On Approximately p-Wright Affine Functions in Ultrametric Spaces
We prove some results for mappings taking values in ultrametric spaces and satisfying approximately a generalization of the equation of p-Wright affine functions. They are motivated by the notion of stability for functional equations.
1. Introduction and Preliminaries
Let and be linear spaces over fields and , respectively, and let be fixed. The functional equation for function , generalizes the equation
For and , solutions of (2) are called the -Wright affine functions, which are both -Wright convex and concave (see ). For information on the -Wright convexity and the -Wright concavity we refer, for example, to [1, 2].
Note also that for (2) becomes the well-known Jensen’s functional equation For (2) takes the form , which has been studied in  (see also ) in connection with some investigations of the generalized -Jordan derivations on Banach algebras. The cases of more arbitrary have been studied in [1, 5] (cf. ).
In this paper we consider (1) in a bit generalized form, with and being some group endomorphisms. Motivated by the notion of the Hyers-Ulam stability (see, e.g., [6–9] for further details), we prove some results concerning the mappings that take values in the ultrametric spaces and satisfy (1) approximately, that is, fulfil inequality (8) (with suitable assumptions on ). Our outcomes correspond to those in , where some issues (analogous as in Theorems 2 and 6) have been considered for functions mapping a classical normed space (real or complex) into a classical Banach space, with being a scalar (real or complex, resp.) and for of the following two forms: with some real and (see also  for similar outcomes but only for ). The main tool in our investigations is a fixed point theorem from  (for information on related results see [12–15]).
Recall that an ultrametric space is a metric space with the metric satisfying the condition for every ; such a metric is called an ultrametric. One of important examples of the ultrametric spaces is a non-Archimedean normed space.
Let us remind yet that a linear space over a field , with a function , is said to be a non-Archimedean normed space provided if and only if and and for all , , where denotes a non-Archimedean valuation in , that is a function such that if and only if and and for .
Clearly, if is a non-Archimedean normed space, then the formula defines an ultrametric in , that is invariant (i.e., for every ).
Any field endowed with a non-Archimedean valuation is said to be a non-Archimedean field. If the valuation is nontrivial (i.e., there is an such that ), then we have and for all (positive integers), where is the neutral element of the semigroup , and for .
The first example of a non-Archimedean field was provided by Hensel in , where he gave a description of the -adic numbers (for each fixed prime number and any non-zero rational number , there exists a unique integer such that , where and are integers not divisible by ; then defines a non-Archimedean valuation in and therefore also a non-Archimedean norm in ).
Let denote the completion of , with respect to the metric . Then (called the -adic number field) can be identified with the set of all formal series , where are integers. The operation of addition and multiplication between any two elements of is defined naturally. The non-Archimedean norm in is defined by ; endowed with it turns out to be a locally compact filed (see ). Let us mention yet that the -adic numbers have gained the interest of physicists because of their connections with some issues in quantum physics, -adic strings and superstrings (see ).
The problem of stability of functional equations was motivated by a question of S.M. Ulam asked in 1940 and an answer to it published by Hyers . Since then numerous papers on this subject have been published, and we refer to [6–9, 13, 15, 20, 21] for more details, some discussions, and further references.
Moslehian and Rassias  (see [26, 27] for some related outcomes) have proved the first stability results for the Cauchy and quadratic functional equations in non-Archimedean normed spaces. Afterwards several stability results for other equations in such spaces have been published, for example, in [28–30] (for further references see [13, 15]).
2. Auxiliary Result
In this section denotes a nonempty set and stands for a complete ultrametric space. The main tool in the proofs of the main theorems of this paper is a fixed point result that can be derived from [11, Theorem 2]. To present it we need to introduce some notions.
For any ( denotes the family of all functions mapping a set into a set ) we write provided for , and we say that an operator is nondecreasing if it satisfies the condition for all with . Moreover, given a sequence in , we write provided for .
We use the following hypothesis concerning operators : for every sequence in satisfying the condition .
Moreover, to simplify some notations, we define by
Now we are in a position to present the mentioned fixed point result.
Theorem 1. Let be a nondecreasing operator satisfying hypothesis . If , , and satisfy then the limit exists for every and the function , defined in this way, is a fixed point of with
3. The Main Results
We say that is an ultrametric group if is a group and is ultrametric in such that the group operation + is continuous with respect to . In what follows is a group (though we use the additive notation for the group operation in , it does not mean that the group must be commutative), is an ultrametric commutative group, and the ultrametric is complete and invariant, unless explicitly stated otherwise. Given a mapping , for simplicity of notation, we write for each .
The following two theorems concern stability of functional equation (1).
Theorem 2. Let and be endomorphisms of with , , and and satisfy the inequalities Then there exists a unique function such that Moreover, is the unique solution of (10) for which there exists a constant with
Proof. Taking in (8) first and next we obtain
Write and for , . Then (13) implies the inequality
where . Define an operator by
Then is non-decreasing, satisfies hypothesis and
Note that by (9)
and analogously for . Consequently
Further, it is easy to show by induction that
Since , this means that for . Moreover, in view of (18) we see that for and , whence
Consequently, by Theorem 1, there is a solution of the equation
such that for . Moreover,
Now, we show that, for every , Clearly, for , (23) reduces to (8). Next, fix and assume that (23) holds for . Then, by (9), Thus we obtain (23) for , which completes the induction. Letting in (23), we obtain that
Write for . Then (11) holds and
It remains to prove the uniqueness of . So, let and a solution to (10) with for . Then and, by (11),
Further, by (9), , whence . This and (28) yield
We show that, for each , The case is just (28). So fix and assume that (30) holds for . Then, from (27), (29), and (18) we obtain Thus we have shown (30). Now, letting in (30) we get .
Remark 3. Let be a normed space (either classical or non-Archimedean) over a field . Clearly, if is classical, then we assume that ; if is non-Archimedean, then is a nontrivial non-Archimedean field. Given , write for . It is easy to see that if , , and has one of the following forms (i), (ii), (iii),
with some and , then the assumptions of the above theorem are fulfilled. Multiplying and/or adding those functions we can obtain numerous further examples.
Note that in the situation when is of form (iii), for , whence in the statement of Theorem 2. This means that (under the assumptions of Theorem 2) with given by (iii), every satisfying (8) must be actually a solution to (1).
In particular, from Theorem 2, we get immediately the following result on stability of (1).
Corollary 4. Let be a normed space over a field , , and, be of one let the forms (i)–(iii). Assume that satisfies the inequality Then the following two statements are valid. If (i) or (ii) holds, there exists a unique solution of (1) such that for . If (iii) holds, then is a solution to (1).
Remark 5. Let be a non-archimedean normed space over a field that is non-Archimedean and . Then results analogous to Corollary 4 cannot be derived from Theorem 2, because we have , which means that, with , we get .
However, if , then , which means that . Consequently, we obtain results analogous to Corollary 4 for and the function is one of the following forms (a) for all and , (b) for all and , (c) for all and for all ,
with some and . Unfortunately, we must define at the point in a bit artificial way in (a) and (b), similar (but even bigger) problem we have in (c). The subsequent modified version of Theorem 2 amends this situation to some extent for (a) and (b).
Theorem 6. Let be monomorphisms with , , and , satisfy for . Then there is a solution of (10) with
Proof. Arguing in the same way as in the proof of Theorem 2, with being replaced by , from Theorem 1, we deduce that there is a solution of the equation
such that for . Moreover, for . Define by for and .
In the same way as in the proof of Theorem 2, we obtain that for all . We need yet to prove that, for every , We show only (38); the proofs for (39) and (40) are analogous.
Clearly, for , (38) follows from (34). Next, if (37) holds for a fixed , then, for every with and , This completes the induction.
Letting in (37)–(40), we obtain that Writing for we obtain that (35) holds and is a solution to (10).
Below we present a theorem that is somewhat complementary to Theorem 2.
Theorem 7. Let and be endomorphisms of with , and be bijective, , and and satisfy the inequalities Assume that satisfies (8). Then there exists a unique solution of (10) such that Moreover, is the unique solution of (10) for which there exists a constant with
Proof. The proof is very similar to that of Theorem 2, but for the convenience of readers we present it here.
Taking in (8) first and next we obtain whence replacing by we derive the following inequality: Since is invariant, this yields with and . Writing and for , , we finally get
Define an operator by for , . Clearly, is non-decreasing and satisfies hypothesis , and, according to (43),
Note that and analogously for . Consequently
It is easy to show by induction that for , . Since , this means that for . Further, by (52), for every and we have , whence Consequently, by Theorem 1, there is a function such that , , , and for . It is easily seen that
Now, we show that, for every , Clearly, for , (55) reduces to (8). Next, fix and assume that (55) holds for every with . Then, by (43), Thus we obtain (55) for , which completes the induction. Letting in (55), we get for . Next, writing for , we obtain that is a solution to (1).
To complete the proof let us yet mention that we show the uniqueness of in the same way as in the proof of Theorem 2.
Corollary 8. Let be a normed space over a field , , and let be one of the forms (i)–(iii). If satisfies inequality (33), then the following two statements are valid. If (i) or (ii) holds, there exists a unique solution of (1) such that for . If (iii) holds, then is a solution to (1).
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