On the stability of pexider functional equation in non-archimedean spaces

  • Reza Saadati1Email author,

    Affiliated with

    • Seiyed Mansour Vaezpour2 and

      Affiliated with

      • Zahra Sadeghi1

        Affiliated with

        Journal of Inequalities and Applications20112011:17

        DOI: 10.1186/1029-242X-2011-17

        Received: 7 January 2011

        Accepted: 24 June 2011

        Published: 24 June 2011


        In this paper, the Hyers-Ulam stability of the Pexider functional equation


        in a non-Archimedean space is investigated, where σ is an involution in the domain of the given mapping f.

        MSC 2010:26E30, 39B52, 39B72, 46S10


        Hyers-Ulam stability of functional equation Non-Archimedean space Quadratic Cauchy and Pexider functional equations


        The stability problem for functional equations first was planed in 1940 by Ulam [1]:

        Let G1 be group and G2 be a metric group with the metric d(·,·). Does, for any ε > 0, there exists δ > 0 such that, for any mapping f : G1G2 which satisfies d(f(xy), f(x)f(y)) ≤ δ for all x, yG1, there exists a homomorphism h : G1G2 so that, for any xG1, we have d(f (x), h(x)) ≤ ε?

        In 1941, Hyers [2] answered to the Ulam's question when G1 and G2 are Banach spaces. Subsequently, the result of Hyers was generalized by Aoki [3] for additive mappings and Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias [4] has provided a lot of influences in the development of the Hyers-Ulam-Rassias stability of functional equations (for more details, see [5] where a discussion on definitions of the Hyers-Ulam stability is provided by Moszner, also [612]).

        In this paper, we give a modification of the approach of Belaid et al. [13] in non-Archimedean spaces. Recently, Ciepliński [14] studied and proved stability of multi-additive mappings in non-Archimedean normed spaces, also see [1522].

        Definition 1.1. The function | · | : K → ℝ is called a non-Archimedean valuation or absolute value over the field K if it satisfies following conditions: for any a, bK,
        1. (1)

          |a| ≥ 0;

        2. (2)

          |a| = 0 if and only if a = 0;

        3. (3)

          |ab| = |a| |b|

        4. (4)

          |a + b| ≤ max{|a|, |b|};

        5. (5)

          there exists a member a 0K such that |a 0| ≠ 0, 1.


        A field K with a non-Archimedean valuation is called a non-Archimedean field.

        Corollary 1.2. |-1| = |1| = 1 and so, for any aK, we have |-a| = |a|. Also, if |a| < |b| for any a, bK, then |a + b| = |b|.

        In a non-Archimedean field, the triangle inequality is satisfied and so a metric is defined. But an interesting inequality changes the usual Archimedean sense of the absolute value. For any n ∈ ℕ, we have |n · 1| ≤ ℝ. Thus, for any aK, n ∈ ℕ and nonzero divisor k ∈ ℤ of n, the following inequalities hold:
        Definition 1.3. Let V be a vector space over a non-Archimedean field K. A non-Archimedean norm over V is a function || · || : V → R satisfying the following conditions: for any αK and u, vV,
        1. (1)

          ||u|| = 0 if and only if u = 0;

        2. (2)

          ||αu|| = |α| ||u||;

        3. (3)

          ||u + v|| ≤ max{||u||, ||v||}.


        Since 0 = ||0|| = ||v - v|| ≤ max{||v||, ||-v||} = ||v|| for any vV, we have ||v|| ≥ 0. Any vector space V with a non-Archimedean norm || · || : V → ℝ is called a non-Archimedean space. If the metric d(u, v) = ||u - v|| is induced by a non-Archimedean norm || · || : V → ℝ on a vector space V which is complete, then (V, || · ||) is called a complete non-Archimedean space.

        Proposition 1.4. ([23]) A sequence http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq1_HTML.gif in a non-Archimedean space is a Cauchy sequence if and only if the sequence http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq2_HTML.gif converges to zero.

        Since any non-Archimedean norm satisfies the triangle inequality, any non-Archimedean norm is a continuous function from its domain to real numbers.

        Proposition 1.5. Let V be a normed space and E be a non-Archimedean space. Let f : VE be a function, continuous at 0 ∈ V such that, for any ×V, f(2x) = 2f(x) (for example, additive functions). Then, f = 0.

        Proof. Since f(0) = 0, for any ε > 0, there exists δ > 0 that, for any xV with ||x|| ≤ δ,
        and, for any xV, there exists n ∈ ℕ that http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq3_HTML.gif and hence

        Since this inequality holds for all ε > 0, it follows that, for any xV, f(x) = 0. This completes the proof.

        The preceding fact is a special case of a general result for non-Archimedean spaces, that is, every continuous function from a connected space to a non-Archimedean space is constant. This is a consequence of totally disconnectedness of every non-Archimedean space (see [23]).

        2. Stability of quadratic and Cauchy functional equations

        Throughout this section, we assume that V1 is a normed space and V2 is a complete non-Archimedean space. Let σ : V1V1 be a continuous involution (i.e., σ (x + y) = σ (x) + σ (y) and σ (σ (x)) = x) and φ : V1 × V1 → ℝ be a function with
        and define a function ϕ : V1 × V1 → ℝ by
        which easily implies
        Theorem 2.1. Suppose that φ satisfies the condition 2.1 and let ϕ is defined by Equation 2.2. If f : V1V2satisfies the inequality
        for all x, yV1, then there exists a unique solution q : V1V2of the functional equation
        such that

        for all xV1.

        Proof. Replacing x and y in Equation 2.4 with http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq4_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq5_HTML.gif , respectively, we obtain
        Replacing x and y in Equation 2.4 with http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq5_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq4_HTML.gif , respectively, we obtain
        Also, replacing both of x, y in Equation 2.4 with http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq5_HTML.gif , we get
        and so, for any n ∈ ℕ, we get
        Similarly, replacing both of x, y in Equation 2.4 with http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq4_HTML.gif , we get
        Replacing x in Equation 2.7 with http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq5_HTML.gif , we obtain
        for all xV1 and so, by assumption Equation 2.1,
        Thus, f(0) = 0 and the inequality Equation 2.10 reduces to
        and so,
        For any n ∈ ℕ, define

        for all x, yV1.

        From Equations (2.9) and (2.11), we get

        and so Proposition 1.4 and the hypothesis Equation 2.1 imply that http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq6_HTML.gif is a Cauchy sequence. Since V2 is complete, the sequence http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq7_HTML.gif converges to a point of V2 which defines a mapping q : V1V2.

        Now, we prove
        for all n ∈ ℕ. Since Equation 2.7 implies
        Assume that ||f(x) -q n (x)|| ≤ ϕ n (x, x) holds for some n ∈ ℕ. Then, we have

        Therefore, by induction on n, Equation 2.13 follows from Equation 2.12. Taking the limit of both sides of Equation 2.13, we prove that q satisfies Equation 2.6.

        For any n ∈ ℕ and x, yV1, we have
        and so, by the continuity of non-Archimedean norm and taking the limit of both sides of the above inequality, we get

        Thus, q is a solution of the Equation 2.5 which satisfies Equation 2.6.

        Then, by replacing x, y with http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq5_HTML.gif in Equation 2.5, we obtain the following identities: for any solution g : V1V2 of the Equation (2.5),
        By induction on n, one can show that

        for all n ∈ ℕ.

        Now, suppose that q' : V1V2 is another solution of 2.5 that satisfies the Equation 2.6. It follows from Equations 2.14 to 2.16 that
        Therefore, since

        we have q(x) = q'(x) for all xV1. This completes the proof.

        In the proof of the next theorem, we need a result concerning the Cauchy functional equation

        which has been established in [20].

        Theorem 2.2. ([20]) Suppose that φ(x, y) satisfies the condition 2.1 and, for a mapping f : V1V2,
        for all x, yV1. Then, there exists a unique solution q : V1V2of the Equation 2.17 such that
        for all xV1, where

        for all x, yV1

        3. Stability of the Pexider functional equation

        In this section, we assume that V1 is a normed space and V2 is a complete non-Archimedean space. For any mapping f : V1V2, we define two mappings F e and F o as follows:
        and also define F(x) = f(x) -f(0). Then, we have obviously
        Theorem 3.1. Let σ : V1V1be a continuous involution and the mappings f i : V1V2for i = 1, 2, 3, 4 and δ > 0, satisfy
        for all x, yV1, then there exists a unique solution q : V1V2of the Equation 2.5 and a mapping v : V1V2which satisfies
        for all x, yV1and exists two additive mappings http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq8_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq9_HTML.gif for i= 1, 2 and, for all xV1,
        Proof. It follows from (3.2) that
        and so, for all x, yV1,
        Similarly, we have

        for all x, yV1.

        Now, first by putting y = 0 in Equation 3.7 and applying Equation 3.2 and second by putting x = 0 in Equation 3.7 and applying Equation 3.2 once again, we obtain
        for all x, yV1 and so these inequalities with Equation 3.7 imply
        Replacing y with σ(y) in Equation 3.11, we get
        It follows from Equations 3.1, 3.11 and 3.12 that
        By Theorem 2.1 of [24], there exists a unique solution q : V1V2 of the functional Equation 2.5 such that

        for all xV1.

        As a result of the inequalities Equations 3.11 and 3.12, we have
        It is easily seen that the mapping v : V1V2 defined by
        is a solution of the functional equation

        for all x, yV1.

        Replacing both of x, y in Equation 3.14 with http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq10_HTML.gif , We get
        for all xV1. Now, Equations 3.13 and 3.15 imply
        Similarly, it follows from the inequalities Equations 3.7, 3.10 and 3.13 that
        Since Equation 3.8 implies
        for all x, yV1, we have
        for all xV1. Now, from Equations 3.8 and 3.20, we obtain
        and so, by interchanging role of x, y in the preceding inequality,
        for all x, yV1. Since y + σ (x) = σ (x + σ (y), it follows from Equations 3.1, 3.24 and 3.25 that
        By Theorem 2.2, there exists a unique additive mapping http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq11_HTML.gif such that

        for all xV1, we deduce http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq12_HTML.gif for all xV1.

        By a similar deduction, Equations 3.8 and 3.21 imply that there exists a unique additive mapping http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq13_HTML.gif such that
        Moreover, we have http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq14_HTML.gif for all xV1. Thus, by Equations 3.16, 3.22, 3.27 and 3.28, we obtain

        This proves Equation 3.3. Similarly, one can prove Equations 3.4 to 3.6.



        The authors would like to thank the referee and area editor Professor Ondrĕj Došlý for giving useful suggestions and comments for the improvement of this paper.

        Authors’ Affiliations

        Department of Mathematics, Science and Research Branch, Islamic Azad University (iau)
        Department of Mathematics and Computer Science, Amirkabir University of Technology


        1. Ulam SM: Problems in Modern Mathematics, Chapter IV, Science Editions. Wiley, New York; 1960.
        2. Hyers DH: On the stability of the linear functional equation. Proc Nat Acad Sci USA 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView Article
        3. Aoki T: On the stability of the linear transformation in Banach spaces. J Math Soc Jpn 1950, 2: 64–66. 10.2969/jmsj/00210064View Article
        4. Rassias THM: On the stability of the linear mapping in Banach spaces. Proc Am Math Soc 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1View Article
        5. Moszner Z: On the stability of functional equations. Aequationes Math 2009, 77: 33–88. 10.1007/s00010-008-2945-7MathSciNetView Article
        6. Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ; 2002.
        7. Hyers DH, Isac G, Rassias THM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998.View Article
        8. Jung SM: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor; 2001.
        9. Rassias TM: On the stability of functional equations and a problem of Ulam. Acta Appl Math 2000, 62: 23–130. 10.1023/A:1006499223572MathSciNetView Article
        10. Rassias THM: Functional Equations, Inequalities and Applications. Kluwer Academic Publishers, Dordrecht; 2003.View Article
        11. Ciepliński K: Generalized stability of multi-additive mappings. Appl Math Lett 2010,23(10):1291–1294. 10.1016/j.aml.2010.06.015MathSciNetView Article
        12. Ciepliński K: Stability of the multi-Jensen equation. J Math Anal Appl 2010,363(1):249–254. 10.1016/j.jmaa.2009.08.021MathSciNetView Article
        13. Bouikhalene B, Elqorachi E, Rassias THM: On the Hyers-Ulam stability of approximately Pexider mappings. Math Inequal Appl 2008, 11: 805–818.MathSciNet
        14. Ciepliński K: Stability of multi-additive mappings in non-Archimedean normed spaces. J Math Anal Appl 2011, 373: 376–383. 10.1016/j.jmaa.2010.07.048MathSciNetView Article
        15. Cho YJ, Park C, Saadati R: Functional inequalities in non-Archimedean in Banach spaces. Appl Math Lett 2010, 60: 1994–2002.MathSciNet
        16. Mirmostafaee AK: Stability of quartic mappings in non-Archimedean normed spaces. Kyungpook Math J 2009, 49: 289–297.MathSciNetView Article
        17. Moslehian MS, Sadeghi GH: A Mazur-Ulam theorem in non-Archimedean normed spaces. Nonlinear Anal 2008, 69: 3405–3408. 10.1016/j.na.2007.09.023MathSciNetView Article
        18. Moslehian MS, Sadeghi GH: Stability of two types of cubic functional equations in non-Archimedean spaces. Real Anal Exch 2008, 33: 375–384.MathSciNet
        19. Moslehian MS, Rassias THM: Stability of functional equations in non-archimedean spaces. Appl Anal Discrete Math 2007, 1: 325–334. 10.2298/AADM0702325MMathSciNetView Article
        20. Moslehian MS, Rassias THM: Stability of functional equations in non-Archimedean spaces. Appl Anal Discrete Math 2007, 1: 325–334. 10.2298/AADM0702325MMathSciNetView Article
        21. Najati A, Moradlou F: Hyers-Ulam-Rassias stability of the Apollonius type quadratic mapping in non-Archimedean spaces. Tamsui Oxf J Math Sci 2008, 24: 367–380.MathSciNet
        22. Saadati R, Cho YJ, Vahidi J: The stability of the quartic functional equation in various spaces. Comput Math Appl 2010, 60: 1994–2002. 10.1016/j.camwa.2010.07.034MathSciNetView Article
        23. Schneider P: Non-Archimedean Functional Analysis. Springer, New York; 2002.View Article
        24. Bouikhalene B, Elqorachi E, Rassias THM: On the generalized Hyers-Ulam stability of the quadratic functional equation with a general involution. Nonlinear Funct Anal Appl 2007,12(2):247–262.MathSciNet


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