scholarly journals Global Structure of Positive Solutions for a Singular Fourth-Order Integral Boundary Value Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Wenguo Shen ◽  
Tao He
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fourth Order ◽  
Boundary Value ◽  
Global Bifurcation ◽  
Global Structure ◽  
Stieltjes Integral ◽  
Integral Boundary ◽  
Integral Boundary Value Problem

We consider fourth-order boundary value problemsu′′′′(t)=λh(t)f(u(t)),  0<t<1,  u(0)=∫01‍u(s)dα(s),  u′(0)=u(1)=u′(1)=0, where∫01‍u(s)dα(s)is a Stieltjes integral withα(t)being nondecreasing andα(t)being not a constant on[0,1];h(t)may be singular att=0andt=1,h∈C((0,1),[0,∞))withh(t)≢0on any subinterval of(0,1);f∈C([0,∞),[0,∞))andf(s)>0for alls>0, andf0=∞,  f∞=0,  f0=lims→0+f(s)/s,  f∞=lims→+∞f(s)/s.We investigate the global structure of positive solutions by using global bifurcation techniques.

scholarly journals Positive Solutions for a Fourth-Order Riemann–Stieltjes Integral Boundary Value Problem

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Yujun Cui ◽  
Donal O’Regan ◽  
Jiafa Xu
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fourth Order ◽  
Existence Theorems ◽  
Boundary Value ◽  
Linear Operators ◽  
Stieltjes Integral ◽  
Integral Boundary ◽  
Integral Boundary Value Problem ◽  
Existence Of Positive Solutions

In this paper, we use the fixed point index to study the existence of positive solutions for the fourth-order Riemann–Stieltjes integral boundary value problem −x4t=ft,xt,x′t,x″t,x″′t, t∈0,1x0=x′0=x″′1=0,x″0=αx″t, where f: 0,1×ℝ+×ℝ+×ℝ+×ℝ+⟶ℝ+ is a continuous function and αx″ denotes a linear function. Two existence theorems are obtained with some appropriate inequality conditions on the nonlinearity f, which involve the spectral radius of related linear operators. These conditions allow ft,z1,z2,z3,z4 to have superlinear or sublinear growth in zi,  i=1,2,3,4.

scholarly journals Positive Solutions of a Nonlinear Fourth-order Integral Boundary Value Problem

2016 ◽  
Vol 54 (1) ◽  
pp. 73-86 ◽  
Author(s):  
Slimane Benaicha ◽  
Faouzi Haddouchi
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fourth Order ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Integral Condition ◽  
Integral Boundary ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Integral Boundary Value Problem ◽  
Existence Of Positive Solutions

Abstract In this paper, the existence of positive solutions for a nonlinear fourth-order two-point boundary value problem with integral condition is investigated. By using Krasnoselskii’s fixed point theorem on cones, sufficient conditions for the existence of at least one positive solutions are obtained.

scholarly journals Global Structure of Positive Solutions for Some Second-Order Multipoint Boundary Value Problems

2017 ◽  
Vol 2017 ◽  
pp. 1-6 ◽  
Author(s):  
Hongyu Li ◽  
Junting Zhang
Keyword(s):  
Boundary Value Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Boundary Value ◽  
Global Bifurcation ◽  
Global Structure ◽  
Second Order ◽  
Bifurcation Method ◽  
Multipoint Boundary ◽  
Multipoint Boundary Value Problems

We investigate in this paper the following second-order multipoint boundary value problem:-(Lφ)(t)=λf(t,φ(t)),0≤t≤1,φ′0=0,φ1=∑i=1m-2βiφηi. Under some conditions, we obtain global structure of positive solution set of this boundary value problem and the behavior of positive solutions with respect to parameterλby using global bifurcation method. We also obtain the infinite interval of parameterλabout the existence of positive solution.

scholarly journals Existence and Multiplicity of Positive Solutions of a Nonlinear Discrete Fourth-Order Boundary Value Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Ruyun Ma ◽  
Yanqiong Lu
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fourth Order ◽  
Boundary Value ◽  
Global Bifurcation ◽  
Main Tool ◽  
Bifurcation Theorem ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

we show the existence and multiplicity of positive solutions of the nonlinear discrete fourth-order boundary value problemΔ4ut-2=λhtfut,t∈T2,u1=uT+1=Δ2u0=Δ2uT=0, whereλ>0,h:T2→(0,∞)is continuous, andf:R→[0,∞)is continuous,T>4,T2=2,3,…,T. The main tool is the Dancer's global bifurcation theorem.

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