Unique Solution for Multi-point Fractional Integro-Differential Equations

Abstract

Abstract We study a fractional integro-differential equation subject to multi-point boundary conditions: D 0 + α u ( t ) + f ( t , u ( t ) , T u ( t ) , S u ( t ) ) = b ,   t ∈ ( 0 , 1 ) , u ( 0 ) = u ′ ( 0 ) = ⋯ = u ( n − 2 ) ( 0 ) = 0 , D 0 + p u ( t ) | t = 1 = ∑ i = 1 m a i D 0 + q u ( t ) | t = ξ i , $$\left\{\begin{array}{l} D^\alpha_{0^+} u(t)+f(t,u(t),Tu(t),Su(t))=b,\ t\in(0,1),\\u(0)=u^\prime(0)=\cdots=u^{(n-2)}(0)=0,\\ D^p_{0^+}u(t)|_{t=1}=\sum\limits_{i=1}^m a_iD^q_{0^+}u(t)|_{t=\xi_i},\end{array}\right.$$ where α ∈ ( n − 1 , n ] ,   n ∈ N ,   n ≥ 3 ,   a i ≥ 0 ,   0 < ξ 1 < ⋯ < ξ m ≤ 1 ,   p ∈ [ 1 , n − 2 ] ,   q ∈ [ 0 , p ] , b > 0 $\alpha\in (n-1,n],\ n\in \textbf{N},\ n\geq 3,\ a_i\geq 0,\ 0<\xi_1<\cdots<\xi_m\leq 1,\ p\in [1,n-2],\ q\in[0,p],b>0$ . By utilizing a new fixed point theorem of increasing ψ − ( h , r ) − $\psi-(h,r)-$ concave operators defined on special sets in ordered spaces, we demonstrate existence and uniqueness of solutions for this problem. Besides, it is shown that an iterative sequence can be constructed to approximate the unique solution. Finally, the main result is illustrated with the aid of an example.