Existence Results on General Integrodifferential Evolution Equations in Banach Space

Affiliation(s)

Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore, India.

Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore, India.

ABSTRACT

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.

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.

Cite this paper

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.

K. Sathiyanathan and T. Gopal, "Existence Results on General Integrodifferential Evolution Equations in Banach Space,"

References

[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

[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