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A generalized nonlinear Gronwall-Bellman inequality with maxima in two variables. (English) Zbl 1266.34027

Summary: This paper deals with a generalized form of nonlinear retarded Gronwall-Bellman type integral inequality in which the maximum of the unknown function of two variables is involved. This form includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals. Requiring neither monotonicity nor separability of given functions, we apply a technique of monotonization to estimate the unknown function. Our result can be used to weaken conditions for some known results. We apply our result to a boundary value problem of a partial differential equation with maxima for uniqueness.

MSC:

34A40 Differential inequalities involving functions of a single real variable
26D10 Inequalities involving derivatives and differential and integral operators
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