Let f : [a, b]→[a,b] be a continuous function that is differentiable on (a, b) , with |f ′ (x)| < 1 for all x ∈ (a,b) .

7.4.1. Let f : [a, b]→[a,b] be a continuous function that is differentiable on (a, b) , with |f ′ (x)| < 1 for all x ∈ (a,b) .

(i) Prove that f has a unique fixed point. (ii) Show that any Picard sequence for f is convergent and converges to the unique fixed point of f .

Solution. (i) By Brouwer’s fixed-point theorem [Exercise 4.4.1 (i)], f has at least one fixed point. Assuming that f has two fixed points x ∗ and x ∗ , then by the Lagrange mean value theorem, there exists ξ ∈ (a,b) such that

|x

∗ −x | = | f (x ∗ ) − f (x )| = | f ( ξ )| · |x ∗ −x | < |x ∗ −x |,

a contradiction. Thus, f has a unique fixed point. We point out that the condition |f ′ (x)| < 1 for all x ∈ (a,b) is essential for the uniqueness of a fixed point. For example, if f : [a, b]→[a,b] is defined by f (x) = x, then f ′ (x) = 1 for all x ∈ [a,b] and every point of [a,b] is a fixed point of f .

(ii) Let x ∗ denote the unique fixed point of f . Consider any x 1 ∈ [a,b] and let (x n ) n ≥1 ⊂ [a,b] be the Picard sequence for f with its initial point x 1 . Fix an integer n ≥ 1. Thus, by the Lagrange mean value theorem, there exists ξ n bet- ween x n and x ∗ such that

x n +1 −x ∗ = f (x n ) − f (x ∗ )=f ′ ( ξ n )(x n −x ∗ ). This implies that |x n +1 −x ∗ | < |x n −x ∗ |. Next, we prove that x n →x ∗ as n →∞. Since

(x n ) n ≥1 is bounded, it suffices to show that every convergent subsequence of (x n ) n ≥1

302 7 Inequalities and Extremum Problems converges to x ∗ . Let x ∈ R and (x n k ) k ≥1

be a subsequence of (x n ) n ≥1 converging to x . Then

+1 −x |→|x − x ∗ | and

k ) − f (x )|→| f (x) − f (x )| as k→∞. It follows that | f (x) − f (x ∗ )| = |x − x ∗

|x ∗ k +1 −x | = | f (x n

∗ , then by the Lagrange mean value theorem, there exists ξ ∈ (a,b) such that

|x − x ∗ | = | f (x) − f (x ∗ )| = | f ′ ( ξ )| · |x − x ∗ | < |x − x ∗ |,

which is a contradiction. This proves that x

We point out that if the condition |f ′ (x)| < 1 for all x ∈ (a,b) is not satisfied, then

f can still have a unique fixed point x ∗ , but the Picard sequence (x n ) n ≥1 with initial point x 1 ∗ may not converge to x ∗ . For example, if f : [−1,1]→R is defined by

f (x) = −x, then f maps [−1,1] into itself, f is differentiable, and | f ′ (x)| = 1 for all x ∈ [−1,1]. Then x ∗ = 0 is the unique fixed point of f , but if x 1 corresponding Picard sequence is x 1 , −x 1 ,x 1 , −x 1 , . . ., which oscillates between x 1 and −x 1 and never reaches the fixed point. In geometric terms, the cobweb that we hope to weave just traces out a square over and over again. When the hypotheses of the Picard convergence theorem are satisfied, a Picard sequence for f : [a, b]→[a,b] with arbitrary x 1 ∈ [a,b] as its initial point converges to a fixed point of f . It is natural to expect that if x 1 is closer to the fixed point, then the convergence rate will be better. A fixed point of f lies not only in the range of

f but also in the ranges of the iterates f ◦ f , f ◦ f ◦ f , and so on. Thus, if R n is the range of the n-fold composite f ◦ ··· ◦ f (n times), then a fixed point is in each R n . If only a single point belongs to ∞ ∩ n =1 R n , then we have found our fixed point. In

fact, the Picard method amounts to starting with any x 1 ∈ [a,b] and considering the image of x 1 under the n-fold composite f ◦ ··· ◦ f . For example, if f : [0,1]→[0,1] is defined by f (x) = (x + 1)/4, then

f ◦ ··· ◦ f (n times) (x) = −1

. Thus, ∞ ∩ n =1 R n = {1/3}; hence 1/3 is the unique fixed point of f . In general, it is not

convenient to determine the ranges R n for all n. So, it is simpler to use the Picard method, but this tool will be more effective if the above observations are used to some extent in choosing the initial point. ⊓ ⊔

7.4 Optimization Problems 303 The following elementary problem is related to the minimization of the

renormalized Ginzburg–Landau energy functional (see [6]). Problems of this type arise in the study of two phenomena in quantum physics: superconductivity (discov- ered in 1911 by the Dutch physicist Heike Kamerlingh Ones (1853–1926), Nobel Prize in Physics 1913 “for his investigations on the properties of matter at low tem- peratures which led, inter alia, to the production of liquid helium”) and superfluids. Superconducting material is used, for example, in magnetic resonance imaging for medical examinations and particle accelerators in physics. Knowledge about super- fluid liquids can give us deeper insight into the ways in which matter behaves in its lowest and most ordered state. We point out that the Nobel Prize in Physics was awarded in 2003 to Alexei Abrikosov, Vitaly Ginzburg, and Anthony Leggett “for pioneering contributions to the theory of superconductors and superfluids.” The next problem gives an idea about the location of vortices (singularities), which have the tendency to be distributed in regular configurations called Abrikosov lattices. We refer to [68] for more details and comments.