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Some new nonlinear inequalities and applications to boundary value problems. (English) Zbl 1094.26011

The author establishes some new retarded Gronwall-Bellman-Ou-Iang type inequalities in two variables with explicit bounds on unknown functions. These results on the one hand generalize several known inequalities and on the other hand furnich a handy tool for the study of qualitative as well as quantitative properties of solutions of differential and integral equations. The obtained new inequalities are applied to study the boundedness, uniqueness and continuous dependence of the solutions of certain initial value problems for hyperbolic partial differential equations.

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B35 Stability in context of PDEs
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