Existence of solutions for Erd\xe9lyi\u2013Kober fractional integral boundary value problems with p ( t )-Laplacian operator

Xiaohui Shen,Tengfei Shen

doi:10.1186/s13662-020-03015-y

Abstract

This paper aims to consider the solvability for Erdélyi–Kober fractional integral boundary value problems with p ( t )-Laplacian operator at resonance. By employing the coincidence degree method, some new results on the existence of solutions are acquired.

Highlights

We always assume that the following condition holds: (H) (a1 + b1) (α) α–1 γ1 + η1 + δ1 + 1
It should be emphasized that the Erdélyi–Kober fractional integral operator is a generalization of the integral of integer order and can convert into Riemann–Liouville fractional integral with a power weight when η = 1 and γ = 0
4 Conclusions This paper is concerned with the solvability for Erdélyi–Kober fractional integral boundary value problems with p(t)-Laplacian operator at resonance

Summary

Introduction

We consider the following fractional integral boundary value problem: ⎧. A1Dα0+–2x(1) + b1Dα0+–1x(1) = ν1I0γ+1,,ηδ x(ξ1), a2Dα0+–2x(0) + b2Dα0+–1x(0) = ν2I0γ+2,,ηδ x(ξ2), (1.1). Where Dβ0+ and Dα0+ are Riemann–Liouville fractional derivatives with 0 < β ≤ 1 and 2 < α ≤ 3, I0γ+i,,δηii is Erdélyi–Kober fractional integral of order δi > 0 with ηi > 0 and γi > 0 in which i = 1, 2, f : [0, 1] × R3 → R is continuous, ai, bi, νi are real numbers in which i = 1, 2,. We always assume that the following condition holds:. = ν1ξ1α–1 α–1 γ1 + η1 + 1 ; a1 (α – 1)

Objectives

Conclusion