Existence of non-spurious solutions to discrete Dirichlet problems with lower and upper solutions

This paper investigates the solvability of discrete Dirichlet boundary value problems by the lower and upper solution method. Here, the second-order difference equation with a nonlinear right hand side f$f$ is studied and f(t,u,v)$f(t,u,v)$ can have a superlinear growth both in u$u$ and in v$v$. Moreover, the growth conditions on f$f$ are one-sided. We compute a priori bounds on solutions to the discrete problem and then obtain the existence of at least one solution. It is shown that solutions of the discrete problem will converge to solutions of ordinary differential equations.     

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.     

The method of upper–lower solutions for nonlinear second order differential inclusions

In this paper we consider a second order differential inclusion driven by the ordinary p-Laplacian, with a subdifferential term, a discontinuous perturbation and nonlinear boundary value conditions. Assuming the existence of an ordered pair of appropriately defined upper and lower solutions φ$\phi$ and ψ$\psi$ respectively, using truncations and penalization techniques and results from nonlinear and multivalued analysis, we prove the existence of solutions in the order interval [ψ,φ]$[\psi ,\phi ]$ and of extremal solutions in [ψ,φ]$[\psi ,\phi ]$. We show that our problem incorporates the Dirichlet, Neumann and Sturm–Liouville problems. Moreover, we show that our method of proof also applies to the periodic problem.     

Periodic solutions to a second order nonlinear neutral functional differential equation in the critical case

In this paper, the authors study the existence of periodic solutions for a second order neutral functional differential equation

(x(t)-cx(t-τ))^″=f(x(t))x^′(t)+g(t,x(t-μ(t)))+e(t)$(x(t)-\mathit{cx}(t-\tau ){)}^{″}=f(x(t))x^{\prime }(t)+g(t,x(t-\mu (t)))+e(t)$