We establish some abstract convergence and Conley index continuation principles for families of singularly perturbed semilinear parabolic equations and apply them to reaction-diffusion equations with nonlinear boundary conditions and localized large diffusion. This extends and refines previous results of [A. Rodriguez-Bernal, Localized spatial homogenization and large diffusion, SIAM J. Math. Anal. 29 (1998), 1361-1380] and [J.M. Arrieta, A.N. Carvalho and A. Rodríguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities, J. Differential Equations 156 (1999), 376-406].

We study the large-interaction limit of an elliptic system modelling the steady states of two species $u$ and $v$ which compete to some extent throughout a domain $\Omega$ but compete strongly on a subdomain $A \subset \Omega$. In the strong-competition limit, $u$ and $v$ segregate on $A$ but not necessarily on $\Omega \setminus A$. The limit problem is a system on $\Omega \setminus A$ and a scalar equation on $A$ and in general admits an interesting range of types of solution, not all of which can be the strong-competition limit of coexistence states of the original system.

We consider the incompressible fluid motion described by the Navier-Stokes equations in a cylindrical domain $\Omega\subset\mathbb R^3$ under the slip boundary conditions. First we prove long time existence of regular solutions such that $v\in W_2^{2,1}(\Omega\times(0,T))$, $\nabla p\in L_2(\Omega\times(0,T))$, where $v$ is the velocity of the fluid and $p$ the pressure. To show this we need smallness of $\|v_{,x_3}(0)\|_{L_2(\Omega)}$ and $\|f_{,x_3}\|_{L_2(\Omega\times(0,T))}$, where $f$ is the external force and $x_3$ is the axis along the cylinder. The above smallness restrictions mean that the considered solution remains close to the two-dimensional solution, which, as is well known, is regular.

Having $T$ sufficiently large and imposing some decay estimates on $\|f(t)\|_{L_2(\Omega)}$ we continue the local solution step by step up to the global one.

We are concerned with the existence, multiplicity and uniqueness of positive solutions for the $2n$-order boundary value problem $$ \begin{cases} (-1)^nu^{(2n)}=f(t,u,u',-u''',\ldots, \\\hskip 1.8cm (-1)^{i-1}u^{(2i-1)},\ldots, (-1)^{n-1}u^{(2n-1)}), \\ u^{(2i)}(0)=u^{(2i+1)}(1)=0, \quad i=0,\ldots,n-1, \end{cases} $$ where $n\geq 2$ and $f\in C([0,1]\times \mathbb{R}_+^{n+1},\mathbb{R}_+)$ $(\mathbb{R}_+:=[0,\infty))$ depends on $u$ and all derivatives of odd orders. Our main hypotheses on $f$ are formulated in terms of the linear function $g(x):=x_1+2\sum_{i=2}^{n+1}x_i$. We use fixed point index theory to establish our main results, based on a priori estimates achieved by utilizing some integral identities and an integral inequality. Finally, we apply our main results to establish the existence, multiplicity and uniqueness of positive symmetric solutions for a Lidostone problem involving an open question posed by P.W. Eloe in 2000.

In this paper, we are concerned with the existence of (component-wise) positive solutions for a semilinear elliptic system, where the nonlinear term is superlinear in one equation and sublinear in the other equation. By constructing a cone $K_1 \times K_2$ which is the Cartesian product of two cones in space $C(\overline{\Omega})$ and computing the fixed point index in $K_1 \times K_2$, we establish the existence of positive solutions for the system. It is remarkable that we deal with our problem on the Cartesian product of two cones, in which the features of two equations can be exploited better.

In this paper, we study the topological structure of solution sets for the first-order differential inclusions with nonlocal conditions: $$ \begin{cases} y'(t) \in F(t,y(t)) &\text{a.e. } t\in [0,b],\\ y(0)+g(y)=y_0, \end{cases} $$ where $F\colon [0,b]\times \mathbb{R}^n\to{\mathcal P}(\mathbb{R}^n)$ is a multivalued map. Also, some geometric properties of solution sets, $R_{\delta}$, $R_\delta$-contractibility and acyclicity, corresponding to Aronszajn-Browder-Gupta type results, are obtained. Finally, we present the existence of viable solutions of differential inclusions with nonlocal conditions and we investigate the topological properties of the set constituted by these solutions.

This paper presents existence results for systems of first order inclusions on time scales with an initial or a periodic boundary value condition. The method of solution-tube is developed for this system.

We consider the initial-boundary value problem for a modified Swift-Hohenberg equation in space dimension $n\leq 7 $. Based on the semigroup theory, we formulate this problem as an abstract evolutionary equation with sectorial operator in the main part. We show that the semigroup generated by this problem admits a global attractor in the phase space $H^2(\Omega)\cap H^{1}_{0}(\Omega)$ and characterize the contents of the attractor.

In this paper, several versions extension of Leggett-Williams norm-type theorem for coincidences are given and proved to obtain the positive solutions of the operator equation $Mx=Nx$, where $M$ is a quasi-linear operator and $N$ is nonlinear. Moreover, as an application, the existence of positive solutions for multi-point boundary value problem with a $p$-Laplacian is obtained by one of those theorems.

In this paper, we shall give some results about best proximity points of cyclic $\varphi$-contractions in ordered metric spaces. These results generalize some known results.