Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the Cauchy additive functional equation in paranormed spaces.
Let S be a set. A function is called a generalized metric on S if m satisfies
-
(1)
if and only if ;
-
(2)
for all ;
-
(3)
for all .
We recall a fundamental result in fixed point theory.
Theorem 2.1 [37, 38]
Let be a complete generalized metric space, and let be a strictly contractive mapping with a Lipschitz constant . Then, for each given element , either
for all nonnegative integers
n
or there exists a positive integer
such that
-
(1)
, ;
-
(2)
the sequence converges to a fixed point of J;
-
(3)
is the unique fixed point of J in the set ;
-
(4)
for all .
In 1996, Isac and Rassias [39] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [40–44]).
Note that for all .
Theorem 2.2
Let
be a function such that there exists an
with
(2.1)
for all . Let be a mapping such that
(2.2)
for all . Then there exists a unique Cauchy additive mapping such that
(2.3)
for all .
Proof Letting in (2.2), we get
and so
(2.4)
for all .
Consider the set
and introduce the generalized metric on S:
where, as usual, . It is easy to show that is complete (see [[45], Lemma 2.1]).
Now we consider the linear mapping such that
for all .
Let be given such that . Since
for all , implies that . This means that
for all .
It follows from (2.4) that .
By Theorem 2.1, there exists a mapping satisfying the following:
-
(1)
A is a fixed point of J, i.e.,
for all . The mapping A is a unique fixed point of J in the set
This implies that A is a unique mapping satisfying (2.5) such that there exists a satisfying
for all ;
-
(2)
as . This implies the equality
for all ;
-
(3)
, which implies the inequality
This implies that the inequality (2.3) holds true.
It follows from (2.1) and (2.2) that
for all . So, for all . Thus is an additive mapping, as desired. □
Corollary 2.3 Let r be a positive real number with , and let be a mapping such that
for all . Then there exists a unique Cauchy additive mapping such that
(2.6)
for all .
Proof Taking for all and choosing in Theorem 2.2, we get the desired result. □
Theorem 2.4
Let
be a function such that
for all . Let be a mapping satisfying (2.2). Then there exists a unique Cauchy additive mapping such that
for all .
Proof The proof is similar to the proof of [[46], Theorem 2.2]. □
Remark 2.5 Let . Letting for all in Theorem 2.4, we obtain the inequality (2.6). The proof is given in [[46], Theorem 2.2].
Theorem 2.6
Let
be a function such that there exists an
with
(2.7)
for all . Let be a mapping such that
(2.8)
for all . Then there exists a unique additive mapping such that
(2.9)
for all .
Proof Letting in (2.8), we get
and so
(2.10)
for all .
Consider the set
and introduce the generalized metric on S:
where, as usual, . It is easy to show that is complete (see [[45], Lemma 2.1]).
Now we consider the linear mapping such that
for all .
Let be given such that . Since
for all , implies that . This means that
for all .
It follows from (2.10) that .
By Theorem 2.1, there exists a mapping satisfying the following:
-
(1)
A is a fixed point of J, i.e.,
(2.11)
for all . The mapping A is a unique fixed point of J in the set
This implies that A is a unique mapping satisfying (2.11) such that there exists a satisfying
for all ;
-
(2)
as . This implies the equality
for all ;
-
(3)
, which implies the inequality
This implies that the inequality (2.9) holds true.
It follows from (2.7) and (2.8) that
for all . So, for all . Thus is an additive mapping, as desired. □
Corollary 2.7 Let r, θ be positive real numbers with , and let be a mapping such that
for all . Then there exists a unique Cauchy additive mapping such that
(2.12)
for all .
Proof Taking for all and choosing in Theorem 2.6, we get the desired result. □
Theorem 2.8
Let
be a function such that
for all . Let be a mapping satisfying (2.8). Then there exists a unique Cauchy additive mapping such that
for all .
Proof The proof is similar to the proof of [[46], Theorem 2.1]. □
Remark 2.9 Let . Letting for all in Theorem 2.8, we obtain the inequality (2.12). The proof is given in [[46], Theorem 2.1].