A new nonlinear integral inequality of Wendroff type with continuous and weakly singular kernel and its application (Q2914883)

The author obtains some new explicit bounds for functions in two variables satisfying a nonlinear integral inequality of Wendroff type, extending previous results proved in [\textit{B. G. Pachpatte}, Soochow J. Math. 31, No. 2, 261--271 (2005; Zbl 1076.26014)].NEWLINENEWLINEThe main technique used in the proof is based on the following estimate: under suitable conditions on \(u\), \(a\), \(b\), \(f\), and \(\omega\), if NEWLINE\[NEWLINE u(x,y)\leq a(x,y)+b(x,y)\int_0^x\int_0^yf(s,t)\omega(u(s,t))\,dtds NEWLINE\]NEWLINE holds, then NEWLINE\[NEWLINE u(x,y)\leq G^{-1}\bigg[G(a(x,y))+b(x,y)\int_0^x\int_0^yf(s,t)\,dtds\bigg], NEWLINE\]NEWLINE where \(G(r)=\int_{r_0}^r\frac{ds}{\omega(s)}.\)