A generalized retarded Gronwall-like inequality in two variables and applications to BVP. (English) Zbl 1193.26014

Summary: We establish a generalized retarded integral inequality of Gronwall-like type in two variables, which includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals without assumption of monotonicity. Using our result we can solve both the integral inequality in [W.S. Cheung, Nonlinear Anal., Theory Methods Appl. 64, No. 9 (A), 2112–2128 (2006; Zbl 1094.26011)] and the one in [R.P. Agarwal, S. Deng, W. Zhang, Appl. Math. Comput. 165, No. 3, 599–612 (2005; Zbl 1078.26010)]. We apply our result to a boundary value problem of a partial differential equation for boundedness, uniqueness and continuous dependence.


26D10 Inequalities involving derivatives and differential and integral operators
35G30 Boundary value problems for nonlinear higher-order PDEs
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