Multiple positive solutions for time scale boundary value problems on infinite intervals
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Multiple positive solutions for time scale boundary value problems on infinite intervals Xiangkui Zhao · Weigao Ge
Received: 3 June 2008 / Accepted: 12 August 2008 / Published online: 3 September 2008 © Springer Science+Business Media B.V. 2008
Abstract The study of dynamic equations on time scales is an area of mathematics. It has been created in order to unify the study of differential and difference equations. In this paper, we consider the time-scale boundary value problems (φp (u (t)))∇ + q(t)f (u(t), u (t)) = 0, t ∈ (0, ∞)T , u(0) = βu (η),
lim
t∈T,t→∞
u (t) = 0,
where T is a time scale. By means of Leggett-Williams fixed point theorem, sufficient conditions are obtained that guarantee the existence of at least three positive solutions to the above boundary value problem. The results obtained are even new for the special cases of difference dynamic equations (when T = Z) and differential dynamic equations (when T = R), as well as in the general time scale setting. Keywords Time scale · Positive solutions · Fixed point theorem · Infinite intervals Mathematics Subject Classification (2000) 39A10
1 Introduction In past years, dynamic equations models of natural occurrences were either entirely continuous or discrete. These models worked well for continuous behavior such as population
Supported by National Natural Sciences Foundation of China (10671012) and the Doctoral Program Foundation of Education Ministry of China (20050007011). X. Zhao () · W. Ge Department of Mathematics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China e-mail: [email protected] X. Zhao Applied Science School, University of Science and Technology Beijing, Beijing 100083, People’s Republic of China
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growth and biological phenomena, and for discrete behavior such as applications of Newton’s method. However, these models are deficient when the behavior is sometimes continuous and sometimes discrete. For example, certain economically important phenomena do not possess solely continuous properties or solely discrete aspects. Rather, these phenomena contain processes that feature elements of both the continuous and the discrete. A simple example of this hybrid continuous-discrete behavior is seen in “seasonally breeding populations in which generations do not overlap. Many natural populations, particularly among temperate-zone insects (including many economically important crop and orchid pests) are of this kind” [1]. These insects lay their eggs just before the generation dies out at the end of the season, with the eggs laying dormant, hatching at the start of the next season giving rise to a new, non-overlapping generation. The continuous-discrete behavior is seen in the fact that during each generation the population varies continuously (due to mortality, resource consumption, predation, interaction etc.), while the population varies in a discrete fashion between the end of one generation and the beginning of the next [2]. The existence of both continuous and discrete behavior c
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