Some weakly singular Volterra integral inequalities with maxima in two variables

Abstract

In this paper, we establish some new weakly singular Volterra type integral inequalities that include the maxima of the unknown function of two variables. We also use the results to research the boundedness of solutions to retarded nonlinear Volterra type integral equations.

New Nonlinear Retarded Integral Inequalities and Their Applications to Nonlinear Retarded Integro-Differential Equations

“…In 2020, Shakoor et al [19] improved the above results, where they generalized inequality (6) to the general form of…”

“…On the other hand, integral inequalities occupy a very privileged position in all mathematical sciences, and they have many applications to questions of the existence, stability, boundedness, uniqueness, and asymptotic behavior of the solutions of nonlinear integro-differential equations. They can be used in various problems involving symmetry (see [1][2][3][4][5][6][7]). In 1919, Gronwall [8] was the first person to introduce the following inequality (which can be used to estimate the solution of a linear differential equation):…”

“…A huge number of useful generalizations of ( 1) and (3) were given by many mathematicians and scientists after the establishment of Gronwall's inequality and the Gronwall-Bellman inequality which can be found in [4][5][6][10][11][12][13][14][15][16][17][18][19][20]. Among them, Abdeldaim and Yakout [16] in 2011 extended the inequality (3) as given below:…”

“…Recently, in 2023, Sun and Xu [6] established new weakly singular Volterra-type integral inequalities that include the maxima of the unknown function of two variables while in [5] the new retarded nonlinear integral inequalities with mixed powers were studied and utilized to study the property of boundedness and the global existence of solutions of the Volterra-type integral equations with delay. Motivated by the inequalities mentioned above, we prove more general integral inequalities with an addition of a differentiable function to replace the constant outside the integral sign.…”