Elastic beam problem with higher order derivatives

Kaufmann, Eric R.; Kosmatov, Nickolai

doi:10.1016/j.nonrwa.2006.03.006

Cited by 13 publications

(12 citation statements)

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“…The nonlinearity f in [6,17] is not affected by the slope and bending moment, and the authors used the method of fixed point index on cone. We also refer to some other articles, for instance, [2,5,7,9,14,19].…”

Section: Introductionmentioning

confidence: 99%

Inequalities of Green’s functions and positive solutions to nonlocal boundary value problems

2020

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In this paper, we discuss the positive solutions of beam equations with the nonlinearities including the slope and bending moment under nonlocal boundary conditions involving Stieltjes integrals. We pose some inequality conditions on nonlinearities and the spectral radius conditions on associated linear operators. These conditions mean that the nonlinearities have superlinear or sublinear growth. The existence of positive solutions is obtained by fixed point index on cones in C 2 [0, 1], and some examples are given for beam equations subject to mixed integral and multi-point boundary conditions with sign-changing coefficients. MSC: Primary 34B18; 34B10; secondary 34B15 Keywords: Positive solution; Fixed point index; Cone; Spectral radius u(t) dB i (t) and α i [u] = 1 0 u(t) dA i (t) © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material.

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Section: Introductionmentioning

confidence: 99%

Inequalities of Green’s functions and positive solutions to nonlocal boundary value problems

2020

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“…Under the conditions that the nonlinearity f (t, x 1 , x 2 , x 3 , x 4 ) may have superlinear or sublinear growth in x 1 , x 2 , x 3 , x 4 , the existence of positive solutions is obtained. We also refer to some previous studies, for instance, [8][9][10][11][12]. Recently the existence of positive solutions was proved in [13] to the following problems:…”

Section: Introductionmentioning

confidence: 99%

Positive solutions of beam equations under nonlocal boundary value conditions

2019

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In this article, we study the fourth-order problem with the first and second derivatives in nonlinearity under nonlocal boundary value conditions2, 3). This equation describes the deflection of an elastic beam. Some inequality conditions on nonlinearity f are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index on a special cone in C 2 [0, 1]. Two examples are provided to support the main results under mixed boundary conditions involving multi-point with sign-changing coefficients and integral with sign-changing kernel. MSC: Primary 34B18; 34B10; secondary 34B15 Keywords: Positive solution; Fixed point index; Cone where β i [u] = 1 0 u(t) dB i (t) is Stieltjes integral with B i of bounded variation (i = 1, 2, 3). This equation describes the deflection of an elastic beam. Alves et al. [1] established the existence of positive solutions for the beam equationu (4) (t) = f t, u(t), u (t) under boundary conditions u(0) = u (0) = 0, u (1) = g u(1) , u (1) = 0 or u (1) = 0,

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

Section: Introductionmentioning

confidence: 99%

“…For fully fourth-order nonlinear BVPs with the boundary condition in BVP (1.1) or other boundary conditions, the existence of solution has discussed by several authors, see [14][15][16][17][18][19][20]. In [14], Kaufmann and Kosmatov considered a symmetric fully fourth-order nonlinear boundary value problem. They used a triple fixed point theorem of cone mapping to obtain existence results of triple positive symmetric solutions when f satisfies some range conditions dependent upon three positive parameters a, b and d. Since they did not give the method to determine these parameters, the range conditions are difficult to verify.…”

Section: Introductionmentioning

confidence: 99%