Dear Colleagues,

                                We continue our Colloquium at the Math
Department

                        Bo Zhang

Department of Mathematics and Computer Science
Fayetteville State University,
Fayetteville, NC 28301 USA
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        Thursday, October 16, EMCS 238, 2:00pm.

        A Fixed Point Theorem of Krasnoselskii Type

Abstract. It is well-known that a combination of the contraction mapping
theorem and Schauder's fixed point theorem (known as Krasnoselskii's
theorem) can be used in the study of a neutral differential equation by
converting it to an integral equation, say

$$x(t) = h(t,x_t) + \int^t_0G(s,x_s)ds$$

with a view of proving $h(t,x_t)$ is a contraction and the integral term
is compact when $G$ is small enough.

In this paper, we prove a fixed point theorem which is a combination of
the contraction mapping theorem and a variant (Browder-Potter theorem)
of the nonlinear alternative of Leray-Schauder degree theory which
requires a less restrictive growth condition on the functions involved.
We use this fixed point theorem to study the existence of periodic
solutions and controllability in a system of neutral differential
equations (NDEs). Due to the topological nature of the approach, the
theorem applies as well to NDEs of mixed type and NDEs with
state-dependent delays for which the fundamental theory of solutions has
not been well developed. Some comparisons between our results and the
existing ones in the literature are also provided.

The presentation will be of interest for physicists and engineers.
Students (especially math majors) are encouraged to attend.


Boris P Belinskiy

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Department of Mathematics, Dept. 6956

University of Tennessee at Chattanooga

Ph. (423) 425-4748