Nonlinear integral inequality in two independent variables (Q1098956)

The authors establish a new generalization of the well known nonlinear integral inequality due to I. Bihari for the case of two independent variables. The main result, which contains several known integral inequalities, can be restated as follows: Theorem. Let \(u(x,y)\in C(J\times K,R_+)\); \(g_ 1\) and \(g_ 1\in C(R_+,R_+)\) (where \(J=[0,X],\quad K=[0,Y],\quad R_+=[0,\infty))\) are nondecreasing and nonnegative, with \(g_ 1\) submultiplicative and \(g_ 2(u)/v\leq g_ 2(u/v)\) for all \(u\in R_+,\) \(v\geq 1.\) If there exists a \(u_ 0>0\) such that \(g_ 1,g_ 2\) are positive when \(u\geq u_ 0,\) then for any nonnegative constants a, b, and c, with \(c\geq 1,\) the inequality \[ u(x,y)\leq c+a\int^{x}_{0}g_ 1(u(s,y))ds+b\int^{y}_{0}g_ 2(u(x,t))dt, \] (x,y)\(\in J\times K\), implies, on a nonempty rectangle containing the point (0,0), the inequality \[ u(x,y)\leq G_ 1^{-1}\{G_ 1(c)+axg_ 1[G_ 2^{- 1}(G_ 2(1)+by)]\}G_ 2^{-1}(G_ 2(1)+by), \] where \(G_ i^{-1}\) denotes the inverse of \(G_ i\), \(i=1,2\), and \[ G_ i(u):=\int^{u}_{u_ 0}\frac{dt}{g_ i(t)},\quad u\geq 0,\quad u_ 0>0. \] An upper bound on the solution of a characteristic initial value problem of a certain nonlinear partial differential equation is also derived with the aid of the last integral inequality. Note that the function \(g_ 2^{-1}\) in the inequality (3.2) of the paper should be replaced by \(G_ 2^{-1}\).