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T/; Ä . t/ Ä c exp t Ã . t/ exp Ã . s/ exp . 112), we have the desired result. 111), corresponding to an estimate in Theorem 1, was obtained in [445]. t; r/ be non-negative and continuous for a Ä r Ä t, and let c1 ; c2 ; and c3 be non-negative. s; r/dr Ä a a Ã Z tÂ Z exp c3 v. / . / C s o ds : w. 120) a Proof The proof is left to the reader as an exercise. Pachpatte [451] showed the next result. s/ s 0 f3 . /x. s/ 0 ÂZ t Ä Z f1 . / C f2 . / exp 0 0 s 0 f3 . /p. s/ s 0 f3 . /x. 122) follows readily.

111), corresponding to an estimate in Theorem 1, was obtained in [445]. t; r/ be non-negative and continuous for a Ä r Ä t, and let c1 ; c2 ; and c3 be non-negative. s; r/dr Ä a a Ã Z tÂ Z exp c3 v. / . / C s o ds : w. 120) a Proof The proof is left to the reader as an exercise. Pachpatte [451] showed the next result. s/ s 0 f3 . /x. s/ 0 ÂZ t Ä Z f1 . / C f2 . / exp 0 0 s 0 f3 . /p. s/ s 0 f3 . /x. 122) follows readily. t/ be also nondecreasing. 2 Linear One-Dimensional Continuous Generalizations on the Gronwall-.

T/ ! 0. t/; then the boundedness of u implies that ı is finite. t/ Ä Â 1 ı. 159) Note that as t ! 159) < ı, this is a contradiction. Hence ı D 0, and as t ! t/ ! 0. s/, then as t ! t/ ! 0. t1 /. 158). In the sequel, we shall consider the following more generalized integral inequalities ! t/ Ä a C e Z Cd ˛t m X e t ! 162) 48 1 Linear One-Dimensional Continuous Integral Inequalities To generalize the above theorem, we need the following three lemmas. t/ WD u. t/ D a C e m X ˛t ! s/; where bN i D .