In this paper we consider the existence, multiplicity, and nonexistence of positive solutions to fractional differential equation with integral boundary conditions. Our analysis relies on the fixed point index.

This paper treats the question of the existence of solutions of a fourth order boundary value problem having the following form: $x^{\left(4\right)}\left(t\right)+f(t,x\left(t\right),x^{\text{'}\text{'}}\left(t\right))=0$, 0 < t < 1, x(0) = x’(0) = 0, x”(1) = 0, $x^{\left(3\right)}\left(1\right)=0$. Boundary value problems of very similar type are also considered. It is assumed that f is a function from the space C([0,1]×ℝ²,ℝ). The main tool used in the proof is the Leray-Schauder nonlinear alternative.

We develop the existence theory for sequential fractional differential equations involving Liouville-Caputo fractional derivative equipped with anti-periodic type (non-separated) and nonlocal integral boundary conditions. Several existence criteria depending on the nonlinearity involved in the problems are presented by means of a variety of tools of the fixed point theory. The applicability of the results is shown with the aid of examples. Our results are not only new in the given configuration...

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.