Motivated by the problem of establishing the existence of periodic solutions of quasi-monotone, periodic differential equations defined on the positive cone in Rn, Krasnosel-skii pioneered the dynamics of sublinear monotone self-mappings of the cone [98]. We will prove Theorem 5.20 below, adapted from the original finite-dimensional version of Krause and Ranft [101].
Let Y denote an ordered Banach space with positive cone Y+. Denote the interior (possibly empty) of Y+ by P . A map T : Y+→ Y+ is sublinear (or “subhomogeneous”)
if
0 < λ < 1 ⇒ λT (x) ≤ T (λx), and strongly sublinear if
0 < λ < 1, x 0 ⇒ λT (x) T (λx)
Strong sublinearity is the strong concavity assumption of Krasnoselskii [98]. It can be verified by using the following result from that monograph:
Lemma 5.18 T : P → Y is strongly sublinear provided T is differentiable and T x T0(x)x for all x 0.
Proof: Let F (s) = s−1T (sx) for s > 0 and some fixed x 0. Then F0(s) = −s−2T (sx)+
s−1T0(sx)x 0 by our hypothesis. So for 0 < λ < 1, we have φ(T x − λ−1T (λx)) = φ(F (1)) − φ(F (λ)) < 0
for every nontrivial φ ∈ Y+∗, the dual cone in Y∗, because dsdφ(F (s)) < 0. The desired conclusion follows from Proposition 3.1.
Corollary 5.19 Assume Y is strongly ordered. A continuous map T : Y+ → Y is sublinear provided T is differentiable in P and T x ≥ T0(x)x for all x 0.
Proof: By continuity it suffices to prove T |P is sublinear. Fix e 0. For each δ > 0 the map P → Y, x 7→ T x + δe is strongly sublinear by Lemma 5.18. Sending δ to zero implies T is sublinear.
Krause and Ranft [101] have results establishing sublinearity of some iterate of T , which is an assumption used in Theorem 5.20 below.
The following theorem demonstrates global convergence properties for order compact maps that are monotone and sublinear in a suitably strong sense.
Theorem 5.20 (Cone Limit Set Trichotomy) Assume T : Y+ → Y+ is continuous and monotone and has the following properties for some r ≥ 1:
(a) Tr is strongly sublinear (b) Trx 0 for all x > 0 (c) Tr is order compact
Then precisely one of the following holds:
(i) each nonzero orbit is order unbounded
(ii) each orbit converges to 0, the unique fixed point of T .
(iii) each nonzero orbit converges to q 0, the unique nonzero fixed point of T . A key tool in the proofs of such results is Hilbert’s projective metric and the related part metric due to Thompson [217]. We define the part metric p(x, y) here in a very limited way, as a metric on P (which is the “part”). For x, y 0, define
p(x, y) := inf{ρ > 0 : e−ρx y eρy}
The family of open order intervals in P forms a base for the topology of the part metric.
It is easy to see that the identity map of P is continuous from the original topology on P to that defined by the part metric.
When Y = IRnwith vector ordering, with P = Int(IRn+), the part metric is isometric to the max metric on IRn, defined by dmax(x, y) = maxi|xi−yi|, via the homeomorphism Int(IRn+) ≈ IRn, x 7→ (log x1, . . . , log xn). Restricted to compact sets in Int(IRn+), the part metric and the max metric are equivalent in the sense that there exist α, β > 0 such that αp(x, y) ≤ |x − y|max≤ βp(x, y).
The usefulness of the part metric in dynamics stems from the following result. Recall map T between metric spaces is a contraction if it has a Lipschitz constant < 1, and it is nonexpansive if it has Lipschitz constant 1. We say T is strictly nonexpansive if p(T x, T y) < p(x, y) whenever x 6= y.
Proposition 5.21 Let T : P → P be a continuous, monotone, sublinear map.
(i) T is nonexpansive for the part metric.
(ii) If T is strongly sublinear, T is strictly nonexpansive for the part metric.
(iii) If T is strongly monotone, A ⊂ P , and no two points of A are linearly dependent, then T |A is strictly nonexpansive for the part metric.
(iv) Under the assumptions of (ii) or (iii), if L ⊂ A is compact (in the norm topology) and T (L) ⊂ L, then the set L∞ =T
n>0Tn(L) is a singleton.
Proof: Fix distinct points x, y ∈ A and set ep(x,y) = λ > 1, so that λ−1x ≤ y ≤ λx and λ is the smallest number with this property. By sublinearity and monotonicity,
λ−1T x ≤ T (λ−1x) ≤ T y ≤ T (λx) ≤ λT x (5.3) which implies p(T x, T y) ≤ p(x, y).
If T is strongly sublinear, the first and last inequalities in (5.3) can be replaced by
, which implies p(T x, T y) < p(x, y).
When xand y are linearly independent, λ−1x < y < λx If also T is strongly monotone, (5.3) is strengthened to
λ−1T x ≤ T (λ−1x) T y T (λx) ≤ λT x
which also implies p(T x, T y) < p(x, y).
To prove (iv), observe first that if L is compact in the norm metric, it is also compact in the part metric. In both (ii) and (iii) T reduces the diameter in the part metric of every compact subset of L. Since T maps L∞ onto itself but reduces its part metric diameter, (iv) follows.
Proof of the Cone Limit Set Trichotomy 5.20. We first work under the assumption that r = 1. In this case Proposition 5.21 shows that every compact invariant set in P reduces to a fixed point, and there is at most one fixed point in P . It suffices to consider the orbits of points x ∈ P , by (b).
Suppose there is a fixed point q 0. There exist numbers 0 < λ < 1 < µ such that x ∈ [λq, µq] ⊂ P . For all n we have
0 λq = λTnq ≤ Tn(λq) ≤ Tnx ≤ Tn(µq) ≤ µTnq = µq
Hence O(x) ⊂ [λq, µq], so O(T x) lies in T ([λp, µq]), which is precompact by (c). There-fore ω(x) is a compact unordered invariant set in P . Proposition 5.21(iii) implies that ω(x) = {q}. This verifies (iii).
Case I: If some orbit O(y) is order unbounded, we prove (i). We may assume y 0.
There exists 0 < γ < 1 such that γy x. Then γTny ≤ Tn(γy) ≤ Tnx, implying O(x) is unbounded.
Case II: If 0 ∈ ω(y) for some y, we prove (ii). We may assume y 0. Fix µ > 1 with x µy. Then 0 ≤ Tnx ≤ Tn(µy) ≤ µTny → 0. Therefore O(x) is compact and Tnx → 0.
Case III: If the orbit closure O(x) ⊂ [a, b] ⊂ P , then (iii) holds. For O(x) is compact by (c), so ω(x) is a nonempty compact invariant set. Because ω(x) ⊂ O(x) ⊂ P , Case I implies (iii).
Cases I, II and III cover all possibilities, so the proof for r = 1 is complete. Now assume r > 1. One of the statements (i), (ii) (iii) is valid for Tr in place of T . If (i) holds for Tr, it obviously holds for T . Assume (ii) holds for Tr. If x > 0 then ω(x) = {0, T (0), . . . , Tr−1(0)}. As this set is compact and Tr invariant, it reduces to {0}, verifying (ii) for T . A similar argument shows that if (iii) holds for Tr, it also holds for T .
The conclusion of the Cone Limit Trichotomy can fail for strongly monotone sub-linear maps— simple sub-linear examples in the plane have a line of fixed points. But the following holds:
Theorem 5.22 Assume:
(a) T : Y+ → Y+ is continuous, sublinear, strongly monotone, and order compact.
(b) for each x > 0 there exists r ∈ N such that Trx 0
Then:
(i) either O(x) is not order bounded for all x > 0, or O(x) converges to a fixed point for all x ≥ 0;
(ii) the set of fixed points > 0 has the form {λe : a ≤ λ ≤ b} where e 0 and 0 ≤ a ≤ b ≤ ∞.
Proof Let y > 0 be arbitrary. If O(y) is not order bounded, or 0 ∈ ω(y), the proof of (i) follows Cases I and II in the proof of the Cone Limit Set Trichotomy 5.20. If O(x) ⊂ [a, b] ⊂ P , then ω(y) is a compact invariant set in P , as in case III of 5.20. As ω(y) is unordered, every pair of its elements are linearly independent. Therefore Proposition 5.21(iv) implies ω(y) reduces to a fixed point, proving (i). The same reference shows that all fixed points lie on a ray R ⊂ Y+ through the origin, which must pass through some e 0 by (b). Suppose p, q are distinct fixed points and 0 p x < q. There exist unique numbers 0 < µ < 1 < ν such that x = µp = µq. Then
T x ≥ µT p = µp = x, T x ≤ νT q = νq = x proving T x = x. This implies (ii).
Papers related to sublinear dynamics and the part metric include Dafermos and Slemrod [35], Krause and Ranft [101], Krause and Nussbaum [102], Nussbaum [145, 146], Smith [183], and Tak´aˇc [207, 214]. For interesting applications of sublinear dynamics to higher order elliptic equations, see Fleckinger and Tak´aˇc [44, 45].