- Research Article
- Open Access
Generalized Ulam-Hyers Stability of Jensen Functional Equation in Šerstnev PN Spaces
© M. Eshaghi Gordji et al. 2010
- Received: 17 November 2009
- Accepted: 1 March 2010
- Published: 11 April 2010
We establish a generalized Ulam-Hyers stability theorem in a Šerstnev probabilistic normed space (briefly, Šerstnev PN-space) endowed with . In particular, we introduce the notion of approximate Jensen mapping in PN-spaces and prove that if an approximate Jensen mapping in a Šerstnev PN-space is continuous at a point then we can approximate it by an everywhere continuous Jensen mapping. As a version of a theorem of Schwaiger, we also show that if every approximate Jensen type mapping from the natural numbers into a Šerstnev PN-space can be approximated by an additive mapping, then the norm of Šerstnev PN-space is complete.
- Additive Mapping
- Unique Additive Mapping
- Probabilistic Normed Space
- Fuzzy Normed Space
- Random Normed Space
Mengerproposed transferring the probabilistic notions of quantum mechanic from physics to the underlying geometry. The theory of probabilistic normed spaces (briefly, PN-spaces) is important as a generalization of deterministic result of linear normed spaces and also in the study of random operator equations. The notion of a probabilistic normed space was introduced by Šerstnev . Alsina, Schweizer, and Skalar gave a general definition of probabilistic normed space based on the definition of Meneger for probabilistic metric spaces in [2, 3].
Theorem 1.1 (see ).
for all .
Theorem 1.2 (see ).
for all . Moreover, if is continuous in for each fixed , then the function is linear.
Theorem 1.2was later extended for all . The stability phenomenon that was presented by Rassias is called the generalized Ulam-Hyers stability. In 1982, Rassias  gave a further generalization of the result of Hyers and proved the following theorem using weaker conditions controlled by a product of powers of norms.
Theorem 1.3 (25).
for all .
The above mentioned stability involving a product of powers of norms is called Ulam-Gavruta-Rassias stability by various authors (see [12–21]). In the last two decades, several forms of mixed type functional equations and their Ulam-Hyers stability are dealt with in various spaces like fuzzy normed spaces, random normed spaces, quasi-Banach spaces, quasi-normed linear spaces, and Banach algebras by various authors like in [6, 9, 14, 22–38].
Let be a mapping between linear spaces. The Jensen functional equation is
It is easy to see that f with satisfies the Jensen equation if and only if it is additive; compare for [39, Theorem ]. Stability of Jensen equation has been studied at first by Kominek  and then by several other mathematicians example, (see [10, 33, 40–42] and references therein).
PN spaces were first defined by Šerstnev in1963(see ). Their definition was generalized in . We recall and apply the definition of probabilistic space briefly as given in , together with the notation that will be needed (see ). A distance distribution function (briefly, a d.d.f.) is a nondecreasing function from into that satisfies and , and is left-continuous on ; here as usual, . The space of d.d.f.'s will be denoted by and the set of all in for which by . The space is partially ordered by the usual pointwise ordering of functions, that is, if and only if for all in . For any , is the d.d.f. given by
The space can be metrized in several ways , but we shall here adopt the Sibley metric . If are d.f.'s and is in , let denote the condition
Then the Sibley metric is defined by
In particular, under the usual pointwise ordering of functions, is the maximal element of . A triangle function is a binary operation on , namely, a function that is associative, commutative, nondecreasing in each place, and has as identity, that is, for all and in :
Moreover, a triangle function is continuous if it is continuous in the metric space .
Typical continuous triangle functions are and . Here is a continuous t-norm, that is, a continuous binary operation on that is commutative, associative, nondecreasing in each variable, and has 1 as identity; is a continuous -conorm, namely, a continuous binary operation on which is related to the continuous -norm through For example, and or and .
Note that for and .
A Probabilistic Normed space (briefly, PN space) is a quadruple , where is a real vector space, and are continuous triangle functions with and is a mapping (the probabilistic norm) from into such that for every choice of and in the following hold:
(N1) if and only if ( is the null vector in ),
(N4) for every .
A PN space is called a Šerstnev space if it satisfies (N1), (N3) and the following condition:
holds for every and . When here is a continuous -norm such that and , the PN space is called Meneger PN space (briefly, MPN space), and is denoted by
Let be an MPN space and let be a sequence in . Then is said to be convergent if there exists such that
for all . In this case is called the limit of .
The sequence in MPN Space is called Cauchy if for each and there exist some such that for all .
Clearly, every convergent sequence in an MPN space is Cauchy. If each Cauchy sequence is convergent in an MPN space , then is called Meneger Probabilistic Banach space (briefly, MPB space).
In this section, we provide a generalized Ulam-Hyers stability theorem in a Šerstnev MPN space.
for each and . Thus satisfies the Jensen equation and so it is additive.
Next, we approximate the difference between and in the Šerstnev MPN space . For every and , by (2.15), for large enough , we have
Hence the right-hand side of the above inequality tends to as . It follows that for all .
One can prove a similar result for the case that . In this case, the additive mapping is defined by .
Moreover, if is a Šerstnev MPN space and is continuous at a point, then is continuous on .
which contradicts (2.29).
This section contains two results concerning the completeness of a Šerstnev MPN space. Those are versions of a theorem of Schwaiger  stating that a normed space is complete if, for each whose Cauchy difference is bounded for all , there exists an additive mapping such that is bounded for all .
for some and all .
In order to prove our next results, we need to put the following conditions on an MPN space.
Clearly a definite MPN space is pseudodefinite.
for all . Then is a Šerstnev MPB-space.
for each . Then is said to be an approximately Jensen-type mapping.
for each . Then is a Šerstnev MPB-space.
Hence the subsequence of the Cauchy sequence converges to . Hence also converges to .
In this work, we have analyzed a generalized Ulam-Hyers theorem in Šerstnev PN spaces endowed with . We have proved that if an approximate Jensen mapping in a Šerstnev PN space is continuous at a point then we can approximate it by an anywhere continuous Jensen mapping. Also, as a version of Schwaiger, we have showed that if every approximate Jensen-type mapping from natural numbers into a Šerstnev PN-space can be approximate by an additive mapping then the norm of Šerstnev PN-space is complete.
The authors are grateful to the referees for their helpful comments. The fourth author was supported by National Research Foundation of Korea (NRF-2009-0070788).
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