Abstract
Positive solutions for a kind of third-order multipoint boundary value problem under the nonresonant conditions and the resonant conditions are considered. In the nonresonant case, by using the Leggett-Williams fixed point theorem, the existence of at least three positive solutions is obtained. In the resonant case, by using the Leggett-Williams norm-type theorem due to O’Regan and Zima, the existence result of at least one positive solution is established. It is remarkable to point out that it is the first time that the positive solution is considered for the third-order boundary value problem at resonance. Some examples are given to demonstrate the main results of the paper.
Highlights
We consider the existence of positive solutions for third-order m-point boundary value problem: x (t) + f (t, x (t)) = 0, t ∈ [0, 1], x (0) = 0, x (0) = 0, m−2 (1)x (1) = ∑ βix, i=1 where 0 < ξ1 < ξ2 < ⋅ ⋅ ⋅ < ξm−2 < 1, 0 ≤ βi ≤ 1, i = 1, 2, . . . , m − 2, ∑mi=−12 βi ≤ 1, and f ∈ C([0, 1] × [0, ∞), R).If condition ∑mi=−12 βi = 1 holds, the problem is called resonant boundary value problem or boundary value problem at resonance
Operator L : dom L ⊂ X → Y is called a Fredholm operator with index zero, that is, Im L is closed and dim Ker L = codim Im L < ∞, which implies that there exist continuous projections P : X → X and Q : Y → Y such that Im P = Ker L and Ker Q = Im L
From the concavity of x(t), we have ξi (x (1) − x (0)) ≤ x − x (0). Multiplying both sides with βi and considering x(1) = ∑mi=−12 βix(ξi), we have m−2 m−2 (1 − ∑ βiξi) x (1) ≥ ∑ βi (1 − ξi) x (0) . (27). This completes the proof of Lemma 7
Summary
We consider the existence of positive solutions for third-order m-point boundary value problem: x (t) + f (t, x (t)) = 0, t ∈ [0, 1] , x (0) = 0, x (0) = 0, m−2. The existence of positive solutions for nonresonant twopoint or three-point boundary value problems (Bvp for short) for nonlinear third-order ordinary differential equations has been studied by several authors. For more existence results of positive solutions for boundary value problems of third-order ordinary differential equations, one can see [5,6,7,8,9,10,11,12] and references therein. No paper deal with the existence result of positive solution for resonant third-order boundary value problems.
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