Higher-order Nonlinear Fractional Differential Equations Research Articles

Uniqueness of solution for higher-order nonlinear fractional differential equations with multi-point and integral boundary conditions

Uniqueness of solution for higher-order nonlinear fractional differential equations with multi-point and integral boundary conditions

Multi-point boundary value problems of higher-order nonlinear fractional differential equations

‎We investigate the existence and uniqueness of solutions for multi-point nonlocal boundary value problems of higher-order nonlinear fractional differential equations by using some well known fixed point theorems‎.

Existence and uniqueness of monotone positive solutions for fractional higher-order BVPs

We discuss the existence and uniqueness of monotone positive solutions for a class of higher-order nonlinear fractional differential equations infinite-point boundary value problems (for short, BVPs) on cones. The existence and uniqueness of solutions are obtained via applying the properties of the Green function and a fixed point theorem. Our analysis is based on the operator equation T ω + S ω = ω on an ordered Banach space. Finally, a example is given to illustrate our results.

Stability in higher-order nonlinear fractional differential equations

We give sufficient conditions to guarantee the asymptotic stability of the zero solution to a kind of higher-order nonlinear fractional differential equations. By using Krasnoselskii's xed point theorem in a weighted Banach space, we establish new results on the asymptotic stability of the zero solution provided that f (t, 0) = 0. The results obtained here generalize the work of Ge and Kou.

Positive Solutions for Higher-Order Nonlinear Fractional Differential Equations

This paper investigates the existence of positive solutions for a class of higher-order nonlinear fractional differential equations with initial conditions given for ordinary as well as fractional derivatives of the unknown function. We assume that the nonlinear term f involves also derivatives of fractional order. The results are established by converting the problem into an equivalent integral equation and applying Guo–Krasnoselskii’s fixed-point theorem in cones.

Existence and uniqueness of solutions for a singular system of higher-order nonlinear fractional differential equations with integral boundary conditions

In this paper, we study the existence and uniqueness of solutions for a singular system of nonlinear fractional differential equations with integral boundary conditions. We obtain existence and uniqueness results of solutions by using the properties of the Green’s function, a nonlinear alternative of Leray–Schauder-type, Guo–Krasnoselskii’s fixed point theorem in a cone and the Banach fixed point theorem. Some examples are included to show the applicability of our results.

Existence of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions and a parameter

In this paper, we study the existence of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions and a parameter. By using the properties of the Green’s function, u 0-positive function and the fixed point index theory, we obtain some existence results of positive solution under some conditions concerning the first eigenvalue with respect to the relevant linear operator. The method of this paper is a unified method for establishing the existence of multiple positive solutions for a large number of nonlinear differential equations of arbitrary order with any allowed number of non-local boundary conditions.

Existence and uniqueness of solutions for a system of higher-order nonlinear fractional differential equations

A system of higher-order nonlinear fractional differential equations is studied in this article, and some sufficient conditions for existence and uniqueness of a solution for the system is established by Schauder fixed point theorem and Banach contraction principle, respectively.

Positive solutions of higher-order nonlinear fractional differential equations with changing-sign measure

Abstract In this article, we consider the existence of positive solutions of the (n - 1, 1) conjugate-type nonlocal fractional differential equation D 0 + α x ( t ) + f ( t , x ( t ) ) = 0 , 0 < t < 1 , n - 1 < α ≤ n , x ( k ) ( 0 ) = 0 , 0 ≤ k ≤ n - 2 , x ( 1 ) = ∫ 0 1 x ( s ) d A ( s ) , where α ≥ 2, D 0 + α is the standard Riemann-Liouville derivative, ∫ 0 1 x ( s ) d A ( s ) is a linear functional given by the Stieltjes integral, A is a function of bounded variation, and dA may be a changing-sign measure, namely the value of the linear functional is not assumed to be positive for all positive x. By constructing upper and lower solutions, some sufficient conditions for the existence of positive solutions to the problem are established utilizing Schauder's fixed point theorem in the case in which the nonlinearities f(t, x) are allowed to have the singularities at t = 0 and (or) 1 and also at x = 0. AMS (MOS) Subject Classification: 34B15; 34B25.

New Existence Results for Higher-Order Nonlinear Fractional Differential Equation with Integral Boundary Conditions

This paper investigates the existence and multiplicity of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions. The results are established by converting the problem into an equivalent integral equation and applying Krasnoselskii's fixed-point theorem in cones. The nonexistence of positive solutions is also studied.

Existence and Uniqueness of Solutions for Coupled Systems of Higher-Order Nonlinear Fractional Differential Equations

AbstractWe study an initial value problem for a coupled Caputo type nonlinear fractional differential system of higher order. As a first problem, the nonhomogeneous terms in the coupled fractional differential system depend on the fractional derivatives of lower orders only. Then the nonhomogeneous terms in the fractional differential system are allowed to depend on the unknown functions together with the fractional derivative of lower orders. Our method of analysis is based on the reduction of the given system to an equivalent system of integral equations. Applying the nonlinear alternative of Leray-Schauder, we prove the existence of solutions of the fractional differential system. The uniqueness of solutions of the fractional differential system is established by using the Banach contraction principle. An illustrative example is also presented.

Existence of Solutions for Nonlocal Boundary Value Problems of Higher-Order Nonlinear Fractional Differential Equations

We study some existence results in a Banach space for a nonlocal boundary value problem involving a nonlinear differential equation of fractional orderqgiven bycDqx(t)=f(t,x(t)),0<t<1,q∈(m−1,m],m∈ℕ,m≥2, x(0)=0, x′(0)=0, x′′(0)=0,…,x(m−2)(0)=0,x(1)=αx(η). Our results are based on the contraction mapping principle and Krasnoselskii's fixed point theorem.