Discussiones Mathematicae Differential Inclusions, Control and Optimization 30 (2010 ) 141–161 THE METHOD OF UPPER AND LOWER SOLUTIONS FOR PARTIAL HYPERBOLIC FRACTIONAL ORDER DIFFERENTIAL INCLUSIONS WITH IMPULSES Sa¨ ıd Abbas Laboratoire de Math´ ematiques Universit´ e de Sa¨ ıda B.P. 138, 20000, Sa¨ ıda, Alg´ erie e-mail: abbas said dz@yahoo.fr and Mouffak Benchohra ∗ Laboratoire de Math´ ematiques Universit´ e de Sidi Bel-Abb` es B.P. 89, 22000, Sidi Bel-Abb` es, Alg´ erie e-mail: benchohra@univ-sba.dz Abstract In this paper we use the upper and lower solutions method to inves- tigate the existence of solutions of a class of impulsive partial hyper- bolic differential inclusions at fixed moments of impulse involving the Caputo fractional derivative. These results are obtained upon suitable fixed point theorems. Keywords and phrases: impulsive hyperbolic differential inclusion, fractional order, upper solution, lower solution, left-sided mixed Riemann-Liouville integral, Caputo fractional-order derivative, fixed point. 2000 Mathematics Subject Classification: 26A33, 34A60. * Corresponding author. DMDICO Page