
General Information
Research Team
Objectives
Results


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, CNCSUEFISCDI

Project Code: PNIIIDPCE201130052 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 timedependent 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, 343390.
(for the semilinear evolution case),
 O. Cârjă, M. Necula, I. I. Vrabie,
Necessary and sufficient conditions for
viability for nonlinear evolution inclusions,
SetValued Analysis, 16(2008), 701731.
(for the fully nonlinear evolution case), and developed in the monograph

O. Cârjă, M. Necula, I. I. Vrabie,
Viability, Invariance and Applications,
NorthHolland 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: BouligandSeveri, 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 reactiondiffusion systems
in abstract spaces.


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 


The aims of the project are:
 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:
 periodic solutions;
 antiperiodic solutions;
 solutions which satisfy generalized initialmean conditions;
 solutions obeying various nonlinear constraints expressed by
another differential equation or inclusion.
 As far as the problem (P2) is concerned, within this project we intend to:
 find easytocheck 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;
 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;
 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 preassigned behavior at infinity, periodic, antiperiodic, etc.
 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
reactiondiffusion systems and nonlinear population model.

 Scientific research report 20112016
 Publications
 Report 2011
 I. I. Vrabie,
Existence for nonlinear evolution inclusions with nonlocal retarded initial conditions,
Nonlinear Analysis,
Vol. 74 (2011), 70477060.
 Report 2012
 M. Burlică, D. Roşu,
A class of nonlinear delay evolution equations with nonlocal initial conditions,
Proc. Amer. Math. Soc.,
142 (2014) 24452458.
 M. Burlică, D. Roşu, I. I. Vrabie,
Continuity with respect to the data for a delay evolution equation with nonlocal initial conditions,
Libertas Mathematica (new series),
32 (2012), 3748.
 M. Necula, M. Popescu,
Viability of a time dependent closed set with respect to a semilinear delay evolution inclusion,
An. Ştiinţ. Al. I. Cuza, Iaşi (N.S.),
61 (2015) 4158
 I. I. Vrabie,
Existence in the large for nonlinear delay evolution inclusions with nonlocal initial conditions,
Journal of Functional Analysis,
262 (2012), 13631391.
 I. I. Vrabie,
Global solutions for nonlinear delay evolution inclusions with nonlocal initial conditions,
SetValued and Variational Analysis,
20 (2012), 477497.
 I. I. Vrabie,
Nonlinear retarded evolution equations with nonlocal initial conditions,
Dynamic Systems and Applications,
21 (2012), 417440.
 Report 2013
 M. Burlică, D. Roşu,
A class of reactiondiffusion systems with nonlocal initial conditions,
An. Ştiinţ. Al. I. Cuza, Iaşi (N.S.),
61 (2015) 5978
 M. Burlică, D. Roşu, I. I. Vrabie,
Abstract reactiondiffusion systems with nonlocal initial conditions,
Nonlinear Analysis, 94 (2014), 107119.
 M. Necula, M. Popescu, I. I. Vrabie,
Nonlinear delay evolution inclusions on graphs, on
System Modeling and Optimization, Proceedings of the 26th IFIP TC 7 Conference, CSMO 2013, Klagenfurt, Austria, September 913, 2013,
Lecture Notes in Computer Science, Barbara Kaltenbacher, Clemens Heuberger, Christian Potze and Franz Rendl Editors, 2014, 207216.
 I. I. Vrabie,
Almost periodic solutions for nonlinear delay
evolutions with nonlocal initial conditions,
J. Evol. Eqn., 13 (2013), 693714.
 I. I. Vrabie,
Delay evolution equations with mixed nonlocal plus local initial conditions,
Commun. Contemp. Math., 17 (2015), 122.
 Report 2014
 M. Necula, M. Popescu, I. I. Vrabie,
Viability for delay evolution equations with nonlocal initial conditions,
Nonlinear Analysis, 121 (2015), 164172.
 M. Necula, I. I. Vrabie,
Nonlinear delay evolution inclusions with general nonlocal initial conditions,
Ann. Acad. Rom. Sci., Math. Appl.,7 (2015), 6797.
 I. I. Vrabie,
Semilinear delay evolution equations with nonlocal initial conditions, in
New Prospects in Direct, Inverse and Control Problems for Evolution Equations,
A. Favini, G. Fragnelli and R. Mininni Editors, Springer INdAM Series, vol. 10, (2014) 421437.
 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/s1023101505356.
 Report 2016
 M. D. Burlica, M. Necula, D. Rosu, I. I. Vrabie,
Delay Differential Evolutions Subjected to Nonlocal Initial Conditions,
Series: Monographs and Research Notes in Mathematics,
Chapman and Hall/CRC, 2016, ISBN 9781498746441.
 M. Burlică, D. Roşu,
An existence result for a class of delay inclusions involving measures, subjected to nonlocal initial data,
Mediterranean Journal of Mathematics, submitted.
 I. I. Vrabie,
A local existence theorem for a class of delay differential equations,
Topol. Methods Nonlinear Anal., DOI: 10.12775/TMNA.2016.023 (2016, first online).
 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 reactiondiffusion 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 reactiondiffusion system with nonlocal initial conditions.

Differential Equations, Inverse Problems and Control Theory,
Cortona, Italy, June 1720, 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 913, 2013:
 Monica Burlică, Viability of a moving set with respect to a semilinear reactiondiffusion 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 (AMMCS2013),
Waterloo, Ontario, Canada, August 2630, 2013:
 Monica Burlică, Viability for a timedependent domain with respect to a reactiondiffusion systems with delay;
 Daniela Roşu, A class of reactiondiffusion systems with mixed initial conditions;

Recent Trends in Nonlinear Partial Differential Equations and Applications (NDPE 2014)
Trieste, Italy, May 2830, 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 711, 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 multivalued reactiondiffusion systems with delay;
 Daniela Roşu, Existence for a nonlinear delay reactiondiffusion 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 1519, 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 VIIth Symposium on Nonlinear Analysis (SNA 2015),
Torun, Poland, September 1418, 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, 2627 October 2012
;

Session of Scientific Communications of the Faculty of Mathematics, Iasi, 2526 October 2013
;
 Differential Equations, Optimization and Optimal Control Seminar,
Faculty of Mathematics, Iaşi (weekly meetings).
Last update: 23.09.2016
