a common fixed point theorem of jungck in ...

Acta Math. Hungar. DOI: 0

A COMMON FIXED POINT THEOREM OF JUNGCK IN RECTANGULAR B-METRIC SPACES ´ 1,∗ and S. RADENOVIC ´2 Z. D. MITROVIC 1

2

Faculty of Electrical Engineering, University of Banja Luka, Patre 5, 78000 Banja Luka, Bosnia and Herzegovina e-mails: [email protected], [email protected]

Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11101 Beograd 35, Serbia e-mail: [email protected] (Received March 21, 2017; accepted June 1, 2017)

Abstract. We give a proof for the common fixed point theorem of Jungck in rectangular b-metric spaces. As a corollary, we obtain well known common fixed point theorems in b-metric spaces.

1. Introduction and preliminaries In 1922, Banach proved the following famous fixed point theorem. Theorem 1.1. Let (X, d) be a complete metric space. Let T be a contractive mapping on X , that is for which there exists λ ∈ [0, 1) satisfying (1.1)

d(T x, T y) ≤ λd(x, y)

for all x, y ∈ X . Then there exists a unique fixed point x ∈ X of T .

This theorem is a forceful tool in nonlinear analysis, has many applications and has been extended by a great number of authors. Jungck [5] proved the following generalization of the contraction mapping principle. Theorem 1.2. Let T and I be a commuting mappings of a complete metric space (X, d) into itself satisfying the inequalty (1.2)

d(T x, T y) ≤ λd(Ix, Iy)

∗ Corresponding

author. Key words and phrases: fixed point, common fixed point, rectangular b-metric space. Mathematics Subject Classification: 47H10.

c 0 Akad´ 0236-5294/$ 20.00 emiai Kiad´ o, Budapest

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´ and S. RADENOVIC ´ Z. D. MITROVIC

for all x, y ∈ X , where 0 < λ < 1. If the range of I contains the range of T and if I is continuous, then T and I have a unique common fixed point.

In [4] the authors introduced the concept of rectangular b-metric space, which is not necessarily Hausdorff and which generalizes the concept of metric space, rectangular metric space (RMS) and b-metric space. Definition 1.3 [4]. Let X be a nonempty set and let the mapping d : X × X → [0, ∞) satisfy: (RbM1) d(x, y) = 0 if and only if x = y; (RbM2) d(x, y) = d(y, x) for all x, y ∈ X;  (RbM3) there  exists a real number s ≥ 1 such that d(x, y) ≤ s d(x, u) + d(u, v) + d(v, y) for all x, y ∈ X and all distinct points u, v ∈ X\{x, y}. Then d is called a rectangular b-metric on X with coefficient s and (X, d, s) is called a rectangular b-metric space (in short RbMS). Note that every rectangular metric space is a rectangular b-metric space (with coeficient s = 1). However the converse of the above implication is not necessarily true. Example 1. Let X = Z, where Z is the set of all integers and  [k]4 = x ∈ Z : x ≡ k (mod 4) , k ∈ {0, 1, 2, 3}.

Define d : X × X → [0, +∞) such that d(x, y) = d(y, x) for all x, y ∈ X and  1, x ∈ [0]4 , y ∈ [1]4 ∪ [2]4 ∪ [3]4 ,      6, x ∈ [1]4 , y ∈ [2]4 ∪ [3]4 , d(x, y) = 1, x ∈ [2]4 , y ∈ [3]4 ,   1, x, y ∈ [k]4 , k ∈ {0, 1, 2, 3}, x 6= y,    0, x = y.

Then it can be easily checked that (X, d) is a rectangular b-metric space (with s = 2) which is not a rectangular metric space since the inequality 6 = d(1, 3) ≤ d(1, 0) + d(0, 2) + d(2, 3) = 3 is not true. Note that (X, d) is not a metric space. Also in [4] the concept of convergence in such spaces is similar to that of standard metric spaces. Definition 1.4 [4]. Let (X, d) be a rectangular b-metric space, {xn } be a sequence in X and x ∈ X. Then (a) The sequence {xn } is said to be convergent in (X, d) and converges to x, if for every ε > 0 there exists n0 ∈ N such that d(xn , x) < ε for all n > n0 and this fact is represented by limn→∞ xn = x or xn → x as n → ∞. Acta Mathematica Hungarica

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(b) The sequence {xn } is said to be a Cauchy sequence in (X, d) if for every ε > 0 there exists n0 ∈ N such that d(xn , xn+p ) < ε for all n > n0 , p > 0 or equivalently, if limn→∞ d(xn , xn+p ) = 0 for all p > 0. (c) (X, d) is said to be a complete rectangular b-metric space if every Cauchy sequence in X converges to some x ∈ X. In Bakhtin [1] and Czerwik [2], the notion of b-metric space has been introduced and some fixed point theorems for single-valued and multi-valued mappings in b-metric spaces were proved. Definition 1.5. Let X be a nonempty set and let s ≥ 1 be a given real number. A function d : X × X → [0, ∞) is said to be a b-metric if and only if for all x, y, z ∈ X the following conditions are satisfied: (1) d(x, y) = 0 if and only if x = y; (2) d(x, y) = d(y, x); (3) d(x, z) ≤ s[d(x, y) + d(y, z)]. A triplet (X, d, s), is called a b-metric space with coefficient s. Note that a metric space is included in the class of b-metric spaces with coefiicent s ≥ 1. Note also that every b-metric space is a rectangular b-metric space (with coefficient s2 ) but the converse is not necessarily true ([4, Examples 2.7]). We have the following diagram where arrows stand for inclusions. The inverse inclusions do not hold. metric space

b-metric space

rectangular metric space

rectangular b-metric space

The aim of this paper is to obtain Theorem 1.2 in rectangular b-metric spaces. 2. Main result Theorem 2.1. Let T and I be commuting mappings of a complete rectangular b-metric space (X, d, s) into itself satisfying the inequality (2.1)

d(T x, T y) ≤ λd(Ix, Iy)

for all x, y ∈ X, where 0 < λ < 1. If the range of I contains the range of T and if I is continuous, then T and I have a unique common fixed point.

Proof. Let x0 ∈ X be arbitrary. Then T x0 and Ix0 are well defined. Since T x0 ∈ I(X), there is an x1 ∈ X such that Ix1 = T x0 . In general, if Acta Mathematica Hungarica

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´ and S. RADENOVIC ´ Z. D. MITROVIC

xn is chosen, then we choose a point xn+1 in X such that Ixn+1 = T xn . We show that {Ixn } is a Cauchy sequence. From (2.1) we have d(Ixm+k , Ixn+k ) = d(T xm+k−1, T xn+k−1 ) ≤ λd(Ixm+k−1 , Ixn+k−1 ). So, (2.2)

d(Ixm+k , Ixn+k ) ≤ λk d(Ixm , Ixn )

for all k ∈ N. Now, we have the following two cases: Case 1: If Ixn = Ixn+1 for some n then T xn = Ixn = ω. We shall show that ω is a unique common fixed point of T and I. Indeed, first T ω = T Ixn = IT xn = Iω. Let d (ω, T ω) > 0. In this case we have d(ω, T ω) = d(T xn , T ω) ≤ λd(Ixn , Iω) = λd(ω, Iω) = λd(ω, T ω) < d(ω, T ω), a contradiction. Since the condition (2.1) implies that T xn = Ixn = ω is a unique common fixed point of T and I the proof of Case 1 is finished. Case 2: If Ixn 6= Ixn+1 for all n ≥ 0, then Ixn 6= Ixn+k for all n ≥ 0, k ≥ 1. Namely, if Ixn = Ixn+k for some n ≥ 0 and k ≥ 1 we have that d(Ixn+1 , Ixn+k+1 ) = d(T xn , T xn+k ) ≤ λd(Ixn , Ixn+k ) = 0. So, Ixn+1 = Ixn+k+1 . Then (2.2) implies that d(Ixn+1 , Ixn ) = d(Ixn+k+1, Ixn+k ) ≤ λk d(Ixn+1, Ixn ) < d(Ixn+1, Ixn ) is a contradiction. Thus we assume that Ixn 6= Ixm for all distinct n, m ∈ N. Note that Ixm+k 6= Ixn+k for all k ∈ N and for all distinct n, m ∈ N and Ixn+k , Ixm+k ∈ X\{Ixn , Ixm }. Since, (X, d, s) is a rectangular b-metric space, from condition (RbM3) (see Definition 1.3) we have   d(Ixm , Ixn ) ≤ s d(Ixm , Ixm+n0 ) + d(Ixm+n0 , Ixn+n0 ) + d(Ixn+n0 , Ixn ) , where n0 ∈ N such that λn0 < 1s . Then,   d(Ixm , Ixn ) ≤ s λm d(Ix0 , Ixn0 ) + λn0 d(Ixm , Ixn ) + λn d(Ix0 , Ixn0 ) , (1 − sλn0 )d(Ixm , Ixn ) ≤ s(λm + λn )d(Ix0 , Ixn0 ).

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From this, we obtain (2.3)

d(Ixm , Ixn ) ≤

s(λm + λn ) d(Ix0 , Ixn0 ). 1 − sλn0

Thus {Ixn } is a Cauchy sequence in X. By completeness of I(X) there exists u ∈ X such that (2.4)

lim Ixn = lim T xn−1 = u.

n→∞

n→∞

Now since I is continuous, (2.1) implies that both I and T are continuous. Since T and I commute we obtain (2.5) Iu = I ( lim T xn ) = lim IT xn = lim T Ixn = T ( lim Ixn ) = T u. n→∞

n→∞

n→∞

n→∞

Let v = Iu = T u, we get (2.6)

T v = T Iu = IT u = Iv.

If T u 6= T v from (2.1) we obtain d(T u, T v) ≤ λd(Iu, Iv) = λd(T u, T v) < d(T u, T v), is contradiction. So we have T u = T v, hence we obtain T v = Iv = v, and v is a common fixed point for T and I. Condition (2.1) implies that v is a unique common fixed point.  Now, we give the following example to illustrate our theorem. Example 2. The space +∞ o n X |xn |p < +∞ , lp = (xn ) ⊂ R :

p ∈ (0, 1),

n=1

together with the function d : lp × lp → R, d(x, y) =

X +∞

n=1

|xn − yn |p

1/p

,

where x = (xn ), y = (yn ) ∈ lp , is a rectangular b-metric space with s = 22/p . Indeed, by an elementary calculation we obtain   d(x, y) ≤ 22/p d(x, u) + d(u, v) + d(v, y) , Acta Mathematica Hungarica

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´ and S. RADENOVIC ´ Z. D. MITROVIC

for all x, y ∈ lp and all distinct points u, v ∈ lp \{x, y}. Let T, I : lp → lp are mapping defined by   x2 x3 x4 T (x1 , x2 , x3 , x4 . . .) = 0, x1 , , , , . . . , 3 3 3  3x x x x  1 2 3 4 I(x1 , x2 , x3 , x4 . . .) = 0, , , , ,... , 2 2 2 2 has a unique common fixed point (0, 0, 0, . . .). Operators T and I are commutative. Further, for fixed x, y ∈ lp , we have 1/p |x2 − y2 |p |x3 − y3 |p + + ··· d(T x, T y) = |x1 − y1 | + 3p 3p 1/p 2 2  3p |x1 − y1 |p |x2 − y2 |p |x3 − y3 |p ≤ + + + · · · = d(Ix, Iy). 3 2p 2p 2p 3 

p

On the other hand, T is not a contraction. For x = (1, 0, 0, . . .) and y = (2, 0, 0, . . .), we have T x = (0, 1, 0, 0, . . .), T y = (0, 2, 0, 0, . . .), d(x, y) = 1, d(T x, T y) = 1. So, d(T x, T y) ≤ λd(x, y) implies λ ≥ 1. From Theorem 2.1 we obtain the following variant of the Banach theorem in rectangular b-metric spaces. Theorem 2.2. Let (X, d, s) be a complete rectangular b-metric space and T : X → X be a mapping satisfying: (2.7)

d(T x, T y) ≤ α d(x, y)

for all x, y ∈ X , where α ∈ [0, 1). Then T has a unique fixed point.

We obtain the following result as a consequence of Theorem 2.1 if we put K = λ1 . Let T be the identity map id(x) = x, x ∈ X. Theorem 2.3. Let I be a continuous onto mappings of a complete rectangular b-metric space (X, d, s). If there exists a K > 1 such that (2.8)

d(Ix, Iy) ≥ K d(x, y)

for all x, y ∈ X , then I has a unique fixed point.

3. A Jungck theorem in b-metric spaces Lemma 3.1. If (X, d) is a b-metric space with coefficient s, then (X, d) is a rectangular b-metric space with coefficient s2 . Acta Mathematica Hungarica

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Proof. Let (X, d) be a b-metric space with coefficient s. Let u and v be distinct points such that u, v ∈ X\{x, y}. Then we have   d(x, y) ≤ s d(x, u) + d(u, y)      ≤ s d(x, u) + s d(u, v) + d(v, y) ≤ s2 d(x, u) + d(u, v) + d(v, y) .

So (X, d) is a rectangular b-metric space with coefficient s2 . 

From Lemma 3.1 and Theorem 2.1 we obtain the next result in b-metric space. Theorem 3.2. Let T and I be a commuting mappings of a complete b-metric space (X, d, s) into itself satisfying the inequalty (3.1)

d(T x, T y) ≤ λd(Ix, Iy)

for all x, y ∈ X , where 0 < λ < 1. If the range of I contains the range of T and if I is continuous, then T and I have a unique common fixed point.

Note that, from Theorem 3.2. we obtain the Banach contraction principle in b-metric spaces. Theorem 3.3 [3, Theorem 2.1]. Let (X, d, s) be a complete b-metric space and let T : X → X be a map such that for all x, y ∈ X and some λ ∈ [0, 1), d(T x, T y) ≤ λd(x, y). Then T has a unique fixed point u and limn→∞ T n x = u for all x ∈ X .

Remark 3.4. Theorem 2.2 provides a complete solution to an open problem raised by George, Radenovi´c, Reshma and Shukla in [4]. References [1] I. A. Bakhtin, The contraction mapping principle in quasimetric spaces, Funct. Anal., Unianowsk Gos. Ped. Inst., 30 (1989), 26–37. [2] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostrav., 1 (1993), 5–11. [3] N. V. Dung and V. T. L. Hang, On relaxations of contraction constants and Caristis theorem in b-metric spaces, J. Fixed Point Theory Appl., 18 (2016), 267–284. [4] R. George, S. Radenovi´c, K. P. Reshma and S. Shukla, Rectangular b-metric space and contraction principles, J. Nonlinear Sci. Appl., 8 (2015), 1005–1013. [5] G. Jungck, Commuting mappings and fixed points, Amer. Math. Monthly, 83 (1976), 261–263.

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