6.7. EXERCISES 181

38. Let φ : Rn→ R be convex, proper lower semicontinuous, and bounded below. Showthat the graph of ∂φ is nonempty. Hint: Just consider ψ (x) = |x|2 + φ (x) andobserve that this is coercive. Then argue using convexity that ∂ψ (x) = ∂φ (x)+2x.(You don’t need to assume that φ is bounded below but it is convenient to assumethis.)

39. Suppose f : [0,T ]×Rn → Rn is continuous and an estimate of the following formholds. (f (t,x) ,x)≤ A+B |x|2 Show that there exists a solution to the initial valueproblem x′ = f (t,x) , x(0) = x0 for t ∈ [0,T ].

40. In the above problem, suppose that −f +αI is monotone for large enough α inaddition to the estimate of that problem. Show that then there is only one solution tothe problem. In fact, show that the solution depends continuously on the initial data.

41. It was shown that if f : X → X is locally Lipschitz where X is a Banach space. Thenthere exists a unique local solution to the IVP

y′ = f (y) , y(0) = y0

If f is bounded, then in fact the solutions exists on [0,T ] for any T > 0. Show that itsuffices to assume that ∥ f (y)∥ ≤ a+b∥y∥.

42. Suppose f (·, ·) :R×Rn→Rn is continuous and also that | f (t,x)| ≤M for all (t,x).Show that there exists a solution to the initial value problem

x′ = f (t,x) , x(0) = x0 ∈ Rn

for t ∈ [0,T ]. Hint: You might consider T : C ([0,T ] ,Rn)→C ([0,T ] ,Rn) given byFx(t)≡ x0+

∫ t0 f (s,x(s))ds. Argue that F has a fixed point using the Schauder fixed

point theorem.

43. Remove the assumption that | f (t,x)| ≤ M at the expense of obtaining only a localsolution.

Hint: You can consider the closed set in Rn B = B(x0,R) where R is some positivenumber. Let P be the projection onto B.

44. In the Schauder fixed point theorem, eliminate the assumption that K is closed. Hint:You can argue that the {yi} in the approximation can be in f (K).

45. Show that there is no one to one continuous function

f : [0,1]→{(x,y) : x2 + y2 ≤ 1

}such that f is onto.