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{\bf Final Report }
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Research project 96-0915 "Qualitative properties of impulsive \\
differential equations with applications to control theory \\
and mathematical biology"
\vskip1mm
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Project Co-ordinator \ professor Mokhtar Kirane

Period: \ October 1, 1997 -- September 30, 1999.
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2. RESEARCH
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2.1. Scientific Objectives

\begin{itemize}
\item What are the main scientific objectives of the project?

The main scientific objectives of the project are the studying and
the development a lot of important aspects of the theory of
impulsive systems, in particular, various qualitative properties of
impulsive evolution systems, impulsive partial differential
equations and delay impulsive differential equations, invariant
sets of impulsive systems, boun\-da\-ry value problems of nonlinear
impulsive systems, applications to control theory and mathematical
biology.
\vskip1mm

2.2. Research Activities:

\item What work has been carried out by each of the Participants?

Participant 1: tasks \# 1, \# 2, \# 3, \# 10, \# 12.

Participant 2: tasks \# 4, \# 5

Participant 3: tasks \#3, \# 4, \# 5, \# 6 \# 12

Participant 4: tasks \# 5, \# 6, \# 7, \# 8, \# 11

Participant 5: tasks \# 3, \# 4, \# 7, \# 9, \# 10, \# 12
\vskip1mm

\item Has the reseach been in accordance with the Work Programme ? Yes

%The research has been in accordance with the Work Programme.
\vskip1mm

2.3. Scientific Results
\item The following main scientific results were obtained:
\vskip1mm

1. Formulae for exact solutions of initial boundary problems for
linear partial differential equations of hyperbolic and parabolic
type were obtained and their analysis were made. Sufficient
conditions for the problem to have periodic solutions are found.

Partial differential equations of parabolic type as well as partial
differential equations with impulse are studied. New conditions of
existence of their global solutions are obtained and their
stability are studied.
The system of two partial differential equations of
reaction-diffusion type with impulse is considered. By using the
Laypunov function approach the sufficient conditions of existence
of uniform bounded solutions are ontained.
%\vskip1mm

2. We studied the following system of ordinary differential equations,
defined on the direct product of a torus and Euclidean space,
subjected to impulse action on a submanifold of the torus of codimension 1
\vskip1mm

\hskip6mm $\displaystyle d\varphi/dt = a (\varphi,
\varepsilon) + b (\varphi, x, \varepsilon),
\quad dx/dt = A(\varphi, \varepsilon)x + f(\varphi, x,
\varepsilon), \
\varphi \in \mathbf{T_m} \setminus \mathbf{\Gamma}, $ \hfill (1)
\vskip1mm

\hskip6mm $\displaystyle
\Delta x|_{\varphi \in \mathbf{\Gamma}} = B(\varphi,
\varepsilon)x +
g(\varphi, x, \varepsilon), $ \hfill (2) \vskip1mm

\noindent where $x\in \mathbf{R^{n}},\ \varphi \in \mathbf{T_{m}},\
\mathbf{T_{m}}$ is an $m$-dimensional torus, $\mathbf{\Gamma }$ is a smooth
compact submanifold of $\mathbf{T_{m}}$ of codimension 1,
$\varepsilon \in \mathbf{R} $ is a parameter. $\Delta x$ stands for
the jump of the function $x$ at the
point $\varphi $ during the motion along the trajectory of first Eq. (1).
It is supposed that $\det (I+B(\varphi,0))=0$ for
some (for all) $\varphi \in \mathbf{\Gamma }.$ Therefore solutions of
linearized system (1), (2) (if $\varepsilon =0,b\equiv 0,f=g\equiv 0,$)
cannot be continued to negative semi-axis $t<0$ or
can be continued ambiguously. For linearized impulsive system a
concept of exponential dichotomy is introduced. The smoothness of
its separatrix manifolds on the set $\mathbf{T_m} \setminus
\mathbf{\Gamma}$ is proved. They have first order discontinuities
on $\mathbf{\Gamma}.$
We proved that the exponential dichotomy is preserved under small
perturbations of the right-hand sides of the linearized system. For
this purpose properties of some piece-wise continuous linear
skew-product semi-flow on the product $\mathbf{R^{n}\times T_{m}}$
are studied.
For the system with linearized system which possess the property
of exponential dichotomy the conditions for the existence of piecewise
smooth invariant toroidal set are obtained.
%\vskip1mm

3. Poincar\`e mapping was constructed for the impulsive systems.
Existence of periodic solutions of the system of differential
equations with impulses has been studied by means of Poincar\`e
mapping. Furthermore, stability of periodic solution has been
studied and examples have been considered.
%\vskip1mm

4. Various applications of the numerical-analytical methods to
impulsive systems of the form
\begin{eqnarray*}
\begin{array}{llllll}
x'' + \omega ^{2}x &=& \varepsilon
f(t, x, x'),\,\,\,x\neq \varepsilon x_{0} \quad
\Delta x'|_{x=\varepsilon x_{0}} &=& \varepsilon I(x').
\end{array}
\end{eqnarray*}
and
\begin{eqnarray*}
\begin{array}{llllll}
x'&=&f(t,x),\ t\neq t_{i},0\leq t\leq T,\ x\in R^{n},\\
\Delta x|_{t_{i}} &=& H_{i}(x),\ i=1,2,...,p;\ 0<t_{1}<.....<t_{p}<T,\\
x(0) & = &x(T).
\end{array}
\end{eqnarray*}
were analyzed. This analysis allowed to indicate sharp conditions
under which nonlinear boundary-value problem for the impulsive
system can be investigated by means of numerical-analytical method.
In particular, we have studied the problem of periodic control of
impulsive systems. It should be noted that several important
results concerning applications of the numerical-analytical method
to the study of boundary value problems for impulsive systems have
been corrected and improved.
%\vskip1mm

5. Delay impulsive differential equations were studied and sufficient
conditions for the existence of periodic solutions systems were
obtained. Results on coexistence (in a sense of Sharkovsky order)
of periodic solutions to delay impulsive differential equations
were obtained. The algorithm of construction approximate solution
of delay impulsive systems with small perturbation is developed.
%\vskip1mm

6. Periodic boundary value problem for a functional differential
equation with impulses at fixed times has been studied. Different
types of functional dependence not only for the impulse action, but
also for the right-hand side of the differential equation have been
considered. A new maximal principle has been proved for a general
problem. It is proposed to construct monotone sequences converging
to the extremal solutions of the periodic problem in a sector by
using the method of upper and lower solutions coupled with the
monotone iterative technique.
%\vskip1mm

7. Some new comparison principles for impulsive differential
equations which improve and complements known results have been
obtained. These principles were applied to study the
existence of solutions to the first order periodic impulsive problem
\begin{eqnarray*}
\begin{array}{llllll}
u'(t)=f(t,u(t)), & a.e. \, t \in J'=J \setminus \{t_1,t_2,...,t_p \}
\\
u(t_k^+)=I_k(u(t_k)), &  k=1,2,...,p\\
u(0)=u(T),
\end{array}
\end{eqnarray*}
and for a second order boundary value problem for ordinary
differential equations with impulses at fixed moments of the form
\begin{eqnarray*}
\begin{array}{ll}
u''(t)=f(t,u(t)), &\mbox{a.e.} \; t \in J'=J\setminus \{t_1,t_2,
\dots, t_p\} \\
\Delta u'(t_k)=L_k(u(t_k),u'(t_k)), & k=1,2,\dots,p\\
\Delta u(t_k)=\tilde{L}_{k}(u(t_k),u'(t_k)), & k=1,2,\dots,p \\
u(0)-u(T)= \lambda_0, \, \, \, \, u'(0)-u'(T)=\lambda_1. &
\end{array}
\end{eqnarray*}
where $J=[0,T]$, $0=t_0<t_1<...<t_p<t_{p+1}=T$, $I_k\colon R \to R$
and  $L_k,\tilde{L}_k \colon R \times R \to R$
are continuous for $k=1,...,p,$
$\Delta u'(t_k)=u'(t_k^+)-u'(t_k^-)$, $\Delta
u(t_k)=u(t_k^+)-u(t_k^-)$, $k=1,\dots,p$, $\lambda_1, \lambda_0 \in R,$
and $f \colon J\times R \to R$ is a Carath\'eodory
function.
Additionally, it is developed the monotone iterative scheme for
this system. The existence of monotone sequences converging to
the maximal and minimal solutions is proved.

8. Existence and uniqueness of solutions of the boundary value
problem
\vskip1mm

\hskip8mm $\displaystyle dx/dt = A(t) x + f(t) +
\int_{\alpha}^t
K(t, s) x(s) ds, \ \ t \not= \theta_i, $ \hfill (3) \vskip1mm

\hskip8mm $\displaystyle
\Delta x |_{t = \theta_i} = B_i x(\theta_i) +
\int_{\alpha}^{\theta_i}
M_i (s) x(s) ds + \sum_{\alpha < \theta_j \le \theta_i} D_{ij}
x(\theta_j) +
I_i, $ \hfill (4) \vskip1mm

\hskip8mm $\displaystyle
x(\alpha )=a,\ x(\beta )=b,\ \ a,b\in R^{n},$ \hfill (5)

\noindent
where $\Delta x |_{t=\theta _{i}}=x(\theta _{j}+0)-x(\theta _{j})$
have been studied.
Integral representation of solutions of system (6) - (8) is
constructed by using fundamental matrices of solutions of the system
conjugated to (3) - (5).
%\vskip2mm

9. Necessary and sufficient conditions of controllability for the
boundary value problem \vskip1mm

$ \displaystyle
 dx/dt = A(t)x + \int_{\alpha}^t K(t,s)x(s)ds+ C(t)u(t) + f(t) +
\mu g(t,x,u,\mu),\;\;  t \neq \zeta_i,$
%\theta_i+\mu\tau_i(x,\mu), $
\vskip2mm
%\hskip10mm $ \displaystyle$\hfill (12) \vskip4mm

$ \displaystyle
\Delta x(\zeta_i ) =  B_i x(\zeta_i) +
\sum_{\alpha<\zeta_j\leq\zeta_i} D_{ij} x(\zeta_j) +
\int_{\alpha}^{\zeta_i}M_i(s)x(s)\,ds +Q_i v_i + J_i+ \mu
W_i(x(\zeta_i),v_i,\mu), $
\vskip1mm

$\displaystyle
x(\alpha )=a,\ x(\beta )=b,\ \ a,b\in R^{n}.$
%\vskip1mm

where $\mu > 0$ is a small parameter, $\zeta_i = \theta_i +
\mu\tau_i(x,\mu), \alpha < \theta_1 < ... < \theta_p < \beta,$
are obtained. In particular, the rank criterion for the linear
systems (if $\mu = 0$) was proposed. The approaching method for the
system of impulsive differential equations with non-fixed moments
of impulses is developed.

10. We studied the following system of differential equations with
impulses
$$dx/dt = f(t, \psi, x, \mu), (\psi, x, \mu) \not\in \Gamma,$$
$$\Delta x \mid_{(x, \psi, \mu) \in \Gamma} = I(x, \psi, \mu),$$
where functions $f$ and $I$ are analytical in $x$ and $\mu$,
continuous and $2\pi$-periodical in $\psi$,
$\psi=\nu_{0}t,\nu_{0}>0$ is a real number, the set $\Gamma$ is
defined by the equation
$$\prod_{i=1}^{p}(\psi-t_{i}^{0}(x,\mu)) = 0,$$ $p$ is a positive
integer, functions $t_{i}^{0}$ are analitical in $x$ and $\mu$.
The problem of existence and stability of periodic solutions
of differential equation with impulses is considered under the
assumption that the generating equation has the family of
quasiperiodic solutions.
%\vskip2mm

\item The following scientific papers and presentations
have resulted directly from this project.

1. M.~Akhmetov and R.~Sejilova, The control of boundary value
problem for linear impulsive integro-differential system. J. Math.
Anal. Appl. 1999, v.236, p.312-326.\\
2. M.~Akhmetov and R.~Sejilova, On the boundary value problem for
impulsive integro-differential system. Izv. Min. nauki - Akad.
nauk Resp. Kazakhstan. Ser. fiz-mat., 1998, no. 5.\\
3. M.~Akhmetov, On the approaching method of impulsive system with
nonfixed impulse moments. Izv. Min. nauki - Akad. nauk Resp.
Kazakhstan. Ser. fiz-mat. 1999, no. 1. \\
4. M.~Akhmetov and A.~Zafer, Successive approximation method for
boundary value problem for quasilinear impulsive systems. Appl.
Math. Letters, 1999. (To appear). \\
5. M.~Akhmetov and R.~Sejilova, On the control of boundary value problem
for linear impulsive integro-differential system. Differential'nye
Uravneniya. (To appear). \\
6. M.~Akhmetov and R.~Nagaev, Periodic solutions of nonlinear impulsive
system in a neiborhood of quasiperiodic solution's family.
Differential'nye Uravneniya. (To appear).\\
7. M.~Akhmetov and R.~Sejilova, Rank criteria of controllability for
boundary value problem of impulsive integro-differential system.
Ukrain. Math. J. (To appear).\\
8. M.~Akhmetov and M.~Kirane, Control problems of quasilinear
integro-differential equations. SIAM, Appl. Math. (Submitted).\\
9. M.U. Akhmetov, A. Zafer and R. Sejilova, The control of the boundary
value problem for linear impulsive integro-differential systems.
Nonlinear analysis. (Submitted).\\
10. M. Kirane and J.I. Kanel, Pointwise a priori bounds for a strongly
coupled system of reaction diffusion equations with a balance law.
Mathem. Models and Methods in the Appl. Scienc. 21(1998),
p.1227-1232.\\
11. M. Kirane and M. Guedda, Diffusion terms in systems of reaction
diffusion equations can lead to blow up. J. Math. Anal. Appl.
218(1998), no. 1, p. 325-327.\\
12. M. Kirane and M. Guedda, A note on nonexistence of global solutions
for a nonlinear integral equation. Bull. Belgian Math. Soc. - Simon
Stevin. (To appear).\\
13. M. Kirane and J.I. Kanel, Existence of travelling wave solutions
for a diffusive epidemic model. Commun. Appl. Analysis. (To appear). \\
14. M. Kirane and J.I. Kanel, Global existence and large time behaviour
of positive solutions to a reaction diffusion system. Integral and
Differ. equat. (To appear).\\
15. M. Kirane and N. Tatar, Global existence and stability of some
semilinear problems. Archivum Mathematicum. (To appear).\\
16. M. Kirane, D. Bainov and E. Minchev, Stability properties of
solutions of impulsive parabolic diffrential-functional equations.
J. Math. Anal. Appl. (Submitted). \\
17. M. Kirane and J.I. Kanel, Global solutions of reaction diffusion
systems with a balance law and nonlinearities of exponential
growth. J. Differ. Equat. (To appear).\\
18. M. Kirane, J.I. Kanel and N. Tatar, Pointwise a priori bounds for a
strongly coupled system of reaction diffusion equations. Intern. J.
Differ. Equat. and Appl. (To appear). \\
19. D. Franco, E. Liz, J.J. Nieto and Yu.V. Rogovchenko, A contribution
to the study of functional differential equations with impulses.
Mathematische Nachrichten. (To appear). \\
20. D. Franco, J.J. Nieto and Yu.V. Rogovchenko, Periodic boundary
value problem for nonlinear first order ordinary differential equations
with impulses at fixed moments. Extracta Mathematicae, 13(1998),
p.313-326. \\
21. A. Cabada, J. Nieto, D. Franco and S. Trofimchuk, A
generalization of the monotone method for second order periodic
boundary value problem with impulses at fixed points, Dynamics of
Continuous, Discrete and Impulsive systems. (To appear). \\
22. E. Liz and S. Trofimchuk, An\'alisis de estabilidad en
ecuaciones diferenciales con m\'aximo, Congreso de Ecuaciones
Diferenciales y Aplicaciones, XVI CEDYA, Las Palmas de Gran
Canaria, 21 al 24 de Septiembre 1999 (Actas del Congreso, p. 357- 364). \\
23. J. Ferreiro, E. Liz and S. Trofimchuk, Soluciones peri\'odicas
de sistemas de ecuaciones diferenciales con m\'aximo, Congreso de
Ecuaciones Diferenciales y Aplicaciones, XVI CEDYA, Las Palmas de
Gran Canaria, 21 al 24 de Septiembre 1999 (Actas del Congreso,
p.277- 283). \\
24. A. Ivanov, E. Liz and S. Trofimchuk, Halanay inequality, Yorke
3/2 stability criterium, and differential equations with maxima.
Tohoku Math. J. (Submitted). \\
25. E.~Liz and S.~Trofimchuk, Existence and stability of almost
periodic solutions for quasilinear delay systems and Halanay
inequality. J. Math. Anal. Appl. (Submitted). \\
26. E.~Liz, C.~Martinez and S.~Trofimchuk, Global attractivity for
infinite delay Mackey-Glass type equations. (In preparation). \\
27. R.P. Agarwal and D. O'Regan, Multiple nonnegative solutions for
second order impulsive differential equations. Appl. Math. Comput.
(Accepted). \\
28. R.P. Agarwal and D. O'Regan, A coupled system of boundary value
problems. Appl. Anal. 69(1998), no.3-4, p.381--385. \\
29. D.~O'Regan, Nonlinear alternatives for multivalued maps with
applications to operator inclusions in abstract spaces. Proc. Amer.
Math. Soc. 127(1999), no.12, p.3557--3564. \\
30. D. O'Regan, Multivalued differential equations in Banach spaces.
Comput. Math. Appl. 38(1999), no.5-6, p.109--116. \\
31. R.P. Agarwal and D. O'Regan, Boundary value problems of nonsingular
type on the semi-infinite interval. Tohoku Math. J. (2) 51(1999),
no.3, p.391-397. \\
32. D. O'Regan, Random fixed point theory for multivalued maps.
Stochastic Anal. Appl. 17(1999), no.4, p.597--607. \\
33. D. O'Regan, Viable solutions of differential equations and
inclusions on proximate retracts in Banach spaces. Panamer. Math.
J. 9(1999), no.3, p.1-15. \\
34. D. O'Regan, A note on multivalued differential equations on
proximate retracts. J. Appl. Math. Stochastic Anal. 12(1999), no.2,
p.169-178. \\
35. D. O'Regan, Coincidence theory for CS maps with applications.
Commun. Appl. Anal. 3(1999), no.3, p.433-446. \\
36. R. Kannan and D. O'Regan, A note on singular boundary value
problems with solutions in weighted spaces. Nonlinear Anal.
37. (1999), no. 6, Ser. A: Theory Methods, p. 791--796. \\
D. O'Regan, Analytic alternatives and minimax inequalities.
Nonlinear Stud. 6 (1999), no. 1, p. 11--20.
38. R.P. Agarwal and D. O'Regan, Right focal singular boundary value
problems. ZAMM Z. Angew. Math. Mech. 79 (1999), no. 6, p. 363--373. \\
39. R.P. Agarwal and D. O'Regan, Superlinear higher order boundary
value problems. Aequationes Math. 57 (1999), no. 2-3, p. 233--240. \\
40. D. O'Regan, Fixed points and random fixed points for
$\alpha$-Lipschitzian maps. Nonlinear Anal. 37 (1999), no. 4,
p. 537--544. \\
41. M. Frigon and D. O'Regan, First order impulsive initial and
periodic problems with variable moments. J. Math. Anal. Appl. 233
(1999), no. 2, p. 730--739. \\
42. R.P. Agarwal and D. O'Regan, Multiple solutions for higher-order
difference equations. Comput. Math. Appl. 37 (1999), no. 9, p.
39--48. \\
43. D. O'Regan, Coincidence principles and fixed point theory for
mappings in locally convex spaces. Rocky Mountain J. Math. 28
(1998), no. 4, p. 1407--1445. \\
44. R.P. Agarwal and D. O'Regan, Twin solutions to singular boundary
value problems. Proc. Amer. Math. Soc. (To appear). \\
45. D. O'Regan, Fixed point theorems and equilibrium points in abstract
economies. Bull. Austral. Math. Soc. 58 (1998), no. 1, p. 33--41. \\
46. D. O'Regan, Random fixed point theory for random operators.
Panamer. Math. J. 9 (1999), no. 1, p. 11--21. \\
47. R.P. Agarwal, D. O'Regan and P.J.Y. Wong, Positive solutions of
differential, difference and integral equations. Kluwer Academic
Publishers, Dordrecht, 1999. \\
48. N.I.~Ronto, A.M.~Samoilenko, S.I.~Trofimchuk,
Theory of numerical-analytic method: achievements and new trends of
development. IV. Ukrain. Math. J. 50(1998), no.12, p.1656-1672. \\
49. A.M.Samoilenko and M.I.Ronto, Numerical-analytic method in the
theory of boundary value problems, World Scientific, Singapore,
1999. \\
50. A.M. Samoilenko, S.I. Trofimchuk, On the concept of solution of
delay differential equation with impulses. (In preparation). \\
51. A.M.Samoilenko, V.Hr.Samoilenko and V.V. Sobchuk,
Periodic solutions of equations of mathematical pendulum with
impulse influence. Ukrain. Math. J. 51 (1999), No. 6, p. 837-846. \\
52. V.Hr. Samoylenko, K.K. Yelgondyev,
On existence of periodical solutions for differential equations
with impulse effectes. Facta Universitatis. Ser. Mechanics,
Automatic control and robotics. 2 (1998), no. 8, p. 635-639. \\
53. V.Hr. Samoilenko and K.K.~Yelgondiev,
Periodical solutions to differential equation with impulse effects.
Abstracts of International conference "Dynamical systems:
stability, control, optimization" Minsk (Belarus) September 28 --
October 4, 1998. v.2, p. 247-248. \\
54. V.Hr. Samoilenko, N.A.~Perestiuk and K.K.~Yelgondyev,
Sequence mapping and periodic solutions of a system of differential
equations with impulse effect. Neliniini kolyvannya (Nonlinear
oscillations), 1998, no.1, p. 44-50. \\
55. V.Hr. Samoilenko, Delay impulsive differential equations and
their periodic solutions (Submitted). \\
56. V.Hr. Samoilenko, On solutions of initial boundary problem for
wave equations with impulsive influence. (Submitted). \\
57. V.Hr. Samoilenko, K.K. Yelgondyev, On heat propagation in
medium under instant change of its temperature. (Submitted). \\
58. V.Hr. Samoylenko and K.K. Yelgondyev, The qualitative behaviour
of solutions to differential equations with impulse effectes.
Differentsial'niye uravneniya, 35 (1999), no.6, p. 852. \\
59. V.Hr. Samoylenko and K.K. Yelgondyev, Coexistence of periodic
solutions to differential equations with impulse effects.
Differentsial'niye uravneniya, 35 (1999), no.6, p. 853-854. \\
60. V.Hr. Samoylenko and V.V. Sobchuk, Bifurcation of periodic
solutions of equation of mathematical pendulum with impulse
effects. Thesis of International Conference ''Dynamical Systems,
Modelling and Stability Investigation''. Kyiv May 25-29, 1999.
p.~52. \\
61. V.Hr. Samoylenko abd V.V. Sobchuk, Periodic solutions of Lienard
equation with impulse influence. Book of Abstracts of International
Conference "Analytical methods of analysis and differential
equations" (AMADE-99). Minsk, September 14  - 19, 1999. \\
62. V.Hr. Samoylenko and V.V. Sobchuk, Periodic solutions for the
Lienard equation with impulse effectes. Thesis of the International
conference "The problems of differential equations, analysis and
algebra". (Aqtyubinsk, September 15-19, 1999), p.48. \\
63. V.Hr. Samoylenko, V.V. Sobchuk and K.K. Yelgondyev, On periodic
solutions to differential equations with impulse effectes.
(Submitted). \\
64. V.Hr. Samoylenko, Specific effects caused by impulse influences in
impulsive systems. Book of Absracts of Interbational Congress on
Industrial and Applied Mathematics (ICIAM 99). Edinburg, July 5 -
9, 1999, p.86. \\
65. V.I. Tkachenko, On the exponential dichotomy of linear almost
periodic pulse systems. Ukrain. Math. J, 50 (1998), N 1, p. 155-163. \\
66. V.I. Tkachenko,
On multi-frequency systems with impulses. Neliniini kolyvannya
(Nonlinear oscillations), Kyiv, 1998, no. 1, p. 108 - 117.\\
67. V.I. Tkachenko, On exponential dichotomy and invariant sets of
impulsive systems. Extended abstracts of Fourth International
Conference on Difference Equations and Applications, Poznan,
Poland, 1998, p.341 - 344. \\
68. V.I. Tkachenko, On exponential dichotomy and invariant sets of
impulsive systems'. Proceedings of the Fourth International
Conference on Difference Equations and Application. New York,
Gordon and Breach Science Publishers. (To appear). \\
69. V.I. Tkachenko, On invariant sets of differential equations
with impulses. Neliniini kolyvannya (Nonlinear oscillations), 2
(1999), no. 4. (To appear). \\
70. V.I. Tkachenko, On reaction-Diffusion Equations with Impulses.
Contributed abstracts of Third European Nonlinear Oscillations
Conference. Technical University of Denmark, Copenhagen, August 8 -
12, 1999.
\vskip2mm

\centerline {\bf Oral presentations and lectures}
\vskip1mm

1. In December 1-7, 1997 and June 15-23, 1998 D. O'Regan visited
Montreal (Canada) where he represent his results on impulsive
system. The joint paper with M. Frigon is the result of these
visits. \\
2. In May 1998 M. Kirane participated at the Delft Meeting on
Functional Analysis and Nonlinear Partial Differential Equations
(Delft, Hollande) where he delivered a lecture on global solutions
and large time behaviour for a strongly coupled system of reaction
diffusion equations.\\
3. In July 1998 M. Kirane visited the Abdus Salam International Center
for Theoretical Physics, Mathematics Section, where he presented
his results on investigation of impulsive partial equations.\\
4. In July 1998 M. Kirane visited the Abdus Salam International Center
for Theoretical Physics, Mathematics Section, where he presented
his results on investigation of impulsive partial equations.\\
5. International Congress of Mathematicians, August 18-27, 1998,
Berlin, Germany.
Communication by A.M. Samoilenko "Investigation of $m$-frequency
oscillatory systems".\\
6. International Congress of Mathematicians, August 18-27, 1998,
Berlin, Germany.
Communication by V.I. Tkachenko "On linear almost periodic systems
with bounded solutions".\\
7. In August, 1998; In March, 1999 and In July, 1999
V.Hr.Samoylenko presented his results on periodic solutions of
impulsive systems at University of Salerno and Second University of
Naples (Italy).\\
8. Fourth international conference on difference equations and
applications, Poznan, Poland, August 27 - 31, 1998.
Communication by V.I. Tkachenko.
"On exponential dichotomy and invariant sets of impulsive
systems".\\
9. International conference "Mathematical modelling in technology"
Kherson (Ukraine) September 3--5, 1998.\\
Plenary lecture by V.Hr.~Samoilenko "Specific effects of impulsive
influences in impulsive systems".\\
10. International conference "Dynamical systems: stability, control,
optimization"  Minsk (Belarus) September 28 -- October 4, 1998.\\
Plenary lecture by V.Hr.~Samoylenko "Systems of differential
equations with impulsive influences".\\
11. In November 1998 S.I. Trofimchuk participated at the International
Conference in honour of J. Hale, Lisbon, Portugal, where he
delivered a lecture "Chaotic motions in a quasilinear functional
equation".\\
12. On December 4, 1998, S.I. Trofimchuk delivered lecture
"Soluciones periodicas multiples y comportamiento caotico en
sistemas con maximos" at University of Santiago de Compostela (Spain).\\
13. On December 12, 1998 S.I. Trofimchuk delivered a lecture on
impulsive systems at Univesity of Vigo (Spain).\\
14. Seminar on qualitative theory of differential equations in Moscow
University.\\
On March 12, 1999 V.Hr. Samoylenko and K.K. Yelgondyev delivered a
lecture ''The qualitative behaviour of solutions to differential
equations with impulse effectes''.\\
On March 26, 1999 V.Hr. Samoylenko and K.K. Yelgondyev delivered a
lecture ''Coexistence of periodic solutions to differential
equations with impulse effects''.\\
15. On March 20-24, 1999 V.Hr. Samoylenko visited Krakow academy
(AGH) and delivered a lecture ''Existence periodic solutions in
impulsive differential equations due to impulsive effectes''.\\
16. Scientific session of Shevchenko Scientific Society in Lviv
(NTSH), March 25-26, 1999.
Plenary lecture by V.Hr. Samoylenko ''Differential equations with impulse
effects as essentially nonlinear system''.\\
17. On April 20-21, 1999 R.D.~Sejilova participate at the
International conference "The modern problems of civil aviation"
and presented communication on controllability of impulsive
systems.\\
18. International conference ''Dynamical systems, modelling and stability
investigation'', Kyiv, May 25-29, 1999.
A presentation by V.Hr. Samoylenko and V.V. Sobchuk ''Bifurcation of
periodic solutions of equation of mathematical pendulum with impulse
effects''.\\
19. International conference "Nonlinear Partial Differential Equations"
dedicatd to J.P.Shauder, Lviv, August 23-29, 1999.
Lecture by V.Hr. Samoylenko "Periodic solutions of nonlinear
differential equations with impulse".\\
20. International conference ``Differential Equations with Impulse
Action'', Kyiv, October 4--6, 1999. The following lectures of
participants of the project were presented:\\
-- A.M. Samoilenko "Asymptotic analysis of oscillatory systems
with $m$ frequencies";\\
-- M.U. Akhmetov, R.D. Seilova
"Control problems for quasilinear impulsive integro-differential
equations";\\
-- J.J. Nieto, S.I. Trofimchuk
"Second order BVP with impulses at fixed points";\\
--  V.Hr. Samoylenko
''Periodic solutions of second-order differential equations with
impulses'';\\
--  V.I. Tkachenko
"On exponential dichotomy and invariant sets of impulsive systems".
\vskip1mm

Ph.D thesis is prepared by R.D.Sejilova (University ''Dunie'',
Aktyubinsk, Kazakhstan).

\item The scientific output is summarized in the table below: \\[2mm]
\begin{tabular}{|c|c|c|c|c|}
\hline
Scientific Output: & published & in press/accepted & submitted & in
preparation \cr
\hline
International Journals	& 24 & 13 & 6 & 3  \cr
\hline
National Journals	& 8 & 3 & 2 & 0  \cr
\hline
Abstracts (proceedings) & 11 & 0 & 0 & 0  \cr
\hline
Book, Monograph & 2 (English) & 0 & 0 & 0  \cr
\hline
Thesis (PhD) & 0 & 0 & 0 & 1 (Russian) \cr
\hline
Oral Presentation	& 24 &	 &   &	  \cr
\hline
\end{tabular}
\vskip2mm

3. MANAGEMENT
\vskip1mm

3.1. Meetings and visits

\item What co-ordination meetings, exchange visits of scientists, or major
field trips took place during the project period? If so, where, when and
who did participate?

On November 29 - December 12, 1998 prof. S.I. Trofimchuk visited
University of Santiago de Compostela (Spain), where he discuss with
prof. J.Nieto and prof. E.Liz the problems connected with
fulfilment of the project.

On May, 15-22, 1998 and October 2-12, 1999 prof. M.Akhmetov visited
the Institute of Mathematics (Kiev).

On August 18-27, 1998 prof. A.M.Samoilenko and prof. V.I.Tkachenko
visited ICM-98 (Berlin, Germany).

On February, 24 - March, 4, 1999 and on July, 19 - August 14, 1999
V.Hr.~Samoylenko visited University of Salerno and Second
University of Naples (Italy).

On September 12-21, 1999 prof. A.M.Samoilenko and prof.
V.Hr.Samoylenko visited University "Dunie" (Aktyubinsk) and
Aktyubinsk State University.

In order to disseminate and to discuss results of the project on
October 4--6, 1999 the International conference (extended seminar)
''Differential Equations with Impulse Actions''was organised in
Kyiv, in the Institute of Mathematics of National Academy of
Sciences of Ukraine. At the conference 36 lectures and
communications were presented.
\vskip1mm

\item Summarise the meetings and visits in the table below:
\vskip1mm

\begin{tabular}{|p{45mm}|p{45mm}|p{45mm}|}
\hline
Visits & Number of scientists & Number of person days  \cr
\hline
West $=>$ East	&  &	   \cr
\hline
East $=>$ West	& 1 & 14   \cr
\hline
West $=>$ West	&   &	   \cr
\hline
East $=>$ East	& 3 & 25   \cr
\hline
\end{tabular}
\vskip1mm

3.2 Collaboration

\item In your opinion, how intense was the collaboration among the
different Participants?
\vskip1mm

\begin{tabular}{|p{50mm}|p{20mm}|p{20mm}|p{20mm}|p{20mm}|}
\hline
Intensity of Collaboration & high & rather high & rather low & low  \cr
\hline
West $<=>$ East  &   & * &  &	\cr
\hline
West $<=>$ West  &   & * &  &	\cr
\hline
East $<=>$ East  & * &	 &  &	\cr
\hline
\end{tabular}
\vskip1mm

\item In this project, did you co-operate to a major extent with additional
(inter)national organisations and institutions not mentioned in Co-operation
Agreement? \ \ No
\vskip1mm

3.3 Time Schedule

\item Has the time planning been in accordance with the Work
Programme? \ \ Yes.
\vskip1mm

3.4 Problems encountered

\item Did you encounter any major problems (e.g. with regard
to quality and quantity of the scientific contributions of the
different Participants, telecommunication, the transfer of funds
and goods, taxation, customs, withholding of overheads)? If yes,
please specify the problems and describe how you solved them. \ \ No.

\item Summarise your experiences in the table below.
\vskip1mm

\begin{tabular}{|p{55mm}|p{20mm}|p{20mm}|p{20mm}|p{20mm}|}
\hline
Problems encountered & major & minor & none & not applicable  \cr
\hline
Co-operation of team members &	&   & * &   \cr
\hline
Transfer of funds	     &	& * &	&   \cr
\hline
Telecommunication	     &	&   & * &   \cr
\hline
Transfer of goods	     &	&   & * &   \cr
\hline
Other			     &	&   &	&   \cr
\hline
\end{tabular}

3.5. Actions required

\item Do you see any need for action from INTAS? \ \ No.
%\vskip2mm

3.6. Manpower invested.

\item On the whole, how many person-years went into the project? Give
a rough estimate. \ \ 20 years

How much of this was due to the funding received from this grant?
\ \ 20 years
\vskip2mm

4. FINANCES
\vskip2mm

4.1. This grant

\item How did you spend the money of this grant? Plezse use the
table below to give for each Participants a breakdown in ECU of the
expenditures actually incurred under the different cost categories.
(Please do not simply copy the fifures from the Work Programme).
\vskip2mm

\noindent
\begin{tabular}{|p{5mm}|p{30mm}|p{15mm}|p{10mm}|p{15mm}|p{15mm}|p{15mm}|
p{15mm}|p{15mm}|}
\hline
N & Name of & Ind. Grants & Over-  &
Travel and & Equip- ment & Consu- & Other & TOTAL \cr
 & Participant & Labour Costs & heards &
Subsis- tence &  & mables  & Costs & (EVRO) \cr
\hline
1 & Univ. Picardie &  0  & 500	& 3.600  &   0 &  300  &  0  & 4.400 \cr
\hline
2 & Univ. Galway   &  0  & 200	& 3.000  &   0 &  100  &  0  & 3.300 \cr
\hline
3 & Univ. Santiago &  0  & 300	& 3.400  &   0 &  100  &  0  & 3.800 \cr
\hline
4 & Univ. 'Dunie'  & 6.500 & 200 &  650  & 1.350 & 200	& 0  & 8.900 \cr
\hline
5 & Inst. Mathem.  & 21.000 & 350 & 3.500  & 3.900 &  350  & 3.000 & 32.100 \cr
\hline
& Total (evro) & 27.500 & 1.550 & 14.150 & 5.250 & 1.050 & 3.000 & 52.500 \cr
\hline
\end{tabular}
\vskip1mm

\item Has the spending been in accordance with the one foreseen
in the Work Programme ?  Yes.

\item Did you spend a major amount on a) travel, b) equipment or
c) other costs (e.g. sub-contracts)?
\vskip1mm

4.2. Other funding

\item Did your project receive substantial funding from other
sources than INTAS? \ \ No
\vskip2mm

5. SUMMARY OF RESULTS AND KEY REFERENCES
\vskip2mm

6. ROLE AND IMPACT OF INTAS
\vskip2mm

\item In your mind (i.e. the Co-ordinator's) how impotent was this
grant for startings and carrying out the project?

\begin{tabular}{|p{74mm}|p{19mm}|p{19mm}|p{19mm}|p{19mm}|}
\hline
Role of INTAS & definitely yes & rather yes & rather not &
definitely not	\cr
\hline
Would the project have beeb started  &	&  &  &   \cr
without funding by INTAS?	     &	&  &  &   \cr
\hline
Would the project have beeb carried out  &  &  &  &   \cr
without funding from INTAS?		 &  &  &  &   \cr
\hline
\end{tabular}
\vskip1mm

\item From your point of view, what were the most important
achievements of the project?
\vskip1mm

\begin{tabular}{|p{62mm}|p{22mm}|p{22mm}|p{22mm}|p{22mm}|}
\hline
Main achivement of the project & very important & quite important
& less important & not important \cr
\hline
exciting science		&  &  &  &   \cr
\hline
new international contacts	&  &  &  &   \cr
\hline
additional prestige for my lab	&  &  &  &   \cr
\hline
additional funds for my lab	&  &  &  &   \cr
\hline
helping scientists in NIS	&  &  &  &   \cr
\hline
other (specify) 		&  &  &  &   \cr
\hline
\end{tabular}
\vskip1mm

\item Will the project continued? \ \ No

\item Will the co-operation among the project participants
continue in the fiture? \ \ Yes
\vskip1mm

7. RECOMMENDATIONS TO INTAS
\vskip1mm

\item What was particularly good and should not be changed?

\item What was particularly bad and should be changed? Please
specify and explain how it could be improved?

\end{itemize}

\end{document}