On Pexider Differences in Topological Vector Spaces

Abstract

Let 𝑋 be a normed space and 𝑌 a sequentially complete Hausdorff topological vector space over the field ℚ of rational numbers. Let 𝐷₁={(𝑥,𝑦)∈𝑋×𝑋∶‖𝑥‖+‖𝑦‖≥𝑑}, and 𝐷₂={(𝑥,𝑦)∈𝑋×𝑋∶‖𝑥‖+‖𝑦‖<𝑑} where 𝑑>0. We prove that the Pexiderized Jensen functional equation is stable for functions defined on 𝐷₁(𝐷₂), and taking values in 𝑌. We consider also the Pexiderized Cauchy functional equation.

1. Introduction

The functional equation (𝜉) is stable if any function 𝑔 satisfying the equation (𝜉) approximately is near to true solution of (𝜉). The stability of functional equations was first introduced by Ulam [1] in 1940. More precisely, Ulam proposed the following problem: given a group 𝐺₁, a metric group (𝐺₂,𝑑), and a positive number 𝜖, does there exist a 𝛿>0 such that if a function 𝑓∶𝐺₁→𝐺₂ satisfies the inequality 𝑑(𝑓(𝑥𝑦),𝑓(𝑥)𝑓(𝑦))<𝛿 for all 𝑥,𝑦∈𝐺₁, then there exists a homomorphism 𝑇∶𝐺₁→𝐺₂ such that 𝑑(𝑓(𝑥),𝑇(𝑥))<𝜖 for all 𝑥∈𝐺₁? As it is mentioned above, when this problem has a solution, we say that the homomorphisms from 𝐺₁ to 𝐺₂ are stable. In 1941, Hyers [2] gave a partial solution of Ulam's problem for the case of approximate additive mappings under the assumption that 𝐺₁ and 𝐺₂ are Banach spaces. Aoki [3] and Rassias [4] provided a generalization of Hyers' theorem for additive and linear mappings, respectively, by allowing the Cauchy difference to be unbounded. During the last decades several stability problems of functional equations have been investigated by several mathematicians. A large list of references concerning the stability of functional equations can be found in [5–8].

2. Hyers-Ulam Stability of Jensen's Functional Equation

Jung investigated the Hyers-Ulam stability for Jensen's equation on a restricted domain [9]. In this section, we prove a local Hyers-Ulam stability of the Pexiderized Jensen functional equation in topological vector spaces. In this section 𝑋 is a normed space and 𝑌 is a sequentially complete Hausdorff topological vector space over the field ℚ of rational numbers.

Theorem 2.1. Let 𝑉 be a nonempty bounded convex subset of 𝑌 containing the origin. Suppose that 𝑓,𝑔,ℎ∶𝑋→𝑌 satisfies 2𝑓𝑥+𝑦2−𝑔(𝑥)−ℎ(𝑦)∈𝑉(2.1) for all 𝑥,𝑦∈𝑋 with ‖𝑥‖+‖𝑦‖⩾𝑑, where 𝑑>0. Then there exists a unique additive mapping 𝐴∶𝑋→𝑌 such that 𝐴(𝑥)−𝐴𝑔(𝑥(𝑥))−+𝑓𝑔((𝑥0))+∈𝑓(0)∈𝑊−𝑊,(2.2)𝐴(𝑥)−ℎ(𝑥)+ℎ(0)∈2(𝑊−𝑊)+𝑊−𝑊,(2.3)2(𝑊−𝑊)+𝑊−𝑊(2.4) for all 𝑥∈ℝ, where 𝑊=3𝑉−2𝑉 and 2(𝑊−𝑊) denotes the sequential closure of 2(𝑊−𝑊).

Proof. Suppose ‖𝑥‖+‖𝑦‖<𝑑. If ‖𝑥‖+‖𝑦‖=0, let 𝑧∈𝑋 with ‖𝑧‖=𝑑, otherwise 𝑧∶=⎧⎪⎨⎪⎩(𝑑+‖𝑥‖)𝑥(𝑑+‖𝑦‖)‖𝑥𝑦‖,if‖𝑥‖⩾‖𝑦‖;‖𝑦‖,if‖𝑦‖⩾‖𝑥‖.(2.5) It is easy to verify that ‖𝑥−𝑧‖+‖𝑦‖+𝑦𝑧+‖𝑧⩾‖𝑑+;‖‖𝑧2‖𝑧‖⩾+𝑑‖;𝑥‖𝑥−‖𝑧+‖‖⩾𝑧‖𝑑;⩾‖𝑦𝑑‖.+‖2𝑧‖⩾𝑑;(2.6) It follows from (2.1) and (2.6) that 2𝑓𝑥+𝑦2−𝑔(𝑥)−ℎ(𝑦)=−2𝑓2𝑓𝑥+2𝑥𝑦+𝑧−𝑔(𝑦+𝑧)−ℎ(𝑥−𝑧)+2𝑓𝑦2+2𝑧−𝑔(2𝑧)−ℎ(𝑥−𝑧)−2𝑓𝑦+22𝑧−𝑔(2𝑧)−ℎ(𝑦)+2𝑓𝑥+2𝑧−𝑔(𝑦+𝑧)−ℎ(𝑧)∈3𝑉−2𝑉2−𝑔(𝑥)−ℎ(𝑧)(2.7) for all 𝑥,𝑦∈𝑋 with ‖𝑥‖+‖𝑦‖<𝑑. Hence, by (2.1) and (2.7), we have 2𝑓𝑥+𝑦2−𝑔(𝑥)−ℎ(𝑦)∈3𝑉−2𝑉(2.8) for all 𝑥,𝑦∈𝑋. Letting 𝑥=0(𝑦=0) in (2.8), we get 2𝑓𝑦2𝑓𝑥2−𝑔(0)−ℎ(𝑦)∈3𝑉−2𝑉,2−𝑔(𝑥)−ℎ(0)∈3𝑉−2𝑉(2.9) for all 𝑥,𝑦∈𝑋. It follows from (2.8) and (2.9) that 2𝑓𝑥+𝑦2−2𝑓𝑥2−2𝑓2𝑦+2𝑓(0)=−2𝑓2𝑓𝑥+2𝑦𝑦−𝑔(𝑥)−ℎ(𝑦)−2𝑓𝑥2−𝑔(𝑥)−ℎ(0)∈2(𝑊−𝑊2)−𝑔(0)−ℎ(𝑦)+[2𝑓(0)−𝑔(0)−ℎ(0)](2.10) for all 𝑥,𝑦∈𝑋, where 𝑊=3𝑉−2𝑉. So we get from (2.10) that 𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)+𝑓(0)∈𝑊−𝑊(2.11) for all 𝑥,𝑦∈𝑋. Setting 𝑦=𝑥 in (2.10), we infer that 𝑓(𝑥)−2𝑓𝑥2+𝑓(0)∈𝑊−𝑊(2.12) for all 𝑥∈𝑋. It is easy to prove that 𝑓2^𝑛+1𝑥2^𝑛+1−𝑓(2^𝑛𝑥)2^𝑛+𝑓(0)2^𝑛+1∈12^𝑛+1𝑓(2_𝑛(𝑊−𝑊)⊆𝑊−𝑊,(2.13)𝑥)2^𝑛−𝑓(𝑥)+^𝑛_𝑘=112^𝑘𝑓(0)∈^𝑛_𝑘=112^𝑘(𝑊−𝑊)⊆𝑊−𝑊(2.14) for all 𝑥∈𝑋 and all integers 𝑛⩾1. Since 𝑉 is a nonempty bounded convex subset of 𝑋 containing the origin, 𝑊−𝑊 is a nonempty bounded convex subset of 𝑋 containing the origin. It follows from (2.13) that 𝑓(2^𝑛𝑥)2^𝑛−𝑓(2^𝑚𝑥)2^𝑚+^𝑛−1_𝑘_=𝑚12^𝑘+1𝑓(0)=^𝑛−1_𝑘_=𝑚𝑓2^𝑘+1𝑥2^𝑘+1−𝑓2^𝑘𝑥2^𝑘+12^𝑘+1∈𝑓(0)_𝑛−1_𝑘_=𝑚12^𝑘+1(𝑊−𝑊)⊆12^𝑚(𝑊−𝑊)(2.15) for all 𝑥∈𝑋 and all integers 𝑛>𝑚⩾0. Let 𝑈 be an arbitrary neighborhood of the origin in 𝑌. Since 𝑊−𝑊 is bounded, there exists a rational number 𝑡>0 such that 𝑡(𝑊−𝑊)⊆𝑈. Choose 𝑛₀∈ℕ such that 2^𝑛0𝑡>1. Let 𝑥∈𝑋 and 𝑚,𝑛∈ℕ with 𝑛⩾𝑚⩾𝑛₀. Then (2.15) implies that 𝑓(2^𝑛𝑥)2^𝑛−𝑓(2^𝑚𝑥)2^𝑚+(2^−𝑚−2^−𝑛)𝑓(0)∈𝑈.(2.16) Thus, the sequence {2^−𝑛𝑓(2^𝑛𝑥)} forms a Cauchy sequence in 𝑌. By the sequential completeness of 𝑌, the limit 𝐴(𝑥)=lim_𝑛→∞2^−𝑛𝑓(2^𝑛𝑥) exists for each 𝑥∈𝑋. So (2.2) follows from (2.14).
To show that 𝐴∶𝑋→𝑌 is additive, replace 𝑥 and 𝑦 by 2^𝑛𝑥 and 2^𝑛𝑦, respectively, in (2.11) and then divide by 2^𝑛 to obtain 𝑓(2^𝑛(𝑥+𝑦))2^𝑛−𝑓(2^𝑛𝑥)2^𝑛−𝑓(2^𝑛y)2^𝑛+𝑓(0)2^𝑛∈12^𝑛(𝑊−𝑊)(2.17) for all 𝑥,𝑦∈𝑋 and all integers 𝑛⩾0. Since 𝑊−𝑊 is bounded, on taking the limit as 𝑛→∞, we get that 𝐴 is additive. It follows from (2.2) and (2.9) that 𝐴(2𝑥)−𝑔(2𝑥)+𝑔(0)=∈−2𝐴𝑔((𝑥2𝑥))−−2𝑓ℎ((0𝑥))−+22𝑓𝑓((00))++𝑔2𝑓(0()𝑥+)ℎ(0)2(𝑊−𝑊)+𝑊−𝑊(2.18) for all 𝑥∈𝑋. So we obtain (2.3). Similarly, we get (2.4).
To prove the uniqueness of 𝐴, assume on the contrary that there is another additive mapping 𝑇∶𝑋→𝑌 satisfying (2.2) and there is an 𝑎∈𝑋 such that 𝑦=𝑇(𝑎)−𝐴(𝑎)≠0. So there is a neighborhood 𝑈 of the origin in 𝑌 such that 𝑦∉𝑈, since 𝑌 is Hausdorff. Since 𝐴 and 𝑇 satisfy (2.2), we get 𝑇(𝑥)−𝐴(𝑥)∈2(𝑊−𝑊) for all 𝑥∈𝑋. Since 𝑊−𝑊 is bounded, there exists a positive integer 𝑚 such that 2(𝑊−𝑊)⊆𝑚𝑈. Therefore, 𝑚𝑦=𝑇(𝑚𝑎)−𝐴(𝑚𝑎)∈𝑚𝑈 which is a contradiction with 𝑦∉𝑈. This completes the proof.

We apply the result of Theorem 2.1 to study the asymptotic behavior of additive mappings.

Theorem 2.2. Suppose that 𝑌 has a bounded convex neighborhood of 0. Let 𝑓,𝑔,ℎ∶𝑋→𝑌 be functions satisfying 2𝑓𝑥+𝑦2−𝑔(𝑥)−ℎ(𝑦)⟶0as‖𝑥‖+‖𝑦‖⟶∞.(2.19) Then 𝑓−𝑓(0),𝑔−𝑔(0),ℎ−ℎ(0) are additive and 𝑓−𝑓(0)=𝑔−𝑔(0)=ℎ−ℎ(0).

Proof. Let 𝑉 be a bounded convex neighborhood of 0 in 𝑌. It follows from (2.19) that there exists an increasing sequence {𝑑_𝑛}_𝑛 such that 2𝑓𝑥+𝑦2−𝑔(𝑥)−ℎ(𝑦)∈1𝑛𝑉(2.20) for all 𝑥,𝑦∈𝑋 with ‖𝑥‖+‖𝑦‖≥𝑑_𝑛. Applying (2.20) and Theorem 2.1, we obtain a sequence {𝐴_𝑛∶𝑋→𝑌} of unique additive mappings satisfying 𝐴_𝑛(𝑥)−𝑓(𝑥)+𝑓(0)∈1𝑛𝐴(𝑊−𝑊),_𝑛(𝑥)−𝑔(𝑥)+𝑔(0)∈3𝑛𝐴(𝑊−𝑊),_𝑛(𝑥)−ℎ(𝑥)+ℎ(0)∈3𝑛(𝑊−𝑊)(2.21) for all 𝑥∈𝑋, where 𝑊=3𝑉−2𝑉. Since (𝑊−𝑊) is convex and 0∈(𝑊−𝑊), we have 𝐴_𝑚(𝑥)−𝑓(𝑥)+𝑓(0)∈1𝑚(𝑊−𝑊)⊆1𝑛𝐴(𝑊−𝑊),_𝑚(𝑥)−𝑔(𝑥)+𝑔(0)∈3𝑚(𝑊−𝑊)⊆3𝑛𝐴(𝑊−𝑊),_𝑚(𝑥)−ℎ(𝑥)+ℎ(0)∈3𝑚(𝑊−𝑊)⊆3𝑛(𝑊−𝑊)(2.22) for all 𝑥∈𝑋 and all 𝑚⩾𝑛. The uniqueness of 𝐴_𝑛 implies 𝐴_𝑚=𝐴_𝑛 for all 𝑚⩾𝑛. Hence, letting 𝑛→∞ in (2.21), we obtain that 𝑓−𝑓(0),𝑔−𝑔(0),ℎ−ℎ(0) are additive and 𝑓−𝑓(0)=𝑔−𝑔(0)=ℎ−ℎ(0).

Theorem 2.3. Let 𝑉 be a nonempty bounded convex subset of 𝑌 containing the origin. Suppose that 𝑓,𝑔,ℎ∶𝑋→𝑌 with 𝑔(0)=ℎ(0)=0 satisfies 2𝑓𝑥+𝑦2−𝑔(𝑥)−ℎ(𝑦)∈‖𝑥‖^𝑝+‖𝑦‖^𝑝𝑉(2.23) for all 𝑥,𝑦∈𝑋 with ‖𝑥‖+‖𝑦‖⩽𝑑, where 𝑑>0 and 𝑝>1. Then there exists a unique additive mapping 𝜑∶𝑋→𝑌 such that 𝜑(𝑥)−𝑓(𝑥)∈2^𝑝2^𝑝−2‖𝑥‖^𝑝𝜑(𝑥)−𝑔(𝑥)∈2_𝑝(𝑉−𝑉),(2.24)2^𝑝−2‖𝑥‖^𝑝𝜑(𝑥)−ℎ(𝑥)∈2_𝑝𝑉−𝑉+𝑉,(2.25)2^𝑝−2‖𝑥‖^𝑝𝑉−𝑉+𝑉(2.26) for all 𝑥∈𝑋 with ‖𝑥‖⩽𝑑/2. Moreover, 𝜑(𝑥)=lim_𝑛→∞2^𝑛𝑓(2^−𝑛𝑥) for all 𝑥∈𝑋.

Proof. Letting 𝑦=𝑥=0 in (2.23), we get 𝑓(0)=0. Letting 𝑦=𝑥 in (2.23), we get 2𝑓(𝑥)−𝑔(𝑥)−ℎ(𝑥)∈2‖𝑥‖^𝑝𝑉(2.27) for all 𝑥∈𝑋 with ‖𝑥‖⩽𝑑/2. If we put 𝑦=0(𝑥=0) in (2.23), we have 2𝑓𝑥2−𝑔(𝑥)∈‖𝑥‖^𝑝𝑉,‖𝑥‖⩽𝑑2𝑓𝑦2,2−ℎ(𝑦)∈‖𝑦‖^𝑝𝑉,‖𝑦‖⩽𝑑2.(2.28) Hence it follows from (2.27) and (2.28) that 2𝑓𝑥2−𝑓(𝑥)∈‖𝑥‖^𝑝(𝑉−𝑉)(2.29) for all 𝑥∈𝑋 with ‖𝑥‖⩽𝑑/2. We can replace 𝑥 by 𝑥/2^𝑛 in (2.29) for all nonnegative integers 𝑛. So we have 2^𝑛+1𝑓2^−𝑛−1𝑥−2^𝑛𝑓(2^−𝑛𝑥)∈22^𝑝^𝑛‖𝑥‖^𝑝(𝑉−𝑉)(2.30) for all 𝑥∈𝑋 with ‖𝑥‖⩽𝑑/2. Therefore, 2^𝑛+1𝑓2^−𝑛−1𝑥−2^𝑚𝑓(2^−𝑚𝑥)=^𝑛_𝑘_=𝑚2^𝑘+1∈𝑓2^−𝑘−1𝑥−2^𝑘𝑓2^−𝑘𝑥_𝑛𝑘_=𝑚22^𝑝^𝑘⊆2‖𝑥‖^𝑝(𝑉−𝑉)_𝑚+𝑝2^𝑚𝑝(2^𝑝−2)‖𝑥‖^𝑝(𝑉−𝑉)(2.31) for all 𝑥∈𝑋 with ‖𝑥‖⩽𝑑/2 and all integers 𝑛⩾𝑚⩾0. Let 𝑥∈𝑋 with ‖𝑥‖⩽𝑑/2, and let 𝑈 be an arbitrary neighborhood of the origin in 𝑌. Since 𝑉−𝑉 is bounded, there exists a rational number 𝑡>0 such that 𝑡(𝑉−𝑉)⊆𝑈. Choose 𝑘∈ℕ such that (2^𝑘𝑝(2^𝑝−2)/2^𝑘+𝑝)𝑡>‖𝑥‖^𝑝. Let 𝑚,𝑛∈ℕ with 𝑛⩾𝑚⩾𝑘. Then (2.31) implies that 2^𝑛+1𝑓2^−𝑛−1𝑥−2^𝑚𝑓(2^−𝑚𝑥)∈𝑈.(2.32) Thus, the sequence {2^𝑛𝑓(2^−𝑛𝑥)} forms a Cauchy sequence in 𝑌 for all 𝑥∈𝑋 with ‖𝑥‖⩽𝑑/2. By the sequential completeness of 𝑌, the limit 𝐴(𝑥)=lim_𝑛→∞2^𝑛𝑓(2^−𝑛𝑥) exists for each ‖𝑥‖⩽𝑑/2. It follows from (2.28) that 𝐴(𝑥)=lim_𝑛→∞2^𝑛𝑓(2^−𝑛𝑥)=lim_𝑛→∞2^𝑛𝑔(2^−𝑛𝑥)=lim_𝑛→∞2^𝑛ℎ(2^−𝑛𝑥)(2.33) for all 𝑥∈𝑋 with ‖𝑥‖⩽𝑑/2. Letting 𝑚=0 and 𝑛→∞ in (2.31), we obtain that 𝐴 satisfies 𝐴(𝑥)−𝑓(𝑥)∈2^𝑝2^𝑝−2‖𝑥‖^𝑝(𝑉−𝑉)(2.34) for all 𝑥∈𝑋 with ‖𝑥‖⩽𝑑/2. It follows from the definition of 𝐴 that 𝐴(0)=0, and we conclude from (2.23) that 𝐴(𝑥+𝑦)=𝐴(𝑥)+𝐴(𝑦) for all 𝑥,𝑦∈𝑋 with ‖𝑥‖,‖𝑦‖,‖𝑥+𝑦‖⩽𝑑/2. Using an extension method of Skof [10], we extend the additivity of 𝐴 to the whole space 𝑋 (see also [11]). Let 𝜑 be the extension of 𝐴 in which 𝜑(𝑥)=𝐴(𝑥) for all 𝑥∈𝑋 with ‖𝑥‖⩽𝑑/2. It follows from (2.34) that 𝜑 satisfies (2.24). To prove (2.25), we have from (2.24) and (2.28) that 𝜙(𝑥)−𝑔(𝑥)∈𝜙(𝑥)−2𝑓𝑥∈22+2𝑓𝑥2−𝑔(𝑥)2^𝑝−2‖𝑥‖^𝑝=2_𝑝(𝑉−𝑉)+‖𝑥‖^𝑝𝑉2^𝑝−2‖𝑥‖^𝑝𝑉−𝑉+𝑉(2.35) for all 𝑥∈𝑋 with ‖𝑥‖⩽𝑑/2. Similarly, we obtain (2.26).

3. Hyers-Ulam Stability of Cauchy's Functional Equation

The following theorems are alternative results for the Pexiderized Cauchy functional equation.

Theorem 3.1. Let 𝑉 be a nonempty bounded convex subset of 𝑌 containing the origin. Suppose that 𝑓,𝑔,ℎ∶𝑋→𝑌 satisfies 𝑓(𝑥+𝑦)−𝑔(𝑥)−ℎ(𝑦)∈𝑉(3.1) for all 𝑥,𝑦∈𝑋 with ‖𝑥‖+‖𝑦‖⩾𝑑, where 𝑑>0. Then there exists a unique additive mapping 𝐴∶𝑋→𝑌 such that 𝐴(𝑥)𝐴−(𝑥𝑔)(−𝑥)𝑓+(𝑔𝑥)(0+)𝑓∈(0)∈2(𝑊−𝑊),𝐴(𝑥)−ℎ(𝑥)+ℎ(0)∈2(𝑊−𝑊)+𝑊−𝑊,2(𝑊−𝑊)+𝑊−𝑊(3.2) for all 𝑥∈ℝ, where 𝑊=3𝑉−2𝑉.

Proof. Using the method in the proof of Theorem 2.1, we get 𝑓(𝑥+𝑦)−𝑔(𝑥)−ℎ(𝑦)∈𝑊(3.3) for all 𝑥,𝑦∈𝑋. Letting 𝑥=0(𝑦=0) in (3.3), we get 𝑓𝑓((𝑥𝑦))−−𝑔𝑔((0𝑥))−−ℎℎ((𝑦0))∈∈𝑊𝑊,(3.4) for all 𝑥,𝑦∈𝑋. It follows from (3.3) and (3.4) that 𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)+𝑓(0)=∈−𝑓2([(𝑊𝑓𝑥(+−𝑦)𝑦𝑊−)−𝑔)𝑔(0()𝑥−)−ℎℎ(𝑦()𝑦])+−[𝑓[𝑓(0(𝑥))−−𝑔𝑔(0(𝑥))−−ℎℎ(0(0)])](3.5) for all 𝑥,𝑦∈𝑋. The rest of the proof is similar to the proof of Theorem 2.1, and we omit the details.

Corollary 3.2. Suppose that 𝑌 has a bounded convex neighborhood of 0. Let 𝑓,𝑔,ℎ∶𝑋→𝑌 be functions satisfying 𝑓(𝑥+𝑦)−𝑔(𝑥)−ℎ(𝑦)⟶0as‖𝑥‖+‖𝑦‖⟶∞.(3.6) Then 𝑓−𝑓(0),𝑔−𝑔(0),ℎ−ℎ(0) are additive and 𝑓−𝑓(0)=𝑔−𝑔(0)=ℎ−ℎ(0).

Theorem 3.3. Let 𝑉 be a nonempty bounded convex subset of 𝑌 containing the origin. Suppose that 𝑓,𝑔,ℎ∶𝑋→𝑌 with 𝑔(0)=ℎ(0)=0 satisfies 𝑓(𝑥+𝑦)−𝑔(𝑥)−ℎ(𝑦)∈‖𝑥‖^𝑝+‖𝑦‖^𝑝𝑉(3.7) for all 𝑥,𝑦∈𝑋 with ‖𝑥‖+‖𝑦‖⩽𝑑, where 𝑑>0 and 𝑝>1. Then there exists a unique additive mapping 𝜑∶𝑋→𝑌 such that 𝜑(𝑥)−𝑓(𝑥)∈22^p−2‖𝑥‖^𝑝𝜑(𝑥)−𝑔(𝑥)∈2_𝑝(𝑉−𝑉),2^𝑝−2‖𝑥‖^𝑝𝜑(𝑥)−ℎ(𝑥)∈2_𝑝𝑉−𝑉+𝑉,2^𝑝−2‖𝑥‖^𝑝𝑉−𝑉+𝑉(3.8) for all 𝑥∈𝑋 with ‖𝑥‖⩽𝑑/2. Moreover, 𝜑(𝑥)=lim_𝑛→∞2^𝑛𝑓(2^−𝑛𝑥) for all 𝑥∈𝑋.

Proof. Using the same method in the proof of Theorem 2.3, we get from (3.7) that 2𝑓(𝑥)−𝑓(2𝑥)∈2(𝑉−𝑉)‖𝑥‖^𝑝 for all 𝑥∈𝑋 with ‖𝑥‖⩽𝑑/2. Therefore, 2𝑓𝑥2−𝑓(𝑥)∈22^𝑝‖𝑥‖^𝑝(𝑉−𝑉)(3.9) for all 𝑥∈𝑋 with ‖𝑥‖⩽𝑑/2. The rest of the proof is similar to the proof of Theorem 2.3, and we omit the details.