Theory and Modern Applications

# Fuzzy stability of a cubic functional equation via fixed point technique

## 1 Introduction, definitions and notations

Fuzzy set theory is a powerful hand set for modeling uncertainty and vagueness in various problems arising in the field of science and engineering. It has a large number of application, for instance, in the computer programming , engineering problems , statistical convergence , nonlinear operator , best approximation  etc. Particularly, fuzzy differential equation is a strong topic with large application areas, for example, in population models , civil engineering  and so on.

By modifying own studies on fuzzy topological vector spaces, Katsaras  first introduced the notion of fuzzy seminorm and norm on a vector space and later on Felbin  gave the concept of a fuzzy normed space (for short, FNS) by applying the notion fuzzy distance of Kaleva and Seikala  on vector spaces. Further, Xiao and Zhu  improved a bit the Felbin's definition of fuzzy norm of a linear operator between FNSs.

Stability problem of a functional equation was first posed by Ulam  which was answered by Hyers  under the assumption that the groups are Banach spaces. Rassias  and Gajda  considered the stability problem with unbounded Cauchy differences. The unified form of the results of Hyers, Rassias, and Gajda is as follows:

Let E and F be real normed spaces with F complete and let f : EF be a mapping such that the following condition holds

${‖f\left(x+y\right)-f\left(x\right)-f\left(y\right)‖}_{F}\le \theta \left({‖x‖}_{E}^{p}+{‖y‖}_{E}^{p}\right),$

for all x, y E, θ ≥ 0 and for some p [0, ∞) | {1}. Then there exists a unique additive function C : E → F such that

$‖f\left(x\right)-C\left(x\right)‖F\le \frac{2\theta }{|2-{2}^{p}|}{‖x‖}_{E}^{p},$

for all x E.

This stability phenomenon is called generalized Hyers-Ulam stability and has been extensively investigated for different functional equations. It is to be noted that almost all proofs used the idea imaginated by Hyers. Namely, the additive function C : E → F is explicitly constructed, starting from the given function f, by the formulae (i) $C\left(x\right)=\underset{n\to \infty }{\text{lim}}\frac{1}{{2}^{n}}f\left({2}^{n}x\right)$, if p < 1; and (ii) $C\left(x\right)=\underset{n\to \infty }{\text{lim}}{2}^{n}f\left(\frac{x}{{2}^{n}}\right)$, if p > 1. This method is called a direct method. It is often used to construct a solution of a given functional equation and is seen to be a powerful tool for studying the stability of many functional equations. Since then several stability problems and its fuzzy version for various functional equations have been investigated in . Recently, Radu  proposed that fixed point alternative method is very useful for obtaining the solution of Ulam problem.

The stability problem for the cubic functional equation was proved by Jun and Kim  for mappings f : X → Y, where X is a real normed space and Y is a Banach space. In this article, we show that the existence of the limit C(x) and the estimation (i) and (ii) can be simply obtained from the alternative of fixed point.

In this section, we recall some notations and basic definitions used in this article.

A fuzzy subset N of X × is called a fuzzy norm on X if the following conditions are satisfied for all x, y X and c ;

1. (a)

N(x, t) = 0 for all non-positive t ,

2. (b)

N(x, t) = 1 for all t + if and only if x = 0,

3. (c)

$N\left(cx,t\right)=N\left(x,\frac{t}{\left|c\right|}\right)$ for all t + and c ≠ 0,

4. (d)

N(x + y, t + s) min{N(x, t), N(y, s)} for all s, t ,

5. (e)

N(x, t) is a non-decreasing function on , and supt N(x, t) = 1.

The pair (X, N) will be referred to as a fuzzy normed space.

Example 1.1. Let (X, ||.||) be a normed linear space. Then

$N\left(x,t\right)=\left\{\begin{array}{cc}\frac{t}{t+∥x∥}\hfill & \mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}t>0,\hfill \\ 0\hfill & \mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}t\le 0,\hfill \end{array}\right\$

is a fuzzy norm on X.

Example 1.2. Let (X, ||.||) be a normed linear space. Then

$N\left(x,t\right)=\left\{\begin{array}{c}1\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}t>∥x∥,\hfill \\ 0\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}t\le ∥x∥,\hfill \end{array}\right\$

is a fuzzy norm on X.

Let (X, N) be a fuzzy normed space. Then, a sequence x = (x k ) is said to be fuzzy convergent to L X if lim N(x k - L, t) = 1, for all t > 0. In this case, we write N-lim x k = L.

Let (X, N) be an fuzzy normed space. Then, x = (x k ) is said to be fuzzy Cauchy sequence if lim N(x k+p - x k , t) = 1 for all t > 0 and p = 1, 2, ....

It is known that every convergent sequence in a fuzzy normed space (X, N) is Cauchy. Fuzzy normed space (X, N) is said to be complete if every fuzzy Cauchy sequence is fuzzy convergent. In this case, (X, N) is called fuzzy Banach space.

## 2 Fixed point technique for Hyers-Ulam stability

In this section, we deal with the stability problem via fixed point method in fuzzy norm space. Before proceeding further, we should recall the following results related to the concept of fixed point.

Theorem 2.1 (Banach's Contraction principle). Let (X, d) be a complete generalized metric space and consider a mapping J : X → X be a strictly contractive mapping, that is

$d\left(Jx,Jy\right)\le Ld\left(x,y\right),\phantom{\rule{2.77695pt}{0ex}}\forall x,y\in X$

for some (Lipschitz constant) L < 1. Then

1. (i)

The mapping J has one and only one fixed point x* = J(x*);

2. (ii)

The fixed point x* is globally attractive, that is

$\underset{n\to \infty }{\text{lim}}{J}^{n}x={x}^{*},$

for any starting point x X;

1. (iii)

One has the following estimation inequalities for all x X and n ≥ 0:

$d\left({J}^{n}x,{x}^{*}\right)\le {L}^{{\phantom{\rule{0.1em}{0ex}}}^{n}}d\left(x,{x}^{*}\right)$
(2.1.1)
$d\left({J}^{n}x,{x}^{*}\right)\le \frac{1}{1-L}d\left({J}^{n}x,{J}^{n+1}x\right)$
(2.1.2)
$d\left(x,{x}^{*}\right)\le \frac{1}{1-L}d\left(x,Jx\right).$
(2.1.3)

Theorem 2.2 (The alternative of fixed point) . Suppose we are given a complete generalized metric space (X, d) and a strictly contractive mapping J : X → X, with Lipschitz constant L. Then, for each given element x X, either

$d\left({J}^{n}x,{J}^{n+1}x\right)=+\infty ,\phantom{\rule{2.77695pt}{0ex}}\forall n\ge 0$
(2.2.1)

or

$d\left({J}^{n}x,\phantom{\rule{2.77695pt}{0ex}}{J}^{n+1}x\right)<+\infty \phantom{\rule{2.77695pt}{0ex}}\forall n\ge {n}_{\circ }$
(2.2.2)

for some natural number n0. Moreover, if the second alternative holds then

1. (i)

The sequence (Jnx) is convergent to a fixed point y* of J;

2. (ii)

y* is the unique fixed point of J in the set $Y=\left\{y\in X,\phantom{\rule{2.77695pt}{0ex}}d\left({J}^{{n}_{\circ }}x,\phantom{\rule{2.77695pt}{0ex}}y\right)<+\infty \right\}$

3. (iii)

$d\left(y,y*\right)\le \frac{1}{1-L}d\left(y,Jy\right),y\in Y$.

We are now ready to obtain our main results.

The functional equation

$f\left(2x+y\right)+f\left(2x-y\right)=2f\left(x+y\right)+2f\left(x-y\right)+12f\left(x\right)$
(2.0.1)

is called the cubic functional equation, since the function f(x) = cx3 is its solution. Every solution of the cubic functional equation is said to be a cubic mapping.

Let φ be a function from X × X to Z. A mapping f : X → Y is said to be φ-approximately cubic function if

$N\left(f\left(2x+y\right)+f\left(2x-y\right)-2f\left(x+y\right)-2f\left(x-y\right)-12f\left(x\right),t\right)\ge {N}^{\prime }\left(\phi \left(x,y\right),t\right)$
(2.0.2)

for all x, y X and t > 0.

Using the fixed point alternative, we can prove the stability of Hyers-Ulam-Rassias type theorem in FNS. First, we prove the following lemma which will be used in our main result.

Lemma 2.1. Let (Z, N') be a fuzzy normed space and φ : X → Z be a function. Let E = {g : X → Y; g(0) = 0} and define

${d}_{M}\left(g,h\right)=\text{inf}\left\{a\in {ℝ}^{+}:N\left(g\left(x\right)-h\left(x\right),at\right)\ge {N}^{\prime }\left(\phi \left(x,0\right),t\right)\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{for}}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{all}}\phantom{\rule{0.3em}{0ex}}x\in X\phantom{\rule{0.3em}{0ex}}\mathsf{\text{and}}\phantom{\rule{0.3em}{0ex}}t>0\right\},$

for all h E. Then d M is a complete generalized metric on E.

Proof. Let g, h, k E, d M (g, h) < ξ1 and d M (h, k) < ξ2. Then

$N\left(g\left(x\right)-h\left(x\right),{\xi }_{1}t\right)\ge {N}^{\prime }\left(\phi \left(x\right),t\right)\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}N\left(h\left(x\right)-k\left(x\right),{\xi }_{2}t\right)\ge {N}^{\prime }\left(\phi \left(x\right),t\right),$

for all x X and t > 0. Thus

$N\left(g\left(x\right)-k\left(x\right),\left({\xi }_{1}+{\xi }_{2}\right)t\right)\ge \text{min}\left\{N\left(g\left(x\right)-h\left(x\right),{\xi }_{1}t\right),N\left(h\left(x\right)-k\left(x\right),{\xi }_{2}t\right)\right\}\ge {N}^{\prime }\left(\phi \left(x\right),t\right),$

for each x X and t > 0. By definition d M (h, k) < ξ1 + ξ2. This proves the triangle inequality for d M . Rest of the proof can be done on the same lines as in (see [, Lemma 2.1]).

Theorem 2.3. Let X be a linear space and (Z, N') be a FNS. Suppose that a function φ : X × XZ satisfying φ(2x, 2y) = αφ(x, y) for all x, y X and α ≠ 0. Suppose that (Y, N) be a fuzzy Banach space and f : XY be a φ-approximately cubic function. If for some 0 < α < 8

${N}^{\prime }\left(\phi \left(2x,0\right),t\right)\ge {N}^{\prime }\left(\alpha \phi \left(x,0\right),t\right),$
(2.3.1)

and

$\underset{n\to \infty }{\text{lim}}{N}^{\prime }\left(\phi \left({2}^{n}x,{2}^{n}y\right),{8}^{n}t\right)=1,$

for all x, y X and t > 0. Then there exists a unique cubic mapping C : X → Y such that

$N\left(C\left(x\right)-f\left(x\right),t\right)\ge {N}^{\prime }\left(\phi \left(x,0\right),2\left(8-\alpha \right)t\right),$

for all x X and all t > 0.

Proof. Put y = 0 in (2.0.2). Then for all x X and t > 0

$N\left(\frac{f\left(2x\right)}{8}-f\left(x\right),\frac{t}{16}\right)\ge {N}^{\prime }\left(\phi \left(x,0\right),t\right).$

Consider the set E = {g : X → Y, g(0) = 0} together with the mapping d M defined on E × E by

${d}_{M}\left(g,h\right)=\text{inf}\left\{a\in {ℝ}^{+}:N\left(g\left(x\right)-h\left(x\right),at\right)\ge {N}^{\prime }\left(\phi \left(x,0\right),t\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{for}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{all}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}x\in X\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{and}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}t>0\right\}.$

It is known that d M (g, h) complete generalized metric space by Lemma 2.1. Now, we define the linear mapping J : EE such that

$Jg\left(x\right)=\frac{1}{8}g\left(2x\right).$

It is easy to see that J is a strictly contractive self-mapping of E with the Lipschitz constant $\frac{\alpha }{8}$. Indeed, let g, h E be given such that d M (g, h) = ε. Then

$N\left(g\left(x\right)-h\left(x\right),\epsilon t\right)\ge {N}^{\prime }\left(\phi \left(x,0\right),t\right),$

for all x X and t > 0. Thus

$\begin{array}{ll}\hfill N\left(Jg\left(x\right)=Jh\left(x\right),\frac{\alpha }{8}\epsilon t\right)& =N\left(\frac{1}{8}g\left(2x\right)-\frac{1}{8}h\left(2x\right),\frac{\alpha }{8}\epsilon t\right)\phantom{\rule{2em}{0ex}}\\ =N\left(g\left(2x\right)-h\left(2x\right),\alpha \epsilon t\right)\ge {N}^{\prime }\left(\phi \left(2x,0\right),\alpha t\right).\phantom{\rule{2em}{0ex}}\end{array}$

It follows from (2.3.1) that

$N\left(Jg\left(x\right)-Jh\left(x\right),\frac{\alpha }{8}\epsilon t\right)\ge {N}^{\prime }\left(\alpha \phi \left(x,0\right),\alpha t\right)={N}^{\prime }\left(\phi \left(x,0\right),t\right),$

for all x X and t > 0. Therefore

${d}_{M}\left(g,h\right)=\epsilon ⇒{d}_{M}\left(Jg,Jh\right)\le \frac{\alpha }{8}\epsilon .$

This means that

${d}_{M}\left(Jg,Jh\right)\le \frac{\alpha }{8}{d}_{M}\left(g,h\right),$

for all g, h E. Next, from

$N\left(\frac{f\left(2x\right)}{8}-f\left(x\right),\frac{t}{16}\right)\ge {N}^{\prime }\left(\phi \left(x,0\right),t\right),$

we have ${d}_{M}\left(f,Jf\right)\le \frac{1}{16}$

Using the fixed point alternative we deduce the existence of a fixed point of J, that is, the existence of a mapping C : X → Y such that C(2x) = 8C(x), for all x X. Moreover, we have d M (Jnf, C) 0, which implies

$N-\underset{n}{\text{lim}}\frac{f\left({2}^{n}x\right)}{{8}^{n}}=C\left(x\right),$

for every x X. Also

${d}_{M}\left(f,C\right)\le \frac{1}{1-L}{d}_{M}\left(f,Jf\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{implies}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{d}_{M}\left(f,C\right)\le \frac{1}{16\left(1-\frac{\alpha }{8}\right)}=\frac{1}{2\left(8-\alpha \right)}.$

This implies that

$N\left(C\left(x\right)-f\left(x\right),\frac{1}{2\left(8-\alpha \right)}t\right)\ge {N}^{\prime }\left(\phi \left(x,0\right),t\right).$

Replacing t by 2(8 - α)t in the above equation, we obtain

$N\left(C\left(x\right)-f\left(x\right),t\right)\ge {N}^{\prime }\left(\phi \left(x,0\right),2\left(8-\alpha \right)t\right),$

for all x X and t > 0.

Let x, y X. Then

$N\left(C\left(2x+y\right)+C\left(2x-y\right)-2C\left(x+y\right)-2C\left(x-y\right)-12C\left(x\right),t\right)\ge {N}^{\prime }\left(\phi \left(x,y\right),t\right).$

Replacing x and y by 2nx and 2ny, respectively, we obtain

$\begin{array}{c}N\left(\frac{C\left({2}^{n}\left(2x+y\right)\right)}{{8}^{n}}+\frac{C\left({2}^{n}\left(2x-y\right)\right)}{{8}^{n}}-\frac{2C\left({2}^{n}\left(x+y\right)\right)}{{8}^{n}}-\frac{2C\left({2}^{n}\left(x-y\right)\right)}{{8}^{n}}-\frac{12C\left({2}^{n}x\right)}{{8}^{n}},t\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\ge {N}^{\prime }\left(\phi \left({2}^{n}x,{2}^{n}y\right),{8}^{n}t\right)\end{array}$

for all x, y X and all t > 0. Since

$\underset{n\to \infty }{\text{lim}}{N}^{\prime }\left(\phi \left({2}^{n}x,{2}^{n}y\right),{8}^{n}t\right)=1,$

we conclude that C fulfills (2.0.1).

The uniqueness of C follows from the fact that C is the unique fixed point of J with the following property that there exists u (0, ∞) such that

$N\left(C\left(x\right)-f\left(x\right),ut\right)\ge {N}^{\prime }\left(\phi \left(x,0\right),t\right),$

for all x X and t > 0.

This completes the proof of the theorem.

By a modification in the proof of Theorem 2.3, one can prove the following:

Theorem 2.4. Let X be a linear space and (Z, N') be a FNS. Suppose that a function φ : X × XZ satisfying

$\phi \left(\frac{x}{2},\frac{y}{2}\right)=\frac{1}{\alpha }\phi \left(x,y\right)$

for all x, y X and α ≠ 0. Suppose that (Y, N) be a fuzzy Banach space and f : XY be a φ-approximately cubic function. If for some α > 8

${N}^{\prime }\left(\phi \left(x/2,0\right),t\right)\ge {N}^{\prime }\left(\phi \left(x,0\right),\alpha t\right),$

and

$\underset{n\to \infty }{\text{lim}}{N}^{\prime }\left({8}^{-n}\phi \left({2}^{-n}x,{2}^{-n}y\right),t\right)=1$

for all x, y X and t > 0. Then there exists a unique cubic mapping C : X → Y such that

$N\left(C\left(x\right)-f\left(x\right),t\right)\ge {N}^{\prime }\left(\phi \left(x,0\right),2\left(\alpha -8\right)t\right),$

for all x X and all t > 0.

The proof of the above theorem is similar to the proof of Theorem 2.3, hence omitted.

## 3 Conclusion

This study indeed presents a relationship between three various disciplines: the theory of fuzzy normed spaces, the theory of stability of functional equations and the fixed point theory. This method is easier than those of previously proved (stability problem) by other authors for fuzzy setting. We established Hyers-Ulam-Rassias stability of a cubic functional equation in fuzzy normed spaces by using fixed point alternative theorem.

## Author' information

Address of both the authors: Department of Mathematics, Faculty of Science, King Abdu-laziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. Email: mohiuddine@gmail.com (S.A. Mohiuddine); mathker11@hotmail.com (A. Alotaibi).

## References

1. Giles R: A computer program for fuzzy reasoning. Fuzzy Sets Syst 1980, 4: 221–234.

2. Hanss M: Applied fuzzy arithmetic: an introduction with engineering applications. Springer-Verlag, Berlin; 2005.

3. Mohiuddine SA, Danish Lohani QM: On generalized statistical convergence in intuitionistic fuzzy normed space. Chaos Solitons Fract 2009, 42: 1731–1737.

4. Mursaleen M, Mohiuddine SA: Statistical convergence of double sequences in intuitionistic fuzzy normed spaces. Chaos Solitons Fract 2009, 41: 2414–2421.

5. Mursaleen M, Mohiuddine SA: On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. J Comput Appl Math 2009, 233(2):142–149.

6. Mursaleen M, Mohiuddine SA, Edely OHH: On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces. Comput Math Appl 2010, 59: 603–611.

7. Mohiuddine SA, Şevli H, Cancan M: Statistical convergence in fuzzy 2-normed space. J Computational Analy Appl 2010, 12(4):787–798.

8. Mursaleen M, Mohiuddine SA: Nonlinear operators between intuitionistic fuzzy normed spaces and Fr'echet differentiation. Chaos Solitons Fract 2009, 42: 1010–1015.

9. Mohiuddine SA: Some new results on approximation in fuzzy 2-normed spaces. Math Comput Model 2011, 53: 574–580.

10. Guo M, Li R: Impulsive functional differential inclusions and fuzzy population models. Fuzzy Sets Syst 2003, 138: 601–615.

11. Oberguggenberger M, Pittschmann S: Differential equations with fuzzy parameters. Math Mod Syst 1999, 5: 181–202.

12. Katsaras AK: Fuzzy topological vector spaces. Fuzzy Sets Syst 1984, 12: 143–154.

13. Felbin C: Finite dimensional fuzzy normed linear spaces. Fuzzy Sets Syst 1992, 48: 239–248.

14. Kaleva O, Seikala S: On fuzzy metric spaces. Fuzzy Sets Syst 1984, 12: 215–229.

15. Xiao JZ, Zhu XH: Fuzzy normed spaces of operators and its completeness. Fuzzy Sets Syst 2003, 133(3):135–146.

16. Ulam SM: A Collection of the Mathematical Problems. In Interscience Publ. New York; 1960.

17. Hyers DH: On the stability of the linear functional equation. Proc Natl Acad Sci USA 1941, 27: 222–224.

18. Rassias TM: On the stability of the linear mapping in Banach spaces. Proc Am Math Soc 1978, 72: 297–300.

19. Gajda Z: On stability of additive mappings. Int J Math Math Sci 1991, 14: 431–434.

20. Alotaibi A, Mohiuddine SA: On the stability of a cubic functional equation in random 2-normed spaces. Adv Diff Equ 2012, 2012: 39.

21. Jun KW, Kim HM: The generalized Hyers-Ulam-Rassias stability of a cubic functional equation. J Math Anal Appl 2002, 274: 867–878.

22. Jung SM, Kim TS: A fixed point approach to the stability of the cubic functional equation. Bol Soc Mat Mexicana 2006, 12(1):51–57.

23. Mohiuddine SA: Stability of Jensen functional equation in intuitionistic fuzzy normed space. Chaos Solitons Fract 2009, 42: 2989–2996.

24. Mohiuddine SA, Şevli H: Stability of Pexiderized quadratic functional equation in intuitionistic fuzzy normed space. J Comput Appl Math 2011, 235: 2137–2146.

25. Mursaleen M, Mohiuddine SA: On stability of a cubic functional equation in intuition-istic fuzzy normed spaces. Chaos Solitons Fract 2009, 42: 2997–3005.

26. Mirmostafaee AK: A fixed point approach to almost quartic mappings in quasi fuzzy normed spaces. Fuzzy Sets Syst 2009, 160: 1653–1662.

27. Radu V: The fixed point alternative and the stability of functional equations. Sem Fixed Point Theory 2003, 4(1):91–96.

28. Diaz JB, Margolis B: A fixed point theorem of the alternative for contractions on generalized complete metric space. Bull Am Math Soc 1968, 126(74):305–309.

29. Miheţ D, Radu V: On the stability of the additive Cauchy functional equation in random normed spaces. J Math Anal Appl 2008, 343: 567–572.

## Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments.

## Author information

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Correspondence to Syed Abdul Mohiuddine.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

Both the authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

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Mohiuddine, S.A., Alotaibi, A. Fuzzy stability of a cubic functional equation via fixed point technique. Adv Differ Equ 2012, 48 (2012). https://doi.org/10.1186/1687-1847-2012-48 