A pexider difference for a pexider functional equation

  • Abbas Najati1Email author,

    Affiliated with

    • Saeid Ostadbashi2,

      Affiliated with

      • Gwang Hui Kim3 and

        Affiliated with

        • Sooran Mahmoudfakhe2

          Affiliated with

          Advances in Difference Equations20122012:26

          DOI: 10.1186/1687-1847-2012-26

          Received: 27 June 2011

          Accepted: 6 March 2012

          Published: 6 March 2012

          Abstract

          We deal with a Pexider difference

          f ( 2 x + y ) + f ( 2 x - y ) - g ( x + y ) - g ( x - y ) - 2 g ( 2 x ) + 2 g ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equa_HTML.gif

          where f and g map be a given abelian group (G, +) into a sequentially complete Hausdorff topological vector space. We also investigate the Hyers-Ulam stability of the following Pexiderized functional equation

          f ( 2 x + y ) + f ( 2 x - y ) = g ( x + y ) + g ( x - y ) + 2 g ( 2 x ) - 2 g ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equb_HTML.gif

          in topological vector spaces.

          Mathematics subject classification (2000): Primary 39B82; Secondary 34K20, 54A20.

          Keywords

          Hyers-Ulam stability additive mapping quadratic mapping topological vector space

          1. Introduction and preliminaries

          In 1940, Ulam [1] proposed the general stability problem: Let G1 be a group, G2 be a metric group with the metric d. Given ε > 0, does there exists δ > 0 such that if a function h: G1 G2 satisfies the inequality
          d h ( x y ) - h ( x ) h ( y ) < δ , ( x , y G 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equc_HTML.gif
          then there is a homomorphism H: G 1 G 2 with
          d ( h ( x ) , H ( x ) ) < ε , ( x G 1 ) ? http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equd_HTML.gif
          Hyers [2] gave a partial affirmative answer to the question of Ulam in the context of Banach spaces. In 1950, Aoki [3] extended the theorem of Hyers by considering the unbounded cauchy difference inequality
          f ( x + y ) - f ( x ) - f ( y ) ε ( x p + y p ) ( ε > 0 , p ε [ 0 , 1 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Eque_HTML.gif

          In 1978, Rassias [4] also generalized the Hyers' theorem for linear mappings under the assumption tf (tx) is continuous in t for each fixed x.

          Recently, Adam and Czerwik [5] investigated the problem of the Hyers-Ulam stability of a generalized quadratic functional equation in linear topological spaces. Najati and Moghimi [6] investigated the Hyers-Ulam stability of the functional equation
          f ( 2 x + y ) + f ( 2 x - y ) = f ( x + y ) + f ( x - y ) + 2 f ( 2 x ) - 2 f ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equf_HTML.gif
          in quasi-Banach spaces. In this article, we prove that the Pexiderized functional equation
          f ( 2 x + y ) + f ( 2 x - y ) = g ( x + y ) + g ( x - y ) + 2 g ( 2 x ) - 2 g ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equg_HTML.gif

          is stable for functions f, g defined on an abelian group and taking values in a topological vector space.

          Throughout this article, let G be an abelian group and X be a sequentially complete Hausdorff topological vector space over the field ℚ of rational numbers.

          A mapping f: GX is said to be quadratic if and only if it satisfies the following functional equation
          f ( x + y ) + f ( x - y ) = 2 f ( x ) + 2 f ( y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equh_HTML.gif
          for all x yG. A mapping f GX is said to be additive if and only if it satisfies f (x + y) = f (x) + f (y) for all x y ε G. For a given f: GX, we will use the following notation
          D f ( x , y ) : = f ( 2 x + y ) + f ( 2 x - y ) - f ( x + y ) - f ( x - y ) - 2 f ( 2 x ) + 2 f ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equi_HTML.gif
          For given sets A BX and a number k ∈ ℝ, we define the well known operations
          A + B : = { a + b : a A , b B } , k A : = { k a : a A } . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equj_HTML.gif
          We denote the convex hull of a set UX by conv(U) and by U ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq1_HTML.gif the sequential closure of U. Moreover it is well know that:
          1. (1)

            If AX are bounded sets, then conv(A) and A ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq2_HTML.gif are bounded subsets of X.

             
          2. (2)

            If A, BX and α β ∈ ℝ, then α conv(A) + β conv(B) = conv(αA + βB).

             
          3. (3)

            Let X1 and X2 be linear spaces over ℝ. If f: X1 X2 is a additive (quadratic) function, then f (rx) = rf (x) (f (rx) = r2f (x)), for all xX1 and all r ∈ ℚ.

             

          2. Main results

          We start with the following lemma.

          Lemma 2.1. Let G be a 2-divisible abelian group and BX be a nonempty set. If the functions f, g: G X satisfy
          f ( 2 x + y ) + f ( 2 x - y ) - g ( x + y ) - g ( x - y ) - 2 g ( 2 x ) + 2 g ( x ) B http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ1_HTML.gif
          (2.1)
          for all x, yG, then
          D f ( x , y ) 2  conv ( B - B ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ2_HTML.gif
          (2.2)
          D g ( x , y ) conv ( B - B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ3_HTML.gif
          (2.3)

          for all x, yG.

          Proof. Putting y = 0 in (2.1), we get
          2 f ( 2 x ) - 2 g ( 2 x ) B http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ4_HTML.gif
          (2.4)
          for all xG. If we replace x by 1 2 x http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq3_HTML.gif in (2.4), then we have
          f ( x ) - g ( x ) 1 2 B http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ5_HTML.gif
          (2.5)
          for all xG. It follows from (2.5) and (2.1) that
          D f ( x , y ) = f ( 2 x + y ) + f ( 2 x - y ) - g ( x + y ) - g ( x - y ) - 2 g ( 2 x ) + 2 g ( x ) - [ f ( x + y ) - g ( x + y ) ] - [ f ( x - y ) - g ( x - y ) ] - [ 2 f ( 2 x ) - 2 g ( 2 x ) ] + [ 2 f ( x ) - 2 g ( x ) ] 2  conv ( B - B ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equk_HTML.gif
          Moreover, we have
          D g ( x , y ) = f ( 2 x + y ) + f ( 2 x - y ) - g ( x + y ) - g ( x - y ) - 2 g ( 2 x ) + 2 g ( x ) - [ f ( 2 x + y ) - g ( 2 x + y ) ] - [ f ( 2 x - y ) - g ( 2 x - y ) ] conv ( B - B ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equl_HTML.gif
          Theorem 2.2. Let G be a 2-divisible abelian group and BX be a bounded set. Suppose that the odd functions f, g: G X satisfy (2 1) for all x, yG. Then there exists exactly one additive function A : G X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq4_HTML.gif such that
          A ( x ) - f ( x ) 4 conv ( B - B ) ¯ , A ( x ) - g ( x ) 2 conv ( B - B ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ6_HTML.gif
          (2.6)
          for all xG. Moreover the function A http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq5_HTML.gif is given by
          A ( x ) = lim n 1 2 n f ( 2 n x ) = lim n 1 2 n g ( 2 n x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equm_HTML.gif

          for all xG. Moreover, the convergence of the sequences are uniform on G.

          Proof. By Lemma 2.1, we get (2.2). Setting y = x, y = 3x and y = 4x in (2.2), we get
          f ( 3 x ) - 3 f ( 2 x ) + 3 f ( x ) 2  conv ( B - B ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ7_HTML.gif
          (2.7)
          f ( 5 x ) - f ( 4 x ) - f ( 2 x ) + f ( x ) 2 conv ( B - B ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ8_HTML.gif
          (2.8)
          f ( 6 x ) - f ( 5 x ) + f ( 3 x ) - 3 f ( 2 x ) + 2 f ( x ) 2 conv ( B - B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ9_HTML.gif
          (2.9)
          for all xG. It follows from (2.7), (2.8), and (2.9) that
          f ( 6 x ) - f ( 4 x ) - f ( 2 x ) 6 conv ( B - B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equn_HTML.gif
          for all xG. So
          f ( 3 x ) - f ( 2 x ) - f ( x ) 6 conv ( B - B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ10_HTML.gif
          (2.10)
          for all xG. Using (2.7) and (2.10), we obtain
          1 2 f ( 2 x ) - f ( x ) 2 conv ( B - B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equo_HTML.gif
          for all xG. Therefore
          1 2 n f ( 2 n x ) - 1 2 m f ( 2 m x ) = k = m n - 1 1 2 k + 1 f ( 2 k + 1 x ) - 1 2 k f ( 2 k x ) k = m n - 1 2 2 k conv ( B - B ) 4 2 m conv ( B - B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ11_HTML.gif
          (2.11)
          for all xG and all integers n > m ≥ 0. Since B is bounded, we conclude that conv(B - B) is bounded. It follows from (2.11) and boundedness of the set conv(B - B) that the sequence { 1 2 n f ( 2 n x ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq6_HTML.gif is (uniformly) Cauchy in X for all xG. Since X is a sequential complete topological vector space, the sequence { 1 2 n f ( 2 n x ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq6_HTML.gif is convergent for all xG, and the convergence is uniform on G. Define
          A 1 : G X , A 1 ( x ) : = lim n 1 2 n f ( 2 n x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equp_HTML.gif
          Since conv(B - B) is bounded, it follows from (2.2) that
          D A 1 ( x , y ) = lim n 1 2 n D f ( 2 n x , 2 n y ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equq_HTML.gif
          for all x yG. So A 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq7_HTML.gif is additive (see [6]). Letting m = 0 and n ∞ in (2.11), we get
          A 1 ( x ) - f ( x ) 4 conv ( B - B ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ12_HTML.gif
          (2.12)
          for all xG. Similarly as before applying (2.3) we have an additive mapping A 2 : G X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq8_HTML.gif defined by A 2 ( x ) : = lim n 1 2 n g ( 2 n x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq9_HTML.gif which is satisfying
          A 2 ( x ) - g ( x ) 2 conv ( B - B ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ13_HTML.gif
          (2.13)

          for all xG. Since B is bounded, it follows from (2.5) that A 1 = A 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq10_HTML.gif. Letting A : = A 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq11_HTML.gif, we obtain (2.6) from (2.12) and (2.13).

          To prove the uniqueness of A http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq5_HTML.gif, suppose that there exists another additive function A http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq12_HTML.gif: G X satisfying (2.6). So
          A ( x ) - A ( x ) = [ A ( x ) - f ( x ) ] + [ f ( x ) - A ( x ) ] 8 conv ( B - B ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equr_HTML.gif
          for all xG. Since A http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq12_HTML.gif and A http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq5_HTML.gif are additive, replacing x by 2 n x implies that
          A ( x ) - A ( x ) 8 2 n conv ( B - B ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equs_HTML.gif

          for all xG and all integers n. Since conv ( B - B ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq13_HTML.gif is bounded, we infer A = A http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq14_HTML.gif. This completes the proof of theorem.

          Theorem 2.3 Let G be a 2, 3-divisible abelian group and BX be a bounded set. Suppose that the even functions f, g: G X satisfy (2 1) for all x, yG. Then there exists exactly one quadratic function Q : G X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq15_HTML.gifsuch that
          Q ( x ) - f ( x ) + f ( 0 ) 4 conv ( B - B ) ¯ , Q ( x ) - g ( x ) + g ( 0 ) 2 conv ( B - B ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equt_HTML.gif
          for all xG. Moreover, the function Q http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq16_HTML.gif is given by
          Q ( x ) = lim n 1 4 n f ( 2 n x ) = lim n 1 4 n g ( 2 n x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equu_HTML.gif

          for all xG. Moreover, the convergence of the sequences are uniform on G.

          Proof. By replacing y by x + y in (2.2), we get
          f ( 3 x + y ) + f ( x - y ) - f ( 2 x + y ) - f ( y ) - 2 f ( 2 x ) + 2 f ( x ) 2 c o n v ( B - B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ14_HTML.gif
          (2.14)
          for all x, yG. Replacing y by - y in (2.14), we get
          f ( 3 x - y ) + f ( x + y ) - f ( 2 x - y ) - f ( y ) - 2 f ( 2 x ) + 2 f ( x ) 2 conv ( B - B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ15_HTML.gif
          (2.15)
          for all x, yG. It follows from (2.2), (2.14), and (2.15) that
          f ( 3 x + y ) + f ( 3 x - y ) - 2 f ( y ) - 6 f ( 2 x ) + 6 f ( x ) 6 conv ( B - B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ16_HTML.gif
          (2.16)
          for all x, yG. By letting y = 0 and y = 3x in (2.16), we get
          2 f ( 3 x ) - 6 f ( 2 x ) + 6 f ( x ) - 2 f ( 0 ) 6 conv ( B - B ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ17_HTML.gif
          (2.17)
          f ( 6 x ) - 2 f ( 3 x ) - 6 f ( 2 x ) + 6 f ( x ) + f ( 0 ) 6 conv ( B - B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ18_HTML.gif
          (2.18)
          for all xG. Using (2.17) and (2.18), we obtain
          f ( 6 x ) - 4 f ( 3 x ) + 3 f ( 0 ) 12 conv ( B - B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ19_HTML.gif
          (2.19)
          for all xG. If we replace x by 1 3 x http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq17_HTML.gif in (2.19), then
          f ( 2 x ) - 4 f ( x ) + 3 f ( 0 ) 12 conv ( B - B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equv_HTML.gif
          for all xG. Therefore
          1 4 n + 1 f ( 2 n + 1 x ) - 1 4 n f ( 2 n x ) + 3 4 n + 1 f ( 0 ) 3 4 n conv ( B - B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ20_HTML.gif
          (2.20)
          for all xG and all integers n. So
          1 4 n f ( 2 n x ) - 1 4 m f ( 2 m x ) = k = m n - 1 1 4 k + 1 f ( 2 k + 1 x ) - 1 4 k f ( 2 k x ) - k = m n - 1 3 4 k + 1 f ( 0 ) + k = m n - 1 3 4 k conv ( B - B ) - k = m n - 1 3 4 k + 1 f ( 0 ) + 1 4 m - 1 conv ( B - B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ21_HTML.gif
          (2.21)

          for all xG and all integers n >m ≥ 0. It follows from (2.21) and boundedness of the set conv(B - B) that the sequence { 1 4 n f ( 2 n x ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq18_HTML.gif is (uniformly) Cauchy in X for all xG. The rest of the proof is similar to proof of of Theorem 2.2.

          Remark 2.4. If the functions f, g: G X satisfy (2.1), where f is even (odd) and g is odd (even), then it is easy to show that f and g are bounded.

          Declarations

          Authors’ Affiliations

          (1)
          Department of Mathematical Sciences, University of Mohaghegh Ardabili
          (2)
          Department of Mathematics, Urmia University
          (3)
          Department of Applied Mathematics, Kangnam University

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          Copyright

          © Najati et al; licensee Springer. 2012

          This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.