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 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 Two-Point Boundary Value Problems for Fourth-Order Differential Equations with Fully Nonlinear Terms

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Yixin Zhang ◽  
Yujun Cui
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Linear Operators ◽  
Continuous Space ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions

In this paper, we consider the existence of positive solutions for the fully fourth-order boundary value problem u 4 t = f t , u t , u ′ t , u ″ t , u ‴ t ,   0 ≤ t ≤ 1 , u 0 = u 1 = u ″ 0 = u ″ 1 = 0 , where f : 0,1 × 0 , + ∞ × − ∞ , + ∞ × − ∞ , 0 × − ∞ , + ∞ ⟶ 0 , + ∞ is continuous. This equation can simulate the deformation of an elastic beam simply supported at both ends in a balanced state. By using the fixed-point index theory and the cone theory, we discuss the existence of positive solutions of the fully fourth-order boundary value problem. We transform the fourth-order differential equation into a second-order differential equation by order reduction method. And then, we examine the spectral radius of linear operators and the equivalent norm on continuous space. After that, we obtain the existence of positive solutions of such BVP.

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