Electron. J. Diff. Equ., Vol. 2011 (2011), No. 135, pp. 1-15.

Boundary-value problems for nonautonomous nonlinear systems on the half-line

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 $t=0$ and with the condition that solutions decay to zero as $t$ 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 $\dot{v}+g(t,w)=f_1(t)$ and $\dot{w}+h(t,v)=f_2(t)$.

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
email: jrmorris@brockport.edu

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