Abstracts - Facultatea de Matematică

Controllability of Damped Second Order Initial Value Problem
for a class of Differential Inclusions with Nonlocal Conditions
on Noncompact Intervals
Dimplekumar N. CHALISHAJAR
Department of Applied Mathematics, Sardar Vallabhbhai Patel Institute of
Technology (SVIT), Gujarat University
Vasad-388 306, DIST: Anand, Gujarat State, INDIA
dipu17370@yahoo.com
In this article, we investigate sufficient conditions for controllability of second-order
semi-linear initial value problem with nonlocal conditions for the class of differential
inclusions in Banach spaces using the theory of strongly continuous cosine families. We
shall rely on a fixed point theorem due to Ma for multi-valued maps. An example is
provided to illustrate the result. This work is motivated by the paper of Benchohra and
Ntouyas [1] and Benchohra, Gatsori and Ntouyas [2].
We have considered the following second order inclusion system with non local conditions
�
y ′′ (t) − Ay(t) ∈ Gy ′ (t) + Bu(t) + F (t, yt, y ′ (t)), t ∈ J
y(0) + g(y) = φ, y ′ (0) = y0.
Here the state y(t) takes values in Banach space E and the control u ∈ L 2 (J, U), the
space of admissible controls, where J = (0, ∞).
Our aim is to study the exact controllability of the above abstract system which will
have applications to many interesting systems including PDE systems. We reduce the
controllability problem (1) to the search for fixed points of a suitable multi-valued map
on a subspace of the Frechet space C(J, E).
References:
[1] Benchohra, M. and Ntouyas, S. K., Controllability for an infinite time horizone
of second order differential inclusions in Banach spaces with Nonlocal conditions, J.
Optim. Theory Appl. 109 (2001), 85–98.
[2] Benchohra, M., Gatsori, E. P. and Ntouyas, S. K., Nonlocal quasilinear damped
differential inclusions, Electronic journal of Differential Equations, Vol.2002 (2002),
No.7, 1–14.
A (p − q) coupled system in elliptic nonlinear boundary value
problems
L. CONSIGLIERI
Department of Mathematics and CMAF
Sciences Faculty of University of Lisbon, 1749-016 Lisboa, Portugal
lcconsiglieri@fc.ul.pt
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