On the existence of nonnegative solutions for a class of fractional boundary value problems (original) (raw)

On the existence of nonnegative solutions of nonlocal boundary value problems for a class of fractional differential equations

Journal of Nonlinear Analysis and Application, 2012

This article studies the existence of nonnegative solutions for a boundary value problem (BVP) of nonlinear fractional differential equations. Some new existence results are obtained by applying Kranoselskii's fixed-point theorem in a cone. First we prove the existence of solutions of an auxiliary BVP formulated by truncating the response function. Then the Arzela-Ascoli theorem is used to take C 1 limits of sequences of such solutions. As an application, we give an examples that illustrate our results.

On the existence of positive solutions for generalized fractional boundary value problems

Boundary Value Problems

The existence of positive solutions is established for boundary value problems defined within generalized Riemann–Liouville and Caputo fractional operators. Our approach is based on utilizing the technique of fixed point theorems. For the sake of converting the proposed problems into integral equations, we construct Green functions and study their properties for three different types of boundary value problems. Examples are presented to demonstrate the validity of theoretical findings.

Positive Solutions to Boundary Value Problems of Nonlinear Fractional Differential Equations

We study the existence of positive solutions for the boundary value problem of nonlinear fractional differential equations D α 0 u t λf u t 0, 0 < t < 1, u 0 u 1 u 0 0, where 2 < α ≤ 3 is a real number, D α 0 is the Riemann-Liouville fractional derivative, λ is a positive parameter, and f : 0, ∞ → 0, ∞ is continuous. By the properties of the Green function and Guo-Krasnosel'skii fixed point theorem on cones, the eigenvalue intervals of the nonlinear fractional differential equation boundary value problem are considered, some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem are established. As an application, some examples are presented to illustrate the main results.

Further results on existence of positive solutions of generalized fractional boundary value problems

Advances in Difference Equations

This paper studies two classes of boundary value problems within the generalized Caputo fractional operators. By applying the fixed point result of α-ϕ-Geraghty contractive type mappings, we derive new results on the existence and uniqueness of the proposed problems. Illustrative examples are constructed to demonstrate the advantage of our results. The theorems reported not only provide a new approach but also generalize existing results in the literature.

Existence of Positive Solutions to a Family of Fractional Two Point Boundary Value Problems

Progress in Fractional Differentiation and Applications, 2016

In this paper we will consider an nth order fractional boundary value problem with boundary conditions that include a fractional derivative at 1. We will develop properties of the Green's Function for this boundary value problem and use these properties along with the Contraction Mapping Principle, and the Schuader's, Krasnozel'skii's, and Legget-Williams fixed point theorems to prove the existence of positive solutions under different conditions. v

Existence of positive solutions for fractional boundary value problems

Journal of Applied Analysis and Computation, 2017

In this paper, by introducing a new operator, improving and generating a p-Laplace operator for some p > 1, we discuss the existence and multiplicity of positive solutions to the four point boundary value problems of nonlinear fractional differential equations. Our results extend some recent works in the literature.