Abstract
Let E be a Banach space with the topological dual E^*. The aim of this paper is two-fold. On the one hand, we prove some basic properties of Hadamard-type fractional integral operators. These results are related to earlier results about integral operators acting on different function spaces, but for the vector-valued case they are of independent interest. Note that we discuss it in a rather general setting. We study Hadamard–Pettis integral operators in both single and multivalued case. On the other hand, we apply these results to obtain the existence of solutions of the fractional-type problem dαx(t)dtα=λf(t,x(t)),α∈(0,1),t∈[1,e],x(1)+bx(e)=h\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\frac{d^\\alpha x(t)}{d t^\\alpha }= \\lambda f(t,x(t)), \\quad \\alpha \\in (0,1),~~t \\in [1,e], \\quad x(1)+ b x(e)=h \\end{aligned}$$\\end{document}with certain constants lambda , b, where h in E and f: [1,e]times E rightarrow E is Pettis integrable function such that, for every varphi in E^*, varphi f lies in an appropriate Orlicz spaces. Here frac{d^alpha }{d t^alpha } stands the Caputo–Hadamard fractional differential operator.
Highlights
Introduction and preliminariesThe issue of fractional calculus for real-valued functions in the context of Orlicz spaces has been studied for the first time by O’Neil [28]
We show that the Hadamard-type fractional integral operator maps the class x ∈ P[I, E]
Because our definition of Ψ differs from typical N -functions studied in the theory of Orlicz spaces, we should present an important property: Proposition 3
Summary
The issue of fractional calculus for real-valued functions in the context of Orlicz spaces has been studied for the first time by O’Neil [28]. We show that the Hadamard-type fractional integral operator maps the class x ∈ P[I , E] Such that φx belongs to some appropriate Orlicz space into the space of (weakly) continuous functions. Let P[I , E] denotes the space of E-valued Pettis integrable functions in the interval I (see [15] or [29] for the definition). Because our definition of Ψ differs from typical N -functions studied in the theory of Orlicz spaces, we should present an important property: Proposition 3 [30] The function Ψ : R+ → R+ defined by (6) is increasing and continuous with Ψ (0) = 0. Let us recall that the Pettis-integrability of strongly measurable functions is strictly related to our families of sets. The problem is much more complicated and a sufficient condition binding α and ψ will be presented in Theorem 2
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