Homework SourseLet D 0 15 times 0 15 Show that the mapping G x12 x22 810
Let D = [0, 1.5] times [0, 1.5]. Show that the mapping G =(x_1^2 + x_2^2 +8/10, x_1x_2^2 +x_1 +8/10)^T: D rightarrow R^2 has a unique fixed point in D.![Let D = [0, 1.5] times [0, 1.5]. Show that the mapping G =(x_1^2 + x_2^2 +8/10, x_1x_2^2 +x_1 +8/10)^T: D rightarrow R^2 has a unique fixed point in D.Solution](/WebImages/22/let-d-0-15-times-0-15-show-that-the-mapping-g-x12-x22-810-1054236-1761549740-0.webp "Let D = [0, 1.5] times [0, 1.5]. Show that the mapping G =(x_1^2 + x_2^2 +8/10, x_1x_2^2 +x_1 +8/10)^T: D rightarrow R^2 has a unique fixed point in D.Solution")
Solution
The solution is given by solving G(x,y)=(x,y), which results in the following two equations:
(x-5)^2+y^2=17 and x(1+y^2)=10y-8.
The first one is clearly a circle. These both need to be plotted, in the given domain D, in order to show that these meet only once in D. That will conclude the result.
