J. Math. Anal. Appl. 414 (2014) 10–20

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Journal of Mathematical Analysis and Applications

www.elsevier.com/locate/jmaa

Fixed point theorems and the Ulam–Hyers stabilityin non-Archimedean cone metric spaces

Nguyen Bich Huy a,∗, Tran Dinh Thanh b

a Department of Mathematics, Ho Chi Minh City University of Pedagogy, 280 An Duong Vuong,Ho Chi Minh City, Viet Namb Department of Mathematics, University of Medicine and Pharmacy, 217 An Duong Vuong,Ho Chi Minh City, Viet Nam

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 August 2013Available online 12 December 2013Submitted byT. Domínguez Benavides

Keywords:Non-Archimedean metricCone metric spaceFixed pointUlam–Hyers stabilityFunctional equation

Let (X, p) be a metric space with a K-valued non-Archimedean metric p. In thispaper, we prove the existence and approximation of a fixed point for operators F :X −→ X satisfying the contractive condition in the form p(F (x), F (y)) � Q[p(x, y)],where Q : K −→ K is an increasing operator. Then, we study the generalizedUlam–Hyers stability of fixed point equations. We next obtain an extension ofthe Krasnoselskii fixed point theorem for the sum of two operators. Finally, anapplication to functional equations is given.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

In a non-Archimedean metric space (or ultra metric space) (X, d) the triangle inequality holds in thestronger form as follows

d(x, y) � max{d(x, z), d(z, y)

}. (1)

Such spaces have many applications in some problems coming from quantum physics, p-adic strings andsuperstrings (see [14]). In recent years, there has been a lot of attention to the problem on the Ulam–Hyersstability for various classes of functional equations in non-Archimedean metric spaces, see for example[3,4,6,5,8,11,15,16,1] and the references therein. The fixed point method is one of the most effective tools instudying these problems.

Cone metric and cone normed spaces were introduced in the middle of the 20th century by using anordered vector space instead of the set of real numbers, as codomain of a metric [12,13,17]. These spaceshave applications in approximation theory, in the fixed point theory and theory of differential equations, see

* Corresponding author.E-mail addresses: huynb@hcmup.edu.vn, nguyenbichhuy@hcm.vnn.vn (N.B. Huy).

0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jmaa.2013.12.020

N.B. Huy, T.D. Thanh / J. Math. Anal. Appl. 414 (2014) 10–20 11

for example [12,13,17]. Recently, the fixed point theory in cone metric spaces (in most cases, theoretically)has been being studied intensively by many mathematicians. For more details and further discussion, werefer to the papers [7,9,10] and the references therein. A natural generalization of property (1) to thecone-valued metric p is

p(x, y) � sup{p(x, z), p(z, y)

}(2)

and a space with such a metric is called a non-Archimedean cone metric space.The aim of this paper is to establish some fixed point results and the Ulam–Hyers stability of fixed point

equations in non-Archimedean cone metric spaces. Our first result is an analog of the classical Banach–Caccioppoli principle. More precisely, we prove the existence and approximation of fixed point for theoperator F : X −→ X, satisfying contractive condition in the form p(F (x), F (y)) � Q[p(x, y)], where p isa metric on X with values in a cone K and Q : K −→ K is an increasing operator. Another our resultis an extension of the Krasnoselskii fixed point theorem for sum of two operators. The exact estimate forapproximation of fixed points enables us to investigate the Ulam–Hyers stability of fixed point equations incone metric spaces. We introduce the generalized Ulam–Hyers stability as follows. Let K be an order conein a locally convex space with intK �= ∅, p be a K-valued metric on X and let F , F1 : X −→ X be givenoperators. We say that the equation F1(x) = F (x) is Ulam–Hyers stable if for each ε ∈ IntK we can findδ ∈ K\{θ} such that for any x′ ∈ X with p(F1(x′), F (x′)) � δ there exists a solution x∗ of the equationsatisfying p(x∗, x

′) � ε. The stability of the equation with respect to perturbations from a class of compactoperators is also investigated.

In order to study the fixed point theory in a cone metric space (X, p), one can use a functional q which isdefined on codomain of metric p (for instance, the Minkowskii functional or the nonlinear scalarization func-tional [7,10]) to reduce to the case of ordinary metric d = q ◦ p. However, in the setting of non-Archimedeancone metric spaces, this method may not be applicable, since the nice property (2) of the metric p is notreflected in the metric d. In the paper, we shall directly work with cone metric to employ property (2) andmany results from the theory of ordered spaces.

Fixed point results in cone metric spaces have been used to study the Ulam–Hyers stability for ordinarydifferential equations (see [2]), and (in implicit form) for functional equations, see [3,4]. Our abstract resultsshed a considerable light on the sources of results of [2] and they are a far-reaching extension of resultsof [3,4].

The paper is organized as follows. In the next section we prove the main results on fixed points and theUlam–Hyers stability in non-Archimedean cone metric spaces. Obtained results will be applied in Section 3to studying a class of functional equations.

2. The main results

2.1. Ordered locally convex spaces

Let E be a real Hausdorff locally convex space whose topology is defined by the family of seminorms(pi)i∈I . Thus, the sets

VJ,ε ={u ∈ E: max

i∈Jpi(u) < ε

}, (3)

where ε > 0, J ⊂ I, J is finite, form a base of neighborhoods of the zero and a net (uα) converges to anelement u iff the net (pi(uα − u)) converges to 0 in R for all i ∈ I. A nonempty subset K ⊂ E is called acone if it is a closed convex and λK ⊂ K for all λ � 0, K ∩ (−K) = {θ}. If K is a cone in E then we definea partial ordering in E with respect to K by u � v if v − u ∈ K and we call the pair (E,K) an ordered

12 N.B. Huy, T.D. Thanh / J. Math. Anal. Appl. 414 (2014) 10–20

locally convex space (OLCS for short). In an OLCS, the notions of upper bound, supremum for a subsetare defined by analogy with the case of R. The cone K is said to be minihedral if for any pair u, v ∈ E

there exists an element sup{u, v}. For simplicity of the notation we also write u ∨ v, u+, u− for sup{u, v},sup{u, θ}, sup{−u, θ} respectively.

Definition 1. We say that the OLCS (E,K) has property (E) if the cone K is minihedral and

(i) ∀i ∈ I the seminorm pi is semi-monotone in the sense that∃Ni > 0: θ � u � v implies pi(u) � Nipi(v);

(ii) ∀i ∈ I ∃mi > 0: pi(u+) � mipi(u) ∀u ∈ E.

It follows from condition (i) of property (E) that if un � vn � wn and lim un = u, limwn = u thenlim vn = u too.

Lemma 1. If the OLCS (E,K) has property (E) then from lim un = u, lim vn = v it follows that lim un∨vn =u ∨ v.

Proof. If lim un = θ then by condition (ii) of (E) we have limu+n = θ. Let lim un = u, we shall prove

lim u+n = u+. Indeed, by

u+n =

((un − u) + u

)+ � (un − u)+ + u+, u+ � (u− un)+ + u+n

we obtain

−(u− un)+ � u+n − u+ � (un − u)+

and hence, lim(u+n −u+) = θ. Finally, we have lim un∨vn = lim[(un−vn)∨θ+vn] = (u−v)∨θ+v = u∨v. �

Definition 2. Given the OLCS (E,K) with property (E) and an operator Q : K −→ K.1. We say that, the operator Q has property (Q) if Q(θ) = θ, Q is continuous at θ and increasing in the

sense that u � v implies Q(u) � Q(v).2. Associated to the operator Q we define the following operators and subsets

Sn(u) = sup{u,Q(u), . . . , Qn−1(u)

}, S(u) = lim

n→∞Sn(u),

D ={u ∈ K: S(u) is defined

}, D0 =

{u ∈ K: lim

n→∞Qn(u) = θ

}.

We denote by D1 the set of all elements u ∈ K such that

limn→∞

Sk

(Qn(u)

)= θ uniformly in k ∈ N

∗. (4)

Since the sets in (3) form a base of neighborhoods of θ, it is easy to prove that condition (4) is equivalentto the following condition

∀i ∈ I ∀ε > 0 ∃n0: ∀n � n0, ∀k ∈ N∗ =⇒ pi

(Sk

(Qn(u)

))< ε. (5)

Lemma 2. Assume that the complete OLCS (E,K) has property (E), the operator Q has property (Q) andu ∈ D1. Then

1. Qm(u) ∈ D1 for all m ∈ N∗ and if θ � v � u then v ∈ D1;

2. Qn(u) ∈ D for all n ∈ N and limn→∞ S(Qn(u)) = θ.

N.B. Huy, T.D. Thanh / J. Math. Anal. Appl. 414 (2014) 10–20 13

Proof. 1. Clearly, Qm(u) ∈ D1 by the definition of D1. By monotonicity of Q we have Sk(Qn(v)) �Sk(Qn(u)) for all k, n ∈ N

∗, which in combination with the semimonotonicity of pi and (5) implies thatv ∈ D1.

2. First, we shall prove that u ∈ D. We observe that if a subset A ⊂ K is finite and A = A1 ∪A2 then

supA = sup{supA1, supA2} � supA1 + supA2.

Consequently,

θ � Sn+k(u) − Sn(u) � Sk

(Qn(u)

)for all n, k ∈ N

∗. (6)

From (4), (6) and the semimonotonicity of seminorms it follows that (Sn(u))n is a Cauchy sequence in E,therefore, it is convergent and u ∈ D. Moreover, we also have Qn(u) ∈ D since Qn(u) ∈ D1, and hence theelements S(Qn(u)) are defined. By letting k −→ ∞ in (5) we see that limn→∞ pi(S(Qn(u))) = 0 ∀i ∈ I orequivalently, limn→∞ S(Qn(u)) = θ. �Remark 1. 1. In the same manner we can prove that if limn→∞ Sk(Qn(u)) = θ uniformly with respect tok ∈ N

∗ and to u ∈ C ⊂ K then limn→∞ S(Qn(u)) = θ uniformly with respect to u ∈ C.2. If (E,K) is an ordered Banach space with property (E) and Q : E −→ E is a positive linear operator

with the spectral radius r(Q) < 1 then D1 = K and one has

S(u) � (I −Q)−1(u). (7)

Indeed, for all k, n ∈ N we have

θ � Sk

(Qn(u)

)�

∞∑i=n

Qi(u),

which implies (4). By letting n = 0 and k −→ ∞ we get (7).

2.2. Fixed point theorems in non-Archimedean cone metric spaces

Definition 3. Let (E,K) be an OLCS.

1. We call p a cone metric (or K-metric) on the set X if p is a mapping from X ×X into K satisfying(i) p(x, y) = θE iff x = y,(ii) p(x, y) = p(y, x) ∀x, y ∈ X,(iii) p(x, y) � p(x, z) + p(z, y) ∀x, y, z ∈ X.Then, the pair (X, p) is called a cone metric space. We endow (X, p) with the topology by defining thefamily of closed subsets as follows. A net (xα) is said to be convergent to element x iff the net (p(xα, x))converges to θ in E. A set A ⊂ X is said to be closed iff the condition that (xα) ⊂ A, lim xα = x impliesx ∈ A.

2. The cone metric space (X, p) is said to be complete in the sense of Kantorovich if the condition that asequence (xn)n satisfies

p(xk, xl) � an ∀k, l � n with (an)n ⊂ K, limn→∞

an = θ

implies that (xn)n converges.

14 N.B. Huy, T.D. Thanh / J. Math. Anal. Appl. 414 (2014) 10–20

Definition 4. If the cone K is minihedral and the cone metric p satisfies

p(x, y) � sup{p(x, z), p(z, y)

}, ∀x, y, z ∈ X

then p is called a non-Archimedean cone metric (or cone ultrametric).

Definition 5. Assume that (E,K) is an OLCS with minihedral cone K and topology of E defined by thefamily of seminorms (pi)i∈I . Let X be a real vector space and p : X −→ K be a mapping satisfying

(i) p(x) = θE ⇐⇒ x = θX , where θE , θX are the zero elements of E and X respectively,(ii) p(λx) = |λ|p(x) ∀λ ∈ R, ∀x ∈ X,(iii) p(x + y) � sup{p(x), p(y)} ∀x, y ∈ X.

Then we call the pair (X, p) a non-Archimedean cone normed space and we endow it with the topologydefined by the family of seminorms (pi ◦ p)i∈I .

Theorem 1. Assume that the complete OLCS (E,K) has property (E) and the non-Archimedean cone metricspace (X, p) is complete in the sense of Kantorovich. Let F : X −→ X be an operator such that

p(F (x), F (y)

)� Q

[p(x, y)

]∀x, y ∈ X, (8)

where the operator Q : K −→ K has property (Q) and there exists an element x0 ∈ X satisfying

(i) Qn[p(x0, F (x0))] ∈ D ∀n ∈ N,(ii) limn→∞ S(Qn[p(x0, F (x0))]) = θ.

Then, the sequence xn = F (xn−1) is convergent and the element x∗ = limn→∞ xn is a fixed point of F ,having the following properties

1. p(xn, x∗) � S(Qn[p(x0, F (x0))]),2. x∗ is a unique fixed point of F in the set {x ∈ X: p(x, x0) ∈ D0}.

Here, the operator S and the sets D, D0 are defined by the operator Q as in Section 2.1.

Proof. Setting u = p(x0, F (x0)) we have

p(xn, xn+1) = p(F (xn−1), F (xn)

)� Q

[p(xn−1, xn)

]� · · ·

� Qn(u)

and

p(xn, xn+k) � sup{p(xn, xn+1), . . . , p(xn+k−1, xn+k)

}

� sup{Qn(u), . . . , Qn+k−1(u)

}= Sk

(Qn(u)

)

� S(Qn(u)

)

because the sequence (Sk(Qn(u)))k is increasing and convergent to S(Qn(u)). Consequently, for k = n + q,l = n + r one has

p(xk, xl) � sup{p(xn+q, xn), p(xn, xn+r)

}� S

(Qn(u)

),

N.B. Huy, T.D. Thanh / J. Math. Anal. Appl. 414 (2014) 10–20 15

which implies the existence of x∗ = limn→∞ xn by hypothesis (ii) and the completeness of (X, p). By lettingk −→ ∞ in the inequality

p(xn, x∗) � sup{p(xn, xn+k), p(xn+k, x∗)

}

� sup{S(Qn(u)

), p(xn+k, x∗)

}

and using Lemma 1 we obtain p(xn, x∗) � S(Qn(u)). To prove that x∗ is a fixed point of F , we write

p(x∗, F (x∗)

)� sup

{p(x∗, F (xn)

), p(F (xn), F (x∗)

)}

� sup{p(x∗, xn+1), Q

[p(xn, x∗)

]}.

Letting n → ∞, by Lemma 1 and the continuity of Q, we obtain x∗ = F (x∗). Finally, if we also havex′ = F (x′) for some x′ such that p(x′, x0) ∈ D0 then

p(xn, x

′) = p(F (xn−1), F

(x′)) � Q

[p(xn−1, x

′)] � · · ·� Qn

(p(x0, x

′)),

which implies limn→∞ xn = x′. Therefore, x′ = x∗. �Combining Theorem 1 with Lemma 2 we get the following corollary.

Corollary 1. Assume that the complete OLCS (E,K) has property (E) and the non-Archimedean conemetric space (X, p) is complete in the sense of Kantorovich. Suppose that the operator F : X −→ X satisfiescondition (8) with Q having property (Q) and there exists an element x0 ∈ X such that p(x0, F (x0)) ∈ D1.Then the conclusions of Theorem 1 hold.

Next, we prove an extension of the Krasnoselskii fixed point theorem for non-Archimedean cone normedspaces.

Theorem 2. Assume that the complete OLCS (E,K) has property (E) and the non-Archimedean cone normedspace (X, p) is complete in the sense of Kantorovich. Let C ⊂ X be a closed convex subset and the operatorsF,G : C −→ X satisfy

(i) F (C) + G(C) ⊂ C,(ii) G is continuous and the set G(C) is compact,(iii) p(F (x) − F (y)) � Q[p(x − y)] ∀x, y ∈ C, where the operator Q : K −→ K has property (Q) and one

of the following hypotheses holds:(a) There exists an operator R : K −→ K such that R(θ) = θ, R is continuous at θ and that

u � sup{v,Q(u)} implies u � R(v),(b) limn→∞ Sk(Qn(u)) = θ uniformly with respect to k ∈ N

∗ and u in any compact subset of K;(iv) There exists an element x0 ∈ C such that p(C − x0) ⊂ D1.

Here the set D1 and the operators {Sk}k have been defined in Section 2.1.Then, the operator F + G has a fixed point in C.

Proof. From hypothesis (i) and the closedness of C it follows that F (C) + G(C) ⊂ C. Consequently, foreach y ∈ G(C) the operator Fy(x) = F (x) + y acts from C into itself. Since p(Fy(x0) − x0) = p(F (x0) +y − x0) ∈ p(C − x0) ⊂ D1, Corollary 1 tells us that the operator Fy has a unique fixed point in the

16 N.B. Huy, T.D. Thanh / J. Math. Anal. Appl. 414 (2014) 10–20

set {x ∈ C: p(x − x0) ∈ D0}, which is equal to C by hypothesis (iv). Thus, there exists the operator(I − F )−1 : G(C) −→ C. We will claim that (I − F )−1 is continuous. Indeed, let a net (yα) ⊂ G(C) beconvergent to y ∈ G(C) and xα = (I−F )−1(yα), x = (I−F )−1(y). Note that xα = F (xα)+yα, x = F (x)+y.It suffices to prove the convergence of the net (xα) to x. For the case (a) we have

p(xα − x) � sup{p(yα − y), p

(F (xα) − F (x)

)}

� sup{p(yα − y), Q

[p(xα − x)

]},

which yields p(xα − x) � R[p(yα − y)] and hence shows that (xα) converges to x. To consider case (b) weput xαn = Tn

yα(x0), xn = Tn

y (x0). It follows that

p(xα − x) � p(xα − xαn) + p(xαn − xn) + p(xn − x). (9)

By Theorem 1 we obtain

θ � p(xα − xαn) � S[Qn

(p(F (x0) − x0 + yα

))],

θ � p(xn − x) � S[Qn

(p(F (x0) − x0 + y

))]. (10)

Since (yα) ⊂ G(C) and G(C) is compact, combining hypothesis (b) with Remark 1 ensures that, the righthand side of (10) tends to θ uniformly with respect to α. Fix i ∈ I and ε > 0. We can find a positiveinteger n0 such that

pi ◦ p(x− xn0) < ε, pi ◦ p(xα − xαn0) < ε ∀α. (11)

By hypothesis (iii) and the continuity of Q it follows that F is continuous. By induction, it is easy to provethe continuity of the operator y �−→ Fn0

y (x0). Hence, the net xαn0 = Fn0yα

(x0) converges to xn0 = Fn0y (x0).

This together with (9) and (11) tells us that we can choose α0 such that pi ◦ p(xα − x) < 3ε ∀α � α0. Thuslim pi ◦ p(xα − x) = 0 ∀i ∈ I or lim xα = x.

Since the operator (I − F )−1 : G(C) −→ C is isomorphism, it follows that (I − F )−1(G(C)) = (I −F )−1(G(C)) and hence, the set (I − F )−1 ◦G(C) is compact. Therefore, the operator (I − F )−1 ◦G has afixed point in C by the Tychonoff theorem. Consequently, the operator F + G has a fixed point in C. �Remark 2. If (E,K) is an ordered Banach space with property (E) and Q : E −→ E is a positive linearoperator with the spectral radius r(Q) < 1 then both hypotheses (a) and (b) in the condition (iii) ofTheorem 2 are satisfied.

Indeed, thanks to u � sup{v,Q(u)} we have that u � v + Q(u) and therefore u � (I − Q)−1(v).Furthermore, observe that

θ � Sk

(Qn(u)

)�

∞∑i=n

Qi(u),

which implies limn→∞ Sk(Qn(u)) = θ uniformly with respect to k ∈ N∗ and u in any bounded subset.

2.3. The generalized Ulam–Hyers stability

Definition 6. Let (E,K) be an OLCS with intK �= ∅, (X, p) be a cone metric space and F, F1 be operatorsfrom X into itself. The equation

F1(x) = F (x) (12)

N.B. Huy, T.D. Thanh / J. Math. Anal. Appl. 414 (2014) 10–20 17

is called to be Ulam–Hyers stable if for each ε ∈ intK we can find δ ∈ K\{θ} such that there exists asolution x∗ of (12) satisfying p(x∗, x

′) � ε whenever p(F (x′), F1(x′)) � δ.

From Theorem 1 and Corollary 1 we have

Theorem 3. Assume that the complete OLCS (E,K) has property (E), intK �= ∅ and the non-Archimedeancone metric space (X, p) is complete in the sense of Kantorovich. Let the operator F : X −→ X satisfycondition (8), where the operator Q : K −→ K has property (Q), Q(K\{θ}) ⊂ K\{θ} and there exists anx0 ∈ X such that p(x0, F (x0)) ∈ D1.

Then the fixed point equation x = F (x) is Ulam–Hyers stable.

Proof. Putting u = p(x0, F (x0)) we have limn→∞ S(Qn(u)) = θ and hence, for ε ∈ intK we find a positiveinteger n0 such that S(Qn0(u)) � ε. Define δ = Qn0(u), by Lemma 2 we have δ ∈ D1 and if p(x′, F (x′)) � δ

then p(x′, F (x′)) ∈ D1 too. Consequently we may apply Corollary 1 with x′ in place of x0 to obtaining afixed point x∗ of F such that

p(x∗, x

′) � S(p(x′, F

(x′))) � S(δ) � ε. �

Now, let us introduce another type of the Ulam–Hyers stability.

Definition 7. Let (E,K) be an OLCS with intK �= ∅, (X, p) be a cone normed space and C ⊂ X. Givenoperators F, F1 : C −→ X and a family ζ of operators from C into X. We say that Eq. (12) is Ulam–Hyersstable with respect to operators from ζ if there holds:

1. for each ε ∈ IntK we can find δ ∈ K so that if G ∈ ζ and p(G(x)) � δ ∀x ∈ C then the equation

F1(x) = F (x) + G(x) (13)

is solvable;2. for any solution x′ of Eq. (13) there exists a solution x∗ of (12) such that p(x∗ − x′) � ε.

As a consequence of Theorem 2 we have

Theorem 4. Assume that the complete OLCS (E,K) and the non-Archimedean cone normed space (X, p)satisfy all conditions of Theorem 2 and, in addition, intK �= ∅. Let the operator F satisfy conditions (iii),(iv) in Theorem 2.

Then the fixed point equation x = F (x) is Ulam–Hyers stable with respect to the family of operators G

satisfying conditions (i), (ii) in Theorem 2.

Proof. For a given ε ∈ intK we apply Theorem 3 to choose δ ∈ K such that if p(y − F (y)) � δ then thereexists a fixed point x∗ of F satisfying p(x∗ − y) � ε. If operator G has properties (i), (ii) and p(G(x)) � δ

∀x ∈ C then the equation x = F (x)+G(x) is solvable by Theorem 2 and for any solution x′ of this equationone has p(x′ − F (x′)) � δ. Consequently, by Theorem 3 there exists a fixed point x∗ of operator F suchthat p(x∗ − x′) � ε. �3. Application to functional equations

In this section we shall apply the obtained results to investigating a class of functional equations, whichgeneralize the equations considered in [3].

Given a set T �= ∅. We denote E = RT , the locally convex space of all functions u : T −→ R whose

topology is defined by the family of seminorms pt(u) = |u(t)|, t ∈ T . With this topology, a net (uα) ⊂ E

18 N.B. Huy, T.D. Thanh / J. Math. Anal. Appl. 414 (2014) 10–20

converges to u iff lim uα(t) = u(t) ∀t ∈ T and E is complete. In E we consider the order cone K ofnonnegative functions. This cone is minihedral, for u, v ∈ E, sup{u, v} is the function sup{u(t), v(t)}. It iseasy to see that the complete OLCS (E,K) has property (E) in Section 2.1. We also have intK = {u ∈ E:inft∈T u(t) > 0}.

Lemma 3. Let Q : K −→ K be an increasing, continuous at θ operator and Q(θ) = θ. Assume that anelement u satisfies limn→∞ Qn(u)(t) = 0 ∀t ∈ T . Then

limn→∞

sup{Qn(u), . . . , Qn+k−1(u)

}= θ in E, uniformly with respect to k ∈ N

∗. (14)

Thus, we have D0 = D1, where the sets D0, D1 are defined in Section 2.

Proof. Putting vn = Qn(u) we have lim vn(t) = 0 ∀t ∈ T and

max{vn(t), . . . , vn+k−1(t)

}� sup

{vm(t): m � n

}−→ 0 as m → ∞.

This together with the definition of the topology of E gives (14). �Let (Y, ‖.‖Y ) be a non-Archimedean Banach space and X = Y T be the vector space of functions x :

T −→ Y . Consider the mapping p : X −→ K that assigns each x ∈ X to the function u(t) = ‖x(t)‖Y . Wehave

p(x + y)(t) =∥∥x(t) + y(t)

∥∥Y

� max{∥∥x(t)

∥∥Y,∥∥y(t)∥∥

Y

}

= max{p(x)(t), p(y)(t)

}∀t ∈ T

or, p(x+y) � sup{p(x), p(y)} in E. Thus, p is a non-Archimedean cone norm on X. According to Definition 5,the topology of (X, p) is defined by the family of seminorms (pt ◦ p)t∈T . With this topology we have

lim xα = x iff lim pt ◦ p(xα − x) = lim∥∥xα(t) − x(t)

∥∥Y

= 0 ∀t ∈ T

or lim xα(t) = x(t) in Y for all t ∈ T . Clearly, (X, p) is complete in the usual sense and hence, in the senseof Kantorovich. By applying Corollary 1 we obtain the following result (cf. [3, Theorem 1]).

Corollary 2. Assume that the operator F : X −→ X satisfies∥∥F (x)(t) − F (y)(t)

∥∥Y� Q

[p(x− y)

](t) ∀x, y ∈ X, ∀t ∈ T, (15)

where Q : K −→ K is increasing, continuous at θ and Q(θ) = θ. Moreover, suppose in addition that thereexist u0 ∈ K and x0 ∈ X satisfying

limn→∞

Qn(u0)(t) = 0,∥∥F (x0)(t) − x0(t)

∥∥Y� u0(t) ∀t ∈ T. (16)

Then the equation x = F (x) is Ulam–Hyers stable.

Proof. Condition (15) means that p(F (x)−F (y)) � Q[p(x−y)]. From (16) we have u0 ∈ D1, p(F (x0)−x0) �u0 and hence p(F (x0) − x0) ∈ D1. Therefore, our assertion follows immediately from Corollary 1. �

Given the functions αi, βj : T −→ T , i = 1, 2, . . . , n; j = 1, 2, . . . ,m and f : T×Y n −→ Y , g : T×Y m −→Y . We consider the equations

N.B. Huy, T.D. Thanh / J. Math. Anal. Appl. 414 (2014) 10–20 19

x(t) = f[t, x

(α1(t)

), . . . , x

(αn(t)

)]:= F (x)(t), (17)

x(t) = f[t, x

(α1(t)

), . . . , x

(αn(t)

)]+ g

[t, x

(β1(t)

), . . . , x

(βm(t)

)]

:= F (x)(t) + G(x)(t),(18)

where x : T −→ Y is an unknown function.Let the function f satisfy the following condition:(f) There exists q ∈ (0, 1) such that

∥∥f(t, y1, . . . , yn) − f(t, z1, . . . , zn)∥∥Y� q max

1�i�n‖yi − zi‖Y ∀(yi), (zi) ∈ Y n, (19)

the function f(t, θ, . . . , θ) is bounded. (20)

Condition (19) implies p(F (x) − F (y)) � Q[p(x − y)] ∀x, y ∈ X with Q : K −→ K given by Q(u)(t) =qmax1�i�n u(αi(t)). From (20) it follows that condition (16) in Corollary 2 holds for the function x0(t) =f(t, θ, . . . , θ) and u0(t) = const.

From definition of the topology of E we see that, if a set V ⊂ K is compact then

∀t ∈ T ∃Mt > 0: 0 � u(t) � Mt ∀u ∈ V. (21)

By (21) we obtain

0 � Qn(u)(t) � qnMt, ∀u ∈ V, ∀t ∈ T,

which yields that limn→∞ Qn(u)(t) = 0 uniformly with respect to u from any compact subset V ⊂ K.Therefore limn→∞ Sk(Qn(u)) = 0 uniformly with respect to k ∈ N

∗ and u in any compact set of K byLemma 3.

For the function g we suppose that(g1) ∀t ∈ T the operator (y1, . . . , ym) �−→ g(t, y1, . . . , ym) is continuous;(g2) ∀t ∈ T the set Yt = g({t} × Y m) is relatively compact in Y .By hypothesis (g1) and the fact that the convergence in (X, p) is equivalent to the pointwise conver-

gence, it follows that the operator G is continuous. Furthermore, combining hypothesis (g2), the relationG(x) ⊂

∏t∈T Yt and the Tychonoff theorem allows us to deduce that the set G(X) is compact. By applying

Theorem 4 we have the last result of the paper.

Corollary 3. Let the function f satisfy condition (f). Then Eq. (17) is Ulam–Hyers stable with respect toclass of operators G given in (18) and satisfying conditions (g1), (g2).

Acknowledgment

The authors would like to thank the referee for his useful comments to improve the paper.

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