Uniform boundedness principle for non-linear operators on cones of functions

We prove an uniform boundedness principle for the Lipschitz seminorm of continuous, monotone, positively homogeneous and subadditive mappings on suitable cones of functions. The result is applicable to several classes of classically non-linear operators.


Introduction and preliminaries
Uniform boundedness principle for bounded linear operators (Banach-Steinhaus theorem) in one of the cornerstones of classical functional analysis (see e.g [12], [1], [14] and the references cited there). In this article we prove a new uniform boundedness principle for monotone, positively homogeneous, subadditive and Lipschitz mappings defined on a suitable cone of functions (Theorem 2.2). This result is applicable to several classes of classically non-linear operators (Examples 2.4, 2.5, and Remarks 2.7, 2.8).
Let Ω be a non-empty set. Throughout the article let X denote a vector space of all functions ϕ : Ω → R or a vector space of all equivalence classes of (almost everywhere equal) real measurable functions on Ω, if (Ω, M, µ) is a measure space. As usually, |ϕ| denotes the absolute value of ϕ ∈ X.
Let Y ⊂ X be a vector space and let Y + denote the positive cone of Y , i.e., the set of all ϕ ∈ Y such that ϕ(ω) ≥ 0 for all (almost all) ω ∈ Ω. The space Y is called an ordered vector space with the partial ordering induced by the cone Y + . If, in addition, Y is a normed space it is called an ordered normed space. The vector space Y ⊂ X is called a vector lattice (or a Riesz space) if for every ϕ, ψ ∈ Y we have a supremum and infimum (greatest lower bound) in Y . If, in addition, Y is a normed space and if |ϕ| ≤ |ψ| implies ϕ ≤ ψ , then Y is a called normed vector lattice (or a normed Riesz space).
Note that in a normed vector lattice Y we have |ϕ| = ϕ for all ϕ ∈ Y . A complete normed vector lattice is called a Banach lattice. Observe that X itself is a vector lattice.
Let Y ⊂ X be a normed space. The cone Y + is called normal if and only if there exists a constant C > 0 such that ϕ ≤ C ψ whenever ϕ ≤ ψ, ϕ, ψ ∈ Y + . A cone Y + is normal if and only if there exists an equivalent monotone norm ||| · ||| on Y , i.e., |||ϕ||| ≤ |||ψ||| whenever 0 ≤ ϕ ≤ ψ (see e.g. [4,Theorem 2.38]). A positive cone of a normed vector lattice is closed and normal. Every closed cone in a finite dimensional Banach space is necessarily normal.
Let Z ⊂ Y be a cone (not necessarily equal to Y + ). A cone Z is said to be complete if it is a complete metric space in the topology induced by Y . In the case when Y is a Banach space this is equivalent to Z being closed in Y .
A mapping A : Z → Z is called positively homogeneous (of degree 1) if A(tϕ) = tA(ϕ) for all t ≥ 0 and ϕ ∈ Z. A mapping A : Z → Z is called Lipschitz if there exists L > 0 such that Aϕ − Aψ ≤ L ϕ − ψ for all ϕ, ψ ∈ Z and we denote If A is Lipschitz and positively homogeneous, then Note also that a Lipschitz and positively homogeneous mapping A on Z is always bounded on Z, i.e., Aϕ is finite and it holds A ≤ A LIP . Moreover, a positively homogeneous mapping A : Z → Z, which is continuous at 0 is bounded on Z .
, which is reflexive, transitive, but not necessary antisymmetric.
If K ⊂ Y is a wedge, then A : K → Y is called subadditive if A(ϕ + ψ) ≤ Aϕ + Aψ for ϕ, ψ ∈ K, and is called monotone (order preserving) if Aϕ ≤ Aψ whenever ϕ ≤ ψ, ϕ, ψ ∈ K. Note that in this definition of subadditivity and monotonicity we consider on Y (and on K) a partial ordering ≤ Y + induced by Y + (not a preordering ≤ K ). One of the reasons for this choice is that, for example, it may happen that a non-linear map is monotone with respect to the ordering ≤ Y + , but it is not monotone with respect to the preordering ≤ K (see, for instance, [24, Section 5] and max-type operators, or [22] and the "renormalization operators" which occur in discussing diffusion on fractals). Moreover, for similar reasons wherever in our article we consider a subcone Z ⊂ Y + we consider on Z a partial ordering ≤ Y + induced by Y + (not a partial ordering ≤ Z ). Observe that in this setting the set Z − Z is a vector subspace in Y and thus a wedge.

Results
We will need the following lemma.
If, in addition, Y is a normed space such that Y + is normal and Z ⊂ Y + is a subcone such that ϕ = |ϕ| for all ϕ = ϕ 1 − ϕ 2 where ϕ 1 , ϕ 2 ∈ Z, and if AZ ⊂ Z and A is bounded on Z, then A is Lipschitz on Z.
Assume that, in addition, Y is a normed space such that Y + is normal (with a normality constant C) and Z ⊂ Y + a subcone such that ϕ 1 − ϕ 2 = |ϕ 1 − ϕ 2 | for all ϕ 1 , ϕ 2 ∈ Z, and that AZ ⊂ Z and A is bounded on Z. It follows from (1) that and thus A is Lipschitz on Z (and A LIP ≤ C A ), which completes the proof.
The following uniform boundedness principle is the central result of this article.
Theorem 2.2. Let Y ⊂ X be a normed space such that Y + is normal and let Z ⊂ Y + be a complete subcone, such that |ϕ − ψ| ∈ Z for all ϕ, ψ ∈ Z and such that ϕ = |ϕ| for all ϕ ∈ Z −Z. Assume that A is a set of subadditive and monotone mappings A : Z −Z → Y such that AZ ⊂ Z and that each A ∈ A is positively homogeneous and continuous on Z.
If the set {Aϕ : A ∈ A} is bounded for each ϕ ∈ Z (i.e., for each ϕ ∈ Z there exists M ϕ > 0 such that Aϕ ≤ M ϕ for all A ∈ A), then there exists M > 0 such that Proof. Since Z is closed and each A ∈ A is continuous on Z the set A n = {ϕ ∈ Z : Aϕ ≤ n for all A ∈ A} is closed in Y for each n ∈ N. Moreover, Z is a complete metric space and Z = ∪ ∞ n=1 A n . By Baire's theorem there exist n 0 ∈ N, ϕ 0 ∈ Z and ε > 0 such that an open ball Let ϕ, ψ ∈ Z such that ϕ − ψ = 1 and A ∈ A. Since Z is a normal cone and A is positively homogeneous on Z, we have by (1) Since A is subadditive and monotone on Z − Z we have which together with (2) implies We also have which proves (3). It follows from (3) that |ε|ϕ − ψ| − ϕ 0 | − ϕ 0 ≤ 3Cε ϕ − ψ = 3Cε and thus |ε|ϕ − ψ| − ϕ 0 | ∈ A n 0 and so A|ε|ϕ − ψ| − ϕ 0 | ≤ n 0 . Therefore Aϕ − Aψ ≤ 2C 2 n 0 ε and so A LIP ≤ 2C 2 n 0 ε .

Remark 2.3. (i) Each
A ∈ A satisfies A ≤ A LIP ≤ C A (see the proof of Lemma 2.1). Therefore we could alternatively set ψ = 0 in the proof above and prove an uniform upper bound A ≤ 2Cn 0 ε for all A ∈ A, which gives the same conclusion.
(ii) In the proofs of Lemma 2.1 and Theorem 2.2 we did not need the assumption Z ∩ (−Z) = {0}, so it suffices to assume that Z is a wedge in this two results (not necessarily a cone).
(iii) Also the assumption on normality of Y + can be slightly weakened in Lemma 2.1 and Theorem 2.2. Instead of normality of Y + it suffices to assume that there exists a constant C > 0 such that ϕ ≤ C ψ whenever ϕ ≤ ψ, ϕ, ψ ∈ Z (where again ϕ ≤ ψ means ϕ ≤ Y + ψ) Our results can be applied to various classes of non-linear operators. In particular, they apply to various max-kernel operators (and their isomorphic versions) appearing in the literature (see e.g. [23], [24], [19], [7] and the references cited there). We point out the following two related examples from [23], [24], [29], [30]. It is clear that for Z = C + [0, a] it holds AZ ⊂ Z. The eigenproblem of these operators arises in the study of periodic solutions of a class of differential-delay equations εy ′ (t) = g(y(t), y(t − τ )), τ = τ (y(t)), with state-dependent delay (see e.g. [23]). The mapping A : Y → Y is subadditive and monotone and is a positively homogeneous and Lipschitz on Z. Moreover, . Cleary, our Theorem 2.2 applies to sets of such mappings.
Consequently, our Theorem 2.2 implies also to isomorphic max-plus mappings (see e.g. [23] and the references cited there) and a Lipschitz seminorm with respect to a suitably induced metric. Note that a related result for uniform boundedness (in fact contractivity) result for a Lipschitz seminorm of semigroups of max-plus mappings was stated in [20], [5]. However, observe that the Lipschitz seminorm there is defined with respect to a different metric than in our case.
We also point out the following related example from [29] and [30]. Together with its isomorphic versions (max-plus algebra and min-plus algebra also known as tropical algebra) it provides an attractive way of describing a class of non-linear problems appearing for instance in manufacturing and transportation scheduling, information technology, discrete event-dynamic systems, combinatorial optimization, mathematical physics, DNA analysis, ... (see e.g. [8], [11], [28], [27], [15], [34] and the references cited there).
Remark 2.7. Our results apply also to more general max-type operators studied in [24,Section 5]. The authors considered there finite sums of more general operators than in Example 2.4 defined on a Banach space of continuous functions and their restrictions to suitable closed cones. The assumptions of our results are satisfied also for these mappings and therefore also for a special case of cone-linear Perron-Frobenius operators studied there.