Find region for which F(x,y) = (x+y)^2 is Lipschitz in y

[jamesb1]

As the title says, I need to find such a region.

Taking any x, and any y1 and y2 I used the expression |F(x,y1) - F(x,y2)| and plugged in the function respectively for y1 and y2.

Now I have to find values for x and y such that the following condition (Lipschitz condition) is satisfied:

| 2x + (y1 + y2) | 0 (indeed after having simplified out the previous expression w.r.t the Lipschitz condition)

My initial idea was to find x for which y = 0 and then the same thing for y1 and y2. This method though is not enough since for x = 0 the region of y1 and y2 for which the condition is satisfied, will have to depend on y1 and y2 directly. It will be better to attain a region such that it will not depend on the variables (obviously). I hope I am not incorrect here.

I cannot seem to find a way to get this region and I would very much appreciate any insight given.

Thank you.

 

[pasmith]

As the title says, I need to find such a region.

Taking any x, and any y1 and y2 I used the expression |F(x,y1) - F(x,y2)| and plugged in the function respectively for y1 and y2.

Now I have to find values for x and y such that the following condition (Lipschitz condition) is satisfied:

| 2x + (y1 + y2) | 0 (indeed after having simplified out the previous expression w.r.t the Lipschitz condition)

My initial idea was to find x for which y = 0 and then the same thing for y1 and y2. This method though is not enough since for x = 0 the region of y1 and y2 for which the condition is satisfied, will have to depend on y1 and y2 directly. It will be better to attain a region such that it will not depend on the variables (obviously). I hope I am not incorrect here.

I cannot seem to find a way to get this region and I would very much appreciate any insight given.

Thank you.

For each x \in \mathbb{R} and each K > 0 there exists an interval L_K(x) = \left[-\tfrac12K - x,\tfrac12K - x\right] such that if y_1 \in L_K(x) and y_2 \in L_K(x) then |F(x,y_1) - F(x,y_2)| \leq K|y_1 - y_2|, so f_x : y \mapsto F(x,y) is lipschitz with respect to y with lipschitz constant K.

Note that you must fix x and then determine the interval; if you want a region of the (x,y)-plane in which F is lipschitz with respect to y with constant K then the condition is
<br /> |F(x_1,y_1) - F(x_2,y_2)| <br /> = |x_1^2 - x_2^2 + 2x_1y_1 - 2x_2y_2 + y_1^2 - y_2^2| \leq K|y_1 - y_2|.<br />

 

[micromass]

Please post such questions in the homework forum in the future I'll move it to there now!

 

[jamesb1]

Shouldn't the interval not contain variables though, to have a definite region? I know you can choose any x and fix it, but it seems indefinite to me.

x should remain fixed instead of having x1 and x2, no? I am unsure how to attain values for both x and y such that the following inequality is satisfied:

| 2x + (y1 + y2) | <= K