TAILIEUCHUNG - Existence of three solutions to a non-homogeneous multi-point BVP of second order differential equations
This paper is concerned with a non-homogeneous multi-point boundary value problem of second order differential equation with one-dimensional p-Laplacian. Using multiple fixed point theorems, sufficient conditions to guarantee the existence of at least three solutions of this kind of BVP are established. Two examples are presented to illustrate the main results. | Turk J Math 35 (2011) , 55 – 86. ¨ ITAK ˙ c TUB doi: Existence of three solutions to a non-homogeneous multi-point BVP of second order differential equations Yuji Liu Abstract This paper is concerned with a non-homogeneous multi-point boundary value problem of second order differential equation with one-dimensional p-Laplacian. Using multiple fixed point theorems, sufficient conditions to guarantee the existence of at least three solutions of this kind of BVP are established. Two examples are presented to illustrate the main results. Key word and phrases: Second order differential equation with p-Laplacian, generalized Sturm-Liouville boundary value problem, fixed point theorem in cone. 1. Introduction In recent years, the solvability of multi-point boundary-value problems (BVPs for short) for second order differential equations or higher order differential equations on finite intervals have been studied by different authors, see papers [1–29]. The methods used in above mentioned papers, are the Guo-Krasnoselskii fixed point theorem, the fixed-point theorem due to Avery and Peterson, the Leggett-Williams fixed point theorem, the five functional fixed point theorem, the monotone iterative techniques and Mawhin coincidence degree theory, et cetera. Ma in [21, 22] studied the following more generalized BVP ⎧ ⎨ [p(t)x (t)] − q(t)x(t) + f(t, x(t)) = 0, αx(0) − βp(0)x (0) = m i=1 ai x(ξi ), ⎩ m γx(1) + δp(1)x (1) = i=1 bi x(ξi ), t ∈ (0, 1), (1) where 0 0 . By using Green s functions (which complicate the studies of BVP(1)) and Guo-Krasnoselskii fixed point theorem, the existence and multiplicity of positive solutions for BVP(1) were given. There has been a large number of papers in which many exciting results concerned with the existence of 2000 AMS Mathematics Subject Classification: 4B10, 34B15, 35B10. Supported by Natural Science Foundation of Guangdong province (No:7004569) and Natural Science Foundation of Hunan province, .
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