Leray-Schauder Alternative Theorem Research Articles

On Higher-Order Nonlinear Fractional Elastic Equations with Dependence on Lower Order Derivatives in Nonlinearity

The paper studied high-order nonlinear fractional elastic equations that depend on low-order derivatives in nonlinearity and established the existence and uniqueness results by using the Leray–Schauder alternative theorem and Perov’s fixed point theorem on an appropriate space under mild assumptions. Examples are given to illustrate the key results.

Investigation on integro-differential equations with fractional boundary conditions by Atangana-Baleanu-Caputo derivative.

We establish, the existence and uniqueness of solutions to a class of Atangana-Baleanu (AB) derivative-based nonlinear fractional integro-differential equations with fractional boundary conditions by using special type of operators over general Banach and Hilbert spaces with bounded approximation numbers. The Leray-Schauder alternative theorem guarantees the existence solution and the Banach contraction principle is used to derive uniqueness solutions. Furthermore, we present an implicit numerical scheme based on the trapezoidal method for obtaining the numerical approximation to the solution. To illustrate our analytical and numerical findings, an example is provided and concluded in the final section.

Total Controllability for a Class of Fractional Hybrid Neutral Evolution Equations with Non-Instantaneous Impulses

This study demonstrates the total control of a class of hybrid neutral fractional evolution equations with non-instantaneous impulses and non-local conditions. The boundary value problem with non-local conditions is created using the Caputo fractional derivative of order 1<α≤2. In order to create novel, strongly continuous associated operators, the infinitesimal generator of the sine and cosine families is examined. Additionally, two approaches are used to discuss the solution’s total controllability. A compact strategy based on the non-linear Leray–Schauder alternative theorem is one of them. In contrast, a measure of a non-compactness technique is implemented using the Sadovskii fixed point theorem with the Kuratowski measure of non-compactness. These conclusions are applied using simulation findings for the non-homogeneous fractional wave equation.

The Existence Theorems of Fractional Differential Equation and Fractional Differential Inclusion with Affine Periodic Boundary Value Conditions

This paper is devoted to investigating the existence of solutions for the fractional differential equation and fractional differential inclusion of order α∈(2,3] with affine periodic boundary value conditions. Applying the Leray–Schauder fixed point theorem, the existence of the solutions for the fractional differential equation is established. Furthermore, for the fractional differential inclusion, we consider two cases: (i) the set-valued function has convex value and (ii) the set-valued function has nonconvex value. The main tools of our research are the Leray–Schauder alternative theorem, Covita and Nadler’s fixed point theorem and some set-valued analysis theories.

STABILITY AND EXISTENCE OF SOLUTIONS FOR A COUPLED SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS

In this paper, we study a coupled system of Caputo type fractional differential equations with integral boundary conditions. By Leray-Schauder alternative theorem, the existence of solutions for the factional differential system are obtained. The Hyers-Ulam stability of solutions is discussed and sufficient conditions for the stability are developed. The main results are well illustrated with examples and numerical simulation graphs. The interesting point of this article is that it not only gives approximate graphs of solution by using the iterative methods, but also verifies the Hyers-Ulam stability of the coupling system by numerical simulation.

Interior approximate controllability of second-order semilinear control systems

In our manuscript, we investigate the interior approximate controllability for the subsequent semilinear second-order system in L 2 ( Ω ) s ″ ( σ ) + A 0 s ( σ ) = B 0 u ( σ ) + ξ ( σ , s ( σ ) ) , 0 ≤ σ ≤ ϱ s ( 0 ) = s 0 , s ′ ( 0 ) = s 1 , where A 0 is the linear and unbounded operator, B 0 is a bounded linear operator and ξ is a nonlinear operator defined on appropriate spaces. The proposed problem can be converted into an equivalent first-order semilinear control system, then the approximate controllability results for the proposed system are obtained from the study of the approximate controllability of the reduced first-order system. The Leray–Schauder alternative theorem and principle of contraction are used in the proof of our main theorems.

On a System of ψ-Caputo Hybrid Fractional Differential Equations with Dirichlet Boundary Conditions

In this article, we investigate sufficient conditions for the existence and stability of solutions to a coupled system of ψ-Caputo hybrid fractional derivatives of order 1<υ≤2 subjected to Dirichlet boundary conditions. We discuss the existence and uniqueness of solutions with the assistance of the Leray–Schauder alternative theorem and Banach’s contraction principle. In addition, by using some mathematical techniques, we examine the stability results of Ulam–Hyers. Finally, we provide one example in order to show the validity of our results.

Existence of general mean square solutions for a sequential random fractional differential system with nonlocal conditions

In this study, we discuss the existence of solutions for a coupled system of two- sequential random fractional differential equations involving nonlocal periodic boundary conditions. Using a version of Leray-Schauder alternative theorem, we present recent results on the existence of coupled solutions for the problem. The obtained solutions as well as their Caputo derivatives satisfy some stochastic processes properties. An illustrative example is also discussed at the end of this paper.

Fractional Evolution Equations with Infinite Time Delay in Abstract Phase Space

In the presented research, the uniqueness and existence of a mild solution for a fractional system of semilinear evolution equations with infinite delay and an infinitesimal generator operator are demonstrated. The generalized Liouville–Caputo derivative of non-integer-order 1<α≤2 and the parameter 0<ρ<1 are used to establish our model. The ρ-Laplace transform and strongly continuous cosine and sine families of uniformly bounded linear operators are adapted to obtain the mild solution. The Leray–Schauder alternative theorem and Banach contraction principle are used to demonstrate the mild solution’s existence and uniqueness in abstract phase space. The results are applied to the fractional wave equation.

Existence and Uniqueness of Solutions for the Neutral Fractional Integro-Differential Equations with Fractional Integral Boundary Conditions

In this paper, we establish sufficient conditions for the existence and uniqueness of solutions for a class of boundary value problems with integral condition for neutral fractional integro-differential equations. The results are established by the application of the contraction mapping principle, Leray-Schauder alternative theorem and Krasnoselskii fixed point theorem. An example is provided to illustrate the main results.

A fractional order control model for Diabetes and COVID-19 co-dynamics with Mittag-Leffler function

The aim of this paper is to present and analyze the fractional optimal control model for COVID-19 and diabetes co-dynamics, using the Atangana-Baleanu derivative. The positivity and boundedness of the solutions was shown by the method of Laplace transform. The existence and uniqueness of the solutions of the proposed model were established using Banach fixed point Theorem and Leray–Schauder alternative Theorem. The fractional model was also shown to be Hyers-Ulam stable. The model was fitted to the cumulative confirmed daily COVID-19 cases for Indonesia. The simulations of the total number of hospitalized individuals co-infected with COVID-19 and diabetes, at different face-mask compliance levels, when vaccination strategy is maintained reveals that the total number of hospitalized co-infection cases decreases with increase in face-mask compliance levels, while maintaining COVID-19 vaccination. The simulation results show that to curtail COVID-19 and diabetes co-infections, policies and measures to enforce mass COVID-19 vaccination and strict face-mask usage in the public must be put in place. To further cut down the spread of COVID-19 and diabetes co-infection, time dependent controls are added into the fractional model, and the obtained optimal control problem investigated via the Pontryagin’s Maximum Principle.

New results for a coupled system of ABR fractional differential equations with sub-strip boundary conditions

<abstract><p>In this article, we investigate sufficient conditions for the existence, uniqueness and Ulam-Hyers (UH) stability of solutions to a new system of nonlinear ABR fractional derivative of order $ 1 < \varrho\leq 2 $ subjected to multi-point sub-strip boundary conditions. We discuss the existence and uniqueness of solutions with the assistance of Leray-Schauder alternative theorem and Banach's contraction principle. In addition, by using some mathematical techniques, we examine the stability results of Ulam-Hyers (UH). Finally, we provide one example in order to show the validity of our results.</p></abstract>

(ω, c)-periodic and asymptotically (ω, c)-periodic mild solutions to fractional Cauchy problems

In this paper, we establish some new properties of -periodic and asymptotically -periodic functions, then we apply them to study the existence and uniqueness of mild solutions of these types to the following semilinear fractional differential equations: (1) and (2) where stands for the Caputo derivative and A is a linear densely defined operator of sectorial type on a complex Banach space and the function is -periodic or asymptotically -periodic with respect to the first variable. Our results are obtained using the Leray–Schauder alternative theorem, the Banach fixed point principle and the Schauder theorem. Then we illustrate our main results with an application to fractional diffusion-wave equations.

Nonlinear singular problems with convection

In this paper we study the existence of a positive solution for a nonlinear Dirichlet problem driven by the p -Laplacian and a reaction which has the competing effects of a singular term and of a convection (gradient dependent) term. Using the “frozen” variable method and eventually the Leray-Schauder Alternative Theorem, we show that the problem has a positive smooth solution. We mention that on the gradient dependent perturbation x ↦ f ( z , x , y ) , we do not impose any bilateral growth condition.

A Fractional-Order Sequential Hybrid System with an Application to a Biological System

With the help of Banach’s fixed-point approach and the Leray–Schauder alternative theorem, we produced existence results for a general class of fractional differential equations in this paper. The proposed problem is more comprehensive and applicable to real-life situations. As an example of how our problem might be used, we have created a fractional-order COVID-19 model whose solution is guaranteed by our results. We employed a numerical approach to solve the COVID-19 model, and the results were compared for different fractional orders. Our numerical results for fractional orders follow the same pattern as the classical example of order 1, indicating that our numerical scheme is accurate.

Dirichlet (p,q)-equations with gradient dependent and locally defined reaction

We consider a Dirichlet (p,q)-equation, with a gradient dependent reaction which is only locally defined. Using truncations, theory of nonlinear operators of monotone type, and fixed point theory (the Leray-Schauder Alternative Theorem), we show the existence of a positive smooth solution.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/34/abstr.html

Existence results to a ψ- Hilfer neutral fractional evolution equation with infinite delay

Abstract In this paper, we prove the existence and uniqueness of a mild solution to the system of ψ- Hilfer neutral fractional evolution equations with infinite delay H 𝔻0 αβ;ψ [x(t) − h(t, xt )] = A x(t) + f (t, x(t), xt ), t ∈ [0, b], b > 0 and x(t) = ϕ(t), t ∈ (−∞, 0]. We first obtain the Volterra integral equivalent equation and propose the mild solution of the system. Then, we prove the existence and uniqueness of solution by using the Banach contraction mapping principle and the Leray-Schauder alternative theorem.

Mild solutions for a coupled system of fractional differential equations with slit-strips type integral boundary conditions

In this present paper, we study the existence and uniqueness of solutions for a coupled system of fractional differential equations with Banach contraction mapping principle and Leray-Schauder alternative theorem. Examples are given to illustrate the applicability of the obtained results.

Global exponential stability of delayed complex-valued neural networks with discontinuous activation functions

This paper studies the global exponential stability of delayed complex-valued neural networks with discontinuous activation functions. By introducing the complex-valued Filippov differential inclusion, we construct the framework of studying the dynamical behaviors of complex-valued neural networks with discontinuous bivariate activation functions. By employing the Leray–Schauder alternative theorem and choosing an appropriate Lyapunov function, we prove the global exponential stability of delayed complex-valued neural networks with discontinuous activation functions. The numerical example provided shows the effectiveness of the obtained results.

Periodicity and stabilization control of the delayed Filippov system with perturbation

By employing Leray-Schauder alternative theorem of set-valued maps and non-Lyapunov method (non-smooth analysis, inequality analysis, matrix theory), this paper investigates the problems of periodicity and stabilization for time-delayed Filippov system with perturbation. Several criteria are obtained to ensure the existence of periodic solution of time-delayed Filippov system by using differential inclusion. By designing appropriate switching state-feedback controller, the asymptotic stabilization and exponential stabilization control of Filippov system are realized. Applying these criteria and control design method to a class of time-delayed neural networks with perturbation and discontinuous activation functions under a periodic environment. The developed theorems improve the existing results and their effectiveness are demonstrated by numerical example.