Positive solutions for third order semipositone boundary value problems

We obtain some sufficient conditions for the existence of positive solutions of a third order semipositone boundary value problem with a multi-point boundary condition. Applications of our results to some special problems are also discussed.     

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 for higher order multi-point boundary value problems with nonhomogeneous boundary conditions

We study nth order boundary value problems with a nonlinear term f(t,x) subject to nonhomogeneous multi-point boundary conditions. Criteria for the existence of positive solutions of such problems are established. Conditions are determined by the relationship between the behavior of f(t,x)/x near 0 and ∞ when compared with the smallest positive characteristic value of an associated linear integral operator. This work improves and extends some recent results in the literature for the second order problems. The results are illustrated with examples.     

Some necessary and sufficient conditions for existence of positive solutions for third order singular super-linear multi-point boundary value problems

We mainly study the existence of positive solutions for the following third order singular super-linear multi-point boundary value problem $$ \left \{ \begin{array}{l} x^{(3)}(t)+ f(t, x(t), x'(t))=0,\quad0 where \(0\leq\alpha_{i}\leq\sum_{i=1}^{m_{1}}\alpha_{i}<1\) , i=1,2,…,m ₁, \(0<\xi_{1}< \xi_{2}< \cdots<\xi_{m_{1}}<1\) , \(0\leq\beta_{j}\leq\sum_{i=1}^{m_{2}}\beta_{i}<1\) , j=1,2,…,m ₂, \(0<\eta_{1}< \eta_{2}< \cdots<\eta_{m_{2}}<1\) . And we obtain some necessary and sufficient conditions for the existence of C ¹[0,1] and C ²[0,1] positive solutions by means of the fixed point theorems on a special cone. Our nonlinearity f(t,x,y) may be singular at t=0 and t=1.     

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:

     

Positive solutions of nonlinear multi-point boundary value problems

This paper deals with the existence of positive solutions of nonlinear differential equation

$$\begin{aligned} u^{\prime \prime }(t)+ a(t) f(u(t) )=0,\quad 0<t <1, \end{aligned}$$

subject to the boundary conditions

$$\begin{aligned} u(0)=\sum _{i=1}^{m-2} a_i u (\xi _i) ,\quad u^{\prime } (1) = \sum _{i=1}^{m-2} b_i u^{\prime } (\xi _i), \end{aligned}$$

where \( \xi _i \in (0,1) \) with \( 0< \xi _1<\xi _2< \cdots<\xi _{m-2} < 1,\) and \(a_i,b_i \) satisfy   \(a_i,b_i\in [0,\infty ),~~ 0< \sum _{i=1}^{m-2} a_i <1,\) and \( \sum _{i=1}^{m-2} b_i <1. \) By using Schauder’s fixed point theorem, we show that it has at least one positive solution if f is nonnegative and continuous. Positive solutions of the above boundary value problem satisfy the Harnack inequality

$$\begin{aligned} \displaystyle \inf _{0 \le t \le 1} u(t) \ge \gamma \Vert u\Vert _\infty . \end{aligned}$$

     

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.     

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.     

Positive solutions of multi-point boundary value problems at resonance

We establish new results on the existence of positive solutions for some multi-point boundary value problems at resonance. Our results are based on a recent Leggett–Williams norm-type theorem due to O’Regan and Zima. We also derive a new result for a three-point problem, previously studied by several authors.     

Positive solutions for singular third-order nonhomogeneous boundary value problems

In this paper, we investigate the existence of positive solutions for singular third-order nonhomogeneous boundary value problems. By using a fixed point theorem of cone expansion-compression type due to Krasnosel??skii, we establish various results on the existence or nonexistence of single and multiple positive solutions to the singular boundary problems in the explicit intervals for the nonhomogeneous term. An example is also given to illustrate some of the main results.     

Positive solutions to a multi-point higher order boundary value problem

The authors consider a higher order multi-point boundary value problem. Some existence and nonexistence results for positive solutions of the problem are obtained by using Krasnosel'skii's fixed point theorem. Examples are included to illustrate the results.     

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 for a system of second-order multi-point boundary value problems

In this paper, we present some results for positive solutions of a system of nonlinear second-order ordinary differential equations subject to multi-point boundary conditions.     

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.     

Positive solutions to certain classes of singular nonlinear second order boundary value problems

This paper is devoted to the problem of existence of solutions to the nonlinear singular two point boundary value problem , withy satisfying either mixed boundary datay(1)=Lim_y0+p(t)y(t)=0 or dirichlet boundary datay(0)=y(1)=0. Throughout our nonlinear termqf is allowed to be singular att=0,t=1,y=0 and/orpy=0.     

Positive solutions for multi-point boundary value problems for nonlinear fractional differential equations

The paper studies the problem of existence of positive solution to the following boundary value problem: $D_{0^ + }^\sigma u''(t) - g(t)f(u(t)) = 0$ , t ∈ (0, 1), u″(0) = u″(1) = 0, au(0) ? bu′(0) = Σ _i=1 ^m?2 a _i u(ξ _i ), cu(1) + du′(1) = Σ _i=1 ^m?2 b _i u(ξ _i ), where $D_{0^ + }^\sigma$ is the Riemann-Liouville fractional derivative of order 1 < σ ≤ 2 and f is a lower semi-continuous function. Using Krasnoselskii’s fixed point theorems in a cone, the existence of one positive solution and multiple positive solutions for nonlinear singular boundary value problems is established.