On Strongly Continuous Resolving Families of Operators for Fractional Distributed Order Equations

Abstract

The aim of this work is to find by the methods of the Laplace transform the conditions for the existence of a strongly continuous resolving family of operators for a linear homogeneous equation in a Banach space with the distributed Gerasimov–Caputo fractional derivative and with a closed densely defined operator A in the right-hand side. It is proved that the existence of a resolving family of operators for such equation implies the belonging of the operator A to the class CW(K,a), which is defined here. It is also shown that from the continuity of a resolving family of operators at t=0 the boundedness of A follows. The existence of a resolving family is shown for A∈CW(K,a) and for the upper limit of the integration in the distributed derivative not greater than 2. As corollary, we obtain the existence of a unique solution for the Cauchy problem to the equation of such class. These results are used for the investigation of the initial boundary value problems unique solvability for a class of partial differential equations of the distributed order with respect to time.

Analytic Resolving Families for Equations with Distributed Riemann–Liouville Derivatives

“…Necessary and sufficient conditions on a closed operator A for the existence of an analytic in a sector resolving operators family are obtained for homogeneous Equation (1) with the Gerasimov-Caputo distributed derivative in [17] with c ∈ (0, 1] and in [18] with c > 1. In [19] analogous result was obtained for Equation (1) with a discretely distributed Gerasimov-Caputo derivative; Reference [20] is devoted to the existence issues for strongly continuous resolving operators family of the homogeneous Equation ( 1) with the Gerasimov-Caputo derivative. The obtained results on resolving operators families allowed, in [17][18][19][20], the research of the unique solvability of inhomogeneous Equation (1) and to investigate some properties of the equation, such as the continuity in the operator norm at zero of a resolving family, conditions for the boundedness of a generating operator A, a perturbation theorem for a class of generators A and others.…”

Integrated Resolving Functions for Equations with Gerasimov–Caputo Derivatives

“…is a mild solution to problem (10) and (11). Its uniqueness can be proven analogously to the homogeneous case.…”

“…Remark 2. Often, the family of operators {S(t) ∈ L(Z ) : t ≥ 0} from Definition 1 is called a solution operator [9] (p. 20, Definition 2.3) or a resolving family of operators [11][12][13][14]. The second option seems more convenient to us.…”

“…By a classical solution of ( 10) and ( 11), we understand a function z (10) hold and equality (11) is valid for all t ∈ [0, T].…”

“…Such advantages include the possibility of studying wide classes of initial boundary value problems for partial differential equations and systems of equations within one class of initial problems for an equation in a Banach space: proving the existence and uniqueness of the solution, obtaining a representation of a solution in the linear case and an approximate solution in the nonlinear one, etc. The technique of resolving families of operators (resolving functions) is used successfully when studying integro-differential equations [5,6], integral evolution equations [7,8], various fractional differential equations [9][10][11][12][13][14].…”