Abstract
This paper is introduced as complementary studies based on fractional Sturm-Liouville problems in a Banach space. We explore the existence results for new considered problems which can be considered as mixture of equations and inclusions. For the sake of that, we use jointly continuous composed functions with multi-valued maps and denote this form by eq-inclusion problems. The form of the solutions is calculated by the rules of Caputo derivative and the corresponding integral. The concept "continuous image of multi-valued maps" is useful to show that the strong results will be under inclusion hypothesis. The argument and fit technicals used here consider both Lipschitz and non-Lipschitz cases with using nonlinear alternative Leray Schauder type and Covitiz and Nadler theorems.