Existence Results on General Integrodifferential Evolution Equations in Banach Space

DOI: 10.4236/am.2013.41025   PDF   HTML   XML   4,163 Downloads   5,912 Views   Citations


In this paper we prove the existence of mild solutions of a general class of nonlinear evolution integrodifferential equation in Banach spaces. Based on the resolvent operator and the Schaefer fixed point theorem, a sufficient condition for the existence of general integrodifferential evolution equations is established.

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K. Sathiyanathan and T. Gopal, "Existence Results on General Integrodifferential Evolution Equations in Banach Space," Applied Mathematics, Vol. 4 No. 1, 2013, pp. 149-154. doi: 10.4236/am.2013.41025.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] A. Pazy, “Semigroups of Linear Operators and Applications to Partial Differential Equations,” Springer-Verlag, New York, 1933.
[2] L. Byszewski, “Theorems about the Existences and Uniqueness of a Solutions of a Semilinear Evolution Nonlocal Cauchy Problem,” Journal of Mathematical Analysis and Application, Vol. 162, No. 2, 1991, pp. 496-505. doi:10.1016/0022-247X(91)90164-U
[3] L. Byszewski, “Applications of Properties of the Right-Hand Sides of Evolution Equations to an Investigation of Nonlocal Evolution Problems,” Nonlinear Analysis, Vol. 33, No. 5, 1998, pp. 413-426. doi:10.1016/S0362-546X(97)00594-4
[4] K. Balachandran and M. Chandrasekaran, “The Nonlocal Cauchy Problem for Semilinear Integrodifferential Equation with Devating Argument,” Proceedings of the Edinburgh Mathematical Society, Vol. 44, No. 1, 2001, pp. 63-70. doi:10.1017/S0013091598001060
[5] K. Balachandran and J. Y. Park, “Existence of Solutions and Controllability of Nonlinear Integrodifferential Systems in Banach Spaces,” Mathematical Problems in Engineering, Vol. 2003, No. 2, 2003, pp. 65-79. doi:10.1155/S1024123X03201022
[6] R. Gimmer, “Resolvent Operators for Integral Equations in a Banach Space,” Transactions of the American Mathematical Society, Vol. 273, 1982, pp. 333-349. doi:10.1090/S0002-9947-1982-0664046-4
[7] J. H. Liu, “Resolvent Operators and Weak Solutions of Integrodifferential Equations,” Differential and Integral Equations, Vol. 7, 1994, pp. 523-534.
[8] J. H. Liu, “Integrodifferential Equations with Nonautonomous Operators,” Dynamic Systems and Applications, Vol. 7, 1998, pp. 427-440.
[9] Y. Lin and J. H. Liu, “Semilinear Integrodifferential Equations with Nonlocal Cauchy Problem,” Nonlinear Analysis; Theory, Methods and Applications, Vol. 26, 1996, pp. 1023-1033.
[10] J. H. Liu and K. Ezzinbi, “Non-Autonomous Integrodifferential Equations with Nonlocal Conditions,” Journal of Integral Equations and Applications, Vol. 15, No. 1, 2003, pp. 79-93. doi:10.1216/jiea/1181074946
[11] L. Byszewski and H. Acka, “Existence of Solutions of a Semilinear Functional Differential Evolution Nonlocal Problem,” Nonlinear Analysis, Vol. 34, No. 1, 1998, pp. 65-72. doi:10.1016/S0362-546X(97)00693-7
[12] K. Balachandran, J. H. Kim and A. Leelamani, “Existence Results for Nonlinear Abstract Neutral Differential Equations with Time Varying Delays,” Applied Mathematics E-Notes, Vol. 6, 2006, pp. 186-193.
[13] H. M. Eduardo, H. R. Henriquez and J. P. C. dos Santos, “Existence Results for Abstract Partial Neutral Integrodifferential Equation with Unbounded Delay,” Electronic Journal of Qualitative Theory of Differential Equations, Vol. 29, 2009, pp. 1-23.
[14] K. Balachandran and R. Ravikumar, “Existence of Solutions of Integrodifferential Evolution Equations with Time Varying Delays,” Applied Mathematics E-Notes, Vol. 7, 2007, pp. 1-8.
[15] H. Schaefer, “Uber Die Methods der a Priori Schranken,” Mathematische Annalem, Vol. 129, No. 1, 1955, pp. 415-416. doi:10.1007/BF01362380

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