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156) holds. 157) by the change of variables t ! t, s ! s. First, we claim that as t ! 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.

D t g. /z. 68) Define Z t g. /z. t/ Ä u0 exp t 0 Z Á Ã f . / C g. / d ÂZ t C Á Ã f . / C g. / d d : t p. t/ u0 exp Z ÂZ t C 0 p. / exp t 0 Ã Œ f . / C g. /d Ã t Œ f . / C g. 61). s/ Ã s 0 g. /u. t/ Z Ä t g. /u. t/ C t g. /Œh. / C p. /v. t/ C 0 t g. /Œh. / C p. /v. t/. t/ Ä 0 ÂZ h. /Œ f . / C g. / exp t Ã p. /Œ f . / C g. 77), we have Z t h h. /Œ f . / C g. p. /Œ f . / C g. 67). s/Œ g. /u. 2 Linear One-Dimensional Continuous Generalizations on the Gronwall-. . s/ exp . f . / C g. s/ 0 s 0 s 0 u.

S/ exp g. /f . s/ s 0 h. /x. s/ Ã s 0 h. /k. /n. 2 Linear One-Dimensional Continuous Generalizations on the Gronwall-. . t/ exp. s/ exp. s 0 h. /k. 59). This completes the proof. Pachpatte [449, 457, 460, 462] showed the following theorem. 15 (Pachpatte [449, 457, 460, 462]) Let u; f ; g and p be non-negative continuous functions defined on RC , and u0 be a non-negative constant. s/ 0 s Ã g. /u. s/ u0 exp. Z p. / exp. Z s 0 Œ f . / C g. /d / s 0 Œ f . / C g. s/ Ã s 0 Œg. /u. / C p. s/ u0 exp Œ f .