1.Introduction

The starting point of studying the stability of functional equations seems to be the famous talk of Ulam (^{29}) in 1940, in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of group homomorphisms.

Let *G*
_{1} be a group and let *G*
_{2} be a metric group with a metric *d*(.,.). Given *ε* > 0, does there exists a δ > 0 such that if a mapping *h*:*G*
_{1} → *G*
_{2} satisfies the inequality *d(h)xy),h(x)h(y))* < *δ* for all *x,y* ∈ _{
G1
} , then there exists a homomorphism *h*:*G*
_{1} → *G*
_{2} with *d(h)x), H(x))< ε* for all *x* ∈ _{
G1
} .

The first partial answer, in the case of Cauchy equation in Banach spaces, to Ulam question was given by Hyers (^{22}). Later, the result of Hyers was first generalized by Aoki (?). And only much later by Rassias (^{27}) and Găvruţa (^{20}). Since then, the stability problems of several functional equations have been extensively investigated.

We say a functional equation is *hyperstable* if any function *f* satisfying the equation approximately (in some sense) must be actually a solution to it. It seems that the first hyperstability result was published in (^{11}) and concerned the ring homomorphisms. However, the term *hyperstability* has been used for the first time in (^{25}). Quite often the hyperstability is confused with superstability, which admits also bounded functions. Numerous papers on this subject have been published and we refer to (^{1}^{)-(}^{5}, ^{8}, ^{15}^{)-(}^{18}, ^{21}, ^{25}, ^{26}, ^{28}).

Throughout this paper, **N** stands for the set of all positive integers,**N**
_{0}:= **N** ∪ {0},
the set of integers ≥ m_{0}, **R**
_{+}: =
0, ∞) and we use the notation *X*
_{0} for the set *X* \ {0}.

Let us recall (see, for instance, (^{24}) some basic definitions and facts concerning non-Archimedean normed spaces.

**Definition 1.1.:** By a Archimedean field we mean a field **K** equipped with a function (*valuation*) |∙|: **K** →
0, ∞) such that for all *r,s* ∈ **K,** the following conditions hold:

1. |*r*| = 0 if and only if *r* = 0,

2.|*rs*| = |*r*| |*s*|,

3.|*r + s*| ≤ *max*{|*r*|,|*s*|}.

The pair (**K**,|.|) *is called a valued field.*

In any non-Archimedean field we have |1| = |-1| = 1 and and |n| ≤ 1 for n ∈ **N**
_{0}. In any field **K** the function |∙|:**K** → **R**
_{+} given by

is a valuation which is called *trivial*, but the most important examples of non-Archimedean fields are *p*-adic numbers which have gained the interest of physicists for their research in some problems coming from quantum physics, *p*-adic strings and superstrings.

**Definition 1.2.** Let *X* be a vector space over a scalar field **K** with a non-Archimedean non-trivial valuation |∙|. A function ||∙||_{*}: X→ **R** is a *Archimedean norm (valuation)* if it satisfies the following conditions:

1 . ||*x*||_{*} = 0 if and only if *x* = 0,

2 . ||*rx*||_{*} = |*r*| ||*x*||_{*}(*r* ∈ **K**, *x* ∈ *X*),

3 .The strong triangle inequality (ultrametric); namely

||*x* + *y*||_{*} ≤ *max*{||*x*||_{*},||*y*||_{*}}*x,y* ∈ *X*.

Then (X,||∙||_{*}) is called *a non-Archimedean normed space or an ultrametric normed space*.

**Definition 1.3.** Let {*x*
_{n}} be a sequence in a non-Archimedean normed space *X*.

1. A sequence
in a non-Archimedean space is a *Cauchy sequence if f* the sequence
converges to zero;

2. The sequence {*x*
_{n}} is said to be *convergent* if, there exists *x* ∈ *X* such that, for any *ε* > 0, there is a positive integer *N* such that ||*x*
_{n} - *x*||_{*} ≤ *ε*, for all *n* ≥ *N*. Then the point *x* ∈ *X* is called the *limit* of the sequence {*x*
_{n}}, which is denoted by *lim*
_{n→∞}
*X*
_{n} = x;

3. If every Cauchy sequence in *X* converges, then the non-Archimedean normed space *X* is called a *non-Archimedean Banach space* or an *ultrametric Banach space*.

Let *X,Y* be normed spaces. A function *f*:*X*→*Y* is Cubic provided it satisfies the functional equation

*f*(2*x* + *y*) + *f*(2*x - y*) = 2*f*(*x + y*) + 2*f*(*x - y*) + 12*f*(*x*) for all *x, y* ∈ *X*,

(1.1)

and we can say that *f*: X→ Y is Cubic on *X*
_{0} if it satisfies (1.1) for all *x, y* ∈ *X*
_{0}, such that x + y ≠ 0.

In 2013, A. Bahyrycz and al. (^{7}) used the fixed point theorem from (^{12}, Theorem 1) to prove the stability results for a generalization of *p*-Wright affine equation in ultrametric spaces. Recently, corresponding results for more general functional equations (in classical spaces) have been proved in (^{9}), (^{10}), (^{30}) and (^{31}).

In this paper, by using the fixed point method derived from (^{8}), (^{15}) and (^{14}), we present some hyperstability results for the equation (1.1) in ultrametric Banach spaces. Before proceeding to the main results, we state Theorem 1.4 which is useful for our purpose. To present it, we introduce the following three hypotheses:

Thanks to a result due to J. Brzdȩk and K. Ciepliñski (^{13}, Remark 2), we state a slightly modified version of the fixed point theorem (^{12}, Theorem 1) in ultrametric spaces. We use it to assert the existence of a unique fixed point of operator *T*: _{
YX
} → _{
YX
} .

**Theorem 1.4.** Let hypotheses (**H1**)-(**H3)** be valid and functions *ε*: *X* → **R**
_{+} and ϕ: *X → Y* fulfil the following two conditions

Then there exists a unique fixed point *ψ* ∈ _{
YX
} of *T* with

2.Main results

In this section, using Theorem 1.4 as a basic tool to prove the hyperstability results of the cubic functional equation in ultrametric Banach spaces.

**Theorem 2.1.** Let (*X*,||∙||) and (*Y*, ||∙||_{*}) be normed space and ultrametric Banach space respectively, *c* ≥ 0, *p,q* ∈ R, *p + q* < 0, and let *f*:*X* → *Y* satisfy

||*f*(2*x + y*) + *f*(*2x - y*) - 2*f*(*x + y*) - 2*f*(*x - y*) - 12*f*(*x*)||_{*} ≤ *c*||*x*||^{p}||*y*||^{q},

(2.1)

for all *x,y* ∈ *X*
_{0}. Then *f* is cubic on *X*
_{0}.

**Proof.** Take *m* ∈ **N** such that

Since *p* + *q* < 0, one of *p,q* must be negative. Assume that *q* < 0 and replace *Y* by *mx* and *x* by
in (2.1). Thus

and write

So, (**H2**) is valid.

By using mathematical induction, we will show that for each *x* ∈ *X*
_{0} we have

This shows that (2.5) holds for *n = k* + 1. Now we can conclude that the inequality (2.5) holds for all *n* ∈ **N**
_{0}. From (2.5), we obtain

for all *x* ∈ *X*
_{0}. Hence, according to Theorem 1.4, there exists a unique solution *C*
_{m}: *X*
_{0} → *Y* of the equation

Moreover,

for all *x* ∈ *X*
_{0}. Now we show that

for every *x,y* ∈ *X*
_{0} such that *x* + *y* ≠ 0. Since the case *n* = 0 is just (2.1), take *k* ∈ **N** and assume that (2.9) holds for *n = k* and every *x,y* ∈ *X*
_{0} such that *x* + *y* ≠ 0. Then

for all *x,y* ∈ *X*
_{0} such that *x + y* ≠ 0. Thus, by induction we have shown that (2.9) holds for every n ∈ **N**
_{0}. Letting *n* → ∞ in (2.9), we obtain that

*C*
_{m}(*2x + y*) + *C*
_{m}(*2x - y*) = 2*C*
_{m}(*x + y*) + 2*C*
_{m}(*x - y*) + 12*C*
_{m}(*x*)

for all *x,y* ∈ *X*
_{0} such that *x + y* ≠ 0. In this way we obtain a sequence
of cubic functions on *X*
_{0} such that

It follows, with *m* → ∞, that *f* is Cubic on *X*
_{0}.

In a similar way we can prove the following theorem.

**Theorem 2.2.**Let (*X*,||∙||) and (*Y*,||∙||_{*}) be normed space and ultrametric Banach space respectively, *c* ≥ 0, *p,q* ∈ **R**, *p + q* > 0 and let *f*: *X* → *Y* satisfy

Write

Moreover, for every *ξ, μ* ∈ _{
YX0, x
} ∈_{
X0
}

So, (**H2**) is valid.

By using mathematical induction, we will show that for each *x* ∈ *X*
_{0} we have

This shows that (2.17) holds for *n = k* + 1. Now we can conclude that the inequality (2.17) holds for all *n* ∈ **N**
_{0}. From (2.17), we obtain

for all *x* ∈ *X*
_{0}. Hence, according to Theorem 1.4, there exists a unique solution *C*
_{m}; *X*
_{0} → *Y* of the equation

for every *x,y* ∈ *X*
_{0} such that *x + y* ≠ 0. Since the case *n* = 0 is just (2.12), take *k* ∈ **N** and assume that (2.20) holds for *n = k* and every *x,y* ∈ *X*
_{0} such that *x + y* ≠ 0. Then

for all *x, y* ∈ *X*
_{0} such that *x* + *y* ≠ 0. Thus, by induction we have shown that (2.20) holds for every *n* ∈ **N**
_{0}.Letting *n* → ∞ in (2.20), we obtain that

*C*
_{m}(2*x + y*) + *C*
_{m}(2*x - y*) = 2*C*
_{m}(*x + y*) + 2*C*
_{m}(*x - y*) + 12*C*
_{m}(*x*)

for all *x, y* ∈ *X*
_{0} such that *x* + *y* ≠ 0. In this way we obtain a sequence
of Cubic functions on *X*
_{0} such that

It follows, with *m* → ∞, that *f* is Cubic on *X*
_{0}.

It easy to show the hyperstability of Cubic equation on the set containing 0. We present the following theorem and we refer to see (^{8}, Theorem 5).

**Theorem 2.3.** Let (*X*,||∙||) and (Y,||∙||_{*}) be normed space and ultrametric Banach space respectively, *c* ≥ 0, *p,q* ∈ **R**, *p, q* > 0 and let *f*:*X* → *Y* satisfy

||*f*(2*x + y*)) + *f*(*2x + y*) - 2*f*(*x + y*) - 2*f*(*x - y*) - 12*f*(*x*)||_{*} ≤ *c*||*x*||^{p}||*y*||^{q},

for all *x,y* ∈ *X*
_{0}. Then *f* is Cubic on *X*
_{0}.

**Proof.** Same proof of last theorem

The above theorems imply in particular the following corollary, which shows their simple application.

**Corollary 2.4.** Let (*X*,||∙||) and (*Y*,||∙||_{*}) be normed space and ultrametric Banach space respectively, *G:X*
^{2}→ *Y* and *G*(*x,y*) ≠ 0 for some *x,y* ∈ *X* and

(2.23) ||*G*(*x,y*)||_{*} ≤ *c*||*x*||^{p}||*y*||^{q},x,y ∈

where c ≥ 0,*p,q* ∈ **R**. Assume that the numbers *p, q* satisfy one of the following conditions:

1. *p + q* < 0, and (2.1) holds for all *x,y* ∈ _{
X0
} ,

2. *p + q* < 0, and (2.12) holds for all *x,y* ∈ _{
X0
} .

Then the functional equation

*f* (2*x + y*) + *f* (2*x + y*) = 2*f* (*x + y*) + 2*f* (*x - y*) + 12*f* (*x*) + *G*(*x,y*), *x, y* ∈ *X*

(2.24)

has no solution in the class of functions *g: X* →*Y*.

In the following theorem, we present a general hyperstability for the Cubic equation where the control function is ϕ(*x*) + ϕ(*y*), which corresponds to the approach introduced in (^{14})

**Theorem 2.5.** Let (*X*,||∙||) be a normed space, (*Y*,||∙||_{*}) be an ultrametric Banach space over a field **K**, and ϕ: X → **R**
_{+} be a function such that

*U*: = {*n* ∈ **N**; *α*
_{n}: = max{λ(*n*), λ(3*n* - 1), λ(-*n* + 1), λ(4*n* - 1)} < 1}

(2.25)

is an infinite set, where λ(*a*): = *inf*{*t* ∈ **R**
_{+}: ϕ(*ax*) ≤ *t*ϕ(*x*) for all x ∈ *X*} for all a ∈ **K**
_{0.} Suppose that

And *f*: *X* → *Y* satisfies the inequality

||*f*(2*x + y*)) + *f*(2*x + y*) -2*f*(*x + y*) - 2*f*(*x - y*) - 12*f*(*x*)||_{*} ≤ ϕ(*x*) + ϕ(*y*),

(2.26)

for all _{
x,y ∈ X0.
} Then *f* is Cubic on *X*
_{0}.

**Proof.** Replacing *x* by *mx* and *Y* by (-2*m* + 1)*x* for *m* ∈ **N** in (2.27) we get

||*f*(*x*)) + *f*((4*m* - 1)x) - 2*f*((-*m* + 1)*x*) -2*f*((3*m* - 1)*x*) - 12*f*(*mx*)||*

(2.27) ≤ *φ* ((-2*m* + 1)*x*) + *φ(mx)*

for all *x* ∈_{
X0.
} For each m ∈ U, we define the operator
by

for all *x* ∈_{
X0
} . Moreover, for every *ξ, μ* ∈ _{
YX0
} , *x* ∈_{
X0
}

So, (**H2**) is valid. By using mathematical induction, we will show that for each *x* ∈_{
X0
} we have

From (2.29), we obtain that the inequality (2.30) holds for *n* = 0. Next, we will assume that (2.30) holds for *n = k*, where *k* ∈ **N**. Then we have

This shows that (2.30) holds for *n* = *k* + 1. Now we can conclude that the inequality (2.30) holds for all *n* ∈ **N**. From (2.30), we obtain

for all *x* ∈ *X*
_{0} and all *m* ∈ *U*. Hence, according to Theorem 1.4, there exists, for each *m* ∈ *U*, a unique solution _{
Cm
}: *X*
_{0} → *Y* of the equation

*C*
_{m}(x) = 12*C*
_{m}(*mx*) + 2*C*
_{m}((3*m* - 1)*x*) + 2*C*
_{m}((*-m* + 1)x) - C_{m}((4*m* - 1)*x*),

(2.30) *x* ∈ *X*
_{0}

such that

for all *x* ∈ *X*
_{0}. Now we show that

for every *x,y* ∈ *X*
_{0} such that *x* + *y* ≠ 0 and *n* ∈ **N.** Since the case *n* = 0 is just (2.27), take *k* ∈ **N** and assume that (2.33) holds for *n* = *k,* where *k* ∈ **N** and every *x,y* ∈ *X*
_{0} such that *x* + *y* ≠ 0. Then

Thus, by induction we have shown that (2.33) holds for every *n* ∈ **N**. Letting *n* → ∞ in (2.33), we obtain that

*C*
_{m}(2*x + y*) + *C*
_{m}(2*x - y*) = 2*C*
_{m} (*x + y*) + 2*C*
_{m}(*x - y*) + 12*C*
_{m}(*x*)

for all *x,y* ∈ *X*
_{0} such that *x + y* ≠ 0. In this way we obtain a sequence {C_{m}}_{m∈U} of Cubic functions on *X*
_{0} such that

Because the precedent inequality holds for over *n* = 0 and _{
αm
} < 1

It follows, with *m* → ∞, that *f* is Cubic on *X*
_{0}.

The following corollary is a particular case of Theorem 2.5 where with ϕ(*x*): = *c* ||*x*||^{
p
} with *c* ≥ 0 and *p* < 0.

**Corollary 2.6.** Let (*X*,||∙||) and (*Y*,||∙||_{*}) be normed space and ultrametric Banach space respectively, *c* ≥ 0, *p* < 0 and let *f*:*X* → *Y* satisfy

||*f*(2*x + y*)) + *f*(2*x + y*) - 2*f*(*x + y*) - 2*f*(*x - y*) - 12*f*(*x*)||_{*}

(2.34) ≤ *c*(||*x*||^{
p
} + ||*y*||^{
p
} ),

for all *x, y* ∈ *X*
_{0}. Then *f* is Cubic on *X*
_{0}.