On recent developments in the theory of abstract differential equations with fractional derivatives

Section: Q X(t) = Ax(t) + F T X(t)mentioning

confidence: 99%

Existence and uniqueness of mild solutions to initial value problems for fractional evolution equations

Sin¹,

In²,

Kim

2018

“…Author details 1 Department of Automation, China University of Petroleum (Beijing), Beijing, 102249, China. 2 School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China.…”

Section: Competing Interestsmentioning

confidence: 99%

Approximate controllability and optimal controls of fractional dynamical systems of order 1 < q < 2 in Banach spaces

Qin¹,

Zuo²,

Liu³

et al. 2015

This paper investigates the approximate controllability and optimal controls of fractional dynamical systems of order 1 < q < 2 in Banach spaces. We research a class of fractional dynamical systems governed by fractional integrodifferential equations with nonlocal initial conditions. Using the Krasnosel'skii fixed point theorem and the Schauder fixed point theorem, the approximate controllability results are obtained under two cases of the nonlinear term. We also present the existence results of optimal pairs of the corresponding fractional control systems with a Bolza cost function. Finally, an application is given to illustrate the effectiveness of our main results. MSC: 26A33; 49J15; 49K27; 93B05; 93C25

“…In recent decades, there has been a lot of interest in this type of problems, its applications and various generalizations (cf. e.g., [8][9][10][11] and references therein). It is significant to study this class of problems, because, in this way, one is more realistic to describe the memory and hereditary properties of various materials and processes (cf.…”

Section: K(t − S)g(s U(s) U(κ 2 (S)))ds T ∈ [0 T] U(0) + H(u)mentioning

confidence: 99%

Abstract fractional integro-differential equations involving nonlocal initial conditions in α-norm

Wang

Chen

2011

In the present paper, we deal with the Cauchy problems of abstract fractional integro-differential equations involving nonlocal initial conditions in a-norm, where the operator A in the linear part is the generator of a compact analytic semigroup. New criterions, ensuring the existence of mild solutions, are established. The results are obtained by using the theory of operator families associated with the function of Wright type and the semigroup generated by A, Krasnoselkii's fixed point theorem and Schauder's fixed point theorem. An application to a fractional partial integrodifferential equation with nonlocal initial condition is also considered. Mathematics subject classification (2000) 26A33, 34G10, 34G20