Multiplicity of solutions to a four-point boundary value problem of a differential system via variational approach

Ge, Weigao; Zhao, Zhihong

doi:10.1186/s13661-016-0559-x

Cited by 2 publications

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References 14 publications

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“…MI-Technique has also been done for four point BVPs. Ge et al [13] studied multiplicity of solution for four point BVPs via the variational approach and MI-Technique. Zhang et al [47] developed the upper and lower solution method and the monotone iterative technique and obtained some new existence results for the following fourth order four point BVPs…”

Section: Introductionmentioning

confidence: 99%

Region of existence of multiple solutions for a class of 4-point BVPs

Verma¹,

Urus²

2019

Preprint

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The aim of this article is to prove the existence of solution and compute the region of existence for a class of 4-point BVPs defined as,where I = [0, 1], 0 < ξ ≤ η < 1 and λ1, λ2 > 0. The non linear source term ψ ∈ C(I ×R×R, R) is one sided Lipschitz in u with Lipschitz constant L1 and Lipschitz in u with Lipschitz function L2(x), where L2(x) : I → R + such that L2(0) = 0 and L 2 (x) ≥ 0. The novelty in this paper allows us to use simplest form of computational iteration and existence is achieved with a restriction that L2 depends on x. We develop monotone iterative technique in well ordered and reverse ordered cases. We prove maximum anti-maximum principle under certain assumptions and use it to show the monotonic behavior of sequences of upper and lower solutions. The conditions derived in this paper are sufficient and have been verified for two examples. The method involves Newton's quasilinearization which involves a parameter k which is equivalent to ∂ψ ∂u . Our aim is to find a range of k so that the iterative technique is convergent.

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Section: Introductionmentioning

confidence: 99%