A fixed point approach to the hyers-ulam stability of a functional equation in various normed spaces
- Hassan Azadi Kenary^{1},
- Sun Young Jang^{2} and
- Choonkil Park^{3}Email author
https://doi.org/10.1186/1687-1812-2011-67
© Kenary et al; licensee Springer. 2011
Received: 2 June 2011
Accepted: 25 October 2011
Published: 25 October 2011
Abstract
Using direct method, Kenary (Acta Universitatis Apulensis, to appear) proved the Hyers-Ulam stability of the following functional equation
in non-Archimedean normed spaces and in random normed spaces, where m, n are different integers greater than 1. In this article, using fixed point method, we prove the Hyers-Ulam stability of the above functional equation in various normed spaces.
2010 Mathematics Subject Classification: 39B52; 47H10; 47S40; 46S40; 30G06; 26E30; 46S10; 37H10; 47H40.
Keywords
1. Introduction
A classical question in the theory of functional equations is the following: "When is it true that a function which approximately satisfies a functional equation must be close to an exact solution of the equation?" If the problem accepts a solution, then we say that the equation is stable. The first stability problem concerning group homomorphisms was raised by Ulam [1] in 1940. In the following year, Hyers [2] gave a positive answer to the above question for additive groups under the assumption that the groups are Banach spaces. In 1978, Rassias [3] proved a generalization of Hyers' theorem for additive mappings. Furthermore, in 1994, a generalization of the Rassias' theorem was obtained by Găvruta [4] by replacing the bound ε (||x|| ^{ p } + ||y|| ^{ p } ) by a general control function ϕ(x, y).
The functional equation f(x + y) + f(x - y) = 2f(x) + 2f(y) is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. In 1983, the Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [5] for mappings f : X → Y, where X is a normed space and Y is a Banach space. In 1984, Cholewa [6] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group and, in 2002, Czerwik [7] proved the Hyers-Ulam stability of the quadratic functional equation.
The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [8–12]).
in various spaces, which was introduced and investigated in [13].
2. Preliminaries
In this section, we give some definitions and lemmas for the main results in this article.
- (a)
|r| = 0 if and only if r = 0;
- (b)
|rs| = |r||s|;
- (c)
|r + s| ≤ |r| + |s|.
A field $K$ is called a valued field if $K$ carries a valuation. The usual absolute values of ℝ and ℂ are examples of valuations.
In 1897, Hensel [14] has introduced a normed space which does not have the Archimedean property.
for all $r,s\in K$ then the function | · | is called a non-Archimedean valuation and the field is called a non-Archimedean field. Clearly, |1| = | -1| = 1 and |n| ≤ 1 for all n ∈ ℕ.
A trivial example of a non-Archimedean valuation is the function | · | taking everything except for 0 into 1 and |0| = 0.
Definition 2.1. Let X be a vector space over a field $K$ with a non-Archimedean valuation | · |. A function || · || : X → [0, ∞) is called a non-Archimedean norm if the following conditions hold:
(a) ||x|| = 0 if and only if x = 0 for all x ∈ X;
(b) ||rx|| = |r| ||x|| for all $r\in K$ and x ∈ X;
for all x, y ∈ X. Then (X, || · ||) is called a non-Archimedean normed space.
Definition 2.2. Let {x_{ n } } be a sequence in a non-Archimedean normed space X.
(a) The sequence {x_{ n } } is called a Cauchy sequence if, for any ε > 0, there is a positive integer N such that ||x_{ n } - x_{ m } || ≤ ε for all n, m ≥ N.
(b) The sequence {x_{ n } } is said to be convergent if, for any ε > 0, there are a positive integer N and x ∈ X such that ||x_{ n } - x|| ≤ ε for all n ≥ N. Then the point x ∈ X is called the limit of the sequence {x_{ n } }, which is denote by lim_{n→∞}x_{ n }= x.
(c) If every Cauchy sequence in X converges, then the non-Archimedean normed space X is called a non-Archimedean Banach space.
for all m, n ≥ 1 with n > m.
In the sequel (in random stability section), we adopt the usual terminology, notions, and conventions of the theory of random normed spaces as in [15].
We can easily show that the maximal element in Γ^{+} is the distribution function H_{0}(t).
Definition 2.3. [15] A function T : [0, 1]^{2} → [0, 1] is a continuous triangular norm (briefly, a t-norm) if T satisfies the following conditions:
(a) T is commutative and associative;
(b) T is continuous;
(c) T(x, 1) = x for all x ∈ [0, 1];
(d) T(x, y) ≤ T(z, w) whenever x ≤ z and y ≤ w for all x, y, z, w ∈ [0, 1].
Three typical examples of continuous t-norms are as follows: T(x, y) = xy, T(x, y) = max{a + b - 1, 0}, and T(x, y) = min(a, b).
Definition 2.4. [16] A random normed space (briefly, RN-space) is a triple (X, μ, T), where X is a vector space, T is a continuous t-norm, and μ : X → D^{+} is a mapping such that the following conditions hold:
(a) μ_{ x } (t) = H_{0}(t) for all x ∈ X and t > 0 if and only if x = 0;
(b) ${\mu}_{\alpha x}\left(t\right)={\mu}_{x}\left(\frac{t}{\left|\alpha \right|}\right)$ for all α ∈ ℝ with α ≠ 0, x ∈ X and t ≥ 0;
(c) μ_{x+y}(t + s) ≥ T (μ_{ x } (t), μ_{ y } (s)) for all x, y ∈ X and t, s ≥ 0.
- (1)
A sequence {x_{ n } } in X is said to be convergent to a point x ∈ X (write x_{ n } → x as n → ∞) if ${lim}_{n\to \infty}\phantom{\rule{2.77695pt}{0ex}}{\mu}_{{x}_{n}-x}\left(t\right)=1$for all t > 0.
- (2)
A sequence {x_{ n } } in X is called a Cauchy sequence in X if ${lim}_{n\to \infty}\phantom{\rule{0.3em}{0ex}}{\mu}_{{x}_{n}-{x}_{m}}\left(t\right)=1$ for all t > 0.
- (3)
The RN-space (X, μ, T) is said to be complete if every Cauchy sequence in X is convergent.
Theorem 2.1. [15]If (X, μ, T) is an RN-space and {x_{ n } } is a sequence such that x_{ n } → x, then${lim}_{n\to \infty}\phantom{\rule{0.3em}{0ex}}{\mu}_{{x}_{n}}\left(t\right)={\mu}_{x}\left(t\right)$.
Definition 2.6. [17]Let X be a real vector space. A function N : X × ℝ → [0, 1] is called a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ ℝ,
(N 1) N(x, t) = 0 for t ≤ 0;
(N 2) x = 0 if and only if N(x, t) = 1 for all t > 0;
(N 3) $N\left(cx,t\right)=N\left(x,\frac{t}{\left|c\right|}\right)$if c ≠ 0;
(N 4) N(x + y, s + t) ≥ min{N(x, s), N(y, t)};
(N 5) N(x,.) is a non-decreasing function of ℝ and lim_{t→∞}N (x, t) = 1;
(N 6) for x ≠ 0, N(x,.) is continuous on ℝ.
The pair (X, N) is called a fuzzy normed vector space. The properties of fuzzy normed vector space are given in [18].
is a fuzzy norm on X.
Definition 2.7. [17]Let (X, N) be a fuzzy normed vector space. A sequence {x_{ n } } in X is said to be convergent or converge if there exists an x ∈ X such that lim_{t→∞}N (x_{ n } - x, t) = 1 for all t > 0. In this case, x is called the limit of the sequence {x_{ n } } in X and we denote it by N - lim_{t→∞}x_{ n }= x.
Definition 2.8. [17]Let (X, N) be a fuzzy normed vector space. A sequence {x_{ n } } in X is called Cauchy if for each ε > 0 and each t > 0 there exists an n_{0} ∈ ℕ such that for all n ≥ n_{0}and all p > 0, we have N (x_{n+p}- x_{ n }, t) > 1 - ε.
It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.
Then (ℝ, N) is a fuzzy Banach space.
We say that a mapping f : X → Y between fuzzy normed vector spaces X and Y is continuous at a point x ∈ X if for each sequence {x_{ n } } converging to x_{0} ∈ X, then the sequence {f(x_{ n } )} converges to f (x_{0}). If f : X → Y is continuous at each x ∈ X, then f : X → Y is said to be continuous on X[19].
Throughout this article, assume that X is a vector space and that (Y, N) is a fuzzy Banach space.
Definition 2.9. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies the following conditions:
(a) d(x, y) = 0 if and only if x = y for all x, y ∈ X;
(b) d(x, y) = d(y, x) for all x, y ∈ X;
(c) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X.
for all non-negative integers n or there exists a positive integer n _{0} such that
(a) d(J^{ n }x, J^{n+1}x) < ∞ for all n_{0} ≥ n_{0};
(b) the sequence {J^{ n }x} converges to a fixed point y* of J;
(c) y* is the unique fixed point of J in the set$Y=\left\{y\in X:d\left({J}^{{n}_{0}}x,y\right)<\infty \right\}$;
(d) $d\left(y,{y}^{*}\right)\le \frac{1}{1-L}d\left(y,Jy\right)$for all y ∈ Y.
3. Non-Archimedean stability of the functional equation (1)
In this section, using the fixed point alternative approach, we prove the Hyers-Ulam stability of the functional equation (1) in non-Archimedean normed spaces.
Throughout this section, let X be a non-Archimedean normed space and Y a complete non-Archimedean normed space. Assume that |m| ≠1.
Lemma 3.1. Let X and Y be linear normed spaces and f : X → Y a mapping satisfying (1). Then f is an additive mapping.
for all x ∈ X and all n ∈ ℕ.□
where inf ∅ = +∞. It is easy to show that (S, d) is complete (see [[22], Lemma 2.1]).
- (1)A is a fixed point of J, that is,$A\left(\frac{x}{m}\right)=\frac{1}{m}A\left(x\right)$(5)
- (2)d(J^{ n }f, A) → 0 as n → ∞. This implies the equality$\underset{n\to \infty}{lim}\phantom{\rule{0.3em}{0ex}}{m}^{n}f\left(\frac{x}{{m}^{n}}\right)=A\left(x\right)$
- (3)$d\left(f,A\right)\le \frac{d\left(f,J\phantom{\rule{2.77695pt}{0ex}}f\right)}{1-L}$ with f ∈ Ω, which implies the inequality$d\left(f,A\right)\le \frac{L}{\left|m\right|-\left|m\right|L}.$
for all x, y ∈ X.
Therefore, the mapping A : X → Y is additive. This completes the proof. □
for all x ∈ X.
for all x, y ∈ X. In fact, if we choose L = |m|^{1-p}, then we get the desired result. □
Proof. The proof is similar to the proof of Theorem 3.1. □
for all x ∈ X.
for all x, y ∈ X. In fact, if we choose L = |2m|^{p-1}, then we get the desired result. □
Then one can easily show that f : Y → Y satisfies (3.5) for the case p = 1 and that there does not exist an additive mapping satisfying (3.6).
4. Random stability of the functional equation (1)
In this section, using the fixed point alternative approach, we prove the Hyers-Ulam stability of the functional equation (1) in random normed spaces.
for all x ∈ X and t > 0.
where inf ∅ = +∞. It is easy to show that (S*, d*) is complete (see [[22], Lemma 2.1]).
for all x ∈ X.
- (1)A is a fixed point of J, that is,$A\left(\frac{x}{m}\right)=\frac{1}{m}A\left(x\right)$(12)
- (2)d*(J^{ n }f, A) → 0 as n → ∞. This implies the equality$\underset{n\to \infty}{lim}\phantom{\rule{2.77695pt}{0ex}}{m}^{n}f\left(\frac{x}{{m}^{n}}\right)=A\left(x\right)$
- (3)${d}^{*}\left(f,A\right)\le \frac{{d}^{*}\left(f,Jf\right)}{1-m\alpha}$ with f ∈ Ω, which implies the inequality${d}^{*}\left(f,A\right)\le \frac{\alpha}{1-m\alpha}$
for all x ∈ X and t > 0. This implies that the inequality (10) holds.
for all x, y ∈ X and t > 0.
Thus the mapping A : X → Y is additive. This completes the proof. □
for all x ∈ X and t > 0.
for all x, y ∈ X and t > 0. In fact, if we choose α = m^{-p}, then we get the desired result. □
for all x ∈ X and t > 0.
Proof. The proof is similar to the proof of Theorem 4.1. □
for all x ∈ X and t > 0.
for all x, y ∈ X and t > 0. In fact, if we choose α = m^{ p } , then we get the desired result. □
Then one can easily show that f : Y → Y satisfies (4.6) for the case p = 1 and that there does not exist an additive mapping satisfying (4.7).
5. Fuzzy stability of the functional equation (1)
Throughout this section, using the fixed point alternative approach, we prove the Hyers-Ulam stability of the functional equation (1) in fuzzy normed spaces.
In the rest of the article, assume that X is a vector space and that (Y, N) is a fuzzy Banach space.
for all x, y ∈ X and all t > 0.
where inf ∅ = +∞. It is easy to show that (S**, d**) is complete (see [[22], Lemma 2.1]).
for all x ∈ X.
The rest of the proof is similar to the proof of Theorem 4.1. □
for all x, y ∈ X. Then we can choose L = m^{1-p}and we get the desired result. □
for all x, y ∈ X and all t > 0.
Proof. The proof is similar to that of the proofs of Theorems 4.1 and 5.1. □
for all x, y, z ∈ X. Then we can choose L = m^{p-1}and we get the desired result. □
Then one can easily show that f : Y → Y satisfies (5.2) for the case p = 1 and that there does not exist an additive mapping satisfying (5.3).
6. Conclusion
We linked here five different disciplines, namely, the random normed spaces, non-Archimedean normed spaces, fuzzy normed spaces, functional equations, and fixed point theory. We established the Hyers-Ulam stability of the functional equation (1) in various normed spaces by using fixed point method.
Declarations
Acknowledgements
The second author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0013211).
Authors’ Affiliations
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