On the Existence of Positive Solutions for a Fourth-Order Boundary Value Problem

Zou, Yumei

doi:10.1155/2017/4946198

Cited by 10 publications

(4 citation statements)

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“…The purpose of this paper is to obtain existence results of solutions to the fully fourthorder BVP (1.1). For fourth-order BVPs with the boundary condition in BVP (1.1) or other boundary conditions, the existence of solutions has been discussed by several authors, see [14][15][16][17][18][19][20][21][22][23][24]. In [14], Kaufmann and Kosmatov considered a symmetric fully fourth-order nonlinear boundary value problem on [-1, 1].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Solvability for fully cantilever beam equations with superlinear nonlinearities

Chen

2019

Bound Value Probl

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This paper deals with the existence of solution for the fully fourth-order boundary value problem u (4) (x) = f (x, u(x), u (x), u (1), u (x)), x ∈ [0, 1], u(0) = u (0) = u (1) = u (1) = 0, which models a statically elastic beam fixed at the left and freed at the right, and it is called cantilever beam in mechanics, where f : [0, 1] × R 4 → R is continuous. Some inequality conditions on f guaranteeing the existence and uniqueness of solutions are presented. The inequality conditions allow f (x, y 0 , y 1 , y 2 , y 3) to grow superlinearly on y 0 , y 1 , y 2 , and y 3 .

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Solvability for fully cantilever beam equations with superlinear nonlinearities

Chen

2019

Bound Value Probl

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“…Nevertheless, the dependence of f on the third derivative u ‴ increases the difficulty for our study, but this is also a fundamental difference from the previous problems. In recent years, the research on the solvability of the elastic beam equation that f involves all lower-order derivatives of deformation function u has become a hot topic (see [5,6,[11][12][13][14][15][16][17][18][19][20][21][22]). For example, the elastic beam equation whose both ends are simply supported (see [14,16,20]):…”

Section: Introductionmentioning

confidence: 99%

Solvability for a Fully Elastic Beam Equation with Left-End Fixed and Right-End Simply Supported

Wei

2021

Mathematical Problems in Engineering

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The aim of the present paper is to consider a fully elastic beam equation with left-end fixed and right-end simply supported, i.e., u 4 t = f t , u t , u ′ t , u ″ t , u ‴ t , t ∈ 0,1 u 0 = u ′ 0 = u 1 = u ″ 1 = 0 , where f : 0,1 × ℝ 4 ⟶ ℝ is a continuous function. By applying Leray–Schauder fixed point theorem of the completely continuous operator, the existence and uniqueness of solutions are obtained under the conditions that the nonlinear function satisfies the linear growth and superlinear growth. For the case of superlinear growth, a Nagumo-type condition is introduced to limit that f t , x 0 , x 1 , x 2 , x 3 is quadratical growth on x 3 at most.

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“…The current analysis of these problems has a great interest and many methods are used to solve such problems. Recently, the study of existence of positive solution to fourth-order boundary value problems has gained much attention and is rapidly growing field, see [1,4,2,5,10,12,6,17,8,22]. However, the approaches used in the literature are usually topological degree theory and fixed-point theorems in cone [7].…”

Section: Introductionmentioning

confidence: 99%