Existence results for impulsive differential equations with nonlocal conditions via measures of noncompactness
-
2096
Downloads
-
3286
Views
Authors
M. Mallika Arjunan
- Department of Mathematics, Karunya University, Karunya Nagar, Coimbatore- 641 114, Tamil Nadu, India.
V. Kavitha
- Department of Mathematics, Karunya University, Karunya Nagar, Coimbatore-641 114, Tamil Nadu, India.
S. Selvi
- Department of Mathematics, Muthayammal College of Arts & Science, Rasipuram- 637408, Tamil Nadu, India.
Abstract
In this paper, we study the existence of integral solutions for impulsive evolution equations with nonlocal
conditions where the linear part is nondensely defined. Some existence results of integral solutions to
such problems are obtained under the conditions in respect of the Hausdorff's measure of noncompactness.
Example is provided to illustrate the main result.
Share and Cite
ISRP Style
M. Mallika Arjunan, V. Kavitha, S. Selvi, Existence results for impulsive differential equations with nonlocal conditions via measures of noncompactness, Journal of Nonlinear Sciences and Applications, 5 (2012), no. 3, 195--205
AMA Style
Arjunan M. Mallika, Kavitha V., Selvi S., Existence results for impulsive differential equations with nonlocal conditions via measures of noncompactness. J. Nonlinear Sci. Appl. (2012); 5(3):195--205
Chicago/Turabian Style
Arjunan, M. Mallika, Kavitha, V., Selvi, S.. "Existence results for impulsive differential equations with nonlocal conditions via measures of noncompactness." Journal of Nonlinear Sciences and Applications, 5, no. 3 (2012): 195--205
Keywords
- Impulsive differential equations
- nondensely defined
- noncompact measures
- nonlocal conditions
- integral solutions
- semigroup theory.
MSC
References
-
[1]
A. Anguraj, M. Mallika Arjunan, Existence and uniqueness of mild and classical solutions of impulsive evolution equations, Electronic Journal of Differential Equations, 111 (2005), 1-8.
-
[2]
A. Anguraj, M. Mallika Arjunan, Existence results for an impulsive neutral integro-differential equations in Banach spaces, Nonlinear Studies, 16(1) (2009), 33-48.
-
[3]
H. Akca, A. Boucherif, V. Covachev, Impulsive functional differential equations with nonlocal conditions, Inter. J. Math. Math. Sci., 29:5 (2002), 251-256.
-
[4]
D. D. Bainov, P. S. Simeonov , Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific and Technical Group, England (1993)
-
[5]
J. Banas, K. Goebel , Measure of Noncompactness in Banach Spaces, in: Lecture Notes in Pure and Appl. Math., 60, Marcel Dekker, New York (1980)
-
[6]
M. Benchohra, S. K. Ntouyas, Existence of mild solutions for certain delay semilinear evolution inclusions with nonlocal condition, Dynam. Systems Appl., 9:3 (2000), 405-412.
-
[7]
M. Benchohra, S. K. Ntouyas, Existence of mild solutions of semilinear evolution inclusions with nonlocal conditions, Georgian Math. J., 7 (2000), 221-230.
-
[8]
M. Benchohra, S. K. Ntouyas, Existence and controllability results for nonlinear differential inclusions with nonlocal conditions, J. Appl. Anal., 8 (2002), 31-46.
-
[9]
L. Byszewski, V. Lakshmikantham, Theorems about the existence and uniqueness of solutions of a nonlocal Cauchy problem in Banach spaces, Appl. Anal., 40 (1990), 11-19.
-
[10]
L. Byszewski, Uniqueness criterian for solution to abstract nonlocal Cauchy problem, J. Appl. Math. Stochastic Anal., 162 (1991), 49-54.
-
[11]
L. Byszewski , Existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, Zesz. Nauk. Pol. Rzes. Mat. Fiz., 18 (1993), 109-112.
-
[12]
T. Cardinali, P. Rubbioni, Impulsive semilinear differential inclusion: Topological structure of the solution set and solutions on non-compact domains, Nonlinear Anal., 14 (2008), 73-84.
-
[13]
Y. K. Chang, A. Anguraj, M. Mallika Arjunan, Existence results for non-densely defined neutral impulsive differential inclusions with nonlocal conditions, J. Appl. Math. Comput., 28 (2008), 79-91.
-
[14]
Y. K. Chang, A. Anguraj, M. Mallika Arjunan, Existence results for impulsive neutral functional differential equations with infinite delay, Nonlinear Anal.: Hybrid Systems, 2(1) (2008), 209-218.
-
[15]
Y. K. Chang, V. Kavitha, M. Mallika Arjunan, Existence results for impulsive neutral differential and integrodifferential equations with nonlocal conditions via fractional operators, Nonlinear Anal.: Hybrid Systems, 4(1) (2010), 32-43.
-
[16]
G. Da Prato, E. Sinestrari , Differential operators with non-dense domain, Ann. Scuola Norm. Sup. Pisa Sci., 14 (1987), 285-344.
-
[17]
Z. Fan, Q. Dong, G. Li, Semilinear didderential equations with nonlocal conditions in Banach spaces, International Journal of Nonlinear Science, 2(3) (2006), 131-139.
-
[18]
Z. Fan, Existence of nondensely defined evolution equations with nonlocal conditions, Nonlinear Anal., 70 (2009), 3829-3836.
-
[19]
Z. Fan, Impulsive problems for semilinear differential equations with nonlocal conditions, Nonlinear Anal., 72(2) (2010), 1104-1109.
-
[20]
M. Kamenskii, V. Obukhovskii, P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Ser: Nonlinear Anal. Appl., 7, de Gruyter, Berlin (2001)
-
[21]
H. Kellerman, M. Heiber, Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180.
-
[22]
V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore (1989)
-
[23]
J. Liang, J. H. Liu, Ti-Jun Xiao, Nonlocal impulsive problems for nonlinear differential equations in Banach spaces, Math. Comput. Model., 49 (2009), 798-804.
-
[24]
J. H. Liu , Nonlinear impulsive evolution equations, Dynam. Contin. Discrete Impuls. Sys., 6 (1999), 77-85.
-
[25]
H. Monch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal., 4 (1980), 985-999.
-
[26]
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, Newyork (1983)
-
[27]
A. M. Samoilenko, N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore (1995)
-
[28]
B. Selvaraj, M. Mallika Arjunan, V. Kavitha, Existence of solutions for impulsive nonlinear differential equations with nonlocal conditions, J. Korean Society for Industrial and Applied Mathematics, 13(3) (2009), 203-215.
-
[29]
X. Xue, Existence of solutions for semilinear nonlocal Cauchy problems in Banach spaces, Electronic Journal of Differential Equations, 64 (2005), 1-7.
-
[30]
K. Yosida, Functional Analysis, 6th edn., Springer, Berlin (1980)