| Summary: | In this paper we initiate the study of boundary value problems for fractional differential equations and inclusions involving <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>k</mi><mo>,</mo><mi>ϕ</mi><mo>)</mo></mrow></semantics></math></inline-formula>-Hilfer fractional derivative of order in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>]</mo></mrow></semantics></math></inline-formula>. In the single-valued case the existence and uniqueness results are established by using classical fixed-point theorems, such as Banach, Krasnoselskiĭ and Leray-Schauder. In the multivalued case we consider both cases, when the right-hand side has convex or non-convex values. In the first case, we apply the Leray–Schauder nonlinear alternative for multivalued maps, and in the second, the Covit–Nadler fixed-point theorem for multivalued contractions. All results are well illustrated by numerical examples.
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