On some Nonlinear Integral Inequalities for Volterra Integral Equations

Abstract

Abstract. In this paper, we have stated and proved some results on nonlinear integral inequalities and its applications which provide an explicit bound on unknown function and can be used as a tool in the study of certain nonlinear retarded Volterra integral equations.Mathematics Subject Classification 2010: 26D10, 34K10, 35R10, 35A05, 35A2, 05C15, 06A12.

Some Gronwall{bellman Type Inequalities on Time Scales for Volterra-Fredholm Dynamic Integral Equations

“…During the past decade a number of dynamic inequalities has been established by some authors which are motivated by some applications, for example, we refer the reader to [23][24][25][26] for contributions, and the references cited therein. In this paper, we present some new nonlinear dynamic inequalities on an arbitrarily time scale T, these dynamic inequalities unify and extend the inequalities presented in [17] and [18]. Our main results will be proved by employing some useful inequalities which will be presented in Section 2.…”

“…Now we are ready to state and prove our main results, which give us the time scales version of the inequalities proved in [17] and [18].…”

“…By taking T = R in Theorem 3.2 and using the relation (2.3), it is easy to observe that the inequality obtained in Theorem 3.2 reduces to the inequality obtained byKender et al in [18, Theorem 2.2].…”

“…A fairly general version of Theorem 1.2 is given in the following theorem by Pachpatte [16]: In [17], Pachpatte established also the following inequality: for all t ∈ [a, b] ⊆ R. Kender et al [18] established the following further generalizations of the inequality (1.5) proved by of Pachpatte in [17] where he replaced the linear term of the unknown function ω by nonlinear term ω p in both sides of the inequality as following…”

On a certain retarded Volterra-Fredholm type integral inequalities in several variables and applications

“…x and q = 1, i = p, then Theorem 2.2 reduces to Theorem 2.1 in [9]. Further for x 1 fixed, c(x, s) = g(x, s) = 0, α(x) = x and q = 1, i = p, then Theorem 2.2 reduces to Theorem 2.4 in [9].…”