Existence of positive solutions for nonlinear m-point boundary value problems on time scales

Zhao Junfang, Lian Hairong, E G, Ge Weigao

Description

In this article, we study the following m -point boundary value problem on time scales, ( Ï• p ( u Δ ( t ) ) ) ∇ + h ( t ) f ( t , u ( t ) ) = 0 , t ∈ ( 0 , T ) T , u ( 0 ) - δ u Δ ( 0 ) = ∑ i = 1 m - 2 . In this article, we study the following m -point boundary value problem on time scales, ( Ï• p ( u Δ ( t ) ) ) ∇ + h ( t ) f ( t , u ( t ) ) = 0 , t ∈ ( 0 , T ) T , u ( 0 ) - δ u Δ ( 0 ) = ∑ i = 1 m - 2 .

Sujets

• Coné
• Boundary value problem
• Time scale

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 9
• Langue: English

Extrait

Zhao et al . Boundary Value Problems 2012, 2012 :4 http://www.boundaryvalueproblems.com/content/2012/1/4

R E S E A R C H Open Access Existence of positive solutions for nonlinear m - point boundary value problems on time scales Junfang Zhao 1* , Hairong Lian 1 and Weigao Ge 2

* Correspondence: zhao [e-mail] _ 1 School of Science, China University of Geosciences, Beijing 100083, P.R. China Full list of author information is available at the end of the article

Abstract In this article, we study the following m -point boundary value problem on time scales, ⎧ ( φ p ( u  ( t ))) ∇ + h ( t ) f ( t , u ( t )) = 0, t ∈ (0, T ) T , ⎪ ⎨ m − 2 ⎪ u (0) − δ u  (0) =  β i u  ( ξ i ), u  ( T ) = 0, i =1 where is a time scale such that 0, T ∈ T , δ , i > 0, i = 1, . . . , m − , j p ( s ) = | s | p-2 s,p > 1 ,h Î C ld (( 0, T ), (0, + ∞ )), and f Î C ([0,+ ∞ ), (0,+ ∞ )), 0 < 1 < 2 < · · · < m − 2 < T ∈ . By using several well-known fixed point theorems in a cone, the existence of at least one, two, or three positive solutions are obtained. Examples are also given in this article. AMS Subject Classification : 34B10; 34B18; 39A10. Keywords: positive solutions, cone, multi-point, boundary value problem, time scale

1 Introduction The study of dynamic equations on time scales goes back to its founder Hilger [1], and is a new area of still theoretical exploration in mathematics. Motivating the subject is the notion that dynamic equations on time scales can build bridges between continu- ous and discrete mathematics. Further, th e study of time scales has led to several important applications, e.g., in the study of insect population models, neural networks, heat transfer, epidemic models, etc. [2]. Multipoint boundary value problems of ord inary differential equations (BVPs for short) arise in a variety of different area s of applied mathematics and physics. For example, the vibrations of a guy wire of a uniform cross section and composed of N parts of different densities can be set up as a multi-point boundary value problem [3]. Many problems in the theory of elastic sta bility can be handled by the method of multi-point problems [4]. Small size bridges are often designed with two supported points, which leads into a standard two-point boundary value condition and large size bridges are sometimes contrived with mult i-point supports, which corresponds to a multi-point boundary value condition [5] . The study of multi-point BVPs for linear second-order ordinary differenti al equations was initiated by Il ’ in and Moiseev [6]. Since then many authors have studied more g eneral nonlinear multi-point BVPs, and

© 2012 Zhao et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.