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A new nonlinear integral inequality of Wendroff type with continuous and weakly singular kernel and its application. (English) Zbl 1254.26024
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 [B. G. Pachpatte, Soochow J. Math. 31, No. 2, 261–271 (2005; Zbl 1076.26014)].
The main technique used in the proof is based on the following estimate: under suitable conditions on \(u\), \(a\), \(b\), \(f\), and \(\omega\), if \[ u(x,y)\leq a(x,y)+b(x,y)\int_0^x\int_0^yf(s,t)\omega(u(s,t))\,dtds \] holds, then \[ u(x,y)\leq G^{-1}\bigg[G(a(x,y))+b(x,y)\int_0^x\int_0^yf(s,t)\,dtds\bigg], \] where \(G(r)=\int_{r_0}^r\frac{ds}{\omega(s)}.\)

MSC:
26D07 Inequalities involving other types of functions
26D15 Inequalities for sums, series and integrals
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