Báo cáo sinh học: Existence of solutions for perturbed abstract measure functional differential equations

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Báo cáo sinh học: " Existence of solutions for perturbed abstract measure functional differential equations"Advances in DifferenceEquations This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Existence of solutions for perturbed abstract measure functional differential equations Advances in Difference Equations 2011, 2011:67 doi:10.1186/1687-1847-2011-67 Xiaojun Wan (wanxiaojun508@sina.com) Jitao Sun (sunjt@sh163.net) ISSN 1687-1847 Article type Research Submission date 5 August 2011 Acceptance date 23 December 2011 Publication date 23 December 2011 Article URL http://www.advancesindifferenceequations.com/content/2011/1/67 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Advances in Difference Equations go to http://www.advancesindifferenceequations.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2011 Wan and Sun ; licensee Springer.This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Existence of solutions for perturbed abstract measure func-tional differential equationsXiaojun Wan and Jitao Sun∗Department of Mathematics, Tongji University, Shanghai 200092, ChinaEmail: XW: wanxiaojun508@sina.com; ∗ sunjt@sh163.net;∗ Corresponding authorAbstractIn this article, we investigate existence of solutions for perturbed abstract measure functional differential equations.Based on the Arzela–Ascoli theorem and the fixed point theorem, we give sufficient conditions for existence of `solutions for a class of perturbed abstract measure functional differential equations. Our system includes thesystems studied in some previous articles as special cases and our sufficient conditions for existence of solutionsare less conservative. An example is given to illustrate the effectiveness of our existence theorem of solutions.1 IntroductionAbstract measure differential equations are more general than difference equations, differential equations,and differential equations with impulses. The study of abstract measure differential equations was initiatedby Sharma [1] in 1970s. From then on, properties of abstract measure differential equations have been 1researched by various authors. But up to now, there were only some limited results on abstract measuredifferential equations can be found, such as existence [2–6], uniqueness [2, 3, 5], and extremal solutions [3,4, 6]. There were also several researches on abstract measure integro-differential equations [7, 8]. The studyon abstract measure differential equations is still rare. Recently, there were a number of focuses on existence problems, for example, see [9–11] and referencestherein, and functional differential equations were also investigated widely, such as work done in [12–14].However, there were only very few results on existence of solutions for abstract measure functional differentialequations. There were some consideration on abstract measure delay differential equations [2] and perturbed abstractmeasure differential equations [4]. However, to the best of authors’ knowledge, there were not any resultsdealing with perturbed abstract measure functional differential equations. In this article, we investigate theexistence of solutions for perturbed abstract measure functional differential equations. This is a problemof difficulty and challenge. Based on the Leray–Schauder alternative involving the sum of two operators[15] and the Arzel`–Ascoli theorem, the existence results of our system is derived. The perturbed abstract ameasure functional differential system researched in this paper includes the systems studied in [2, 4] asspecial cases. Additionally, considering appropriate degeneration, ou ...