MONOTONIC SOLUTIONS OF SYSTEMS OF NONLINEAR DIFFERENTIAL EQUATIONS
Myers, Tyler J.
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Systems of nonseparable nonlinear di↵erential equations can be hard to work with, but with certain additional conditions, we can acquire answers about continuability, existence, and boundedness of monotonic solutions. Specifically, we study systems of nonseparable first order nonlinear di↵erential equations of the form x0(t) = F(t, y(t)), y0(t) = G(t, x(t)). First, we start by proving that all solutions of this system are eventually monotonic and can be sorted into one of four classes. Next, we discuss the continuability of all solutions using a further set of assumptions. Then we prove that all solutions are bounded provided that |x0(t)| p(t)f(y(t)), |y0(t)| q(t)g(x(t)) for some functions p, f, q, and g. Finally, we look at existence of solutions for two subclasses of such systems using the Schauder Fixed-Point Theorem.