For an infinite Toeplitz matrix T with nonnegative real entries we find the conditions under which the equation $\boldsymbol {x}=T\boldsymbol {x}$ , where $\boldsymbol {x}$ is an infinite vector column, has a nontrivial bounded positive solution. The problem studied in this paper is associated with the asymptotic behaviour of convolution-type recurrence relations and can be applied to problems arising in the theory of stochastic processes and other areas.

We investigate the approximate controllability of a size- and space-structured population model, for which the control function acts on a subdomain and corresponds to the migration of individuals. We establish the main result via the unique continuation property of the adjoint system. The desired controller is the minimizer of an infinite-dimensional optimization problem.

Let $X$ be a nonempty subset of a normed space such that $0\notin X$ and $X$ is symmetric with respect to $0$ and let $Y$ be a Banach space. We study the generalised hyperstability of the Drygas functional equation

$$\begin{eqnarray}f(x+y)+f(x-y)=2f(x)+f(y)+f(-y),\end{eqnarray}$$

$f$

$X$

$Y$

$x,y\in X$

$x+y,x-y\in X$

In this paper we present a simple (fixed point) method that yields various results concerning approximate solutions of some difference equations. The results are motivated by the notion of Ulam stability.

In this paper, we consider the Neumann boundary value problem with a parameter λ∈(0,∞): By using fixed point theorems in a cone, we obtain some existence, multiplicity and nonexistence results for positive solutions in terms of different values of λ. We also prove an existence and uniqueness theorem and show the continuous dependence of solutions on the parameter λ.

In the context of self-stabilizing processes, that is processes attracted by their ownlaw, living in a potential landscape, we investigate different properties of the invariantmeasures. The interaction between the process and its law leads to nonlinear stochasticdifferential equations. In [S. Herrmann and J. Tugaut. Electron. J. Probab.15 (2010) 2087–2116], the authors proved that, for linearinteraction and under suitable conditions, there exists a unique symmetric limit measureassociated to the set of invariant measures in the small-noise limit. The aim of thisstudy is essentially to point out that this statement leads to the existence, as the noiseintensity is small, of one unique symmetric invariant measure for the self-stabilizingprocess. Informations about the asymmetric measures shall be presented too. The main keyconsists in estimating the convergence rate for sequences of stationary measures usinggeneralized Laplace’s method approximations.

In this paper we analyse an approximate controllability result for a nonlinear population dynamics model. In this model the birth term is nonlocal and describes the recruitment process in newborn individuals population, and the control acts on a small open set of the domain and corresponds to an elimination or a supply of newborn individuals. In our proof we use a unique continuation property for the solution of the heat equation and the Kakutani-Fan-Glicksberg fixed point theorem.

A class of first-order impulsive functional differential equations with forcing terms is considered. It is shown that, under certain assumptions, there exist positive T-periodic solutions, and under some other assumptions, there exists no positive T-periodic solution. Applications and examples are given to illustrate the main results.

In this paper, some new coincidence point theorems in continuous function spaces are presented. We show the hybrid mapping version and multivalued version of both Lou’s fixed point theorem (Proc. Amer. Math. Soc.127 (1999)) and de Pascale and de Pascale’s fixed point theorem (Proc. Amer. Math. Soc.130 (2002)). Our new results encompass a number of previously known generalizations of the theorems. Two examples are presented.

In this paper, existence criteria for multiple solutions of periodic boundary value problems for the first-order difference equation are established by using the Leggett–Williams multiple fixed point theorem and fixed point theorem of cone expansion and compression. Two examples are also given to illustrate the main results.

Matching theorems, fixed point theorems and minimax inequalities are obtained in H-spaces which generalize the corresponding results of Bae-Kim-Tan, Browder, Fan, Horvath, Kim, Ko-Tan, Shih-Tan, Takahashi, Tan and Tarafdar to non-compact and/or non-convex settings.