Existence results for fractional order differential equation with nonlocal Erd\xe9lyi\u2013Kober and generalized Riemann\u2013Liouville type integral boundary conditions at resonance

Qiao Sun,Shuman Meng,Yujun Cui

doi:10.1186/s13662-018-1668-x

Abstract

In this paper, we discuss a nonlinear fractional order boundary value problem with nonlocal Erdélyi–Kober and generalized Riemann–Liouville type integral boundary conditions. By using Mawhin continuation theorem, we investigate the existence of solutions of this boundary value problem at resonance. It is shown that the boundary value problem \t\t\tDqcx(t)=f(t,x(t),x′(t)),t∈[0,T],1<q≤2,x(0)=αIηγ,δx(ζ),x(T)=βρIpx(ξ),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{gathered} {}^{c}D^{q}x(t)=f \\bigl(t, x(t),x'(t) \\bigr),\\quad t \\in[0,T], 1< q\\leq2, \\\\ x(0)=\\alpha I^{\\gamma,\\delta}_{\\eta}x(\\zeta),\\qquad x(T)=\\beta{}^{\\rho }I^{p}x( \\xi),\\end{gathered} $$\\end{document} has at least one solution under some suitable conditions, where alpha, betainmathbb{R}, 0<zeta, xi<T.

Highlights

In this paper, we intend to discuss the following boundary value problem at resonance: cDqx(t) = f (t, x(t), x (t)), t ∈ [0, T], x(0) = αIηγ,δx(ζ ), x(T) = βρIpx(ξ ), 0 < ζ, η ≤ T, (1)where cDq is the Caputo fractional derivative of order 1 < q ≤ 2, Iηγ,δ is a Erdélyi–Kober type integral of order δ > 0 with η > 0 and γ ∈ R, ρIp denotes the generalized Riemann– Liouville type integral of order p > 0, ρ > 0, and α, β ∈ R.Boundary value problems at resonance have aroused people’s interest these days
In [17], Jiang and Qiu studied the existence of solutions for the following (k, n – k) conjugate boundary value problem at resonance: (–1)n–ky(n)(t) = f t, y(t), y (t), . . . , y(n–1)(t), t ∈ [0, 1], Sun et al Advances in Difference Equations (2018) 2018:243 y(i)(0) = y(j)(1) = 0, 0 ≤ i ≤ k – 1, 0 ≤ j ≤ n – k – 2, m y(n–1)(1) = αiy(n–1)(ξi), i=1 where 1 ≤ k ≤ n – 1, 0 < ξ1 < ξ2 < · · · < ξm < 1
In [5], Zhang and Bai investigated the existence of solutions for the following m-point boundary value problems: Dα0+ u(t) = f t, u(t), Dα0+–1u(t) + e(t), t ∈ (0, 1), α ∈ (1, 2], I0α+ u(t) t=0 = 0, m–2

Summary

Introduction

Definition 2.6 ([26]) Assume that X and Y are real Banach spaces, L : dom L ⊂ X → Y is a Fredholm operator of index zero if the following conditions hold: (1) Im L is a closed subspace of Y ; (2) dim Ker L = co dim Im L < ∞. 3 Main results Assume that the following conditions hold in this paper: (H1) f : [0, 1] × R2 → R is a continuous function.

Results

Conclusion