On the Stability of a Parametric Additive Functional Equation in Quasi-Banach Spaces

Abstract

We investigate the generalized Hyers-Ulam stability of the following functional equation ∑𝑚𝑖=1𝑓(𝑥𝑖)=1/2𝑚[∑𝑚𝑖=1𝑓(𝑚𝑥𝑖+∑𝑚𝑗=1,𝑗≠𝑖𝑥𝑗)+𝑓(∑𝑚𝑖=1𝑥𝑖)] for a fixed positive integer 𝑚 with 𝑚≥2 in quasi-Banach spaces.

1. Introduction

It is of interest to consider the concept of stability for a functional equation arising when we replace the functional equation by an inequality which acts as a perturbation of the equation.

The first stability problem was raised by Ulam [1] during his talk at the University of Wisconsin in 1940. The stability question of functional equations is that how do the solutions of the inequality differ from those of the given functional equation? If the answer is affirmative, we would say that the equation is stable.

In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. Let 𝑓∶𝐸→𝐸′ be a mapping between Banach spaces such that ‖𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)‖≤𝛿,(1.1) for all 𝑥, 𝑦∈𝐸, and for some 𝛿>0. Then there exists a unique additive mapping 𝑇∶𝐸→𝐸′ such that ‖𝑓(𝑥)−𝑇(𝑥)‖≤𝛿,(1.2) for all 𝑥∈𝐸. Moreover, if 𝑓(𝑡𝑥) is continuous in 𝑡∈ℝ for each fixed 𝑥∈𝐸, then 𝑇 is linear. Aoki [3], Bourgin [4] considered the stability problem with unbounded Cauchy differences. In 1978, Rassias [5] provided a generalization of Hyers’ theorem by proving the existence of unique linear mappings near approximate additive mappings. It was shown by Gajda [6], as well as by Rassias and Šemrl [7] that one cannot prove a stability theorem of the additive equation for a specific function. Găvruţa [8] obtained generalized result of Rassias’ theorem which allows the Cauchy difference to be controlled by a general unbounded function. Isac and Rassias [9] generalized the Hyers’ theorem by introducing a mapping 𝜓∶ℝ+→ℝ+ subject to the conditions:(1)lim𝑡→∞𝜓(𝑡)/𝑡=0, (2)𝜓(𝑡𝑠)≤𝜓(𝑡)𝜓(𝑠); 𝑠,𝑡>0,(3)𝜓(𝑡)<𝑡; 𝑡>1.

These stability results can be applied in stochastic analysis [10], financial, and actuarial mathematics, as well as in psychology and sociology.

In 1987 Gajda and Ger [11] showed that one can get analogous stability results for subadditive multifunctions. In 1978 Gruber [12] remarked that Ulam’s problem is of particular interest in probability theory and in the case of functional equations of different types. We refer the readers to [2, 5–32] and references therein for more detailed results on the stability problems of various functional equations.

We recall some basic facts concerning quasi-Banach space. A quasi-norm is a real-valued function on 𝑋 satisfying the following.(1)‖𝑥‖≥0 for all 𝑥∈𝑋 and ‖𝑥‖=0 if and only if 𝑥=0.(2)‖𝜆⋅𝑥‖=|𝜆|⋅‖𝑥‖ for all 𝜆∈ℝ and all 𝑥∈𝑋.(3) There is a constant 𝐾≥1 such that ‖𝑥+𝑦‖≤𝐾(‖𝑥‖+‖𝑦‖) for all 𝑥, 𝑦∈𝑋. The pair (𝑋,‖⋅‖) is called a quasi-normed space if ‖⋅‖ is a quasi-norm on 𝑋. A quasi-Banach space is a complete quasi-normed space. A quasi-norm ‖⋅‖ is called a 𝑝-norm (0<𝑝≤1) if‖𝑥+𝑦‖𝑝≤‖𝑥‖𝑝+‖𝑦‖𝑝,(1.3) for all 𝑥, 𝑦∈𝑋. In this case, a quasi-Banach space is called a 𝑝-Banach space. Given a 𝑝-norm, the formula 𝑑(𝑥,𝑦)∶=‖𝑥−𝑦‖𝑝 gives us a translation invariant metric on 𝑋. By the Aoki-Rolewicz theorem (see [33]), each quasi-norm is equivalent to some 𝑝-norm. Since it is much easier to work with 𝑝-norms, henceforth we restrict our attention mainly to 𝑝-norms. In this paper, we consider the generalized Hyers-Ulam stability of the following functional equation:𝑚∑𝑖=1𝑓(𝑚𝑥𝑖+𝑚∑𝑗=1,𝑗≠𝑖𝑥𝑗)+𝑓(𝑚∑𝑖=1𝑥𝑖)=2𝑚𝑚∑𝑖=1𝑓(𝑥𝑖),(1.4) for a fixed positive integer 𝑚 with 𝑚≥2 in quasi-Banach spaces.

Throughout this paper, assume that 𝑋 is a quasi-normed space with quasi-norm ‖⋅‖𝑋 and that 𝑌 is a 𝑝-Banach space with 𝑝-norm ‖⋅‖𝑌.

2. Stability of Functional Equation (1.4) in Quasi-Banach Spaces

For simplicity, we use the following abbreviation for a given mapping 𝑓∶𝑋→𝑌:𝐷𝑓(𝑥1,𝑥2,…,𝑥𝑚)=𝑚∑𝑖=1𝑓(𝑚𝑥𝑖+𝑚∑𝑗=1,𝑗≠𝑖𝑥𝑗)+𝑓(𝑚∑𝑖=1𝑥𝑖)−2𝑚𝑚∑𝑖=1𝑓(𝑥𝑖),(2.1) for all 𝑥𝑗∈𝑌(1≤𝑗≤𝑚).

We start our work with the following theorem which can be regard as a general solution of functional equation (1.4).

Theorem 2.1. Let 𝑉 and 𝑊 be real vector spaces. A mapping 𝑓∶𝑉→𝑊 satisfies in (1.4) if and only if 𝑓 is additive.

Proof. Setting 𝑥𝑗=0 in (1.4) (1≤𝑗≤𝑚), we obtain (𝑚+1)𝑓(0)=2𝑚2𝑓(0).(2.2) Since 𝑚≥2, we have 𝑓(0)=0.(2.3) Setting 𝑥1=𝑥,𝑥𝑗=0(2≤𝑗≤𝑚) in (1.4), we obtain 𝑓(𝑚𝑥)=𝑚𝑓(𝑥).(2.4) Putting 𝑥1=𝑥,𝑥2=𝑦,𝑥𝑗=0(3≤𝑗≤𝑚), we get 𝑓(𝑚𝑥+𝑦)+𝑓(𝑚𝑦+𝑥)+(𝑚−1)𝑓(𝑥+𝑦)=2𝑚(𝑓(𝑥)+𝑓(𝑦)).(2.5) Putting 𝑥1=𝑥,𝑥𝑗=𝑦/(𝑚−1)(2≤𝑗≤𝑚), we get 𝑓(𝑚𝑥+𝑦)+(𝑚−1)𝑓(2𝑦+𝑥)+𝑓(𝑥+𝑦)=2𝑚(𝑓(𝑥)+(𝑚−1)𝑓(𝑦𝑚−1)).(2.6) Let 𝑥=0 in (2.6), we obtain 2𝑚(𝑚−1)𝑓(𝑦𝑚−1)=2𝑓(𝑦)+(𝑚−1)𝑓(2𝑦).(2.7) So, (2.6) turns to the following: 𝑓(𝑚𝑥+𝑦)−(𝑚−1)𝑓(2𝑦+𝑥)+(𝑚−2)𝑓(𝑥+𝑦)=(2𝑚−2)𝑓(𝑦)−(𝑚−1)𝑓(2𝑦).(2.8) From (2.5) and (2.8), we have 𝑓(𝑚𝑥+𝑦)+(𝑚−1)𝑓(2𝑦+𝑥)+𝑓(𝑥+𝑦)=2𝑚𝑓(𝑥)+2𝑓(𝑦)+(𝑚−1)𝑓(2𝑦).(2.9) Replacing 𝑥 by 𝑦 and 𝑦 by 𝑥 in (2.8) and comparing it with (2.9), we get (𝑚−1)[𝑓(2𝑥+𝑦)+𝑓(2𝑦+𝑥)]−(𝑚−3)𝑓(𝑥+𝑦)=2[𝑓(𝑥)+𝑓(𝑦)]+(𝑚−1)[𝑓(2𝑥)+𝑓(2𝑦)].(2.10) Letting 𝑥=𝑦 in (2.5), (2.8), and (2.10), respectively, we obtain 2𝑓((𝑚+1)𝑥)+(𝑚−1)𝑓(2𝑥)=4𝑚𝑓(𝑥),𝑓((𝑚+1)𝑥)+(𝑚−1)𝑓(3𝑥)=(2𝑚+2)𝑓(𝑥)+(𝑚−2)𝑓(2𝑥),𝑓(3𝑥)=𝑓(2𝑥)+𝑓(𝑥).(2.11) From (2.11) we have 𝑓(2𝑥)=2𝑓(𝑥).(2.12) Replacing 𝑓(2𝑥) and 𝑓(2𝑦) by their equivalents by using (2.12) in (2.10), we get (𝑚−1)[𝑓(2𝑥+𝑦)+𝑓(2𝑦+𝑥)]−(𝑚−3)𝑓(𝑥+𝑦)=2𝑚𝑓(𝑥+𝑦).(2.13) Replacing 𝑦 by −𝑥 in (2.13), we get 𝑓(𝑥)=−𝑓(𝑥).(2.14) Replacing 𝑥 by 𝑥−𝑦 in (2.13), we get (𝑚−1)[𝑓(2𝑥−𝑦)+𝑓(𝑥+𝑦)]−(𝑚−3)𝑓(𝑥)=2𝑚(𝑓(𝑥−𝑦)+𝑓(𝑦)).(2.15) Similarly, replacing 𝑦 by 𝑦−𝑥 in (2.13), we obtain (𝑚−1)[𝑓(2𝑦−𝑥)+𝑓(𝑥+𝑦)]−(𝑚−3)𝑓(𝑦)=2𝑚(𝑓(𝑥)+𝑓(𝑦−𝑥)).(2.16) Replacing 𝑦 by −𝑦 and 𝑥 by −𝑥 in (2.15) and (2.16), respectively, we obtain (𝑚−1)[𝑓(2𝑥+𝑦)+𝑓(𝑥−𝑦)]−(𝑚−3)𝑓(𝑥)=2𝑚(𝑓(𝑥+𝑦)+𝑓(−𝑦)),(𝑚−1)[𝑓(2𝑦+𝑥)+𝑓(𝑦−𝑥)]−(𝑚−3)𝑓(𝑦)=2𝑚(𝑓(𝑥+𝑦)+𝑓(−𝑥)).(2.17) Adding both sides of (2.17) and using (2.14), we get (𝑚−1)[𝑓(2𝑥+𝑦)+𝑓(2𝑦+𝑥)]+(𝑚+3)[𝑓(𝑥)+𝑓(𝑦)]=4𝑚𝑓(𝑥+𝑦).(2.18) Comparing (2.18) and (2.13), we obtain 𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦),(2.19) for all 𝑥,𝑦∈𝑉. So, if a mapping 𝑓 satisfying (1.4) it must be additive. Conversely, let 𝑓∶𝑉→𝑊 be additive, it is clear that 𝑓 satisfying (1.4), and the proof is complete.

Now, we investigate the generalized Hyers-Ulam stability of functional equation (1.4) in quasi-Banach spaces.

Theorem 2.2. Let 𝜙∶𝑋×⋯×𝑋⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝑚-times→[0,∞) be a function satisfying Φ(𝑥)=∞∑𝑖=1(1𝑚)𝑖𝑝(𝜙(𝑚𝑖−1𝑥,0,…,0))𝑝<∞,(2.20) for all 𝑥∈𝑋, and lim𝑛→∞1𝑚𝑛𝜙(𝑚𝑛𝑥1,…,𝑚𝑛𝑥𝑚)=0,(2.21) for all 𝑥𝑗∈𝑋(1≤𝑗≤𝑚). Suppose that a function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality: ‖𝐷𝑓(𝑥1,…,𝑥𝑚)‖≤𝜙(𝑥1,…,𝑥𝑚),(2.22) for all 𝑥𝑗∈𝑋(1≤𝑗≤𝑚). Then there exists a unique additive mapping 𝑇 defined by 𝑇(𝑥)=lim𝑛→∞1𝑚𝑛𝑓(𝑚𝑛𝑥),(2.23) for all 𝑥∈𝑋 and the mapping 𝑇∶𝑋→𝑌 satisfies the inequality: ‖𝑓(𝑥)−𝑇(𝑥)‖≤[Φ(𝑥)]1/𝑝,(2.24) for all 𝑥∈𝑋.

Proof. Putting 𝑥1=𝑥 and 𝑥𝑗=0(2≤𝑗≤𝑚) in (2.22) and using 𝑓(0)=0, we obtain ‖𝑓(𝑚𝑥)−𝑚𝑓(𝑥)‖≤𝜙(𝑥,0,…,0),(2.25) for all 𝑥∈𝑋. By a simple induction we can prove that ‖𝑓(𝑥)−1𝑚𝑛𝑓(𝑚𝑛𝑥)‖𝑝≤𝑛∑𝑖=1(1𝑚)𝑖𝑝(𝜙(𝑚𝑖−1𝑥,0,…,0))𝑝,(2.26) for all 𝑥∈𝑋 and 𝑛∈ℕ. Thus ‖1𝑚𝑙𝑓(𝑚𝑙𝑥)−1𝑚𝑛+𝑙𝑓(𝑚𝑙+𝑛𝑥)‖𝑝≤𝑛∑𝑖=1+(1𝑚)𝑖𝑝(𝜙(𝑚𝑖−1𝑥,0,…,0))𝑝,(2.27) for all 𝑥∈𝑋 and all 𝑙∈ℕ(𝑙≤𝑛). (2.20) and (2.27) show the sequence {(1/𝑚𝑛)𝑓(𝑚𝑛𝑥)} is a Cauchy sequence in 𝑌 for all 𝑥∈𝑋. Since 𝑌 is complete, the sequence {(1/𝑚𝑛)𝑓(𝑚𝑛𝑥)} converges in 𝑌 for all 𝑥∈𝑋. Hence we can define the mapping 𝑇∶𝑋→𝑌 by 𝑇(𝑥)=lim𝑛→∞1𝑚𝑛𝑓(𝑚𝑛𝑥),(2.28) for all 𝑥∈𝑋. Letting 𝑛→∞ in (2.26), we obtain (2.24). Now we show that the mapping 𝑇 is additive. We conclude from (2.21), (2.22), and (2.28) ‖𝐷𝑇(𝑥1,𝑥2,…,𝑥𝑚)‖=lim𝑛→∞1𝑚𝑛‖𝐷𝑓(𝑚𝑛𝑥1,…,𝑚𝑛𝑥𝑚)‖≤lim𝑛→∞1𝑚𝑛𝜙(𝑚𝑛𝑥1,…,𝑚𝑛𝑥𝑚)=0,(2.29) for all 𝑥𝑗∈𝐺(1≤𝑗≤𝑚). So 𝑚∑𝑖=1𝑇(𝑚𝑥𝑖+𝑚∑𝑗=1,𝑗≠𝑖𝑥𝑗)+𝑇(𝑚∑𝑖=1𝑥𝑖)=2𝑚𝑚∑𝑖=1𝑇(𝑥𝑖).(2.30) Hence, by Theorem 2.1, the mapping 𝑇∶𝑋→𝑌 is additive. Now we prove the uniqueness assertion of 𝑇, by this mean let 𝑇′∶𝑋→𝑌 be another mapping satisfies (2.24). It follows from (2.24) ‖𝑇(𝑥)−𝑇′(𝑥)‖=𝑚−𝑗‖𝑇(𝑚𝑗𝑥)−𝑇′(𝑚𝑗𝑥)‖≤𝑚−𝑗‖𝑇(𝑚𝑗𝑥)−𝑓(𝑚𝑗𝑥)‖+‖𝑓(𝑚𝑗𝑥)−𝑇′(𝑚𝑗𝑥)‖≤2𝑚−𝑗(Φ(𝑚𝑗𝑥))1/𝑝=2𝑚−𝑗(∞∑𝑖=1(1𝑚)𝑖𝑝(𝜙(𝑚𝑖−1𝑥,0,…,0))𝑝)1/𝑝=2∞∑𝑖=1+𝑗((1𝑚)𝑖𝑝(𝜙(𝑚𝑖−1𝑥,0,…,0))𝑝)1/𝑝,(2.31) for all 𝑥∈𝑋. The right-hand side tends to zero as 𝑗→∞, hence 𝑇(𝑥)=𝑇′(𝑥) for all 𝑥∈𝑋. This show the uniqueness of 𝑇.

Corollary 2.3. Let 𝜃,𝑟𝑗(1≤𝑗≤𝑚) be nonnegative real numbers such that 0<𝑟𝑗<1. Suppose that a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality: ‖𝐷𝑓(𝑥1,…,𝑥𝑚)‖𝑌≤𝜃𝑚∑𝑗=1‖𝑥𝑗‖𝑟𝑗𝑋,(2.32) for all 𝑥𝑗∈𝑋(1≤𝑗≤𝑚). Then there exists a unique additive mapping 𝑇∶𝑋→𝑌 such that ‖𝑓(𝑥)−𝑇(𝑥)‖𝑌≤𝜃‖𝑥‖𝑟1𝑚𝑟1{𝑚(1−𝑟1)𝑝𝑚(1−𝑟1)𝑝−1}1/𝑝,(2.33) for all 𝑥∈𝑋.
Proof. This is a simple consequence of Theorem 2.2.

The following corollary is Hyers-Ulam-type stability for the functional equation (1.4).

Corollary 2.4. Let 𝜃 be nonnegative real number. Suppose that a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality: ‖𝐷𝑓(𝑥1,…,𝑥𝑚)‖𝑌≤𝜃,(2.34) for all 𝑥𝑗∈𝑋(1≤𝑗≤𝑚). Then there exists a unique additive mapping 𝑇∶𝑋→𝑌 such that ‖𝑓(𝑥)−𝑇(𝑥)‖𝑌≤𝜃{1𝑚𝑝−1}1/𝑝,(2.35) for all 𝑥∈𝑋.

Proof. In Theorem 2.2, let 𝜙(𝑥1,𝑥2,…,𝑥𝑚)∶=𝜃,(2.36) for all 𝑥𝑖∈𝑋(1≤𝑖≤𝑚).

The following corollary is Isac-Rassias-type stability for the functional equation (1.4).

Corollary 2.5. Let 𝜓∶ℝ+→ℝ+ be a mapping such that lim𝑡→∞𝜓(𝑡)𝑡=0,𝜓(𝑡𝑠)≤𝜓(𝑡)𝜓(𝑠)𝑠,𝑡>0,𝜓(𝑡)<𝑡𝑡>1.(2.37) Let 𝜃,𝑟𝑗(1≤𝑗≤𝑚) be nonnegative real numbers. Suppose that a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality ‖𝐷𝑓(𝑥1,…,𝑥𝑚)‖𝑌≤𝜃𝑚∑𝑗=1𝜓(‖𝑥𝑗‖𝑋),(2.38) for all 𝑥𝑗∈𝑋(1≤𝑗≤𝑚). Then there exists a unique additive mapping 𝑇∶𝑋→𝑌 such that ‖𝑓(𝑥)−𝑇(𝑥)‖𝑌≤𝑘𝜃𝜓(𝑚−1)𝜓(‖𝑥‖),(2.39) for all 𝑥∈𝑋, where 𝑘=𝜓(𝑚)/(𝑚−𝜓(𝑚)).
Proof. The proof follows from Theorem 2.2 by taking 𝜙(𝑥1,…,𝑥𝑚)∶=𝜃𝑚∑𝑗=𝑖𝜓(‖𝑥𝑗‖𝑋),(2.40) for all 𝑥𝑗∈𝑋(1≤𝑗≤𝑚).

Remark 2.6. In Theorem 2.2, If we replace control function by 𝜃∏𝑚𝑗=1‖𝑥𝑗‖𝑟𝑗, then 𝑇=𝑓. Therefore in this case, 𝑓 is superstable.

Theorem 2.7. Let 𝜙∶𝑋×⋯×𝑋⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝑚-times→[0,∞) be a mapping such that lim𝑛→∞𝑚𝑛𝜙(𝑥1𝑚𝑛,…,𝑥𝑚𝑚𝑛)=0,(2.41) for all 𝑥𝑗∈𝑋(1≤𝑗≤𝑚) and Φ(𝑥)=∞∑𝑖=0𝑚𝑖𝑝𝜙(𝑥𝑚𝑖+1,0,…,0)𝑝<∞,(2.42) for all 𝑥∈𝑋. Suppose that a function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality: ‖𝐷𝑓(𝑥1,…,𝑥𝑚)‖≤𝜙(𝑥1,…,𝑥𝑚),(2.43) for all 𝑥𝑗∈𝑋(1≤𝑗≤𝑚). Then there exists a unique additive mapping 𝑇 defined by 𝑇(𝑥)=lim𝑛→∞𝑚𝑛𝑓(𝑥𝑚𝑛),(2.44) for all 𝑥∈𝑋 and the mapping 𝑇∶𝑋→𝑌 satisfies the inequality: ‖𝑓(𝑥)−𝑇(𝑥)‖≤[Φ(𝑥)]1/𝑝,(2.45) for all 𝑥∈𝑋.

Proof. Putting 𝑥1=𝑥 and 𝑥𝑗=0(2≤𝑗≤𝑚) in (2.43) we obtain ‖𝑓(𝑚𝑥)−𝑚𝑓(𝑥)‖≤𝜙(𝑥,0,…,0),(2.46) for all 𝑥∈𝑋. Replacing 𝑥 by 𝑥/𝑚𝑛+1 in (2.46) and multiplying both sides of (2.46) to 𝑚𝑛, we get ‖𝑚𝑛𝑓(𝑥𝑚𝑛)−𝑚𝑛+1𝑓(𝑥𝑚𝑛+1)‖≤𝑚𝑛𝜙(𝑥𝑚𝑛+1,0,…,0),(2.47) for all 𝑥∈𝑋 and all 𝑛∈ℕ∪{0}. Since 𝑌 is a 𝑝-Banach space, we have ‖𝑚𝑛+1𝑓(𝑥𝑚𝑛+1)−𝑚𝑟𝑓(𝑥𝑚𝑟)‖𝑝≤𝑛∑𝑖=𝑟‖𝑚𝑖+1𝑓(𝑥𝑚𝑖+1)−𝑚𝑖𝑓(𝑥𝑚𝑖)‖≤𝑛∑𝑖=𝑟𝑚𝑖𝑝(𝜙(𝑥𝑚𝑖+1,0,…,0))𝑝,(2.48) for all 𝑥∈𝑋 and all nonnegative 𝑛 and 𝑟 with 𝑛≥𝑟. Therefore, we have from (2.42) and (2.48) that the sequence {𝑚𝑛𝑓(𝑥/𝑚𝑛)} is a Cauchy in 𝑌 for all 𝑥∈𝑋. Because of 𝑌 is complete, the sequence {𝑚𝑛𝑓(𝑥/𝑚𝑛)} converges for all 𝑥∈𝑋. Hence, we can define the mapping 𝑇∶𝑋→𝑌 by 𝑇(𝑥)=lim𝑛→∞𝑚𝑛𝑓(𝑥𝑚𝑛),(2.49) for all 𝑥∈𝑋. Putting 𝑟=0 and passing the limit 𝑛→∞ in (2.48), we obtain (2.45). Showing the additivity and uniqueness of 𝑇 is similar to Theorem 2.2, and the proof is complete.

Corollary 2.8. Let 𝜃,𝑟𝑗(1≤𝑗≤𝑚) be nonnegative real numbers such that 𝑟𝑗>1(1≤𝑗≤𝑚). Suppose that a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality: ‖𝐷𝑓(𝑥1,…,𝑥𝑚)‖𝑌≤𝜃𝑚∑𝑗=1‖𝑥𝑗‖𝑟𝑗𝑋,(2.50) for all 𝑥𝑗∈𝑋(1≤𝑗≤𝑚). Then there exists a unique additive mapping 𝑇∶𝑋→𝑌 such that ‖𝑓(𝑥)−𝑇(𝑥)‖𝑌≤𝜃‖𝑥‖𝑟1𝑚𝑟1{𝑚(𝑟1−1)𝑝1−𝑚(𝑟1−1)𝑝}1/𝑝,(2.51) for all 𝑥∈𝑋.
Proof. This is a simple consequence of Theorem 2.7.

Remark 2.9. We can formulate similar statement to Corollaries 2.4 and 2.5 for Theorem 2.7. Moreover, In Theorem 2.7, If we replace control function by 𝜃∏𝑚𝑗=1‖𝑥𝑗‖𝑟𝑗, then 𝑇=𝑓. Therefore in this case, 𝑓 is superstable.

Now, we apply a fixed point method and prove the generalized Hyers-Ulam stability of functional equation (1.4).

We recall a fundamental result in fixed point theory.

Theorem 2.10 (see [34]). Let (𝒳,𝑑) be a complete generalized metric space and let 𝐽∶𝒳→𝒳 be a strictly contractive mapping with Lipschits constant 𝐿∈(0,1). Then, for a given element 𝑥∈𝒳, exactly one of the following assertions is true:
either(1)𝑑(𝐽𝑛𝑥,𝐽𝑛+1𝑥)=∞ for all 𝑛≥0 or(2)there exists 𝑛0 such that 𝑑(𝐽𝑛𝑥,𝐽𝑛+1𝑥)<∞ for all 𝑛≥𝑛0.Actually, if(𝑎2) holds, then the sequence𝐽𝑛𝑥 is a convergent to a fixed point𝑥∗ of𝐽 and(3)𝑥∗ is the unique fixed point of 𝐽 in Λ∶={𝑦∈𝒳,𝑑(𝐽𝑛0𝑥,𝑦)<∞};(4)𝑑(𝑦,𝑥∗)≤𝑑(𝑦,𝐽𝑦)/(1−𝐿) for all 𝑦∈Λ.

Theorem 2.11. Let 𝑓∶𝑋→𝑌 with 𝑓(0)=0 be a mapping for which there exists a function 𝜙∶𝑋×⋯×𝑋⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝑚-times→[0,∞) such that ‖𝐷𝑓(𝑥1,…,𝑥𝑚)‖≤𝜙(𝑥1,…,𝑥𝑚),(2.52)lim𝑛→∞𝑚𝑛𝜙(𝑥1𝑚𝑛,…,𝑥𝑚𝑚𝑛)=0,(2.53) for all 𝑥𝑗∈𝑋(1≤𝑗≤𝑚). If there exists an 𝐿<1 such that 𝜙(𝑥1,…,𝑥𝑗)≤(𝐿/𝑚)𝜙(𝑚𝑥1,…,𝑚𝑥𝑗)(1≤𝑗≤𝑚), then there exists a unique additive mapping 𝑇∶𝑋→𝑌 satisfying ‖𝑓(𝑥)−𝑇(𝑥)‖≤𝐿𝑚−𝑚𝐿𝜙(𝑥,0,…,0),(2.54) for all 𝑥∈𝑋.

Proof. Putting 𝑥1=𝑥 and 𝑥𝑗=0(2≤𝑗≤𝑚) in (2.52) and using 𝑓(0)=0, we obtain ‖𝑓(𝑚𝑥)−𝑚𝑓(𝑥)‖≤𝜙(𝑥,0,…,0),(2.55) for all 𝑥∈𝑋. Hence, ‖𝑓(𝑥)−𝑚𝑓(𝑥𝑚)‖≤𝜙(𝑥𝑚,0,…,0)≤𝐿𝑚𝜙(𝑥,…,0),(2.56) for all 𝑥∈𝑋.
Let 𝐸∶={𝑔∶𝑋→𝑌}. We introduce a generalized metric on 𝐸 as follows: 𝑑(𝑔,ℎ)∶=inf{𝐶∈ℝ+∶‖𝑔(𝑥)−ℎ(𝑥)‖≤𝐶𝜙(𝑥,0,…,0)∀𝑥∈𝑋}.(2.57) It is easy to show that (𝐸,𝑑) is a generalized complete metric space.
Now we consider the mapping 𝐽∶𝐸→𝐸 defined by (𝐽𝑔)(𝑥)=𝑚𝑔(𝑥𝑚),(2.58) for all 𝑔∈𝐸 and all 𝑥∈𝑋. Let 𝑔,ℎ∈𝐸 and let 𝐶∈ℝ+ ba an arbitrary constant with 𝑑(𝑔,ℎ)≤𝐶. From the definition of 𝑑, we have ‖𝑔(𝑥)−ℎ(𝑥)‖≤𝐶𝜙(𝑥,0,…,0),(2.59) for all 𝑥∈𝑋. By the assumption and last inequality, we have ‖(𝐽𝑔)(𝑥)−(𝐽ℎ)(𝑥)‖=𝑚‖𝑔(𝑥𝑚)−ℎ(𝑥𝑚)‖≤𝑚𝐶𝜙(𝑥𝑚,0,…,0),(2.60) for all 𝑥∈𝑋. So 𝑑(𝐽𝑔,𝐽ℎ)≤𝐿𝑑(𝑔,ℎ) for all 𝑔,ℎ∈𝐸. It follows from (2.56) that 𝑑(𝐽𝑓,𝑓)≤𝐿/𝑚. Therefore, according to Theorem 2.10, the sequence {𝐽𝑛𝑓} converges to a fixed point 𝑇 of 𝐽, that is, 𝑇∶𝑋⟶𝑌,𝑇(𝑥)=lim𝑛→∞(𝐽𝑛𝑓)(𝑥)=lim𝑛→∞𝑚𝑛𝑓(𝑥𝑚𝑛),(2.61) and 𝑇(𝑚𝑥)=𝑚𝑇(𝑥) for all 𝑥∈𝑋. Also 𝑇 is the unique fixed point of 𝐽 in the set 𝐸𝜙={𝑔∈𝐸∶𝑑(𝑓,𝑔)<∞} and 𝑑(𝑇,𝑓)≤11−𝐿𝑑(𝐽𝑓,𝑓)≤𝐿𝑚(1−𝐿),(2.62) that is, inequality (2.54) holds true for all 𝑥∈𝑋. It follows from the definition of 𝑇, and (2.52) and (2.53) that 𝑚∑𝑖=1𝑇(𝑚𝑥𝑖+𝑚∑𝑗=1,𝑗≠𝑖𝑥𝑗)+𝑇(𝑚∑𝑖=1𝑥𝑖)=2𝑚𝑚∑𝑖=1𝑇(𝑥𝑖).(2.63) Hence, by Theorem 2.1, the mapping 𝑇∶𝑋→𝑌 is additive.

Corollary 2.12. Let 𝑟𝑗∈(1,∞)(1≤𝑗≤𝑚) and 𝜃 be real numbers. Let 𝑓∶𝑋→𝑌 with 𝑓(0)=0 such that ‖𝐷𝑓(𝑥1,…,𝑥𝑚)‖≤𝜃𝑚∑𝑗=1‖𝑥𝑗‖𝑟𝑗,(2.64) for all 𝑥𝑗∈𝑋(1≤𝑗≤𝑚). Then there exists a unique additive mapping 𝑇∶𝑋→𝑌 satisfies the inequality: ‖𝑓(𝑥)−𝑇(𝑥)‖≤𝜃𝑚𝑟1−𝑚‖𝑥‖,(2.65) for all 𝑥∈𝑋.

Proof. Setting 𝜙(𝑥1,𝑥2,…,𝑥𝑚)∶=𝜃∑𝑚𝑗=1‖𝑥𝑗‖𝑟𝑗 for all 𝑥𝑗∈𝑋(1≤𝑗≤𝑚) in Theorem 2.11. Then by 𝐿=𝑚1−𝑟1, we get the desired result.

Remark 2.13. We can formulate similar statement to Theorem 2.11 in which we can define the sequence 𝑇(𝑥)∶=lim𝑛→∞(1/𝑚𝑛)𝑓(𝑚𝑛𝑥) under suitable conditions on the function 𝜙 and then obtain similar result to Corollary 2.12.

Remark 2.14. We can formulate similar statements for stability of (1.4) on Banach spaces.