Generalized fractional calculus in Banach spaces and applications to existence results for boundary value problems

Abstract In this paper, we present the definitions of fractional integrals and derivatives a Pettis integrable function with respect to another function. This concept follows idea Stieltjes-type operators should allow us study using methods known from measure differential equations in abstract spaces. We will show that some well-known properties calculus for space Lebesgue functions also hold true particular, prove general Goebel–Rzymowski lemma De Blasi weak noncompactness our integrals. suggest new definition Caputo derivative function, which allows investigate existence solutions Caputo-type boundary value problems. As deal functions, main tool utilized considerations is based on technique measures Mönch’s fixed-point theorem. Finally, encompass full scope research, examples illustrating results are given.