Existence of solutions for fractional differential inclusions with integral boundary value conditions. (English) Zbl 1499.34128
Summary: In this paper, we investigate the existence of solutions for fractional boundary value problems with integral boundary value conditions in the autonomous case: \[\begin{cases}^cD^\alpha_{0^+}y(t)\in F(y(t)), \quad t\in(0,1),\\ y(0)+y'(0)=g(y),\quad \int^1_0y(t)dt=m,\\ y^{\prime \prime}(0)=y^{\prime\prime\prime}(0)=\dots=y^{(n-1)}(0)=0,\end {cases}\] where \(^cD^\alpha_{0^+}\) is the Caputo fractional derivatives, \(F:\mathbb{R}\to\mathcal{P}(\mathbb{R})\) is a multivalued map, and \(g:C([0,1],\mathbb{R})\to\mathbb{R}\) is a continuous function and \(m\in\mathbb{R}\), \(n-1<\alpha<n,\) \(n\ge 2\). By means of some standard fixed point theorems, sufficient conditions for the existence of solutions for the fractional differential inclusions with integral boundary value problems are presented. An example is presented to illustrate our main result. Our result generalizes the single known results to the multi-valued ones.
MSC:
34A60 | Ordinary differential inclusions |
34A08 | Fractional ordinary differential equations |
34B15 | Nonlinear boundary value problems for ordinary differential equations |
47N20 | Applications of operator theory to differential and integral equations |
34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
26A33 | Fractional derivatives and integrals |