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Discrete & Continuous Dynamical Systems - B

2014 , Volume 19 , Issue 8

Special issue in honor of Avner Friedman

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Preface
Urszula Ledzewicz, Marek Galewski, Andrzej Nowakowski, Andrzej Swierniak, Agnieszka Kalamajska and Ewa Schmeidel
2014, 19(8): i-ii doi: 10.3934/dcdsb.2014.19.8i +[Abstract](327) +[PDF](447.1KB)
Abstract:
Most mathematicians who in their professional career deal with differential equations, PDEs, dynamical systems, stochastic equations and a variety of their applications, particularly to biomedicine, have come across the research contributions of Avner Friedman to these fields. However, not many of them know that his family background is actually Polish. His father was born in the small town of Włodawa on the border with Belarus and lived in another Polish town, Łomza, before he emigrated to Israel in the early 1920's (when it was still the British Mandate, Palestine). His mother came from the even smaller Polish town Knyszyn near Białystok and left for Israel a few years earlier. In May 2013, Avner finally had the opportunity to visit his father's hometown for the first time accompanied by two Polish friends, co-editors of this volume. His visit in Poland became an occasion to interact with Polish mathematicians. Poland has a long tradition of research in various fields related to differential equations and more recently there is a growing interest in biomedical applications. Avner visited two research centers, the Schauder Center in Torun and the Department of Mathematics of the Technical University of Lodz where he gave a plenary talk at a one-day conference on Dynamical Systems and Applications which was held on this occasion. In spite of its short length, the conference attracted mathematicians from the most prominent research centers in Poland including the University of Warsaw, the Polish Academy of Sciences and others, and even some mathematicians from other countries in Europe. Avner had a chance to get familiar with the main results in dynamical systems and applications presented by the participants and give his input in the scientific discussions. This volume contains some of the papers related to this meeting and to the overall research interactions it generated. The papers were written by mathematicians, mostly Polish, who wanted to pay tribute to Avner Friedman on the occasion of his visit to Poland.

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Fractional variational principle of Herglotz
Ricardo Almeida and Agnieszka B. Malinowska
2014, 19(8): 2367-2381 doi: 10.3934/dcdsb.2014.19.2367 +[Abstract](447) +[PDF](236.5KB)
Abstract:
The aim of this paper is to bring together two approaches to non-conservative systems -- the generalized variational principle of Herglotz and the fractional calculus of variations. Namely, we consider functionals whose extrema are sought, by differential equations that involve Caputo fractional derivatives. The Euler--Lagrange equations are obtained for the fractional variational problems of Herglotz-type and the transversality conditions are derived. The fractional Noether-type theorem for conservative and non-conservative physical systems is proved.
A singularly perturbed age structured SIRS model with fast recovery
Jacek Banasiak and Rodrigue Yves M'pika Massoukou
2014, 19(8): 2383-2399 doi: 10.3934/dcdsb.2014.19.2383 +[Abstract](438) +[PDF](408.7KB)
Abstract:
Age structure of a population often plays a significant role in the spreading of a disease among its members. For instance, childhood diseases mostly affect the juvenile part of the population, while sexually transmitted diseases spread mostly among the adults. Thus, it is important to build epidemiological models which incorporate the demography of the affected populations. Doing this we must be careful as many diseases act on a time scale different from that of the vital processes. For many diseases, e.g. measles, influenza, the typical time unit is one day or one week, whereas the proper time unit for the vital processes is the average lifespan in the population; that is, 10-100 years. In such a case, the epidemiological model with vital dynamics becomes a multiple time scale model and thus it often can be significantly simplified by various asymptotic methods. The presented paper is concerned with an SIRS type disease spreading in a population with a continuous age structure modelled by the McKendrick-von Foerster equation. We consider a disease with a quick recovery rate in a large population. Though it is not too surprising that in such a model the introduced disease quickly vanishes, the result is mathematically interesting as the error estimates are uniform on the whole infinite time interval, in contrast to the typical results based on the Tikhonov theorem and classical asymptotic expansions.
Systems described by Volterra type integral operators
Dorota Bors, Andrzej Skowron and Stanisław Walczak
2014, 19(8): 2401-2416 doi: 10.3934/dcdsb.2014.19.2401 +[Abstract](419) +[PDF](378.4KB)
Abstract:
In the paper we consider a nonlinear Volterra integral operator defined on some subspace of absolutely continuous function. Some sufficient conditions for the operator considered to be a diffeomorphism are formulated. The proof of main result relies in essential way on variational method. Applications of results to control systems with feedback and a specific nonlinear Volterra equation are presented.
A nonlocal problem describing spherical system of stars
Marcin Bugdoł and Tadeusz Nadzieja
2014, 19(8): 2417-2423 doi: 10.3934/dcdsb.2014.19.2417 +[Abstract](301) +[PDF](284.7KB)
Abstract:
We prove in this note the existence and uniqueness of solutions of a nonlocal problem appearing as a model of galaxy in early stage of evolution. Some properties of solutions are also given.
Analysis of two quasistatic history-dependent contact models
Xiaoliang Cheng, Stanisław Migórski, Anna Ochal and Mircea Sofonea
2014, 19(8): 2425-2445 doi: 10.3934/dcdsb.2014.19.2425 +[Abstract](374) +[PDF](440.6KB)
Abstract:
We consider two mathematical models which describe the evolution of a viscoelastic and viscoplastic body, respectively, in contact with a piston or a device, the so-called obstacle or foundation. In both models the contact process is assumed to be quasistatic and the friction is described with a Clarke subdifferential boundary condition. The novelty of the models consists in the constitutive laws as well as in the contact conditions we use, which involve a memory term. We derive a variational formulation of the problems which is in the form of a system coupling a nonlinear integral equation with a history--dependent hemivariational inequality. Then, we prove the existence of a weak solution and, under additional assumptions, its uniqueness. The proof is based on a result on history--dependent hemivariational inequalities obtained in [18].
Existence of unbounded solutions of a linear homogenous system of differential equations with two delays
Josef Diblík, Radoslav Chupáč and Miroslava Růžičková
2014, 19(8): 2447-2459 doi: 10.3934/dcdsb.2014.19.2447 +[Abstract](370) +[PDF](408.9KB)
Abstract:
Behavior of solutions of a linear homogeneous system of differential equations with deviating arguments in the form \begin{equation*} \dot y(t)=\beta(t)\left[y(t-\delta)-y(t-\tau)\right] \end{equation*} is discussed for $t\to\infty$. It is assumed that $y$ is an $n$-dimensional column vector, $n\geq 1$ is an integer, $\delta,\tau\in{\mathbb{R}}$, $\tau>\delta>0$, and $\beta(t)$ is an $n\times n$ matrix defined for $t\geq t_{0}$, $t_{0}\in\mathbb{R}$, and such that its elements are nonnegative, continuous functions and in every row of this matrix at least one element is nonzero. The existence of solutions in an exponential form under certain assumptions is proved. Sufficient conditions for the existence of unbounded solutions and the estimations for a solution are derived. A comparison with the known results and an illustrative example are given.
An explicit coefficient criterion for the existence of positive solutions to the linear advanced equation
Josef Diblík, Klara Janglajew and Mária Kúdelčíková
2014, 19(8): 2461-2467 doi: 10.3934/dcdsb.2014.19.2461 +[Abstract](427) +[PDF](305.1KB)
Abstract:
The paper is devoted to the investigation of a linear differential equation with advanced argument $\dot y(t)=c(t)y(t+\tau),$ where $\tau>0$, and the function $c\colon [t_0,\infty)\to (0,\infty)$, $t_0\in \mathbb{R}$ is bounded and locally Lipschitz continuous. New explicit coefficient criterion for the existence of a positive solution in terms of $c$ and $\tau$ is derived.
Modeling DNA thermal denaturation at the mesoscopic level
Marina Dolfin and Mirosław Lachowicz
2014, 19(8): 2469-2482 doi: 10.3934/dcdsb.2014.19.2469 +[Abstract](391) +[PDF](394.5KB)
Abstract:
In this paper a mesoscopic approach is proposed to describe the process of breaking of hydrogen bonds during the DNA thermal denaturation, also known as DNA melting. A system of integro-differential equations describing the dynamic of the variable which characterizes the opening of the base pairs is proposed. In the derivation of the model non linear effects arising from the collective behavior, namely the interactions, of base pairs are taken into account. Solutions of the mesoscopic model show significative analogies with the experimental S-shaped curves describing the fraction of broken bonds as a function of temperature at the macroscopic level, althought we instead simulate the variation in time. With this respect a research perspective connecting the theoretical results to the experimental one is proposed.
Existence of weak solutions for non-local fractional problems via Morse theory
Massimiliano Ferrara, Giovanni Molica Bisci and Binlin Zhang
2014, 19(8): 2483-2499 doi: 10.3934/dcdsb.2014.19.2483 +[Abstract](448) +[PDF](442.4KB)
Abstract:
In this paper, we study the existence of non-trivial solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions. Non-trivial solutions are obtained by computing the critical groups and Morse theory. Our results extend some classical theorems for semilinear elliptic equations to the non-local fractional setting.
Two-stage model of carcinogenic mutations with the influence of delays
Urszula Foryś and Beata Zduniak
2014, 19(8): 2501-2519 doi: 10.3934/dcdsb.2014.19.2501 +[Abstract](459) +[PDF](2547.5KB)
Abstract:
In the paper we make an attempt to study the influence of time delays combined with diffusion on the dynamics of two-stage carcinogenic mutations model. Included delays represent time needed for transformation from one type of cells to the other one. In the presented analysis we focus on possible stability switches due to increasing delays and diffusion driven instability. It occurs that diffusion has no significant impact on asymptotic behaviour of the model solutions, while one of the present delays has destabilising effect in most of cases we study. Analytical results are illustrated by numerical examples of the model dynamics.
Parameter estimation of systems with delays via structural sensitivity analysis
Krzysztof Fujarewicz and Krzysztof Łakomiec
2014, 19(8): 2521-2533 doi: 10.3934/dcdsb.2014.19.2521 +[Abstract](381) +[PDF](599.0KB)
Abstract:
This article presents a method for sensitivity analysis of non-linear continuous-time models with delays and its application to parameter estimation. The method is universal and may be used for sensitivity analysis of any system given as a block diagram with arbitrary structure and any number of delays. The method gives sensitivity functions of model trajectories with respect to all model parameters, including delay times, and both forward and adjoint sensitivity analysis may be performed. Two examples application of the method are presented: identification of a Wiener model with delay and identification of a model of JAK-STAT cell signal transduction mechanism.
Multiple periodic solutions to a discrete $p^{(k)}$ - Laplacian problem
Marek Galewski and Renata Wieteska
2014, 19(8): 2535-2547 doi: 10.3934/dcdsb.2014.19.2535 +[Abstract](347) +[PDF](376.2KB)
Abstract:
We investigate the existence of multiple periodic solutions to the anisotropic discrete system. We apply the linking method and a new three critical point theorem which we provide.
A global implicit function theorem and its applications to functional equations
Dariusz Idczak
2014, 19(8): 2549-2556 doi: 10.3934/dcdsb.2014.19.2549 +[Abstract](410) +[PDF](345.4KB)
Abstract:
The main result of the paper is a global implicit function theorem. In the proof of this theorem, we use a variational approach and apply Mountain Pass Theorem. An assumption guarantying existence of an implicit function on the whole space is a Palais-Smale condition. Some applications to differential and integro-differential equations are given.
On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points
Rafał Kamocki and Marek Majewski
2014, 19(8): 2557-2568 doi: 10.3934/dcdsb.2014.19.2557 +[Abstract](404) +[PDF](393.7KB)
Abstract:
In the paper we consider a Dirichlet problem for a fractional differential equation. The main goal is to prove an existence and continuous dependence of solution on functional parameter $u$ for the above problem. To prove it we use a variational method.
The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion
Piotr Kowalski
2014, 19(8): 2569-2580 doi: 10.3934/dcdsb.2014.19.2569 +[Abstract](413) +[PDF](204.1KB)
Abstract:
We consider a differential inclusion which is somehow analogous to the classical Duffing's equation with Dirichlet boundary condition. We prove the existence of a solution using two-steps approach. Firstly we consider an auxiliary problem where we substitute $h := \frac{dx} {dt} \in L^2(0,1)$. Next, using the method of pseudomonotone and coercive operators, we prove the existence of a solution to the auxiliary problem. Finally we prove that under suitable assumptions, an iterative scheme converges to the solution of our inclusion, which appears to be unique.
Multiple solutions for Dirichlet nonlinear BVPs involving fractional Laplacian
Tadeusz Kulczycki and Robert Stańczy
2014, 19(8): 2581-2591 doi: 10.3934/dcdsb.2014.19.2581 +[Abstract](337) +[PDF](355.6KB)
Abstract:
The existence of at least two solutions to superlinear integral equation in cone is proved using the Krasnosielskii Fixed Point Theorem. The result is applied to the Dirichlet BVPs with the fractional Laplacian.
Periodic solutions to differential equations with a generalized p-Laplacian
Adam Lipowski, Bogdan Przeradzki and Katarzyna Szymańska-Dębowska
2014, 19(8): 2593-2601 doi: 10.3934/dcdsb.2014.19.2593 +[Abstract](385) +[PDF](323.7KB)
Abstract:
The existence of a periodic solution to nonlinear ODEs with $\varphi$-Laplacian is proved under conditions on functions given in the equation (not on the unknown solutions). The results are applied to a relativistic pendulum equation in a general form.
Variational approach to stability of semilinear wave equation with nonlinear boundary conditions
Andrzej Nowakowski
2014, 19(8): 2603-2616 doi: 10.3934/dcdsb.2014.19.2603 +[Abstract](363) +[PDF](402.3KB)
Abstract:
We discuss solvability for the semilinear equation of the vibrating string $ x_{tt}(t,y)-\Delta x(t,y)=F_{x}(t,y,x(t,y))-G_{x}(t,y,x(t,y))$ in bounded domain and same type of nonlinearity on the boundary. To this effect we derive new variational methods one for the boundary equation the second for interior equation. Next we discuss stability of solutions with respect to initial conditions basing on variational approach.
Generalized fractional isoperimetric problem of several variables
Tatiana Odzijewicz
2014, 19(8): 2617-2629 doi: 10.3934/dcdsb.2014.19.2617 +[Abstract](453) +[PDF](347.1KB)
Abstract:
This work deals with the generalized fractional calculus of variations of several variables. Precisely, we prove a sufficient optimality condition for the fundamental problem and a necessary optimality condition for the isoperimetric problem. Our results cover important particular cases of problems with constant and variable order fractional operators.
A note on the existence and properties of evanescent solutions for nonlinear elliptic problems
Aleksandra Orpel
2014, 19(8): 2631-2639 doi: 10.3934/dcdsb.2014.19.2631 +[Abstract](301) +[PDF](361.8KB)
Abstract:
Basing ourselves on the subsolution and supersolution method we investigate the existence and properties of solutions of the following class of elliptic differential equations $div(a(||x||)\nabla u(x)) + f(x,u(x)) + g(||x||)k(x\cdot\nabla u(x)) = 0,$ $x\in\mathbb{R}^{n},||x||>R.$ Our main result concernes the behavior of solution at infinity.
On a nonlinear age-structured model of semelparous species
Ryszard Rudnicki and Radosław Wieczorek
2014, 19(8): 2641-2656 doi: 10.3934/dcdsb.2014.19.2641 +[Abstract](352) +[PDF](429.9KB)
Abstract:
We study a nonlinear age-structured model of a population such that individuals may give birth only at a given age. Properties of measure-valued periodic solutions of this system are investigated. We show that in some cases the age profile of the population tends to a Dirac measure, which means that the population asymptotically consists of individuals at the same age. This phenomenon is observed in nature in some insects populations.
Sufficient conditions for strong local optimality in optimal control problems with $L_{2}$-type objectives and control constraints
Heinz Schättler, Urszula Ledzewicz and Helmut Maurer
2014, 19(8): 2657-2679 doi: 10.3934/dcdsb.2014.19.2657 +[Abstract](400) +[PDF](650.5KB)
Abstract:
We consider the optimal control problem of minimizing an objective function that is quadratic in the control over a fixed interval for a multi-input bilinear dynamical system in the presence of control constraints. Such models are motivated by and applied to mathematical models for cancer chemotherapy over an a priori specified fixed therapy horizon. The necessary conditions for optimality of the Pontryagin maximum principle are easily evaluated and give a functional description of optimal controls as continuous functions of states and multipliers. However, there is no a priori guarantee that a numerically computed extremal controlled trajectory is locally optimal. In this paper, we formulate sufficient conditions for strong local optimality that are based on the existence of a bounded solution to a matrix Riccati differential equation. The theory is applied to a $3$-compartment model for multi-drug cancer chemotherapy with cytotoxic and cytostatic agents. The numerical results are compared with those for a corresponding optimal control problem when the objective is taken linear in the controls.
On the existence of weighted asymptotically constant solutions of Volterra difference equations of nonconvolution type
Ewa Schmeidel, Karol Gajda and Tomasz Gronek
2014, 19(8): 2681-2690 doi: 10.3934/dcdsb.2014.19.2681 +[Abstract](337) +[PDF](373.4KB)
Abstract:
A Volterra difference equation of the form $$x(n+2)=a(n)+b(n)x(n+1)+c(n)x(n)+\sum\limits^{n+1}_{i=1}K(n,i)x(i)$$ where $a, b, c, x \colon\mathbb{N} \to\mathbb{R}$ and $K \colon \mathbb{N}\times\mathbb{N}\to \mathbb{R}$ is studied. For every admissible constant $C \in \mathbb{R}$, sufficient conditions for the existence of a solution $x \colon\mathbb{N} \to\mathbb{R}$ of the above equation such that \[ x(n)\sim C \, n \, \beta(n), \] where $\beta(n)= \frac{1}{2^n}\prod\limits_{j=1}^{n-1}b(j)$, are presented. As a corollary of the main result, sufficient conditions for the existence of an eventually positive, oscillatory, and quickly oscillatory solution $x$ of this equation are obtained. Finally, a conditions under which considered equation possesses an asymptotically periodic solution are given.
Asymptotically zero solution of a class of higher nonlinear neutral difference equations with quasidifferences
Ewa Schmeidel and Robert Jankowski
2014, 19(8): 2691-2696 doi: 10.3934/dcdsb.2014.19.2691 +[Abstract](421) +[PDF](303.8KB)
Abstract:
A class of higher order nonlinear neutral difference equations with quasidifferences is studied. Sufficient conditions under which considered equation has a solution which converges to zero are presented.
The meaning of sensitivity functions in signaling pathways analysis
Jaroslaw Smieja, Malgorzata Kardynska and Arkadiusz Jamroz
2014, 19(8): 2697-2707 doi: 10.3934/dcdsb.2014.19.2697 +[Abstract](389) +[PDF](365.5KB)
Abstract:
The paper deals with local sensitivity analysis of signaling pathway models, based on sensitivity functions. Though the methods are well known, their application to various models is always based on the assumption that the output of the system whose sensitivity is analyzed is given by absolute quantitative data. In signaling pathways models, however, this data is always normalized. In this paper we show what are the implications of the way the signaling pathways models are built for the interpretation of sensitivity functions and parameter rankings based on them. The reasoning is illustrated using simple first- and second order systems as well as an example of a simple regulatory module.

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