- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 724854, 8 pages
An Existence Result for Nonlocal Impulsive Second-Order Cauchy Problems with Finite Delay
School of Mathematics, Yunnan Normal University, Kunming 650092, China
Received 1 October 2012; Revised 7 December 2012; Accepted 9 December 2012
Academic Editor: Toka Diagana
Copyright © 2013 Fang Li and Huiwen Wang. 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.
- H. R. Henríquez and C. H. Vásquez, “Differentiability of solutions of second-order functional differential equations with unbounded delay,” Journal of Mathematical Analysis and Applications, vol. 280, no. 2, pp. 284–312, 2003.
- F. Li, “Solvability of nonautonomous fractional integrodifferential equations with infinite delay,” Advances in Difference Equations, vol. 2011, Article ID 806729, 18 pages, 2011.
- F. Li, “An existence result for fractional differential equations of neutral type with infinite delay,” Electronic Journal of Qualitative Theory of Differential Equations, no. 52, pp. 1–15, 2011.
- J. Liang and T. J. Xiao, “Solvability of the Cauchy problem for infinite delay equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 58, no. 3-4, pp. 271–297, 2004.
- T. J. Xiao and J. Liang, “Blow-up and global existence of solutions to integral equations with infinite delay in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e1442–e1447, 2009.
- H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, vol. 108 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 1985.
- J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs, Oxford University Press, New York, NY, USA, 1985.
- J. Kisyński, “On cosine operator functions and one-parameter groups of operators,” Studia Mathematica, vol. 44, no. 1, pp. 93–105, 1972.
- J. Liang and T. J. Xiao, “A characterization of norm continuity of propagators for second order abstract differential equations,” Computers & Mathematics with Applications, vol. 36, no. 2, pp. 87–94, 1998.
- J. Liang, R. Nagel, and T. J. Xiao, “Approximation theorems for the propagators of higher order abstract Cauchy problems,” Transactions of the American Mathematical Society, vol. 360, no. 4, pp. 1723–1739, 2008.
- C. C. Travis and G. F. Webb, “Second order differential equations in Banach space,” in Nonlinear Equations in Abstract Spaces (Proc. Internat. Sympos., Univ. Texas, Arlington, Tex., 1977), pp. 331–361, Academic Press, New York, NY, USA, 1978.
- T. J. Xiao and J. Liang, “Second order linear differential equations with almost periodic solutions,” Acta Mathematica Sinica (New Series), vol. 7, no. 4, pp. 354–359, 1991.
- T. J. Xiao and J. Liang, “Differential operators and -wellposedness of complete second order abstract Cauchy problems,” Pacific Journal of Mathematics, vol. 186, no. 1, pp. 167–200, 1998.
- T. J. Xiao and J. Liang, The Cauchy Problem for Higher-Order Abstract Differential Equations, vol. 1701 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1998.
- T. J. Xiao and J. Liang, “Higher order abstract Cauchy problems: their existence and uniqueness families,” Journal of the London Mathematical Society, vol. 67, no. 1, pp. 149–164, 2003.
- N. U. Ahmed, “Optimal feedback control for impulsive systems on the space of finitely additive measures,” Publicationes Mathematicae Debrecen, vol. 70, no. 3-4, pp. 371–393, 2007.
- G. Arthi and K. Balachandran, “Controllability of second-order impulsive functional differential equations with state-dependent delay,” Bulletin of the Korean Mathematical Society, vol. 48, no. 6, pp. 1271–1290, 2011.
- M. Benchohra, J. Henderson, and S. Ntouyas, Impulsive Differential Equations and Inclusions, vol. 2 of Contemporary Mathematics and Its Applications, Hindawi Publishing Corporation, New York, NY, USA, 2006.
- T. Cardinali and P. Rubbioni, “Impulsive semilinear differential inclusions: topological structure of the solution set and solutions on non-compact domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 1, pp. 73–84, 2008.
- T. Cardinali and P. Rubbioni, “Impulsive mild solutions for semilinear differential inclusions with nonlocal conditions in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 2, pp. 871–879, 2012.
- E. Hernández, K. Balachandran, and N. Annapoorani, “Existence results for a damped second order abstract functional differential equation with impulses,” Mathematical and Computer Modelling, vol. 50, no. 11-12, pp. 1583–1594, 2009.
- E. Hernández M., H. R. Henríquez, and M. A. McKibben, “Existence results for abstract impulsive second-order neutral functional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 7, pp. 2736–2751, 2009.
- E. Hernández M. and S. M. T. Aki, “Global solutions for abstract impulsive differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1280–1290, 2010.
- J. Liang, J. H. Liu, and T. J. Xiao, “Nonlocal impulsive problems for nonlinear differential equations in Banach spaces,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 798–804, 2009.
- J. H. Liu, “Nonlinear impulsive evolution equations,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 6, no. 1, pp. 77–85, 1999.
- Y. V. Rogovchenko, “Impulsive evolution systems: main results and new trends,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 3, no. 1, pp. 57–88, 1997.
- S. T. Zavalishchin, “Impulse dynamic systems and applications to mathematical economics,” Dynamic Systems and Applications, vol. 3, no. 3, pp. 443–449, 1994.
- J. Liang, J. Liu, and T. J. Xiao, “Nonlocal Cauchy problems governed by compact operator families,” Nonlinear Analysis: Theory, Methods & Applications, vol. 57, no. 2, pp. 183–189, 2004.
- J. Liang and T. J. Xiao, “Semilinear integrodifferential equations with nonlocal initial conditions,” Computers & Mathematics with Applications, vol. 47, no. 6-7, pp. 863–875, 2004.
- T. J. Xiao and J. Liang, “Existence of classical solutions to nonautonomous nonlocal parabolic problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 5–7, pp. e225–e232, 2005.
- J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, vol. 60 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980.
- D. Bothe, “Multivalued perturbations of -accretive differential inclusions,” Israel Journal of Mathematics, vol. 108, pp. 109–138, 1998.
- M. Kamenskii, V. Obukhovskii, and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, vol. 7 of De Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter, Berlin, Germany, 2001.
- T. Cardinali and P. Rubbioni, “On the existence of mild solutions of semilinear evolution differential inclusions,” Journal of Mathematical Analysis and Applications, vol. 308, no. 2, pp. 620–635, 2005.
- A. Ambrosetti, “Un teorema di esistenza per le equazioni differenziali negli spazi di Banach,” Rendiconti del Seminario Matematico della Università di Padova, vol. 39, pp. 349–361, 1967.