Abstract
Using the method of lower and upper solutions, we study the following singular nonlinear three-point boundary value problems: , where K ∈ C[0,1] ,0 α η λ is a positive parameter and present the existence, uniqueness, and the dependency on parameters of the positive solutions under various assumptions. Our result improves those in the previous literatures.
Highlights
Introduction and Main ResultsIn this paper, we consider the three-point boundary value problem= x(0) 0=, x(1) ax(η), (1.1)where K ∈ C[0,1], 0 < a < 1, 0 < η < 1, and λ is a positive parameter
Using the method of lower and upper solutions, we study the following singular nonlinear three-point boundary value problems:
The m-point boundary value problem for linear second-order ordinary differential equations was initiated by Ilin and Moiseev [1] [2]
Summary
Where K ∈ C[0,1] , 0 < a < 1 , 0 < η < 1 , and λ is a positive parameter. The m-point boundary value problem for linear second-order ordinary differential equations was initiated by Ilin and Moiseev [1] [2]. The method of upper and lower solutions is very important for the study of the boundary value problems, see [8]-[18]. Establishing the method of upper and lower solutions for three-point boundary value problems is necessary and important. Wei ([15]) constructed the method of upper and lower solutions for three-point boundary value problems and gave the sufficient and necessary conditions for the existence of positive solutions of the problem. An interesting result comes from [25], in which, using method of upper and lower solutions, Shi and Yao discussed the following problem. Under various appropriate assumptions on p, q and K (t) , we will obtain the existence and uniqueness of positive solution of problem (1.1) for λ in different circumstances.
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