Proportional Caputo Fractional Differential Inclusions in Banach Spaces

In this work, we introduce the notion of a (weak) proportional Caputo fractional derivative order α∈(0,1) for continuous (locally integrable) function u:[0,∞)→E, where E is complex Banach space. our definition, do not require that u(·) continuously differentiable, which enables us to consider wellposedness corresponding relaxation problems in much better theoretical way. More precisely, systematically investigate several new classes (degenerate) solution operator families connected with use type derivatives, obeying multivalued linear approach abstract Volterra integro-differential inclusions. The quasi-periodic properties integrals as well existence and uniqueness almost periodic-type solutions various differential inclusions spaces are also considered.