Upper and Lower Solutions for -Point Impulsive BVP with One-Dimensional -Laplacian

Upper and lower solutions theories are established for a kind of -point impulsive boundary value problems with -Laplacian. By using such techniques and the Schauder fixed point theorem, the existence of solutions and positive solutions is obtained. Nagumo conditions play an important role in the nonlinear term involved with the first-order derivatives. 1. Introduction In this paper, we study the following -point impulsive boundary value problem with one-dimensional -Laplacian: where , . , ( , where is a fixed positive integer) are fixed points with . . , . , , where and represent the right-hand limit and left-hand limit of at , respectively. The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. Processes with such a character arise naturally and often, especially in phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics. For an introduction of the basic theory of impulsive differential equations in , see Lakshmikantham et al. [1], Ba？nov and Simeonov [2], Samoilenko and Perestyuk [3], and the references therein. The theory of impulsive differential equations has become an important area of investigation in recent years and is much richer than the corresponding theory of differential equations (see for instance [4–9] and the references therein). At the same time, it is well known that the method of lower and upper solutions is a powerful tool for proving the existence results for a large class of boundary value problems. For a few of such works, we refer the readers to [10–14]. In [15], Cabada and Pouso considered by using the upper and lower method, the authors get the existence of solution to the above BVP. In [16], Lü et al. studied by giving conditions on involving pairs of lower and upper solutions, they get the existence of at least three solutions to the above BVP. In [12], Shen and Wang studied they prove the existence of solutions to the problem under the assumption that there exist lower and upper solutions associated with the problem. Motivated by the works mentioned above, in this paper, we considered BVP (1), the main tool is upper and lower method, and the Sch？uder fixed point theorem. We not only get the existence of solutions, but also the existence of positive solutions. The main structure of this paper is as follows. In Section 2, we give the preliminary and present some lemmas in order to prove our main results. Section 4 presents the main theorems of this paper, and at the end of Section 4, we give an example