148. A special class of systems.

by Virgil Nicula, Oct 7, 2010, 9:47 AM

Lemma. If $f$ , $g$ are strict increasing $\nearrow$ dynamic function, i.e. $f,g:I\rightarrow I$ , where $I$ is an interval, then $\left\{\begin {array}{c} f(x)=g(y)\\\\ f(y)=g(z)\\\\ f(z)=g(x)\end{array}\right\|\ \Longrightarrow\ x=y=z$ .

Proof. $\left\{\begin{array}{cc} \blacktriangleright & \underline{x<y}\Longleftrightarrow f(x)<f(y)\Longleftrightarrow g(y)<g(z)\Longleftrightarrow\underline{y<z}\Longleftrightarrow\\\\ & f(y)<f(z)\Longleftrightarrow g(z)<g(x)\Longleftrightarrow\underline{z<x}\Longleftrightarrow \mathrm{\ absurd}\ .\\\\ \blacktriangleright & \underline{x>y}\ \Longleftrightarrow f(x)>f(y)\Longleftrightarrow g(y)>g(z)\Longleftrightarrow\underline{y>z}\Longleftrightarrow\\\\ & f(y)>f(z)\Longleftrightarrow g(z)>g(x)\Longleftrightarrow\underline{z>x}\Longleftrightarrow \mathrm{\ absurd}\ .\end{array}\right\|$

Examples. Solve the following systems :

$1.\blacktriangleright\ \left\{\begin{array}{c} 2x=y+\frac 2y\\\\ 2y=z+\frac 2z\\\\ 2z=x+\frac 2x\end{array}\right\|$ ...... $2.\blacktriangleright\ \left\{\begin{array}{c} x= y^3 + y - 8\\\\ y = z^3 + z - 8\\\\ z = x^3 + x - 8\end{array}\right\|$
This post has been edited 7 times. Last edited by Virgil Nicula, Dec 1, 2015, 10:44 AM

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