# Integral and Discrete Inequalities and Their Applications: by Yuming Qin

By Yuming Qin

This e-book specializes in one- and multi-dimensional linear essential and discrete Gronwall-Bellman type inequalities. It offers an invaluable assortment and systematic presentation of identified and new effects, in addition to many functions to differential (ODE and PDE), distinction, and essential equations. With this paintings the writer fills a spot within the literature on inequalities, providing an amazing 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 device in classical research and play a vital position in constructing the well-posedness of the similar equations, i.e., differential, distinction and quintessential equations.

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Extra info for Integral and Discrete Inequalities and Their Applications: Volume I: Linear Inequalities

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As in Kong and Zhang [317], to see the difference among the three results, we next give an example. 1 C 3x C 2x2 /2 Then Z A1 . 2 Linear One-Dimensional Continuous Generalizations on the Gronwall-. . 13 (Theorem 1 in [182]), we have Â Z 8 ˆ 1 1 ˆ ˆ < E . x/ Ä E2 . 1 C 3s C 2s2 /2 Obviously, E2 . ex The difference among the estimates are quite large. 12. s/ 0 Ã s 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/ Ã s 0 h. /k. /n. 2 Linear One-Dimensional Continuous Generalizations on the Gronwall-.

S/ exp t ! 2. 2 Linear One-Dimensional Continuous Generalizations on the Gronwall-. . s/. 0 s 0 g. /u. 45) where u0 is a non-negative constant. s/ exp Ã s . f . / C g. 45). t/ t g. /u. t/ C Ã t 0 g. /v. t/ C t 0 g. /v. t/ Ä u0 exp t 0 Ã . t/ exp Z t 0 Á . 46). To prove the next theorem, we need the following lemma. 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 Œxy: 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 ˛ Ã biC1 Ai ŒqiC1 d s biC1 Ai Œx exp Z t biC1 Ai Œx C y exp Ã biC1 Ai ŒqiC1 d ÂZ t biC1 Ai Œy exp ds ds Ã biC1 Ai ŒqiC1 d ds s D AiC1 Œx C AiC1 Œy; ÂZ t Ã 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 Ã biC1 Ai ŒqiC1 d ds y s D AiC1 Œxy: This proves that (1) and (2) are true for k D i C 1.

2 Linear One-Dimensional Continuous Generalizations on the Gronwall-. . s/ exp t Ã b. s/ exp t Ã b. 103). 3). 105) ˛ In 1962, Bykov an Salpagarov [120] proved the next two results. t; r; x/ be non-negative functions for all t r x a and c1 ; c2 , and c3 be non-negative constants not all zero. 106). t/, we may complete the proof. t/ be non-negative continuous functions in a real interval I D Œa; b. s/ exp . 110). 2 Linear One-Dimensional Continuous Generalizations on the Gronwall-. . t/; Ä . t/ Ä c exp t Ã .

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