Multiple solutions for an impulsive boundary value problems on the halfline via Morse theory

Karima Ait-Mahiout, Smail Djebali, Toufik Moussaoui



In this paper, Morse theory is used to establish the existence of multiple solutions for an impulsive boundary value problem posed on the half-line.


Impulsive boundary value problem; half-line; critical point; critical group; Morse theory

Full Text:



A. Ambrosetti and Ph. 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, Universitext. Springer, London, 2011. x+199 pp.

D.D. Bainov and P.S. Simeonov, Systems with Impulse Effect. Stability, theory and applications, Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1989, 255 pp.

M. Briki, S. Djebali and T. Moussaoui, Solvability of an impulsive boundary value problem on the half-line via critical point theory, Bull. Iran. Math. Soc., to appear.

K.C. Chang, Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math. 34 (1981), no. 5, 693–712.

K.C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Differential Equations and their Applications, 6. Birkh¨auser Boston, Inc., Boston, MA, 1993, x+312 pp.

K.C. Chang, Methods in Nonlinear Analysis, Springer Monographs in Mathematics. Springer–Verlag, Berlin, 2005. x+439 pp.

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.

K. Deimling, Nonlinear Functional Analysis, Springer–Verlag, Berlin, 1985, xiv+450 pp.

D. Guo, Variational approach to a class of impulsive differential equations, Bound. Value Probl. 2014, doi:10.1186/1687-2770-2014-37, 10 pp.

G. Han and Z. Xu, Multiple solutions of some nonlinear fourth-order beam equations, Nonlinear Anal. 68 (2008), no. 12, 3646–3656.

S. Liu and S. Li, Existence of solutions for asymptotically ‘linear’ p-Laplacian equations, Bull. London Math. Soc. 36 (2004), no. 1, 81–87.

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, 74, Springer–Verlag, New York, 1989, xiv+277 pp.

D. Motreanu, V.V. Motreanu and N. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014, xii+459 pp.

J.J. Nieto and Z. Luo, Multiple positive solutions of the singular boundary value problem for second-order impulsive differential equations on the half-line, Bound. Value Probl. 2010, Art. ID 281908, 13 pp.

J.J. Nieto and D. O’Regan, Variational approach to impulsive differential equations, Nonlinear Anal. Real World Appl. 10 (2009), no. 2, 680–690.

K. Perera, R. Agarwal and D. O’Regan, Morse Theoretic Aspects of p-Laplacian Type Operators, Mathematical Surveys and Monographs, 161, American Mathematical Society, Providence, RI, 2010, xx+141 pp.

Ph. Rabinowitz, Variational Methods for Nonlinear Eigenvalue Problems, Eigenvalues of non-linear problems (Centro Internaz. Mat. Estivo (C.I.M.E.), III Ciclo, Varenna, 1974), pp. 139–195, Edizioni Cremonese, Rome, 1974.

C. L. Tang, Multiplicity of periodic solutions for second-order systems with a small forcing term, Nonlinear Anal. Ser. A: Theory Methods 38 (1999), no. 4, 471–479.

L. Yan, J. Liu and Z. Luo, Existence and multiplicity of solutions for second-order impulsive differential equations on the half-line, Adv. Difference Equ. 2013, 2013:293, 12 pp.

W. Zou and S. Li, Infinitely many solutions for Hamiltonian systems, J. Differential Equations 186 (2002), no. 1, 141–164.


  • There are currently no refbacks.

Partnerzy platformy czasopism