Existence results for fractional order differential equation with nonlocal Erdélyi–Kober and generalized Riemann–Liouville type integral boundary conditions at resonance

• Main Authors: Qiao Sun, Shuman Meng, Yujun Cui
• Format: Article
• Language: English
• Published: SpringerOpen 2018-07-01
• Series: Advances in Difference Equations
• Subjects: Boundary value problem; Resonance; Integral conditions
• Online Access: http://link.springer.com/article/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 Dqcx(t)=f(t,x(t),x′(t)),t∈[0,T],1<q≤2,x(0)=αIηγ,δx(ζ),x(T)=βρIpx(ξ), $$\begin{gathered} {}^{c}D^{q}x(t)=f \bigl(t, x(t),x'(t) \bigr),\quad t \in[0,T], 1< q\leq2, \\ x(0)=\alpha I^{\gamma,\delta}_{\eta}x(\zeta),\qquad x(T)=\beta{}^{\rho }I^{p}x( \xi),\end{gathered} $$ has at least one solution under some suitable conditions, where α,β∈R $\alpha, \beta\in\mathbb{R}$, 0<ζ,ξ<T $0<\zeta, \xi<T$.