Existence and uniqueness of solutions of initial value problems for nonlinear langevin equation involving two fractional orders

The solvability of initial value problems for nonlinear Langevin equation involving two fractional orders are discussed in this paper. An existence result for the solution is obtained using the Leray–Schauder nonlinear alternative. In addition, sufficient conditions for unique solution are established under the Banach contraction principle. The existence results for the initial value problems of nonlinear classical Langevin equation follow as a special case of our results.     

Solvability for impulsive fractional Langevin equation

We investigate impulsive fractional Langevin equation involving two fractional Caputo derivatives with boundary value conditions. By Banach contraction mapping principle and Krasnoselskii"s fixed point theorem, some results on the existence and uniqueness of solution are obtained.     

Existence results of solutions for anti-periodic fractional Langevin equation

Recently, Khalili and Yadollahzadeh [ Y. Khalili and M. Yadollahzadeh, Existence results for a new class of nonlinear Langevin equations of fractional orders, Iranian J. Sci. Tech., Trans. A: Sci., 2019, 43(5), 2335–2342] have investigated the uniqueness and existence of solution $u(t),~t\in[0,1]$ for a class of nonlocal boundary conditions to fractional Langevin equation. The authors used the boundary condition $u"(0)=0$ by incorrect method. In the current contribution, we show the correct method for using this condition and study the existence and uniqueness of solution for the same class of equation in slightly different form with anti-periodic and nonlocal integral boundary conditions as well as the boundary condition $u"(0)=0$. An exemplar is provided to illustrate our results.     

Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order

In this paper, we prove some existence and uniqueness of mild solutions for a semilinear integrodifferential equation of fractional order with nonlocal initial conditions in α-norm. We assume that the linear part generates a noncompact analytic semigroup. Our results cover the cases that the nonlinearity F takes values in different spaces such as X,X_α and X_β, where α≤β(0,1). Finally, some practical consequences are also obtained.     

Fractional Langevin Equation at Resonance

The study of fractional Langevin equation has obtained abundant results in recent years. However, there are few studies on resonant fractional Langevin equation. In this paper, we investigate boundary value problems for fractional Langevin equation at resonance. By virtue of Banach contraction mapping principle and Leray-Schauder fixed point theorem, we obtain the uniqueness and existence of solutions. In addition, we get different stability results, including Ulam-Hyres stability and generalized Ulam-Hyres stability. Finally, give relevant examples to demonstrate the main results.     

Controllability of nonlinear Hilfer fractional Langevin dynamical system

The main objective of this article is to prove the controllability result of a nonlinear Hilfer fractional Langevin dynamical system in an abstract-weighted space. This paper initiates with the persistence of the control formula for the Hilfer fractional Langevin dynamical system. Furthermore, As ensue is extended to the controllability result of Hilfer fractional Langevin dynamical system by the use of Krasnoselskii fixed point theorem. Ultimately, an example is furnished to demonstrate our outcomes.     

分数阶朗之万方程的非局部狄利克雷边值问题解的存在性

讨论了分数阶Langevin方程的非局部狄利克雷边值问题,利用Leray-Schauder's和压缩映像原理,分别得到了方程的解的存在及唯一性.     

On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order

In this paper, we prove the existence and uniqueness of solutions for a four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional order q ∈ (1, 2] by applying some standard fixed point theorems. An illustrative example is also presented.     

On quadratic integral equation of fractional orders

We present an existence theorem for a nonlinear quadratic integral equations of fractional orders, arising in the queuing theory and biology, in the Banach space of real functions defined and continuous on a bounded and closed interval. The concept of measure of noncompactness and a fixed point theorem due to Darbo are the main tool in carrying out our proof.     

The unique solution for a fractional q-difference equation with three-point boundary conditions

The aim of this paper is to investigate the existence and uniqueness of solutions for nonlinear fractional $q$-difference equations with three-point boundary conditions. Our approach relies on a new fixed point theorem of increasing $\psi ?(h,r)?$concave operators defined on ordered sets. Further, we can construct a monotone explicit iterative scheme to approximate the unique solution. Finally, the main results are illustrated with the aid of two interesting examples.     

Fractional Differential Equations with Anti-Periodic Boundary Conditions

We are concerned with the existence of solutions of a class of fractional differential equations with anti-periodic boundary conditions involving the Caputo fractional derivative. We give two results: the first is based on Banach's fixed-point theorem, and the second is based on Schauder's fixed-point theorem.     

Positive solutions to singular fractional differential system with coupled boundary conditions

By using the fixed point theory in cone and constructing some available integral operators together with approximating technique, the existence of positive solution for a singular nonlinear semipositone fractional differential system with coupled boundary conditions is established. Two examples are then given to demonstrate the validity of our main results.     

A generalized Gronwall inequality and its application to a fractional differential equation

This paper presents a generalized Gronwall inequality with singularity. Using the inequality, we study the dependence of the solution on the order and the initial condition of a fractional differential equation.     

Existence-uniqueness result for a nonlinear n-term fractional equation

We prove the existence-uniqueness of the solution to the nonlinear n-term time-fractional differential equation with constant coefficients in the Banach space C([0,T]),

(1)     

Initial value problems for arbitrary order fractional differential equations with delay

By fixed point theory the nonlinear alternative of Leray–Schauder type, and the properties of absolutely continuous functions space, we study the existence and uniqueness of initial value problems for nonlinear higher fractional equations with delay, and obtain some new results involving local and global solutions.     

The existence and uniqueness theorem of the solution to a class of nonlinear fractional order system with time delay

In this paper, we investigate the existence and uniqueness of the solution to a class of nonlinear fractional order system with delay. The estimate value of the above solution is also obtained by using the generalized Gronwall inequality.     

Existence and uniqueness of positive solution for nonlinear fractional q-difference equation with integral boundary conditions

This paper studies a class of nonlinear fractional $q$-difference equations with integral boundary conditions. By exploiting the properties of Green"s function and two fixed point theorems for a sum operator, the existence and uniqueness of positive solutions for the boundary value problem are established. Iterative schemes for approximating the solutions are also obtained. Explicit examples are given to illustrate main results.