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Common fixed points for single-valued and multi-valued maps satisfying a generalized contraction in G-metric spaces

Abstract

In this article, we establish some common fixed point theorems for a hybrid pair {g, T} of single valued and multi-valued maps satisfying a generalized contractive condition defined on G-metric spaces. Our results unify, generalize and complement various known comparable results from the current literature.

2000 MSC: 54H25; 47H10; 54E50.

1. Introduction and preliminaries

Nadler [1] initiated the study of fixed points for multi-valued contraction mappings and generalized the well known Banach fixed point theorem. Then after, many authors studied many fixed point results for multi-valued contraction mappings see [213].

Mustafa and Sims [14] introduced the G-metric spaces as a generalization of the notion of metric spaces. Mustafa et al. [1519] obtained some fixed point theorems for mappings satisfying different contractive conditions. Abbas and Rhoades [20] initiated the study of common fixed point in G-metric spaces. While Saadati et al. [21] studied some fixed point theorems in generalized partially ordered G-metric spaces. Gajić and Crvenković [22, 23] proved some fixed point results for mappings with contractive iterate at a point in G-metric spaces. For other studies in G-metric spaces, we refer the reader to [2438]. Consistent with Mustafa and Sims [14], the following definitions and results will be needed in the sequel.

Definition 1.1. (See [14]). Let X be a non-empty set, G : X × X × X → + be a function satisfying the following properties

(G1) G(x, y, z) = 0 if x = y = z,

(G2) 0 < G(x, x, y) for all x, y X with x ≠ y,

(G3) G(x, x, y) ≤ G(x, y, z) for all x, y, z X with y ≠ z,

(G4) G(x, y, z) = G(x, z, y) = G(y, z, x) = ... (symmetry in all three variables),

(G5) G(x, y, z) ≤ G(x, a, a) + G(a, y, z) for all x, y, z, aX (rectangle inequality).

Then the function G is called a generalized metric, or, more specially, a G-metric on X, and the pair (X, G) is called a G-metric space.

Definition 1.2. (See [14]). Let (X, G) be a G-metric space, and let (x n ) be a sequence of points of X, therefore, we say that (x n ) is G-convergent to x X if lim n , m + G x , x n , x m =0, that is, for any ε > 0, there exists N such that G(x, x n , x m ) < ε, for all n, m ≥ N. We call x the limit of the sequence and write x n → x or lim n + x n =x.

Proposition 1.1. (See [14]). Let (X, G) be a G-metric space. The following statements are equivalent:

  1. (1)

    (x n ) is G-convergent to x,

  2. (2)

    G(x n , x n , x) 0 as n → +∞,

  3. (3)

    G(x n , x, x) 0 as n → +∞,

  4. (4)

    G(x n , x m , x) 0 as n, m → +.

Definition 1.3. (See [14]). Let (X, G) be a G-metric space. A sequence (x n ) is called a G-Cauchy sequence if for any ε > 0, there is N such that G(x n , x m , x l ) < ε for all m, n, l ≥ N, that is, G(x n , x m , x l ) → 0 as n, m, l → +.

Proposition 1.2. (See [14]). Let (X, G) be a G-metric space. Then the following statements are equivalent:

  1. (1)

    the sequence (x n ) is G-Cauchy,

  2. (2)

    for any ε > 0, there exists N such that G(x n , x m , x m ) < ε, for all m, n ≥ N.

Definition 1.4. (See [14]). A G-metric space (X, G) is called G-complete if every G-Cauchy sequence is G-convergent in (X,G).

Every G-metric on X defines a metric d G on X given by

d G x , y = G x , y , y + G y , x , x , for all x , y X .
(1)

Recently, Kaewcharoen and Kaewkhao [34] introduced the following concepts. Let X be a G-metric space. We shall denote CB(X) the family of all nonempty closed bounded subsets of X. Let H(.,.,.) be the Hausdorff G-distance on CB(X), i.e.,

H G A , B , C = max sup x A G x , B , C , sup x B G x , C , A , sup x C G x , A , B ,

where

G x , B , C = d G x , B + d G B , C + d G x , C , d G x , B = inf d G x , y , y B , d G A , B = inf d G a , b , a A , b B .

Recall that G(x, y, C) = inf {G(x, y, z), z C}. A mapping T : X → 2X is called a multi-valued mapping. A point x X is called a fixed point of T if x Tx.

Definition 1.5. Let X be a given non empty set. Assume that g : XX and T : X → 2X.

If w = gx Tx for some x X, then x is called a coincidence point of g and T and w is a point of coincidence of g and T.

Mappings g and T are called weakly compatible if gx Tx for some x X implies gT(x) Tg(x).

Proposition 1.3. (see [34]). Let X be a given non empty set. Assume that g : XX and T : X → 2X are weakly compatible mappings. If g and T have a unique point of coincidence w = gx Tx, then w is the unique common fixed point of g and T.

In this article, we establish some common fixed point theorems for a hybrid pair {g,T} of single valued and multi-valued maps satisfying a generalized contractive condition defined on G-metric spaces. Also, an example is presented.

2. Main results

We start this section with the following lemma, which is the variant of the one given in Nadler [1] or Assad and Kirk [4]. Its proof is a simple consequence of the definition of the Hausdorff G-distance H G (A, B, B).

Lemma 2.1. If A, B CB(X) and a A, then for each ε > 0, there exists b B such that G(a,b,b) ≤ H G (A, B, B) + ε.

The main result of the article is the following.

Theorem 2.1. Let (X, G) be a G-metric space. Set g : XX and T : XCB(X). Assume that there exists a function α : [0,+∞) → [0,1) satisfying lim sup r t + α ( r ) < 1 for every t ≥ 0 such that

H G T x , T y , T z α G g x , g y , g z G g x , g y , g z ,
(2)

for all x, y, z X. If for any x X, Tx g(X) and g(X) is a G-complete subspace of X, then g and T have a point of coincidence in X. Furthermore, if we assume that gp Tp and gq Tq implies G(gq, gp, gp) ≤ H G (Tq, Tp, Tp), then

(i) g and T have a unique point of coincidence.

(ii) If in addition g and T are weakly compatible, then g and T have a unique common fixed point.

Proof. Let x0 be arbitrary in X. Since Tx0 g(X), choose x1 X such that gx1 Tx0. If gx1= gx0, we finished. Assume that gx0 ≠ gx1, so G(gx0, gx1, gx1) > 0. We can choose a positive integer n1 such that

α n 1 G g x 0 , g x 1 , g x 1 1 - α G g x 0 , g x 1 , g x 1 G g x 0 , g x 1 , g x 1 .

By Lemma 2.1 and the fact that Tx1 g(X), there exists gx2 Tx1 such that

G g x 1 , g x 2 , g x 2 H G T x 0 , T x 1 , T x 1 + α n 1 G g x 0 , g x 1 , g x 1 .

Using the two above inequalities and (2), it follows that

G g x 1 , g x 2 , g x 2 H G T x 0 , T x 1 , T x 1 + α n 1 G g x 0 , g x 1 , g x 1 α G g x 0 , g x 1 , g x 1 G g x 0 , g x 1 , g x 1 + 1 - α G g x 0 , g x 1 , g x 1 G g x 0 , g x 1 , g x 1 = G g x 0 , g x 1 , g x 1 .

If gx1 = gx2, we finished. Assume that gx1 ≠ gx2. Now we choose a positive integer n2> n1 such that

α n 2 G g x 1 , g x 2 , g x 2 1 - α G g x 1 , g x 2 , g x 2 G g x 2 , g x 2 , g x 2 .

Since Tx2 CB(X) and the fact that Tx2 g(X), we may select gx3 Tx2 such that from Lemma 2.1

G g x 2 , g x 3 , g x 3 H G T x 1 , T x 2 , T x 2 + α n 2 G g x 1 , g x 2 , g x 2 ,

and then, similarly to the previous case, we have

G g x 2 , g x 3 , g x 3 H G T x 1 , T x 2 , T x 2 + α n 2 G g x 1 , g x 2 , g x 2 α G g x 1 , g x 2 , g x 2 G g x 1 , g x 2 , g x 2 + 1 - α G g x 1 , g x 2 , g x 2 G g x 1 , g x 2 , g x 2 = G g x 1 , g x 2 , g x 2 .

By repeating this process, for each k *, we may choose a positive integer n k such that

α n k G g x k - 1 , g x k , g x k 1 - α G g x k - 1 , g x k , g x k G g x k - 1 , g x k , g x k .

Again, we may select gx k+1 Tx k such that

G g x k , g x k + 1 , g x k + 1 H G T x k - 1 , T x k , T x k + α n k G g x k - 1 , g x k , g x k .
(3)

The last two inequalities together imply that

G ( g x k , g x k + 1 , g x k + 1 ) G ( g x k - 1 , g x k , g x k ) ,

which shows that the sequence of nonnegative numbers {d k }, given by d k = G(gx k-1 , gx k , gx k ), k = 1, 2,. . ., is non-increasing. This means that there exists d ≥ 0 such that

lim k + d k = d .

Let now prove that the {gx k } is a G-Cauchy sequence.

Using the fact that, by hypothesis for t = d, lim sup r d + α ( t ) < 1 , it results that there exists a rank k0 such that for k ≥ k0, we have α(d k ) < h, where

lim sup t d + α ( t ) <h<1.

Now, by (3) we deduce that the sequence {d k } satisfies the following recurrence inequality

d k + 1 H G T x k - 1 , T x k , T x k + α n k ( d k ) α ( d k ) d k + α n k ( d k ) , k 1 .
(4)

By induction, from (4), we get

d k + 1 i = 1 k α ( d i ) d 1 + m = 1 k - 1 i = m + 1 k α ( d i ) α n m ( d m ) + α n k ( d k ) , k 1 ,

which, by using the fact that α < 1, can be simplified to

d k + 1 i = 1 k α ( d i ) d 1 + m = 1 k - 1 i = max { k 0 , m + 1 } k α ( d i ) α n m ( d m ) + α n k ( d k ) , k 1 ,

Referring to the proof of Theorem 2.1 in [11] or Lemma 3.2 in [12], we may obtain

i = 1 k α ( d i ) d 1 + m = 1 k - 1 i = max { k 0 , m + 1 } k α ( d i ) α n m ( d m ) + α n k ( d k ) c h k ,

where c is a positive constant. We deduce that

d k + 1 = G ( g x k , g x k + 1 , g x k + 1 ) c h k .

Now for k ≥ k0 and m is a positive arbitrary integer, we have using the property (G4)

G ( g x k , g x k + m , g x k + m ) G ( g x k , g x k + 1 , g x k + 1 ) + G ( g x k + 1 , g x k + 2 , g x k + 2 ) + + G ( g x k + m - 2 , g x k + m - 1 , g x k + m - 1 ) + G ( g x k + m - 1 , g x k + m , g x k + m ) c h k + h k + 1 + + h k + m - 1 c h k 1 - h 0 as k + ,

since 0 < h < 1. This shows that the sequence {gx n } is G-Cauchy in the complete subspace g(X). Thus, there exists q g(X) such that, from Proposition 1.1

lim n + G ( g x n , g x n , q ) = lim n + G ( g x n , q , q ) = 0 .
(5)

Since q g(X), then there exists p X such that q = gp. From (5), we have

lim n + G ( g x n , g x n , g p ) = lim n + G ( g x n , g p , g p ) = 0 .
(6)

We claim that gp Tp. Indeed, from (2), we have

G ( g x n + 1 , T p , T p ) H G ( T x n , T p , T p ) α ( G ( g x n , g p , g p ) ) G ( g x n , g p , g p ) .
(7)

Letting n → +∞ in (7) and using (6), we get

G ( g p , T p , T p ) = lim n + G ( g x n + 1 , T p , T p ) = 0 ,

that is, gp Tp. That is T and g have a point of coincidence. Now, assume that if gp Tp and gq Tq, then G(gq, gp, gp) ≤ H G (Tq, Tp, Tp). We will prove the uniqueness of a point of coincidence of g and T. Suppose that gp Tp and gq Tq. By (2) and this assumption, we have

G ( g q , g p , g p ) H G ( T q , T p , T p ) α ( G ( g q , g p , g p ) ) G ( g q , g p , g p ) ,
(8)

and since α(G(gq, gp, gp)) < G(gq, gp, gp), so necessarily from (8), we have G(gq, gp, gp) = 0, i.e., gp = gq. In view of

H G ( T q , T p , T p ) α ( G ( g q , g p , g p ) ) G ( g q , g p , g p ) = 0 ,

we get Tq = Tp. Thus, T and g have a unique point of coincidence. Suppose that g and T are weakly compatible. By applying Proposition 1.3, we obtain that g and T have a unique common fixed point.

Corollary 2.1. Let (X,G) be a complete G-metric space. Assume that T : XCB(X) satisfies the following condition

H G ( T x , T y , T z ) α ( G ( x , y , z ) ) G ( x , y , z ) ,
(9)

for all x, y, z X, where α : [0,+∞) → [0,1) satisfies lim sup r t + α ( r ) < 1 for every t ≥ 0. Then T has a fixed point in X. Furthermore, if we assume that p Tp and q Tq implies G(q, p, p) ≤ H G (Tq, Tp, Tp), then T has a unique fixed point.

Proof. It follows by taking g the identity on X in Theorem 2.1.

Corollary 2.2. Let (X, G) be a G-metric space. Assume that g : XX and T : XCB(X) satisfy the following condition

H G ( T x , T y , T z ) k G ( g x , g y , g z ) ,
(10)

for all x, y, z X, where k [0,1). If for any x X, Tx g(X) and g(X) is a G-complete subspace of X, then g and T have a point of coincidence in X. Furthermore, if we assume that gp Tp and gq Tq implies G(gq, gp, gp) ≤ H G (Tq, Tp, Tp), then

(i) g and T have a unique point of coincidence.

(ii) If in addition g and T are weakly compatible, then g and T have a unique common fixed point.

Proof. It follows by taking α(t) = k, k [0,1), in Theorem 2.1.

In the case of single-valued mappings, that is, if T : XX, (i.e., Tx = {Tx} for any x X), it is obviously that

H G ( T x , T y , T z ) = G ( T x , T y , T z ) , x , y , z X .

Furthermore, if gp Tp (i.e., gp = Tp) and gq Tq (i.e., gq = Tq), then clearly,

G ( g q , g p , g p ) = G ( T q , T p , T p ) = H G ( T q , T p , T p ) ,

that is, the assumption given in Theorem 2.1 is verified.

Also, the single-valued mappings T, g : XX are said weakly compatible if Tgx = gTx whenever Tx = gx for some x X.

Now, we may state the following corollaries from Theorem 2.1 and the precedent corollaries:

Corollary 2.3. Let (X, G) be a complete G-metric space. Assume that T : XX satisfies the following condition

G ( T x , T y , T z ) α ( G ( x , y , z ) ) G ( x , y , z )
(11)

for all x, y, z X, where α : [0, +∞) → [0, 1) satisfies lim sup r t + α ( r ) < 1 for every t ≥ 0. Then, T has a unique fixed point.

Corollary 2.4. Let (X, G) be a G-metric space. Assume that g : XX and T : XX satisfy the following condition

G ( T x , T y , T z ) α ( G ( g x , g y , g z ) ) G ( g x , g y , g z )
(12)

for all x, y, z X, where α : [0, +∞) → [0, 1) satisfies lim sup r t + α ( r ) < 1 for every t ≥ 0. If T(X) g(X) and g(X) is a G-complete subspace of X, then

(i) g and T have a unique point of coincidence.

(ii) Furthermore, if g and T are weakly compatible, then g and T have a unique common fixed point.

Now, we introduce an example to support the useability of our results.

Example 2.1. Let X = [0, 1]. Define T : XCB(X) by Tx= 0 , 1 16 x and define g : X → X by gx= x . Define a G-metric on X by G(x, y, z) = max{|x-y|, |x-z|, |y-z|}. Also, define α : [0, +∞) → [0, 1) by α ( t ) = 1 2 Then:

  1. (1)

    Tx g(X) for all x X.

  2. (2)

    g(X) is a G-complete subspace of X.

  3. (3)

    g and T are weakly compatible.

  4. (4)

    H G (Tx, Ty, Tz) ≤ α(G(gx, gy, gz))G(gx, gy, gz) for all x, y, z X.

Proof. The proofs of (1), (2), and (3) are clear. By (1), we have

d G x , y = G x , y , y + G y , x , x = 2 x - y  for all  x , y X .

To prove (4), let x, y, z X. If x = y = z = 0, then

H G T x , T y , T z = 0 α G g x , g y , g z G g x , g y , g z .

Thus, we may assume that x, y, and z are not all zero. With out loss of generality, we assume that x ≤ y ≤ z. Then

H G T x , T y , T z = H G 0 , 1 16 x , 0 , 1 16 y , 0 , 1 16 z = max sup 0 a 1 16 x G a , 0 , 1 16 y , 0 , 1 16 z , sup 0 b 1 16 y G b , 0 , 1 16 z , 0 , 1 16 x , sup 0 c 1 16 z G c , 0 , 1 16 x , 0 , 1 16 y .

Since x ≤ y ≤ z, so 0 , 1 16 x 0 , 1 16 y 0 , 1 16 z This implies that

d G ( [ 0 , 1 16 x ] , [ 0 , 1 16 y ] ) = d G ( [ 0 , 1 16 y ] , [ 0 , 1 16 z ] ) = d G ( [ 0 , 1 16 x ] , [ 0 , 1 16 z ] ) = 0 .

For each 0a 1 16 x, we have

G a , 0 , 1 16 y , 0 , 1 16 z = d G a , 0 , 1 16 y + d G 0 , 1 16 y , 0 , 1 16 z + d G a , 0 , 1 16 z = 0 .

Also, for each 0b 1 16 y, we have

G ( b , [ 0 , 1 16 z ] , [ 0 , 1 16 x ] ) = d G ( b , [ 0 , 1 16 z ] ) + d G ( [ 0 , 1 16 z ] , [ 0 , 1 16 x ] ) + d G ( b , [ 0 , 1 16 x ] ) = { 0 if b x 16 2 b x 8 if b x 16 .

This yields that

sup 0 b 1 16 y G b , 0 , 1 16 z , 0 , 1 16 x = y 8 - x 8 .

Moreover, for each 0c 1 16 z, we have

G c , 0 , 1 16 x , 0 , 1 16 y = d G c , 0 , 1 16 x + d G 0 , 1 16 x , 0 , 1 16 y + d G c , 0 , 1 16 y = 0 if c x 16 2 c - x 8 if x 16 c y 16 4 c - x 8 - y 8 if c y 16 .

This yields that

sup 0 c 1 16 z G c , 1 16 c , 0 , 1 16 y = z 4 - x 8 - y 8 .

We deduce that

H G ( T x , T y , T z ) = z 4 - x 8 - y 8 1 4 ( z - x ) = 1 2 1 2 ( z - x ) 1 2 z - x x + z = 1 2 ( z - x )

On the other hand, it is obvious that all other hypotheses of Theorem 2.1 are satisfied and so g and T have a unique common fixed point, which is u = 0.

Remark 1 • Theorem 2.1 improves Kaewcharoen and Kaewkhao [[34], Theorem 3.3] (in case b = c = d = 0).

  • Corollary 2.3 generalizes Mustafa [[15], Theorem 5.1.7] and Shatanawi [[35], Corollary 3.4].

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Tahat, N., Aydi, H., Karapinar, E. et al. Common fixed points for single-valued and multi-valued maps satisfying a generalized contraction in G-metric spaces. Fixed Point Theory Appl 2012, 48 (2012). https://doi.org/10.1186/1687-1812-2012-48

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