Abstract
This paper deals with the existence and uniqueness of PC-mild solutions for fractional impulsive evolution equation involving nonlocal conditions and sectorial operators. We also study the nonlocal controllability of the control system governed by fractional impulsive evolution equation.
Highlights
Where J = [0, a], a > 0 is a constant, CDqt denotes the Caputo fractional derivative of order q ∈ (0, 1), A : D(A) ⊂ X → X is a sectorial operator in X, tk (k = 1, 2, . . . , m) are the constants where the impulses occur, 0 = t0 < t1 < t2 < · · · < tm < tm+1 = a, x|t=tk = x(tk+) – x(tk–), x(tk+) and x(tk–) denote the right and left limits of x at t = tk, and f, g, and Ik are given functions, which will be specified later
It is well known that the fractional derivatives are valuable tools for description of memory and hereditary properties of various materials and processes, which cannot be characterized by integer-order derivatives
The theory of fractional differential equations has emerged as an active branch of applied mathematics
Summary
The existence of mild solutions for abstract evolution equations or inclusions involving sectorial operators has been studied by many authors. Agarwal et al [1], in finite-dimensional spaces, discussed the existence of mild solutions for fractional nonlocal evolution inclusions without impulses when A is a sectorial operator. They studied the dimension of the set of mild solutions. Wang et al [37] investigated the existence of PC-mild solutions for fractional impulsive evolution inclusions with nonlocal initial conditions when A is a sectorial operator. In this paper, motivated by the results mentioned, we consider the existence and controllability of fractional nonlocal impulsive problem for abstract evolution equation (1.1).
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