Jason R. Morris
A method is presented for proving the existence of solutions for boundary-value 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 boundary-value 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; half-line; 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
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