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 of nonlinear elliptic boundary value problems

This paper considers the problem of finding positive vector-valued solutions U of the nonlinear elliptic boundary value problem L(U) + f(x, U) = 0 on a bounded region Ω, $\text{U +}\text{?U}\text{?v}\text{= 0}$ on ?Ω. The operator L is uniformly elliptic and in divergence form, and f is, roughly speaking, superlinear; by the positivity of U is meant the positivity of each component of U on Ω. Under certain growth conditions on f and some further technical assumptions, the existence of a positive solution is proved, an a priori bound on all positive solutions is obtained, and a certain fixed point index is proved equal to ? 1. As an example, information about fixed point indices is used to allow perturbations of the form ?h(x, U, DU). In the final section, an essentially best possible theorem is given for Ω a ball and for radially symmetric solutions of the Laplacian with Dirichlet boundary conditions.     

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 even order system periodic boundary value problems

We study a class of even order system boundary value problems with periodic boundary conditions. A series of criteria are obtained for the existence of one, two, any arbitrary number, and even a countably infinite number of positive solutions. Criteria for the nonexistence of positive solutions are also derived. As for the second order case, our results extend, improve, and supplement those in the literature for scalar and system boundary value problems. Several examples are given to demonstrate the applications. Moreover, we obtain conditions for system periodic boundary value problems of a different form to have nontrivial solutions by transforming our main results to such problems.     

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 for nonlinear periodic problems

We consider a nonlinear periodic problem driven by the scalar p-Laplacian and with a nonsmooth potential. Using the degree map for multivalued perturbations of (S)₊-operators and the spectrum of a weighted eigenvalue problem for the scalar periodic p-Laplacian, we prove the existence of a strictly positive solution. Michael E. Filippakis: Researcher supported by a grant of the National Scholarship Foundation of Greece (I.K.Y.)     

Positive solutions for systems of nonlinear discrete boundary value problems

Existence of eigenvalues yielding positive solutions for systems of second order discrete boundary value problems (two-point, three-point and generalized three-point) are established. The results are obtained by the use of a Guo–Krasnoselskii fixed point theorem in cones.     

Triple positive periodic solutions of nonlinear singular second-order boundary value problems

This paper studies the positive solutions of the nonlinear second-order periodic boundary value problem u″(t) + λ(t)u(t) = f(t,u(t)),a.e.t ∈ [0,2π],u(0) = u(2π),u′(0) = u′(2π),where f(t,u) is a local Carath′eodory function.This shows that the problem is singular with respect to both the time variable t and space variable u.By applying the Leggett–Williams and Krasnosel'skii fixed point theorems on cones,an existence theorem of triple positive solutions is established.In order to use these theorems,the exact a priori estimations for the bound of solution are given,and some proper height functions are introduced by the estimations.     

Positive solutions of second-order boundary value problems with sign-changing nonlinear terms

In this paper the existence results of positive solutions are obtained for second-order boundary value problem

−u″=f(t,u),t∈(0,1),u(0)=u(1)=0,     

Positive solutions of multi-point boundary value problems with multivalued operators at resonance

We establish new results on the existence of positive solutions for a kind multi-point boundary value problem with multivalued operator. Our results are based on a recent Leggett-Williams theorem for coincidences of multivalued operators due to O’Regan and Zima. The most interesting point is the acquisition of positive solutions for the resonance case. And an example is constructed to show that our result here is valid.