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.
“…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.…”
Section: Introductionmentioning
confidence: 65%
“…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].…”
Section: )mentioning
confidence: 91%
“…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].…”
mentioning
confidence: 84%
“…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…”
In this paper, we prove several new explicit estimations for the solutions of some classes of nonlinear dynamic inequalities of Gronwall-Bellman-Pachpatte type on time scales. Our results formulate some integral and discrete inequalities discussed in the literature as special cases and extend some known dynamic inequalities on time scales. The inequalities given here can be used in the analysis of the qualitative properties of certain classes of dynamic equations on time scales. Some examples are presented to demonstrate the applications of our results.
“…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.…”
Section: Introductionmentioning
confidence: 65%
“…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].…”
Section: )mentioning
confidence: 91%
“…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].…”
mentioning
confidence: 84%
“…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…”
In this paper, we prove several new explicit estimations for the solutions of some classes of nonlinear dynamic inequalities of Gronwall-Bellman-Pachpatte type on time scales. Our results formulate some integral and discrete inequalities discussed in the literature as special cases and extend some known dynamic inequalities on time scales. The inequalities given here can be used in the analysis of the qualitative properties of certain classes of dynamic equations on time scales. Some examples are presented to demonstrate the applications of our results.
“…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].…”
In the present paper some new explicit bounds on solutions to a class of new nonlinear retarded integral inequalities of Volterra-Fredholm type of several variables are established. The results obtained extend some known results in the literature and can be used as useful tools in the analysis of the qualitative properties of solutions of certain integral and differential equations. An application is given to illustrate the usefulness of our results to the boundedness and the uniqueness of the solutions of Volterra-Fredholm type integral equation with delay in n variables.
MSC 2010: 26D15; 45B05; 45D05.
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