Krasnoselskii Fixed Point Theorem Research Articles

New existence results for nonlinear functional hybrid differential equations involving the $\psi-$Caputo fractional derivative

In this manuscript, we are concerned with the existence result of nonlinear hybrid differential equations involving $\psi-$Caputo fractional derivatives of an arbitrary order $\alpha\in(0,1)$. By applying Krasnoselskii fixed point theorem and some fractional analysis techniques, we prove our main result. As application, a nontrivial example is given to demonstrate the effectiveness of our theoretical result.

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.

Neutral functional sequential differential equations with Caputo fractional derivative on time scales

In this paper, we establish the existence and uniqueness of a solution for a class of initial value problems for implicit fractional differential equations with Caputo fractional derivative. The arguments are based upon the Banach contraction principle, the nonlinear alternative of Leray–Schauder type and Krasnoselskii fixed point theorem. As applications, two examples are included to show the applicability of our results.

Controllability of neutral impulsive stochastic functional integrodifferential equations driven by a fractional Brownian motion with infinite delay via resolvent operator

This paper is concerned with the controllability results of neutral impulsive stochastic functional integrodifferential equations driven by a fractional Brownian motion with infinite delay in a real separable Hilbert space. The controllability results are obtained using stochastic analysis, the theory of resolvent operator in the sense of Grimmer and Krasnoselskii fixed point theorem. An example is provided to illustrate the obtained theory.

Solutions of mixed-type functional differential equations with state-dependence

In this paper we study existence and asymptotic behaviors of solutions for mixed-type functional differential equation with state-dependence x˙(t)=f(t,x(T1(t,x(t))),...,x(Tn(t,x(t)))). Because of complexity of state-dependent terms, it is so difficult to discuss its stability by Lyapunov direct method. Through an order-preserving transformation, we generalize Gronwall inequality to obtain boundedness and asymptotics of solutions. It can be regarded as an extension to Lyapunov indirect method. Moreover, we indicate that Krasnoselskii fixed point theorem also can be used to study existence of sign-changing solutions. Finally, we apply our method to iterative functional differential equations and provide two concrete examples.

Existence of mild solutions to semilinear fractional evolution equation using Krasnoselskii fixed point theorem

This paper is devoted to study the existence and stability of mild solutions for semilinear fractional evolution equations with a nonlocal final condition. The analysis is based on analytic semigroup theory, Krasnoselskii fixed point theorem, and a special probability density function. An application to a time fractional diffusion equation with nonlocal final condition is also given.

Controllability of nonlinear ordinary differential equations with non-instantaneous impulses

<abstract><p>In this paper, we consider controllability of the initial value problem with non-instantaneous impulse on ordered Banach spaces. We firstly give a solution expression for initial value problems with non-instantaneous impulses in ordered Banach Spaces by using Schauder fixed point theorem. Sufficient conditions for controllability results are obtained by Krasnoselskii's fixed point theorem in the infinite-dimensional spaces. An example is also given to illustrate the feasibility of our theoretical results.</p></abstract>

Ulam stability for nonlinear implicit differential equations with Hilfer-Katugampola fractional derivative and impulses

<abstract><p>In this paper, we investigate the existence, uniqueness and stability results for a class of nonlinear impulsive Hilfer-Katugampola problems. Our reasoning is founded on the Banach contraction principle and Krasnoselskii's fixed point theorem. In addition, an example is provided to demonstrate the effectiveness of the main results.</p></abstract>

Investigation of Caputo proportional fractional integro-differential equation with mixed nonlocal conditions with respect to another function

<abstract><p>In this manuscript, we analyze the existence, uniqueness and Ulam's stability for Caputo proportional fractional integro-differential equation involving mixed nonlocal conditions with respect to another function. The uniqueness result is proved via Banach's fixed point theorem and the existence results are established by using the Leray-Schauder nonlinear alternative and Krasnoselskii's fixed point theorem. Furthermore, by using the nonlinear analysis techniques, we investigate appropriate conditions and results to study various different types of Ulam's stability. In addition, numerical examples are also constructed to demonstrate the application of the main results.</p></abstract>

Existence of mild solutions for nonlocal ψ−Caputo-type fractional evolution equations with nondense domain

Abstract The main crux of this manuscript is to establish the existence and uniqueness of solutions for nonlocal fractional evolution equations involving ψ−Caputo fractional derivatives of an arbitrary order α ∈ (0, 1) with nondense domain. The mild solutions of our proposed model are constructed by employing generalized ψ−Laplace transform and some new density functions. The proofs are based on Krasnoselskii fixed point theorem and some basic techniques of ψ−fractional calculus. As application, a nontrivial example is given to illustrate our theoritical results.

Krasnoselskii-type fixed point theorem in ordered Banach spaces and application to integral equations

In this paper, we present a variant of Krasnoselskii's fixed point theorem in the case of ordered Banach spaces, where the order is generated by a normal and minihedral cone. In such a structure, (...)

On the solution and Ulam-Hyers-Rassias stability of a Caputo fractional boundary value problem.

In this paper, we investigate a class of boundary value problems involving Caputo fractional derivative $ {{}^C\mathcal{D}^{\alpha}_{a}} $ of order $ \alpha \in (2, 3) $, and the usual derivative, of the form $ \begin{equation*} ({{}^C\mathcal{D}^{\alpha}_{a}}x)(t)+p(t)x'(t)+q(t)x(t) = g(t), \quad a\leq t\leq b, \end{equation*} $ for an unknown $ x $ with $ x(a) = x'(a) = x(b) = 0 $, and $ p, \; q, \; g\in C^2([a, b]) $. The proposed method uses certain integral inequalities, Banach's Contraction Principle and Krasnoselskii's Fixed Point Theorem to identify conditions that guarantee the existence and uniqueness of the solution (for the problem under study) and that allow the deduction of Ulam-Hyers and Ulam-Hyers-Rassias stabilities.

Existence of solutions for nonlinear integro-dynamic equations with mixed perturbations of the second type via Krasnoselskii's fixed point theorem

Existence of solutions for nonlinear integro-dynamic equations with mixed perturbations of the second type via Krasnoselskii's fixed point theorem

Sequential fractional order Neutral functional Integro differential equations on time scales with Caputo fractional operator over Banach spaces

<abstract><p>In this work, we scrutinize the existence and uniqueness of the solution to the Integro differential equations for the Caputo fractional derivative on the time scale. We derive the solution of the neutral fractional differential equations along the finite delay conditions. The fixed point theory is demonstrated, and the solution depends upon the fixed point theorems: Banach contraction principle, nonlinear alternative for Leray-Schauder type, and Krasnoselskii fixed point theorem.</p></abstract>

Study of implicit-impulsive differential equations involving Caputo-Fabrizio fractional derivative

<abstract><p>This article is devoted to investigate a class of non-local initial value problem of implicit-impulsive fractional differential equations (IFDEs) with the participation of the Caputo-Fabrizio fractional derivative (CFFD). By means of Krasnoselskii's fixed-point theorem and Banach's contraction principle, the results of existence and uniqueness are obtained. Furthermore, we establish some results of Hyers-Ulam (H-U) and generalized Hyers-Ulam (g-H-U) stability. Finally, an example is provided to demonstrate our results.</p></abstract>

New fractional results for Langevin equations through extensive fractional operators

<abstract><p>Fractional Langevin equations play an important role in describing a wide range of physical processes. For instance, they have been used to describe single-file predominance and the behavior of unshackled particles propelled by internal sounds. This article investigates fractional Langevin equations incorporating recent extensive fractional operators of different orders. Nonperiodic and nonlocal integral boundary conditions are assumed for the model. The Hyres-Ulam stability, existence, and uniqueness of the solution are defined and analyzed for the suggested equations. Also, we utilize Banach contraction principle and Krasnoselskii fixed point theorem to accomplish our results. Moreover, it will be apparent that the findings of this study include various previously obtained results as exceptional cases.</p></abstract>

Positive solutions for fractional differential equations with non-separated type nonlocal multi-point and multi-term integral boundary conditions

In this paper, we investigate a class of nonlinear fractional differential equations that contain both the multi-term fractional integral boundary condition and the multi-point boundary condition. By the Krasnoselskii fixed point theorem we obtain the existence of at least one positive solution. Then, we obtain the existence of at least three positive solutions by the Legget-Williams fixed point theorem. Two examples are given to illustrate our main results.

Optimal controls for fractional stochastic differential systems driven by Rosenblatt process with impulses

AbstractThe objective of this article is to consider a new class of fractional stochastic differential systems driven by the Rosenblatt process with impulses. We used fractional calculus, stochastic analysis, and Krasnoselskii's fixed point theorem to study the existence of piecewise continuous mild solutions for the proposed system. Further, we discussed the existence of optimal controls for the considered system. Our main results are well supported by an illustrative example.

Some Variants of Krasnoselskii-Type Fixed Point Results for Equiexpansive Mappings with Applications

In this paper, we investigate the Krasnoselskii-type fixed point results for the operator F of two variables by assuming that the family F x , . : x is equiexpansive. The results may be considered as variants of the Krasnoselskii fixed point theorem in a general setting. We use our main results to obtain the existence of solutions of a fractional evolution differential equation. An example of a controlled system is given to illustrate the application.

On solvability of differential equations with the Riesz fractional derivative

We consider the solvability of fractional differential equations involving the Riesz fractional derivative. Our approach basically relies on the reduction of the problem considered to the equivalent nonlinear mixed Volterra and Cauchy‐type singular integral equation and on the theory of fractional calculus. By establishing a compactness property of the Riemann–Liouville fractional integral operator on Lebesgue spaces and using the well‐known Krasnoselskii's fixed point theorem, an existence of at least one solution is gleaned. An example is finally included to show the applicability of the theory.