A note on multi-point boundary value problems

Multi-point boundary value problems for a second-order ordinary differential equation are considered in this note. An existence result is obtained with the help of coincidence degree theory.     

Second order singular boundary value problems with integral boundary conditions

We study the second order singular boundary value problem

     

Conditions for the existence of positive solutions covering a class of boundary value problems in a uniform way

In this paper, we will give conditions which will guarantee the existence of positive solutions for a variety of second order boundary value problems, using the well-known Krasnoselskii fixed point theorem. We will deal with a specific differential equation meeting a specific initial condition and use a general boundary condition, involving a not necessarily linear functional. Our purpose is to pose conditions on that functional, which will guarantee that the Krasnoselskii fixed point theorem can be applied. It is important to notice that only the values of this functional on two specific functions are involved in the conditions we pose. This paper unifies the way we deal with a wide variety of boundary value problems and provides results which, to the best of our knowledge, are new.     

Higher order boundary value problems with nonhomogeneous boundary conditions

We study the following nonlinear boundary value problem with nonhomogeneous multi-point boundary condition

     

Necessary and sufficient conditions for the existence of positive solutions of fourth order multi-point boundary value problems

This paper investigates the existence of positive solutions of singular multi-point boundary value problems of fourth order ordinary differential equation with p-Laplacian. A necessary and sufficient condition for the existence of C²[0,1] positive solution as well as pseudo-C³[0,1] positive solution is given by means of the fixed point theorems on cones.     

Existence and nonexistence of solutions of second-order nonlinear boundary value problems

We study the nonlinear boundary value problem consisting of the equation −y^″+q(t)y=w(t)f(y) on [a,b]$-y^{″}+q\left(t\right)y=w\left(t\right)f\left(y\right)\text{ on }[a,b]$ and a general separated homogeneous linear boundary condition. By comparing this problem with a corresponding linear Sturm–Liouville problem we obtain conditions for the existence and nonexistence of solutions of this problem. More specifically, let _λn,n=0,1,2,…${\lambda }_n,n=0,1,2,\dots$, be the n$n$-th eigenvalues of the corresponding linear Sturm–Liouville problem. Then under certain assumptions, the boundary value problem has a solution with exactly n$n$ zeros in (a,b)$(a,b)$ if _λn${\lambda }_n$ is in the interior of the range of f(y)/y,y∈(0,∞)$f\left(y\right)/y,y\in (0,\infty )$; and does not have any solution with exactly n$n$ zeros in (a,b)$(a,b)$ if _λn${\lambda }_n$ is outside of the range of f(y)/y,y∈(0,∞)$f\left(y\right)/y,y\in (0,\infty )$. These conditions become necessary and sufficient when f(y)/y$f\left(y\right)/y$ is monotone. The existences of multiple and even an infinite number of solutions are derived as consequences. We also discuss the changes of the number and the types of nontrivial solutions as the interval [a,b]$[a,b]$ shrinks, as q$q$ increases in a given direction, and as the boundary condition changes.     

Positive solutions for semi-positone three-point boundary value problems

Using a fixed point theorem of generalized cone expansion and compression we present in this paper criteria which guarantee the existence of at least two positive solutions for semi-positone three-point boundary value problems with parameter λ>0$\lambda >0$ belonging to a certain interval.     

Eigenvalues of semi-positone Hammerstein integral equations and applications to boundary value problems

Eigenvalue intervals and the existence of finitely many positive eigenfunctions for semi-positone Hammerstein integral equations are obtained. The positive characteristic values and their upper and lower bounds of the corresponding linear Hammerstein integral operators are studied. Applications of the results are given to third-order differential equations with three-point boundary conditions.