A new exploration on existence of Sobolev‐type Hilfer fractional neutral integro‐differential equations with infinite delay

Abstract

This article is primarily focusing on the existence of Sobolev-type Hilfer fractional neutral integro-differential systems via measure of noncompactness. We study our primary outcomes by employing fractional calculus, measure of noncompactness and fixed point technique. First, we discuss the existence of mild solution for the fractional evolution system. Then, we extend our results to discuss the system with nonlocal conditions. Finally, we provide theoretical and practical applications to illustrate the obtained theory.

A note on approximate controllability of the Hilfer fractional neutral differential inclusions with infinite delay

“…Inspired by this article, many researchers discussed the existence, exact controllability, and approximate controllability of the Hilfer fractional differential systems with or without finite delay, one can refer previous studies 18,45,46 and the cited articles therein. Very recently, the researchers focusing the existence, exact controllability, and approximate controllability of the Hilfer fractional system with infinite delay, the readers can verify 37,38,40,47,48 . To the best of our knowledge, the approximate controllability of the Hilfer fractional neutral differential inclusions with infinite delay discussed in this article is an untreated topic in the literature and it gives the additional motivation for writing this paper.…”

“…Very recently, the researchers focusing the existence, exact controllability, and approximate controllability of the Hilfer fractional system with infinite delay, the readers can verify. 37,38,40,47,48 To the best of our knowledge, the approximate controllability of the Hilfer fractional neutral differential inclusions with infinite delay discussed in this article is an untreated topic in the literature and it gives the additional motivation for writing this paper.…”

A discussion concerning the existence results for the Sobolev-type Hilfer fractional delay integro-differential systems

“…In recent years, mathematical modeling has been upheld by fractional calculus, with a few outcomes, and fractional operators were demonstrated to be a fantastic instrument to depict the hereditary characteristics of different patterns. As of late, this blend has acquired a lot of significance, basically because fractional differential equations have become amazing assets for displaying a few complex wonders in various assorted and boundless fields of science and engineering; readers are referred to [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and articles [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37]. Hilfer [38] initiated another kind of derivative, along with Riemann-Liouville and Caputo fractional derivative.…”

A New Approach to Hyers-Ulam Stability of r -Variable Quadratic Functional Equations

“…It is worth noting that among the stability problem of functional equations, the study of the Ulam stability of different types of quadratic functional equations is an important and interesting topic, and it has attracted many scholars [13][14][15][16][17][18]. In addition, very recently, authors studied various types of stability results and have been discussed with differential equation [19][20][21][22][23][24][25][26][27][28][29]. To the best of the author's knowledge, a new approach to Hyers-Ulam stability of r-variable quadratic functional equations has not been studied so far, which motivates the present study.…”