Multiplicity Of Positive Solutions To Impulsive Equations With Sturm-Liouville Boundary Value Problems

In mordern technology areas of practial problems, impulse as an instantaneaus catastrophe phenomenon is universal existence. Recentlly some new technology achievement has been proved that impulsive system universally existed in aeromau-tics , informatics , cybernetics , communicats , biology , medicine , economics areasThis paper is composed of two parts. In the first chapter, we introduce the historical background of the problems which will be investigated and the main results of this paper. In the second chapter, by employing Krasnoselskii fixed point theorem, it was proved the existance of multiple posive solutions of Sturm-Liouville problems for second order impulsive differential equations.here Lu = (p(x)u')' + q(x)u is Sturm-Liouville operator, I = [0,1], I' = I \ {x1,x2,...,xm} and 0 < x1 < x2 < ...< xm < 1, In recent years, boundary problems of second-order differential equations with impulses have been studied extensively in the literature (see for instance [1,2,5,6,7,9,10,11] and their refer-ences),where the most common cases are p(x) = 1, q(x) = 0 .However,for the case p(x) ≠ 1, q(x)≠ 0 main results have not been improved and generalized yet,in addition few literature have been studied the existence of posive solution.Liu and Li [13] have applied Krassnoselskii fixed point theorem and fixed index theorem in cones to establish the existentence of multiple posibve solutions of ODE, whereI_k;= 0, p(x) = 1, q(x) = 0.our results in this paper improve those in [13]. The proof is based on fixed point index theroy in cones [3].