Solvability of an impulsive boundary value problem on the half-line via critical point theory

Document Type: Research Paper


1 Laboratory of Fixed Point Theory and Applications‎, Ecole Normale Superieure‎, ‎Kouba‎, ‎Algiers‎, ‎Algeria‎.

2 Department of Mathematics‎, ‎Faculty of Sciences‎, ‎Al Imam Mohammad Ibn Saud Islamic University (IMSIU)‎. ‎P.O‎. ‎Box 90950‎, ‎Riyadh 11623‎, ‎Saudi Arabia and Department of Mathematics, Ecole Normale Superieure, Kouba.


In this paper, an impulsive boundary value problem on the half-line is considered and existence of solutions is proved using Minimization Principal and Mountain Pass Theorem.


Main Subjects

A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge Studies in Advanced Mathematics, 34, Cambridge Univ. Press, Cambridge, 1995.

A. Ambrosetti and P.H. Rabinowitz, Dual variational method in critical point theory and applications, J. Funct. Anal. 14 (1973) 349--381.

M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners, Existence Results via the Variational Approach, Universitext, Springer, London, 2011.

D.D. Bainov and P.S. Simeonov, Systems with Impulse Effect. Stability, Theory and Applications, Ellis Horwood Ltd. Chichester; Halsted Press/John Wiley & Sons, New York, 1989.

G. Bonanno, B. Di Bella and J. Henderson, Existence of solutions to second-order boundary-value problems with small perturbations of impulses, Electron. J. Differential Equations 2013 (2013), no. 126, 14 pages.

H. Brezis, Analyse Fonctionnelle, Th_eorie et Applications, Masson, Paris, 1983.

H. Chen and J. Sun, An application of variational method to second-order impulsive differential equation on the half-line, Appl. Math. Comput. 217 (2010), no. 5, 1863--1869.

C. Corduneanu, Integral Equations and Stability of Feedback Systems, Academic Press, New York, 1973.

A. Friedman, Foundations of Modern Analysis, Reprint of the 1970 original, Dover Publications, New York, 1982.

S.D. Guo, Variational approach to a class of impulsive differential equations, Bound. Value Probl. 2014 (2014), no. 37, 10 pages.

O. Kavian, Introduction _a la Th_eorie des Points Critiques et Applications aux Problemes Elliptiques (French), Mathematics & Applications 13, Springer-Verlag, Paris, 1993.

J. Liu and Z. Zhao, Existence of positive solutions to a singular boundary-value problem using variational methods, Electron. J. Differential Equations, 2014 (2014), no. 135, 9 pages.

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences 74, Springer-Verlag, New York, 1989.

P.H. Rabinowitz, Variational methods for nonlinear eigenvalue problems, in: Eigenvalues of Non-Linear Problems (Centro Internaz. Mat. Estivo (C. I. M. E.), III Ciclo, Varenna, 1974) pp. 139--195, Edizioni Cremonese, Rome, 1974.

P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics 65, Amer. Math. Soc. Providence, RI, 1986.

M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergebnisse der Mathematik und ihrer Grenzgebiete 3, A Series of Modern Surveys in Mathematics 34, Springer-Verlag, 2nd edition, Berlin, 1996.

Y. Tian, W. Ge and D. Yang, Existence results for second-order system with impulse effects via variational methods, J. Appl. Math. Comput. 31 (2009), no. 1-2, 255--265.