Keywords

Mathematics Subject Classification (2010):

1 Introduction

In what follows \(\mathbb{N}_{0}\), \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{Q}\), \(\mathbb{R}\) and \(\mathbb{R}_{+}\) denote, as usual the sets of nonnegative integers, positive integers, integers, rationals, reals and nonnegative reals, respectively. Moreover, let \(a,b \in \mathbb{R}\setminus \{0\}\) with \(ab^{-1}\notin \mathbb{Q}\) and ab < 0 be fixed. Montel [13] (see also [14] and [11, p. 228]) proved that a function \(f: \mathbb{R} \rightarrow \mathbb{R}\), that is continuous at a point and satisfies the system of functional inequalities

$$\displaystyle{ f(x + a) \leq f(x),\qquad f(x + b) \leq f(x)\,,\qquad x \in \mathbb{R}\;, }$$
(1)

has to be constant. A similar (but more abstract) result for measurable functions has been proved in [2].

In [7–9] (see also [10]) the result of Montel has been generalized and extended in several ways. In particular, motivated by some problem arising in a characterization of L p norm, Krassowska and Matkowski [8] (cf. also [7]) have proved that if \(\alpha,\beta \in \mathbb{R}\) and α b ≤ β a, then a continuous at a point function \(f: \mathbb{R} \rightarrow \mathbb{R}\) satisfies the following two functional inequalities

$$\displaystyle{ f(x + a) \leq f(x)+\alpha,\qquad f(x + b) \leq f(x)+\beta \,,\qquad x \in \mathbb{R}\;, }$$
(2)

if and only if α b = β a; moreover f has to be of the form \(f(x) = cx + d\) for \(x \in \mathbb{R}\), with some \(c,d \in \mathbb{R}\).

In this paper we investigate the possibility to obtain results analogous to those in [2, 8, 13] for functions taking values in Riesz spaces. Moreover, we consider the system (2) in a conditional form and almost everywhere. We obtain outcomes that correspond somewhat to the results in [3, 4] and to the problem of stability of functional equations and inequalities (for some further information concerning that problem we refer to, e.g., [1, 5, 6]).

2 Preliminaries

For the readers convenience we present the definition and some basic properties of Riesz spaces (see [12]).

Definition 1 (cf. [12, Definitions 11.1 and 22.1]).

We say that a real linear space L, endowed with a partial order ≤   ⊂ L 2, is a Riesz space if  sup {x, y} exists for all x, y ∈ L and

$$\displaystyle{ax + y \leq az + y\,,\qquad x,y,z \in X,x \leq z,a \in \mathbb{R}_{+}\;;}$$

we define the absolute value of x ∈ L by the formula \(\vert x\vert:=\sup \;\{ x,-x\} \geq 0.\) Next, we write x < z provided x ≤ z and x ≠ z.

A Riesz space L is called Archimedean if, for each x ∈ L, the inequality x ≤ 0 holds whenever the set \(\{nx: n \in \mathbb{N}\}\) is bounded from above.

In the following it will be assumed that L is an Archimedean Riesz space. It is easily seen that α u ≤ β u for every \(u \in L_{+}:=\{ x \in L: x > 0\}\) and \(\alpha,\beta \in \mathbb{R}\), α ≤ β. Moreover, given u ∈ L + we can define an extended (i.e., admitting the infinite value) norm \(\|\cdot \|_{u}\) on L by

$$\displaystyle{\|v\|_{u}:=\inf \;\{\lambda \in \mathbb{R}_{+}: \vert v\vert \leq \lambda u\}\,,\qquad v \in L\;,}$$

where it is understood that \(\inf \;\varnothing = +\infty \) and \(0 \cdot (+\infty ) = 0.\)

Let us yet recall some further necessary definitions.

Definition 2.

Let \(E \subset \mathbb{R}\) be nonempty and let \(\mathcal{I}\subset 2^{\mathbb{R}}\). We say that a property p(x) (x ∈ E) holds \(\mathcal{I}\)-almost everywhere in E (abbreviated in the sequel to \(\mathcal{I}\)-a.e. in E) provided there exists a set \(A \in \mathcal{I}\) such that p(x) holds for all \(x \in E\setminus A\).

Definition 3.

\(\mathcal{I}\subset 2^{\mathbb{R}}\) is a σ-ideal provided \(2^{A} \subset \mathcal{I}\;\) for \(A \in \mathcal{I}\) and

$$\displaystyle{\bigcup _{n\in \mathbb{N}}A_{n} \in \mathcal{I}\,,\qquad \{A_{n}\}_{n\in \mathbb{N}} \subset \mathcal{I}\;.}$$

Moreover, if \(\mathcal{I}\neq 2^{\mathbb{R}}\), then we say that \(\mathcal{I}\) is proper; if \(\mathcal{I}\neq \{\varnothing \}\), then we say that \(\mathcal{I}\) is nontrivial. Finally, \(\mathcal{I}\) is translation invariant (abbreviated to t.i. in the sequel) if \(x + A \in \mathcal{I}\) for \(A \in \mathcal{I}\) and \(x \in \mathbb{R}\).

We have the following (see [3, Propositions 2.1 and 2.2]).

Proposition 1.

Let \(\mathcal{I}\subset 2^{\mathbb{R}}\) be a proper t.i. σ-ideal and let \(U \subset \mathbb{R}\) be open and nonempty. Then

$$\displaystyle{ \mathrm{int}\,\left [(U\setminus T) - V \right ]\neq \varnothing \,,\qquad V \in 2^{\mathbb{R}}\setminus \mathcal{I},\;T \in \mathcal{I}\;, }$$
(3)

where \((U\setminus T) - V = \left \{u - v: u \in U\setminus T,v \in V \right \}\)  .

3 The Main Result

Let us start with an auxiliary result.

Theorem 1.

Let P be a dense subset of  \(\mathbb{R}\), \(\mathcal{J} \subset 2^{\mathbb{R}}\) be a proper t.i. \(\sigma\) -ideal and let E be a subset of a nontrivial interval \(I \subset \mathbb{R}\) with \(H:= I\setminus E \in \mathcal{J}\) . We assume that v: I → L satisfies

$$\displaystyle{ v(p + x) \leq v(x)\,,\qquad x \in E \cap (E - p),p \in P\;. }$$
(4)

If there exists u ∈ L + such that v is continuous at a point x 0 ∈ I, with respect to the extended norm \(\|\cdot \|_{u}\)  , then v(x) = v(x 0 ) \(\mathcal{J}\) -a.e. in I .

Proof.

Note that (4) yields

$$\displaystyle{ v(y) \leq v(y + q)\,,\qquad y \in E \cap (E - q),q \in -P\;, }$$
(5)

where \(-P:=\{ -p: p \in P\}\). Since \(\mathcal{J}\) is proper and t.i., we deduce that \(I\not\in \mathcal{J}\), whence \(E\not\in \mathcal{J}\).

For each \(n \in \mathbb{N}\) we write

$$\displaystyle{D_{n}:=\Big\{ z \in I:\| v(z) - v(x_{0})\|_{u} < \frac{1} {n}\Big\}\;,}$$
$$\displaystyle{E'_{n}:=\Big\{ z \in E: v(z) - v(x_{0}) < \frac{1} {n}u\Big\}\;,}$$
$$\displaystyle{F'_{n}:=\Big\{ z \in E: v(x_{0}) - v(z) < \frac{1} {n}u\Big\}\;,}$$

\(C_{n}:= D_{n}\setminus H\), \(E_{n}:= E\setminus E'_{n}\) and \(F_{n}:= E\setminus F'_{n}\). Clearly, \(\mathrm{int}\;D_{n}\neq \varnothing \) for \(n \in \mathbb{N}\), because v is continuous at x 0.

Suppose that there exists \(k \in \mathbb{N}\) with \(E_{k}\notin \mathcal{J}\). Then, on account of Proposition 1, there is p ∈ P such that \(-p \in \mathrm{ int}\,(C_{k} - E_{k})\), whence \(p + c = e \in E_{k} \subset E\) with some c ∈ C k and e ∈ E k . Hence, by (4),

$$\displaystyle{v(e) - v(x_{0}) = v(p + c) - v(x_{0}) \leq v(c) - v(x_{0}) < \frac{1} {k}u\;.}$$

This is a contradiction.

Next, suppose that \(F_{k}\notin \mathcal{J}\) for some \(k \in \mathbb{N}\). Then, on account of Proposition 1, there is q ∈ −P with \(-q \in \mathrm{ int}\,\left (C_{k} - F_{k}\right )\), whence \(q + c = e \in F_{k} \subset E\) with some c ∈ C k and e ∈ F k . Hence, by (5),

$$\displaystyle{v(x_{0}) - v(e) = v(x_{0}) - v(c + q) \leq v(x_{0}) - v(c) < \frac{1} {k}u\;.}$$

This is a contradiction, too.

In this way we have shown that \(G_{k}:= E_{k} \cup F_{k} \in \mathcal{J}\) for \(k \in \mathbb{N}\). Clearly

$$\displaystyle\begin{array}{rcl} V &:=& \,v^{-1}(L\setminus \{v(x_{ 0})\}) = I\setminus \bigcap _{n\in \mathbb{N}}D_{n} {}\\ & =& \,\bigcup _{n\in \mathbb{N}}I\setminus D_{n} \subset H \cup \bigcup _{k\in \mathbb{N}}G_{k} \in \mathcal{J} {}\\ \end{array}$$

and v(x) = v(x 0) for x ∈ I∖ V. □ 

The next theorem is the main result of this paper.

Theorem 2.

Let I be a real infinite interval, \(\mathcal{J} \subset 2^{\mathbb{R}}\) be a proper t.i. σ-ideal, L be an Archimedean Riesz space, \(v: I \rightarrow L\), \(a_{1},a_{2},\alpha _{1},\alpha _{2} \in \mathbb{R}\), \(a_{1} < 0 < a_{2}\), \(a_{1}a_{2}^{-1}\not\in \mathbb{Q}\) and

$$\displaystyle\begin{array}{rcl} c_{i}:= \frac{1} {a_{i}}\alpha _{i},\qquad i = 1,2\;.& &{}\end{array}$$
(6)

If \(c_{1} \geq c_{2}\) and there exist ω,u ∈ L + such that \(\|\omega \|_{u} < \infty \) , v is continuous at some point x 0 ∈ I, with respect to the extended norm \(\|\cdot \|_{u}\) , and the following two conditional inequalities

$$\displaystyle{ \mbox{ if }\ a_{1} + x \in I,\ \mbox{ then }\ v(a_{1} + x) - v(x) \leq \alpha _{1}\omega \;, }$$
(7)
$$\displaystyle{ \mbox{ if }\ a_{2} + x \in I,\ \mbox{ then }\ v(a_{2} + x) - v(x) \leq \alpha _{2}\omega }$$
(8)

are valid \(\mathcal{J}\) -a.e. in I, then \(c_{2} = c_{1}\) and

$$\displaystyle{ v(x) = c_{1}(x - x_{0})\,\omega + v(x_{0})\,,\qquad \mathcal{J}\mbox{ -a.e. in }I\;. }$$
(9)

Conversely, if \(c_{1} \leq c_{2}\) and(9)holds for some x 0 ∈ I, then v satisfies inequalities(7)and(8) \(\mathcal{J}\) -a.e. in I.

Proof.

Since \(\mathcal{J}\) is a proper and t.i. σ-ideal, it is easily seen that we have the following property

$$\displaystyle{ \mathrm{int}\;T = \varnothing \,,\qquad T \in \mathcal{J}. }$$
(10)

Next, there is a set \(T \in \mathcal{J}\) such that conditions (7) and (8) hold for \(x \in F:= I\setminus T\).

Let

$$\displaystyle{w_{i}(x):= v(x) - c_{i}x\,\omega \,,\qquad i = 1,2,x \in I\;.}$$

Clearly w i is continuous at x 0 with respect to \(\|\cdot \|_{u}\). Further, for every i, j ∈ { 1, 2}, we have \(\alpha _{j} \leq c_{i}a_{j}\) and consequently

$$\displaystyle\begin{array}{rcl} w_{i}(x + a_{j})& =& \,v(x + a_{j}) - c_{i}(x + a_{j})\omega \\ & \leq & \,v(x) +\alpha _{j}\omega - c_{i}x\omega - c_{i}a_{j}\omega \\ & \leq & \,w_{i}(x)\,,\qquad x \in F \cap (F - a_{j})\;.{}\end{array}$$
(11)

Let E: = I∖ H, where

$$\displaystyle{H:=\bigcup _{m,n\in \mathbb{Z}}(T + na_{1} + ma_{2}) \in \mathcal{J}\;.}$$

If we write \(P:=\{ na_{1} + ma_{2}: n,m \in \mathbb{N}_{0}\},\) then

$$\displaystyle\begin{array}{rcl} H + p = H\,,\qquad p \in P\;,& &{}\end{array}$$
(12)

the set P is dense in \(\mathbb{R}\) (see, e.g., [7–9]) and, in view of (11) and (12), it is easy to notice that

$$\displaystyle{ w_{i}(x + p) \leq w_{i}(x)\,,\qquad x \in E \cap (E - p),p \in P,i = 1,2\;. }$$
(13)

Hence, on account of Theorem 1, there are \(V _{1},V _{2} \in \mathcal{J}\) such that

$$\displaystyle{w_{i}(x) = w_{i}(x_{0})\,,\qquad x \in E\setminus V _{i},i = 1,2\;,}$$

which implies (9).

Further, observe that, by (10), we have \(\mathrm{int}(H \cup V _{1} \cup V _{2}) = \varnothing \) and

$$\displaystyle{v(x_{0}) - c_{i}x_{0}\omega = v(x) - c_{i}x\omega \,,\qquad x \in E_{0}:= I\setminus (H \cup V _{1} \cup V _{2}),i = 1,2\;.}$$

Hence

$$\displaystyle{(c_{1} - c_{2})x\omega = (c_{1} - c_{2})x_{0}\omega \,,\qquad x \in E_{0}\;,}$$

whence we get \(c_{1} = c_{2}\).

The converse is easy to check. □ 

Remark 1.

Let \(a_{1},a_{2} \in \mathbb{R}\) and \(\alpha _{1},\alpha _{2} \in (0,\infty )\). Then every function \(v: I \rightarrow \mathbb{R}\) with

$$\displaystyle{\sup _{x\in \mathbb{R}}\;\vert v(x)\vert \leq \frac{1} {2}\min \,\{\alpha _{1},\alpha _{2}\}}$$

fulfils (7) and (8) for each real interval I. This shows that some assumptions concerning \(a_{1},a_{2},c_{1},c_{2}\) are necessary in Theorem 2.

Taking \(\mathcal{J} =\{ \varnothing \}\) in Theorem 2 we obtain the following corollary.

Corollary 1.

Let \(a_{1},a_{2},\alpha _{1},\alpha _{2} \in \mathbb{R}\) be such that \(a_{1} < 0 < a_{2}\), \(a_{1}a_{2}^{-1}\not\in \mathbb{Q}\) and \(c_{1} \geq c_{2}\) , where \(c_{1},c_{2}\) are given by (6) . Let I be a real infinite interval, L be an Archimedean Riesz space, u,ω ∈ L + and \(\|\omega \|_{u} < \infty \) . Then a function v: I → L, that is continuous (with respect to the extended norm \(\|\cdot \|_{u}\)  ) at a point x 0 ∈ I, satisfies the inequalities

$$\displaystyle{ \mbox{ if }\ a_{1} + x \in I,\ \mbox{ then }\ v(a_{1} + x) - v(x) \leq \alpha _{1}\omega \;, }$$
$$\displaystyle{ \mbox{ if }\ a_{2} + x \in I,\ \mbox{ then }\ v(a_{2} + x) - v(x) \leq \alpha _{2}\omega }$$

if and only if \(c_{2} = c_{1}\) and

$$\displaystyle{ v(x) = c_{1}(x - x_{0})\,\omega + v(x_{0})\,,\qquad x \in I\;. }$$