Integral inequalities in two independent variables for retarded Volterra equations☆
Introduction
Integral inequalities play an important role in the qualitative analysis of differential and integral equations. Many retarded inequalities have been discovered (see [1], [2], [3], [5]) recently Lipovan [6] presented a retarded inequality which has very well characters. Others results may be found in [4], [7] now let us first recall the main results of [6] as follows: Theorem A Let with . Assume in addition that nondecreasing and for . If satisfiesthen Theorem B Let and assume that is nondecreasing with for . If satisfiesthen Theorem C Let a, be as in Theorem A. Assume are nondecreasing function with for and If satisfiesthenwhere . Theorem D Let and assume that are nondecreasing function with for . Let also be a nondecreasing function such that for and . If satisfiesthenwhere .
In this paper,we will generalize above retarded Volterra integral inequalities to more general integral inequalities with two independent variables,which can be used as tools in the theory of partial differential and integral equations with time delays.
Section snippets
Main results
Theorem 2.1 If are real valued nonnegative continuous functions defined for be continuous nondecreasing in x and y for each . are real valued continuous functions defined for . satisfiesthen Proof Let then our
Simple applications
In this section we will indicate some applications for our results to obtain the bounds on the solution of some partial differential and integral equations with time delay. To illustrate the application of our results, we establish the bound on the solution of partial integral equations of the form:andwhere all the functions are continuous on their respective domains of their definitions and
References (7)
On retarded integral inequalities and their applications
J. Math. Anal. Appl.
(2005)A retarded Gronwall-like inequality and its applications
J. Math. Anal. Appl.
(2000)- et al.
On retarded integral inequalities in two independent variables and their applications
Appl. Math. Comput.
(2006)
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2023, Australian Journal of Mathematical Analysis and ApplicationsOn some non-linear integral inequalities for two-variable functions with delay involving integration over infinite intervals
2018, WSEAS Transactions on Mathematics
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This research was supported by the NNSF of China (10771118).