A Characterization of Multi-Mixed Additive-Quadratic Mappings and a Fixed Point Application
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aracterization of Multi-Mixed Additive-Quadratic Mappings and a Fixed Point Application S. Falihi1* , A. Bodaghi2** , and B. Shojaee1*** 1 2
Karaj Branch, Islamic Azad University, Karaj, Iran
Garmsar Branch, Islamic Azad University, Garmsar, Iran
Received March 30, 2019; revised September 19, 2019; accepted October 11, 2019
Abstract—In this paper, we introduce n-variables mappings which are mixed additive-quadratic in each variable. We show that such mappings can be described by a equation, namely, by a multi-mixed additive-quadratic functional equation. The main goal is to extend the applications of a fixed point method to establish the Hyers-Ulam stability for the multi-mixed additive-quadratic mappings. MSC2010 numbers : 39B82, 39B52, 47H10. DOI: 10.3103/S1068362320040044 Keywords: Banach space; generalized Hyers-Ulam stability; multi-mixed additive-quadratic mapping; fixed point method
1. INTRODUCTION Throughout the paper we use the following notation. By N we denote the set of all positive integers, N0 := N ∪ {0}, R+ := [0, ∞), and n ∈ N. For any l ∈ N0 , m, s ∈ N with s ≥ 3, t = (t1 , · · · , tm ) ∈ {−s, −1, 1, s}m and x = (x1 , · · · , xm ) ∈ V m we write lx := (lx1 , · · · , lxm ) and tx := (t1 x1 , · · · , tm xm ), where ra stands for the rth power of an element a of the commutative group V . The additive (Cauchy) equation A(x + y) = A(x) + A(y) and the quadratic (Jordan-von Neumann) equation Q(x + y) + Q(x − y) = 2Q(x) + 2Q(y) are the well-known equations in mathematics which play a remarkable role in the algebra and analysis. Some information about the solutions, stability and applications of these equations can be found in [26] and [36]. Let V be a commutative group, W be a linear space, and n ≥ 2 be an integer. Recall that a mapping f : V n −→ W is called multi-additive if it is additive (that is, satisfies Cauchy functional equation) in each variable (see [22]). Some facts on such mappings can be found, for instance, in [28]. Furthermore, f is said to be multi-quadratic if it is quadratic in each variable (see [21]). In Zhao et al. [40] it was proved that a mapping f : V n −→ W is multi-quadratic if and only if the following relation holds: f (x1 + tx2 ) = 2n f (x1j1 , x2j2 , · · · , xnjn ), (1.1) j1 ,j2 ,··· ,jn ∈{1,2}
t∈{−1,1}n
where xj = (x1j , x2j , · · · , xnj ) ∈ V with j ∈ {1, 2}. In 1940, Ulam [37] raised the first stability problem for functional equations. Specifically, he proposed a question whether there exists an exact homomorphism near an approximate homomorphism. One year later, an answer to this problem was given by Hyers [25] in the setting of Banach spaces. Later on, the stability problems have been extensively investigated for a variety of functional equations and spaces. For instance, various generalizations and extensions of Ulam’s problem and Hyers’ result were ascertained n
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E-mail: [email protected] E-mail: [email protected] *** E-mail: [email protected] **
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by Th.M. Rassias [34], Gajda [24], Aoki [1], J.M. Rassias [33] and Skof [
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