Existence of Mild Solutions to Nonlocal Fractional Cauchy Problems via Compactness

Abstract

We obtain characterizations of compactness for resolvent families of operators and as applications we study the existence of mild solutions to nonlocal Cauchy problems for fractional derivatives in Banach spaces. We discuss here simultaneously the Caputo and Riemann-Liouville fractional derivatives in the cases 0 < < 1 and 1 < < 2.

Existence and Optimal Controls for Fractional Stochastic Evolution Equations of Sobolev Type Via Fractional Resolvent Operators

“…In the following, we recall some results on fractional resolvent operator family {S E α,β (t)} t≥0 , which can be found in details in [17,18].…”

Аттракторы, Затенение И Аппроксимация Абстрактных Полулинейных Дифференциальных Уравнений

“…Теорема 5.3.1 (см. [109,175,192,224]). Пусть A генератор аналитического α-разрешающего семейства S α (•, A) на банаховом пространстве E. Предположим, что функция f (•,…”

Existence Results of Mild Solutions for the Fractional Stochastic Evolution Equations of Sobolev Type

“…In the following, we establish the concept of the fractional resolvent family which is a basic concept in our main results, see [4,8] for more details. Let {Π(t)} t≥0 be a strongly continuous family of B(X).…”

“…Since fractional differential equations can describe many problems in the fields of physical, biological and chemical and so on, some properties of solutions for the fractional differential equations have been considered by many authors, see [1][2][3][4][5][6][7][8]. In [2], when the nonlinearity satisfies non-Lipschitz conditions, Wang studied the existence of mild solutions of α ∈ (0, 1)-order fractional stochastic evolution equations with Caputo derivative in abstract spaces.…”

“…By virtue of the characteristic solution operators, they obtained the exact controllability results via Schauder fixed point theorem. In [8], by using the characterizations of compact resolvent families, Ponce investigated the Cauchy problem for a class of fractional evolution equations. Furthermore, the stochastic perturbation is unavoidable in the natural systems.…”