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

doi:10.1016/S0252-9602(14)60124-7

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

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

韦忠礼

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

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/Abstract

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) −∑^m1_i=1α_ix(ξ_i) = 0, x′(0) −∑^m2_i=1β_ix′(η_i) = 0, x′(1) = 0,
where 0 ≤α_i ≤∑^m1_i=1α_i < 1, i = 1, 2, …, m₁, 0 < ξ₁ < ξ₂ < … < ξ_m₁ < 1, 0 ≤β_j ≤∑^m2_i=1β_i <1, j = 1, 2, … , m₂, 0 < η₁ < η₂ < … < η_m₂ < 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) −∑^m1_i=1α_ix(ξ_i) = 0, x′(0) −∑^m2_i=1β_ix′(η_i) = 0, x′(1) = 0,
where 0 ≤α_i ≤∑^m1_i=1α_i < 1, i = 1, 2, …, m₁, 0 < ξ₁ < ξ₂ < … < ξ_m₁ < 1, 0 ≤β_j ≤∑^m2_i=1β_i <1, j = 1, 2, … , m₂, 0 < η₁ < η₂ < … < η_m₂ < 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

韦忠礼. SOME NECESSARY AND SUFFICIENT CONDITIONS FOR EXISTENCE OF POSITIVE SOLUTIONS FOR THIRD ORDER SINGULAR SUBLINEAR MULTI-POINT BOUNDARY VALUE PROBLEMS[J]. 数学物理学报(英文版), 2014, 34(6): 1795-1810.

WEI Zhong-Li. SOME NECESSARY AND SUFFICIENT CONDITIONS FOR EXISTENCE OF POSITIVE SOLUTIONS FOR THIRD ORDER SINGULAR SUBLINEAR MULTI-POINT BOUNDARY VALUE PROBLEMS[J]. Acta mathematica scientia,Series B, 2014, 34(6): 1795-1810.

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

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

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