Abstract
This paper is concerned with the existence and uniqueness of mild solution of the fractional differential equations with nonlocal conditions , in a Banach space , where . General existence and uniqueness theorem, which extends many previous results, are given.
Highlights
The fractional differential equations can be used to describe many phenomena arising in engineering, physics, economy, and science, so they have been studied extensively see, e.g., 1–8 and references therein
The nonlocal condition x 0 g x x0 can be applied in physics with better effect than that of the classical initial condition x 0 x0
In this paper, motivated by 1–7, 9–15 especially the estimating approach given by Xiao and Liang 14, we study the semilinear fractional differential equations with nonlocal condition 1.1 in a Banach space X, assuming that the nonlinear map f is defined on 0, T × Xα × Xα and g is defined on C 0, T, Xα where Xα D Aα, for 0 < α < 1, the domain of the fractional power of A
Summary
−Ax t f t, x t , Gx t , t ∈ 0, T , 1.1 x 0 g x x0, where 0 < q < 1, T > 0, and −A generates an analytic compact semigroup {S t }t≥0 of uniformly bounded linear operators on a Banach space X. In this paper, motivated by 1–7, 9–15 especially the estimating approach given by Xiao and Liang 14 , we study the semilinear fractional differential equations with nonlocal condition 1.1 in a Banach space X, assuming that the nonlinear map f is defined on 0, T × Xα × Xα and g is defined on C 0, T , Xα where Xα D Aα , for 0 < α < 1, the domain of the fractional power of A.
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