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Existence of solution for function given inequality

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October 21, 2010
functioninequalitiesalgebrapolynomialalgebra unsolved

Problem Statement

Let p(x,y)p(x, y) and q(x,y)q(x, y) be polynomials in two variables such that for x≥0,y≥0x \ge 0, y \ge 0 the following conditions hold: (i)p(x,y)(i) p(x, y) and q(x,y)q(x, y) are increasing functions of xx for every fixed yy. (ii)p(x,y)(ii) p(x, y) is an increasing and q(x)q(x) is a decreasing function of yy for every fixed xx. (iii)p(x,0)=q(x,0)(iii) p(x, 0) = q(x, 0) for every xx and p(0,0)=0p(0, 0) = 0. Show that the simultaneous equations p(x,y)=a,q(x,y)=bp(x, y) = a, q(x, y) = b have a unique solution in the set x≥0,y≥0x \ge 0, y \ge 0 for all a,ba, b satisfying 0≤b≤a0 \le b \le a but lack a solution in the same set if a<ba < b.
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