Abstract
Positive solutions for a kind of third-order multipoint boundary value problem under the non-resonant conditions and the resonant conditions are considered. In the nonresonant case, by using Leggett-Williams fixed-point theorem, the existence of at least three positive solutions is obtained. In the resonant case, by using Leggett-Williams norm-type theorem due to O’Regan and Zima, existence result of at least one positive solution is established. The results obtained are valid and new for the problem discussed. Two examples are given to illustrate the main results.
Highlights
We consider the third-order m-point boundary value problem given by x (t) + f (t, x (t)) = 0, t ∈ [0, 1]
We consider the positive solution for the nonresonant case with the condition 0 < ∑mi=−01 βi < 1 and we always suppose that f ∈ C([0, 1] × [0, ∞), [0, ∞))
We consider the third-order m-point boundary value problem given by x (t) + y (t) = 0, t ∈ [0, 1]
Summary
For more existence results of positive solutions for boundary value problems of third-order ordinary differential equations, one can see [6,7,8,9,10,11,12] and references therein. For boundary value problems of second-order or higherorder differential equations at resonance, many existence results of solutions have been established; see [13,14,15,16,17,18,19,20,21,22,23,24,25]. Few paper dealt with the existence result of positive solution for resonant third-order boundary value problems. (2) It is the first time that the positive solution is considered for third-order boundary value problem at resonance
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