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Some inequalities of Gronwall type. (English) Zbl 0553.26004

The authors prove one theorem on the integral Gronwall type inequality which unifies some recent results of B. G. Pachpatte [Indian J. Pure Appl. Math. 6, 769-772 (1975; Zbl 0385.34003), J. Math. Phys. Sci. 9, 405-416 (1975; Zbl 0321.26009), Indian J. Pure Appl. Math. 8, 1093- 1107 (1977; Zbl 0402.26008), J. Math. Anal. Appl. 51, 141-150 (1975; Zbl 0305.26010)]. The considered inequality is of the form \[ u(x)\leq a(x)+b(x)\sum^{m}_{r=1}E_ r(x,u), \] where \[ E_ r(x,u)=\int^{x}_{0}f_{r1}(x_ 1)\int^{x_ 1}_{0}f_{r2}(x_ 2)...\int^{x_{r-1}}_{0}f_{rr}(x_ r)u(x_ r)dx_ r...dx_ 1 \] \(f_{ii}(x)=f_ i(x)\), \(i\leq m\); \(f_{ji}(x)=g_ i(x)\), \(1\leq i\leq m-1\), \(i<j\leq m\). All the functions u, a, b, \(f_ i\), \(g_ i\) \((1\leq i\leq m)\) are real-valued, nonnegative and continuous on \([0,\infty).\) Both in the statement and the proof appears the imprudently defined symbol \(\sum^{k}_{r=1}b(x)f_ r(x)\cup^{t}_{i=1}g_ i(x).\) It seems that a better estimation of the unknown function u we can obtain by replacing this expression by \[ h_{k,t}(x)=\sup \{\sum^{k}_{r=1}b(x)f_ r(x),g_ 1(x),...,g_ t(x)\}. \] The authors state (no proof) Theorem 2, which is a discrete analogue of Theorem 1 under additional assumptions. For similar results see the authors’ papers [Bull. Inst. Math., Acad. Sin. 9, 235-248 (1981; Zbl 0474.26009), Appl. Math. Comput. 7, 205-224 (1980; Zbl 0455.26010)].
Reviewer: J.Popenda

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
34A40 Differential inequalities involving functions of a single real variable
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