Existence And Multipicity Of Positive Solutions For Boundary Value Problems Of Impulsive Differential Equation

In recent years, the theory of impulsive differential equations[l-3] is an important area of investigation. It represents a more natural framework for mathematical modeling of many real world phenomena than differential equations. It shows some processes which distinguished at certain instants that their state is subjected to a sudden change, for example, in population dynamics, physics, ecology, biological systems, etc.In this paper, we mainly consider the existence and multiple results for positive solutions of nonlinear impulsive boundary value problems. This paper has three sections. It is organized as follows.In section 1, we use a generalization of Leggett Williams fixed point([4]) to establish the existence of solutions to second order impulsive boundary value problemwe extended the corresponding conclusions of paper([5]). In this section, it should be noted that our result remove the continuous condition on / as is customary in the literature([5]). Meanwhile, the existence results about the triple positive solutions were done under the assumption that the first order derivative is involved explicitly in the nonlinear term. That is to say, we are consider the multiplicity of positive solutions with dependence on derivatives.In section 2, we firstly discuss the nonlinearity f{t,x) is followed to change sign on [0,1]. When the impulsive is absent, this problem (i.e. ODE) has been widely considered. Compared with other type differential equation, to my best knowledge, there are very few works considering the relative results. we consider the existence of positive solutions of second order impulsive boundary value problemsby constructing the appropriate homotopy operator and applying the theory of fixed point index in cones. our results contain the conclusion when f ≥ 0 and J_k = 0([6]).Secondly, we discuss multiple positive solutions of the following impulsive boundary...