APPROXIMATELY QUINTIC MAPPINGS IN NON-ARCHIMEDEAN 2-NORMED SPACES BY FIXED POINT THEOREM

Abstract

In this paper, using the fixed point method, we investigate the generalized Hyers-Ulam stability of the system of quintic functional equation $f(x_1+x_2,y)+f(x_1-x_2,y)=2f(x_1,y)+2f(x_2,y)\;f(x,2_{y1}+y_2)+f(x,2_{y1}-y_2)=f(x,y_1-2_{y2})+f(x,y_1+y_2)\;-f(x,y_1-y_2)+15f(x,y_1)+6f(x,y_2)$ in non-Archimedean 2-Banach spaces.

1. Introduction and preliminaries

In 1940, Ulam [22] posed the following problem concerning the stability of functional equations:

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(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?

Hyers [8] solved the Ulam's problem for the case of approximately additive functions in Banach spaces. Since then, the stability of several functional equations has been extensively investigated by several mathematicians [3,5,9,10,11,14,17]. The Hyers-Ulam stability for the quadratic functional equation

was proved by Skof [21] for a function f : E1 → E2, where E1 is a normed space and E2 is a Banach space and later by Jung [13] on unbounded domains.

Rassias [20] investigated the stability for the following cubic functional equation

and Jun and Kim [12] investigated the stability for the following cubic funtional equation

A valuation is a function | · | from a field K into [0,∞) such that for any r, s ∈ K, the following conditions hold: (i) |r| = 0 if and only if r = 0, (ii) |rs| = |r║s|, and (iii) |r + s| ≤ |r| + |s|. A field K is called a valued field if K carries a valuation. The usual absolute values of ℝ and ℂ are examples of valuations. If the triangle inequality is replaced by |r + s| ≤ max{|r|, |s|} for all r, s ∈ K, then the valuation | · | is called a non-Archimedean valuation and the field with a non-Archimedean valuation is called a non-Archimedean field. If | · | is a non-Archimedean valuation on K, then clearly, |1| = | − 1| and |n| ≤ 1 for all n ∈ ℕ.

Definition 1.1. Let X be a vector space over a non-Archimedean field K. A function ║ · ║ : X → ℝ is called a non-Archimedean norm if it satisfies the following conditions:

(a) ║x║ = 0 if and only if x = 0,

(b) ║rx║ = |r|║x║, and

(c) ║x + y║ ≤ max{║x║, ║y║} for all x, y ∈ X and all r ∈ K.

If ║ · ║ is a non-Archimedean norm, then (X, ║ · ║) is called a non-Archimedean normed space.

Let (X, ║ · ║) be a non-Archimedean normed space and {xn} a sequence in X. Then {xn} is said to be convergent in (X, ║ · ║) if there exists an x ∈ X such that limn→∞ ║xn − x║ = 0. In case, x is called the limit of the sequence {xn}, and one denotes it by limn→∞ xn = x. A sequence {xn} is said to be Cauchy in (X, ║ · ║) if limn→∞ ║xn+p − xn║ = 0 for all p ∈ ℕ. By (c) in Definition 1.1,

a sequence {xn} is Cauchy in (X, ║ · ║) if and only if {xn+1 − xn} converges to zero in (X, ║ · ║). By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent.

Gähler [6,7] has introduced the concept of 2-normed spaces and White [23] introduced the concept of 2-Banach spaces. In 1999 to 2003, Lewandowska published a series of papers on 2-normed sets and generalized 2-normed spaces [15,16].

Definition 1.2. Let X be a linear space over a non-Archimedean field K with dim X > 1 and ║·, ·║ : X × X → ℝ a function satisfying the following properties

(N A1) ║x, y║ = 0 if and only if x and y are linearly dependent,

(N A2) ║x, y║ = ║y, x║,

(N A3) ║x, ay║ = |a|║x, y║ , and

(N A4) ║x, y + z║ ≤ max{║x, y║, ║x, z║}

for all x, y, z ∈ X and all a ∈ K. Then ║· , ·║ is called a non-Archimedean 2-norm and (X, ║· , ·║) is called a non-Archimedean 2-normed spaces.

Definition 1.3. A sequence {xn} in a non-Archimedean 2-normed space (X, ║· , ·║) is called a Cauchy sequence if

for all x ∈ X.

Definition 1.4. A sequence {xn} in a non-Archimedean 2-normed space (X, ║· , ·║) is called convergent if

for all y ∈ X and for some x ∈ X. In case, x is called the limit of the sequence {xn}, and we denoted by xn → x as n → ∞ or limn→∞ xn = x.

Let {xn} be a sequence in a non-Archimedean 2-normed space (X, ║· , ·║). It follows from (NA4) that

for all y ∈ X and so a sequence {xn} is a Cauchy sequence in (X, ║· , ·║) if and only if {xm+1 − xm} converges to zero in (X, ║· , ·║).

A non-Archimedean 2-normed space (X, ║· , ·║) is called a non-Archimedean 2-Banach space if every Cauchy sequence in (X, ║· , ·║) is convergent. Now, we state the following results as lemma [18].

Lemma 1.5. Let (X, ║· , ·║) be a non-Archimedean 2-normed space. Then we have the following :

(1) |║x, z║ − ║y, z║| ≤ ║x − y, z║ for all x, y, z ∈ X,

(2) ║x, z║ = 0 for all z ∈ X if and only if x = 0, and

(3) for any convergent sequence {xn} in (X, ║· , ·║),

for all z ∈ X.

In 2003, Radu [19] proposed a new method for obtaining the existence of exact solutions and error estimations, based on the fixed point alternative (see also [1,2]).

We recall the following theorem by Margolis and Diaz.

Theorem 1.6 ([4]). Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with some Lipschitz constant L with 0 < L < 1. Then for each given element x ∈ X, either d(Jnx, Jn+1x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that

(1) d(Jnx, Jn+1x) < ∞ for all n ≥ n0

(2) the sequence {Jnx} converges to a fixed point y∗ of J

(3) y∗ is the unique fixed point of J in the set Y = {y ∈ X | d(Jn0x, y) < ∞}

(4) d(y, y∗) ≤ for all y ∈ Y .

In this paper, we investigate the following cubic functional equation

and using fixed point method, we inverstigate the generalized Hyers-Ulam stability for the system of the quintic functional equation

and prove the generalized Hyers-Ulam stability for (3) in non-Archimedean 2-Banach spaces. In this paper, we will assume that (X, ║·║) is a a non-Archimedean norm space and (Y, ║· , ·║) is a non-Archimedean 2-Banach space.

 

2. Stability of quintic mappings

In this section, using the fixed point method, we investigate the generalized Hyers-Ulam stability for the system of quintic functional equation (3) in non-Archimedean 2-Banach spaces. We start the following lemma.

Lemma 2.1. Let f : X → Y be a mapping with (2). Then f is a cubic mapping.

Proof. Suppose that f satisfies (2). Letting x = y = 0 in (2), we have f(0) = 0 and letting y = 0 in (2), we have

for all x ∈ X. Letting x = 0 in (2), by (4), we have f(y) = −f(−y) for all y ∈ X and so f is odd. Letting y = −y in (2), we have

for all x, y ∈ X and by (2) and (5), we have

for all x, y ∈ X. Interching x and y in (6), since f is odd, f satisfies (1) and hence f is cubic. □

The function f : ℝ × ℝ → ℝ given by f(x, y) = cx2y3 is a solution of (3). In partcular, letting y = x in (3), we get a quintic function g : ℝ → ℝ in one variable given by g(x) = f(x, x) = cx5.

Proposition 2.2. If a mapping f : X2 → Y satisfies (3), then f(λx, μy) = λ2μ3f(x, y) for all x, y ∈ X and all rational numbers λ, μ.

Theorem 2.3. Let ϕ1, ϕ2 : X3 × Y → [0,∞) be functions such that

for all x, y, z ∈ X, w ∈ Y and some L with 0 < L < 1. Suppose that f : X2 → Y is a mapping such that f(x, 0) = f(0, x) = 0 for all x ∈ X,

and

for all w ∈ Y and all x, y, x1, x2, y1, y2 ∈ X. Then there exists a unique quintic mapping T : X2 → Y satisfying (3) and

for all w ∈ Y and all x, y ∈ X, where

Proof. Putting y2 = 0 and y1 = y in (9), we get

for all w ∈ Y and all x, y ∈ X. Putting x1 = x2 = x in (8), we get

for all w ∈ Y and all x, y ∈ X. Thus by (11) and (12), we have

for all w ∈ Y and all x, y ∈ X. It follows from (13) that

for all w ∈ Y and all x, y ∈ X.

Consider the set S = {h | h : X × X → Y with h(x, 0) = h(0, x) = 0, ∀x ∈ X} and the generalized metric d on S defined by

Then (S, d) is a complete metric space [2]. Define a mapping J : S → S by Jg(x, y) = 2−5g(2x, 2y) for all x, y ∈ X and all g ∈ S. Let g, h ∈ S and d(g, h) ≤ ε for some non-negative real number ε. Then by (7), we have

and so d(Jg, Jh) ≤ εL. This mean that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S and so J is a strictly contractive mapping. By (14), we get d(Jf, f) ≤ 1 < ∞. By Theorem 1.6, there exists a mapping T : X2 → Y which is a fixed point of J such that d(Jnf, T) → 0 as n → ∞, which implies the equality T(x, y) = limn→∞ 2−5n f(2nx, 2ny). Since d(Jf, f) ≤ 1 < ∞, by (4) in Theorem 1.6, we have (10). By (8) and (9), we get

and

for all w ∈ Y and all x, y, x1, x2, y1, y2 ∈ X. Hence T satisfies (3).

To prove the uniquness of T, assume that T1 : X2 → Y is another solution of (3) satisfying (10). Then T1 is a fixed point of J and by (10),

By (3) in Theorem 1.6, we have T = T1. □

Theorem 2.4. Let ϕ1, ϕ2 : X3 × Y → [0,∞) be functions such that

for all x, y, z ∈ X, w ∈ Y and some L with 0 < L < 1. Suppose that f : X2 → Y is a mapping satisfying f(x, 0) = f(0, x) = 0 for all x ∈ X, (8) and (9). Then there exists a unique quintic mapping T : X2 → Y satisfying (3) and

for all w ∈ Y and all x, y ∈ X, where

Proof. Putting y2 = 0 and in (9), we get

for all w ∈ Y and all x, y ∈ X. Putting in (8), we get

for all w ∈ Y and all x, y ∈ X. Thus by (17) and (18), we have

for all x, y ∈ X and all w ∈ Y . That is, we have

for all x, y ∈ X and all w ∈ Y .

Consider the set S = {h | h : X × X → Y with h(x, 0) = h(0, x) = 0, ∀x ∈ X} and the generalized metric d on S defined by

Then (S, d) is a complete metric space([2]). Define a mapping J : S → S by for all x, y ∈ X and all g ∈ S. Let g, h ∈ S and d(g, h) ≤ ε for some non-negative real number ε. Then by (15), we have

and so d(Jg, Jh) ≤ εL. This mean that d(Jg, Jh) ≤ L d(g, h) for all g, h ∈ S and so J is a strictly contractive mapping. By (19), we get d(Jf, f) ≤ L < ∞. By Theorem 1.6, there exists a mapping T : X2 → Y which is a fixed point of J such that d(Jnf, T) → 0 as n → ∞, which imples the equality T(x, y) = limn→∞ . Since d(Jf, f) ≤ L, by (4) in Theorem 1.6, we have (16) and by (8) and (9), we get

and

for all w ∈ Y and all x, y, x1, x2, y1, y2 ∈ X. Hence T satisfies (3).

To prove the uniquness of T, assume that T1 : X2 → Y is another solution of (3) satisfying (16). Then T1 is a fixed point of J and by (16),

By (3) in Theorem 1.6, we have T = T1. □

As example of ϕ1(x1, x2, y,w) and ϕ2(x, y1, y2,w) in Theorem 2.3 and Theorem 2.4, we can take ϕ1(x1, x2, y,w) = θ (║x1║p + ║x2║p + ║y║p)║w║ and ϕ2(x, y1, y2,w) = |2|4 θ (║x║p +║y1║p +║y2║p)║w║ for all x, y, x1, x2, y1, y2 ∈ X, all w ∈ Y and some positive real number θ. Then we have the following corollary.

Corollary 2.5. Let θ, p be positive real numbers with p ≠ 5. Suppose that f : X2 → Y is a mapping satisfying f(x, 0) = f(0, x) = 0,

and

for all w ∈ Y and all x, y, x1, x2, y1, y2 ∈ X. Then there exists a unique quintic mapping T : X2 → Y satisfying

for all w ∈ Y and x, y ∈ X.

Proof. Let ϕ1(x1, x2, y,w) = θ (║x1║p +║x2║p +║y║p)║w║ and ϕ2(x, y1, y2,w) = |2|4 θ (║x║p + ║y1║p + ║y2║p)║w║. Note that

So if p > 5, by Theorem 2.3, we have (20). Note that

So if p < 5, by Theorem 2.4, we have (20). □

As another example of ϕ1(x, y, z,w) and ϕ2(x, y, z,w) in Theorem 2.3 and Theorem 2.4, we can take ϕ1(x, y, z,w) = ϕ2(x, y, z,w) = θ ║x║p║y║q║z║r║w║ for all x, y, z ∈ X, all w ∈ Y and some positive real number p, q, r, θ. Then we have the following corollary:

Corollary 2.6. Let p, q, r and θ be positive real numbers with p + q + r ≠ 5. Suppose that f : X2 → Y is a mapping satisfying f(x, 0) = 0,

and

for all w ∈ Y and all x, y, x1, x2, y1, y2 ∈ X. Then there exists a unique quintic mapping T : X2 → Y satisfying

for all w ∈ Y and all x, y ∈ X.

Proof. Let ϕ1(x, y, z,w) = ϕ2(x, y, z,w) = θ ║x║p║y║q║z║r║w║. Then we have

Hence if p + q + r > 5, by Theorem 2.3, we have (21). Note that

Thus if p + q + r < 5, by Theorem 2.4, we have (21). □