Properties of Green\u2019s function and the existence of different types of solutions for nonlinear fractional BVP with a parameter in integral boundary conditions

Wenxia Wang

doi:10.1186/s13661-019-1184-2

Abstract

This paper is concerned with the impact of the parameter on the existence of different types of solutions for a class of nonlinear fractional integral boundary value problems with a parameter that causes the sign of Green’s function associated with the BVP to change. By using the Guo–Krasnoselskii fixed point theorem, the Leray–Schauder nonlinear alternative, and the analytic technique, we give the range of the parameter for the existence of strong positive solutions, strong negative solutions, negative solutions, and sign-changing solutions for the boundary value problem. Some examples are given to illustrate our main results.

Highlights

1 Introduction and preliminaries Fractional differential equations are recognized as adequate mathematical models to study some materials and processes that have memory and hereditary properties
Because of the extensive application in mathematics and the applied science, fractional boundary value problems with parameters have attracted considerable attention and obtained some interesting results; see, for instance, the works of Bai [19], Song and Bai [20], Jiang [21], Sun et al [22], Zhai and Xu [23], and Zhang and Liu [24] on the eigenvalue problems; the works of Jia and Liu [25], Wang and Liu [26], Su et al [27], and Li et al [28] on the problems with disturbance parameters in the boundary conditions; and the work of Wang and Guo [29] on the eigenvalue problems with a disturbance parameter in the boundary conditions. We notice that another type of fractional integral boundary value problems with μ in the boundary conditions has received much attention; see [30,31,32,33] and the references therein
We give the range of the parameter μ on the existence of strong positive solutions, strong negative solutions, negative solutions, and sign-changing solutions for boundary value problem (BVP) (1). These results show the impact of the parameter μ on the existence of different types of solutions

Summary

When μ

For any μ = 1, it is clear by Lemma 2.1 that xμ is a solution of BVP (1) ⇔ xμ is a fixed point of Tμ in E. It is clear from Lemma 2.3(i) that Tμ(E) ⊂ P. Lemma 3.1 If xμ is a solution of BVP (1) for μ = 1, xμ(t) is decreasing with respect to t for t ∈ [0, 1]. (t – s)α–2f s, xμ(s) ds ≤ 0, t ∈ [0, 1], which implies that the solution xμ(t) is decreasing on [0, 1] It follows from Lemma 1.4 that BVP (1) has at least one solution xμ ∈ P1μ with r1 ≤ xμ ≤ R1.

This means that

This implies that