Hyers-Ulam Rassias stability of Pexiderized Cauchy functional equation in 2-Banach spaces

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1 Available online at J. Nonlinear Sci. Appl. 5 (202), Research Article Hyers-Ulam Rassias stability of Pexiderized Cauchy functional equation in 2-Banach spaces G. Zamani Eskandani a,, P. Gavruta b a Faculty of Mathematical Science University of Tabriz, Tabriz, Iran. b Department of Mathematics, University Politehnica of Timisoara, Piata Victoriei No. 2, Timisoara, Romania. Dedicated to George A Anastassiou on the occasion of his sixtieth birthday Communicated by Professor R. Saadati Abstract In this paper, we investigate stability of the Pexiderized Cauchy functional equation in 2-Banach spaces and pose an open problem. copyright 202 NGA. All rights reserved. Keywords: Linear 2-normed space, Generalized Hyers-Ulam stability, Pexiderized Cauchy functional equation. 200 MSC: Primary 46BXX, 39B72, 47Jxx. Introduction and preliminaries In 940, S.M. Ulam [34] asked the first question on the stability problem for mappings. In 94, D. H. Hyers [4] solved the problem of Ulam. This result was generalized by Aoki [] for additive mappings and by Th. M. Rassias [24] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias has provided a lot of influence in the development of what we now call Hyers-Ulam- Rassias stability of functional equations. In 994, a further generalization was obtained by P. Găvruta [3]. During the last two decades, a number of papers and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappings (see [5]-[2],[6]-[20],[22]-[23], [25]-[29], [32],[33]). We also refer the readers to the books: P. Czerwik [4] and D.H. Hyers, G. Isac and Th.M. Rassias [5]. In the 960s, S. Gahler [8, 9] introduced the concept of linear 2-normed spaces. Corresponding author addresses: zamani@tabrizu.ac.ir (G. Zamani Eskandani ), pgavruta@yahoo.com (P. Gavruta) Received

2 G.Z. Eskandani, P. Gavruta, J. Nonlinear Sci. Appl. 5 (202), Definition.. Let X be a linear space over R with dimx > and let.,. : X X R be a function satisfying the following properties: (a) x, y = 0 if and only if x and y are linearly dependent, (b) x, y = y, x, (c) λx, y = λ x, y, (d) x, y + z x, y + x, z for all x, y, z X and λ R. Then the function.,. is called a 2-norm on X and the pair (X,.,. ) is called a linear 2-normed space. Sometimes the condition (d) called the triangle inequality. It follows from (d), that x+y, z x, z + y, z and x, z y, z x y, z. Hence the functions x x, y are continuous functions of X into R for each fixed y X. Lemma.2. ([2]) Let (X,.,. ) be a linear 2-normed space. If x X and x, y = 0 for all y X, then x = 0. Definition.3. A sequence {x n } in a linear 2-normed space X is called a Cauchy sequence if there are two points y, z X such that y and z are linearly independent, lim x m x n, y = 0 and lim x m x n, z = 0. m, m, Definition.4. A sequence {x n } in a linear 2-normed space X is called a convergent sequence if there is an x X such that lim x n x, y = 0, for all y X. If {x n } converges to x, write x n x as n and call x the limit of {x n }. In this case, we also write lim x n = x. Definition.5. A linear 2-normed space in which every Cauchy sequence is a convergent sequence is called a 2-Banach space. Lemma.6. ([2]) For a convergent sequence {x n } in a linear 2-normed space X, for all y X. lim x n, y = lim x n, y, In [2] Won-Gil Park has investigated approximate additive mappings, approximate Jensen mappings and approximate quadratic mappings in 2-Banach spaces. In this paper, we investigate stability of the Pexiderized Cauchy functional equation in 2-Banach spaces and pose an open problem. 2. Stability of the Pexiderized Cauchy functional equation Throughout this paper, let X be a normed linear space, Y be a 2-Banach space with dimy > and k is a fixed integer greater than. Theorem 2.. Let ϕ : X X X [0, + ) be a function such that lim k n ϕ(kn x, k n y, z) = 0, (2.) for all x, y, z X. Suppose that f, g, h : X Y be mappings with f(0) = g(0) = h(0) = 0 and f(x + y) g(x) h(y), z ϕ(x, y, z), (2.2)

3 G.Z. Eskandani, P. Gavruta, J. Nonlinear Sci. Appl. 5 (202), and M k (x, z) := k n=0 i= M(k n x, ik n x, z) k n, (2.3) exist for all x, y, z X, where M(x, y, z) := ϕ(x, y, z)+ϕ(0, y, z)+ϕ(x, 0, z). Then there is a unique additive mapping A k : X Y such that for all x, z X. Proof. Replacing y = 0 in (2.2), we get for all x, z X. Replacing x = 0 in (2.2), we get for all y, z X. By using (2.5) and (2.6), we get for all x, y, z X, where By induction on k, we show that f(x) A k (x), z k M k (x, z) (2.4) f(x) g(x), z ϕ(x, 0, z), (2.5) f(y) h(y), z ϕ(0, y, z), (2.6) f(x + y) f(x) f(y), z M(x, y, z), (2.7) M(x, y, z) := ϕ(x, y, z) + ϕ(x, 0, z) + ϕ(0, y, z). f(kx) kf(x), z M k (x, z), (2.8) k for all x, z X, where M k (x, z) := M(x, ix, z). Letting y = x in (2.7), we get for all x, z X. So we get (2.8) for k = 2. Assume that (2.8) holds for k. Letting y = kx in (2.7), we get i= for all x, z X. It follows from (2.8) and (2.0) that f(2x) 2f(x), z M(x, x, z), (2.9) f ( (k + )x ) f(x) f(kx), z M(x, kx, z) (2.0) f ( (k + )x ) (k + )f(x), z f ( (k + )x ) f(x) f(kx), z + f(kx) kf(x), z M k+ (x, z), This completes the induction argument. Replacing x by k n x in (2.8) and dividing both sides of (2.8) by k n+, we get k n+ f(kn+ x) k n f(kn x), z k n+ M k(k n x, z), (2.) for all x, y X and all non-negative integers n. Hence k n+ f(kn+ x) n k m f(km x), z k i+ f(ki+ x) k i f(ki x), z i=m n k k i M k(k i x, z), i=m (2.2)

4 G.Z. Eskandani, P. Gavruta, J. Nonlinear Sci. Appl. 5 (202), for all x, z X and all non-negative integers m and n with n m. Therefore, we conclude from (2.3) and (2.2) that the sequence { k n f(kn x)} is a Cauchy sequence in Y for all x X. Since Y is complete the sequence { k n f(kn x)} converges in Y for all x X. So one can define the mapping A k : X Y by: for all x X. That is A k (x) := lim k n f(kn x), (2.3) lim k n f(kn x) A k (x), y = 0, for all x, y X. Letting m = 0 and passing the limit n in (2.2), we get (2.4). Now, we show that A k : X Y is an additive mapping. It follows from (2.), (2.7), (2.3) and Lemma.6 that A k (x + y) A k (x) A k (y), z = lim lim k n f(kn x + k n y) f(k n x) f(k n y), z k n M(kn x, k n y, z) = 0, for all x, y, z X. By Lemma.2, A k (x + y) A k (x) A k (y) = 0 for all x, y X. So the mapping A k : X Y is additive. To prove the uniqueness of A k, let T : X Y be another additive mapping satisfying (2.4). Then A k (x) T (x), z = lim lim Ak k n (k n x) T (k n x), z k M n+ k (k n x, z) = 0, for all x, z X. By Lemma.2, A k (x) f(x) = 0 for all x, y X. So A k = T. Theorem 2.2. Let ψ : [0, ) [0, ) be a function such that ψ(0) = 0 and. ψ(ts) ψ(t)ψ(s), 2. ψ(t) < t for all t >. Suppose that f, g, h : X Y be mappings with f(0) = g(0) = h(0) = 0 and f(x + y) g(x) h(y), z ψ( x X ) + ψ( y X ) + ψ( z X ), (2.4) for all x, y, z X. Then there is a unique additive mapping A k : X Y satisfying f(x) A k (x), z ( + ψ(i)) k 2 i= k ψ(k) ψ( x X ) + 3ψ( z X ), (2.5) for all x, z X. Moreover, A k = A 2 for all k 2. Proof. Let ϕ(x, y, z) = ψ( x X ) + ψ( y X ) + ψ( z X ), for all x, y, z X. It follows from () that ψ(k n ) ( ψ(k) ) n and ϕ(k n x, k n y, z) (ψ(k) ) n( ψ( x X ) + ψ( y X ) ) + ψ( z X )

5 G.Z. Eskandani, P. Gavruta, J. Nonlinear Sci. Appl. 5 (202), By using Theorem 2., we can get (2.5). Now, we show that A k = A 2. It follows from Theorem 2. that A k (x) = lim k n f(k n x). Replacing x by 2 n x in (2.5) and dividing both sides of (2.5) by 2 n, we get f(2n x) 2 n A k (x), z ( + ψ(i)) k 2 i= k 2 k ψ(k) ( + ψ(i)) i= k ψ(k) ψ( 2 n x X ) 2 n + 3ψ( z X) 2 n for all x, z X. By passing the limit n in (2.6), we get A k = A 2. ψ( x X ) ψ(2n ) 2 n + 3ψ( z X) 2 n (2.6) Theorem 2.3. Let p, q be non-negative real numbers such that p > 0, q < and H : [0, ) [0, ) [0, ) be a homogeneous function of degree q. Suppose that f, g, h : X Y be mappings with f(0) = g(0) = h(0) = 0 and f(x + y) g(x) h(y), z H( x X, y X ) + z p X, for all x, y, z X. Then thereis a unique additive mapping A k : X Y such that f(x) A k (x), z σ k(h) k k q x q X + 3 z p X (2.7) k for all x X, where σ k (H) := (k )H(, 0) + H(, i) + H(0, i). Moreover, A k = A 2 for all k 2. Proof. Let i= ϕ(x, y, z) = H( x X, y X ) + z p X for all x, y, z X. By using Theorem 2., we can get (2.7). Now, we show that A k = A 2. It follows from Theorem 2. that A k (x) = lim k f(k n x). Replacing x by 2 n x in (2.7) and dividing both sides of (2.7) n by 2 n, we get f(2n x) 2 n A k (x), z σ k(h) 2 n x q X k k q 2 n + 3 z p X (2.8) 2 n for all x X. By passing the limit n in (2.8), we get x) lim f(2n 2 n A k (x), z = 0, lim f(2 n x) 2 n A k (x), z = 0, for all x, z X. By Lemma.2, lim f(2 n x) 2 n A k (x) = 0, so A k = A 2. Theorem 2.4. Let p, q be non-negative real numbers such that p > 0, q < and H : [0, ) [0, ) [0, ) be a homogeneous function of degree q. Suppose that f, g, h : X Y be mappings with f(0) = g(0) = h(0) = 0 and f(x + y) g(x) h(y), z H( x X, y X ) z p X, for all x, y, z X. Then there is a unique additive mapping A k : X Y such that f(x) A k (x), z σ k(h) k k q x q X z p X (2.9) k for all x, z X, where σ k (H) := (k )H(, 0) + H(, i) + H(0, i). Moreover, A k = A 2 for all k 2. i=

6 G.Z. Eskandani, P. Gavruta, J. Nonlinear Sci. Appl. 5 (202), Corollary 2.5. Let p be real number such that 0 < p <. Suppose that f, g, h : X Y be mappings with f(0) = g(0) = h(0) = 0 and f(x + y) g(x) h(y), z x p X + y p X + z p X, for all x, y, z X. Then there is a unique additive mapping A k : X Y such that f(x) A k (x), z 2(k ) + 2(p + 2 p (k ) p ) k k p x p X + 3 z p X for all x, z X, Moreover, A k = A 2 for all k 2. Corollary 2.6. Let r, s, p be real numbers such that p > 0, r + s <. Suppose that f, g, h : X Y be mappings with f(0) = g(0) = h(0) = 0 and f(x + y) g(x) h(y), z x r X y s X z p X for all x, y, z X. Then there is a unique additive mapping A k : X Y such that f(x) A k (x), z s + 2 s (k ) s k k r+s for all x, z X, Moreover, A k = A 2 for all k 2 and for all x X. f(x) = g(x) = h(x), x r+s X z p X Open problem: What is the best possible value of k in Corollaries 2.5 and 2.6? References [] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (950) [2] J. Aczél and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, 989. [3] Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, vol., Colloq. Publ., vol. 48, Amer. Math. Soc., Providence, RI, [4] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, New Jersey, London, Singapore, Hong Kong, [5] G.Z. Eskandani, On the Hyers-Ulam-Rassias stability of an additive functional equation in quasi-banach spaces, J. Math. Anal. Appl. 345 (2008) [6] G. Z. Eskandani, P. Gavruta, J. M. Rassias, and R. Zarghami, Generalized Hyers-Ulam stability for a general mixed functional equation in quasi-β normed spaces,. Mediterr. J. Math. 8 (20), [7] G.Z. Eskandani, H. Vaezi and Y.N. Dehghan, Stability of mixed additive and quadratic functional equation in non-archimedean Banach modules, Taiwanese J. Math. 4 (200) [8] S. Ghler, 2-metrische Rume und ihre topologische Struktur, Math. Nachr. 26 (963) [9] S. Ghler, Lineare 2-normierte Rumen, Math. Nachr. 28 (964) -43. [0] L. Găvruta, P. Găvruta and G.Z.Eskandani, Hyers Ulam stability of frames in Hilbert spaces, Bul. Stiint. Univ. Politeh. Timis. Ser. Mat. Fiz. 55(69), 2 (200) [] P. Găvruta, On a problem of G. Isac and Th. M. Rassias concerning the stability of mappings, J. Math. Anal. Appl. 26 (200), [2] P. Găvruta, An answer to question of John M. Rassias concerning the stability of Cauchy equation, Advanced in Equation and Inequality (999) [3] P. Găvruta, A generalization of the Hyers Ulam Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 84 (994) [4] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. 27 (94), [5] D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Basel, 998. [6] S. M. Jung, Hyers Ulam Rassias Stability of Functional Equations in Mathimatical Analysis, Hadronic Press, Palm Harbor, 200. [7] S. M. Jung, Asymptotic properties of isometries, J. Math. Anal. Appl. 276 (2002)

7 G.Z. Eskandani, P. Gavruta, J. Nonlinear Sci. Appl. 5 (202), [8] Pl. Kannappan, Quadratic functional equation and inner product spaces, Results Math. 27 (995), [9] A. Najati and G.Z. Eskandani, Stability of a mixed additive and cubic functional equation in quasi Banach spaces, J. Math. Anal. Appl. 342 (2008) [20] C. Park, Homomorphisms between Poisson JC -algebras, Bull. Braz. Math. Soc. 36 (2005), [2] W.G. Park, Approximate additive mappings in 2-Banach spaces and related topics, J. Math. Anal. Appl. 376 (20) [22] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, Journal of Functional Analysis, vol. 46, no., pp , 982. [23] J. M. Rassias, On approximation of approximately linear mappings by linear mappings, Bulletin des Sciences Mathematiques, vol. 08, no. 4, pp , 984. [24] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (978), [25] Th.M. Rassias, On a modified Hyers Ulam sequence, J. Math. Anal. Appl. 58 (99) [26] Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 25 (2000) [27] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 () (2000) [28] Th.M. Rassias and P. Semrl, On the behaviour of mappings which do not satisfy HyersUlam stability, Proc. Amer. Math. Soc. 4 (992) [29] Th.M. Rassias and J. Tabor, What is left of Hyers Ulam stability?, J. Natural Geometry, (992), [30] K. Ravi, M. Arunkumar and J. M. Rassias, Ulam stability for the orthogonally general Euler-Lagrange type functional equation, Intern. J. Math. Stat. 3 (A08)(2008), [3] S. Rolewicz, Metric Linear Spaces, PWN-Polish Sci. Publ., Warszawa, Reidel, Dordrecht, 984. [32] R. Saadati, Y.J. Cho and J. Vahidi, The stability of the quartic functional equation in various spaces, Comput. Math. Appl. (200) doi:0.06/j.camwa [33] R. Saadati, S.M. Vaezpour, Y. Cho, A note on the On the stability of cubic mappings and quadratic mappings in random normed spaces, J. Inequal. Appl. (2009). Article ID [34] S.M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, (960),