Existence Results for Tempered-Hilfer Fractional Differential Problems on Hölder Spaces

Abstract

This paper considers a nonlinear fractional-order boundary value problem  HDa,gα1,β,μx(t)+f(t,x(t),HDa,gα2,β,μx(t))=0, for t∈[a,b], α1∈(1,2], α2∈(0,1], β∈[0,1] with appropriate integral boundary conditions on the Hölder spaces. Here, f is a real-valued function that satisfies the Hölder condition, and  HDa,gα,β,μ represents the tempered-Hilfer fractional derivative of order α>0 with parameter μ∈R+ and type β∈[0,1]. The corresponding integral problem is introduced in the study of this issue. This paper addresses a fundamental issue in the field, namely the circumstances under which differential and integral problems are equivalent. This approach enables the study of differential problems using integral operators. In order to achieve this, tempered fractional calculus and the equivalence problem of the studied problems are introduced and studied. The selection of an appropriate function space is of fundamental importance. This paper investigates the applicability of these operators on Hölder spaces and provides a comprehensive rationale for this choice.