数学物理学报(英文版) ›› 2011, Vol. 31 ›› Issue (4): 1569-1582.doi: 10.1016/S0252-9602(11)60343-3

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NONEXISTENCE AND EXISTENCE OF POSITIVE SOLUTIONS FOR 2nTH-ORDER SINGULAR SUPERLINEAR PROBLEMS WITH#br# STRUM-LIOUVILLE BOUNDARY CONDITIONS

赵增勤   

  1. School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
  • 收稿日期:2008-12-05 修回日期:2010-03-23 出版日期:2011-07-20 发布日期:2011-07-20
  • 基金资助:

    Research supported by the National Natural Science Foundation of China (10871116), the Natural Science Foundation of Shandong Province of China (ZR2010AM005) and the Doctoral Program Foundation of Education Ministry of China (200804460001).

NONEXISTENCE AND EXISTENCE OF POSITIVE SOLUTIONS FOR 2nTH-ORDER SINGULAR SUPERLINEAR PROBLEMS WITH#br# STRUM-LIOUVILLE BOUNDARY CONDITIONS

 ZHAO Zeng-Qin   

  1. School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
  • Received:2008-12-05 Revised:2010-03-23 Online:2011-07-20 Published:2011-07-20
  • Supported by:

    Research supported by the National Natural Science Foundation of China (10871116), the Natural Science Foundation of Shandong Province of China (ZR2010AM005) and the Doctoral Program Foundation of Education Ministry of China (200804460001).

摘要:

This paper investigates a class of 2nth-order singular superlinear problems with Strum-Liouville boundary conditions. We obtain a necessary and sufficient condition for the existence of C2n−2[0, 1] positive solutions, and a sufficient condition, a necessary condition for the existence of C2n−1[0, 1] positive solutions. Relations between the positive solutions and the Green’s functions are depicted. The results are used to judge nonexistence or existence of positive solutions for given boundary value problems.

关键词: 2nth-order differential equation, singular boundary value problem, fixed point theorem, positive solution

Abstract:

This paper investigates a class of 2nth-order singular superlinear problems with Strum-Liouville boundary conditions. We obtain a necessary and sufficient condition for the existence of C2n−2[0, 1] positive solutions, and a sufficient condition, a necessary condition for the existence of C2n−1[0, 1] positive solutions. Relations between the positive solutions and the Green’s functions are depicted. The results are used to judge nonexistence or existence of positive solutions for given boundary value problems.

Key words: 2nth-order differential equation, singular boundary value problem, fixed point theorem, positive solution

中图分类号: 

  • 34B15