Existence of Three Positive Solutions of Three-Order with m -Point Impulsive Boundary Value Problems
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Existence of Three Positive Solutions of Three-Order with m-Point Impulsive Boundary Value Problems Sihua Liang · Jihui Zhang
Received: 15 September 2008 / Accepted: 15 December 2008 / Published online: 20 December 2008 © Springer Science+Business Media B.V. 2008
Abstract This paper deals with the existence of three positive solutions for the following impulsive boundary value problem ⎧ (ϕ(−u )(t)) + a(t)f (u(t)) = 0, t = tk , 0 < t < 1, ⎪ ⎪ ⎪ ⎨ k = 1, 2, . . . , N, u|t=tk = Ik (u(tk )), m−2 ⎪ u(0) = i=1 αi u(ξi ), ⎪ ⎪ ⎩ u (0) = 0, u (1) = 0, where ϕ : R → R is the increasing homeomorphism (see Definition 1.1) and positive homomorphism and ϕ(0) = 0. Using the five functionals fixed point theorem, we provide sufficient conditions for the existence of three positive solutions for the above boundary value problems. Keywords Impulsive boundary value problems · The five functionals fixed-point theorem · Positive solutions · Cone Mathematics Subject Classification (2000) 34B18 · 34B40
S. Liang () · J. Zhang Institute of Mathematics, School of Mathematics and Computer Sciences, Nanjing Normal University, 210097 Jiangsu, People’s Republic of China e-mail: [email protected] J. Zhang e-mail: [email protected] S. Liang College of Mathematics, Changchun Normal University, Changchun 130032, Jilin, People’s Republic of China
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S. Liang, J. Zhang
1 Introduction Mathematics is successfully applied in numerous technical problems, biology, economics, control theory, etc. In general, taking into account different properties and effects intrinsic of a considered system, it gives us the possibility to study processes and phenomena by means of mathematical simulation more exactly. For instance, many areas of industry deal with technical systems being subjected to shocks, impacts or impulses. Thus, in cases when an external perturbation for the system is of a short-term (shock, impact or impulse) nature and its duration could be disregarded while formulating the corresponding mathematical model we have to study dynamical system with discontinuous trajectories. A classical example of that problem is the model of a clock mechanism. Impulsive differential equations describe processes which experience a sudden change of their state at certain moments. The theory of impulsive differential equations has become important in recent years in mathematical models of real processes rising in phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics; see [2]. There has been a significant development in impulsive theory especially in the area of impulsive differential equations with fixed moments [4]. In this paper, we introduce a new operator, which improves and generates a p-Laplace operator for some p > 1, and we study the existence of three positive solutions for the following three-order with m-point nonlinear impulsive boundary value problem of the form (ϕ(−u )(t)) + a(t)f (u(t)) = 0, t = tk , 0 < t < 1, (1) k = 1, 2, . . . , N, u|t=tk = Ik (u(tk )), with the following boundar
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