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2014 | OriginalPaper | Chapter

On the Generalized Hyers–Ulam Stability of the Pexider Equation on Restricted Domains

Authors : Youssef Manar, Elhoucien Elqorachi, Themistocles M. Rassias

Published in: Handbook of Functional Equations

Publisher: Springer New York

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Abstract

Let σ: $E\longrightarrow E$ be an involution of the normed space E and let p, M, d be nonnegative real numbers, such that $0<p<1$. In this chapter, we investigate the Hyers–Ulam–Rassias stability of the Pexider functional equations

$$\begin{aligned} f(x+y) = &\, g(x)+h(y),\ f(x+y)+g(x-y)=h(x)+k(y),\\ & f(x+y)+g(x+\sigma(y))=h(x)+k(y), x,y\in{E}\end{aligned}$$

on restricted domains $\mathcal{B}=\{(x,y)\in{E^{2}}: \|x\|^{p}+\|y\|^{p}\geq M^{p}\}$ and $\mathcal{C}=\{(x,y)\in{E}^{2}:\|x\|\geq d\; or\; \|y\|\geq d\}.$

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Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Japan. 2, 64–66 (1950)CrossRefMATHMathSciNet

Akkouchi, M.: Stability of certain functional equations via a fixed point of Ćirić. Filomat 25, 121–127 (2011)CrossRefMATHMathSciNet

Akkouchi, M.: Hyers–Ulam–Rassias stability of Nonlinear Volterra integral equation via a fixed point approach. Acta. Univ. Apulensis Math. Inform. 26, 257–266 (2011)MATHMathSciNet

Bae, J.-H.: On the stability of 3-dimensional quadratic functional equation. Bull. Korean Math. Soc. 37(3), 477–486 (2000)MATHMathSciNet

Bae, J.-H., Jun, K.-W.: On the generalized Hyers–Ulam–Rassias stability of a quadratic functional equation. Bull. Korean Math. Soc. 38(2), 325–336 (2001)MATHMathSciNet

Bouikhalene, B., Elqorachi, E., Rassias, Th.M.: On the Hyers–Ulam stability of approximately Pexider mappings. Math. Inequal. Appl. 11, 805–818 (2008)MATHMathSciNet

Brzd\cek, J.: On a method of proving the Hyers–Ulam stability of functional equations on restricted domains. Aust. J. of Math. Anal. Appl. 6(1), 1–10, Article 4, (2009)

Brzd\cek, J.: A note on stability of the Popoviciu functional equation on restricted domain. Dem. Math. XLIII(3), 635–641 (2010)

Brzd\cek, J., Pietrzyk, A.: A note on stability of the general linear equation. Aequ. Math. 75, 267–270 (2008)

10.

C\vadariu, L., Radu, V.:Fixed points and the stability of Jensens functional equation. J. Inequal. Pure Appl. Math. 4(1), Article 4 (2003)

11.

C\vadariu, L., Radu, V.: On the stability of the Cauchy functional equation: A fixed point approach. Grazer Math. Berichte. 346, 43–52 (2003)

12.

Charifi, A., Bouikhalene, B., Elqorachi, E.: Hyers–Ulam–Rassias stability of a generalized Pexider functional equation. Banach J. Math. Anal. 1(2), 176–185 (2007)CrossRefMATHMathSciNet

13.

Chung, J.-Y.: A generalized Hyers–Ulam stability of a Pexiderized logarithmic functional equation in restricted domains. J. Ineq. Appl. 2012, 1–5 (2012)CrossRef

14.

Chung, J.-Y.: Stability of functional equations on restricted domains in a group and their asymptotic behaviors. Comp. Math. Appl. 60(9), 2653–2665 (2010)CrossRefMATH

15.

Chung, J., Sahoo, P.K..: Stability of a logarithmic functional equation in distributions on a restricted domain. Abstr. Appl. Anal. 9 p, Article ID 751680, (2013)

16.

Cieplisński, K.: Applications of fixed point theorems to the Hyers–Ulam stability of functional equations. A survey. Ann. Funct. Anal. 3, 151–164 (2012)CrossRefMathSciNet

17.

Dales, H.G., Moslehian, M.S.: Stability of mappings on multi-normed spaces. Glasgow Math. J. 49, 321–332 (2007)CrossRefMATHMathSciNet

18.

Elqorachi, E., Manar, Y., Rassias, Th.M: Hyers–Ulam stability of the quadratic functional equation. Int. J. Nonlin. Anal. Appl. 1(2), 11–20 (2010)MathSciNet

19.

Elqorachi, E., Manar, Y., Rassias, Th.M: Hyers-Ulam stability of the quadratic and Jensen functional equations on unbounded domains. J. Math. Sci. Adv. Appl. 4(2), 287–301 (2010)MATHMathSciNet

20.

Forti, G.L.: Hyers-Ulam stability of functional equations in several variables. Aequ. Math. 50, 143–190 (1995)CrossRefMATHMathSciNet

21.

Forti, G.-L., Sikorska, J.: Variations on the Drygas equation and its stability. Nonlin. Anal.: Theory Meth. Appl. 74, 343–350 (2011)CrossRefMATHMathSciNet

22.

G\vavruta, P.: A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)

23.

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

24.

Ger, R.: On a factorization of mappings with a prescribed behaviour of the Cauchy difference. Ann. Math. Sil. 8 141–155 (1994)

25.

Hyers, D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U. S. A. 27, 222–224 (1941)CrossRefMathSciNet

26.

Hyers, D.H., Rassias, Th.M.: Approximate homomorphisms. Aequ. Math. 44, 125–153 (1992)CrossRefMATHMathSciNet

27.

Hyers, D.H., Isac, G., Rassias, T.M.: On the asymptoticity aspect of Hyers–Ulam stability of mappings. Proc. Amer. Math. Soc. 126, 425–430 (1998)CrossRefMATHMathSciNet

28.

Hyers, D.H., Isac, G., Rassias, T.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Basel (1998)CrossRefMATH

29.

Jung, S.-M.: On the Hyers-Ulam stability of the functional equation that have the quadratic property. J. Math. Anal. Appl. 222, 126–137 (1998)CrossRefMATHMathSciNet

30.

Jung, S.-M.: Hyers–Ulam–Rassias stability of Jensen’s equation and its application. Proc. Amer. Math. Soc. 126(11), 3137–3143 (1998)CrossRefMATHMathSciNet

31.

Jung, S.-M.: Stability of the quadratic equation of Pexider type. Abh. Math. Sem. Univ. Hamburg 70, 175–190 (2000)CrossRefMATHMathSciNet

32.

Jung, S.-M.: Hyers–Ulam–Rassias stability of isometries on restricted domains. Nonlin. Stud. 8(1), 12–5 (2001)

33.

Jung, S.-M.: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, New York (2011)CrossRefMATH

34.

Jung, S.-M., Kim, B.: Local stability of the additive functional equation and its applications. Int. J. Math. Math. Sci. 2003, 15–26 (2003)

35.

Jun, K.-W., Lee, Y.-H.: A generalization of the Hyers-Ulam-Rassias stability of Jensen’s equation. J. Math. Anal. Appl. 238, 305–315 (1999)CrossRefMATHMathSciNet

36.

Jung, S.-M., Sahoo, P.K.: Hyers–Ulam stability of the quadratic equation of Pexider type. J. Korean Math. Soc. 38(3), 645–656 (2001)MATHMathSciNet

37.

Kannappan, Pl.: Functional Equations and Inequalities with Applications. Springer, New York, 2009CrossRefMATH

38.

Kim, B.: Local stability of the additive equation. Int. J. Pure Appl. Math., 5 (1), 65–74 (2003)

39.

Kim, G.H., Lee, S.H.: Stability of the d’Alembert type functional equations. Nonlin. Funct. Anal. Appl. 9, 593–604 (2004)MATH

40.

Kim, G.H., Lee, Y.-H.: Hyers–Ulam stability of a Bi-Jensen functional equation on a punctured domain. J. Ineq. Appl. 2010, 15 p, Article ID 476249, (2010)

41.

Lee, Y.H., Jung, K.W.: A generalization of the Hyers-Ulam-Rassias stability of the pexider equation. J. Math. Anal. Appl. 246, 627–638 (2000)CrossRefMATHMathSciNet

42.

Lee, J.R., Shin, D.Y., Park C.: Hyers-Ulam stability of functional equations in matrix normed spaces. J. Inequal. Appl. 2013, 11 p (2013)

43.

Manar, Y., Elqorachi, E., Rassias, Th.M.: Hyers–Ulam stability of the Jensen functional equation in quasi-Banach spaces. Nonlin. Funct. Anal. Appl. 15(4), 581–603 (2010)MATHMathSciNet

44.

Manar, Y., Elqorachi, E., Rassias, Th.M: On the Hyers–Ulam stability of the quadratic and Jensen functional equations on a restricted domain. Nonlin. Funct. Anal. Appl. 15(4), 647–655 (2010)MATHMathSciNet

45.

Moghimi, M.B., Najati, A., Park, C.: A functional inequality in restricted domains of Banach modules. Adv. Diff. Eq. 2009, 14–p, Article ID 973709, doi:10.1155/2009/973709

46.

Moslehian, M.S.: The Jensen functional equation in non-Archimedean normed spaces. J. Funct. Spaces Appl. 7, 13–24 (2009)CrossRefMATHMathSciNet

47.

Moslehian, M.S., Najati, A.: Application of a fixed point theorem to a functional inequality. Fixed Point Theory 10, 141–149 (2009)MATHMathSciNet

48.

Moslehian, M.S., Sadeghi, G.: Stability of linear mappings in quasi-Banach modules. Math. Inequal. Appl. 11, 549–557 (2008)MATHMathSciNet

49.

Najati, A.: On the stability of a quartic functional equation. J. Math. Anal. Appl. 340, 569–574 (2008)CrossRefMATHMathSciNet

50.

Najati, A., Jung, S.M.: Approximately quadratic mappings on restricted domains, J. Ineq. Appl. 10 p, Article ID 503458, (2010)

51.

Najati, A., Moghimi, M.B.: Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces. J. Math. Anal. Appl. 337, 399–415 (2008)CrossRefMATHMathSciNet

52.

Najati, A., Park, C.: Hyers–Ulam–Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation. J. Math. Anal. Appl. 335, 763–778 (2007)CrossRefMATHMathSciNet

53.

Pardalos, P.M., Georgiev, P.G., Srivastava, H.M. (eds.): Nonlinear Analysis - Stability, Approximation, and Inequalities. In honor of Th.M. Rassias on the occasion of his 60th birthday, Springer, Berlin (2012)

54.

Park, C.G.: On the stability of the linear mapping in Banach modules. J. Math. Anal. Appl. 275, 711–720 (2002)CrossRefMATHMathSciNet

55.

Pourpasha, M.M., Rassias, J.M., Saadati, R., Vaezpour, S.M.: A fixed point approach to the stability of Pexider quadratic functional equation with involution, J. Ineq. Appl. Article ID 839639, (2010). doi:10.1155/2010/839639

56.

Rahimi, A., Najati, A., Bae J.-H.: On the asymptoticity aspect of Hyers–Ulam stability of quadratic mappings. J. Ineq. Appl. 2010, 14–p (2010)MathSciNet

57.

Rassias, Th.M.: On the stability of linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72, 297–300 (1978)CrossRefMATHMathSciNet

58.

Rassias, J.M.: On approximation of approximately linear mappings by linear mappings. J. Funct. Anal. 46, 126–130 (1982)CrossRefMATHMathSciNet

59.

Rassias, J.M.: Solution of a problem of Ulam. J. Approx. Theory 57, 268–273 (1989)CrossRefMATHMathSciNet

60.

Rassias, Th.M.: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl. 251(1), 264–284 (2000)CrossRefMATHMathSciNet

61.

Rassias, Th.M.: The problem of S. M. Ulam for approximately multiplicative mappings. J. Math. Anal. Appl. 246, 352–378 (2000)CrossRefMATHMathSciNet

62.

Rassias, Th.M.: On the stability of the functional equations and a problem of Ulam. Acta Appl. Math. 62, 23–130 (2000)CrossRefMATHMathSciNet

63.

Rassias, Th.M., Brzd\cek, J.: Functional Equations in Mathematical Analysis. Springer, New York (2011)

64.

Rassias, J.M., Rassias, M.J.: On the Ulam stability of Jensen type mappings on restricted domains. J. Math. Anal. Appl. 281, 516–524 (2003)CrossRefMATHMathSciNet

65.

Rassias, Th.M., $\check{S}$emrl, P.: On the behavior of mappings which do not satisfy Hyers-Ulam stability. Proc. Amer. Math. Soc. 114, 989–993 (1992)

66.

Sikorska, J.: Exponential functional equation on spheres. Appl. Math. Lett. 23, 156–160 (2010)CrossRefMATHMathSciNet

67.

Skof, F.: Approssimazione di funzioni δ-quadratic su dominio restretto. Atti. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 118, 58–70 (1984)MATHMathSciNet

68.

Skof, F.: Sull’approssimazione delle applicazioni localmente δ-additive. Atti. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 117, 377–389 (1983)MATHMathSciNet

69.

Stetk\aer, H.: Functional equations on abelian groups with involution. Aequ. Math. 54, 144–172 (1997)CrossRef

70.

Ulam, S.M.: A Collection of Mathematical Problems. Interscience, New York, (1961); Problems in Modern Mathematics, Wiley, New York, (1964)

71.

Xiang, S.-H.: Hyers-Ulam-Rassias stability of approximate isometries on restricted domains. J. Cent. S. Univ. Tech. 9(4), 289–292 (2002)CrossRef

72.

Yang, D.: Remarks on the stability of Drygas` equation and the Pexider-quadratic equation. Aequ. Math. 68, 108–116 (2004)MATH

Title: On the Generalized Hyers–Ulam Stability of the Pexider Equation on Restricted Domains
Authors: Youssef Manar
Elhoucien Elqorachi
Themistocles M. Rassias
Publisher: Springer New York
Book: Handbook of Functional Equations
Print ISBN: 978-1-4939-1285-8

Electronic ISBN: 978-1-4939-1286-5

Copyright Year: 2014
DOI: https://doi.org/10.1007/978-1-4939-1286-5_13

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