Generalized Hyers-Ulam Stability of a Mixed Type Functional Equation

Abstract

We investigate the stability of a functional equation 𝑓(𝑥+𝑦+𝑧) + 𝑓(𝑥−𝑦+𝑧) + 𝑓(𝑥+𝑦−𝑧)+𝑓(−𝑥+𝑦+𝑧)=3𝑓(𝑥)+𝑓(−𝑥)+3𝑓(𝑦)+𝑓(−𝑦)+3𝑓(𝑧)+𝑓(−𝑧) by applying the direct method in the sense of Hyers and Ulam.

1. Introduction

In 1940, Ulam [1] gave a wide ranging talk before the mathematics club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the question 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 function ℎ∶𝐺₁→𝐺₂ satisfies the inequality 𝑑(ℎ(𝑥𝑦),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?

The Ulam’s problem for the Cauchy additive functional equation was solved by Hyers under the assumption that 𝐺₁ and 𝐺₂ are Banach spaces. Indeed, Hyers [2] proved that every solution of the inequality ‖𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)‖≤𝜀 (for all 𝑥 and 𝑦) can be approximated by an additive function. In this case, the Cauchy additive functional equation, 𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦), is said to satisfy the Hyers-Ulam stability.

Thereafter, Rassias [3] attempted to weaken the condition for the bound of norm of the Cauchy difference as follows: 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜀‖𝑥‖^𝑝+𝑦^𝑝, (1) and he proved that Hyers’ theorem is also true for this case. Indeed, Rassias proved the generalized Hyers-Ulam stability (or the Hyers-Ulam-Rassias stability) of the Cauchy additive functional equation between Banach spaces. We here remark that a paper of Aoki [4] was published concerning the generalized Hyers-Ulam stability of the Cauchy functional equation earlier than Rassias’ paper.

The stability concept that was introduced by Rassias’ theorem provided a large influence to a number of mathematicians to develop the notion of what is known today with the term generalized Hyers-Ulam stability of functional equations. Since then, the stability problems of several functional equations have been extensively investigated by several mathematicians (e.g., see [5–10] and the references therein).

Almost all subsequent proofs in this very active area have used the Hyers’ method presented in [2]. Namely, starting from the given mapping 𝑓 that approximately satisfies a given functional equation, a solution 𝐹 of the functional equation is explicitly constructed by using the formula: 𝐹(𝑥)∶=lim_𝑛→∞12^𝑛𝑓2^𝑛𝑥or𝐹(𝑥)∶=lim_𝑛→∞2^𝑛𝑓𝑥2^𝑛, (2) which approximates the mapping 𝑓. This method of Hyers is called the direct method.

We remark that another method for proving the Hyers-Ulam stability of various functional equations was introduced by Baker [11], which is called the fixed-point method. This method is very powerful technique of proving the stability of functional equations (see [12, 13]).

Now we consider the following functional equation: 𝑓(𝑥)()𝑦(𝑧)(),𝑥+𝑦+𝑧+𝑓𝑥−𝑦+𝑧+𝑓𝑥+𝑦−𝑧+𝑓−𝑥+𝑦+𝑧=3𝑓+𝑓−𝑥+3𝑓+𝑓−𝑦+3𝑓+𝑓−𝑧 (3) which is called the mixed type functional equation. The mapping 𝑓(𝑥)=𝑎𝑥²+𝑏𝑥 is a solution of this functional equation, where 𝑎,𝑏 are real constants. Every solution of (3) will be called a quadratic-additive mapping.

In 1998, Jung [14] proved the stability of (3) by decomposing 𝑓 into the odd and even parts. In his proof, using the direct method, an additive mapping 𝐴 and a quadratic mapping 𝑄 are separately constructed from the odd and even parts of 𝑓, and then 𝐴 and 𝑄 are combined to provide a quadratic-additive mapping 𝐹 which is close to 𝑓.

In this paper, we will prove the generalized Hyers-Ulam stability of (3) by making use of the direct method. In particular, we will approximate the given mapping 𝑓 by a solution 𝐹 of (3) without decomposing 𝑓 into its odd and even parts, while in the Jung’s paper [14] the mapping 𝑓 was decomposed into the odd and even parts, and each of them was separately approximated by the corresponding part of a solution 𝐹 of (3).

2. Main Results

Throughout this paper, let 𝑋 be a (real or complex) normed space and 𝑌 a Banach space. For an arbitrary 𝑝∈ℝ, we define 𝑠∶=sign(2−𝑝) and 𝑡∶=sign(1−𝑝).

For a given mapping 𝑓∶𝑋→𝑌, we use the following abbreviations: 𝑓_𝑜(𝑥)𝑓(𝑥)()∶=−𝑓−𝑥2,𝑓_𝑒(𝑥)𝑓(𝑥)()∶=+𝑓−𝑥2,𝐽_𝑛𝑓(𝑥)9∶=^−𝑠𝑛2𝑓3^𝑠𝑛𝑥+𝑓−3^𝑠𝑛𝑥+3^−𝑡𝑛2𝑓3^𝑡𝑛𝑥−𝑓−3^𝑡𝑛𝑥,(𝑥)𝑦,(𝑥)𝑦,(𝑥)()𝑦(𝑧)()𝐴𝑓𝑥,𝑦∶=𝑓𝑥+𝑦−𝑓−𝑓𝑄𝑓𝑥,𝑦∶=𝑓𝑥+𝑦+𝑓𝑥−𝑦−2𝑓−2𝑓𝐷𝑓𝑥,𝑦,𝑧∶=𝑓𝑥+𝑦+𝑧+𝑓𝑥−𝑦+𝑧+𝑓𝑥+𝑦−𝑧+𝑓−𝑥+𝑦+𝑧−3𝑓−𝑓−𝑥−3𝑓−𝑓−𝑦−3𝑓−𝑓−𝑧 (4) for all 𝑥,𝑦,𝑧∈𝑋.

As we stated in the previous section, 𝑓 is called a quadratic-additive mapping provided that 𝑓 satisfies the functional equation 𝐷𝑓(𝑥,𝑦,𝑧)=0 for all 𝑥,𝑦,𝑧∈𝑋.

Proposition 1. A mapping 𝑓∶𝑋→𝑌 is a solution of (3) if and only if 𝑓_𝑒 is a quadratic mapping and 𝑓_𝑜 is an additive mapping.

Proof. Assume that 𝑓∶𝑋→𝑌 is a solution of (3). Then we have 𝑄𝑓_𝑒𝑥,𝑦,𝑧=𝐷𝑓_𝑒𝑥,𝑦,02=0,𝐴𝑓_𝑜𝑥,𝑦,𝑧=𝐷𝑓_𝑜𝑥,𝑦,02=0 (5) for all 𝑥,𝑦,𝑧∈𝑋, that is, 𝑓_𝑒 is a quadratic mapping and 𝑓_𝑜 is an additive mapping.
Conversely, assume that 𝑓_𝑒 is a quadratic mapping and 𝑓_𝑜 is an additive mapping. Then we get 𝐷𝑓𝑥,𝑦,𝑧=𝐷𝑓_𝑒𝑥,𝑦,𝑧+𝐷𝑓_𝑜𝑥,𝑦,𝑧=𝑄𝑓_𝑒𝑥+𝑦,𝑧+𝑄𝑓_𝑒𝑥−𝑦,𝑧+2𝑄𝑓_𝑒𝑥,𝑦+𝐴𝑓_𝑜𝑥+𝑦,𝑧+𝐴𝑓_𝑜𝑥+𝑦,−𝑧+𝐴𝑓_𝑜𝑥−𝑦,𝑧+𝐴𝑓_𝑜−𝑥+𝑦,𝑧+2𝐴𝑓_𝑜𝑥,𝑦=0 (6) for all 𝑥,𝑦,𝑧∈𝑋; that is, 𝑓 is a solution of (3).

We first prove the following lemma.

Lemma 2. If a mapping 𝑓∶𝑋→𝑌 satisfies 𝐷𝑓(𝑥,𝑦,𝑧)=0 for all 𝑥,𝑦,𝑧∈𝑋⧵{0} and 𝑓(0)=0, then 𝑓 is a quadratic-additive mapping.

Proof. Using the hypothesis, we have 𝑓()(𝑥)()=2𝑥−3𝑓−𝑓−𝑥11()()()()()()−3112𝐷𝑓4𝑥,3𝑥,𝑥−𝐷𝑓4𝑥,2𝑥,2𝑥−𝐷𝑓2𝑥,2𝑥,2𝑥+2𝐷𝑓2𝑥,𝑥,𝑥+3𝐷𝑓𝑥,𝑥,𝑥+𝐷𝑓−𝑥,−𝑥,−𝑥()()()()()()112𝐷𝑓−4𝑥,−3𝑥,−𝑥−𝐷𝑓−4𝑥,−2𝑥,−2𝑥−𝐷𝑓−2𝑥,−2𝑥,−2𝑥+2𝐷𝑓−2𝑥,−𝑥,−𝑥+3𝐷𝑓−𝑥,−𝑥,−𝑥+𝐷𝑓𝑥,𝑥,𝑥=0 (7) for all 𝑥∈𝑋⧵{0}. Furthermore, by the last equality, we get 𝑦𝐷𝑓𝑥,𝑦,0=𝐷𝑓𝑥,2,𝑦2𝑥+𝐷𝑓𝑦,2,𝑥2(𝑥)𝑥−2𝑓+6𝑓2−𝑥+2𝑓2𝑦𝑦−2𝑓+6𝑓2−𝑦+2𝑓2=0 (8) for all 𝑥,𝑦∈𝑋⧵{0}. Since 𝐷𝑓(𝑥,𝑦,𝑧) is invariant with respect to the permutation of (𝑥,𝑦,𝑧), it holds that 𝐷𝑓(0,𝑦,𝑧)=𝐷𝑓(𝑥,0,𝑧)=0 for all 𝑥,𝑦,𝑧∈𝑋⧵{0}. It is also easy to show that 𝐷𝑓(𝑥,0,0)=0, 𝐷𝑓(0,𝑦,0)=0, 𝐷𝑓(0,0,𝑧)=0, and 𝐷𝑓(0,0,0)=0 for all 𝑥,𝑦,𝑧∈𝑋⧵{0} as we desired.

In the following theorem, we can prove the generalized Hyers-Ulam stability of the functional equation (3) by making use of the direct method.

Theorem 3. If a mapping 𝑓∶𝑋→𝑌 satisfies 𝑓(0)=0 and 𝐷𝑓𝑥,𝑦,𝑧≤‖𝑥‖^𝑝+𝑦^𝑝+‖𝑧‖^𝑝 (9) for all 𝑥,𝑦,𝑧∈𝑋⧵{0} with a real constant 𝑝∉{1,2}, then there exists a unique quadratic-additive mapping 𝐹∶𝑋→𝑌 such that 𝑓(𝑥)(𝑥≤3‖𝑥‖−𝐹)_𝑝3^𝑝,3−9𝑓𝑜𝑟𝑝>29−3^𝑝+33^𝑝‖𝑥‖−3^𝑝,3‖𝑥‖𝑓𝑜𝑟1<𝑝<2_𝑝3−3^𝑝𝑓𝑜𝑟0≤𝑝<1 (10) for all 𝑥∈𝑋⧵{0}. Moreover, if 𝑝<0, then 𝑓 itself is a quadratic-additive mapping.

Proof. Let us define 𝜏_𝑠,𝑛∶=𝑠(𝑛+1/2)−1/2, where 𝑠∈{−1,1}. From the definitions of 𝐽_𝑛𝑓(𝑥) and 𝐷𝑓(𝑥,𝑦,𝑧), we have 𝐽_𝑛𝑓(𝑥)−𝐽_𝑛+1𝑓(𝑥)1=−29^{𝜏−𝑠,𝑛}𝐷𝑓3^{𝜏𝑠,𝑛}𝑥,3^{𝜏𝑠,𝑛}𝑥,3^{𝜏𝑠,𝑛}𝑥+𝐷𝑓−3^{𝜏𝑠,𝑛}𝑥,−3^{𝜏𝑠,𝑛}𝑥,−3^{𝜏𝑠,𝑛}𝑥𝑠+3^{𝜏−𝑡,𝑛}𝐷𝑓3^{𝜏𝑡,𝑛}𝑥,3^{𝜏𝑡,𝑛}𝑥,3^{𝜏𝑡,𝑛}𝑥−𝐷𝑓−3^{𝜏𝑡,𝑛}𝑥,−3^{𝜏𝑡,𝑛}𝑥,−3^{𝜏𝑡,𝑛}𝑥𝑡 (11) for all 𝑥∈𝑋⧵{0} and 𝑛∈ℕ₀. It follows from (9) and (11) that 𝐽_𝑛𝑓(𝑥)−𝐽_𝑛+𝑚𝑓(𝑥=_{𝑛+𝑚−1𝑗=𝑛})𝐽_𝑗𝑓(𝑥)−𝐽_𝑗+1𝑓(𝑥≤1)2^{𝑛+𝑚−1}_𝑗=𝑛9^{𝜏−𝑠,𝑗}𝐷𝑓3^{𝜏𝑠,𝑗}𝑥,3^{𝜏𝑠,𝑗}𝑥,3^{𝜏𝑠,𝑗}𝑥𝑠+3^{𝜏−𝑡,𝑗}𝐷𝑓3^{𝜏𝑡,𝑗}𝑥,3^{𝜏𝑡,𝑗}𝑥,3^{𝜏𝑡,𝑗}𝑥𝑡+9^{𝜏−𝑠,𝑗}𝐷𝑓−3^{𝜏𝑠,𝑗}𝑥,−3^{𝜏𝑠,𝑗}𝑥,−3^{𝜏𝑠,𝑗}𝑥𝑠−3^{𝜏−𝑡,𝑗}𝐷𝑓−3^{𝜏𝑡,𝑗}𝑥,−3^{𝜏𝑡,𝑗}𝑥,−3^{𝜏𝑡,𝑗}𝑥𝑡≤^{𝑛+𝑚−1}_𝑗∑_=𝑛3^−𝑗3^𝑗𝑥^𝑝for𝑝<1_{𝑛+𝑚−1},_𝑗∑_=𝑛3^−2𝑗−13^𝑗𝑥^𝑝+3^𝑗+13^−𝑗−1𝑥^𝑝for1<𝑝<2_{𝑛+𝑚−1},_𝑗∑_=𝑛3^2𝑗+13^−𝑗−1𝑥^𝑝≤3for𝑝>2_𝑛𝑝‖𝑥‖^𝑝3^𝑛−13−3^𝑝,3for𝑝<1_𝑛𝑝‖𝑥‖^𝑝3^2𝑛−19−3^𝑝+3^𝑛+1‖𝑥‖^𝑝3^𝑛𝑝3^𝑝,3−3for1<𝑝<2_2𝑛+1‖𝑥‖^𝑝3^𝑛𝑝3^𝑝−9for𝑝>2 (12) for all 𝑥∈𝑋⧵{0}. So, it is easy to show that the sequence {𝐽_𝑛𝑓(𝑥)} is a Cauchy sequence for all 𝑥∈𝑋⧵{0}.
Since 𝑌 is complete and 𝑓(0)=0, the sequence {𝐽_𝑛𝑓(𝑥)} converges for all 𝑥∈𝑋. Hence, we can define a mapping 𝐹∶𝑋→𝑌 by 𝐹(𝑥)∶=lim_𝑛→∞𝐽_𝑛𝑓(𝑥) (13) for all 𝑥∈𝑋. Moreover, if we put 𝑛=0 and let 𝑚→∞ in (12), we obtain the inequality (10).
From the definition of 𝐹, we get 𝐷𝐹𝑥,𝑦,𝑧=lim_𝑛→∞9^−𝑠𝑛2𝐷𝑓3^𝑠𝑛𝑥,3^𝑠𝑛𝑦,3^𝑠𝑛𝑧+𝐷𝑓−3^𝑠𝑛𝑥,−3^𝑠𝑛𝑦,−3^𝑠𝑛𝑧+lim_𝑛→∞3^−𝑡𝑛2𝐷𝑓3^𝑡𝑛𝑥,3^𝑡𝑛𝑦,3^𝑡𝑛𝑧−𝐷𝑓−3^𝑡𝑛𝑥,−3^𝑡𝑛𝑦,−3^𝑡𝑛𝑧≤lim_𝑛→∞3^{−𝑠(2−𝑝)𝑛}+3^{−𝑡𝑛(1−𝑝)}‖𝑥‖^𝑝+𝑦^𝑝+‖𝑧‖^𝑝=0 (14) for all 𝑥,𝑦,𝑧∈𝑋⧵{0}. By Lemma 2, 𝐹 is a quadratic-additive mapping.
Now, we will show that 𝐹 is uniquely determined. Let 𝐹^′∶𝑋→𝑌 be another quadratic-additive mapping satisfying (10). It is easy to show that 𝐹^′(0)=0 for all quadratic-additive mapping 𝐹^′. It follows from (11) that 𝐹^′(𝑥)−𝐽_𝑛𝐹^′(𝑥)=^𝑛−1_𝑗=0𝐽_𝑗𝐹^′(𝑥)−𝐽_𝑗+1𝐹^′(𝑥)1=−2^𝑛−1_𝑗=09^{𝜏−𝑠,𝑗}𝐷𝐹^′3^{𝜏𝑠,𝑗}𝑥,3^{𝜏𝑠,𝑗}𝑥,3^{𝜏𝑠,𝑗}𝑥+𝐷𝐹^′−3^{𝜏𝑠,𝑗}𝑥,−3^{𝜏𝑠,𝑗}𝑥,−3^{𝜏𝑠,𝑗}𝑥𝑠+3^{𝜏−𝑡,𝑗}𝐷𝐹^′3^{𝜏𝑡,𝑗}𝑥,3^{𝜏𝑡,𝑗}𝑥,3^{𝜏𝑡,𝑗}𝑥−𝐷𝐹^′−3^{𝜏𝑡,𝑗}𝑥,−3^{𝜏𝑡,𝑗}𝑥,−3^{𝜏𝑡,𝑗}𝑥𝑡=0 (15) for all 𝑛∈ℕ and 𝑥∈𝑋. Since 𝐹 and 𝐹^′ are quadratic-additive, if we replace 𝑥 with 3^𝑛𝑥 in (10), then we have 𝐹(𝑥)−𝐹^′(𝑥=𝐽_𝑛)𝐹(𝑥)−𝐽_𝑛𝐹^′(𝑥≤9_−𝑠𝑛)2𝐹−𝑓3^𝑠𝑛𝑥+𝑓−𝐹^′3^𝑠𝑛𝑥+𝐹−𝑓−3^𝑠𝑛𝑥+𝑓−𝐹^′−3^𝑠𝑛𝑥+2^−𝑡𝑛2𝐹−𝑓3^𝑡𝑛𝑥+𝐹^′−𝑓3^𝑡𝑛𝑥+𝐹−𝑓−3^𝑡𝑛𝑥+𝐹^′−𝑓−3^𝑡𝑛𝑥≤69−3^𝑝+63^𝑝3−3^{−𝑠𝑛(2−𝑝)}+3^{−𝑡𝑛(1−𝑝)}‖𝑥‖^𝑝 (16) for all 𝑥∈𝑋⧵{0} and 𝑛∈ℕ. Taking the limit in the above inequality as 𝑛→∞, we can conclude that 𝐹(𝑥)=𝐹^′(𝑥) for all 𝑥∈𝑋, which proves the uniqueness of 𝐹.
Since 𝑓(𝑥)(𝑥≤)))+𝐹)𝑥))𝑥+𝑓)𝑥))𝑥+𝑓)𝑥))𝑥𝑓()(𝑓()(≤()−𝐹)𝐷𝑓((2𝑘−1𝑥,𝑘𝑥,𝑘𝑥−𝐷𝐹((2𝑘−1𝑥,𝑘𝑥,𝑘𝑥)((4𝑘−1−𝑓((4𝑘−1)((2𝑘−1−𝐹((2𝑘−1)((1−2𝑘−𝐹((1−2𝑘)+2−𝑘𝑥−𝐹−𝑘𝑥)+6𝑘𝑥−𝐹𝑘𝑥)2𝑘−1^𝑝+2𝑘^𝑝+3(4𝑘−1)^𝑝+2(2𝑘−1)^𝑝+8𝑘^𝑝3−3^𝑝×‖𝑥‖^𝑝 (17) for all 𝑥∈𝑋⧵{0} and 𝑘∈ℕ, if 𝑝<0, then we conclude that 𝑓(𝑥)=𝐹(𝑥) for all 𝑥∈𝑋⧵{0} by letting 𝑘→∞ in the previous inequality. From the fact that 𝑓(0)=0, 𝑓 is a quadratic-additive mapping.

Theorem 4. If a mapping 𝑓∶𝑋→𝑌 satisfies 𝐷𝑓𝑥,𝑦,𝑧≤‖𝑥‖^𝑝+𝑦^𝑝+‖𝑧‖^𝑝 (18) for all 𝑥,𝑦,𝑧∈𝑋 and for a nonnegative real constant 𝑝∉{1,2}, then there exists a unique quadratic-additive mapping 𝐹∶𝑋→𝑌 such that 𝑓(𝑥)(𝑥≤‖𝑥‖−𝐹)_𝑝2^𝑝,1−4𝑓𝑜𝑟𝑝>24−2^𝑝+12^𝑝‖𝑥‖−2^𝑝,‖𝑥‖𝑓𝑜𝑟1<𝑝<2_𝑝2−2^𝑝𝑓𝑜𝑟0<𝑝<1,2516𝑓𝑜𝑟𝑝=0 (19) for all 𝑥∈𝑋.

Proof. Since 𝑓(0=1)8(≤3‖0‖𝐷𝑓0,0,0)^𝑝8, (20) we get 𝑓(0)=0 for 𝑝∉{0,1,2} and ‖𝑓(0)‖≤3/8 for 𝑝=0. From the definitions of 𝐽_𝑛𝑓(𝑥) and 𝐷𝑓(𝑥,𝑦,𝑧), we have 𝐽_𝑛𝑓(𝑥)−𝐽_𝑛+1𝑓(𝑥)1=−44^{𝜏−𝑠,𝑛}𝐷𝑓2^{𝜏𝑠,𝑛}𝑥,2^{𝜏𝑠,𝑛}𝑥,0+𝐷𝑓−2^{𝜏𝑠,𝑛}𝑥,−2^{𝜏𝑠,𝑛}𝑥,0𝑠+2^{𝜏−𝑡,𝑛}𝐷𝑓2^{𝜏𝑡,𝑛}𝑥,2^{𝜏𝑡,𝑛}𝑥,0−𝐷𝑓−2^{𝜏𝑡,𝑛}𝑥,−2^{𝜏𝑡,𝑛}𝑡+1𝑥,024^{𝜏−𝑠,𝑛}𝑓(0) (21) for all 𝑥∈𝑋 and 𝑛∈ℕ₀, where 𝜏_𝑠,𝑛 is defined by 𝜏_𝑠,𝑛=𝑠(𝑛+1/2)−1/2 and 𝑠∈{−1,1}.
It follows from (18) and (21) that 𝐽_𝑛𝑓(𝑥)−𝐽_𝑛+𝑚𝑓(𝑥=_{𝑛+𝑚−1𝑗=𝑛})𝐽_𝑗𝑓(𝑥)−𝐽_𝑗+1𝑓(𝑥≤1)4^{𝑛+𝑚−1}_𝑗=𝑛4^{𝜏−𝑠,𝑗}𝐷𝑓2^{𝜏𝑠,𝑗}𝑥,2^{𝜏𝑠,𝑗}𝑥,0𝑠+2^{𝜏−𝑡,𝑗}𝐷𝑓2^{𝜏𝑡,𝑗}𝑥,2^{𝜏𝑡,𝑗}𝑡+4𝑥,0_{𝜏−𝑠,𝑗}𝐷𝑓−2^{𝜏𝑠,𝑗}𝑥,−2^{𝜏𝑠,𝑗}𝑥,0𝑠−2^{𝜏−𝑡,𝑗}𝐷𝑓−2^{𝜏𝑡,𝑗}𝑥,−2^{𝜏𝑡,𝑗}𝑥,0𝑡+2⋅4^{𝜏−𝑠,𝑗}𝑓(0≤_{𝑛+𝑚−1})_𝑗∑_=𝑛32^𝑗+2+3⋅2^−2𝑗−38for𝑝=0_{𝑛+𝑚−1},_𝑗∑_=𝑛2^−𝑗−12^𝑗𝑥^𝑝for0<𝑝<1_{𝑛+𝑚−1},_𝑗∑_=𝑛2^−2𝑗−22^𝑗𝑥^𝑝+2^𝑗2^−𝑗−1𝑥^𝑝for1<𝑝<2_{𝑛+𝑚−1},_𝑗∑_=𝑛2^2𝑗2^−𝑗−1𝑥^𝑝≤3for𝑝>22^𝑛+1+12^2𝑛+4,2for𝑝=0_𝑛𝑝‖𝑥‖^𝑝2^𝑛2−2^𝑝,2for0<𝑝<1_𝑛𝑝‖𝑥‖^𝑝4^𝑛4−2^𝑝+2^𝑛‖𝑥‖^𝑝2^𝑛𝑝2^𝑝,4−2for1<𝑝<2_𝑛‖𝑥‖^𝑝2^𝑛𝑝2^𝑝−4for𝑝>2 (22) for all 𝑥∈𝑋. So, it is easy to show that the sequence {𝐽_𝑛𝑓(𝑥)} is a Cauchy sequence for all 𝑥∈𝑋.
Since 𝑌 is complete, the sequence {𝐽_𝑛𝑓(𝑥)} converges for all 𝑥∈𝑋. Hence, we can define a mapping 𝐹∶𝑋→𝑌 by 𝐹(𝑥)∶=lim_𝑛→∞𝐽_𝑛𝑓(𝑥) (23) for all 𝑥∈𝑋. Moreover, putting 𝑛=0 and letting 𝑚→∞ in (22), we get the inequality (19).

Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2012R1A1A4A01002971).