scholarly journals Existence of three weak solutions for fourth-order elastic beam equations on the whole space

2020 ◽  
Vol 72 (12) ◽  
pp. 1697-1707
Author(s):  
M. R. H. Tavani
Keyword(s):  
Critical Point ◽  
Fourth Order ◽  
Elastic Beam ◽  
Point Theory ◽  
Order Problem ◽  
Multiplicity Results ◽  
Nonlinear Anal ◽  
Beam Equations ◽  
Whole Space ◽  
Fourth Order Problem

UDC 517.9 Multiplicity results for a perturbed fourth-order problem on the real line with a perturbed nonlinear term depending on one real parameter is investigated. Our approach is based on variational methods and critical point theory which are obtained in [G. Bonanno, <em>A critical point theorem via the Ekeland variational principle</em>, Nonlinear Anal., <strong>75</strong>, 2992-3007 (2012)].

scholarly journals Two solutions to Kirchhoff-type fourth-order implusive elastic beam equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jian Liu ◽  
Wenguang Yu
Keyword(s):  
Iterative Method ◽  
Variational Methods ◽  
Critical Point ◽  
Fourth Order ◽  
Elastic Beam ◽  
Kirchhoff Type ◽  
Newton Iterative Method ◽  
Beam Equations ◽  
Critical Point Theorems ◽  
Elastic Beam Equations

AbstractIn this paper, the existence of two solutions for superlinear fourth-order impulsive elastic beam equations is obtained. We get two theorems via variational methods and corresponding two-critical-point theorems. Combining with the Newton-iterative method, an example is presented to illustrate the value of the obtained theorems.

scholarly journals Positive solutions of beam equations under nonlocal boundary value conditions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Shenglin Wang ◽  
Jialong Chai ◽  
Guowei Zhang
Keyword(s):  
Positive Solutions ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Elastic Beam ◽  
Mixed Boundary Conditions ◽  
Order Problem ◽  
Beam Equations ◽  
Existence Of Positive Solutions ◽  
Fourth Order Problem

AbstractIn this article, we study the fourth-order problem with the first and second derivatives in nonlinearity under nonlocal boundary value conditions $$\begin{aligned}& \left \{ \textstyle\begin{array}{l}u^{(4)}(t)=h(t)f(t,u(t),u'(t),u''(t)),\quad t\in(0,1),\\ u(0)=u(1)=\beta_{1}[u],\qquad u''(0)+\beta_{2}[u]=0,\qquad u''(1)+\beta_{3}[u]=0, \end{array}\displaystyle \right . \end{aligned}$$ {u(4)(t)=h(t)f(t,u(t),u′(t),u″(t)),t∈(0,1),u(0)=u(1)=β1[u],u″(0)+β2[u]=0,u″(1)+β3[u]=0, where $f: [0,1]\times\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{R}_{-}\to \mathbb{R}_{+}$f:[0,1]×R+×R×R−→R+ is continuous, $h\in L^{1}(0,1)$h∈L1(0,1) and $\beta_{i}[u]$βi[u] is Stieltjes integral ($i=1,2,3$i=1,2,3). 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 $C^{2}[0,1]$C2[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.

scholarly journals The Existence of Symmetric Positive Solutions of Fourth-Order Elastic Beam Equations

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 121 ◽  
Author(s):  
Münevver Tuz
Keyword(s):  
Fixed Point ◽  
Positive Solutions ◽  
Fourth Order ◽  
Eigenvalue Problems ◽  
Fixed Point Theory ◽  
Elastic Beam ◽  
Point Theory ◽  
Beam Equations ◽  
Symmetric Positive Solutions ◽  
Elastic Beam Equations

In this study, we consider the eigenvalue problems of fourth-order elastic beam equations. By using Avery and Peterson’s fixed point theory, we prove the existence of symmetric positive solutions for four-point boundary value problem (BVP). After this, we show that there is at least one positive solution by applying the fixed point theorem of Guo-Krasnosel’skii.

scholarly journals S-Shaped Connected Component for Nonlinear Fourth-Order Problem of Elastic Beam Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Jinxiang Wang ◽  
Ruyun Ma ◽  
Jin Wen
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fourth Order ◽  
Boundary Value ◽  
Elastic Beam ◽  
Connected Components ◽  
Beam Equation ◽  
Connected Component ◽  
Order Problem ◽  
Fourth Order Problem

We investigate the existence of S-shaped connected component in the set of positive solutions of the fourth-order boundary value problem: u′′′′x=λhxfux, x∈(0,1),u(0)=u(1)=u′′0=u′′1=0, where λ>0 is a parameter, h∈C[0,1], and f∈C[0,∞) with f0≔lims→0⁡(f(s)/s)=∞. We develop a bifurcation approach to deal with this extreme situation by constructing a sequence of functions f[n] satisfying f[n]→f and (f[n])0→∞. By studying the auxiliary problems, we get a sequence of unbounded connected components C[n], and, then, we find an unbounded connected component C in the set of positive solutions of the fourth-order boundary value problem which satisfies 0,0∈C⊂lim⁡sup⁡C[n] and is S-shaped.

Further study of a fourth-order elliptic equation with negative exponent

2011 ◽  
Vol 141 (3) ◽  
pp. 537-549 ◽  
Author(s):  
Zongming Guo ◽  
Zhongyuan Liu
Keyword(s):  
Fourth Order ◽  
Blow Up ◽  
Smooth Domain ◽  
Regular Solutions ◽  
Order Problem ◽  
Bounded Smooth Domain ◽  
Main Technique ◽  
Fourth Order Elliptic Equation ◽  
Fourth Order Problem ◽  
Bounded Smooth

We continue to study the nonlinear fourth-order problem TΔu – DΔ2u = λ/(L + u)2, –L < u < 0 in Ω, u = 0, Δu = 0 on ∂Ω, where Ω ⊂ ℝN is a bounded smooth domain and λ > 0 is a parameter. When N = 2 and Ω is a convex domain, we know that there is λc > 0 such that for λ ∊ (0, λc) the problem possesses at least two regular solutions. We will see that the convexity assumption on Ω can be removed, i.e. the main results are still true for a general bounded smooth domain Ω. The main technique in the proofs of this paper is the blow-up argument, and the main difficulty is the analysis of touch-down behaviour.

Some nonexistence theorems for semilinear fourth-order equations

2018 ◽  
Vol 149 (03) ◽  
pp. 761-779 ◽  
Author(s):  
M. Á. Burgos-Pérez ◽  
J. García-Melián ◽  
A. Quaas
Keyword(s):  
Fourth Order ◽  
Exterior Domains ◽  
Order Problem ◽  
Previous Condition ◽  
Fourth Order Problem ◽  
Radially Symmetric Solutions ◽  
Image Position ◽  
Fourth Order Equations

AbstractIn this paper, we analyse the semilinear fourth-order problem ( − Δ)2 u = g(u) in exterior domains of ℝN. Assuming the function g is nondecreasing and continuous in [0, + ∞) and positive in (0, + ∞), we show that positive classical supersolutions u of the problem which additionally verify − Δu &gt; 0 exist if and only if N ≥ 5 and $$\int_0^\delta \displaystyle{{g(s)}\over{s^{(({2(N-2)})/({N-4}))}}} {\rm d}s \lt + \infty$$ for some δ &gt; 0. When only radially symmetric solutions are taken into account, we also show that the monotonicity of g is not needed in this result. Finally, we consider the same problem posed in ℝN and show that if g is additionally convex and lies above a power greater than one at infinity, then all positive supersolutions u automatically verify − Δu &gt; 0 in ℝN, and they do not exist when the previous condition fails.

scholarly journals Infinitely many solutions for nonlocal problems with variable exponent and nonhomogeneous Neumann conditions

2019 ◽  
Vol 38 (4) ◽  
pp. 71-96 ◽  
Author(s):  
Shapour Heidarkhani ◽  
Anderson Luis Albuquerque de Araujo ◽  
Amjad Salari
Keyword(s):  
Variational Methods ◽  
Critical Point ◽  
Critical Point Theory ◽  
Variable Exponent ◽  
Point Theory ◽  
Infinitely Many Solutions ◽  
Nonlocal Problems ◽  
Multiplicity Results

In this article we will provide new multiplicity results of the solutions for nonlocal problems with variable exponent and nonhomogeneous Neumann conditions. We investigate the existence of infinitely many solutions for perturbed nonlocal problems with variable exponent and nonhomogeneous Neumann conditions. The approach is based on variational methods and critical point theory.

scholarly journals Periodic Solutions for Semilinear Fourth-Order Differential Inclusions via Nonsmooth Critical Point Theory

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Bian-Xia Yang ◽  
Hong-Rui Sun
Keyword(s):  
Critical Point ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Critical Point Theory ◽  
Differential Inclusions ◽  
Point Theory ◽  
Nonsmooth Critical Point Theory ◽  
Critical Point Theorem ◽  
Related Literature ◽  
Nonsmooth Critical Point

Three periodic solutions with prescribed wavelength for a class of semilinear fourth-order differential inclusions are obtained by using a nonsmooth version critical point theorem. Some results of previous related literature are extended.

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