Exploratory research project

PN-II-ID-PCE-2011-3-0052

Project leader: Prof. dr. Ioan I. VRABIE

   

  General Information

  Research Team

  Objectives

  Results



I. GENERAL INFORMATION

Project Title : Retarded Functional Differential Inclusions in Banach Spaces

Supported by: Romanian National Research Council and Executive Agency for Higher Education, Research, Development and Innovation, CNCS-UEFISCDI

Project Code: PN-II-ID-PCE-2011-3-0052 nr. 240/05.10.2011

Thematic fields : Main Sciences: Mathematics

Summary:

(P1) The first problem we will study within the project consists in obtaining very general sufficient conditions for existence concerning functional differential inclusions with delay subjected to initial nonlocal conditions in an infinite dimensional real Banach space.

(P2) An extremely important theoretical problem, in differential equations, functional differential equations and inclusions is that of viability. Essentially, this problem consists in establishing sufficient and even necessary conditions which both the vector field and a part, K, of its domain of definition - with possible empty interior or containing boundary points - must satisfy, in order to exist at least one state of the system which do not leave that part of the domain. And this, should happen for any initial time and initial state whatsoever. If this happens, we say that K is viable with respect to the driving vector field.
Within this project, we are interested in getting new viability results for time-dependent sets with respect to functional differential inclusions and evolutions, linear or not, expressed by means of the concept of tangent set to another set at some of its points. This tangency concept is very recent. More precisely, it was introduced in the papers
  • O. Cârjă, M. Necula, I. I. Vrabie, Necessary and sufficient conditions for viability for semilinear differential inclusions, Trans. Amer. Math. Soc, 361(2009) No. 1, 343-390.
(for the semilinear evolution case),
  • O. Cârjă, M. Necula, I. I. Vrabie, Necessary and sufficient conditions for viability for nonlinear evolution inclusions, Set-Valued Analysis, 16(2008), 701-731.
(for the fully nonlinear evolution case), and developed in the monograph
  • O. Cârjă, M. Necula, I. I. Vrabie, Viability, Invariance and Applications, North-Holland Mathematics Studies 207, Elsevier, 2007.
In the above mentioned papers, it was proved that, unlike differential equations, in the case of differential inclusions and evolutions, the tangency conditions (used in the study of viability problems) can be expressed by means of the concept of tangent set more naturally and efficiently than by means of the one of tangent vector (no matter in which sense: Bouligand-Severi, Bony, Federer, Clarke, etc.).

(P3) A third problem we will focus our attention consists in obtaining very precise information concerning viability of a given set with respect to a class of reaction-diffusion systems in abstract spaces.

II. RESEARCH TEAM

 No.   Family and given names   Birth year   Scientific title   PhD 
 1  Ioan I. Vrabie  1951  principal investigator  yes
 2  Mihai Necula  1959  senior researcher  yes
 3  Daniela Roşu  1968  researcher  yes
 4  Monica Dana Burlică  1971  researcher  yes
 5  Marius Popescu  1975  researcher  yes


III. OBJECTIVES

The aims of the project are:

  1. Concerning the problem (P1), we intend to prove substantial extensions of the above cited results to the case of functional differential inclusions of retarded type in possibly nonreflexive Banach spaces. We also focus our attention on the hypotheses used in order to obtain as consequences, existence results for:
    1. periodic solutions;
    2. anti-periodic solutions;
    3. solutions which satisfy generalized initial-mean conditions;
    4. solutions obeying various nonlinear constraints expressed by another differential equation or inclusion.

  2. As far as the problem (P2) is concerned, within this project we intend to:
    1. find easy-to-check sufficient, or even necessary and sufficient conditions, in order that a certain set be viable with respect to a semilinear functional differential equation or inclusion;
    2. find easy to check sufficient, or even necessary and sufficient conditions, in order that a certain set be invariant with respect to a fully nonlinear functional differential equation or inclusion;
    3. to use the abstract viability a nd invariance results in order to obtain information on the existence of solutions enjoying some special properties as: positivity, boundedness, with a pre-assigned behavior at infinity, periodic, anti-periodic, etc.

  3. Referring to problem (P3), our intention is to exploit the specific form of the system in order to deduce various deep viability results under very general appropriate conditions taking into account the possible interplay between the hypotheses acting on X and those on Y, as for instance: the compactness of the semigroup generated by A coupled with the compactness of the mapping G, or the Lipschitz continuity of G. Of course, the hypotheses imposed will be chosen such that to fit the particular cases of reaction-diffusion systems and nonlinear population model.


IV. RESULTS

Scientific research report 2011-2016

Publications

Report 2011
  •   I. I. Vrabie, Existence for nonlinear evolution inclusions with nonlocal retarded initial conditions, Nonlinear Analysis, Vol. 74 (2011), 7047-7060.

Report 2012
Report 2013
Report 2014
Report 2015
  •  M. Burlică, D. Roşu, Nonlinear delay systems with nonlocal initial conditions having affine growth, Topol. Methods Nonlinear Anal., DOI: 10.12775/TMNA.2016.023 (2016, first online)
  •  I. Benedetti, L. Malaguti, V. Taddei, I. I. Vrabie, Semilinear delay evolution equations with measures subjected to nonlocal initial conditions, Ann. Mat. Pura Appl. DOI 10.1007/s10231-015-0535-6.

Report 2016

Mobilities

The 9th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Orlando, Florida, USA, July, 1 - 5, 2012:
  • Monica Burlică, Existence for a class of nonlinear delay reaction-diffusion systems;
  • Marius Popescu, Viability of a time dependent closed set with respect to a semilinear delay evolution inclusion;
  • Daniela Roşu, Global existence and exponential stability for a nonlinear delay evolution equation with nonlocal initial condition;
  • Ioan I. Vrabie, Nonlinear delay evolution inclusions with nonlocal conditions on the initial history.

The 20th Conference on Applied and Industrial Mathematics, Chişinău, Republic of Moldova, August 22 - 25, 2012:
  • Monica Burlică, Daniela Roşu, Solutions for a delay reaction-diffusion system with nonlocal initial conditions.

Differential Equations, Inverse Problems and Control Theory, Cortona, Italy, June 17-20, 2013:
  • Ioan I. Vrabie, Source Identification in a Semilinear Evolution Equation with Delay.

Joint International Meeting of the American Mathematical Society and the Romanian Mathematical Society, Alba Iulia, Romania, June 27 - 30, 2013:
  • Ioan I. Vrabie, Evolution delay equations with nonlocal initial conditions.

The 26th IFIP TC 7 Conference on System Modeling and Optimization, Klagenfurt, Austria, September 9-13, 2013:
  • Monica Burlică, Viability of a moving set with respect to a semilinear reaction-diffusion system with delay;
  • Marius Popescu, Nonlinear delay evolution inclusions on graphs;
  • Daniela Roşu, Continuity with respect to the data for delay equations subjected to nonlocal conditions;
  • Ioan I. Vrabie, Almost periodic solutions for nonlinear delay evolutions with nonlocal initial conditions.

Applied Mathematics, Modeling and Computer Science (AMMCS-2013), Waterloo, Ontario, Canada, August 26-30, 2013:
  • Monica Burlică, Viability for a time-dependent domain with respect to a reaction-diffusion systems with delay;
  • Daniela Roşu, A class of reaction-diffusion systems with mixed initial conditions;

Recent Trends in Nonlinear Partial Differential Equations and Applications (NDPE 2014) Trieste, Italy, May 28-30, 2014, Celebrating Enzo Mitidieri's 60th birthday:
  • Ioan I. Vrabie, Delay evolution equations with implicit nonlocal initial conditions.

The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Madrid, Spain, July 7-11, 2014.
  • Special session on "Evolution equations and inclusions with applications to control, mathematical modeling and mechanics" organized by N. U. Ahmed, S. Migorski and I. I. Vrabie.
  • Monica Burlică, Nonlinear multi-valued reaction-diffusion systems with delay;
  • Daniela Roşu, Existence for a nonlinear delay reaction-diffusion system subjected to nonlocal initial conditions;
  • Ioan I. Vrabie, A viability result for delay evolution equations with implicit nonlocal initial conditions.

PDE's, Control Theory and Inverse problems, Bologna Italy, September 15-19, 2014, Dedicated to the Memory of Alfredo Lorenzi:
  • Ioan I. Vrabie, Nonlinear delay evolution inclusions with general nonlocal implicit initial conditions.

Topological methods in the theory of differential equations and applications, Lyon, France, July 5 - 12, 2015:
  • Ioan I. Vrabie, Delay evolution equations with measures and nonlocal initial conditions. (jointly with I. Benedetti, L. Malaguti, V. Taddei)

The VII-th Symposium on Nonlinear Analysis (SNA 2015), Torun, Poland, September 14-18, 2015:
  • Ioan I. Vrabie, A local existence theorem for functional delay differential equations in Banach spaces.

Other activities

Session of Scientific Communications of the Faculty of Mathematics, Iasi, 26-27 October 2012 ;

Session of Scientific Communications of the Faculty of Mathematics, Iasi, 25-26 October 2013 ;

Differential Equations, Optimization and Optimal Control Seminar, Faculty of Mathematics, Iaşi (weekly meetings).



Last update: 23.09.2016