数学物理学报(英文版) ›› 2014, Vol. 34 ›› Issue (6): 1795-1810.doi: 10.1016/S0252-9602(14)60124-7

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SOME NECESSARY AND SUFFICIENT CONDITIONS FOR EXISTENCE OF POSITIVE SOLUTIONS FOR THIRD ORDER SINGULAR SUBLINEAR MULTI-POINT BOUNDARY VALUE PROBLEMS

韦忠礼   

  1. Department of Mathematics, Shandong Jianzhu University, Jinan 250101, China;
    School of Mathematics, Shandong University, Jinan 250100, China
  • 收稿日期:2013-08-07 出版日期:2014-11-20 发布日期:2014-11-20
  • 基金资助:

    Research supported by the National Science Foundation of Shandong Province (ZR2009AM004).

SOME NECESSARY AND SUFFICIENT CONDITIONS FOR EXISTENCE OF POSITIVE SOLUTIONS FOR THIRD ORDER SINGULAR SUBLINEAR MULTI-POINT BOUNDARY VALUE PROBLEMS

 WEI Zhong-Li   

  1. Department of Mathematics, Shandong Jianzhu University, Jinan 250101, China;
    School of Mathematics, Shandong University, Jinan 250100, China
  • Received:2013-08-07 Online:2014-11-20 Published:2014-11-20
  • Supported by:

    Research supported by the National Science Foundation of Shandong Province (ZR2009AM004).

摘要:

We mainly study the existence of positive solutions for the following third order singular multi-point boundary value problem
x(3)(t) + f(t, x(t), x′(t)) = 0,     0 < t < 1,
x(0) −∑m1i=1αix(ξi) = 0, x′(0) −∑m2i=1βix′(ηi) = 0, x′(1) = 0,
where 0 ≤αi ≤∑m1i=1αi < 1, i = 1, 2, …, m1, 0 < ξ1ξ2 < … < ξm1 < 1, 0 ≤βj ≤∑m2i=1βi <1, j = 1, 2, … , m2, 0 < η1η2 < … < ηm2 < 1. And we obtain some necessary and sufficient conditions for the existence of C1[0, 1] and C2[0, 1] positive solutions by constructing lower and upper solutions and by using the comparison theorem. Our nonlinearity f(t, x, y) may be singular at x, y, t = 0 and/or t = 1.

关键词: boundary value problems, positive solutions, lower and upper solutions, com-parison theorem

Abstract:

We mainly study the existence of positive solutions for the following third order singular multi-point boundary value problem
????
x(3)(t) + f(t, x(t), x′(t)) = 0,     0 < t < 1,
x(0) −∑m1i=1αix(ξi) = 0, x′(0) −∑m2i=1βix′(ηi) = 0, x′(1) = 0,
where 0 ≤αi ≤∑m1i=1αi < 1, i = 1, 2, …, m1, 0 < ξ1ξ2 < … < ξm1 < 1, 0 ≤βj ≤∑m2i=1βi <1, j = 1, 2, … , m2, 0 < η1η2 < … < ηm2 < 1. And we obtain some necessary and sufficient conditions for the existence of C1[0, 1] and C2[0, 1] positive solutions by constructing lower and upper solutions and by using the comparison theorem. Our nonlinearity f(t, x, y) may be singular at x, y, t = 0 and/or t = 1.

Key words: boundary value problems, positive solutions, lower and upper solutions, com-parison theorem

中图分类号: 

  • 34B16