A fixed point theorem and the Hyers-Ulam stability in Riesz spaces
© Batko and Brzdęk; licensee Springer 2013
Received: 8 January 2013
Accepted: 23 April 2013
Published: 14 May 2013
We prove a fixed point theorem and show its applications in investigations of the Hyers-Ulam type stability of some functional equations (in single and many variables) in Riesz spaces.
The Hyers-Ulam stability for functional, but also for difference, differential and integral equations, is a very quickly growing area of investigations (for more details and further references, see, e.g., [1–7]; examples of some recent results can be found in [8–20]). It is related to the notions of shadowing (see, e.g., [21–23]) as well as to the theories of perturbation (see, e.g., [24, 25]) and optimization. The first known result on such stability is due to Pólya and Szegö  and reads as follows.
For every real sequence with , there is a real number ω such that . Moreover, .
But it was Ulam (cf. [27, 28]) who in 1940 gave the main motivation for the study of that subject, at the University of Wisconsin, where he presented some unsolved problems and in particular the following one.
Let be a group and a metric group. Given , does there exist such that if satisfies for all , then a homomorphism exists with for all ?
By now we express that result simply saying that the Cauchy functional equation (1) is Hyers-Ulam stable (or has the Hyers-Ulam stability) in the class of functions .
The Hyers-Ulam stability in Riesz spaces have already been studied in  (with a direct method) and in  (with an application of the spectral representation theorem). The main motivation for this kind of investigations follows from the pretty natural concept to pose the stability problem for a given functional equation in the settings of an ordered structure as an alternative for the topological or metric ones. To preserve the spirit of the Hyers-Ulam stability, we do that in a way which allows to sustain the full coincidence with the classical notion in the simplest case of real functions.
The fixed point approach has been already applied in the investigation of the Hyers-Ulam stability, e.g., in [8, 12, 13, 30–37] and it seems that Baker (see ) has used this tool for the first time in this field; for a survey on this subject, see . In this paper we continue this direction following the approach presented in [12, 13, 31]. We start our paper with a natural fixed point theorem and next derive some stability results from it.
In this paper, as usual, ℕ, , ℝ and denote the sets of all positive integers, nonnegative integers, real numbers and nonnegative real numbers, respectively.
For the readers’ convenience, we quote basic definitions and properties concerning Riesz spaces (see ).
Definition 1 (cf. , Definition 11.1 and Definition 22.1)
we define the absolute value of by the formula .
A Riesz space L is called Archimedean if, for each , the inequality holds whenever the set is bounded from above.
It is easily seen that in a Riesz space L we have and for every , , and , .
There are several types of convergence that may be defined according to the order structure. One of them is the relatively uniform convergence defined as follows.
Definition 2 (cf. , Definition 39.1)
Let L be a Riesz space and . A sequence in L is said to converge e-uniformly to an element whenever, for every , there exists a positive integer such that holds for all . A sequence in L is called e-uniform Cauchy sequence whenever, for every , there exists a positive integer such that holds for all .
Let us point out (see [, p.252]) that in a Riesz space L that is Archimedean, the e-uniform limit of a sequence in it, if exists, is unique and the fact that converges e-uniformly to f will be denoted by (in particular, if and for , then ).
Definition 3 The series in a Riesz space L is said to converge e-uniformly (for a given ) to an element whenever the sequence converges e-uniformly to S. If L is Archimedean, then we write .
Definition 4 (cf. , Definition 39.3)
A Riesz space L is called e-uniformly complete (with a given ) whenever every e-uniform Cauchy sequence has an e-uniform limit.
There is a large class of spaces satisfying the above conditions. In particular, every Dedekind σ-complete space (that is such that any non-empty at most countable subset which is bounded from above has a supremum) is an Archimedean e-uniformly complete space for every (see [, pp.125, 252, 253]).
3 Main result
In what follows X is a nonempty set and Y is an Archimedean u-uniformly complete Riesz space for some given .
We start that part with a lemma that is quite elementary. However, since we use it several times, we present it with a proof for the convenience of readers.
Lemma 1 Let L be an Archimedean Riesz space and . The sequence of rests of an e-uniformly convergent series in L is e-uniformly convergent to 0.
Proof Let us observe that . Consequently, as converges e-uniformly to S and, therefore, is well defined. Moreover, we have . □
The following two hypotheses will be useful.
Observe that Λ is monotone with respect to the pointwise ordering (because is nonnegative). Our main theorem reads as follows.
Proof The proof is somewhat similar to the proof of [, Theorem 1], but since there are also significant differences between them, we present it here.
which results in (6) as Y is Archimedean.
which means that as Y is Archimedean.
Thus we have proved (15). Letting and taking into account Lemma 1, we infer that . □
Remark 1 If in (2) we take , and for , then Theorem 1 becomes a result corresponding to the classical Banach contraction principle.
From Theorem 1 we can easily deduce the following corollary.
Moreover, ψ is given by (7).
Proof Define by and for . Next let be given by (3). Then it is easily seen that (4) and (5) are valid. Consequently, in view of Theorem 1, has a unique fixed point such that (6) holds; moreover, ψ is given by (7). Clearly, (6) and (19) are equivalent, whence we derive the statement concerning the uniqueness of ψ. □
4 The Hyers-Ulam stability
in the class of functions , with given and .
Moreover, ψ is given by (7) with for , .
Proof Use Theorem 1 with , , , and for , . □
Example 1 If there is such that (in Corollary 2) for , then clearly we get for , provided is u-uniformly convergent in Y. Moreover, in the case where for all and some with (e.g., when and ε is a constant function), for . This means that Corollary 2 implies some analogues of the results in  for the Riesz spaces.
Analogously, from Corollary 1 we obtain the following.
Moreover, ψ is given by (7).
We also can use Theorem 1 and Corollary 1 to deal with the stability of a bit more involved equations with any number of unknown functions, e.g., as follows.
where for , .
Proof It is enough to use Theorem 1 with . □
and (18) holds with given by (16). Then there exists a unique satisfying inequality (27) and equation (29). Moreover, ψ is given by (30).
Proof It is enough to use Corollary 1 with . □
In the next part we supply an example of a result concerning the Hyers-Ulam stability of functional equations in several variables.
5 Stability of the equation of homomorphism for the square symmetric groupoids
(for instance, with some fixed , and for functions ) is also an example of a homomorphism equation for some square symmetric groupoids.
such that condition (36) is fulfilled. Moreover, Φ is given by (37).
This means that Φ satisfies (35) (in view of (34) and (37)).
Since is the unique solution of functional equation (41) fulfilling (36), and every solution of (35) satisfies (41), we also obtain the statement concerning the uniqueness of Φ. □
Moreover, Φ is given by (37).
Proof We argue as in the proof of Corollary 1. Namely, define and by and for . Then (42)-(44) imply (32)-(34). Hence, by Corollary 6, there exists a unique solution of functional equation (35) such that (36) holds; in particular, Φ is given by (37). Clearly (36) and (45) are equivalent, so we also obtain the statement on the uniqueness of Φ. □
In the very particular case where and functions χ and are constant, Corollaries 6 and 7 yield analogues of the classical result of Hyers .
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