The existence of countably many positive solutions for singular multipoint boundary value problems

In this paper, we study the existence of countably many positive solutions for a singular multipoint boundary value problem. By using fixed-point index theory and the Leggett-Williams’ fixed-point theorem, sufficient conditions for the existence of countably many positive solutions are established.     

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.     

Positive solutions for second-order four-point boundary value problems with alternating coefficient

In this paper, we investigate the existence of positive solutions for a class of nonlinear second-order four-point boundary value problems with alternating coefficient. Our approach relies on the Krasnosel’skii fixed point theorem. The result of this paper is new and extent the previously known result.     

Triple positive solutions for boundary value problems on infinite intervals

This paper deals with the existence of triple positive solutions for Sturm–Liouville boundary value problems of second-order nonlinear differential equation on a half line. By using a fixed point theorem in a cone due to Avery–Peterson, we show the existence of at least three positive solutions with suitable growth conditions imposed on the nonlinear term.     

Positive solutions for 2p-order and 2q-order systems of singular boundary value problems with integral boundary conditions

In this paper, we study the existence, multiplicity and nonexistence of positive solutions for 2p-order and 2q-order systems of singular boundary value problems with integral boundary conditions. The results are based upon the fixed-point theorem of cone expansion and compression type due to Krasnosel’skill. Moreover, it generalizes and includes some known results.     

The existence of multiple positive solutions to boundary value problems of nonlinear delay differential equations with countably many singularities on infinite interval

In this paper, we consider the existence of countably many positive solutions to a boundary value problem of a nonlinear delay differential equation with countably many singularities on infinite interval

     

Second-order boundary value problems with integral boundary conditions

In this paper we investigate the existence of positive solutions of nonlocal second-order boundary value problems with integral boundary conditions.     

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.     

Existence of multiple positive solutions of singular nonlinear boundary value problems

For a given positive integer N, we provide conditions on the nonlinear function f which guarantee that the boundary value problem

     

Existence and uniqueness for a class of quasilinear elliptic boundary value problems

We prove existence and uniqueness of positive solutions for the boundary value problem

(rN−1φ(u′))′=−λrN−1f(u),u′(0)=u(1)=0,     

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.