The Stability of Multi-Coefficients Pexider Additive Functional Inequalities in Banach Spaces

Liu, Yang; Lyu, Gang; Jin, Yuanfeng; Yang, Jiangwei

doi:https://doi.org/10.1155/2024/6931488

Journal of Mathematics

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Abstract Introduction and Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 6931488 | https://doi.org/10.1155/2024/6931488

The Stability of Multi-Coefficients Pexider Additive Functional Inequalities in Banach Spaces

Yang Liu,¹Gang Lyu,²Yuanfeng Jin,¹and Jiangwei Yang²

Academic Editor: Ali Jaballah

Received30 May 2022

Revised10 Aug 2022

Accepted27 Mar 2024

Published12 Apr 2024

Abstract

The Hyers–Ulam stability of multi-coefficients Pexider additive functional inequalities in Banach spaces is investigated. In order to do this, the fixed point method and the direct method are used.

1. Introduction and Preliminaries

For an object possessing some properties only approximately in mathematics and in many other scientific investigations, can one find the special object satisfied them truly? One of the effective methods to solve this problem is to use the concept of generalized Hyers–Ulam stability.

Let us review the definition of Hyers–Ulam stability. In a class of mappings, if each mapping of this class fulfilling the equation approximately is “near” to its real solution or stable approximate solution, then the equation is said to be Hyers–Ulam stability.

The stability problem of functional equations is from a question of Ulam [1] in 1940, that is, the stability of metric group homomorphisms. In 1941, Hyers [2] gave the first affirmative answer to the question of Ulam for Banach spaces about the Cauchy functional equation. Hyers’ method of proof is called the “direct method.” The functional equationis called an additive functional equation. More generalizations and applications of the Hyers–Ulam stability to a number of functional equations and mappings can be found in [3–10].

In 2013, Li et al. [11] investigated the generalized Hyers–Ulam stability of the following function inequalities:in quasi-Banach spaces. In the paper, assume that is a linear space over the field , and is a linear space over the field . Let and be given scalars.

The functional equationis called a Pexider additive functional equation (for more details, see [12–23]). In the paper, we introduce and investigate the following functional equation:where . The stability problems of several functional inequalities have been extensively investigated by a number of authors (see [24–47]).

In order to find the stability of (4), the following fixed point theory would be applied.

Theorem 1 (see [48, 49]). Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . Then for each given element , eitherfor all nonnegative integers or there exists a positive integer such that(1), for all (2)The sequence converges to a fixed point of (3) is the unique fixed point of in the set (4) for all

2. Hyers–Ulam Stability of Functional Inequality (4): A Fixed Point Method

Theorem 2. Suppose that is a Banach space and is a function such that there exists an with

If are mappings satisfying andthen there exists a unique solution of (4) such that

Proof. Letting in (7), we get . Letting in (7), we obtainfor all . Thus,Letting in (7), we haveThus,Next, replacing by and by in (7), we getThus,Letting in (16), we getfor all .
Consider the setand introduce the generalized metric on :Then will be proved to be complete. Let and ; by the definition of and property of infimum, satisfies the triangle inequality. Suppose that is -Cauchy sequence on . That is, for any , , , such that . By the definition of , it is easy to see that is a Cauchy sequence in . Since is complete, there exist and . Taking the limit as , we get , for all , such that is -convergent, i.e., is a complete generalized metric (for more details, we refer to [48]).
Now, we consider the linear mapping such thatfor all .
Let be given such that . Then,for all . Hence,for all . So, implies that . This means thatfor all .
It follows from (17) thatfor all . So, .
By Theorem 1, there exists a mapping satisfying the following:(1) is a fixed point of , i.e., for all . The mapping is a unique fixed point of in the set This implies that is a unique mapping satisfying (45) such that there exists a satisfying for all .(2) as . This implies the equality for all .(3), which impliesfor all .
It follows from (16) and (28) thatSo, the mapping is additive. Next, by (8), (29) can be proved. Similarly, we can obtain inequalities (9) and (10).

Corollary 3. Let and be nonnegative real numbers and be mappings satisfyingfor all and . Then there exists a unique additive mapping such that

Theorem 4. Let be a function such that there exists an withfor all . Let be mappings satisfying (7) for all and . Then there exists a unique additive mapping such that

Corollary 5. Let and be nonnegative real numbers and be mappings satisfying (31) for all and . Then there exists a unique additive mapping such that

3. Hyers–Ulam Stability of Functional Inequality (4): A Direct Method

Using the direct method, we prove the Hyers–Ulam stability of functional inequality (4).

Theorem 6. Assume that is a Banach space and with satisfy the inequalitywhere satisfiesfor all . Then there exists a unique additive mapping such that

Proof. Letting in (38), we get . So, .
Letting in (38), we getfor all . Thus,for all .
Letting in (38), we getfor all . In (43), replacing by , we getBy the same way, from (38), we have the following inequality:From (42), (44), and (45), it follows thatwhere .
It follows from (46) thatfor all nonnegative integers and with and all . It means that the sequence is a Cauchy sequence for all . Since is complete, the sequence converges. We define the mapping byfor all . Moreover, letting and passing to the limit , we getSimilarly, there exists a mapping such that andfor all .
We also obtain a mapping such that , andNext, we show that is an additive mapping.for all . Thus, the mapping is additive.
Now, we prove the uniqueness of . Assume that is another additive mapping satisfying (40). We obtainwhich tends to zero as for all . Then we can conclude that for all . In fact, by (42), we get . Similarly, we obtain .

Corollary 7. Let and be positive real numbers with . Let be mappings with satisfyingfor all . Then there exists a unique additive mapping such that

4. Conclusion

In this paper, we have investigated the Hyers–Ulam stability of general Pexider function inequalities in Banach spaces by using the fixed point method and the direct method.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The authors equally conceived the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript. Y. Liu conceptualized the study, developed the research question, oversaw the overall progress of the study, managed the collaboration, wrote the initial draft of the manuscript, and integrated feedback from co-authors. G. Lyu assisted in developing the methodology and research design, conducted the literature review and contributed significantly to the theoretical framework, performed data collection and carried out preliminary data analysis, and reviewed and provided substantive edits to subsequent versions of the manuscript. Y. Jin designed and implemented the statistical models for data analysis, interpreted the results, wrote the results section, participated in drafting the discussion and conclusion sections, and secured funding for the project. J. Yang proposed the application of innovative techniques used in the study, contributed to writing the methods section, supported data validation, and critically reviewed the manuscript for intellectual content and clarity.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11761074), Project of Jilin Science and Technology Development for Leading Talent of Science and Technology Innovation in Middle and Young and Team Project (no. 20200301053RQ), Natural Science Foundation of Jilin Province (no. YDZJ202101ZYTS136), and Scientific Research Project of Guangzhou College of Technology and Business in 2023 (no. KYPY2023012).

References

S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ, New York, NY, USA, 1960.
D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences, vol. 27, no. 4, pp. 222–224, 1941.
View at: Publisher Site | Google Scholar
J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge, UK, 1989.
S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing, River Edge, NJ, USA, 2002.
D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Boston, UK, 1998.
S. Jung, D. Popa, and M. T. Rassias, “On the stability of the linear functional equation in a single variable on complete metric spaces,” Journal of Global Optimization, vol. 59, pp. 13–16, 2014.
View at: Google Scholar
S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, NY, USA, 2011.
P. Kannappan, Functional Equations and Inequalities with Applications, Springer, New York, NY, USA, 2009.
T. Kim, C. Park, and S. Park, “An AQ-functional equation in paranormed spaces,” Journal of Computational Analysis and Applications, vol. 15, pp. 1467–1475, 2013.
View at: Google Scholar
Y. Lee, S. Jung, and M. T. Rassias, “Uniqueness theorems on functional inequalities concerning cubic-quadratic-additive equation,” Journal of Mathematical Inequalities, vol. 12, no. 1, pp. 43–61, 2018.
View at: Publisher Site | Google Scholar
L. Li, G. Lu, C. Park, and D. Shin, “Additive functional inequalities in in quasi-banach spaces,” Journal of Computational Analysis and Applications, vol. 15, pp. 1165–1175, 2013.
View at: Google Scholar
H. Kenary, C. Park, and D. Shin, “Orthogonal stability of an additive functional equation in Banach modules over a C^∗-algebra,” Journal of Computational Analysis and Applications, vol. 15, pp. 1057–1068, 2013.
View at: Google Scholar
M. Gordji, M. Ghanifard, H. Khodaei, and C. Park, “Fixed points and the random stability of a mixed type cubic, quadratic and additive functional equation,” Journal of Computational Analysis and Applications, vol. 15, pp. 612–621, 2013.
View at: Google Scholar
G. Lu and C. Park, “Hyers-Ulam stability of additive set-valued functional equations,” Applied Mathematics Letters, vol. 24, no. 8, pp. 1312–1316, 2011.
View at: Publisher Site | Google Scholar
J. Lee, C. Park, Y. Cho, and D. Shin, “Orthogonal stability of A cubic-quartic functional equation in non-archimedean spaces,” Journal of Computational Analysis and Applications, vol. 15, pp. 572–583, 2013.
View at: Google Scholar
C. Park, “Homomorphisms between Poisson JC^∗-algebra,” Bulletin of the Brazilian Mathematical Society, vol. 36, pp. 79–97, 2005.
View at: Google Scholar
Y. Jin, C. Park, and M. T. Rassias, “Hom-derivations in C^∗-ternary algebras,” Acta Mathematica Sinica, English Series, vol. 36, no. 9, pp. 1025–1038, 2020.
View at: Publisher Site | Google Scholar
Y. Lee and K. Jun, “A generalization of the hyers–ulam–rassias stability of the pexider equation,” Journal of Mathematical Analysis and Applications, vol. 246, no. 2, pp. 627–638, 2000.
View at: Publisher Site | Google Scholar
K. Jun and Y. Lee, “A generalization of the Hyers-Ulam-Rassias stability of the Pexiderized quadratic equations,” Journal of Mathematical Analysis and Applications, vol. 297, no. 1, pp. 70–86, 2004.
View at: Publisher Site | Google Scholar
M. S. Moslehian, “On the orthogonal stability of the Pexiderized quadratic equation,” Journal of Difference Equations and Applications, vol. 11, pp. 999–1004, 2005.
View at: Publisher Site | Google Scholar
M. S. Moslehian, “On the stability of the orthogonal Pexiderized Cauchy equation,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 211–223, 2006.
View at: Publisher Site | Google Scholar
S. A. Mohiuddine, “Stability of Jensen functional equation in intuitionistic fuzzy normed space,” Chaos, Solitons & Fractals, vol. 42, no. 5, pp. 2989–2996, 2009.
View at: Publisher Site | Google Scholar
S. A. Mohiuddine and H. Sevli, “Stability of Pexiderized quadratic functional equation in intuitionistic fuzzy normed space,” Journal of Computational and Applied Mathematics, vol. 235, no. 8, pp. 2137–2146, 2011.
View at: Publisher Site | Google Scholar
B. Belaid, E. Elhoucien, and T. M. Rassias, “On the Hyers-Ulam stability of approximately Pexider mappings,” Mathematical Inequalities and Applications, vol. 11, no. 4, pp. 805–818, 2008.
View at: Publisher Site | Google Scholar
D. G. Bourgin, “Classes of transformations and bordering transformations,” Bulletin of the American Mathematical Society, vol. 57, no. 4, pp. 223–237, 1951.
View at: Publisher Site | Google Scholar
Y. Cho, C. Park, and R. Saadati, “Functional inequalities in non-Archimedean Banach spaces,” Applied Mathematics Letters, vol. 23, no. 10, pp. 1238–1242, 2010.
View at: Publisher Site | Google Scholar
Y. Ding, “Ulam-Hyers stability of fractional impulsive differential equations,” The Journal of Nonlinear Science and Applications, vol. 11, no. 8, pp. 953–959, 2018.
View at: Publisher Site | Google Scholar
A. Ebadian, I. Nikoufar, T. M. Rassias, and N. Ghobadipour, “Stability of generalized derivations on Hilbert C^∗-modules associated with a Pexiderized Cauchy-Jensen type functional equation,” Acta Mathematica Scientia, vol. 32, no. 3, pp. 1226–1238, 2012.
View at: Publisher Site | Google Scholar
D. H. Hyers and T. M. Rassias, “Approximate homomorphisms,” Aequationes Mathematicae, vol. 44, no. 2-3, pp. 125–153, 1992.
View at: Publisher Site | Google Scholar
Y. Lee and S. Jung, “A fixed point approach to the stability of a general quartic functional equation,” The Journal of Mathematics and Computer Science, vol. 20, no. 3, pp. 207–215, 2019.
View at: Publisher Site | Google Scholar
G. Lu, J. Xin, Y. Jin, and C. Park, “Approximation of general Pexider functional inequalities in fuzzy Banach spaces,” The Journal of Nonlinear Science and Applications, vol. 12, no. 4, pp. 206–216, 2018.
View at: Publisher Site | Google Scholar
G. Lu and C. Park, “Additive functional inequalities in Banach spaces,” Journal of Inequalities and Applications, vol. 2012, no. 1, 2012.
View at: Google Scholar
Z. Lu, G. Lu, Y. Jin, and C. Park, “The stability of additive (α,β)-functional equations,” Journal of Applied Analysis & Computation, vol. 9, no. 6, pp. 2295–2307, 2019.
View at: Publisher Site | Google Scholar
Y. Manar, E. Elqorachi, and T. M. Rassias, “On the generalized Hyers-Ulam stability of the Pexider equation on restricted domains,” in Handbook of Functional Equations-Functional Inequalities, pp. 279–299, Springer, New York, NY, USA, 2014.
View at: Google Scholar
A. Najati and T. M. Rassias, “Stability of the Pexiderized Cauchy and Jensen’s equations on restricted domains,” Communications in Mathematical Analysis, vol. 8, no. 2, pp. 125–135, 2010.
View at: Google Scholar
C. Park and M. T. Rassias, “Additive functional equations and partial multipliers in C^∗-algebras,” Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, vol. 113, no. 3, pp. 2261–2275, 2019.
View at: Publisher Site | Google Scholar
C. Park, “Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras,” Bulletin des Sciences Mathematiques, vol. 132, no. 2, pp. 87–96, 2008.
View at: Publisher Site | Google Scholar
C. Park, “Set-valued additive ρ-functional inequalities,” Journal of Fixed Point Theory and Applications, vol. 20, 2018.
View at: Google Scholar
C. Park, Y. Cho, and M. Han, “Functional inequalities associated with Jordan-von Neumann type additive functional equations,” Journal of Inequalities and Applications, vol. 2007, Article ID 41820, pp. 1–14, 2007.
View at: Publisher Site | Google Scholar
C. Park, J. Lee, and X. Zhang, “Additive s-functional inequality and home-derivations in Banach algebras,” Journal of Fixed Point Theory and Applications, vol. 21, no. 1, 2019.
View at: Google Scholar
T. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978.
View at: Publisher Site | Google Scholar
N. Sene, “Exponential form for Lyapunov function and stability analysis of the fractional differential equations,” The Journal of Mathematics and Computer Science, vol. 18, no. 4, pp. 388–397, 2018.
View at: Publisher Site | Google Scholar
M. Gordji, G. Kim, J. Lee, and C. Park, “Generalized ternary Bi-derivations on ternary BanachAlgebras:A fixed point approach,” Journal of Computational Analysis and Applications, vol. 15, no. 1, pp. 45–54, 2013.
View at: Google Scholar
G. Lu, Y. Jiang, and C. Park, “Additive functional equation in frechet spaces,” Journal of Computational Analysis and Applications, vol. 15, pp. 269–373, 2013.
View at: Google Scholar
I. Chang and Y. Lee, “Satbility for an n-dimensional functional equation of quadrati-additive type with the fixed point approach,” Journal of Computational Analysis and Applications, vol. 15, pp. 1096–1103, 2013.
View at: Google Scholar
S. Yang and C. Park, “Additive functional inequalities in paranormed spaces,” Journal of Computational Analysis and Applications, vol. 16, pp. 165–171, 2014.
View at: Google Scholar
J. Rassias and H. Kim, “Approximate (m,n)-Cauchy-Jensen mappings in quasi-β-Normed spaces,” Journal of Computational Analysis and Applications, vol. 16, pp. 346–358, 2014.
View at: Google Scholar
L. Cădariu and V. Radu, “Fixed points and the stability of Jensen’s functional equation,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 1, 2003.
View at: Google Scholar
J. B. Diaz and B. Margolis, “A fixed point theorem of the alternative, for contractions on a generalized complete metric space,” Bulletin of the American Mathematical Society, vol. 74, no. 2, pp. 305–309, 1968.
View at: Publisher Site | Google Scholar

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Copyright © 2024 Yang Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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