In this paper we establish the existence of single and multiple solutions to the positone discrete Dirichlet boundary value problem $$\left\{\begin{array}{c}\Delta \left[\phi \right(\Delta u(t-1)\left)\right]+q\left(t\right)f(t,u(t\left)\right)=0,t\in \{1,2,\cdots ,T\}\hfill \\ u\left(0\right)=u(T+1)=0,\hfill \end{array}\right.$$ where $\phi \left(s\right)={\left|s\right|}^{p-2}s$, $p>1$ and our nonlinear term $f(t,u)$ may be singular at $u=0$.

This paper studies the existence of solutions to the singular boundary value problem $$\left\}\begin{array}{c}-u^{\text{'}\text{'}}=g(t,u)+h(t,u),t\in (0,1),u\left(0\right)=0=u\left(1\right),\hfill \end{array}\right.$$ where $g(0,1)\times (0,\infty )\to ℝ$ and $h(0,1)\times [0,\infty )\to [0,\infty )$ are continuous. So our nonlinearity may be singular at $t=0,1$ and $u=0$ and, moreover, may change sign. The approach is based on an approximation method together with the theory of upper and lower solutions.

Dorsey, Di Bartolo and Dolgert (Di Bartolo et al., 1996; 1997) have constructed asymptotic matched solutions at order two for the half-space Ginzburg-Landau model, in the weak-$\kappa$ limit. These authors deduced a formal expansion for the superheating field in powers of ${\kappa }^{\frac{1}{2}}$ up to order four, extending the formula by De Gennes (De Gennes, 1966) and the two terms in Parr’s formula (Parr, 1976). In this paper, we construct asymptotic matched solutions at all orders leading to a complete expansion in powers...

Dorsey, Di Bartolo and Dolgert (Di Bartolo et al., 1996; 1997) have constructed asymptotic matched solutions at order two for the half-space Ginzburg-Landau model, in the weak-κ limit. These authors deduced a formal expansion for the superheating field in powers of ${\kappa }^{\frac{1}{2}}$ up to order four, extending the formula by De Gennes (De Gennes, 1966) and the two terms in Parr's formula (Parr, 1976). In this paper, we construct asymptotic matched solutions at all orders leading to a complete expansion...

In this paper we consider periodic and Dirichlet problems for second order vector differential inclusions. First we show the existence of extremal solutions of the periodic problem (i.e. solutions moving through the extreme points of the multifunction). Then for the Dirichlet problem we show that the extremal solutions are dense in the $C^1(T,R^N)$-norm in the set of solutions of the “convex” problem (relaxation theorem).

In this paper we study semilinear second order differential inclusions involving a multivalued maximal monotone operator. Using notions and techniques from the nonlinear operator theory and from multivalued analysis, we obtain “extremal” solutions and we prove a strong relaxation theorem.

We consider a simple boundary value problem at resonance for an ordinary differential equation. We employ a shift argument and construct a regular fixed point operator. In contrast to current applications of coincidence degree, standard fixed point theorems are applied to give sufficient conditions for the existence of solutions. We provide three applications of fixed point theory. They are delicate and an application of the contraction mapping principle is notably missing. We give a partial explanation...

We formulate nonuniform nonresonance criteria for certain third order differential systems of the form $X^{{}^{\text{'}\text{'}\text{'}}}+A{X}^{{}^{\text{'}\text{'}}}+G(t,X^{{}^{\text{'}}})+CX=P\left(t\right)$, which further improves upon our recent results in [12], given under sharp nonresonance considerations. The work also provides extensions and generalisations to the results of Ezeilo and Omari [5], and Minhós [9] from the scalar to the vector situations.