Abstract
We obtained results on the existence and uniqueness of a mild solution for a fractional-order semi-linear differential inclusion in a Hilbert space whose right-hand side contains an unbounded linear monotone operator and a Carathéodory-type multivalued nonlinearity satisfying some monotonicity condition in the phase variables. We used the Yosida approximations of the linear part of the inclusion, the method of a priori estimates of solutions, and the topological degree method for condensing vector fields. As an example, we considered the existence and uniqueness of a solution to the Cauchy problem for a system governed by a perturbed subdifferential inclusion of a fractional diffusion type.
Highlights
Fractional DerivativeWe will recall some notions and definitions that we will need (details can be found, e.g., in [1,2,3,28])
Research Center for Interneural Computing, China Medical University Hospital, China Medical University, Abstract: We obtained results on the existence and uniqueness of a mild solution for a fractional-order semi-linear differential inclusion in a Hilbert space whose right-hand side contains an unbounded linear monotone operator and a Carathéodory-type multivalued nonlinearity satisfying some monotonicity condition in the phase variables
In the fifth section, we present the existence and uniqueness of a solution to the Cauchy problem for a system governed by a perturbed subdifferential inclusion of a fractional diffusion type as an example
Summary
We will recall some notions and definitions that we will need (details can be found, e.g., in [1,2,3,28]). The Caputo fractional derivative of the order q ∈ (0, 1) of a continuous function q g : [0, a] → E is the function C D0 g defined in the following way: D0 g(t) = D q ( g(·) − g(0)) (t), provided that the right-hand side of this equality is well defined. By a solution of this problem, we mean a continuous function x : [0, T ] → R satisfying condition (11) whose fractional derivative C D q x is continuous and satisfies Equation (10). It is known (see [1], Example 4.9).
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