Abstract
In this paper, we study the existence of nontrivial solution for the fourth-order three- point boundary value problem having the following formu(4) (t) + f (t, u(t)) = 0, 0 < t < 1,u(0) = α(η), u'(0) = u''(0) = 0, u(1) = βu(η),where η ∈ (0, 1), α, β ∈ R, f ∈ C ([0, 1] × R, R). We give sufficient conditions that allow us to obtain the existence of a nontrivial solution. And by using the Leray-Schauder nonlinear alternative we prove the existence of at least one solution of the posed problem. As an application, we also given some examples to illustrate the results obtained.
Highlights
Is a parameter, The authors presented the existence of positive solutions by using the Krasnosel’skii fixed point theorem
Motivated by the above works, the aim of this paper is to establish some sufficient conditions for the existence of solution for the fourth-order three-point boundary value problem (BVP)
Remark: We can give examples similar in relation to the Corollary 3.3, Theorem 3.4, and Corollary 3.5
Summary
Proof of this Corollary 3.3 is the same method in the proof Theorem 3.2. If one of the following conditions is fulfilled (1) There exists a constant p > 1 such that. Let M be given as in the proof of Theorem 3.1. To prove Theorem 3.4, we only need to prove that M < 1. Since α > 0, β > 0, and ζ < 0, we have Proof of this Theorem 3.4 is the same method in the proof Theorem 3.2. If one of the following conditions is holds (1) There exists a constant p > 1 such that.
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