Multiplicity of Positive Solutions for a Singular Second-Order Three-Point Boundary Value Problem with a Parameter

This paper is concerned with the following second-order three-point boundary value problem u″t+β2ut+λqtft,ut=0, t∈0 , 1, u0=0, u(1)=δu(η), where β∈(0,π/2), δ>0, η∈(0,1), and λ is a positive parameter. First, Green’s function for the associated linear boundary value problem is constructed, and then some useful properties of Green’s function are obtained. Finally, existence, multiplicity, and nonexistence results for positive solutions are derived in terms of different values of λ by means of the fixed point index theory

On existence of positive solutions of coupled integral boundary value problems for a nonlinear singular superlinear differential system

By constructing a special cone and using fixed point index theory, this paper investigates the existence of positive solutions of singular superlinear coupled integral boundary value problems for differential systems $$\left\{ \begin{array}{ll} -x''(t)=f_1(t,x(t),y(t)),\ \ t\in (0,1),&\\ -y''(t)=f_2(t,x(t),y(t)),\ \ t\in (0,1),&\\ x(0)=y(0)=0, x(1)=\alpha[y], y(1)=\beta[x],& \end{array} \right.$$ where $\alpha[x], \beta[x]$ are bounded linear functionals on $C[0,1]$ given by $$\alpha[x]=\int_0^1x(t)dA(t), beta[x]=\int_0^1x(t)dB(t)$$ with $A,B$ functions of bounded variation with positive measures

Nonexistence of positive solutions of nonlinear boundary value problems

We discuss the nonexistence of positive solutions for nonlinear boundary value problems. In particular, we discuss necessary restrictions on parameters in nonlocal problems in order that (strictly) positive solutions exist. We consider cases that can be written in an equivalent integral equation form which covers a wide range of problems. In contrast to previous work, we do not use concavity arguments, instead we use positivity properties of an associated linear operator which uses ideas related to the $u_0$-positive operators of Krasnosel'skii