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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Existence Results for a Fractional Equation with State-Dependent Delay | Hindawi Publishing Corporation Advances in Difference Equations Volume 2011 Article ID 642013 15 pages doi 10.1155 2011 642013 Research Article Existence Results for a Fractional Equation with State-Dependent Delay Jose Paulo Carvalho dos Santos 1 Claudio Cuevas 2 and Bruno de Andrade3 1 Instituto de Ciencias Exatas Universidade Federal de Alfenas 37130-000 Alfenas MG Brazil 2 Departamento de Matematica Universidade Federal de Pernambuco 50540-740 Recife PE Brazil 3 Departamento de Matemứtica ICMC USP-Sao Carlos 13569-970 Sao Carlos SP Brazil Correspondence should be addressed to Bruno de Andrade bruno00luis@gmail.com Received 26 August 2010 Revised 14 January 2011 Accepted 7 March 2011 Academic Editor Dumitru Baleanu Copyright 2011 Jose Paulo Carvalho dos Santos et al. 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. We provide sufficient conditions for the existence of mild solutions for a class of abstract fractional integrodifferential equations with state-dependent delay. A concrete application in the theory of heat conduction in materials with memory is also given. 1. Introduction In the last two decades the theory of fractional calculus has gained importance and popularity due to its wide range of applications in varied fields of sciences and engineering as viscoelasticity electrochemistry of corrosion chemical physics optics and signal processing and so on. The main object of this paper is to provide sufficient conditions for the existence of mild solutions for a class of abstract partial neutral integrodifferential equations with statedependent delay described in the form D xit Ax t f B t - s x s ds f i xpụ xt a e 1 2 0 X0 f eB x 0 0 1.1 1.2 where t e I 0 b A B t t 0 are closed linear operators defined on a common domain which is dense in a Banach space X II II and Dtạhự represent the Caputo .