Proportional Caputo Fractional Differential Inclusions in Banach Spaces
Date Issued
2022-09-18
Author(s)
A. Rahmani, W-S. Du, M. T. Khalladi, M. Kostic, D. Velinov
DOI
https://doi.org/10.3390/ sym14091941
Abstract
In this work, we introduce the notion of a (weak) proportional Caputo fractional derivative
of order a 2 (0, 1) for a continuous (locally integrable) function u : [0,¥) ! E, where E is a complex
Banach space. In our definition, we do not require that the function u( ) is continuously differentiable,
which enables us to consider the wellposedness of the corresponding fractional relaxation problems
in a much better theoretical way. More precisely, we systematically investigate several new classes of
(degenerate) fractional solution operator families connected with the use of this type of fractional
derivatives, obeying the multivalued linear approach to the abstract Volterra integro-differential
inclusions. The quasi-periodic properties of the proportional fractional integrals as well as the
existence and uniqueness of almost periodic-type solutions for various classes of proportional Caputo
fractional differential inclusions in Banach spaces are also considered.
of order a 2 (0, 1) for a continuous (locally integrable) function u : [0,¥) ! E, where E is a complex
Banach space. In our definition, we do not require that the function u( ) is continuously differentiable,
which enables us to consider the wellposedness of the corresponding fractional relaxation problems
in a much better theoretical way. More precisely, we systematically investigate several new classes of
(degenerate) fractional solution operator families connected with the use of this type of fractional
derivatives, obeying the multivalued linear approach to the abstract Volterra integro-differential
inclusions. The quasi-periodic properties of the proportional fractional integrals as well as the
existence and uniqueness of almost periodic-type solutions for various classes of proportional Caputo
fractional differential inclusions in Banach spaces are also considered.
Subjects