I have a family of positive, convex, parameterized functions by $ lambda in (0, infty) $. Let us denote the functional corresponding to $ lambda $ how $ C λ (f) $.

Existence and uniqueness of minimizer for each of these functionalities in a set $ S $ It has already been tested. We will denote the minimizer of $ C λ (f) $ on the whole $ S $ how $ f lambda $

Now I don't know what functional you should choose to minimize, since each has its own advantages and disadvantages. For some reason, I choose to minimize the functional corresponding to that $ lambda = lambda_0 $for which $ | f _ { lambda} | $ is maximum over $ lambda in (0, infty) $.

Fortunately, I can show that $ lim limits _ { lambda a 0} | f _ { lambda} | = $ 0 Y $ lim limits _ { lambda to infty} | f _ { lambda} | = $ 0, so that it establishes the existence of $ lambda_0 $.

Now I need to demonstrate the uniqueness of $ lambda_0 $.

I am able to prove that the equation $ frac { partial { | f- nabla C _ { lambda} (f) |}} { partial { lambda}} = 0 $, solving for $ lambda $, has a unique solution in $ (0, infty) $ For any $ f in S $. In fact, this equation is linear in $ lambda $.

where $ nabla C _ { lambda} (f) $ denotes the gradient of the functional $ C _ lambda $ in some $ f in S $. It may be called a functional derivative with an appropriate name, but that is not the subject of interest.

Is there any way to use this fact to conclude that $ lambda_0 $ It's unique?