Existence of positive solutions for integral boundary value problems of fractional differential equations on infinite interval

Abstract

In this paper, we consider a class of nonlinear fractional differential equations on the infinite interval D0+αu(t)+ft,u(t),D0+α−1u(t)=0,t∈(0,+∞), with the integral boundary conditions u(0)=0,D0+α−1u(∞)=∫0τg1(s)u(s)ds+a,D0+α−2u(0)=∫0τg2(s)u(s)ds+b. By using Krasnoselskii fixed point theorem, the existence results of positive solutions for the boundary value problem in three cases τ=0,τ∈(0,+∞) and τ=+∞, are obtained, respectively. We also give out two corollaries as applications of the existence theorems and some examples to illustrate our results. Copyright © 2016 John Wiley & Sons, Ltd.

Existence results for fractional integral boundary value problem involving fractional derivatives on an infinite interval

“…Li et al used the Guo‐Krasnoselskii fixed point theorem to show the existence of positive solutions for fractional differential equation with integral boundary conditions on the infinite interval where 2 < α ≤ 3, f : J × J × J → J satisfies the Caratheodory conditions, are nonnegative, τ = 0, τ ∈ (0, + ∞ ), or τ = + ∞ , a , b are positive parameters.…”

“…It should be mentioned that most of the results on fractional calculus are devoted to the solvability of fractional differential equations on finite interval. Recently, there are few papers concerning the fractional differential equations with various boundary conditions on infinite interval, for example, see other references …”

Existence of Solutions for Fractional Multi-Point Boundary Value Problems on an Infinite Interval at Resonance

“…In recent years, the discussion of fractional initial value problems (IVPs) and BVPs have attracted the attention of many scholars and valuable results have been obtained (see ). Various methods have been utilized to study fractional IVPs and BVPs such as the Banach contraction map principle (see [8][9][10][11]), fixed point theorems (see [12][13][14][15][16][17][18]), monotone iterative method (see [19][20][21]), variational method (see [22][23][24]), fixed point index theory (see [17][18][19][20][21][22][23][24][25]), coincidence degree theory (see [26][27][28][29]), and numerical methods [30,31]. For instance, Jiang (see [26]) studied the existence of solutions using coincidence degree theory for the following fractional BVP:…”

“…Numerous papers discuss BVPs of integer-order differential equations on infinite intervals (see [35][36][37][38]). Naturally, BVPs of fractional differential equations on infinite intervals have received some attention (see [8,12,[14][15][16][18][19][20]27,29,32]). For example, Wang et al [8] considered the following fractional BVPs on an infinite interval:…”

“…The main difficulties in solving the present BVP are: Constructing suitable Banach spaces for BVP (1); Since [0, +∞) is noncompact, it is difficult to prove that operator N is L-compact; The theory of Mawhin's continuation theorem is characterized by higher dimensions of the kernel space on resonance BVPs, therefore, constructing projections P and Q is difficult; Estimating a priori bounds of the resonance problem on an infinite interval with dim KerL = 2 (see Section 3, Lemmas [11][12][13][14][15][16].…”