Approximate Quartic and Quadratic Mappings in Quasi-Banach Spaces

Abstract

we establish the general solution for a mixed type functional equation of aquartic and a quadratic mapping in linear spaces. In addition, we investigate the generalized Hyers-Ulam stability in 𝑝-Banach spaces.

1. Introduction and Preliminaries

The stability problem of functional equations originated from a question of Ulam [1] in 1940, concerning the stability of group homomorphisms. Let (𝐺₁,⋅) be a group, and let (𝐺₂,∗) be a metric group with the metric 𝑑(⋅,⋅). Given 𝜖>0, does there exist a 𝛿>0 such that if a mapping ℎ∶𝐺₁→𝐺₂ satisfies the inequality 𝑑(ℎ(𝑥⋅𝑦),ℎ(𝑥)∗ℎ(𝑦))<𝛿 for all 𝑥,𝑦∈𝐺₁, then there exists a homomorphism 𝐻∶𝐺₁→𝐺₂ with 𝑑(ℎ(𝑥),𝐻(𝑥))<𝜖 for all 𝑥∈𝐺₁? In other words, under what condition does there exists a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. Let 𝑓∶𝑋→𝑋^′ be a mapping between Banach spaces such that 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝛿, (1.1) for all 𝑥,𝑦∈𝑋, and for some 𝛿>0. Then, there exists a unique additive mapping 𝑇∶𝑋→𝑋^′ such that 𝑓(𝑥)−𝑇(𝑥)≤𝛿, (1.2) for all 𝑥∈𝑋.

The result of Hyers was generalized by Aoki [3] for approximate additive function and by Rassias [4] for approximate linear function by allowing the difference Cauchy equation ‖𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)‖ to be controlled by 𝜀(‖𝑥‖^𝑝+‖𝑦‖^𝑝). Taking into consideration a lot of influence of Ulam, Hyers and Rassias on the development of stability problems of functional equations, the stability phenomenon that was proved by Rassias may be called the Hyers-Ulam-Rassias stability (see [5, 6]). In 1994, a generalization of Rassias theorem was obtained by Găvruta [7], who replaced 𝜀(‖𝑥‖^𝑝+‖𝑦‖^𝑝) by a general control function 𝜑(𝑥,𝑦). The functional equation𝑓𝑥+𝑦+𝑓𝑥−𝑦=2𝑓(𝑥)+2𝑓𝑦 (1.3) is related to a symmetric biadditive function [8–10]. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.3) is said to be a quadratic function. It is well known that a function 𝑓 between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive function 𝐵₁ such that 𝑓(𝑥)=𝐵₁(𝑥,𝑥) for all 𝑥 in the vector space. The biadditive function 𝐵₁ is given by𝐵₁=1𝑥,𝑦4𝑓𝑥+𝑦−𝑓𝑥−𝑦. (1.4) A Hyers-Ulam stability problem for the quadratic functional equation (1.3) was proved by Skof for functions 𝑓∶𝑋→𝑌, where 𝑋 is normed space and 𝑌 is Banach space (see [11]). In the paper [12], Czerwik proved the Hyers-Ulam-Rassias stability of (1.3).

Lee et al. [13] considered the following functional equation:𝑓2𝑥+𝑦+𝑓2𝑥−𝑦=4𝑓𝑥+𝑦+4𝑓𝑥−𝑦+24𝑓(𝑥)−6𝑓𝑦. (1.5) In fact, they proved that a function 𝑓 between two real vector spaces 𝑋 and 𝑌 is a solution of (1.5) if and only if there exists a unique symmetric biquadratic function 𝐵₂∶𝑋×𝑋→𝑌 such that 𝑓(𝑥)=𝐵₂(𝑥,𝑥) for all 𝑥∈𝑋. The biquadratic function 𝐵₂ is given by𝐵₂=1𝑥,𝑦𝑓(𝑥)𝑦.12𝑥+𝑦+𝑓𝑥−𝑦−2𝑓−2𝑓 (1.6) It is easy to show that the function 𝑓(𝑥)=𝑎𝑥⁴ satisfies the functional equation (1.5), which is called the quartic functional equation (see also [14]).

Jun and Kim [15] have obtained the generalized Hyers-Ulam stability for a mixed type of cubic and additive functional equation. In addition, the generalized Hyers-Ulam stability for a mixed type of cubic, quadratic, and additive functional equation has been investigated by Gordji and Khodaei [16] (see also [17, 18]). The stability problems for several mixed types of functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem [19–27].

In this paper, we deal with the following functional equation derived from quartic and quadratic functions:𝑓𝑘𝑥+𝑦+𝑓𝑘𝑥−𝑦=𝑘²𝑓𝑥+𝑦+𝑘²𝑓+𝑘𝑥−𝑦²𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)−2𝑘²−1𝑓𝑦 (1.7) for fixed integers 𝑘≠0, ±1. It is easy to see that the function 𝑓(𝑥)=𝑎𝑥⁴+𝑏𝑥² is a solution of the functional equation (1.7). In the sequel, we investigate the general solution of functional equation (1.7) when 𝑓 is a function between vector spaces, and then we prove the generalized Hyers-Ulam stability of (1.7) in the spirit of Hyers, Ulam, and Rassias using the direct method.

We recall some basic facts concerning quasi-Banach spaces and some preliminary results.

Definition 1.1 (see [28, 29]). Let 𝑋 be a real linear space. A quasinorm 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 𝑀≥1 such that ‖𝑥+𝑦‖≤𝑀(‖𝑥‖+‖𝑦‖) for all 𝑥,𝑦∈𝑋.
The pair (𝑋,‖⋅‖) is called a quasinormed space if ‖⋅‖ is a quasinorm on 𝑋.

The smallest possible 𝑀 is called the modulus of concavity of ‖⋅‖. A quasi-Banach space is a complete quasinormed space. A quasinorm ‖⋅‖ is called a 𝑝-norm (0<𝑝≤1) if 𝑥+𝑦^𝑝≤‖𝑥‖^𝑝+𝑦^𝑝, (1.8) 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 [29], each quasinorm is equivalent to some 𝑝-norm (see also [28]). Since it is much easier to work with 𝑝-norms, henceforth we restrict our attention mainly to 𝑝-norms.

Lemma 1.2 (see [17]). Let 𝑥₁,𝑥₂,…,𝑥_𝑛 be nonnegative real numbers. Then, one has ^𝑛_𝑖=1𝑥_𝑖^𝑝≤^𝑛_𝑖=1𝑥_𝑖^𝑝, (1.9) for a positive real number 𝑝 with 𝑝≤1.

2. General Solution

We here present the general solution of (1.7).

Theorem 2.1. Let both 𝑋 and 𝑌 be real vector spaces. A function 𝑓∶𝑋→𝑌 satisfies (1.7) for all 𝑥,𝑦∈𝑋 if and only if there exists a unique symmetric biquadratic function 𝐵₂∶𝑋×𝑋→𝑌 and a unique symmetric biadditive function 𝐵₁∶𝑋×𝑋→𝑌 such that 𝑓(𝑥)=𝐵₂(𝑥,𝑥)+𝐵₁(𝑥,𝑥), (2.1) for all 𝑥∈𝑋.

Proof. Let 𝑓 satisfy (1.7) and let 𝑔,ℎ∶𝑋→𝑌 be functions defined by 𝑔(𝑥)∶=𝑓(2𝑥)−16𝑓(𝑥),ℎ(𝑥)∶=𝑓(2𝑥)−4𝑓(𝑥), (2.2) for all 𝑥∈𝑋. We claim that the functions 𝑔 and ℎ are quadratic and quartic, respectively.
Letting 𝑥=𝑦=0 in (1.7), we have 𝑓(0)=0. By putting 𝑥=0 in (1.7), one leads to the evenness 𝑓(−𝑦)=𝑓(𝑦) of 𝑓. Replacing 𝑦 by 𝑥+𝑦 in (1.7), we have𝑓(𝑘+1)𝑥+𝑦+𝑓(𝑘−1)𝑥−𝑦=𝑘²𝑓2𝑥+𝑦+𝑘²𝑓+𝑘−𝑦²𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)+21−𝑘²𝑓𝑥+𝑦, (2.3) for all 𝑥,𝑦∈𝑋. Replacing 𝑦 by −𝑦 in (2.3), we obtain 𝑓(𝑘+1)𝑥−𝑦+𝑓(𝑘−1)𝑥+𝑦=𝑘²𝑓2𝑥−𝑦+𝑘²𝑓𝑦+𝑘²𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)+21−𝑘²𝑓𝑥−𝑦, (2.4) for all 𝑥,𝑦∈𝑋. Adding (2.3) to (2.4), we get by evenness of 𝑓, 𝑓(𝑘+1)𝑥+𝑦+𝑓(𝑘+1)𝑥−𝑦+𝑓(𝑘−1)𝑥+𝑦+𝑓(𝑘−1)𝑥−𝑦=𝑘²𝑓2𝑥+𝑦+𝑓2𝑥−𝑦+2𝑘²𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)+21−𝑘²𝑓𝑥+𝑦+𝑓𝑥−𝑦+2𝑘²𝑓𝑦, (2.5) for all 𝑥,𝑦∈𝑋. From the substitution 𝑦=𝑘𝑥+𝑦 in (1.7), we have by evenness of 𝑓, 𝑓2𝑘𝑥+𝑦+𝑓𝑦=𝑘²𝑓(𝑘+1)𝑥+𝑦+𝑘²𝑓()+𝑘𝑘−1𝑥+𝑦₂𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)+21−𝑘²𝑓𝑘𝑥+𝑦, (2.6) for all 𝑥,𝑦∈𝑋. Replacing 𝑦 by −𝑦 in (2.6), we get 𝑓2𝑘𝑥−𝑦+𝑓−𝑦=𝑘²𝑓(𝑘+1)𝑥−𝑦+𝑘²𝑓()+𝑘𝑘−1𝑥−𝑦₂𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)+21−𝑘²𝑓𝑘𝑥−𝑦, (2.7) for all 𝑥,𝑦∈𝑋. Adding (2.6) to (2.7), we get by evenness of 𝑓, 𝑓2𝑘𝑥+𝑦+𝑓2𝑘𝑥−𝑦=𝑘²𝑓()()()()+𝑘+1𝑥+𝑦+𝑓𝑘+1𝑥−𝑦+𝑓𝑘−1𝑥+𝑦+𝑓𝑘−1𝑥−𝑦2𝑘²𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)+21−𝑘²𝑓𝑘𝑥+𝑦+𝑓𝑘𝑥−𝑦−2𝑓𝑦, (2.8) for all 𝑥,𝑦∈𝑋. By using (1.7) and (2.5), it follows from (2.8) that 𝑓2𝑘𝑥+𝑦+𝑓2𝑘𝑥−𝑦=𝑘²𝑘²𝑓2𝑥+𝑦+𝑓2𝑥−𝑦+2𝑘²𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)+21−𝑘²𝑓𝑥+𝑦+𝑓𝑥−𝑦+2𝑘²𝑓𝑦+21−𝑘²𝑘²𝑓𝑥+𝑦+𝑘²𝑓+𝑘𝑥−𝑦²𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)+21−𝑘²𝑓𝑦+2𝑘²𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)−2𝑓𝑦, (2.9) for all 𝑥,𝑦∈𝑋. If we replace 𝑥 by 2𝑥 in (1.7), then we get that 𝑓2𝑘𝑥+𝑦+𝑓2𝑘𝑥−𝑦=𝑘²𝑓2𝑥+𝑦+𝑘²𝑓+𝑘2𝑥−𝑦²𝑘²−16𝑓(4𝑥)−4𝑓(2𝑥)+21−𝑘²𝑓𝑦, (2.10) for all 𝑥,𝑦∈𝑋. It follows from (2.9) and (2.10) that 𝑘²𝑘²𝑓2𝑥+𝑦+𝑓2𝑥−𝑦+2𝑘²𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)+21−𝑘²𝑓𝑥+𝑦+𝑓𝑥−𝑦+2𝑘²𝑓𝑦+21−𝑘²𝑘²𝑓𝑥+𝑦+𝑘²𝑓+𝑘𝑥−𝑦²𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)+21−𝑘²𝑓𝑦+2𝑘²𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)−2𝑓𝑦=𝑘²𝑓2𝑥+𝑦+𝑘²𝑓+𝑘2𝑥−𝑦²𝑘²−16𝑓(4𝑥)−4𝑓(2𝑥)+21−𝑘²𝑓𝑦, (2.11) for all 𝑥,𝑦∈𝑋. On the other hand, putting 𝑦=0 in (1.7), we get 𝑓(𝑘𝑥)=𝑘²𝑓(𝑥)+𝑘²𝑘²−1𝑓()(𝑥),122𝑥−4𝑓 (2.12) for all 𝑥∈𝑋. Putting 𝑦=𝑥 in (1.7), we get 𝑓((𝑘+1)𝑥)+𝑓((𝑘−1)𝑥)=𝑘²𝑓()+𝑘2𝑥²𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)+21−𝑘²𝑓(𝑥), (2.13) for all 𝑥∈𝑋. Putting 𝑦=𝑘𝑥 in (1.7) and using the evenness of 𝑓, we obtain 𝑓(2𝑘𝑥)=𝑘²𝑓)𝑥))𝑥)+𝑘((𝑘+1+𝑓((𝑘−1²𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)+21−𝑘²𝑓(𝑘𝑥), (2.14) for all 𝑥∈𝑋. Letting 𝑦=0 in (2.10), we have 𝑓(2𝑘𝑥)=𝑘²𝑓()+𝑘2𝑥²𝑘²−1𝑓()(),124𝑥−4𝑓2𝑥 (2.15) for all 𝑥∈𝑋. It follows from (2.14) and (2.15) that 𝑘²𝑘²−1𝑓()()124𝑥−4𝑓2𝑥=𝑘²𝑓)𝑥))𝑥)+𝑘((𝑘+1+𝑓((𝑘−1²𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)+21−𝑘²𝑓(𝑘𝑥)−𝑘²𝑓(2𝑥), (2.16) for all 𝑥∈𝑋. Now, by using (2.12), (2.13) and (2.16), we lead to 𝑘²𝑘²−1𝑓()()124𝑥−4𝑓2𝑥=𝑘²𝑘²𝑓()+𝑘2𝑥²𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)+21−𝑘²𝑓(𝑥)+21−𝑘²𝑘²𝑓(𝑥)+𝑘²𝑘²−1𝑓()(𝑥)+𝑘122𝑥−4𝑓₂𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)−𝑘²𝑓(2𝑥), (2.17) for all 𝑥∈𝑋. Finally, comparing (2.11) with (2.17), then we conclude that 𝑓2𝑥+𝑦+𝑓2𝑥−𝑦=4𝑓𝑥+𝑦+4𝑓𝑥−𝑦+2𝑓(2𝑥)−4𝑓(𝑥)−6𝑓𝑦, (2.18) for all 𝑥,𝑦∈𝑋. Replacing 𝑦 by 2𝑦 in (2.18), we get 𝑓2𝑥+2𝑦+𝑓2𝑥−2𝑦=4𝑓𝑥+2𝑦+4𝑓𝑥−2𝑦+2𝑓(2𝑥)−4𝑓(𝑥)−6𝑓2𝑦, (2.19) for all 𝑥,𝑦∈𝑋. Interchanging 𝑥 with 𝑦 in (2.18), one gets 𝑓𝑥+2𝑦+𝑓𝑥−2𝑦=4𝑓𝑥+𝑦+4𝑓𝑥−𝑦+2𝑓2𝑦−4𝑓𝑦−6𝑓(𝑥), (2.20) for all 𝑥,𝑦∈𝑋. It follows from (2.19) and (2.20) that 𝑓22𝑓()(𝑥)𝑓𝑦,𝑥+𝑦−16𝑓𝑥+𝑦+𝑓𝑥−𝑦−16𝑓𝑥−𝑦=22𝑥−16𝑓+22𝑦−16𝑓 (2.21) for all 𝑥,𝑦∈𝑋. This means that 𝑔𝑥+𝑦+𝑔𝑥−𝑦=2𝑔(𝑥)+2𝑔𝑦, (2.22) for all 𝑥,𝑦∈𝑋. So the function 𝑔∶𝑋→𝑌 defined by 𝑔(𝑥)∶=𝑓(2𝑥)−16𝑓(𝑥) is quadratic.
To prove that ℎ∶𝑋→𝑌 defined by ℎ(𝑥)∶=𝑓(2𝑥)−4𝑓(𝑥) is quartic, we need to show thatℎ2𝑥+𝑦+ℎ2𝑥−𝑦=4ℎ𝑥+𝑦+4ℎ𝑥−𝑦+24ℎ(𝑥)−6ℎ𝑦, (2.23) for all 𝑥,𝑦∈𝑋. Replacing 𝑥 and 𝑦 by 2𝑥 and 2𝑦 in (2.18), respectively, we obtain 𝑓22𝑥+𝑦+𝑓22𝑥−𝑦=4𝑓2𝑥+𝑦+4𝑓2𝑥−𝑦+2𝑓(4𝑥)−4𝑓(2𝑥)−6𝑓2𝑦, (2.24) for all 𝑥,𝑦∈𝑋. But, since 𝑔(2𝑥)=4𝑔(𝑥) for all 𝑥∈𝑋, where 𝑔∶𝑋→𝑌 is a quadratic function defined above, we see that 𝑓(4𝑥)=20𝑓(2𝑥)−64𝑓(𝑥), (2.25) for all 𝑥∈𝑋. Hence, according to (2.24) and (2.25), we get 𝑓22𝑥+𝑦+𝑓22𝑥−𝑦=4𝑓2𝑥+𝑦+4𝑓2𝑥−𝑦+32𝑓(2𝑥)−4𝑓(𝑥)−6𝑓2𝑦, (2.26) for all 𝑥,𝑦∈𝑋. By multiplying 4 on both sides of (2.18), we get that 4𝑓2𝑥+𝑦+4𝑓2𝑥−𝑦=16𝑓𝑥+𝑦+16𝑓𝑥−𝑦+8𝑓(2𝑥)−4𝑓(𝑥)−24𝑓𝑦, (2.27) for all 𝑥,𝑦∈𝑋. If we subtract the last equation from (2.26), then we arrive at 𝑓22𝑓2𝑓2𝑓()(𝑥)𝑓𝑦,2𝑥+𝑦−4𝑓2𝑥+𝑦+𝑓2𝑥−𝑦−4𝑓2𝑥−𝑦=4𝑥+𝑦−4𝑓𝑥+𝑦+4𝑥−𝑦−4𝑓𝑥−𝑦+242𝑥−4𝑓−62𝑦−4𝑓 (2.28) for all 𝑥,𝑦∈𝑋. This means that ℎ satisfies (2.23) and, therefore, the function ℎ∶𝑋→𝑌 is quartic. Thus, there exists a unique symmetric biquadratic function 𝐵₂∶𝑋×𝑋→𝑌 and a unique symmetric biadditive function 𝐵₁∶𝑋×𝑋→𝑌 such that ℎ(𝑥)=12𝐵₂(𝑥,𝑥) and 𝑔(𝑥)=−12𝐵₁(𝑥,𝑥) for all 𝑥∈𝑋 (see [8, 13]). Therefore, we obtain from (2.2) that 𝑓(𝑥)=1ℎ(𝑥)−112𝑔(𝑥)12=𝐵₂(𝑥,𝑥)+𝐵₁(𝑥,𝑥), (2.29) for all 𝑥∈𝑋.
The proof of the converse is trivial.

3. Generalized Hyers-Ulam Stability

From this point on, assume that 𝑋 is a quasinormed space with quasinorm ‖⋅‖_𝑋 and that 𝑌 is a 𝑝-Banach space with 𝑝-norm ‖⋅‖_𝑌. Let 𝑀 be the modulus of concavity of ‖⋅‖_𝑌.

Before taking up the main subject, given a mapping 𝑓∶𝑋→𝑌, we define the difference operator 𝐷_𝑓∶𝑋×𝑋→𝑌 by 𝐷_𝑓𝑥,𝑦∶=𝑓𝑘𝑥+𝑦+𝑓𝑘𝑥−𝑦−𝑘²𝑓𝑥+𝑦−𝑘²𝑓−𝑘𝑥−𝑦₂𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)+2𝑘²−1𝑓𝑦, (3.1) for all 𝑥,𝑦∈𝑋. Let 𝜑^𝑝(𝑥,𝑦)∶=(𝜑(𝑥,𝑦))^𝑝 for notational convenience.

Theorem 3.1. Let 𝑗∈{−1,1} be fixed and let 𝜑_𝑞∶𝑋×𝑋→[0,∞) be a function such that lim_𝑛→∞4^𝑛𝑗𝜑_𝑞𝑥2^𝑛𝑗,𝑦2^𝑛𝑗=0, (3.2) for all 𝑥,𝑦∈𝑋 and ^∞_{𝑖=(1+𝑗)/2}4^𝑖𝑝𝑗𝜑^𝑝_𝑞𝑢2^𝑖𝑗,𝑦2^𝑖𝑗<∞, (3.3) for all (𝑢,𝑦)∈{(𝑥,0),(2𝑥,0),(𝑥,𝑥),(𝑥,𝑘𝑥)∶𝑥∈𝑋}. Suppose that an even function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality 𝐷_𝑓𝑥,𝑦_𝑌≤𝜑_𝑞𝑥,𝑦, (3.4) for all 𝑥,𝑦∈𝑋. Then, there exists a unique quadratic function 𝑄∶𝑋→𝑌 such that 𝑓(2𝑥)−16𝑓(𝑥)−𝑄(𝑥)_𝑌≤𝑀²4𝜓_𝑞(𝑥)^1/𝑝, (3.5) for all 𝑥∈𝑋, where 𝜓_𝑞(𝑥)∶=^∞_{𝑖=(1+𝑗)/2}4^𝑖𝑝𝑗𝑘^2𝑝𝑘²−1^𝑝12𝑘^2𝑝𝜑^𝑝_𝑞𝑥2^𝑖𝑗,𝑥2^𝑖𝑗+12𝑘²−1^𝑝𝜑^𝑝_𝑞𝑥2^𝑖𝑗,0+6^𝑝𝜑^𝑝_𝑞2𝑥2^𝑖𝑗,0+12^𝑝𝜑^𝑝_𝑞𝑥2^𝑖𝑗,𝑘𝑥2^𝑖𝑗. (3.6)

Proof. Let 𝑗=1. Setting 𝑦=0 in (3.4), we have 2𝑓(𝑘𝑥)−2𝑘²𝑓(𝑥)−𝑘²𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)_𝑌≤𝜑_𝑞(𝑥,0), (3.7) for all 𝑥∈𝑋. Putting 𝑦=𝑥 in (3.4), we obtain 𝑓((𝑘+1)𝑥)+𝑓((𝑘−1)𝑥)−𝑘²𝑓()−𝑘2𝑥²𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)+2𝑘²−1𝑓(𝑥)_𝑌≤𝜑_𝑞(𝑥,𝑥), (3.8) for all 𝑥∈𝑋. Replacing 𝑥 by 2𝑥 in (3.7), we see that 2𝑓(2𝑘𝑥)−2𝑘²𝑓()−𝑘2𝑥²𝑘²−16𝑓(4𝑥)−4𝑓(2𝑥)_𝑌≤𝜑_𝑞(2𝑥,0), (3.9) for all 𝑥∈𝑋. Setting 𝑦 by 𝑘𝑥 in (3.4) and using the evenness of 𝑓, we get 𝑓(2𝑘𝑥)−𝑘²𝑓((𝑘+1)𝑥)−𝑘²𝑓((𝑘−1)𝑥)+2𝑘²𝑓()−𝑘−1𝑘𝑥²𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)_𝑌≤𝜑_𝑞(𝑥,𝑘𝑥), (3.10) for all 𝑥∈𝑋. It follows from (3.9) and (3.10) that 𝑘²𝑓()+𝑘2𝑥²𝑘²−1𝑓()()124𝑥−4𝑓2𝑥−𝑘²𝑓((𝑘+1)𝑥)−𝑘²𝑓)𝑥)𝑘((𝑘−1+2²𝑓()−𝑘−1𝑘𝑥²𝑘²−16𝑓(2𝑥)−4𝑓(𝑥)_𝑌1≤𝑀2𝜑_𝑞(2𝑥,0)+𝜑_𝑞(𝑥,𝑘𝑥), (3.11) for all 𝑥∈𝑋. Also, it follows from (3.7) and (3.8) that 𝑘²𝑓((𝑘+1)𝑥)+𝑘²𝑓((𝑘−1)𝑥)−2𝑘²−1𝑓(𝑘𝑥)−𝑘⁴𝑓()−𝑘2𝑥₂𝑘²−16(𝑓(2𝑥)−4𝑓(𝑥))+4𝑘²𝑘²−1𝑓(𝑥)_𝑌≤𝑀𝑘²𝜑_𝑞(𝑥,𝑥)+𝑘²−1𝜑_𝑞(𝑥,0), (3.12) for all 𝑥∈𝑋. Finally, using (3.11) and (3.12), we obtain that 𝑓(4𝑥)−20𝑓(2𝑥)+64𝑓(𝑥)_𝑌≤𝑀²𝑘²𝑘²−112𝑘²𝜑_𝑞(𝑥,𝑥)+12𝑘²−1𝜑_𝑞(𝑥,0)+6𝜑_𝑞(2𝑥,0)+12𝜑_𝑞(𝑥,𝑘𝑥)=𝑀²𝜓_𝑞(𝑥), (3.13) where 𝜓_𝑞(𝑥)1∶=𝑘²𝑘²−112𝑘²𝜑_𝑞(𝑥,𝑥)+12𝑘²−1𝜑_𝑞(𝑥,0)+6𝜑_𝑞(2𝑥,0)+12𝜑_𝑞(𝑥,𝑘𝑥), (3.14) for all 𝑥∈𝑋. Let 𝑔∶𝑋→𝑌 be a function defined by 𝑔(𝑥)∶=𝑓(2𝑥)−16𝑓(𝑥) for all 𝑥∈𝑋. From (3.13), we conclude that ‖𝑔(2𝑥)−4𝑔(𝑥)‖_𝑌≤𝑀²𝜓_𝑞(𝑥), (3.15) for all 𝑥∈𝑋. If we replace 𝑥 in (3.15) by 𝑥/2^𝑛+1 and multiply both sides of (3.15) by 4^𝑛, then we get 4^𝑛+1𝑔𝑥2^𝑛+1−4^𝑛𝑔𝑥2^𝑛_𝑌≤𝑀²4^𝑛𝜓_𝑞𝑥2^𝑛+1, (3.16) for all 𝑥∈𝑋 and all non-negative integers 𝑛. Since 𝑌 is a 𝑝-Banach space, the inequality (3.16) gives 4^𝑛+1𝑔𝑥2^𝑛+1−4^𝑚𝑔𝑥2^𝑚𝑝_𝑌≤^𝑛_𝑖=𝑚4^𝑖+1𝑔𝑥2^𝑖+1−4^𝑖𝑔𝑥2^𝑖𝑝_𝑌≤𝑀^𝑛2𝑝_𝑖=𝑚4^𝑖𝑝𝜓^𝑝_𝑞𝑥2^𝑖+1, (3.17) for all nonnegative integers 𝑛 and 𝑚 with 𝑛≥𝑚 and all 𝑥∈𝑋. Since 0<𝑝≤1, by Lemma 1.2 and (3.14), we conclude that 𝜓^𝑝_𝑞(𝑥)≤1𝑘^2𝑝𝑘²−1^𝑝12𝑘^2𝑝𝜑^𝑝_𝑞(𝑥,𝑥)+12𝑘²−1^𝑝𝜑^𝑝_𝑞(𝑥,0)+6^𝑝𝜑^𝑝_𝑞(2𝑥,0)+12^𝑝𝜑^𝑝_𝑞(𝑥,𝑘𝑥), (3.18) for all 𝑥∈𝑋. Therefore, it follows from (3.3) and (3.18) that ^∞_𝑖=14^𝑖𝑝𝜓^𝑝_𝑞𝑥2^𝑖<∞, (3.19) for all 𝑥∈𝑋. It follows from (3.17) and (3.19) that the sequence {4^𝑛𝑔(𝑥/2^𝑛)} is a Cauchy for all 𝑥∈𝑋. Since 𝑌 is complete, the sequence {4^𝑛𝑔(𝑥/2^𝑛)} converges for all 𝑥∈𝑋. So one can define a function 𝑄∶𝑋→𝑌 by 𝑄(𝑥)=lim_𝑛→∞4^𝑛𝑔𝑥2^𝑛, (3.20) for all 𝑥∈𝑋. Letting 𝑚=0 and passing the limit 𝑛→∞ in (3.17), we get 𝑔(𝑥)−𝑄(𝑥)^𝑝_𝑌≤𝑀^∞2𝑝_𝑖=04^𝑖𝑝𝜓^𝑝_𝑞𝑥2^𝑖+1=𝑀^2𝑝4^𝑝∞_𝑖=14^𝑖𝑝𝜓^𝑝_𝑞𝑥2^𝑖, (3.21) for all 𝑥∈𝑋. Thus (3.5) follows from (3.18) and (3.21). Now we show that 𝑄 is quadratic. It follows from (3.16), (3.19) and (3.20) that ‖𝑄(2𝑥)−4𝑄(𝑥)‖_𝑌=lim_𝑛→∞4^𝑛𝑔𝑥2^𝑛−1−4^𝑛+1𝑔𝑥2^𝑛_𝑌=4lim_𝑛→∞4^𝑛−1𝑔𝑥2^𝑛−1−4^𝑛𝑔𝑥2^𝑛_𝑌≤𝑀²lim_𝑛→∞4^𝑛𝜓_𝑞𝑥2^𝑛=0, (3.22) for all 𝑥∈𝑋. So, 𝑄(2𝑥)=4𝑄(𝑥), (3.23) for all 𝑥∈𝑋. On the other hand, it follows from (3.2), (3.4) and (3.20) that 𝐷_𝑄𝑥,𝑦_𝑌=lim_𝑛→∞4^𝑛𝐷_𝑔𝑥2^𝑛,𝑦2^𝑛_𝑌=lim_𝑛→∞4^𝑛𝐷_𝑓𝑥2^𝑛−1,𝑦2^𝑛−1−16𝐷_𝑓𝑥2^𝑛,𝑦2^𝑛_𝑌≤𝑀lim_𝑛→∞4^𝑛𝐷_𝑓𝑥2^𝑛−1,𝑦2^𝑛−1_𝑌+16𝐷_𝑓𝑥2^𝑛,𝑦2^𝑛_𝑌≤𝑀lim_𝑛→∞4^𝑛𝜑_𝑞𝑥2^𝑛−1,𝑦2^𝑛−1+16𝜑_𝑞𝑥2^𝑛,𝑦2^𝑛=0, (3.24) for all 𝑥,𝑦∈𝑋. Hence the function 𝑄 satisfies (1.7). Thus, by Theorem 2.1, the function 𝑥⇝𝑄(2𝑥)−16𝑄(𝑥) is quadratic. Therefore, (3.23) implies that the function 𝑄 is quadratic.
Now, to prove the uniqueness property of 𝑄, let 𝑄^′∶𝑋→𝑌 be another quadratic function satisfying (3.5). It follows from (3.3) thatlim_𝑛→∞4^∞𝑛𝑝_𝑖=14^𝑖𝑝𝜑^𝑝_𝑞𝑢2^𝑛+𝑖,𝑦2^𝑛+𝑖=lim^∞_{𝑛→∞𝑖=𝑛+1}4^𝑖𝑝𝜑^𝑝_𝑞𝑢2^𝑖,𝑦2^𝑖=0, (3.25) for all (𝑢,𝑦)∈{(𝑥,0),(2𝑥,0),(𝑥,𝑥),(𝑥,𝑘𝑥)∶𝑥∈𝑋}. Hence, lim_𝑛→∞4^𝑛𝑝𝜓_𝑞𝑥2^𝑛=0, (3.26) for all 𝑥∈𝑋. It follows from (3.5), (3.20) and (3.26) that 𝑄(𝑥)−𝑄^′(𝑥)^𝑝_𝑌=lim_𝑛→∞4^𝑛𝑝𝑔𝑥2^𝑛−𝑄^′𝑥2^𝑛𝑝_𝑌≤𝑀^2𝑝4^𝑝lim_𝑛→∞4^𝑛𝑝𝜓_𝑞𝑥2^𝑛=0, (3.27) for all 𝑥∈𝑋. So 𝑄=𝑄^′.
For 𝑗=−1, we can prove the theorem by a similar argument.

Corollary 3.2. Let 𝜃,𝑟,𝑠 be nonnegative real numbers such that 𝑟, 𝑠>2 or 𝑟, 𝑠<2. Suppose that an even function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality 𝐷_𝑓𝑥,𝑦_𝑌≤𝜃‖𝑥‖^𝑟_𝑋+𝑦^𝑠_𝑋, (3.28) for all 𝑥,𝑦∈𝑋. Then there exists a unique quadratic function 𝑄∶𝑋→𝑌 satisfying 𝑓(2𝑥)−16𝑓(𝑥)−𝑄(𝑥)_𝑌≤𝑀²𝜃𝑘²𝑘²𝛾−1^𝑞(𝑥), (3.29) for all 𝑥∈𝑋, where 𝛾_𝑞(𝑥)=12^𝑝𝑘^2𝑝+(𝑘²−1)^𝑝+2^𝑟−1𝑝+14^𝑝−2^𝑟𝑝‖𝑥‖_𝑋^𝑟𝑝+12^𝑝𝑘^2𝑝+𝑘^𝑠𝑝4^𝑝−2^𝑠𝑝‖𝑥‖_𝑋^{𝑠𝑝1/𝑝}. (3.30)

Proof. In Theorem 3.1, putting 𝜑_𝑞(𝑥,𝑦)∶=𝜃(‖𝑥‖^𝑟_𝑋+‖𝑦‖^𝑠_𝑋) for all 𝑥,𝑦∈𝑋, we get the desired result.

Corollary 3.3. Let 𝜃≥0 and 𝑟,𝑠>0 be real numbers such that 𝜆∶=𝑟+𝑠≠2. Suppose that an even function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality 𝐷_𝑓𝑥,𝑦_𝑌≤𝜃‖𝑥‖^𝑟_𝑋𝑦^𝑠_𝑋, (3.31) for all 𝑥,𝑦∈𝑋. Then there exists a unique quadratic function 𝑄∶𝑋→𝑌 satisfying 𝑓(2𝑥)−16𝑓(𝑥)−𝑄(𝑥)_𝑌≤𝑀²𝜃𝑘²𝑘²−112^𝑝𝑘^2𝑝+𝑘^𝑠𝑝4^𝑝−2^{𝜆𝑝1/𝑝}‖𝑥‖^𝜆_𝑋, (3.32) for all 𝑥∈𝑋.

Proof. In Theorem 3.1, taking 𝜑_𝑞(𝑥,𝑦)∶=𝜃‖𝑥‖^𝑟_𝑋‖𝑦‖^𝑠_𝑋, for all 𝑥,𝑦∈𝑋, we arrive at the desired result.

Theorem 3.4. Let 𝑗∈{−1,1} be fixed and let 𝜑_𝑣∶𝑋×𝑋→[0,∞) be a function such that lim_𝑛→∞16^𝑛𝑗𝜑_𝑣𝑥2^𝑛𝑗,𝑦2^𝑛𝑗=0, (3.33) for all 𝑥,𝑦∈𝑋 and ^∞_{𝑖=(1+𝑗)/2}16^𝑖𝑝𝑗𝜑^𝑝_𝑣𝑢2^𝑖𝑗,𝑦2^𝑖𝑗<∞, (3.34) for all (𝑢,𝑦)∈{(𝑥,0),(2𝑥,0),(𝑥,𝑥),(𝑥,𝑘𝑥)∶𝑥∈𝑋}. Suppose that an even function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality 𝐷_𝑓(𝑥,y)_𝑌≤𝜑_𝑣𝑥,𝑦, (3.35) for all 𝑥,𝑦∈𝑋. Then there exists a unique quartic function 𝑉∶𝑋→𝑌 such that 𝑓(2𝑥)−4𝑓(𝑥)−𝑉(𝑥)_𝑌≤𝑀²𝜓16^𝑣(𝑥)^1/𝑝, (3.36) for all 𝑥∈𝑋, where 𝜓_𝑣(𝑥)∶=^∞_{𝑖=(1+𝑗)/2}16^𝑖𝑝𝑗𝑘^2𝑝𝑘²−1^𝑝12𝑘^2𝑝𝜑^𝑝_𝑣𝑥2^𝑖𝑗,𝑥2^𝑖𝑗+12𝑘²−1^𝑝𝜑^𝑝_𝑣𝑥2^𝑖𝑗,0+6^𝑝𝜑^𝑝_𝑣2𝑥2^𝑖𝑗,0+12^𝑝𝜑^𝑝_𝑣𝑥2^𝑖𝑗,𝑘𝑥2^𝑖𝑗. (3.37)

Proof. Being similar to the proof of Theorem 3.1, we omit its proof.

Corollary 3.5. Let 𝜃,𝑟,𝑠 be nonnegative real numbers such that 𝑟,𝑠>4 or 𝑟,𝑠<4. Suppose that an even function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality (3.28) for all 𝑥,𝑦∈𝑋. Then there exists a unique quartic function 𝑉∶𝑋→𝑌 satisfying 𝑓(2𝑥)−4𝑓(𝑥)−𝑉(𝑥)_𝑌≤𝑀²𝜃𝑘²𝑘²𝛾−1^𝑣(𝑥), (3.38) for all 𝑥∈𝑋, where 𝛾_𝑣(𝑥)=12^𝑝𝑘^2𝑝+𝑘²−1^𝑝+2^𝑟−1𝑝+116^𝑝−2^𝑟𝑝‖𝑥‖_𝑋^𝑟𝑝+12^𝑝𝑘^2𝑝+𝑘^𝑠𝑝16^𝑝−2^𝑠𝑝‖𝑥‖_𝑋^{𝑠𝑝1/𝑝}, (3.39) for all 𝑥∈𝑋.

Corollary 3.6. Let 𝜃≥0 and 𝑟,𝑠>0 be real numbers such that 𝜆∶=𝑟+𝑠≠4. Suppose that an even function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality (3.31) for all 𝑥,𝑦∈𝑋. Then, there exists a unique quartic function 𝑉∶𝑋→𝑌 satisfying 𝑓(2𝑥)−4𝑓(𝑥)−𝑉(𝑥)_𝑌≤𝑀²𝜃𝑘²𝑘²−112^𝑝𝑘^2𝑝+𝑘^𝑠𝑝16^𝑝−2^{𝜆𝑝1/𝑝}‖𝑥‖^𝜆_𝑋, (3.40) for all 𝑥∈𝑋.

Now, we are ready to prove the main theorem concerning the stability problem for (1.7).

Theorem 3.7. Let 𝑗∈{−1,1} be fixed and let 𝜑∶𝑋×𝑋→[0,∞) be a function such that lim_𝑛→∞1−𝑗24^𝑛𝑗𝜑𝑥2^𝑛𝑗,𝑦2^𝑛𝑗+1+𝑗216^𝑛𝑗𝜑𝑥2^𝑛𝑗,𝑦2^𝑛𝑗=0, (3.41) for all 𝑥,𝑦∈𝑋 and ^∞_{𝑖=(1+𝑗)/2}1−𝑗24^𝑖𝑝𝑗𝜑^𝑝𝑢2^𝑖𝑗,𝑦2^𝑖𝑗+1+𝑗216^𝑖𝑝𝑗𝜑^𝑝𝑢2^𝑖𝑗,𝑦2^𝑖𝑗<∞, (3.42) for all (𝑢,𝑦)∈{(𝑥,0),(2𝑥,0),(𝑥,𝑥),(𝑥,𝑘𝑥)∶𝑥∈𝑋}. Suppose that an even function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality 𝐷_𝑓(𝑥,𝑦)_𝑌≤𝜑𝑥,𝑦, (3.43) for all 𝑥,𝑦∈𝑋. Then, there exists a unique quadratic function 𝑄∶𝑋→𝑌 and a unique quartic function 𝑉∶𝑋→𝑌 such that 𝑓(𝑥)−𝑄(𝑥)−𝑉(𝑥)_𝑌≤𝑀³4𝜓192^𝑞(𝑥)^1/𝑝+𝜓_𝑣(𝑥)^1/𝑝, (3.44) for all 𝑥∈𝑋, where 𝜓_𝑞(𝑥)∶=^∞_{𝑖=(1+𝑗)/2}4^𝑖𝑝𝑗𝑘^2𝑝𝑘²−1^𝑝12𝑘^2𝑝𝜑^𝑝𝑥2^𝑖𝑗,𝑥2^𝑖𝑗+12𝑘²−1^𝑝𝜑^𝑝𝑥2^𝑖𝑗,0+6^𝑝𝜑^𝑝2𝑥2^𝑖𝑗,0+12^𝑝𝜑^𝑝𝑥2^𝑖𝑗,𝑘𝑥2^𝑖𝑗,𝜓_𝑣(𝑥)∶=^∞_{𝑖=(1+𝑗)/2}16^𝑖𝑝𝑗𝑘^2𝑝𝑘²−1^𝑝12𝑘^2𝑝𝜑^𝑝𝑥2^𝑖𝑗,𝑥2^𝑖𝑗+12𝑘²−1^𝑝𝜑^𝑝𝑥2^𝑖𝑗,0+6^𝑝𝜑^𝑝2𝑥2^𝑖𝑗,0+12^𝑝𝜑^𝑝𝑥2^𝑖𝑗,𝑘𝑥2^𝑖𝑗. (3.45)

Proof. By Theorems 3.1 and 3.4, there exists a quadratic function 𝑄₀∶𝑋→𝑌 and a quartic function 𝑉₀∶𝑋→𝑌 such that 𝑓(2𝑥)−16𝑓(𝑥)−𝑄₀(𝑥)_Y≤𝑀²4𝜓_𝑞(𝑥)^1/𝑝,𝑓(2𝑥)−4𝑓(𝑥)−𝑉₀(𝑥)_𝑌≤𝑀²𝜓16^𝑣(𝑥)^1/𝑝, (3.46) for all 𝑥∈𝑋. Therefore, it follows from (3.46) that 𝑓(𝑥)+1𝑄12⁰(𝑥)−1𝑉12⁰(𝑥)_𝑌≤𝑀³4𝜓192^𝑞(𝑥)^1/𝑝+𝜓_𝑣(𝑥)^1/𝑝, (3.47) for all 𝑥∈𝑋. Thus we obtain (3.44) by letting 𝑄(𝑥)=−(1/12)𝑄₀(𝑥) and 𝑉(𝑥)=(1/12)𝑉₀(𝑥) for all 𝑥∈𝑋.
To prove the uniqueness property of 𝑄 and 𝑉, let 𝑄^′,𝑉^′∶𝑋→𝑌 be another quadratic and quartic functions satisfying (3.44). Let 𝑄=𝑄−𝑄^′ and 𝑉=𝑉−𝑉^′. Hence,𝑄(𝑥)+𝑉(𝑥)_𝑌≤𝑀𝑓(𝑥)−𝑄(𝑥)−𝑉(𝑥)_𝑌+𝑓(𝑥)−𝑄^′(𝑥)−𝑉^′(𝑥)_𝑌≤𝑀⁴4𝜓96^𝑞(𝑥)^1/𝑝+𝜓_𝑣(𝑥)^1/𝑝, (3.48) for all 𝑥∈𝑋. Since lim_𝑛→∞4^𝑛𝑝𝑗𝜓_𝑞(𝑥/2^𝑛)=lim_𝑛→∞16^𝑛𝑝𝑗𝜓_𝑣(𝑥/2^𝑛)=0, for all 𝑥∈𝑋, we figure out that lim_𝑛→∞16^𝑛𝑄𝑥2^𝑛+𝑉𝑥2^𝑛_𝑌=0, (3.49) for all 𝑥∈𝑋. Therefore, we get 𝑉=0 and then 𝑄=0.

Corollary 3.8. Let 𝜃,𝑟,𝑠 be nonnegative real numbers such that 𝑟,𝑠>4 or 2<𝑟, 𝑠<4 or 𝑟,𝑠<2. Suppose that an even function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality (3.28), for all 𝑥,𝑦∈𝑋. Then, there exists a unique quadratic function 𝑄∶𝑋→𝑌 and a unique quartic function 𝑉∶𝑋→𝑌 such that 𝑓(𝑥)−𝑄(𝑥)−𝑉(𝑥)_𝑌≤𝑀³𝜃12𝑘²𝑘²𝛾−1^𝑞(𝑥)+𝛾_𝑣(𝑥), (3.50) for all 𝑥∈𝑋, where 𝛾_𝑞(𝑥) and 𝛾_𝑣(𝑥) are defined as in Corollaries 3.2 and 3.5.

Corollary 3.9. Let 𝜃≥0 and 𝑟,𝑠>0 be non-negative real numbers such that 𝜆∶=𝑟+𝑠∈(0,2)∪(2,4)∪(4,∞). Suppose that an even function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality (3.31) for all 𝑥,𝑦∈𝑋. Then there exist a unique quadratic function 𝑄∶𝑋→𝑌 and a unique quartic function 𝑉∶𝑋→𝑌 such that 𝑓(𝑥)−𝑄(𝑥)−𝑉(𝑥)_𝑌≤𝑀³𝜃12𝑘²𝑘²−112^𝑝𝑘^2𝑝+𝑘^𝑠𝑝4^𝑝−2^{𝜆𝑝1/𝑝}+12^𝑝𝑘^2𝑝+𝑘^𝑠𝑝16^𝑝−2^{𝜆𝑝1/𝑝}‖𝑥‖^𝜆_𝑋, (3.51) for all 𝑥∈𝑋.

Corollary 3.10. Suppose that an even function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality 𝐷_𝑓𝑥,𝑦_𝑋≤𝜀, (3.52) for all 𝑥,𝑦∈𝑋 where 𝜀>0. Then there exist a unique quadratic function 𝑄∶𝑋→𝑌 and a unique quartic function 𝑉∶𝑋→𝑌 such that 𝑓(𝑥)−𝑄(𝑥)−𝑉(𝑥)_𝑌≤𝑀³𝜀𝑘²𝑘²𝑘−1^2𝑝+𝑘²−1^𝑝+2^−𝑝+14^𝑝−1^1/𝑝+𝑘^2𝑝+(𝑘²−1)^𝑝+2^−𝑝+116^𝑝−1^1/𝑝, (3.53) for all 𝑥∈𝑋.

Acknowledgment

Hark-Mahn Kim was supported by Basic Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (no. 2011-0002614).