On a Caputo conformable inclusion problem with mixed Riemann–Liouville conformable integro-derivative conditions
Advances in Difference Equations
We discuss some existence criteria for a new category of the Caputo conformable differential inclusion furnished with four-point mixed Riemann-Liouville conformable integro-derivative boundary conditions. In this way, we employ some analytical techniques on α-ψ-contractive mappings and operators having the approximate endpoint property to reach desired theoretical results. Finally, we provide an example to illustrate our last main result. MSC: Primary 34A08; secondary 34A12 Keywords:
... eywords: α-ψ-contractive mappings; Approximate endpoint property; Mixed integro-derivative conditions; The Caputo conformable derivative Introduction From a long time ago, human beings have been thinking about finding the secrets and phenomena of the world around in order to be able to answer some questions. For this reason, by increasing its knowledge, the mankind invoked new logical and computational tools. The mathematical operators are one of these useful tools for modeling natural processes in the world. Over the years, mathematicians have introduced various operators for different models, but since fractional order modelings are more accurate than integer order ones, new fractional operators have been defined for this purpose today. In the meantime, the Caputo and the Riemann-Liouville fractional operators have been used more than other operators for complicated fractional modelings (see, for example,      ). Lately, the Hadamard and Caputo-Hadamard fractional operators have been introduced by some researchers and then different modelings have been studied using these operators (see, for instance,     ). In 2015, Caputo and Fabrizio  presented a new fractional derivative without singular kernel entitled fractional Caputo-Fabrizio operator, and in the same year, Losada and Nieto  investigated some properties of this new fractional operator. Some flexible properties of this nonsingular operator led to numerous papers on the various fractional modelings in this regard (see, for example,    ). Following this path, Abdeljawad  developed the concepts introduced in  and investigated some properties of the well-behaved conformable fractional derivatives. In a © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material.