Electron. J. Diff. Equ.,
Vol. 2011 (2011), No. 135, pp. 115.
Boundaryvalue problems for nonautonomous nonlinear systems
on the halfline
Jason R. Morris
Abstract:
A method is presented for proving the existence of solutions for
boundaryvalue problems on the half line. The problems under study
are nonlinear, nonautonomous systems of ODEs with the possibility
of some prescribed value at
and with the condition that
solutions decay to zero as
grows large. The method relies
upon a topological degree for proper Fredholm maps.
Specific conditions are given to ensure that the boundaryvalue
problem corresponds to a functional equation that involves an
operator with the required smoothness, properness, and Fredholm
properties (including a calculable Fredholm index).
When the Fredholm index is zero and the solutions are bounded
a priori, then a solution exists. The method is applied
to obtain new existence results for systems of the form
and
.
Submitted October 4, 2011. Published October 17, 2011.
Math Subject Classifications: 34B40, 34B15, 34D09, 46E15, 47H11, 47N20.
Key Words: Ordinary differential equation; halfline; infinite interval;
boundary and initial value problem; Fredholm operator;
degree theory; exponential dichotomy; properness; a priori bounds.
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Jason R. Morris
Department of Mathematics, The College at Brockport
State University of New York
350 New Campus Drive, Brockport, NY 14420, USA
email: jrmorris@brockport.edu

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