Positive solutions to singular fractional differential system with coupled boundary conditions

This paper investigates the existence of at least two positive solutions for the following high-order fractional semipositone boundary value problem (SBVP, for short) with coupled integral boundary value conditions:where n − 1 < α n, n 3, 0 < η 1 , η 2 1, λ, λ 1 , λ 2 are parameters and satisfy λ 1 λ 2 (η 1 η 2 ) α < Γ 2 (α + 1), D α 0 + is the standard Riemann-Liouville derivative, and f, g are continuous and semipositone. By using the nonlinear alternative of Leray-Schauder type, Krasnoselskii's fixed point theorems, and the theory of fixed point index on cone, we establish some existence results of multiple positive solutions to the considered fractional SBVP. As applications, two examples are presented to illustrate our main results. c 2017 All rights reserved.Keywords: Fractional differential equations, semipositone boundary value problem, coupled integral boundary value conditions, fixed point index. 2010 MSC: 34A08, 34B15, 34B18.

Section: Introductionmentioning

confidence: 86%

Twin solutions to semipositone boundary value problems for fractional differential equations with coupled integral boundary conditions

Deng-feng¹,

Liu²

2017

“…The main features are as follows: First, the system we discuss here is different from [2,6,13,17,20,22,23,27,28,29,30,31,33], where the nonlinear terms involved two unknown functions u, v and the fractional derivative of unknown functions u, v, and the case is more general and difficult than the above papers. Secondly, by using a new inequality of fractional order form, the restrictive conditions a ik (t) in (H 1 ), (H 2 ), (H 4 ) and (H 6 ) are applicable for more general problems than the conditions a ik in the mentioned documents.…”

Section: Introductionmentioning

confidence: 99%

“…At the same time, there are many papers concerned with solvability of coupled systems of nonlinear fractional differential equations, because of their wide applications. For details, see [2,6,13,17,20,22,23,27,28,29,30,31,33] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Existence and uniqueness of solutions for a coupled system of nonlinear fractional differential equations with fractional integral boundary conditions

Zhang¹,

Li²,

Wei³

2016

“…Specially, the study of coupled systems of fractional order differential equations has been addressed extensively by several researchers, see [3,5,18,20,21,29,33,36,40] and the references cited therein. For instance, By applying some standard fixed point theorems, Jiang et al [23] and Yuan et al [41] considered the existence of positive solutions to the four-point coupled boundary value problems for systems of nonlinear semipositone fractional differential equations under different conditions, respectively. In [20], Hao and Zhai studied the existence of at least one positive solution to a coupled system of fractional boundary value problems by using Schauder fixed point theorem.…”

Section: Introductionmentioning

confidence: 99%

Positive solutions for singular coupled integral boundary value problems of nonlinear Hadamard fractional differential equations

Yang¹

2015

In this paper, we study the existence of positive solutions for a class of coupled integral boundary value problems of nonlinear semipositone Hadamard fractional differential equationswhere λ, µ, ν are three parameters with 0 < µ < β and 0 < ν < α, α, β ∈ (n − 1, n] are two real numbers and n ≥ 3, D α , D β are the Hadamard fractional derivative of fractional order, and f, g are sign-changing continuous functions and may be singular at t = 1 or/and t = e. First of all, we obtain the corresponding Green's function for the boundary value problem and some of its properties. Furthermore, by means of the nonlinear alternative of Leray-Schauder type and Krasnoselskii's fixed point theorems, we derive an interval of λ such that the semipositone boundary value problem has one or multiple positive solutions for any λ lying in this interval. At last, several illustrative examples were given to illustrate the main results. c 2015 All rights reserved.