Proportional Caputo Fractional Differential Inclusions in Banach Spaces

In this work, we introduce the notion of a (weak) proportional Caputo fractional derivative of order (Formula presented.) for a continuous (locally integrable) function (Formula presented.), where E is a complex Banach space. In our definition, we do not require that the function (Formula presented.) 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.