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This article considers elliptic problems of Kirchhoff-type. We give some new definitions of lower and upper solutions for the problem and establish the method of lower and upper solutions when the upper and lower solutions are well ordered, i.e., the lower solution is less than the upper one, and we also consider the case when the upper and lower solutions have opposite ordering. In addition we use the relation between the topological degree and strict upper and lower solutions in both cases and using this we obtain multiplicity results for nonlinear Kirchhoff-type elliptic problems.

In this article, using the sub-supersolution method and Rabinowitz-type global bifurcation theory, we prove some results on existence, uniqueness and multiplicity of positive solutions for some singular nonlocal elliptic problems.

In this article, we consider the singular elliptic boundary-value problem $$ -\Delta u+f(u)-u^{-\gamma} =\lambda u \text{ in } \Omega,\quad u>0\text{ in } \Omega,\quad u=0 \text{ on } \partial\Omega. $$ Using the upper-lower solution method, we show the existence and uniqueness of the solution. Also we study the boundary behavior and asymptotic behavior of the positive solutions.

In this article, we study the structure of global solutions to the boundary-value problem $$\displaylines{ -x''(t)+f(t,x)=\lambda ax(t),\quad t\in(0,1),\; t\neq\frac{1}{2},\cr \Delta x|_{t=1/2}=\beta_1 x(\frac{1}{2}),\quad \Delta x'|_{t=1/2}=-\beta_{2} x(\frac{1}{2}),\cr x(0)=x(1)=0, }$$ where $\lambda\neq0$, $\beta_1\geq\beta_{2}\geq0$, $\Delta x|_{t=1/2}=x(\frac{1}{2}+0)-x(\frac{1}{2})$, $\Delta x'|_{t=1/2}=x'(\frac{1}{2}+0)-x'(\frac{1}{2}-0)$, and $f:[0,1]\times\mathbb{R}\to\mathbb{R}$, $a:[0,1]\to(0,+\infty)$ are continuous. By a comparison principle and spectral properties of the corresponding linear equations, we prove the existence of solutions by using Rabinowitz-type global bifurcation theorems, and obtain results on the behavior of positive solutions for large $\lambda$ when $f(x)=x^{p+1}$.

This article presents global properties and existence of multiple solutions for a class of boundary value problems of impulsive differential equations. We first show that the spectral properties of the linearization of these problems are similar to the well-know properties of the standard Sturm-Liouville problems. These spectral properties are then used to prove two Rabinowitz-type global bifurcation theorems. Finally, we use the global bifurcation theorems to obtain multiple solutions for the above problems having specified nodal properties.

In this article, we study the existence and uniqueness of the positive solution for a second-order singular three-point boundary-value problem with sign-changing nonlinearities. Our main tool is a fixed-point theorem.

Using the fixed point theorem in cones, this paper shows the existence of multiple positive solutions for the singular $m$-point boundary-value problem $$displaylines{ x''(t)+a(t)f(t,x(t),x'(t))=0,quad 0

Using the theory of fixed point index, this paper discusses the existence of at least one positive solution and the existence of multiple positive solutions for the singular three-point boundary value problem: $$displaylines{ y''(t)+a(t)f(t,y(t),y'(t))=0,quad 0

Using a fixed point theorem in cones, this paper shows the existence of positive solutions for the singular three-point boundary-value problem $$displaylines{ x''(t)+a(t)f(t,x(t),x'(t))=0,quad 0

In this paper, Krasnoselskii's theorem and the fixed point theorem of cone expansion and compression are improved. Using the results obtained, we establish the existence of multiple positive solutions for the singular second-order boundary-value problems with derivative dependance on finite and infinite intervals.

Using a specially constructed cone and the fixed point index theory, this paper shows the existence of multiple positive solutions for a class of nonresonant singular boundary-value problem of second-order differential equations. The nonexistence of positive solution is also studied.

nUsing the fixed point theorem in cones, this paper shows the existence of multiple positive solutions for the singular m-point boundary-value problem x"(t)+a(t)f(t,x(t), x' (t)) = 0, 0 ∑i=1m-2ai < 1 and f maybe singular at x = 0 and x' = 0. [ABSTRACT FROM AUTHOR]

Using a fixed point theorem in cones, this paper shows the existence of positive solutions for the singular three-point boundary-value problem x″(t) + a(t)f(t, x(t), x′(t)) = 0, 0 < t < 1, x′(0) = 0, x(1) = αx(η), where 0 < &alpah; < 1, 0 < η < 1, and f may change sign and may be singular at x = 0 and x′ = 0. [ABSTRACT FROM AUTHOR]

Using the theory of fixed point index, this paper discusses the existence of at least one positive solution and the existence of multiple positive solutions for the singular three-point boundary value problem: y"(t) + a(t)f(t, y(t), y'(t)) = 0, 0 < t < 1, y'(0) = 0, y(1) = αy(η), where 0 < α < 1, 0 < η < 1, and f may be singular at y = 0 and y' = 0. [ABSTRACT FROM AUTHOR]