A generalized retarded Gronwall-like inequality in two variables and applications to BVP

In this paper, 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, Some new nonlinear inequalities and applications to boundary value problems, Nonlinear Anal. 64 (2006) 2112–2128] and the one in [R.P. Agarwal, S. Deng, W. Zhang, Generalization of a retarded Gronwall-like inequality and its applications, Appl. Math. Comput. 165 (2005) 599–612]. We apply our result to a boundary value problem of a partial differential equation for boundedness, uniqueness and continuous dependence.