On the Solvability of Equations with a Distributed Fractional Derivative Given by the Stieltjes Integral

Abstract

Linear equations in Banach spaces with a distributed fractional derivative given by the Stieltjes integral and with a closed operator A in the right-hand side are considered. Unlike the previously studied classes of equations with distributed derivatives, such kinds of equations may contain a continuous and a discrete part of the integral, i.e., a standard integral of the fractional derivative with respect to its order and a linear combination of fractional derivatives with different orders. Resolving families of operators for such equations are introduced into consideration, and their properties are studied. In terms of the resolvent of the operator A, necessary and sufficient conditions are obtained for the existence of analytic resolving families of the equation under consideration. A perturbation theorem for such a class of operators is proved, and the Cauchy problem for the inhomogeneous equation with a distributed fractional derivative is studied. Abstract results are applied for the research of the unique solvability of initial boundary value problems for partial differential equations with a distributed derivative with respect to time.

A Class of Quasilinear Equations with Distributed Gerasimov–Caputo Derivatives

“…is the resolvent set of A. Define a class A W (θ 0 , a 0 ) (see [29]) of all operators A ∈ Cl(Z ), such that: (i) there exist θ 0 ∈ (π/2, π], a 0 ≥ 0, such that W(λ) ∈ ρ(A) for every λ ∈ S θ 0 ,a 0 ; (ii) for every θ ∈ (π/2, θ 0 ), a > a 0 there exists K(θ, a) > 0, such that for all λ ∈ S θ,a…”

“…were obtained in [28]. All these results were generalized and combined in general formulations with the Riemann-Stieltjes integral in the definition of the distributed derivative [29].…”

“…Consequently, the left-hand side of (2) has the form Each result in the listed works [23][24][25][26][27][28][29] on the linear homogeneous equation is accompanied by theorems on the solvability of the corresponding linear inhomogeneous equation. Here, such theorems are used for the study of the Cauchy problem (1) to quasilinear Equation (2).…”