Results for Retarded Nonlinear Integral Inequalities with Mixed Powers and Their Applications to Delay Integrodifferential Equations

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

“…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:…”

“…In these perspectives, retarded integral inequalities were introduced, where non-retarded argument r is modified into retarded argument ϑ(r). In 2015, the following inequality was studied in [12] in which the retarded case of inequality (5) was obtained by replacing r by a function ϑ(r):…”

“…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.…”