Dear Colleagues,

We continue our Colloquium in the Math Department.

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Lingju Kong*

Department of Mathematical Sciences

Northern Illinois University

Tuesday, March 1, EMCS 422, 3:00 pm.

Nonlinear Boundary Value Problems

Abstract. Presented in this talk are results on nonlinear multi-point
boundary value problems with emphasis on those consisting of the equation

$$u'' + f(t,u,u') = 0, t\in (0,1),$$
and one of the two-parameter boundary conditions

$$u'(0) =\lambda_1,\;u(1) - \sum_{i=1}^m\,b_i u(t_i) =\lambda_2$$
and

$$u(0)-\sum_{i=1}^m\,a_i u(t_i) =\lambda_1,\;u(1)-\sum_{i=1}^m\,b_i
u(t_i)=\lambda_2$$

Sufficient conditions are found for the existence of solutions of the above
problems based on the existence of a pair of lower and upper solutions. For
each of the two problems, under some assumptions, explicit ranges of values
of $\lambda_1$ and $\lambda_2$ are obtained such that the problem has a
solution, has a positive solution, and has no solution, respectively.
Furthermore, the theoretical structure of the $(\lambda_1,\lambda_2)$-plane
is characterized, more precisely, it is proved that
$(\lambda_1,\lambda_2)$-plane can be divided by some continuous curve
$\Gamma$ into two disjoint connected regions $\Lambda^E$ and $\Lambda^N$
such that the problem has a solution for $(\lambda_1,\lambda_2)\in
\Lambda^E$ and has no solution for $(\lambda_1,\lambda_2)\in \Lambda^N.$
Existence of solutions for $(\lambda_1,\lambda_2)\in \Gamma$ is also
discussed. Some work on higher even order boundary value problems will also
be addressed.

*Lingju Kong is a candidate for a position at our Department.

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Boris P Belinskiy

Department of Mathematics, Dept. 6956

University of Tennessee at Chattanooga

Ph. (423) 425-4748