On Gronwall like integral inequalities (Q1183955)

The present paper is devoted to the following Gronwall type nonlinear integral inequalities \[ u(t)\leq u_ 0+\int^ t_ 0p_ 1(t_ 1)\int^{t_ 1}_ 0p_ 2(t_ 2)\cdots\int_ 0^{t_{n-2}}p_{n- 1}(t_{n-1}) \int_ 0^{t_{n-1}}p(t_ n) \tag{A} \] \[ \times u(t_ n)g[\log u(t_ n)]dt_ n\cdots dt_ 2 dt_ 1 \] and \[ z(x,y)\leq z_ 0+\int^ x_ 0\int_ 0^{s_{n-1}}\cdots\int_ 0^{s_ 1}\int^ y_ 0\int_ 0^{t_{m-1}}\cdots\int _ 0^{t_ 1}h(s,t)z(s,t) \tag{B} \] \[ \times g[\log z(s,t)]dt dt_ 1\cdots dt_{m-1} ds ds_ 1\cdots ds_{n-1} \] \((s_ 0=x,t_ 0=y)\), where \(g:[0,\infty)\to[0,\infty)\) is differentiable with \(g(0)=0\) and \(g'\geq 0\); \(u:[0,T]\to[1,\infty)\); \(z:[0,a]\times[0,b]\to[1,\infty)\), and \(u_ 0\), \(z_ 0\) are constants with \(u_ 0,z_ 0\geq 1\). Under suitable conditions, explicit upper bounds on solutions of (A) and (B) are obtained. These results obtained generalise the known integral inequalities due to \textit{H. Engler} [Math. Z. 202, No. 2, 251-259 (1989; Zbl 0697.73033)] and \textit{A. Haraux} [``Nonlinear evolution equations --- global behavior of solutions'', Lect. Notes Math. 841 (1981; Zbl 0461.35002)]. Some misprints are in the right members of (2), (7) and (11). We note also that \(M[x,y,k]=M[y,x,k]\) does not hold in general unless \(m=n\).