Theoretical framework for linear stability of an anomalous sub-diffusive activator-inhibitor system is set. Generalized Turing instability conditions are found to depend on anomaly exponents of various species. In addition to monotonous instability, known from normal diffusion, in an anomalous system oscillatory modes emerge. For equal anomaly exponents for both species the type of unstable modes is determined by the ratio of the reactants' diffusion coefficients. When the ratio exceeds its normal...

Fundamental results concerning Stieltjes integrals for functions with values in Banach spaces have been presented in [5]. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. [3]). It is known that the Kurzweil theory leads to the (non-absolutely convergent) Perron-Stieltjes integral in the finite dimensional case. Here basic results concerning equations of the form x(t) = x(a) +at [A(s)]x(s) +f(t) - f(a) are presented on the basis of...

This paper is a continuation of [9]. In [9] results concerning equations of the form x(t) = x(a) +at [A(s)]x(s) +f(t) - f(a) were presented. The Kurzweil type Stieltjes integration in the setting of [6] for Banach space valued functions was used. Here we consider operator valued solutions of the homogeneous problem (t) = I +dt [A(s)](s) as well as the variation-of-constants formula for the former equation.

Given a second order differential equation on a manifold we find necessary and sufficient conditions for the existence of a coordinate system in which the system is linear. The main tool to be used is a linear connection defined by the system of differential equations.

The aim of the paper is to provide a linearization approach to the $\mathrm{See\; PDF}$-control problems. We begin by proving a semigroup-type behaviour of the set of constraints appearing in the linearized formulation of (standard) control problems. As a byproduct we obtain a linear formulation of the dynamic programming principle. Then, we use the $\mathrm{See\; PDF}$ approach and the associated linear formulations. This seems to be the most appropriate tool for treating $\mathrm{See\; PDF}$ problems in continuous and lower semicontinuous setting.

The aim of the paper is to provide a linearization approach to the ${𝕃}^{\infty }$See PDF-control problems. We begin by proving a semigroup-type behaviour of the set of constraints appearing in the linearized formulation of (standard) control problems. As a byproduct we obtain a linear formulation of the dynamic programming principle. Then, we use the ${𝕃}^p$See PDF approach and the associated linear formulations. This seems to be the most appropriate tool for treating ${𝕃}^{\infty }$See PDF problems in continuous and lower semicontinuous...

A class of nonlinear neutral differential equations with variable coefficients and delays is considered. Conditions for the existence of eventually positive solutions are obtained which extend some of the criteria existing in the literature. In particular, a linearized comparison theorem is obtained which establishes a connection between our nonlinear equations and a class of linear neutral equations with constant coefficients.

This paper is concerned with the nonlinear advanced difference equation with constant coefficients $$x_{n+1}-x_n+\sum _{i=1}^m{p}_i{f}_i\left(x_{n-k_i}\right)=0,n=0,1,\cdots$$ where $p_i\in (-\infty ,0)$ and $k_i\in \{\cdots ,-2,-1\}$ for $i=1,2,\cdots ,m$. We obtain sufficient conditions and also necessary and sufficient conditions for the oscillation of all solutions of the difference equation above by comparing with the associated linearized difference equation. Furthermore, oscillation criteria are established for the nonlinear advanced difference equation with variable coefficients $$x_{n+1}-x_n+\sum _{i=1}^m{p}_{in}{f}_i\left(x_{n-k_i}\right)=0,n=0,1,\cdots$$ where $p_{in}\le 0$ and $k_i\in \{\cdots ,-2,-1\}$ for $i=1,2,\cdots ,m$.