Positive solutions to singular semipositone boundary value problems of second order coupled differential systems

We establish the existence of positive solutions for the second order singular semipositone coupled Dirichlet systems $$\left\{ \begin{aligned} &x{''} +f_1 \bigl(t,y(t)\bigr)+e_1(t)=0, \\ &y{''} +f_2\bigl(t,x(t) \bigr)+e_2(t)=0, \\ &x(0)=x(1)=0,\qquad y(0)=y(1)=0. \end{aligned} \right. $$ The proof relies on Schauder’s fixed point theorem.     

Positive solutions for nonlinear singular boundary value problems

Some sufficient conditions for the existence of positive solutions to Dirichlet boundary value problems of a class of nonlinear second order differential equations are given.     

Existence of solutions to the nonlinear,singular second order Bohr boundary value problems

This paper investigates the existence of solutions for nonlinear systems of second order, singular boundary value problems (BVPs) with Bohr boundary conditions. A key application that arises from this theory is the famous Thomas–Fermi equations for the model of the atom when it is in a neutral state. The methodology in this paper uses an alternative and equivalent BVP, which is in the class of resonant singular BVPs, and thus this paper obtains novel results by implementing an innovative differential inequality, Lyapunov functions and topological techniques. This approach furnishes new results in the area of singular BVPs for a priori bounds and existence of solutions, where the BVP has unrestricted growth conditions and subject to the Bohr boundary conditions. In addition, the results can be relaxed and hold for the non-singular case too.     

Positive solutions for third order multi-point singular boundary value problems

We study a third order singular boundary value problem with multi-point boundary conditions. Sufficient conditions are obtained for the existence of positive solutions of the problem. Recent results in the literature are significantly extended and improved. Our analysis is mainly based on a nonlinear alternative of Leray-Schauder.     

Positive solutions for higher order singular p-Laplacian boundary value problems

This paper investigates 2m-th (m ≥ 2) order singular p-Laplacian boundary value problems, and obtains the necessary and sufficient conditions for existence of positive solutions for sublinear 2m-th order singular p-Laplacian BVPs on closed interval.     

Positive solutions to superlinear singular boundary value problems

New existence results are presented for the second-order equation y″ + f(t,y) = 0, 0<t<1 with Dirichlet or mixed boundary data. In our theory the nonlinearity f is allowed to change sign. Singularities at y = 0, t = 0 and t = 1 are discussed.     

Extremal solutions of second order nonlinear periodic boundary value problems

We study a periodic boundary value problem for a nonlinear ordinary differential equation of second order when the nonlinearity is given by a Carathéodory function. We generalize the monotone iterative method to cover the fully nonlinear case.     

Positive solutions of a singular nonlinear third order two-point boundary value problem

In this paper some existence results of positive solutions for the following singular nonlinear third order two-point boundary value problem:

     

On the approximation of nonlinear singular self-adjoint second order boundary value problems

We investigate the approximation of the solutions of a class of nonlinear second order singular boundary value problems with a self-adjoint linear part. Our strategy involves two ingredients. First, we take advantage of certain boundary condition functions to obtain well behaved functions of the solutions. Second, we integrate the problem over an interval that avoids the singularity. We are able to prove a uniform convergence result for the approximate solutions. We describe how the approximation is constructed for the various values of the deficiency index associated with the differential equation. The solution of the nonlinear problem is obtained by a globally convergent iterative method.     

Positive solutions for second order boundary value problems with derivative terms

This paper weals with the existence of positive solutions of the fully second‐order boundary value problem where is continuous. Under the conditions that the nonlinearity may be superlinear or sublinear growth on x and y, the existence results of positive solutions are obtained. For the superlinear case, a Nagumo‐type condition is presented to restrict the growth of f on y. The superlinear and sublinear growth of the nonlinearity f are described by inequality conditions instead of the usual upper and lower limits conditions. Our discussion is based on the fixed point index theory in cones.     

Positive solutions of higher-order singular boundary value problems

Some results of existence of positive solutions for singular boundary value problem $$\left\{\begin{array}{l}\displaystyle (-1)^{m}u^{(2m)}(t)=p(t)f(u(t)),\quad t\in(0,1),\\[2mm]\displaystyle u^{(i)}(0)=u^{(i)}(1)=0,\quad i=0,\ldots,m-1,\end{array}\right.$$ are given, where the function p(t) may be singular at t=0,1. Our analysis relies on the variational method.     

Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations

In this paper, we study the three-point boundary value problems for systems of nonlinear second order ordinary differential equations of the form

Under some conditions, we show the existence and multiplicity of positive solutions of the above problem by applying the fixed point index theory in cones.     

Positive solutions for singular semi-positone boundary value problems

In this paper, some new results about the existence of positive solutions for singular semi-positone boundary value problems are obtained. The results of this paper partially improve the former corresponding work.