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Symmetric Positive Solutions for a Singular Second-Order Three-Point Boundary Value Problem

Authors:

Yong-ping Sun

Affiliation:

(1) Department of Mathematics, Qufu Normal University, Qufu, 273165, China;(2) Department of Fundamental Courses, Hangzhou Radio & TV University, Hangzhou 310012, China

Abstract:

Abstract In this paper, we consider the following second-order three-point boundary value problem

$$
begin{array}{*{20}c}
{{{u}ifmmode{'}else$'$fi{left( t right)} + a{left( t right)}f{left( {u{left( t right)}} right)} = 0,}} & {{0 < t < 1,}}
{{u{left( 0 right)} - u{left( 1 right)} = 0,}} & {{{u}ifmmode{'}else$'$fi{left( 0 right)} - {u}ifmmode{'}else$'$fi{left( 1 right)} = u{left( {1/2} right)},}}

end{array}
$$

where a : (0, 1) → [0, ∞) is symmetric on (0, 1) and may be singular at t = 0 and t = 1, f : [0, ∞) → [0, ∞) is continuous. By using Krasnoselskii’s fixed point theorem in a cone, we get some existence results of positive solutions for the problem. The associated Green’s function for the three-point boundary value problem is also given. Supported by the National Natural Science Foundation of China (No.10471075), National Natural Science Foundation of Shandong Province of China (No.Y2003A01) and Foundation of Education Department of Zhejiang Province of China (No.20040495, No.20051897)

Keywords:

Symmetric positive solution three-point boundary value problem fixed point theorem existence

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