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Consider the inequality. Note that Open image in new window satisfy Open image in new window for Open image in new window. Using Theorem 2. Then use the fact that Open image in new window and we get Corollary 3.

If Open image in new window where Open image in new window , then 3. If Open image in new window , Open image in new window , Open image in new window , 3. Equation 3. In this section, we apply Theorem 2. Then 4. Appling Theorem 2. This paper was supported by Guangdong Provincial natural science Foundation The authors would like to thank Professor Boling Guo for his great help. This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Cobweb Model As an Application of Difference Equation - Economics Live

Download PDF. Advances in Difference Equations December , Cite as. Open Access. First Online: 21 January This process is experimental and the keywords may be updated as the learning algorithm improves. Introduction Gronwall-Bellman inequalities and their various linear and nonlinear generalizations play very important roles in the discussion of existence, uniqueness, continuation, boundedness, and stability properties of solutions of differential equations and difference equations. In particular, the book [ 21 ] written by Pachpatte considered three types of discrete inequalities: Open image in new window.

Assume that C 1 Open image in new window is nonnegative for Open image in new window and Open image in new window ; C 2 Open image in new window Open image in new window are nondecreasing for Open image in new window , the range of each Open image in new window belongs to Open image in new window , and Open image in new window ; C 3 all Open image in new window Open image in new window are nonnegative for Open image in new window ; C 4 all Open image in new window Open image in new window are continuous and nondecreasing functions on Open image in new window and are positive on Open image in new window.

Suppose that Open image in new window — Open image in new window hold and Open image in new window is a nonnegative function for Open image in new window satisfying 1. Then Open image in new window. Open image in new window , Open image in new window , Open image in new window Identity , and Open image in new window is the largest positive integer such that Open image in new window.

Remark 2. By the definitions of Open image in new window and Open image in new window , it is easy to check that they are nonnegative and nondecreasing in Open image in new window , and Open image in new window and Open image in new window for each fixed Open image in new window where Open image in new window. Open image in new window in Open image in new window implies that Open image in new window for all Open image in new window.

Clearly, Open image in new window.

Sharp estimation for the solutions of delay differential and Halanay type inequalities

Assume that Open image in new window is nondecreasing in Open image in new window. Take any arbitrary positive integer Open image in new window and consider the auxiliary inequality Open image in new window. Claim that Open image in new window in 2. Let Open image in new window for Open image in new window and Open image in new window. It is clear that Open image in new window is nonnegative and nondecreasing. Observe that 2. Since Open image in new window is nondecreasing and Open image in new window , we have Open image in new window.

The definition of Open image in new window in Theorem 2. Thus the monotonicity of Open image in new window implies Open image in new window. Assume that 2. Consider Open image in new window. Let Open image in new window and Open image in new window. Then Open image in new window is nonnegative and nondecreasing and satisfies Open image in new window for Open image in new window.

Moreover, we have Open image in new window. Since Open image in new window and Open image in new window are nondecreasing in their arguments and Open image in new window , we have by the assumption Open image in new window Open image in new window. Therefore, Open image in new window.

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From the assumption Open image in new window , each Open image in new window , Open image in new window , is continuous and nondecreasing on Open image in new window and is positive on Open image in new window since Open image in new window is continuous and nondecreasing on Open image in new window. Moreover, Open image in new window. By the inductive assumption, we have Open image in new window. Note that Open image in new window. Thus, we have from 2. It implies that Open image in new window.

Thus, 2. Since 2. If Open image in new window for all Open image in new window , then Open image in new window. Let Open image in new window where Open image in new window is given in Open image in new window. Using the same arguments as in 2. Consider the inequality Open image in new window. Suppose that Open image in new window — Open image in new window hold. If Open image in new window in 3. Let Open image in new window. Then 3. Suppose that Open image in new window and the functions Open image in new window and Open image in new window in 4.

If Open image in new window is a solution of 4. Using 4. Clearly, Open image in new window for all Open image in new window since Open image in new window. For positive constants Open image in new window , we have Open image in new window. It is obvious that Open image in new window and Open image in new window satisfy Open image in new window. Applying Theorem 2. Let Open image in new window and Open image in new window be two solutions of 4.

From 4. Acknowledgments This paper was supported by Guangdong Provincial natural science Foundation Agarwal RP: On an integral inequality in Open image in new window independent variables. Journal of Mathematical Analysis and Applications , 85 1 Elsevier, Amsterdam Kiryakova, Generalized Fractional Calculus and Applications. Pitman Res. Notes in Math. Liu, N. Loi, V.

Sharp estimation for the solutions of delay differential and Halanay type inequalities

Obukhovskii, Existence and global bifurcation of periodic solutions to a class of differential variational inequalities. Miller and B. Obukhovskii and J. Yao, Some existence results for fractional functional differential equations. Fixed Point Theory 11, No 1 , Pang, D.

Discrete Halanay-type inequalities and applications

Steward, Differential variational inequalities. A , Podlubny, Fractional Differential Equations. Seidman, Invariance of the reachable set under nonlinear perturbations. SIAM J. Control Optim. Wang, D. Chena, T. Xiao, Abstract fractional Cauchy problems with almost sectorial operators.

Differential Equations , Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations. Export Citation. Here you can find all Crossref-listed publications in which this article is cited. User Account Log in Register Help. Search Close Advanced Search Help. My Content 1 Recently viewed 1 Decay solutions for a Show Summary Details. More options …. Fractional Calculus and Applied Analysis. Editor-in-Chief: Kiryakova, Virginia.

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Decay solutions for a class of fractional differential variational inequalities. Abstract Our aim is to study a new class of differential variational inequalities involving fractional derivatives. References [1] J. About the article Received : Published Online : Published in Print : Citing Articles Here you can find all Crossref-listed publications in which this article is cited. Asymptotically periodic solutions for Caputo type fractional evolution equations. DOI: Related Content Loading General note: By using the comment function on degruyter. A respectful treatment of one another is important to us.

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