By Yuming Qin
This e-book specializes in one- and multi-dimensional linear critical and discrete Gronwall-Bellman type inequalities. It presents an invaluable assortment and systematic presentation of recognized and new effects, in addition to many functions to differential (ODE and PDE), distinction, and fundamental equations. With this paintings the writer fills a niche within the literature on inequalities, providing a terrific resource for researchers in those topics.
The current quantity is a component 1 of the author’s two-volume paintings on inequalities.
Integral and discrete inequalities are an important software in classical research and play a very important position in developing the well-posedness of the similar equations, i.e., differential, distinction and necessary equations.
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Additional info for Integral and Discrete Inequalities and Their Applications: Volume I: Linear Inequalities
Example text
T/ is non-decreasing. Proof Clearly, conclusions (1) and (2) are true for k D 0. , Ai Œx C y D Ai Œx C Ai Œy; Ai Œxy Ä Ai Œxy: Then Z AiC1 Œx C y D Ai Œx C y C Ai ŒqiC1 Z D Ai Œx C Ai ŒqiC1 t ˛ t ˛ ÂZ ÂZ t s CAi Œy C Ai ŒqiC1 t ˛ à biC1 Ai ŒqiC1 d s biC1 Ai Œx exp Z t biC1 Ai Œx C y exp à biC1 Ai ŒqiC1 d ÂZ t biC1 Ai Œy exp ds ds à biC1 Ai ŒqiC1 d ds s D AiC1 Œx C AiC1 Œy; ÂZ t à Z t AiC1 Œxy D Ai Œxy C Ai ŒqiC1 biC1 Ai Œxy exp biC1 Ai ŒqiC1 d ds ˛ Z Ä Ai Œx C Ai ŒqiC1 t ˛ ÂZ s t biC1 Ai Œx exp à biC1 Ai ŒqiC1 d ds y s D AiC1 Œxy: This proves that (1) and (2) are true for k D i C 1.
S/ exp 0 à s f . /d 0 Ä ÂZ à Z s 1C g. / exp Œg. / C h. s/ 0 ÂZ s g. /p. / ÂZ t 0 0 h. /u. /d à s g. /u. s/ exp f . /p. /d Z k. /Œ f . / C g. / C g. /p. / ÂZ 0 k. /Œ f . / C g. / C h. / 0 à p. /Œ f . / C g. / C h. 15. 2). 86) ˛ where a is a constant. 88) ˛ where b 0 and a are constants. 2 Linear One-Dimensional Continuous Generalizations on the Gronwall-. . s/ds is a non-decreasing function in J. 4. 12). s/ exp Rt ˛ ÂZ ˛Ätġ à b. 93). t/ exp ˛ b. /d Z ÂZ t C à t b. t/ Ä a exp ˛ b. /d Z ÂZ t C b.
12. s/ 0 às h. /x. 55) where x0 is a non-negative constant. s/ exp s h. /k. s/ exp g. /f . s/ s 0 h. /x. s/ às 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 à g. /u. s/ u0 exp. Z p. / exp.