Positive Solutions of Fourth-Order Problem Subject to Nonlocal Boundary Conditions

2020 ◽  
Vol 43 (5) ◽  
pp. 3675-3691
Author(s):  
Lin Han ◽  
Guowei Zhang ◽  
Hongyu Li
Keyword(s):  
Boundary Conditions ◽  
Positive Solutions ◽  
Fourth Order ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Order Problem ◽  
Fourth Order Problem

scholarly journals Fourth-order elliptic problems with critical nonlinearities by a sublinear perturbation

2021 ◽  
Vol 26 (2) ◽  
pp. 227-240
Author(s):  
Lin Li ◽  
Donal O’Regan
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Fourth Order ◽  
Local Minimum ◽  
Elliptic Problems ◽  
Order Problem ◽  
Critical Nonlinearities ◽  
Fourth Order Problem ◽  
Minimum Theorem ◽  
Sublinear Perturbation

In this paper, we get the existence of two positive solutions for a fourth-order problem with Navier boundary condition. Our nonlinearity has a critical growth, and the method is a local minimum theorem obtained by Bonanno.

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.

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