Rather than invent new modifications of the standard Lyapunov function al method to overcome the difficulties, the authors demonstrate by various examples that the contraction mapping principle can do the magic in many circumstances. Fixed Point Theory and Applications 2007, 2007:-9. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved which improves and generalizes some well-known results. Of course, there are issues with the alternative method as well, mostly centering on finding the right way to express the problem as a fixed-point problem. Replacing with in , we see there is a such that implies that the unique continuous solution agreeing with on satisfies for all. An example is given to illustrate our theory. T A Burton Six years ago I obtained a transformation which mapped integral equations into a form very suitable for fixed point methods.
In 2001, Burton and Furumochi 2001 observed some difficulties that occur in studying the stability theory of ordinary and functional differential equations by the Lyapunov's second direct method. In this paper we introduce stability by fixed point theory. An example is given to illustrate our main results. Then, the condition ii of Theorem 2. Proceedings of the Royal Society of Edinburgh. By the contraction mapping principle, has a unique fixed point in , which is a solution of 1.
At the end of the paper I give a long list of papers which used the transformation and with each paper I list the kind of equation it covers. Hence, is a contraction mapping in with constant. We obtain their differences about the stability of the zero solution, and we also improve and generalize the special case. It contains an extensive collection of new and classical examples worked in detail and presented in an elementary manner. Two examples is also given to illustrate our results.
I do not have a copy of the published version yet but I should have one in a day or two. An application to fractional partial differential equations subject to impulsive effects is given to illustrate our results. The material is highly accessible to upper-level undergraduate students in the mathematical sciences, as well as working biologists, chemists, economists, engineers, mathematicians, physicists, and other scientists using differential equations. We focus on an elementary example which is suitable for undergraduate students, a more sophisticated example with many real-world applications, and an example of a type not previously seen in stability theory. Suppose that the following conditions are satisfied: i there exists a constant such that , ii there exists a constant such that , , satisfy a Lipschitz condition with constant on an interval , iii and are odd, increasing on. Abstract In this paper, we study the mean square asymptotic stability of a class of generalized nonlinear neutral stochastic differential equations with variable time delays by using fixed point theory. Also, by ii , is a contraction mapping with contraction constant.
We deal with the Cauchy problem associated with integro-differential inclusions of diffusion-wave type involving infinite delays. Synopsis This book is the first general introduction to stability of ordinary and functional differential equations by means of fixed point techniques. In this paper we begin a study of stability theory for ordinary and functional differential equations by means of fixed point theory. Then it is available on your computer or on a flash drive. In the past years, many researchers claimed that the fixed point theory has an important advantage over the Lyapunov's direct method. Because, while the Lyapunov 's direct method usually requires pointwise conditions, fixed point theory needs average conditions, see Burton 2005 , Burton 2006 and Burton and Furumochi 2001.
We consider two nonlinear scalar delay differential equations with variable delays and give some new conditions for the boundedness and stability by means of the contraction mapping principle. Nonlinear Analysis: Theory, Methods and Applications 2005, 63 5—7 :e233-e242. Burton proposes to attack the problem in a different way, via fixed-point theory for transformations of metric spaces. Functional Differential Equations 1997, 4 3-4 :279-293. Finally, two examples are given to illustrate our results. Thus, when I look through the bibliography I can find the examples and theorem which I want. Then I can click on a file and bring it up without going to the literature.
It also introduces many research problems that promise to remain of ongoing interest. This study is mainly restricted to contraction mappings. And , then the condition iii and iv of Theorem 2. Writing in Mathematical Reviews, A. Throughout the book, Burton not only develops the theory using fixed-point methods but also compares it with the Lyapunov approach.
Let , then the condition i of Theorem 2. The examples are taken from papers which have previously appeared in the literature. As a consequence, the weak stability of zero solution is proved in case the resolvent operator is asymptotically stable. In order to prove stability at , let be given. Clearly this is why he decided to publish Stability by Fixed Point Theory with them.
It also introduces many research problems that promise to remain of ongoing interest. In Part I we obtain asymptotic stability. Because, while the Lyapunov 's direct method usually requires pointwise conditions, fixed point theory needs average conditions, see Burton 2005 , Burton 2006 and Burton and Furumochi 2001. In Dynamic Systems and Applications. Suppose that the following conditions are satisfied: 2.
Journal of Mathematical Analysis and Applications 2007, 334 1 :431-440. We study a generalized Cauchy problem associated with a class of impulsive fractional differential inclusions of Sobolev type in Banach spaces. Dynamic Systems and Applications 2001, 10 1 :89-116. In a series of papers we have studied stability properties of functional differential equations by means of fixed point theory. Dynamic Systems and Applications 2004, 13 3—4 :459—477. Suitable for advanced undergraduates and graduate students, it contains an extensive collection of new and classical examples, all worked in detail and presented in an elementary manner.