternative, while the uniqueness of its solutions is accomplished by applying the Banach fixed point theorem. Examples are constructed for the illustration of the main results. Some special cases arising from the present study are discussed.</p></abstract>

Nonlinear analysis in p-vector spaces for single-valued 1-set contractive mappings

The goal of this paper is to develop some fundamental and important nonlinear analysis for single-valued mappings under the framework of p-vector spaces, in particular, for locally p-convex spaces for 0 < p leq 1. More precisely, based on the fixed point theorem of single-valued continuous condensing mappings in locally p-convex spaces as the starting point, we first establish best approximation results for (single-valued) continuous condensing mappings, which are then used to develop new results for three classes of nonlinear mappings consisting of 1) condensing; 2) 1-set contractive; and 3) semiclosed 1-set contractive mappings in locally p-convex spaces. Next they are used to establish the general principle for nonlinear alternative, Leray–Schauder alternative, fixed points for nonself mappings with different boundary conditions for nonlinear mappings from locally p-convex spaces, to nonexpansive mappings in uniformly convex Banach spaces, or locally convex spaces with the Opial condition. The results given by this paper not only include the corresponding ones in the existing literature as special cases, but are also expected to be useful tools for the development of new theory in nonlinear functional analysis and applications to the study of related nonlinear problems arising from practice under the general framework of p-vector spaces for 0< p leq 1.Finally, the work presented by this paper focuses on the development of nonlinear analysis for single-valued (instead of set-valued) mappings for locally p-convex spaces. Essentially, it is indeed the continuation of the associated work given recently by Yuan (Fixed Point Theory Algorithms Sci. Eng. 2022:20, 2022); therein, the attention is given to the study of nonlinear analysis for set-valued mappings in locally p-convex spaces for 0 < p leq 1.

On the Generalized Liouville–Caputo Type Fractional Differential Equations Supplemented with Katugampola Integral Boundary Conditions

In this study, we examine the existence and Hyers–Ulam stability of a coupled system of generalized Liouville–Caputo fractional order differential equations with integral boundary conditions and a connection to Katugampola integrals. In the first and third theorems, the Leray–Schauder alternative and Krasnoselskii’s fixed point theorem are used to demonstrate the existence of a solution. The Banach fixed point theorem’s concept of contraction mapping is used in the second theorem to emphasise the analysis of uniqueness, and the results for Hyers–Ulam stability are established in the next theorem. We establish the stability of Ulam–Hyers using conventional functional analysis. Finally, examples are used to support the results. When a generalized Liouville–Caputo (ρ) parameter is modified, asymmetric results are obtained. This study presents novel results that significantly contribute to the literature on this topic.

Existence and Ulam–Hyers Stability Analysis for Coupled Differential Equations of Fractional-Order with Nonlocal Generalized Conditions via Generalized Liouville–Caputo Derivative

In this paper, we investigate the existence and Hyers–Ulam stability of a coupled differential equations of fractional-order with multi-point (discrete) and integral boundary conditions that are related to Katugampola integrals. This manuscript can be categorized into four parts: The Leray–Schauder alternative and Krasnoselskii’s fixed point theorems are used to prove the existence of a solution in the first and third section. The second section emphasizes the analysis of uniqueness, which is based on the Banach fixed point theorem’s concept of contraction mapping, and the fourth section establishes the Hyers–Ulam stability results. We demonstrate Hyers–Ulam stability using the traditional functional analysis technique. Finally, the consequences are validated using examples.

Existence and Hyers–Ulam stability of solutions for nonlinear three fractional sequential differential equations with nonlocal boundary conditions

Abstract In this paper, we analyses the existence and Hyers–Ulam stability of a coupled system of three sequential fractional differential equations with coupled integral boundary conditions. This manuscript can be categorized into three parts: The Leray–Schauder alternative is used to prove the existence of a solution in the first section. The second section emphasizes the analysis of uniqueness, which is based on the Banach fixed point theorem’s concept of contraction mapping, and the third section establishes the Hyers–Ulam stability results. In addition, we provide examples to demonstrate our findings.

Investigation of a Nonlinear Coupled (k, ψ)–Hilfer Fractional Differential System with Coupled (k, ψ)–Riemann–Liouville Fractional Integral Boundary Conditions

This paper is concerned with the existence of solutions for a new boundary value problem of nonlinear coupled (k,ψ)–Hilfer fractional differential equations subject to coupled (k,ψ)–Riemann–Liouville fractional integral boundary conditions. We prove two existence results by applying the Leray–Schauder alternative, and Krasnosel’skiĭ’s fixed-point theorem under different criteria, while the third result, concerning the uniqueness of solutions for the given problem, relies on the Banach’s contraction mapping principle. Examples are included for illustrating the abstract results.

Existence Results for Nonlinear Coupled Hilfer Fractional Differential Equations with Nonlocal Riemann–Liouville and Hadamard-Type Iterated Integral Boundary Conditions

We introduce and study a new class of nonlinear coupled Hilfer differential equations with nonlocal boundary conditions involving Riemann–Liouville and Hadamard-type iterated fractional integral operators. By applying the Leray–Schauder alternative and Krasnosel’skiĭ’s fixed point theorem, two results presenting different criteria for the existence of solutions to the given problem are proven. The third result provides a sufficient criterion for the existence of a unique solution to the problem at hand. Numerical examples are constructed to demonstrate the application of the results obtained. Two graphs show asymmetric solutions when a Hilfer parameter is varied. The work presented in this paper is novel and significantly enriches the literature on the topic.

Uniqueness and stability of coupled sequential fractional differential equations with boundary conditions

This paper is concerned with the existence and uniqueness of solutions for a coupled system of Caputo-type sequential fractional differential equations supplemented with boundary conditions. The existence of solutions is derived by applying the Leray–Schauder alternative, while the uniqueness of solutions is established via Banach’s contraction principle. Moreover, some necessary conditions for the Hyers–Ulam-type stability to the solutions of the boundary value problem are developed. Finally the results are supported by example.