- Research
- Open access
- Published:
A Pexider system of additive functional equations in Banach algebras
Journal of Inequalities and Applications volume 2024, Article number: 27 (2024)
Abstract
In this paper, we solve the system of functional equations
and we investigate the stability of g-derivations in Banach algebras.
1 Introduction
Let \(\mathcal{B}\) be a complex Banach algebra and let \(g :\mathcal{B}\rightarrow \mathcal{B}\) be a C-linear mapping. Mirzavaziri and Moslehian [1] introduced the concept of g-derivation \(f: \mathcal{B}\to \mathcal{B}\) as follows:
for all \(x, y \in \mathcal{B}\). Park et al. [2] introduced the concept of hom-derivation on \(\mathcal{B}\), i.e., \(g :\mathcal{B}\rightarrow \mathcal{B}\) is a homomorphism and f satisfies (1.1) for all \(x, y \in \mathcal{B}\).
The stability problem of functional equations originated from a question of Ulam [3] concerning the stability of group homomorphisms. Hyers [4] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki [5] for additive mappings and by Th.M. Rassias [6] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by Găvruta [7] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. Rassias’ approach. Recently, Lee et al. [8, 9] extended more general functional equations, which were mixed types of additive, quadratic and cubic functional equations in Banach spaces, and Park and Rassias [10] applied the functional equation theory to study partial multipliers in \(C^{*}\)-algebras. Many mathematicians developed the Hyers results in various directions [11–18].
The method provided by Hyers [4] which produces the additive function will be called a direct method. This method is the most significant and strongest tool to study the stability of different functional equations. That is, the exact solution of the functional equation is explicitly constructed as a limit of a sequence, starting from the given approximate solution [19, 20]. The other significant method is the fixed point theorem, that is, the exact solution of the functional equation is explicitly created as a fixed point of some certain map (see [21, 22]).
We consider a fixed point alternative theorem.
Theorem 1.1
[23] Assume that \((\mathcal{B} ,d)\) is a complete generalized metric space and \(\mathcal{I} : \mathcal{B} \rightarrow \mathcal{B}\) is a strictly contractive mapping, that is,
for all \(u,v\in \mathcal{B}\) and a Lipschitz constant \(L<1\). Then for each given element \(u\in \mathcal{B}\), either
or
for some positive integer \(n_{0}\). Furthermore, if the second alternative holds, then:
-
(i)
the sequence \((\mathcal{I}^{n}u)\) is convergent to a fixed point p of \(\mathcal{I}\);
-
(ii)
p is the unique fixed point of \(\mathcal{I}\) in the set \(V: = \{v\in \mathcal{B} , d(\mathcal{I}^{n_{0}} u,v)<+\infty \}\);
-
(iii)
\(d(v,p)\leq \frac{1}{1-L}d(v,\mathcal{I}v)\) for all \(u,v\in V\).
In this paper, we consider the following system of functional equations:
for all \(x,y\in \mathcal{B}\).
The aim of the present paper is to solve the system of functional equations (1.2) and prove the Hyers–Ulam stability of g-derivations in complex Banach algebras by using the fixed point method.
Throughout this paper, we assume that \(\mathcal{B}\) is a complex Banach algebra.
2 Stability of the system of functional equations (1.2)
We solve and investigate the system of additive functional equations (1.2) in complex Banach algebras.
Lemma 2.1
[24, Theorem 2.1] Let \(\mathcal{B}\) be a complex Banach algebra and let \(\mathcal{F} :\mathcal{B} \rightarrow \mathcal{B}\) be an additive mapping such that \(\mathcal{F}(\alpha x)=\alpha \mathcal{F}(x)\) for all \(\alpha \in {\mathbf{T}}^{1} := \{\zeta \in {\mathbf{C}}: |\zeta |= 1\}\) and all \(x\in \mathcal{B}\). Then \(\mathcal{F}\) is C-linear.
Lemma 2.2
Let \(f,g : \mathcal{B} \rightarrow \mathcal{B}\) be mappings satisfying (1.2) for all \(x,y\in \mathcal{B}\). Then the mappings \(f,g : \mathcal{B} \rightarrow \mathcal{B}\) are additive.
Proof
Letting \(x=y=0\) in (1.2), we get
Putting \(y=x\) in (1.2), we have
for all \(x\in \mathcal{B}\). Setting \(y=0\) in (1.2), we obtain
for all \(x\in \mathcal{B}\).
Replacing \(g(y-x)\) by \(f(x-y)\) in (1.2), we have
for all \(x,y\in \mathcal{B}\). Hence the mapping \(f: \mathcal{B} \rightarrow \mathcal{B}\) is additive and thus by (2.1) the mapping \(g: \mathcal{B} \rightarrow \mathcal{B}\) is additive. □
Using the fixed point technique, we prove the Hyers–Ulam stability of the system of the additive functional equations (1.2) in complex Banach algebras.
Theorem 2.3
Suppose that \(\Delta : \mathcal{B}^{2}\rightarrow [0,\infty )\) is a function such that there exists an \(L<1\) with
for all \(x,y\in \mathcal{B}\). Let \(f,g : \mathcal{B} \rightarrow \mathcal{B}\) be mappings satisfying
for all \(x,y\in \mathcal{B}\). Then there exist unique additive mappings \(F, G : \mathcal{B}\rightarrow \mathcal{B}\) such that
for all \(x\in \mathcal{B}\).
Proof
Putting \(x=y=0\) in (2.3), we get
so \(f(0)=g(0)=0\).
Letting \(y=x\) in (2.3), we obtain
Let \(\Gamma =\{\gamma : \mathcal{B}\rightarrow \mathcal{B}: \gamma (0)=0 \}\). We define a generalized metric \(d: \Gamma \times \Gamma \rightarrow [0,\infty ]\) by
and we consider \(\inf \emptyset =+\infty \). Then d is a complete generalized metric on Γ (see [25]).
Now, we define the mapping \(\mathcal{J}:(\Gamma ,d) \rightarrow (\Gamma ,d)\) such that
for all \(x\in \mathcal{B}\).
Actually, let \(\delta , \gamma \in (\Gamma ,d)\) be given such that \(d(\delta , \gamma )=\mu \). Then
for all \(x\in \mathcal{B}\). Hence
for all \(x\in \mathcal{B}\). It follows that \(d( \mathcal{J}\delta , \mathcal{J}\gamma )\leq L\mu \). So
for all \(x\in \mathcal{B}\) and all \(\delta ,\gamma \in \Gamma \).
Using (2.4), we obtain
for all \(x\in \mathcal{B}\), which imply that \(d(f,\mathcal{J} f)\leq \frac{L}{2}\) and \(d(g,\mathcal{J}g)\leq \frac{L}{2}\).
Using the fixed point alternative, we deduce the existence of unique fixed points of \(\mathcal{J}\), that is, the existence of mappings \(F,G : \mathcal{B}\rightarrow \mathcal{B}\), respectively, such that
with the following property: there exist \(\mu , \eta \in (0,\infty )\) satisfying
for all \(x\in \mathcal{B}\).
Since \(\lim_{n\rightarrow \infty}d(\mathcal{J}^{n}f,F)=0\) and \(\lim_{n\rightarrow \infty}d(\mathcal{J}^{n}g,G)=0\),
for all \(x\in \mathcal{B}\).
Next, \(d(f,F)\leq \frac{1}{1-L}d(f,\mathcal{J}f)\) and \(d(g,G)\leq \frac{1}{1-L}d(g,\mathcal{J}g)\), which imply
for all \(x\in \mathcal{B}\).
Using (2.2) and (2.3), we conclude that
and
for all \(x,y\in \mathcal{B}\), since \(L<1\). Hence
for all \(x,y\in \mathcal{B}\), since \(L<1\). Therefore by Lemma 2.2, the mappings \(F,G: \mathcal{B}\rightarrow \mathcal{B}\) are additive. □
Corollary 2.4
Let η, p be nonnegative real numbers with \(p\geq 1\) and let \(f,g : \mathcal{B} \rightarrow \mathcal{B}\) be mappings satisfying
for all \(x,y\in \mathcal{B}\). Then there exist unique additive mappings \(F, G : \mathcal{B}\rightarrow \mathcal{B}\) such that
for all \(x\in \mathcal{B}\).
Proof
The proof follows from Theorem 2.3 by taking \(L=2^{1-p}\) and \(\Delta (x,y)=\eta (\Vert x\Vert ^{p}+\Vert y\Vert ^{p})\) for all \(x,y\in \mathcal{B}\). □
3 Stability of G-derivations in Banach algebras
In this section, by using the fixed point technique, we prove the Hyers–Ulam stability of g-derivations in complex Banach algebras.
Lemma 3.1
Let \(f,g : \mathcal{B} \rightarrow \mathcal{B}\) be mappings satisfying
for all \(x,y\in \mathcal{B}\) and all \(\lambda \in {\mathbf{T}}^{1}\). Then the mappings \(f,g : \mathcal{B} \rightarrow \mathcal{B}\) are C-linear.
Proof
If we put \(\lambda =1\) in (3.1), then f and g are additive by Lemma 2.2.
Letting \(y=x\) in (3.1), we have
for all \(x\in \mathcal{B}\) and all \(\lambda \in {\mathbf{T}}^{1}\). Since the mappings f and g are additive,
for all \(x\in \mathcal{B}\) and all \(\lambda \in {\mathbf{T}}^{1}\). So by Lemma 2.1 the mappings f and g are C-linear. □
Theorem 3.2
Suppose that \(\Delta : \mathcal{B}^{2}\rightarrow [0,\infty )\) is a function such that there exists an \(L<1\) with
for all \(x,y\in \mathcal{B}\). Let \(f,g : \mathcal{B} \rightarrow \mathcal{B}\) be mappings satisfying
for all \(x,y\in \mathcal{B}\) and all \(\lambda \in {\mathbf{T}}^{1}\). Then there exist unique C-linear mappings \(F,G: \mathcal{B} \rightarrow \mathcal{B}\) such that F is a G-derivation and
for all \(x\in \mathcal{B}\).
Proof
Let \(\lambda =1\) in (3.3). By Theorem 2.3, there are unique additive mappings \(F,G: \mathcal{B} \rightarrow \mathcal{B}\) satisfying (3.5) and (3.6) given by
for all \(x\in \mathcal{B}\).
Using (3.2) and (3.3), we conclude that
and
for all \(x,y\in \mathcal{B}\), since \(L<1\). Hence
for all \(x,y\in \mathcal{B}\) and all \(\lambda \in {\mathbf{T}}^{1}\), since \(L<1\). Therefore by Lemma 3.1, the mappings \(F,G: \mathcal{B}\rightarrow \mathcal{B}\) are C-linear.
It follows from (3.4) that
for all \(x,y\in \mathcal{B}\). So
for all \(x,y\in \mathcal{B}\). Thus the C-linear mapping F is a G-derivation. □
Corollary 3.3
Let p, q, η be nonnegative real numbers with \(p+q>2\) and let \(f,g : \mathcal{B} \rightarrow \mathcal{B}\) be mappings satisfying
and
for all \(x,y\in \mathcal{B}\) and all \(\lambda \in {\mathbf{T}}^{1}\). Then there exist unique C-linear mappings \(F,G: \mathcal{B} \rightarrow \mathcal{B}\) such that F is a G-derivation and
for all \(x\in \mathcal{B}\).
Proof
The proof follows from Theorem 3.2 by taking \(\Delta (x,y)=\eta \Vert x\Vert ^{p}\Vert y\Vert ^{q}\) for all \(x,y\in \mathcal{B}\). Choosing \(L=2^{2-p-q}\), we obtain the desired result. □
4 Conclusion
We solved the system of functional equations (1.2) and we proved the Hyers–Ulam stability of g-derivations in Banach algebras.
Data availability
Not applicable.
References
Mirzavaziri, M., Moslehian, M.S.: Automatic continuity of σ-derivations on \(C^{*}\)-algebras. Proc. Am. Math. Soc. 134, 3319–3327 (2006)
Park, C., Lee, J., Zhang, X.: Additive s-functional inequality and hom-derivations in Banach algebras. J. Fixed Point Theory Appl. 21, 18 (2019)
Ulam, S.M.: A Collection of the Mathematical Problems. Interscience Publ., New York (1960)
Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941)
Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64–66 (1950)
Rassias, T.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)
Găvruta, P.: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)
Lee, Y., Jung, S., Rassias, M.T.: On an n-dimensional mixed type additive and quadratic functional equation. Appl. Math. Comput. 228, 13–16 (2014)
Lee, Y., Jung, S., Rassias, M.T.: Uniqueness theorems on functional inequalities concerning cubic-quadratic-additive equation. J. Math. Inequal. 12(1), 43–61 (2018)
Park, C., Rassias, M.T.: Additive functional equations and partial multipliers in \(C^{*}\)-algebras. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(3), 2261–2275 (2019)
Dehghanian, M., Modarres, S.M.S.: Ternary γ-homomorphisms and ternary γ-derivations on ternary semigroups. J. Inequal. Appl. 2012, 34 (2012)
Dehghanian, M., Modarres, S.M.S., Park, C., Shin, D.: \(C^{*}\)-Ternary 3-derivations on \(C^{*}\)-ternary algebras. J. Inequal. Appl. 2013, 124 (2013)
Dehghanian, M., Park, C.: \(C^{*}\)-Ternary 3-homomorphisms on \(C^{*}\)-ternary algebras. Results Math. 66(3), 385–404 (2014)
Czerwik, S.: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge (2002)
Jung, S.: On the Hyers-Ulam stability of the functional equations that have the quadratic property. J. Math. Anal. Appl. 222, 126–137 (1998)
Eshaghi Gordji, M., Hayati, B., Kamyar, M., Khodaei, H.: On stability and nonstability of systems of functional equations. Quaest. Math. 44(4), 557–567 (2020)
EL-Fassi, Iz.: Approximate solution of a generalized multi-quadratic type functional equation in Lipschitz space. J. Math. Anal. Appl. 519(2), 126840 (2023)
EL-Fassi, Iz.: Generalized hyperstability of a Drygas functional equation on a restricted domain using Brzdȩk’s fixed point theorem. J. Fixed Point Theory Appl. 19(4), 2529–2540 (2017)
Dehghanian, M., Sayyari, Y., Park, C.: Hadamard homomorphisms and Hadamard derivations on Banach algebras. Miskolc Math. Notes 24(1), 81–91 (2023)
Sayyari, Y., Dehghanian, M., Park, C., Lee, J.: Stability of hyper homomorphisms and hyper derivations in complex Banach algebras. AIMS Math. 7(6), 10700–10710 (2022)
Dutta, H., Kumar, B.V.S., Al-Shaqsi, K.: Approximation of Jensen type reciprocal mappings via fixed point technique. Miskolc Math. Notes 23(2), 607–619 (2022)
Lu, G., Park, C.: Hyers–Ulam stability of general Jensen-type mappings in Banach algebras. Results Math. 66, 87–98 (2014)
Diaz, J.B., Margolis, B.: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 74(2), 305–309 (1968)
Park, C.: Homomorphisms between Poisson \(JC^{*}\)-algebras. Bull. Braz. Math. Soc. 36, 79–97 (2005)
Mihet, D., Radu, V.: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl. 343, 567–572 (2008)
Acknowledgements
We would like to express our sincere gratitude to the anonymous referee for his/her helpful comments, which helped to improve the quality of the manuscript.
Author information
Authors and Affiliations
Contributions
M.D. and Y.S. wrote the main manuscript and S.P. and C.P. revised the manuscript. All authors reviewed the manuscript.
Corresponding authors
Ethics declarations
Ethics approval and consent to participate
Not applicable.
Consent for publication
The authors declare that they have no competing interests.
Competing interests
The authors declare no competing interests.
Additional information
Abbreviations
Not applicable.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Dehghanian, M., Sayyari, Y., Donganont, S. et al. A Pexider system of additive functional equations in Banach algebras. J Inequal Appl 2024, 27 (2024). https://doi.org/10.1186/s13660-024-03104-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-024-03104-6