Research Article Some Weakly Singular Integral Inequalities and Their Applications to Tempered Fractional Differential Equations Abdellatif Ben Makhlouf , 1 Djalal Boucenna , 2 A.M. Nagy , 3,4 and Lassaad Mchiri 5 1 Department of Mathematics, College of Science, Jouf University, P.O. Box: 2014, Sakaka, Saudi Arabia 2 High School of Technological Teaching, Enset, Skikda, Algeria 3 Faculty of Science, Department of Mathematics, Benha University, Benha, Egypt 4 Mathematics Department, Faculty of Science, Kuwait University, Safat 13060, Kuwait 5 Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Sfax, Tunisia Correspondence should be addressed to A.M. Nagy; abdelhameed_nagy@yahoo.com Received 7 February 2022; Revised 14 March 2022; Accepted 17 March 2022; Published 27 April 2022 Academic Editor: Ching-Feng Wen Copyright © 2022 Abdellatif Ben Makhlouf et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we establish several new weakly singular integral inequalities that generalize several previously known ones. Several applications for fractional differential equations in the Caputo context have been derived using tempered fractional differential equations. 1. Introduction Integral inequalities are important in the qualitative study of differential and integral equations, see [1, 2]. e Gronwall inequality defines explicit bounds on the solutions of a class of integral inequalities. is inequality has been refined and used in a variety of contexts [1, 3–5]. However, none of the preceding inequalities are directly applicable to the analysis of integral equations with weakly singular kernels. e author in [6] proposed a new approach for solving linear Integral Inequalities with weakly Singular Kernels (IISKs) in 1981. Medved developed a novel technique for solving Henry–Gronwall type integral inequalities and their Bihari version [7] as well as global solutions to semilinear evolution equations [8]. IISKs have recently received increased re- search attention due to the growth of fractional differential equations, see for example [3, 5–12]. Indeed, Medved and Ma as well as Pecaric have investigated the following integral inequalities in [7, 11]: g(ξ ) ≤ a(ξ )+ b(ξ)  ξ 0 (ξ - κ) c- 1 v(κ)g(κ)dκ. (1) Medved has studied the Henry-type integral inequalities in [7, 8]: g(ξ ) ≤ a(ξ )+  ξ 0 (ξ - κ) c- 1 v(κ)w(g(κ))dκ. (2) Later, in [5], Zhu has described the integral inequalities (1) and (2) and, as an application, the existence of solutions to fractional differential equations with Caputo derivative has been studied. Tempered fractional calculus is a broadening of frac- tional calculus defined by integrals in which the kernel begins a fractional power function multiplied by an expo- nential factor [13, 14]. ere are more applications for tempered differential equations and tempered fractional derivatives, such as finance [15], poroelasticity [16] and geophysical flows [17]. e main contribution of this work, which was inspired by Zhu’s technique [5], is the description of certain weakly singular integral inequalities as well as the Ulam stability of the following issue as an application, Hindawi Journal of Mathematics Volume 2022, Article ID 1682942, 9 pages https://doi.org/10.1155/2022/1682942