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A fixed point theorem and the Hyers-Ulam stability in Riesz spaces
Advances in Difference Equations volume 2013, Article number: 138 (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 ?
Let be a commutative semigroup, Y be a Banach space and . Then for every with there is a unique such that and
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 say that a real linear space L, endowed with a partial order , is a Riesz space if exists for all and
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.
(H1) Mappings and are given and is an operator satisfying the inequality
(H2) is a linear operator given by
Observe that Λ is monotone with respect to the pointwise ordering (because is nonnegative). Our main theorem reads as follows.
Theorem 1 Let us assume that hypotheses (H1) and (H2) are fulfilled and that functions satisfy the conditions
Then has a unique fixed point with
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.
First we show by induction that
For , (8) and (4) coincide. So, let us suppose that (8) holds for a fixed . Using (2) and the definition of Λ (i.e., (3)), we have
which completes the proof of (8). Therefore, for , we get
This, according to (5) and Lemma 1, means that is a u-uniform Cauchy sequence for any , and therefore u-uniformly convergent, as Y is u-uniformly complete. Let us denote by its relative uniform limit. Using (9) with , we have
Write for and . By the triangle inequality and (10), for any and , we obtain
From the u-uniform convergence of to and to , for , we derive
which results in (6) as Y is Archimedean.
We will show that ψ is a fixed point of . Having applied (2) with and , for any , one has
Next, using the triangle inequality (for , ), we infer that
which means that as Y is Archimedean.
For the proof of the uniqueness of ψ, assume that are two fixed points of satisfying
At first let us observe that for every , we get
In fact, for inequality (15) follows directly from (14). Now assume that (15) holds for a given . Then, by (12), for every , we have
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.
Remark 2 We can associate to a given an extended (i.e., admitting the infinite value) norm on Y denoted by and defined by
From Theorem 1 we can easily deduce the following corollary.
Corollary 1 Let hypothesis (H1) be fulfilled, be a linear operator given by
and functions and satisfy the conditions
Then has a unique fixed point with
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 simplest case , Theorem 1 yields at once the following corollary concerning the Hyers-Ulam stability of a quite general functional equation
in the class of functions , with given and .
Corollary 2 Let , , , ,
Then there is a unique solution of functional equation (20) such that
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.
Corollary 3 Let , , , , (21) holds,
Then there is a unique solution of equation (20) such that
Moreover, ψ is given by (7).
Example 2 If (in Corollaries 2 and 3) for , with some , and , then we get stability results for the equation
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.
Corollary 4 Let , , , , and
Suppose that functions satisfy the inequality
and (5) holds with Λ given by (3). Then there exists a unique satisfying inequality (24), which solves the equation
where for , .
Proof It is enough to use Theorem 1 with . □
Corollary 5 Let , , , , and satisfy (28). Suppose that functions and are such that
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
Let T be a nonempty set. A binary operation is square symmetric provided
is square symmetric. So, it is easily seen that the following two very well-known functional equations
are particular examples of homomorphism equations for square symmetric groupoids. Moreover, we have a very particular case of a square symmetric operation when T is a linear space over a field and ∗ is defined by
with some fixed and . So, the general linear functional equation
(for instance, with some fixed , and for functions ) is also an example of a homomorphism equation for some square symmetric groupoids.
Corollary 6 Let , , , , , be a square symmetric binary operation, and let be given by for ,
Then there exists a unique solution of the functional equation
Proof Take in (32). Then we get
Hence (21) and (23) hold with and
for , . Now, if is defined by (22), then it is easily seen that (33) holds. Consequently, according to Corollary 2, there exists a unique solution of the equation
such that condition (36) is fulfilled. Moreover, Φ is given by (37).
Now we show that Φ is a solution of equation (35). To this end, fix and . Then , whence by (32),
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. □
Corollary 7 Let , , , , , be a square symmetric binary operation, let be given by for ,
Then there is a unique solution of equation (35) such that
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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The authors declare that they have no competing interests.
All authors contributed equally to the manuscript. All authors have read and approved the final version of the manuscript.
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Batko, B., Brzdęk, J. A fixed point theorem and the Hyers-Ulam stability in Riesz spaces. Adv Differ Equ 2013, 138 (2013). https://doi.org/10.1186/1687-1847-2013-138
- Hyers-Ulam type stability
- fixed point
- Riesz space
- square symmetric groupoid