Semilinear Fractional Differential Equations Research Articles

On the stability and stabilization of some semilinear fractional differential equations in Banach spaces

In this paper, we prove the existence and uniqueness of a mild solution to a class of semilinear fractional differential equation in an infinite Banach space with Caputo derivative order 0 < α ≤ 1. Furthermore, we establish the stability conditions and then prove that the considered initial value problem is exponentially stabilizable when the stabilizer acts linearly on the control system.

A New Generalized Gronwall Inequality with a Double Singularity and Its Applications to Fractional Stochastic Differential Equations

In many cases, the existence and uniqueness of the solution of a differential equation are proved using fixed point theory. In this paper, we utilize the theory of operators and ingenious techniques to investigate the well-posedness of mild solution to semilinear fractional stochastic differential equations. We first discuss some properties of a class of Volterra integral operators and then establish a new generalized Gronwall integral inequality with a double singularity. Finally, we use the properties and integral inequality to study the well-posedness of mild solution to the semilinear fractional stochastic differential equations. One sees that it is concise and effectiveness using the previous results to investigate the well-posedness of the mild solution.

Semilinear fractional differential equations with infinite delay and non-instantaneous impulses

In this article, we study the existence of mild solutions for a new class of impulsive semilinear fractional differential equations with infinite delay and non-instantaneous impulses in Banach spaces. The new results are obtained using suitable fixed point theorems and the technique of measures of noncompactness. An example is also given to illustrate the obtained theory.

Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

This paper is concerned with the existence of asymptotically almost automorphic mild solutions to a class of abstract semilinear fractional differential equations Dtαxt=Axt+Dtα-1Ft,xt,Bxt, t∈R, where 1&lt;α&lt;2, A is a linear densely defined operator of sectorial type on a complex Banach space X and B is a bounded linear operator defined on X, F is an appropriate function defined on phase space, and the fractional derivative is understood in the Riemann-Liouville sense. Combining the fixed point theorem due to Krasnoselskii and a decomposition technique, we prove the existence of asymptotically almost automorphic mild solutions to such problems. Our results generalize and improve some previous results since the (locally) Lipschitz continuity on the nonlinearity F is not required. The results obtained are utilized to study the existence of asymptotically almost automorphic mild solutions to a fractional relaxation-oscillation equation.

Nonlocal Conditions for Semilinear Fractional Differential Equations With Hilfer Derivative

This paper studies the existence of solutions and some topological proprieties of solution sets for nonlocal semi fractional differential equations of Hilfer type in Banach space by using noncompact measure method in the weighted space of continuous functions. The main result is illustrated with the aid of an example

On S‐asymptotically ω‐periodic and Bloch periodic mild solutions to some fractional differential equations in abstract spaces

We are concerned with the existence and uniqueness of S‐asymptotically ω‐periodic and Bloch periodic mild solutions to the semilinear fractional differential equation of the form where is the Riemann‐Liouville derivative, 1 &lt; α &lt; 2 and A is a (generally unbounded) linear operator, which generates a bounded α‐resolvent family Tα(t) on a (complex) Banach space X. We use some classical fixed‐point theorems to reach our results. The results are new even in the context of asymptotically periodic solutions.

Translation invariance and related properties of \u03bc-pseudo almost automorphic (periodic) functions with application

It is well known that many important properties of μ-pseudo almost automorphic (periodic) functions are based on the translation invariance. In this paper we give a new result of the translation invariance with conditions weaker than the known one. Then some more properties are studied, including the uniqueness of the decomposition and the convolution of the space. Moreover, we present a result on the equality of two spaces of this kind of functions, which extends the well-known results to the case when the measures of the spaces are not equivalent. As an application, we give an existence and uniqueness theorem of μ-pseudo almost automorphic mild solution for a semilinear fractional differential equation.

Polynomial Decay of Mild Solutions to Semilinear Fractional Differential Equations with Nonlocal Initial Conditions

This paper deals with a class of two-term time fractional differential equations with nonlocal initial conditions. We establish the existence of mild solutions with explicit decay rate of polynomial type. To illustrate the abstract results, an example is also given.

Stepanov-like almost automorphic mild solutions for semilinear fractional differential equations

This work is concerned with the existence and uniqueness of Stepanov-like almost automorphic mild solutions for a class of semilinear fractional differential equations&#x0D; Dtα x(t) = Ax(t) + Dtα-1F(t,x(t)), t ∈ ℝ,&#x0D; where 1 &lt; α &lt; 2, A is a linear densely defined operator of sectorial type of ω &lt; 0 on a complex Banach space X and F is an appropriate function defined on phase space. The fractional derivative is understood in the Riemann-Liouville sense. The results obtained are utilized to study the existence and uniqueness of Stepanov-like almost automorphic mild solutions for a fractional relaxation-oscillation equation.

Finite-Time Attractivity for Semilinear Fractional Differential Equations

The aim of our work is to establish some sufficient conditions ensuring the finite-time attractivity of solutions to semilinear fractional differential equations, in which the nonlinearity has a superlinear growth. We obtain the results by employing the fractional resolvent theory and deriving local estimates of solutions. An application to semilinear subdiffusion equations will be shown.

A Class of Nonlocal Fractional Evolution Equations and Optimal Controls

In this paper we study the existence of solutions for a class of semilinear fractional differential equations with nonlocal conditions and involving abstract Volterra operators. The existence of an optimal solution for a class of fractional control problem involving Caputo fractional derivatives is obtained. An example is presented to illustrate our main result.

Nonlocal semi-linear fractional-order boundary value problems with strip conditions

This paper is concerned with the question of existence of solutions for one-dimensional higher-order semi-linear fractional differential equations supplemented with nonlocal strip type boundary conditions. The nonlocal strip condition addresses a situation where the linear combination of the values of unknown function at two nonlocal points, located to the left and right hand sides of the strip, respectively, is proportional to its strip value. The case of Stieltjes type strip condition is also discussed. Our results, relying on some standard fixed point theorems are supported with illustrative examples.

Optimal Controls of Systems Governed by Semilinear Fractional Differential Equations with Not Instantaneous Impulses

This paper is concerned on optimal control problems for systems governed semilinear fractional differential equations with not instantaneous impulses in the infinite dimensional spaces. We utilize fractional calculus, semigroup theory and fixed point approach to present the solvability of the corresponding control system by using the new introduced concept of mild solutions. Next, we give the existence result of optimal controls for Lagrange problem under the suitable conditions. Finally, an example is given to illustrate the effectiveness of our results.

Analysis of the compact difference scheme for the semilinear fractional partial differential equation with time delay

In the paper, a linearized compact finite difference scheme is presented for the semilinear fractional delay convection-reaction–diffusion equation. Firstly, the equation is transformed into an equivalent semilinear fractional delay reaction–diffusion equation by using a special transformation. Then, the temporal Caputo derivative is discreted by using approximation and the second-order spatial derivative is approximated by the compact finite difference scheme. The solvability, unconditional stability, and convergence in the sense of - and - norms are proved rigorously. Finally, numerical examples are carried out extensively to support our theoretical analysis.

Pseudo almost automorphy of semilinear fractional differential equations in Banach Spaces

Abstract In this paper, we investigate the existence and uniqueness of (μ,ν)-pseudo almost automorphic mild solutions to semilinear fractional differential equation with Riemann-Liouville derivative in Banach space, where the nonlinear perturbation is of (μ,ν)-pseudo almost automorphic type or Stepanov-like (μ,ν)-pseudo almost automorphic type. As application, we explore the same topic for a fractional relaxation-oscillation equation.

On the compactness of fractional resolvent operator functions

We study and characterize the compactness of resolvent families of operators associated to fractional differential equations. We show an application in the study of existence of mild solutions for a class of semilinear fractional differential equations with non-local initial conditions.

Existence and Asymptotic Behavior of Positive Solutions for a Coupled System of Semilinear Fractional Differential Equations

In this paper, we study a coupled system of semilinear Riemann–Liouville type fractional differential equations. Applying the Schauder fixed point theorem, we prove the existence of positive solutions of the fractional differential system. Furthermore, the asymptotic behavior of the solutions is given by using Karamata regular variation theory.

DEGENERATE DISTRIBUTED CONTROL SYSTEMS WITH FRACTIONAL TIME DERIVATIVE

The existence of a unique strong solution for the Cauchy problem to semilinear nondegenerate fractional differential equation and for the generalized Showalter–Sidorov problem to semilinear fractional differential equation with degenerate operator at the Caputo derivative in Banach spaces is proved. These results are used for search of solution existence conditions for a class of optimal control problems to a system described by the degenerate semilinear fractional evolution equation. Abstract results are applied to the research of an optimal control problem solvability for the equations system of Kelvin–Voigt fractional viscoelastic fluids.

Pseudo Almost Automorphic Solution of Semilinear Fractional Differential Equations with the Caputo Derivatives

Abstract In this paper, we deal with existence and uniqueness of (μ, ν)-pseudo almost automorphic mild (classical) solution to semilinear fractional differential equations with the Caputo derivatives. The main results are obtained by means of the fixed point theory, Leray-Schauder alternative theorem and fractional powers of operators. Moreover, an application to fractional predator-prey system with diffusion is given.

Mild solutions for semi-linear fractional order functional stochastic differential equations with impulse effect

This paper is concerned with the existence results of mild solution for an impulsive fractional order stochastic differential equation with infinite delay subject to nonlocal conditions. The results are obtained by using the fixed point techniques and solution operator generated by sectorial operator on a Hilbert space.