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.
The presented article is deduced about the positive solutions of the fractional differential inclusion at resonance on the half line. The fractional derivative used is in the sense of Riemann–Liouville and the problem is supplemented by unseparated conditions. The existence results are illustrated in view of Leggett–Williams theorem due to O’Regan and Zima on unbounded domain.
The monotonicity of multi-valued operators serves as a guideline to prove the existence of the results in this article. This theory focuses on the existence of solutions without continuity and compactness conditions. We study these results for the (k,n−k) conjugate fractional differential inclusion type with λ>0,1≤k≤n−1.
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