1. Introduction and Preliminaries Uniform boundedness principle for bounded linear operators (Banach-Steinhaus theorem) is one of the cornerstones of classical functional analysis (see, e.g., [1–3] and the references cited therein). 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). This result is applicable to several classes of classically nonlinear operators (Examples 4 and 5 and Remarks 7 and 8).

Let Ω be a nonempty 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 equal everywhere) real measurable functions on Ω, if (Ω,M,μ) is a measure space. As usual, φ denotes the absolute value of φ∈X.

Let Y⊂X be a vector space and let Y+ denote the positive cone of Y, that is, 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≤Cψ whenever φ≤ψ, φ,ψ∈Y+. A cone Y+ is normal if and only if there exists an equivalent monotone norm |||·||| on Y; that is, |||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≤Lφ-ψ for all φ,ψ∈Z and we denote(1)ALIP=supφ,ψ∈Z,φ≠ψ Aφ-Aψφ-ψ.If A is Lipschitz and positively homogeneous, then(2)ALIP=supφ,ψ,∈Z,φ-ψ=1 Aφ-Aψ.Note also that a Lipschitz and positively homogeneous mapping A on Z is always bounded on Z; that is, (3)A=supφ∈Z,φ≠0 Aφφ=supφ∈Z,φ=1 Aφis finite and it holds that A≤ALIP. Moreover, a positively homogeneous mapping A:Z→Z, which is continuous at 0, is bounded on Z.

A set K⊂Y is called a wedge if K+K⊂K and if tK⊂K for all t≥0. A wedge K induces on Y a vector preordering ≤K (φ≤Kψ if and only if ψ-φ∈K), which is reflexive and transitive, but not necessarily 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 nonlinear map is monotone with respect to the ordering ≤Y+, but it is not monotone with respect to the preordering ≤K (see, e.g., [5, Section 5] and max-type operators, or [6] 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.

In our main result (Theorem 2) we will consider a normed space Y⊂X with a normal cone Y+ and a complete subcone Z⊂Y+ that satisfies φ-ψ∈Z for all φ,ψ∈Z and such that φ=φ for all φ=φ1-φ2, where φ1,φ2∈Z. Since X itself is a vector lattice the above assumptions make sense. Note also that a positive cone Z=Y+ of each Banach lattice Y or, in particular, of each Banach function space (see, e.g., [2, 7–11] and the references cited therein) satisfies these properties. For the theory of cones, wedges, linear and nonlinear operators on cones and wedges, Banach ordered spaces, Banach function spaces, vector and Banach lattices, and applications, for example, in financial mathematics, we refer the reader to [2, 4, 5, 7, 8, 12–19] and the references cited therein.

2. Results We will need the following lemma.

Lemma 1. Let Y⊂X be a vector space and let Z⊂Y+ be a subcone such that φ-ψ∈Z for all φ,ψ∈Z. If A:Z-Z→Y is a subadditive and monotone mapping, then(4)Aφ-Aψ≤Aφ-ψ,for all φ,ψ∈Z.

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=φ1-φ2, where φ1,φ2∈Z, and if AZ⊂Z and A is bounded on Z, then A is Lipschitz on Z.

Proof. Let φ,ψ∈Z. Since A:Z-Z→Y is a subadditive, we have(5)Aφ=Aφ-ψ+ψ≤Aφ-ψ+Aψ.It follows that Aφ-Aψ≤A(φ-ψ)≤Aφ-ψ, since A is monotone and φ-ψ≤φ-ψ. Similarly one obtains that Aψ-Aφ≤Aφ-ψ, which proves (4).

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 φ1-φ2=φ1-φ2 for all φ1,φ2∈Z and that AZ⊂Z and A is bounded on Z. It follows from (4) that(6)Aφ-Aψ=Aφ-Aψ≤CAφ-ψ≤CAφ-ψ,and thus A is Lipschitz on Z (and ALIP≤CA), which completes the proof.

The following uniform boundedness principle is the central result of this article.

Theorem 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 Mφ>0 such that Aφ≤Mφ for all A∈A), then there exists M>0 such that ALIP≤M for all A∈A.

Proof. Since Z is closed and each A∈A is continuous on Z the set(7)An=φ∈Z: Aφ≤n ∀A∈Ais closed in Y for each n∈N. Moreover, Z is a complete metric space and Z=⋃n=1∞An. By Baire’s theorem there exist n0∈N, φ0∈Z, and ε>0 such that an open ball O(φ0,3εC)=φ∈Z: φ-φ0<3εC⊂An0, where C is the normality constant of Y+.

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 (4)(8)Aφ-Aψ=Aφ-Aψ≤CAφ-ψ=CεAεφ-ψ.Since A is subadditive and monotone on Z-Z we have(9)Aεφ-ψ=Aφ0+εφ-ψ-φ0≤Aφ0+Aεφ-ψ-φ0≤Aφ0+Aεφ-ψ-φ0,which together with (8) implies(10)Aφ-Aψ≤C2εAφ0+Aεφ-ψ-φ0≤C2εn0+Aεφ-ψ-φ0.We also have(11)εφ-ψ-φ0-φ0<3εφ-ψ.

Indeed, if εφω-ψω-φ0(ω)≤0, then(12)εφω-ψω-φ0ω-φ0ω=εφω-ψω,and if ε|φ(ω)-ψ(ω)|-φ0(ω)>0, then(13)εφω-ψω-φ0ω-φ0ω=εφω-ψω-2φ0ω≤εφω-ψω+2φ0ω<3εφω-ψω,which proves (11).

It follows from (11) that εφ-ψ-φ0-φ0≤3Cεφ-ψ=3Cε and thus |ε|φ-ψ-φ0∈An0 and so A|ε|φ-ψ|-φ0|≤n0. Therefore(14)Aφ-Aψ≤2C2n0ε,and so ALIP≤2C2n0/ε.

Remark 3. (i) Each A∈A satisfies A≤ALIP≤CA (see the proof of Lemma 1). Therefore we could alternatively set ψ=0 in the proof above and prove a uniform upper bound A≤2Cn0/ε for all A∈A, which gives the same conclusion.

(ii) In the proofs of Lemma 1 and Theorem 2 we did not need the assumption Z∩(-Z)={0}, so it suffices to assume that Z is a wedge in these two results (not necessarily a cone).

(iii) Also the assumption on normality of Y+ can be slightly weakened in Lemma 1 and Theorem 2. Instead of normality of Y+ it suffices to assume that there exists a constant C>0 such that φ≤Cψ whenever φ≤ψ, φ,ψ∈Z (where again φ≤ψ means φ ≤Y+ψ).

Our results can be applied to various classes of nonlinear operators. In particular, they apply to various max-kernel operators (and their isomorphic versions) appearing in the literature (see, e.g., [5, 19–21] and the references cited therein). We point out the following two related examples from [5, 17–19].

Example 4. Given a>0, let Y=C[0,a] be Banach lattice of continuous functions on [0,a] equipped with ·∞ norm. Consider the following max-type kernel operators A:C[0,a]→C[0,a] of the form(15)Aφs=maxt∈αs,βs ks,tφt,where φ∈C[0,a] and α,β:[0,a]→[0,a] are given continuous functions satisfying α≤β. The kernel k:S→[0,∞) is a given nonnegative continuous function, where S denotes the compact set(16)S=s,t∈0,a×0,a: t∈αs,βs.It is clear that for Z=C+[0,a] it holds that AZ⊂Z. The eigenproblem of these operators arises in the study of periodic solutions of a class of differential-delay equations(17)εy′t=gyt,yt-τ, τ=τyt,with state-dependent delay (see, e.g., [19]).

The mapping A:Y→Y is subadditive and monotone and is positively homogeneous and Lipschitz on Z. Moreover, ALIP=A=max(s0,s1)∈S1k(s0,s1), where S1={(s0,s1): s0∈[0,a], s1∈[α(s0),β(s0)]}. Clearly, Theorem 2 applies to sets of such mappings.

Consequently, Theorem 2 applies also to isomorphic max-plus mappings (see, e.g., [19] and the references cited therein) 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 [22, 23]. However, observe that the Lipschitz seminorm there is defined with respect to a different metric than that in our case.

We also point out the following related example from [17, 18].

Example 5. Let M be a nonempty set and let Y be the set of all bounded real functions on M. With the norm f∞=supft: t∈M and natural operations, Y is a Banach lattice. Let Z=Y+ and let k:M×M→[0,∞) satisfy supkt,s: t,s∈M<∞. Let A:Y→Y be defined by (Af)(s)=supks,tft: t∈M. Then A:Y→Y is subadditive and monotone mapping that satisfies AZ⊂Z and is positively homogeneous and Lipschitz on Z; therefore Theorem 2 applies to sets of such mappings. It also holds that ALIP=A=supkt,s: t,s∈M. In particular, if M is the set of all natural numbers N, our results apply to infinite bounded nonnegative matrices k=[k(i,j)] (i.e., k(i,j)≥0 for all i,j∈N and k∞=supi,j∈ Nk(i,j)<∞). In this case, Y=l∞ and Z=l+∞ and ALIP=A=k∞.

Remark 6. The special case of Example 5 when M={1,…,n} for some n∈N is well known and studied under the name max-algebra (an analogue of linear algebra). 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 nonlinear problems appearing, for instance, in manufacturing and transportation scheduling, information technology, discrete event-dynamic systems, combinatorial optimization, mathematical physics, and DNA analysis (see, e.g., [24–29] and the references cited therein).

Remark 7. Our results apply also to more general max-type operators studied in [5, Section 5]. The authors considered there finite sums of more general operators than in Example 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.

Remark 8. Theorem 2 applies to several classes of nonlinear integral operators (under suitable assumptions on the kernels and on the defining nonlinearities) including Hammerstein type operators (see, e.g., [12, Chapter 12, p. 338] and [30, 31]), Uryson type operators (see, e.g., [12, Chapter 12, p. 339] and [32]), and Hardy-Littlewood type operators (see, e.g., [33–36]).