On a novel fractional Caputo-derivative Orlicz space

TL;DR

This paper introduces a new fractional Caputo-derivative Orlicz space, explores its properties, and applies it to establish the existence of solutions for fractional problems using variational methods.

Contribution

It defines a novel fractional Orlicz space based on Caputo derivatives and investigates its fundamental properties and embeddings, extending classical fractional calculus results.

Findings

The space is reflexive, complete, and separable.

Established continuous embeddings into Orlicz and continuous function spaces.

Proved existence of nontrivial solutions for fractional problems using mountain-pass theorem.

Abstract

In this work, we aim to explore whether a novel type of fractional space can be defined using Orlicz spaces and fractional calculus. This inquiry is fruitful, as extending classical results to new contexts can lead to a better and deeper understanding of those classical results. Our main objective is to introduce a new fractional Caputo-derivative Orlicz space, denoted by $\mathcal{O}^{\alpha,G}(\Lambda, \mathbb{R}) $. We are interested in several qualitative properties of this space, such as reflexivity, completeness, and separability. Additionally, we establish a continuous embedding results of this space into a suitable Orlicz space and the space of continuous functions. As an application, we use the mountain-pass theorem (MPT) to ensure the existence of a nontrivial weak solution for a new class of fractional-type problems in Orlicz space.