On Approximately p-Wright Affine Functions in Ultrametric Spaces

Abstract

We prove some results for mappings taking values in ultrametric spaces and satisfying approximately a generalization of the equation of p-Wright affine functions. They are motivated by the notion of stability for functional equations.

1. Introduction and Preliminaries

Let 𝑋₁ and 𝑋₂ be linear spaces over fields 𝔽₁ and 𝔽₂, respectively, and let 𝑝,𝑝∈𝔽₁ be fixed. The functional equation 𝑓𝑝𝑥+𝑝𝑦+𝑓𝑝𝑥+𝑝𝑦=𝑓(𝑥)+𝑓𝑦, (1) for function 𝑓∶𝑋₁→𝑋₂, generalizes the equation 𝑓𝑝𝑥+1−𝑝𝑦+𝑓1−𝑝𝑥+𝑝𝑦=𝑓(𝑥)+𝑓𝑦. (2)

For 𝔽₁=ℝ and 𝑝∈(0,1), solutions of (2) are called the 𝑝-Wright affine functions, which are both 𝑝-Wright convex and concave (see [1]). For information on the 𝑝-Wright convexity and the 𝑝-Wright concavity we refer, for example, to [1, 2].

Note also that for 𝑝=1/2 (2) becomes the well-known Jensen’s functional equation 𝑓𝑥+𝑦2=𝑓(𝑥)+𝑓𝑦2. (3) For 𝑝=1/3 (2) takes the form 𝑓(2𝑥+𝑦)+𝑓(𝑥+2𝑦)=𝑓(3𝑥)+𝑓(3𝑦), which has been studied in [3] (see also [4]) in connection with some investigations of the generalized (𝜎,𝜏)-Jordan derivations on Banach algebras. The cases of more arbitrary 𝑝 have been studied in [1, 5] (cf. [2]).

In this paper we consider (1) in a bit generalized form, with 𝑝 and 𝑝 being some group endomorphisms. Motivated by the notion of the Hyers-Ulam stability (see, e.g., [6–9] for further details), we prove some results concerning the mappings that take values in the ultrametric spaces and satisfy (1) approximately, that is, fulfil inequality (8) (with suitable assumptions on 𝜃). Our outcomes correspond to those in [10], where some issues (analogous as in Theorems 2 and 6) have been considered for functions mapping a classical normed space (real or complex) into a classical Banach space, with 𝑝 being a scalar (real or complex, resp.) and for 𝜃 of the following two forms: 𝜃𝑥,𝑦=𝐴‖𝑥‖^𝑘+𝑦^𝑘,𝜃𝑥,𝑦=𝐴‖𝑥‖^𝑘𝑦^𝑘, (4) with some real 𝐴>0 and 𝑘>0 (see also [4] for similar outcomes but only for 𝑝=1/3). The main tool in our investigations is a fixed point theorem from [11] (for information on related results see [12–15]).

Recall that an ultrametric space is a metric space (𝑌,𝑑) with the metric 𝑑 satisfying the condition 𝑑(𝑥,𝑧)≤max{𝑑(𝑥,𝑦),𝑑(𝑦,𝑧)} for every 𝑥,𝑦,𝑧∈𝑌; such a metric 𝑑 is called an ultrametric. One of important examples of the ultrametric spaces is a non-Archimedean normed space.

Let us remind yet that a linear space 𝑋 over a field 𝕂, with a function ‖⋅‖∶𝑋→[0,∞), is said to be a non-Archimedean normed space provided ‖𝑥‖=0 if and only if 𝑥=0 and ‖𝑟𝑥‖=|𝑟|‖𝑥‖ and ‖𝑥+𝑦‖≤max{‖𝑥‖,‖𝑦‖} for all 𝑥,𝑦∈𝑋, 𝑟∈𝕂, where |⋅| denotes a non-Archimedean valuation in 𝕂, that is a function |⋅|∶𝕂→[0,∞) such that |𝑟|=0 if and only if 𝑟=0 and |𝑟𝑠|=|𝑟||𝑠| and |𝑟+𝑠|≤max{|𝑟|,|𝑠|} for 𝑟,𝑠∈𝕂.

Clearly, if 𝑋 is a non-Archimedean normed space, then the formula 𝑑(𝑥,𝑦)∶=‖𝑥−𝑦‖ defines an ultrametric 𝑑 in 𝑋, that is invariant (i.e., 𝑑(𝑥+𝑧,𝑦+𝑧)=𝑑(𝑥,𝑦) for every 𝑥,𝑦,𝑧∈𝑋).

Any field endowed with a non-Archimedean valuation is said to be a non-Archimedean field. If the valuation is nontrivial (i.e., there is an 𝑟₀∈𝕂 such that 0≠|𝑟₀|≠1), then we have |𝟏|=|−𝟏|=1 and |𝑛×𝟏|≤1 for all 𝑛∈ℕ (positive integers), where 𝟏 is the neutral element of the semigroup (𝕂,⋅), 1×𝟏=𝟏 and (𝑛+1)×𝟏=(𝑛×𝟏)+𝟏 for 𝑛∈ℕ.

The first example of a non-Archimedean field was provided by Hensel in [16], where he gave a description of the 𝑝-adic numbers (for each fixed prime number 𝑝 and any non-zero rational number 𝑥, there exists a unique integer 𝑛_𝑥 such that 𝑥=(𝑎/𝑏)𝑝^𝑛𝑥, where 𝑎 and 𝑏 are integers not divisible by 𝑝; then |𝑥|_𝑝∶=𝑝^−𝑛𝑥 defines a non-Archimedean valuation in ℚ and therefore also a non-Archimedean norm in ℚ).

Let ℚ_𝑝 denote the completion of ℚ, with respect to the metric 𝑑(𝑥,𝑦)=|𝑥−𝑦|_𝑝. Then ℚ_𝑝 (called the 𝑝-adic number field) can be identified with the set of all formal series 𝑥=∑^∞_{𝑘≥𝑛𝑥}𝑎_𝑘𝑝^𝑘, where |𝑎_𝑘|≤𝑝−1 are integers. The operation of addition and multiplication between any two elements of ℚ_𝑝 is defined naturally. The non-Archimedean norm ‖⋅‖_𝑝 in ℚ_𝑝 is defined by ‖∑^∞_{𝑘≥𝑛𝑥}𝑎_𝑘𝑝^𝑘‖_𝑝=𝑝^−𝑛𝑥; ℚ_𝑝 endowed with it turns out to be a locally compact filed (see [17]). Let us mention yet that the 𝑝-adic numbers have gained the interest of physicists because of their connections with some issues in quantum physics, 𝑝-adic strings and superstrings (see [18]).

The problem of stability of functional equations was motivated by a question of S.M. Ulam asked in 1940 and an answer to it published by Hyers [19]. Since then numerous papers on this subject have been published, and we refer to [6–9, 13, 15, 20, 21] for more details, some discussions, and further references.

Let us mention yet that the issue of stability of functional equations is related to the notions of shadowing (see [22, 23]), the theory of perturbation (see [24]), and optimization.

Moslehian and Rassias [25] (see [26, 27] for some related outcomes) have proved the first stability results for the Cauchy and quadratic functional equations in non-Archimedean normed spaces. Afterwards several stability results for other equations in such spaces have been published, for example, in [28–30] (for further references see [13, 15]).

2. Auxiliary Result

In this section 𝑋 denotes a nonempty set and (𝑌,𝑑) stands for a complete ultrametric space. The main tool in the proofs of the main theorems of this paper is a fixed point result that can be derived from [11, Theorem 2]. To present it we need to introduce some notions.

For any 𝛿₁,𝛿₂∈ℝ₊^𝑋 (𝐴^𝐵 denotes the family of all functions mapping a set 𝐵≠∅ into a set 𝐴≠∅) we write 𝛿₁≤𝛿₂ provided 𝛿₁(𝑥)≤𝛿₂(𝑥) for 𝑥∈𝑋, and we say that an operator Λ:ℝ₊^𝑋→ℝ₊^𝑋 is nondecreasing if it satisfies the condition Λ𝛿₁≤Λ𝛿₂ for all 𝛿₁,𝛿₂∈ℝ₊^𝑋 with 𝛿₁≤𝛿₂. Moreover, given a sequence (𝑔_𝑛)_𝑛∈ℕ in ℝ₊^𝑋, we write lim_𝑛→∞𝑔_𝑛=0 provided lim_𝑛→∞𝑔_𝑛(𝑥)=0 for 𝑥∈𝑋.

We use the following hypothesis concerning operators Λ∶ℝ₊^𝑋→ℝ₊^𝑋: (𝒞)  lim_𝑛→∞Λ𝛿_𝑛=0 for every sequence (𝛿_𝑛)_𝑛∈ℕ in ℝ₊^𝑋 satisfying the condition lim_𝑛→∞𝛿_𝑛=0.

Moreover, to simplify some notations, we define Δ∶(𝑌^𝑋)²→ℝ₊^𝑋 by Δ𝜉,𝜇(𝑥):=𝑑𝜉(𝑥),𝜇(𝑥),𝜉,𝜇∈𝑌^𝑋,𝑥∈𝑋. (5)

Now we are in a position to present the mentioned fixed point result.

Theorem 1. Let Λ∶ℝ₊^𝑋→ℝ₊^𝑋 be a nondecreasing operator satisfying hypothesis (𝒞). If 𝒯∶𝑌^𝑋→𝑌^𝑋, 𝜀∶𝑋→ℝ₊, and 𝜑∶𝑋→𝑌 satisfy 𝑑(𝒯𝜉)(𝑥),𝒯𝜇(𝑥)≤ΛΔ𝜉,𝜇(𝑥),𝜉,𝜇∈𝑌^𝑋𝑑(𝑥)(𝑥)(𝑥),𝑥∈𝑋,𝒯𝜑,𝜑≤𝜀,𝑥∈𝑋,lim_𝑛→∞Λ^𝑛𝜀=0, (6) then the limit lim_𝑛→∞(𝒯^𝑛𝜑)(𝑥)=∶𝜓(𝑥) exists for every 𝑥∈𝑋 and the function 𝜓∈𝑌^𝑋, defined in this way, is a fixed point of 𝒯 with 𝑑𝜑(𝑥),𝜓(𝑥)≤sup_𝑛∈ℕ0Λ^𝑛𝜀(𝑥),𝑥∈𝑋. (7)

3. The Main Results

We say that (𝑌,+,𝑑) is an ultrametric group if (𝑌,+) is a group and 𝑑 is ultrametric in 𝑌 such that the group operation + is continuous with respect to 𝑑. In what follows (𝑋,+) is a group (though we use the additive notation for the group operation in 𝑋, it does not mean that the group must be commutative), (𝑌,+,𝑑) is an ultrametric commutative group, and the ultrametric 𝑑 is complete and invariant, unless explicitly stated otherwise. Given a mapping 𝑟∶𝑋→𝑋, for simplicity of notation, we write 𝑟𝑥:=𝑟(𝑥) for each 𝑥∈𝑋.

The following two theorems concern stability of functional equation (1).

Theorem 2. Let 𝑝 and 𝑝 be endomorphisms of 𝑋 with 𝑝∘𝑝=𝑝∘𝑝, 𝐵∈(0,1), and 𝑔∶𝑋→𝑌 and 𝜃∶𝑋²→[0,∞) satisfy the inequalities 𝑑𝑔(𝑥)𝑦,𝜃𝑝𝑥+𝑝𝑦+𝑔𝑝𝑥+𝑝𝑦,𝑔+𝑔≤𝜃𝑥,𝑦𝑥,𝑦∈𝑋,max𝑝𝑥,𝑝𝑦,𝜃𝑝𝑥,𝑝𝑦≤𝐵𝜃𝑥,𝑦,𝑥,𝑦∈𝑋. (8)(9) Then there exists a unique function 𝐺∶𝑋→𝑌 such that 𝐺(𝑥)𝑦𝑑𝑔(𝑥)(𝑥){𝜃()(𝑝𝑥+𝑝𝑦+𝐺𝑝𝑥+𝑝𝑦=𝐺+𝐺,𝑥,𝑦∈𝑋,,𝐺≤min𝑥,0,𝜃0,𝑥)},𝑥∈𝑋. (10)(11) Moreover, 𝐺 is the unique solution of (10) for which there exists a constant 𝑀∈(0,∞) with 𝑑𝑔(𝑥),𝐺(𝑥)≤𝑀min{𝜃(𝑥,0),𝜃(0,𝑥)},𝑥∈𝑋. (12)

Proof. Taking in (8) first 𝑦=0 and next 𝑥=0 we obtain 𝑑𝑔(𝑥)(0)()𝑑𝑔(𝑥)(0)()𝑝𝑥+𝑔𝑝𝑥,𝑔+𝑔≤𝜃𝑥,0,𝑥∈𝑋,𝑝𝑥+𝑔𝑝𝑥,𝑔+𝑔≤𝜃0,𝑥,𝑥∈𝑋. (13) Write 𝑔₀(𝑥):=𝑔(𝑥)−𝑔(0) and 𝒯𝜉(𝑥)∶=𝜉(𝑝𝑥)+𝜉(𝑝𝑥) for 𝑥∈𝑋, 𝜉∈𝑌^𝑋. Then (13) implies the inequality 𝑑𝒯𝑔₀(𝑥),𝑔₀(𝑥)≤𝜀(𝑥),𝑥∈𝑋, (14) where 𝜀(𝑥)∶=min{𝜃(𝑥,0),𝜃(0,𝑥)}. Define an operator Λ∶ℝ₊^𝑋→ℝ₊^𝑋 by Λ𝜂(𝑥):=max𝜂𝑝𝑥,𝜂𝑝𝑥,𝜂∈ℝ₊^𝑋,𝑥∈𝑋. (15) Then Λ is non-decreasing, satisfies hypothesis (𝒞) and 𝑑(𝑥)(𝑥)𝜉𝑑𝜉𝜉Δ(𝑥)𝒯𝜉,𝒯𝜇=𝑑𝑝𝑥+𝜉𝑝𝑥,𝜇𝑝𝑥+𝜇𝑝𝑥≤max𝑝𝑥,𝜇𝑝𝑥,𝑑𝑝𝑥,𝜇𝑝𝑥=max𝜉,𝜇𝑝𝑥,Δ𝜉,𝜇𝑝𝑥=ΛΔ𝜉,𝜇,𝜉,𝜇∈𝑌^𝑋,𝑥∈𝑋. (16) Note that by (9) 𝜀𝜃{()({𝜃()((𝑥)𝑝𝑥=min𝑝𝑥,0,𝜃0,𝑝𝑥≤min𝐵𝜃𝑥,0,𝐵𝜃0,𝑥)}=𝐵min𝑥,0,𝜃0,𝑥)}=𝐵𝜀,𝑥∈𝑋, (17) and analogously 𝜀(𝑝𝑥)≤𝐵𝜀(𝑥) for 𝑥∈𝑋. Consequently max𝜀𝑝𝑥,𝜀𝑝𝑥≤𝐵𝜀(𝑥),𝑥∈𝑋. (18) Further, it is easy to show by induction that Λ^𝑛𝜀(𝑥)≤𝐵^𝑛𝜀(𝑥),𝑥∈𝑋,𝑛∈ℕ. (19) Since 0<𝐵<1, this means that lim_𝑛→∞(Λ^𝑛𝜀)(𝑥)=0 for 𝑥∈𝑋. Moreover, in view of (18) we see that Λ^𝑛+1𝜀(𝑥)≤Λ^𝑛𝜀(𝑥) for 𝑛∈ℕ₀ and 𝑥∈𝑋, whence sup_𝑛∈ℕ0Λ^𝑛𝜀(𝑥)=Λ⁰𝜀(𝑥)=𝜀(𝑥),sup_𝑛∈ℕ0Λ^𝑛+1𝜀(𝑥)=Λ𝜀(𝑥). (20) Consequently, by Theorem 1, there is a solution 𝐺₀∶𝑋→𝑌 of the equation 𝐺₀(𝑥)=𝐺₀𝑝𝑥+𝐺₀𝑝𝑥 (21) such that 𝑑(𝑔₀(𝑥),𝐺₀(𝑥))≤𝜀(𝑥) for 𝑥∈𝑋. Moreover, 𝐺₀(𝑥):=lim_𝑛→∞𝒯^𝑛𝑔₀(𝑥),𝑥∈𝑋. (22)
Now, we show that, for every 𝑛∈ℕ₀, 𝑑𝒯^𝑛𝑔₀𝑝𝑥+𝑝𝑦+𝒯^𝑛𝑔₀,𝒯𝑝𝑥+𝑝𝑦_𝑛𝑔₀(𝑥)+𝒯^𝑛𝑔₀𝑦≤𝐵^𝑛𝜃𝑥,𝑦,𝑥,𝑦∈𝑋. (23) Clearly, for 𝑛=0, (23) reduces to (8). Next, fix 𝑚∈ℕ₀ and assume that (23) holds for 𝑛=𝑚. Then, by (9), 𝑑𝒯^𝑚+1𝑔₀𝑝𝑥+𝑝𝑦+𝒯^𝑚+1𝑔₀,𝒯𝑝𝑥+𝑝𝑦_𝑚+1𝑔₀(𝑥)+𝒯^𝑚+1𝑔₀𝑦=𝑑𝒯^𝑚𝑔₀𝑝𝑝𝑥+𝑝𝑦+𝒯^𝑚𝑔₀𝑝𝑝𝑥+𝑝𝑦+𝒯^𝑚𝑔₀𝑝𝑝𝑥+𝑝𝑦+𝒯^𝑚𝑔₀𝑝,𝒯𝑝𝑥+𝑝𝑦_𝑚𝑔₀𝑝𝑥+𝒯^𝑚𝑔₀𝑝𝑥+𝒯^𝑚𝑔₀𝑝𝑦+𝒯^𝑚𝑔₀𝑑𝒯𝑝𝑦≤max^𝑚𝑔₀𝑝𝑝𝑥+𝑝𝑝𝑦+𝒯^𝑚𝑔₀,𝒯𝑝𝑝𝑥+𝑝𝑝𝑦_𝑚𝑔₀𝑝𝑥+𝒯^𝑚𝑔₀,𝑑𝒯𝑝𝑦_𝑚𝑔₀𝑝𝑝𝑥+𝑝𝑝𝑦+𝒯^𝑚𝑔₀𝑝,𝒯𝑝𝑥+𝑝𝑝𝑦_𝑚𝑔₀𝑝𝑥+𝒯^𝑚𝑔₀𝐵𝑝𝑦≤max^𝑚𝜃𝑝𝑥,𝑝𝑦,𝐵^𝑚𝜃𝑝𝑥,𝑝𝑦≤𝐵^𝑚+1𝜃𝑥,𝑦,𝑥,𝑦∈𝑋. (24) Thus we obtain (23) for 𝑛=𝑚+1, which completes the induction. Letting 𝑛→∞ in (23), we obtain that 𝐺₀𝑝𝑥+𝑝𝑦+𝐺₀𝑝𝑥+𝑝𝑦=𝐺₀(𝑥)+𝐺₀𝑦,𝑥,𝑦∈𝑋. (25)
Write 𝐺(𝑥):=𝐺₀(𝑥)+𝑔(0) for 𝑥∈𝑋. Then (11) holds and 𝐺𝑝𝑥+𝑝𝑦+𝐺𝑝𝑥+𝑝𝑦=𝐺(𝑥)+𝐺𝑦,𝑥,𝑦∈𝑋. (26)
It remains to prove the uniqueness of 𝐺. So, let 𝑀∈(0,∞) and 𝐺₁∶𝑋→𝑌 a solution to (10) with 𝑑(𝑔(𝑥),𝐺₁(𝑥))≤𝑀𝜀(𝑥) for 𝑥∈𝑋. Then 𝐺₁𝑝𝑥+𝐺₁𝑝𝑥=𝐺₁(𝑥)+𝐺₁(0)𝐺(𝑥)(0),𝑥∈𝑋,𝑝𝑥+𝐺𝑝𝑥=𝐺+𝐺,𝑥∈𝑋, (27) and, by (11), 𝑑𝐺(𝑥),𝐺₁(𝑥)≤max{1,𝑀}𝜀(𝑥),𝑥∈𝑋. (28)
Further, by (9), 𝜃(0,0)=0, whence 𝜀(0)=0. This and (28) yield 𝐺(0)=𝐺₁(0). (29)
We show that, for each 𝑛∈ℕ₀, 𝑑𝐺(𝑥),𝐺₁(𝑥)≤max{1,𝑀}𝜀(𝑥)sup_𝑗≥𝑛𝐵^𝑗,𝑥∈𝑋. (30) The case 𝑛=0 is just (28). So fix 𝑚∈ℕ₀ and assume that (30) holds for 𝑛=𝑚. Then, from (27), (29), and (18) we obtain 𝑑𝐺(𝑥),𝐺₁(𝑥)=𝑑𝐺𝑝𝑥+𝐺𝑝𝑥,𝐺₁𝑝𝑥+𝐺₁𝑑𝐺𝑝𝑥≤max𝑝𝑥,𝐺₁𝑝𝑥,𝑑𝐺𝑝𝑥,𝐺₁{}𝜀𝑝𝑥≤maxmax1,𝑀𝑝𝑥sup_𝑗≥𝑚𝐵^𝑗,max{1,𝑀}𝜀𝑝𝑥sup_𝑗≥𝑚𝐵^𝑗=max{1,𝑀}sup_𝑗≥𝑚𝐵^𝑗𝜀{}(𝑥)max𝑝𝑥,𝜀𝑝𝑥≤max1,𝑀𝐵𝜀sup_𝑗≥𝑚𝐵^𝑗=max{1,𝑀}𝜀(𝑥)sup_{𝑗≥𝑚+1}𝐵^𝑗,𝑥∈𝑋. (31) Thus we have shown (30). Now, letting 𝑛→∞ in (30) we get 𝐺₁=𝐺.

Remark 3. Let 𝑋 be a normed space (either classical or non-Archimedean) over a field 𝔽. Clearly, if 𝑋 is classical, then we assume that 𝔽∈{ℝ,ℂ}; if 𝑋 is non-Archimedean, then 𝔽 is a nontrivial non-Archimedean field. Given 𝑟∶𝑋→𝑋, write ‖𝑟‖:=inf{𝜉∈(0,∞)∶‖𝑟𝑥−𝑟𝑦‖≤𝜉‖𝑥−𝑦‖ for 𝑥,𝑦∈𝑋}. It is easy to see that if ‖𝑝‖<1, ‖𝑝‖<1, and 𝜃 has one of the following forms (i)𝜃(𝑥,𝑦)=𝐴(‖𝑥‖^𝑘+‖𝑦‖^𝑙)^𝑚, (ii)𝜃(𝑥,𝑦)=𝐴(max{‖𝑥‖^𝑘,‖𝑦‖^𝑙})^𝑚, (iii)𝜃(𝑥,𝑦)=𝐴‖𝑥‖^𝑘‖𝑦‖^𝑙,
with some 𝐴∈(0,∞) and 𝑘,𝑙,𝑚∈ℕ, then the assumptions of the above theorem are fulfilled. Multiplying and/or adding those functions we can obtain numerous further examples.
Note that in the situation when 𝜃 is of form (iii), 𝜃(𝑥,0)=𝜃(0,𝑥)=0 for 𝑥∈𝑋, whence 𝑔=𝐺 in the statement of Theorem 2. This means that (under the assumptions of Theorem 2) with 𝜃 given by (iii), every 𝑔∶𝑋→𝑌 satisfying (8) must be actually a solution to (1).
In particular, from Theorem 2, we get immediately the following result on stability of (1).

Corollary 4. Let 𝑋 be a normed space over a field 𝔽∈{ℝ,ℂ}, 𝑝∈𝔽, max𝑝,1−𝑝<1, (32) and, 𝜃∶𝑋²→[0,∞) be of one let the forms (i)–(iii). Assume that 𝑔∶𝑋→𝑌 satisfies the inequality 𝑑𝑔𝑝𝑥+1−𝑝𝑦+𝑔1−𝑝𝑥+𝑝𝑦,𝑔(𝑥)+𝑔𝑦≤𝜃𝑥,𝑦,𝑥,𝑦∈𝑋. (33) Then the following two statements are valid. (𝛼) If (i) or (ii) holds, there exists a unique solution 𝐺∶𝑋→𝑌 of (1) such that 𝑑(𝑔(𝑥),𝐺(𝑥))≤𝐴(min{‖𝑥‖^𝑘,‖𝑥‖^𝑙})^𝑚 for 𝑥∈𝑋. (𝛽) If (iii) holds, then 𝑔 is a solution to (1).

Remark 5. Let 𝑋 be a non-archimedean normed space over a field 𝔽 that is non-Archimedean and 𝑝∈𝔽. Then results analogous to Corollary 4 cannot be derived from Theorem 2, because we have 1=|𝑝+1−𝑝|≤max{|𝑝|,|1−𝑝|}, which means that, with |𝑝|<1, we get 1≤|1−𝑝|≤max{|𝑝|,1}=1.
However, if |𝑝|>1, then 1<|𝑝|=|1−𝑝−1|≤max{1,|1−𝑝|}, which means that |1−𝑝|>1. Consequently, we obtain results analogous to Corollary 4 for |𝑝|>1 and the function 𝜃 is one of the following forms (a)𝜃(𝑥,𝑦)=𝐴(‖𝑥‖^𝑘+‖𝑦‖^𝑙)^𝑚 for all (𝑥,𝑦)∈𝑋²⧵{(0,0)} and 𝜃(0,0)=0, (b)𝜃(𝑥,𝑦)=𝐴(max{‖𝑥‖^𝑘,‖𝑦‖^𝑙})^𝑚 for all (𝑥,𝑦)∈𝑋²⧵{(0,0)} and 𝜃(0,0)=0, (c)𝜃(𝑥,𝑦)=𝐴(‖𝑥‖^𝑘‖𝑦‖^𝑙)^𝑚 for all 𝑥,𝑦∈𝑋⧵{0} and 𝜃(𝑥,0)=𝜃(0,𝑥)=0 for all 𝑥∈𝑋,
with some 𝐴,𝑘,𝑙∈(0,∞) and 𝑚∈(−∞,0). Unfortunately, we must define 𝜃 at the point (0,0) in a bit artificial way in (a) and (b), similar (but even bigger) problem we have in (c). The subsequent modified version of Theorem 2 amends this situation to some extent for (a) and (b).

Theorem 6. Let 𝑝,𝑝∶𝑋→𝑋 be monomorphisms with 𝑝∘𝑝=𝑝∘𝑝, 𝐵∈(0,1), and 𝑔∶𝑋→𝑌, 𝜃∶𝑋²⧵{(0,0)}→[0,∞) satisfy 𝑑𝑔(𝑥)𝑦,𝜃𝑝𝑥+𝑝𝑦+𝑔𝑝𝑥+𝑝𝑦,𝑔+𝑔≤𝜃𝑥,𝑦max𝑝𝑥,𝑝𝑦,𝜃𝑝𝑥,𝑝𝑦≤𝐵𝜃𝑥,𝑦 (34) for (𝑥,𝑦)∈𝑋²⧵{(0,0)}. Then there is a solution 𝐺∶𝑋→𝑌 of (10) with 𝑑𝑔(𝑥),𝐺(𝑥)≤min{𝜃(𝑥,0),𝜃(0,𝑥)},𝑥∈𝑋⧵{0}. (35)

Proof. Arguing in the same way as in the proof of Theorem 2, with 𝑋 being replaced by 𝑋₀∶=𝑋⧵{0}, from Theorem 1, we deduce that there is a solution 𝐺₁∶𝑋₀→𝑌 of the equation 𝐺₁(𝑥)=𝐺₁𝑝𝑥+𝐺₁𝑝𝑥 (36) such that 𝑑(𝑔₀(𝑥),𝐺₁(𝑥))≤min{𝜃(𝑥,0),𝜃(0,𝑥)} for 𝑥∈𝑋₀. Moreover, 𝐺₁(𝑥)∶=lim_𝑛→∞(𝒯^𝑛𝑔₀)(𝑥) for 𝑥∈𝑋₀. Define 𝐺₀∶𝑋→𝑌 by 𝐺₀(𝑥)=𝐺₁(𝑥) for 𝑥∈𝑋₀ and 𝐺₀(0)=0.
In the same way as in the proof of Theorem 2, we obtain that 𝑑𝒯^𝑛𝑔₀𝑝𝑥+𝑝𝑦+𝒯^𝑛𝑔₀,𝒯𝑝𝑥+𝑝𝑦_𝑛𝑔₀(𝑥)+𝒯^𝑛𝑔₀𝑦≤𝐵^𝑛𝜃𝑥,𝑦,𝑥,𝑦∈𝑋₀,𝑝𝑥+𝑝𝑦,𝑝𝑥+𝑝𝑦∈𝑋₀ (37) for all 𝑛∈ℕ₀. We need yet to prove that, for every 𝑛∈ℕ₀, 𝑑𝒯^𝑛𝑔₀𝑝𝑥+𝑝𝑦,𝒯^𝑛𝑔₀(𝑥)+𝒯^𝑛𝑔₀𝑦≤𝐵^𝑛𝜃𝑥,𝑦,𝑥,𝑦∈𝑋₀𝑑𝒯,𝑝𝑥+𝑝𝑦=0,𝑝𝑥+𝑝𝑦≠0,_𝑛𝑔₀𝑝𝑥+𝑝𝑦,𝒯^𝑛𝑔₀(𝑥)+𝒯^𝑛𝑔₀𝑦≤𝐵^𝑛𝜃𝑥,𝑦,𝑥,𝑦∈𝑋₀,𝑝𝑥+𝑝𝑦≠0,𝑝𝑥+𝑝𝑦=0, (38)(39)𝑑0,𝒯^𝑛𝑔₀(𝑥)+𝒯^𝑛𝑔₀𝑦≤𝐵^𝑛𝜃𝑥,𝑦,𝑥,𝑦∈𝑋₀,𝑝𝑥+𝑝𝑦=𝑝𝑥+𝑝𝑦=0. (40) We show only (38); the proofs for (39) and (40) are analogous.
Clearly, for 𝑛=0, (38) follows from (34). Next, if (37) holds for a fixed 𝑛∈ℕ, then, for every 𝑥,𝑦∈𝑋₀ with 𝑝𝑥+𝑝𝑦=0 and 𝑝𝑥+𝑝𝑦≠0, 𝑑𝒯^𝑚+1𝑔₀𝑝𝑥+𝑝𝑦,𝒯^𝑚+1𝑔₀(𝑥)+𝒯^𝑚+1𝑔₀𝑦=𝑑𝒯^𝑚𝑔₀𝑝𝑝𝑥+𝑝𝑦+𝒯^𝑚𝑔₀𝑝,𝒯𝑝𝑥+𝑝𝑦_𝑚𝑔₀𝑝𝑥+𝒯^𝑚𝑔₀𝑝𝑥+𝒯^𝑚𝑔₀𝑝𝑦+𝒯^𝑚𝑔₀𝑑𝒯𝑝𝑦≤max^𝑚𝑔₀,𝒯𝑝𝑝𝑥+𝑝𝑝𝑦_𝑚𝑔₀𝑝𝑥+𝒯^𝑚𝑔₀,𝑑𝒯𝑝𝑦_𝑚𝑔₀𝑝,𝒯𝑝𝑥+𝑝𝑝𝑦_𝑚𝑔₀𝑝𝑥+𝒯^𝑚𝑔₀𝐵𝑝𝑦≤max^𝑚𝜃𝑝𝑥,𝑝𝑦,𝐵^𝑚𝜃𝑝𝑥,𝑝𝑦≤𝐵^𝑚+1𝜃𝑥,𝑦. (41) This completes the induction.
Letting 𝑛→∞ in (37)–(40), we obtain that 𝐺₀𝑝𝑥+𝑝𝑦+𝐺₀𝑝𝑥+𝑝𝑦=𝐺₀(𝑥)+𝐺₀𝑦,𝑥,𝑦∈𝑋. (42) Writing 𝐺(𝑥)∶=𝐺₀(𝑥)+𝑔(0) for 𝑥∈𝑋 we obtain that (35) holds and 𝐺 is a solution to (10).

Below we present a theorem that is somewhat complementary to Theorem 2.

Theorem 7. Let 𝑝 and 𝑝 be endomorphisms of 𝑋 with 𝑝∘𝑝=𝑝∘𝑝, and 𝑝 be bijective, 𝐵∈(0,1), and 𝑔∶𝑋→𝑌 and 𝜃∶𝑋²→[0,∞) satisfy the inequalities max𝜃𝑥,𝑦,𝜃𝑝𝑥,𝑝𝑦≤𝐵𝜃𝑝𝑥,𝑝𝑦,𝑥,𝑦∈𝑋. (43) Assume that 𝑔∶𝑋→𝑌 satisfies (8). Then there exists a unique solution 𝐺∶𝑋→𝑌 of (10) such that 𝑑𝑔(𝑥),𝐺(𝑥)≤min𝜃𝑝⁻¹𝑥,0,𝜃0,𝑝⁻¹𝑥,𝑥∈𝑋. (44) Moreover, 𝐺 is the unique solution of (10) for which there exists a constant 𝑀∈(0,∞) with 𝑑𝑔(𝑥),𝐺(𝑥)≤𝑀min𝜃𝑝⁻¹𝑥,0,𝜃0,𝑝⁻¹𝑥,𝑥∈𝑋. (45)

Proof. The proof is very similar to that of Theorem 2, but for the convenience of readers we present it here.
Taking in (8) first 𝑦=0 and next 𝑥=0 we obtain 𝑑𝑔𝑝𝑥+𝑔𝑝𝑥,𝑔(𝑥)+𝑔(0)≤min{𝜃(𝑥,0),𝜃(0,𝑥)},𝑥∈𝑋, (46) whence replacing 𝑥 by 𝑝⁻¹𝑥 we derive the following inequality: 𝑑𝑔(𝑥)+𝑔𝑝𝑝⁻¹𝑥,𝑔𝑝⁻¹𝑥(0𝜃𝑝+𝑔)≤min⁻¹𝑥,0,𝜃0,𝑝⁻¹𝑥,𝑥∈𝑋. (47) Since 𝑑 is invariant, this yields 𝑑𝑔(𝑥)−𝑔(0),𝑔𝑞𝑥−𝑔𝑝𝑞𝑥≤𝜀(𝑥),𝑥∈𝑋, (48) with 𝑞∶=𝑝⁻¹ and 𝜀(𝑥)∶=min{𝜃(𝑞𝑥,0),𝜃(0,𝑞𝑥)}. Writing 𝑔₀(𝑥)∶=𝑔(𝑥)−𝑔(0) and 𝒯𝜉(𝑥)∶=𝜉(𝑞𝑥)−𝜉(𝑝𝑞𝑥) for 𝑥∈𝑋, 𝜉∈𝑌^𝑋, we finally get 𝑑𝑔₀(𝑥),𝒯𝑔₀(𝑥)≤𝜀(𝑥),𝑥∈𝑋. (49)
Define an operator Λ∶ℝ₊^𝑋→ℝ₊^𝑋 by Λ𝜂(𝑥)∶=max{𝜂(𝑞𝑥),𝜂(𝑝𝑞𝑥)} for 𝜂∈ℝ₊^𝑋, 𝑥∈𝑋. Clearly, Λ is non-decreasing and satisfies hypothesis (𝒞), and, according to (43), 𝑑(𝑥)(𝑥)𝜉𝑑𝜉𝜉Δ(𝑥)𝒯𝜉,𝒯𝜇=𝑑𝑞𝑥−𝜉𝑝𝑞𝑥,𝜇𝑞𝑥−𝜇𝑝𝑞𝑥≤max𝑞𝑥,𝜇𝑞𝑥,𝑑𝑝𝑞𝑥,𝜇𝑝𝑞𝑥=max𝜉,𝜇𝑞𝑥,Δ𝜉,𝜇𝑝𝑞𝑥=ΛΔ𝜉,𝜇,𝜉,𝜇∈𝑌^𝑋,𝑥∈𝑋. (50)
Note that 𝜀𝜃𝜃(𝑥)𝑞𝑥=min𝑞𝑞𝑥,0,𝜃0,𝑞𝑞𝑥≤min𝐵𝜃𝑞𝑥,0,𝐵𝜃0,𝑞𝑥=𝐵min𝑞𝑥,0,𝜃0,𝑞𝑥=𝐵𝜀,𝑥∈𝑋, (51) and analogously 𝜀(𝑝𝑞𝑥)≤𝐵𝜀(𝑥) for 𝑥∈𝑋. Consequently max𝜀𝑞𝑥,𝜀𝑝𝑞𝑥≤𝐵𝜀(𝑥),𝑥∈𝑋. (52)
It is easy to show by induction that Λ^𝑛𝜀(𝑥)≤𝐵^𝑛𝜀(𝑥) for 𝑥∈𝑋, 𝑛∈ℕ. Since 0<𝐵<1, this means that lim_𝑛→∞(Λ^𝑛𝜀)(𝑥)=0 for 𝑥∈𝑋. Further, by (52), for every 𝑛∈ℕ₀ and 𝑥∈𝑋 we have Λ^𝑛+1𝜀(𝑥)≤Λ^𝑛𝜀(𝑥), whence sup_𝑛∈ℕ0Λ^𝑛𝜀(𝑥)=Λ⁰𝜀(𝑥)=𝜀(𝑥),sup_𝑛∈ℕ0Λ^𝑛+1𝜀(𝑥)=Λ𝜀(𝑥). (53) Consequently, by Theorem 1, there is a function 𝐺₀∶𝑋→𝑌 such that 𝑑(𝑔₀(𝑥), 𝐺₀(𝑥))≤𝜀(𝑥), 𝐺₀(𝑥)=𝐺₀(𝑞𝑥)−𝐺₀(𝑝𝑞𝑥), and 𝐺₀(𝑥)=lim_𝑛→∞(𝒯^𝑛𝑔₀)(𝑥) for 𝑥∈𝑋. It is easily seen that 𝐺₀(𝑥)=𝐺₀𝑝𝑥+𝐺₀𝑝𝑥,𝑥∈𝑋. (54)
Now, we show that, for every 𝑛∈ℕ₀, 𝑑𝒯^𝑛𝑔₀𝑝𝑥+𝑝𝑦+𝒯^𝑛𝑔₀,𝒯𝑝𝑥+𝑝𝑦_𝑛𝑔₀(𝑥)+𝒯^𝑛𝑔₀𝑦≤𝐵^𝑛𝜃𝑥,𝑦,𝑥,𝑦∈𝑋. (55) Clearly, for 𝑛=0, (55) reduces to (8). Next, fix 𝑚∈ℕ₀ and assume that (55) holds for every 𝑥,𝑦∈𝑋 with 𝑛=𝑚. Then, by (43), 𝑑𝒯^𝑚+1𝑔₀𝑝𝑥+𝑝𝑦+𝒯^𝑚+1𝑔₀,𝒯𝑝𝑥+𝑝𝑦_𝑚+1𝑔₀(𝑥)+𝒯^𝑚+1𝑔₀𝑦=𝑑𝒯^𝑚𝑔₀𝑞𝑝𝑥+𝑝𝑦−𝒯^𝑚𝑔₀𝑝𝑞𝑝𝑥+𝑝𝑦+𝒯^𝑚𝑔₀𝑞𝑝𝑥+𝑝𝑦−𝒯^𝑚𝑔₀,𝒯𝑝𝑞𝑝𝑥+𝑝𝑦_𝑚𝑔₀𝑞𝑥−𝒯^𝑚𝑔₀𝑝𝑞𝑥+𝒯^𝑚𝑔₀𝑞𝑦−𝒯^𝑚𝑔₀𝑑𝒯𝑝𝑞𝑦≤max^𝑚𝑔₀𝑝𝑞𝑥+𝑝𝑞𝑦+𝒯^𝑚𝑔₀,𝒯𝑝𝑞𝑥+𝑝𝑞𝑦_𝑚𝑔₀𝑞𝑥+𝒯^𝑚𝑔₀,𝑑𝒯𝑞𝑦_𝑚𝑔₀𝑝𝑝𝑞𝑥+𝑝𝑝𝑞𝑦+𝒯^𝑚𝑔₀𝑝,𝒯𝑝𝑞𝑥+𝑝𝑝𝑞𝑦_𝑚𝑔₀𝑝𝑞𝑥+𝒯^𝑚𝑔₀𝐵𝑝𝑞𝑦≤max^𝑚𝜃𝑞𝑥,𝑞𝑦,𝐵^𝑚𝜃𝑝𝑞𝑥,𝑝𝑞𝑦≤𝐵^𝑚+1𝜃𝑥,𝑦,𝑥,𝑦∈𝑋. (56) Thus we obtain (55) for 𝑛=𝑚+1, which completes the induction. Letting 𝑛→∞ in (55), we get 𝐺₀(𝑝𝑥+𝑝𝑦)+𝐺₀(𝑝𝑥+𝑝𝑦)=𝐺₀(𝑥)+𝐺₀(𝑦) for 𝑥,𝑦∈𝑋. Next, writing 𝐺(𝑥)∶=𝐺₀(𝑥)+𝑔(0) for 𝑥∈𝑋, we obtain that 𝐺 is a solution to (1).
To complete the proof let us yet mention that we show the uniqueness of 𝐺 in the same way as in the proof of Theorem 2.

Now, it is easily seen that Theorem 7 yields the subsequent corollary, which is an analogue of Corollary 4.

Corollary 8. Let 𝑋 be a normed space over a field 𝔽∈{ℝ,ℂ}, 𝑝∈𝔽, max1,1−𝑝<𝑝, (57) and let 𝜃∶𝑋²→[0,∞) be one of the forms (i)–(iii). If 𝑔∶𝑋→𝑌 satisfies inequality (33), then the following two statements are valid. (𝛼) If (i) or (ii) holds, there exists a unique solution 𝐺∶𝑋→𝑌 of (1) such that 𝑑(𝑔(𝑥),𝐺(𝑥))≤𝐴(min{‖𝑝⁻¹𝑥‖^𝑘,‖𝑝⁻¹𝑥‖^𝑙})^𝑚 for 𝑥∈𝑋. (𝛽) If (iii) holds, then 𝑔 is a solution to (1).