On Common Fixed Points in the Context of Brianciari Metric Spaces ...

Results. Math. Online First c 2015 Springer International Publishing  DOI 10.1007/s00025-015-0516-5

Results in Mathematics

On Common Fixed Points in the Context of Brianciari Metric Spaces Hassen Aydi, Erdal Karapınar and Dong Zhang Abstract. In this paper, we introduce the concept of generalized (α, ψ)contractions and generalized (α, ψ)-Meir–Keeler-contractions in the setting of Brianciari metric spaces. We prove some common fixed point results for such contractions. An example is presented making effective the new concepts and results. Mathematics Subject Classification. 47H10, 54H25. Keywords. Brianciari metric space, α-admissible pairs of mapping, common fixed point.

1. Introduction Variant generalizations and extensions of the Banach contraction principle [12] have been done. One of these generalizations has been discussed by Brianciari [13] who introduced the concept of a generalized metric space (Brianciari metric space) by replacing the triangle inequality by a rectangular one. For more fixed point results in the class of Brianciari metric spaces, see [7–11,19]. In the following, we recall the notion of a generalized metric space. Definition 1. [13] Let X be a nonempty set and let d : X × X −→ [0, ∞) satisfy the following conditions for all x, y ∈ X and all distinct u, v ∈ X each of which is different from x and y. (BM S1) d(x, y) = 0 if and only if x = y (BM S2) d(x, y) = d(y, x) (BM S3) d(x, y) ≤ d(x, u) + d(u, v) + d(v, y).

(1)

H. Aydi et al.

Results. Math.

Then, the map d is called a Brianciari metric space. Here, the pair (X, d) is called a Brianciari metric space and abbreviated as BMS. In some sources, BMS is known also as generalized metric space. The concepts of convergence, Cauchy sequence, completeness and continuity on a BMS are defined below. Definition 2. (1) A sequence {xn } in a BMS (X, d) is BMS convergent to a limit x if and only if d(xn , x) → 0 as n → ∞. (2) A sequence {xn } in a BMS (X, d) is BMS Cauchy if and only if for every ε > 0 there exists positive integer N (ε) such that d(xn , xm ) < ε for all n > m > N (ε). (3) A BMS (X, d) is called complete if every BMS Cauchy sequence in X is BMS convergent. (4) A mapping T : (X, d) → (X, d) is continuous if for any sequence {xn } in X such that d(xn , x) → 0 as n → ∞, we have d(T xn , T x) → 0 as n → ∞. Proposition 1. [16] Suppose that {xn } is a Cauchy sequence in a BMS (X, d) with limn→∞ d(xn , u) = 0 where u ∈ X. Then, limn→∞ d(xn , z) = d(u, z) for all z ∈ X. In particular, the sequence {xn } does not converge to z if z = u. Recall that Samet et al. [20] introduced the following concept. Definition 3. [20] For a nonempty set X, let T : X → X and α : X × X → [0, ∞) be mappings. We say that T is α−admissible if for all x, y ∈ X, we have α(x, y) ≥ 1 =⇒ α(T x, T y) ≥ 1.

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The following papers [1,4–6,14,17] used some concepts of α-admissible mappings to prove (common) fixed points in variant metric spaces. Based on the above definition, we introduce new classes of α-admissible (pair of) mappings and new variant generalized contractions in the setting of Brianciari metric spaces. We also prove many (common) fixed point theorems for such contractions. An example is also provided.

2. Main Results First, denote N the set of positive integers and Ψ the set of functions ψ : [0, ∞) → [0, ∞) satisfying: (ψ1 ) ψ is upper semi-continuous, (ψ2 ) ψ(t) ∞ < tnfor any t > 0, + n th iterate of ψ. (ψ3 ) n=1 ψ (t) < ∞ for each t ∈ R , where ψ is the n Due to the special character of the rectangular inequality d(x, y) ≤ d(x, u) + d(u, v) + d(v, y) for all x, y ∈ X and all distinct u, v ∈ X each of which is different from x and y, there are few results in literature dealing with common fixed point results in the setting of Brianciari metric spaces. See for example, [3,15,18]. To our knowledge, the common fixed point results

On Common Fixed Points in the Context of Brianciari Metric Spaces

presented in this paper are the first ones using the concept of α-admissible mappings. Our main results concern generalized (α, ψ)-contractions and generalized (α, ψ)-Meir–Keeler-contractions on Brianciari metric spaces. We start with the following subsection in which the existing of a common fixed point will be discussed via generalized (α, ψ)-contractions. 2.1. A Common Fixed Point Theorem for Generalized (α, ψ)-Contractions on Brianciari Metric Spaces As in [2], we introduce the following definition. Definition 4. For a nonempty set X, let T : X → X, S : X → X and α : X × X → [0, +∞) be mappings. We say that the pair (T, S) is α-admissible if, for all x, y ∈ X, we have α(x, y) ≥ 1 implies min{α(T x, T y), α(Sx, T y), α(T x, Sy), α(Sx, Sy)} ≥ 1. We call T an α-admissible mapping if (T, T ) is α-admissible. We also introduce the following. Definition 5. Let (X, d) be a Brianciari metric space and T, S : X → X be two given mappings. We say that (T, S) is a generalized (α, ψ)-contractive mapping-pair if there exist two functions α : X × X → [0, +∞) and ψ ∈ Ψ such that α(x, y)d(T x, Sy) ≤ ψ(MT,S (x, y)) and α(x, y)d(Sx, T y) ≤ ψ(MS,T (x, y)), for any x, y ∈ X, where MT,S (x, y) = max{d(x, y), d(x, T x), d(y, Sy)}, MS,T (x, y) = max{d(x, y), d(x, Sx), d(y, T y)}. Our first main result is Theorem 1. Let (X, d) be a complete Brianciari metric space and T, S : X → X be two mappings. Suppose that: (C1) (T, S) is a generalized (α, ψ)-contractive mapping pair; (C2) There exists a point x0 ∈ X such that α(x0 , Sx0 ) ≥ 1, α(x0 , T x0 ) ≥ 1 and α(x0 , T Sx0 ) ≥ 1; (C3) T, S are continuous and (T S)n x0 = (ST )n x0 for any sufficiently large positive integer n. Then, there exists a point u ∈ X such that T u = u = Su. Proof. By assumption (C2), there exists a point x0 ∈ X such that α(x0 , Sx0 ) ≥ 1 and α(x0 , T Sx0 ) ≥ 1. We define a sequence {xn } in X such that x1 = Sx0 , x2 = T x1 , . . . , x2n = T x2n−1 and x2n+1 = Sx2n , . . . , n = 1, 2, . . . .

H. Aydi et al.

Results. Math.

Since (T, S) is α-admissible, observe that α(x0 , x1 ) = α(x0 , Sx0 ) ≥ 1 ⇒ α(x1 , x2 ) = α(Sx0 , T Sx0 ) ≥ 1. By repeating the process above, we derive α(xn , xn+1 ) ≥ 1,

for all n = 0, 1, . . .

Again, from Definition 4 α(x0 , x2 ) = α(x0 , T Sx0 ) ≥ 1 ⇒ α(x1 , x3 ) = α(Sx0 , S(T Sx0 )) ≥ 1. The same procedure yields that α(xn , xn+2 ) ≥ 1,

for all n = 0, 1, . . .

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If x2n0 = x2n0 +1 for some n0 , then u = x2n0 = x2n0 +1 = Sx2n0 = Su. Moreover d(x2n0 +1 , x2n0 +2 ) ≤ α(x2n0 , x2n0 +1 )d(x2n0 +1 , x2n0 +2 ) ≤ ψ(MS,T (x2n0 , x2n0 +1 )) = ψ(max{d(x2n0 , x2n0 +1 ), d(x2n0 +1 , x2n0 +2 )}) = ψ(d(x2n0 +1 , x2n0 +2 )). So d(x2n0 +1 , x2n0 +2 ) = 0, i.e., u = x2n0 +1 = x2n0 +2 = T x2n0 +1 = T u. Hence, u is a common fixed point of S and T . Similarly, if x2n0 −1 = x2n0 for some n0 , we can get that u = x2n0 is a common fixed point of S and T . Consequently, from now on, we assume that xn = xn+1 We denote



M (xn , xm ) =

MT,S (xn , xm ), MS,T (xn , xm ),

for all n. if n is odd and m is even if n is even and m is odd

Step 1: We shall prove lim d(xn , xn+1 ) = 0.

n→+∞

Note that d(x2n+1 , x2n+2 ) = d(Sx2n , T x2n+1 ) ≤ ψ(M (x2n , x2n+1 )) for all n ≥ 0, where M (x2n , x2n+1 ) = max{d(x2n , x2n+1 ), d(x2n+1 , T x2n+1 ), d(x2n , Sx2n )} = max{d(x2n , x2n+1 ), d(x2n+1 , x2n+2 ), d(x2n , x2n+1 )} = max{d(x2n , x2n+1 ), d(x2n+1 , x2n+2 )}. If for some n, M (x2n , x2n+1 ) = d(x2n+1 , x2n+2 ), then d(x2n+1 , x2n+2 ) = d(Sx2n , T x2n+1 ) ≤ ψ(M (x2n , x2n+1 )) = ψ(d(x2n+1 , x2n+2 )) < d(x2n+2 , x2n+1 ),

(4)

On Common Fixed Points in the Context of Brianciari Metric Spaces

which is a contradiction. Thus, M (x2n , x2n+1 ) = d(x2n , x2n+1 ) for any n. Therefore, for any n d(x2n+1 , x2n+2 ) ≤ ψ(d(x2n , x2n+1 ))

(5)

Again, d(x2n , x2n+1 ) = d(T x2n−1 , Sx2n ) ≤ ψ(M (x2n−1 , x2n )) for all n ≥ 1, where M (x2n−1 , x2n ) = max{d(x2n−1 , x2n ), d(x2n−1 , T x2n−1 ), d(x2n , Sx2n )} = max{d(x2n−1 , x2n ), d(x2n−1 , x2n ), d(x2n , x2n+1 )} = max{d(x2n−1 , x2n ), d(x2n , x2n+1 )}. If for some n, M (x2n−1 , x2n ) = d(x2n , x2n+1 ), then d(x2n , x2n+1 ) = d(T x2n−1 , Sx2n ) ≤ ψ(M (x2n−1 , x2n )) = ψ(d(x2n , x2n+1 )) < d(x2n , x2n+1 ), which is a contradiction. Thus, M (x2n−1 , x2n ) = d(x2n−1 , x2n ) for any n. We get for any n d(x2n , x2n+1 ) ≤ ψ(d(x2n−1 , x2n )). (6) By the above discussions, we find M (xn−1 , xn ) = d(xn−1 , xn )

∀ n = 1, 2, . . . .

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Combining (5) and (6), we have d(xn , xn+1 ) ≤ ψ(d(xn−1 , xn ))

for any n.

By a property of ψ, we derive that d(xn , xn+1 ) < d(xn−1 , xn ) for all n.

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Since d(xn , xn+1 ) > 0, so there exists r ≥ 0 such that lim d(xn , xn+1 ) = r. n→+∞

If r > 0, then r = lim sup d(xn , xn+1 ) ≤ lim sup ψ(d(xn−1 , xn )) n→+∞

n→+∞

≤ ψ( lim d(xn−1 , xn )) = ψ(r) < r, n→+∞

which is a contradiction. Thus, we conclude that r = 0, that is, lim d(xn , xn+1 ) = 0.

(9)

n→+∞

Step 2: We shall prove that x2n = x2m

and

x2n+1 = x2m+1 ,

n = m.

We argue it by contradiction. Without loss of generality, we may assume that: Case 2.1: there exist m, n such that m > n, x2m = x2n . Case 2.2: there exist m, n such that m > n, x2m+1 = x2n+1 .

H. Aydi et al.

Results. Math.

In Case 2.1, regarding (8) (that is, the sequence {d(xn , xn+1 )} is decreasing), we find d(x2n , x2n+1 ) = d(x2n , Sx2n ) = d(x2m , Sx2m ) = d(x2m , x2m+1 ) < d(x2n , x2n+1 ) which is a contradiction. In Case 2.2, taking (8) into account, we get d(x2n+2 , x2n+1 ) = d(T x2n+1 , x2n+1 ) = d(T x2m+1 , x2m+1 ) = d(x2m+2 , x2m+1 ) < d(x2n+2 , x2n+1 ), a contradiction. Step 2 implies an useful property: (P1): For any n, there is at most one m = n such that xm = xn . In fact, if there are distinct three numbers k, m, n such that xk = xm = xn , then there are two distinct numbers i, j ∈ {k, m, n} such that i ≡ j mod 2. Assume i = k and j = m, then xk = xm by Step 2, which is a contradiction. Step 3: We shall prove that {xn } is a BMS Cauchy sequence. We distinguish two cases: Case 3.1: For any N > 0, there exist n, m ∈ N with m > n ≥ N such that xn = xm . We first shall prove d(xn , xn+2p+1 ) ≤

2p+1 

d(xn+i−1 , xn+i ) for any n ≥ N,

(10)

i=1

where N is a positive integer number up to {xn }. We argue by induction. Obviously, (10) is true for p = 0. Assume (10) is true for p = q, then when p = q + 1, we distinguish three subcases. Case 3.1.1: xn = xn+2(q+1)−1 . By (4), we have xn+2(q+1)−1 = xn+2(q+1) and xn+2(q+1) = xn+2(q+1)+1 . From Step 2, xn+2(q+1)−1 = xn+2(q+1)+1 and xn = xn+2(q+1) , so we deduce that d(xn , xn+2(q+1)+1 ) ≤ d(xn , xn+2(q+1)−1 ) + d(xn+2(q+1)−1 , xn+2(q+1) ) + d(xn+2(q+1) , xn+2(q+1)+1 ). The hypothesis of induction yields that d(xn , xn+2(q+1)−1 ) ≤ d(xn+i−1 , xn+i ). Thus, 2(q+1)+1

d(xn , xn+2(q+1)+1 ) ≤

 i=1

Hence, (10) is proved in this case. Case 3.1.2: xn+1 = xn+2(q+1) .

d(xn+i−1 , xn+i ).

2q+1 i=1

On Common Fixed Points in the Context of Brianciari Metric Spaces

Again, by (4), xn = xn+1 and xn+2(q+1) = xn+2(q+1)+1 . From Step 2, xn = xn+2(q+1) and xn+1 = xn+2(q+1)+1 , so we deduce that d(xn , xn+2(q+1)+1 ) ≤ d(xn , xn+1 ) + d(xn+1 , xn+2(q+1) ) + d(xn+2(q+1) , xn+2(q+1)+1 ). 2q+1 Then, by mathematical induction, d(xn+1 , xn+2(q+1) ) ≤ i=1 d(xn+1+i−1 , xn+1+i ). Thus 2(q+1)+1

d(xn , xn+2(q+1)+1 ) ≤



d(xn+i−1 , xn+i ).

i=1

Case 3.1.3: xn = xn+2(q+1)−1 and xn+1 = xn+2(q+1) . In this case, d(xn , xn+1 ) = d(xn+2(q+1)−1 , xn+2(q+1) ) < d(xn , xn+1 ), which is a contradiction. So, we have only two cases: Case 3.1.1 and Case 3.1.2. Thus, (10) is true for any p ≥ 0. Now, using (7) and (10) d(xn , xn+2p+1 ) ≤

2p+1 

d(xn+i−1 , xn+i ) ≤

2p+1 

i=1

≤

2p+1 

α(xn+i−2 , xn+i−1 )d(xn+i−1 , xn+i )

i=1

ψ(M (xn+i−2 , xn+i−1 ))

i=1


0, from (11), there exists an integer N such that for any n > N and p, q, r ≥ 0, d(xn , xn+2q+1 )
0 such that xn = xm for any m > n ≥ N . In this case, by rectangular inequality and (9), again as (11), we have ∀p ≥ 0.

lim d(xn , xn+2p+1 ) = 0,

n→∞

(13)

Now, take an = d(xn , xn+2 ). We have |an − am | → 0 when n, m → +∞ and |n − m| is odd. We can also prove that |an − am | → 0 as n, m → +∞. So, {an } is a Cauchy sequence in ([0, ∞), |.|) where |.| is the standard metric. Then, there exists a ≥ 0 such that {an } converges to a in ([0, ∞), |.|). To prove that a = 0, we only need to prove a2n = d(x2n , x2n+2 ) → 0 as n → +∞. Suppose to the contrary that a > 0. Consider a new sequence {yn } defined by y0 = x0 ,

y1 = T y 0 ,

y2n+1 = T y2n , . . . ,

y2 = Sy1 ,

. . . y2n = Sx2n−1

and

n = 1, 2, . . . .

By the condition (C3), we can easily check that x2n = (T S)n x0 = (ST ) x0 = (ST )n y0 = y2n for sufficiently large positive integer n. Thus, we have the following commutative diagram: n

x0

y0

S

T

/x /y

T 1

1

S

/x /y

S 2

2

T

/ ···

S

/ ···

T

/x /y

2n−1

2n−1

T

S

/x /y

S

/x

T

/y

2n

2n

2n+1

2n+1

T

/x

S

/y

2n+2

2n+2

S

/

···

T

/

···

Similar to d(xn , xn+1 ) → 0 stated in Step 1, we can symmetrically obtain (P4): d(yn , yn+1 ) → 0, n → +∞. (Note that the condition α(x0 , Sx0 ) ≥ 1 and α(x0 , T x0 ) ≥ 1 in (C2) are symmetric, and accordingly there is no difficulty to verify that such proofs are the same.) In addition, by (3), we have α(x2n−2 , x2n ) ≥ 1 and from Definition 4 α(x2n−1 , y2n+1 ) = α(Sx2n−2 , T y2n ) = α(Sx2n−2 , T x2n ) ≥ 1.

On Common Fixed Points in the Context of Brianciari Metric Spaces

Consequently, together with Step 1 and (P4), for sufficiently large positive integer n, we have d(x2n , x2n+2 ) = d(x2n , y2n+2 ) = d(T x2n−1 , Sy2n+1 ) ≤ α(x2n−1 , y2n+1 )d(T x2n−1 , Sy2n+1 ) ≤ ψ(MT,S (x2n−1 , y2n+1 )) = ψ(max{d(x2n−1 , y2n+1 ), d(x2n−1 , T x2n−1 ), d(y2n+1 , Sy2n+1 )}) = ψ(max{d(x2n−1 , y2n+1 ), d(x2n−1 , x2n ), d(y2n+1 , y2n+2 )}) = ψ(d(x2n−1 , y2n+1 )) < d(x2n−1 , y2n+1 ) = d(Sx2n−2 , T y2n ) = d(Sx2n−2 , T x2n ) ≤ α(x2n−2 , x2n )d(Sx2n−2 , T x2n ) ≤ ψ(MS,T (x2n−2 , x2n )) = ψ(max{d(x2n , x2n−2 ), d(x2n−2 , Sx2n−2 ), d(x2n , T x2n )}) = ψ(max{d(x2n , x2n−2 ), d(x2n−2 , x2n−1 ), d(y2n , y2n+1 )}) = ψ(d(x2n , x2n−2 )). Thus, using the property (ψ1 ) of ψ, we get a = lim d(x2n , x2n+2 ) ≤ lim sup ψ(d(x2n , x2n−2 )) n→+∞

n→+∞

≤ ψ( lim d(x2n , x2n−2 )) = ψ(a) < a, n→+∞

which is a contradiction. So lim d(xn , xn+2 ) = 0.

(14)

n→+∞

Therefore, under the setting of Case 3.2, for any ε > 0, there exists N  > 0 such that for any n > N  and p ∈ N, xn , xn+2p−1 , xn+2p and xn+2p+1 are pairwise different, and there holds d(xn+2p , xn+2p+1 ) < 3ε (from (9)), d(xn , xn+2p−1 ) < ε ε 3 (from (13)) and d(xn+2p−1 , xn+2p+1 ) < 3 (from (14)). Hence, we deduce d(xn , xn+2p ) ≤ d(xn , xn+2p−1 )+d(xn+2p−1 , xn+2p+1 )+d(xn+2p+1 , xn+2p ) < ε, that is, d(xn , xn+2p ) → 0 as n → +∞, Combining (13) to (15), we conclude that d(xn , xn+p ) → 0

as n → +∞,

∀p ≥ 0.

(15)

∀p ≥ 0,

(16)

that is, {xn } is a BMS Cauchy sequence. Since (X, d) is a complete BMS, there exists u ∈ X such that lim d(xn , u) = 0.

n→+∞

Since T is continuous, we obtain that lim d(x2n , T u) = d(T x2n−1 , T u) = 0,

n→+∞

H. Aydi et al.

that is,

lim x2n = T u. We also have

n→+∞

Results. Math.

lim x2n = u. Taking Proposition

n→+∞

1 into account, we conclude that T u = u, that is, u is a fixed point of T . Similarly, the continuity of S asserts that u is again a fixed point of S. The proof is completed.  Corollary 2.1. Theorem 1 is a generalization of Theorem 15 in [8]. Proof. It is sufficient to take T = S in Theorem 1.



Remark 1. T S = ST implies (T S)n x0 = (T S)n x0 for any sufficiently large positive integer n, but the converse is not true. Remark 2. The condition α(x0 , T Sx0 ) ≥ 1 given in Theorem 1 can not be removed. The following example confirms our assertion. Example 1. Let X = {0, 1, 2, . . . , n, . . .}. Take d : X × X → [0, +∞) such that: ⎧ if p = q, ⎨0 if p − q = 2n + 2, d(q, p) = d(p, q) = 1 ⎩ 1 if p − q = 2n + 1. q 3 In the above definition, it is clear that p ≥ q, p, q ∈ X and n ∈ N. We can easily check that (X, d) is a complete Brianciari metric space. Let α : X ×X → [0, +∞) be such that α(n, n + 1) = α(n + 1, n) = 1, α(n, n + k) = α(n + k, n) = 0 for any n ∈ X, k = 2, 3, . . . Consider S, T : X → X such that S(n) = T (n) = n + 1. Take ψ(t) = 2t for any t ∈ [0, ∞). Mention that T and S has no common fixed point, but T and S satisfy all the conditions (C1)–(C3) except that α(x0 , T Sx0 ) ≥ 1 for any x0 ∈ X. If we remove the condition for MS,T in Theorem 1, then we should add the symmetry condition for α, i.e., α(x, y) = α(y, x). We have Theorem 2. Let (X, d) be a complete Brianciari metric space and T, S : X → X be two mappings. Suppose that: (C’1) α(x, y)d(T x, Sy) ≤ ψ(MT,S (x, y)), for any x, y ∈ X; (C’2) There exists a point x0 ∈ X such that α(x0 , Sx0 ) ≥ 1, α(x0 , T x0 ) ≥ 1 and α(x0 , T Sx0 ) ≥ 1; (C’3) α(x, y) = α(y, x) for any x, y ∈ X; (C’4) T, S are continuous and (T S)n x0 = (T S)n x0 for any sufficiently large positive integer n. Then, there exists a point u ∈ X such that T u = u = Su. Now, we introduce the following.

On Common Fixed Points in the Context of Brianciari Metric Spaces

Definition 6. For a nonempty set X, let T : X → X, S : X → X and α : X × X → [0, +∞) be mappings. We say that (T, S) is strong α-admissible if, for all x, y ∈ X, we have α(x, y) ≥ 1 ⇒ α(x, T y) ≥ 1, α(T x, y) ≥ 1, α(x, Sy) ≥ 1, α(Sx, y) ≥ 1 We call T a strong α-admissible mapping if (T, T ) is strong α-admissible. Theorem 3. Let (X, d) be a complete Brianciari metric space and T, S : X → X be two mappings. Suppose that: (D1) (T, S) is generalized (α, ψ)-contractive and strong α-admissible, where ψ ∈ Ψ, (D2) There exists a point x0 ∈ X such that α(x0 , Sx0 ) ≥ 1, α(x0 , T x0 ) ≥ 1 and α(x0 , T Sx0 ) ≥ 1; (D3) T, S are continuous. Then, there exists u ∈ X such that T u = u = Su. Proof. By the conditions of α, we can easily get that α(xn , xm ) ≥ 1. The rest of the proof is similar to the proof of Theorem 1, except the proof of that {xn } is a BMS Cauchy sequence. We argue by contradiction. Suppose to the contrary that there exists ε, for all n there exists rn > qn ≥ n such that d(xrn , xqn ) ≥ ε. Note that qn is the least one such that rn − qn is the possible smallest odd number, that is, d(xrn −2 , xqn ) < ε, . . .. Since lim d(xqn +1 , xqn ) = 0 and lim d(xqn +2 , xqn ) = 0, we have n→+∞

n→+∞

rn ≥ qn + 3 for sufficiently large n. So in three cases, we have d(xrn , xqn ) → ε, n → +∞. If xrn −1 = xqn −1 for sufficiently large n, then ε = lim d(xrn , xqn ) n→+∞

≤ lim sup ψ(max{d(xrn −1 , xqn −1 ), d(xrn −1 , xrn ), d(xqn −1 , xqn )}) n→+∞

≤ lim sup max{d(xrn −1 , xqn −1 ), d(xrn −1 , xrn ), d(xqn −1 , xqn )} n→+∞

= 0, which is a contradiction. Hence, xrn −1 = xqn −1 for infinite many n ∈ N+ . We still denote them by xrn −1 and xqn −1 , then xrn , xrn −1 , xqn and xqn −1 are distinct, so we have d(xrn , xqn ) ≤ d(xrn , xrn −1 ) + d(xrn −1 , xqn −1 ) + d(xqn −1 , xqn ) and d(xrn −1 , xqn −1 ) ≤ d(xrn −1 , xrn ) + d(xrn , xqn ) + d(xqn , xqn −1 ). Thus |d(xrn , xqn ) − d(xrn −1 , xqn −1 )| ≤ d(xqn , xqn −1 ) + d(xrn −1 , xrn ) → 0.

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We can get that d(xrn −1 , xqn −1 ) → ε, n → +∞. Therefore, ε = lim d(xrn , xqn ) n→+∞

≤ lim sup ψ(max{d(xrn −1 , xqn −1 ), d(xrn −1 , xrn ), d(xqn −1 , xqn )}) n→+∞

≤ ψ( lim max{d(xrn −1 , xqn −1 ), d(xrn −1 , xrn ), d(xqn −1 , xqn }) n→+∞

= ψ(ε) < ε, 

which is a contradiction.

2.2. A Fixed Point Theorem for Generalized (α, ψ)-Meir–Keeler-Contractions on Brianciari Metric Spaces We introduce the following. Definition 7. Let (X, d) be a Brianciari metric space and T : X → X be a given mapping. The mapping T is called a generalized (α, ψ)-Meir–Keeler contractive mapping if there exist two functions ψ ∈ Ψ and α : X ×X → [0, +∞) satisfying the following condition: for each ε > 0, there exists δ > 0 such that ε ≤ ψ(M (x, y)) < ε + δ implies α(x, y)d(T x, T y) < ε, where

  1 M (x, y) = max d(x, y), d(x, T x), d(y, T y), (d(x, T y) + d(y, T x)) . 2 If T : X → X is a generalized (α, ψ)-Meir–Keeler contractive mapping,

then α(x, y)d(T x, T y) ≤ ψ(M (x, y)),

for any x, y ∈ X.

The first main result of this section is Theorem 4. Let (X, d) be a generalized complete metric space and T : X → X be a generalized-(α, ψ)-Meir–Keeler contractive mapping, where ψ ∈ Ψ satisfying 2ψ( 2t ) ≤ ψ(t) for sufficiently small t > 0. Suppose that (i) T is α-admissible; (ii) there exists x0 ∈ X such that α(x0 , T x0 ) ≥ 1 and α(x0 , T 2 x0 ) ≥ 1; (iii) T is continuous. Then, there exists u ∈ X such that T u = u. Proof. Let x0 ∈ X be such that α(x0 , T x0 ) ≥ 1. Note that such point x0 exists due to condition (ii). We define the sequence {xn } in X by xn+1 = T xn for all n ≥ 0. If xn0 = xn0 +1 for some n0 , then clearly xn0 is a fixed point of T . Hence, throughout the proof, we suppose that xn = xn+1 for all n ∈ N. Since T is α-admissible, we have α(x0 , x1 ) = α(x0 , T x0 ) ≥ 1 ⇒ α(x1 , x2 ) = α(T x0 , T 2 x0 ) ≥ 1.

On Common Fixed Points in the Context of Brianciari Metric Spaces

Since T is α-admissible, so by repeating the process above, we derive α(xn , xn+1 ) ≥ 1,

for all n = 0, 1, . . .

By using the same technique above, we get α(x0 , x2 ) = α(x0 , T 2 x0 ) ≥ 1 ⇒ α(x1 , x3 ) = α(T x0 , T 3 x0 ) ≥ 1. The expression above yields α(xn , xn+2 ) ≥ 1,

for all n = 0, 1, . . . .

Step 1: We shall prove that xn = xm ,

for all n = m.

Suppose the contrary. We assume that exist n0 ≥ 0 and k ≥ 2 such that xn0 = xn0 +k . Then, xn = xn+k for any n ≥ n0 and {xn } has only finite distinct points. For simplicity, we assume n0 = 0, that is, xn = xn+k , for any n ∈ N. Case 1.1: k = 2. In this case, we have d(x0 , x1 ) = d(x2 , x1 ) = d(T x1 , T x0 ) = d(T x0 , T x1 ) ≤ α(x0 , x1 )d(T x0 , T x1 ) ≤ ψ(M (x0 , x1 ))  

1 = ψ max d(x0 , x1 ), d(x0 , x1 ), d(x1 , x2 ), (d(x0 , x2 ) + d(x1 , x1 )) 2 = ψ(d(x0 , x1 )) < d(x0 , x1 ), which is a contradiction. Case 1.2: k ≥ 3. In this case, for i = 1, 2, . . ., we have d(xi , xi+1 ) = d(T xi−1 , T xi ) ≤ α(xi−1 , xi )d(T xi−1 , T xi ) ≤ ψ(M (xi−1 , xi ))  1 = ψ max d(xi−1 , xi ), d(xi−1 , xi ), d(xi , xi+1 ), (d(xi−1 , xi+1 ) 2 + d(xi , xi ))})  

1 = ψ max d(xi−1 , xi ), d(xi , xi+1 ), d(xi−1 , xi+1 ) 2   1 < max d(xi−1 , xi ), d(xi , xi+1 ), d(xi−1 , xi+1 ) . 2 If max{d(xi−1 , xi ), 12 d(xi−1 , xi+1 )} ≤ d(xi , xi+1 ), then   1 d(xi , xi+1 ) < max d(xi−1 , xi ), d(xi , xi+1 ), d(xi−1 , xi+1 ) ≤ d(xi , xi+1 ), 2

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which is a contradiction. So max{d(xi−1 , xi ), 12 d(xi−1 , xi+1 )} > d(xi , xi+1 ). Hence, we have   1 d(xi , xi+1 ) < max d(xi−1 , xi ), d(xi , xi+1 ), d(xi−1 , xi+1 ) 2   1 = max d(xi−1 , xi ), d(xi−1 , xi+1 ) . 2 For i = 1, 2, . . ., we have d(xi , xi+2 ) = d(T xi−1 , T xi+1 ) ≤ α(xi−1 , xi+1 )d(T xi−1 , T xi+1 ) ≤ ψ(M (xi−1 , xi+1 )) 1 = ψ(max{d(xi−1 , xi+1 ), d(xi−1 , xi ), d(xi+1 , xi+2 ), (d(xi−1 , xi+2 ) 2 + d(xi , xi+1 ))}) 1 < max{d(xi−1 , xi+1 ), d(xi−1 , xi ), d(xi+1 , xi+2 ), (d(xi−1 , xi+2 ) 2 + d(xi , xi+1 ))}. Since xi = xi+1 , xi = xi+2 , xi−1 = xi+1 , xi−1 = xi and xi+1 = xi+2 , we have d(xi−1 , xi+2 ) ≤ d(xi−1 , xi ) + d(xi , xi+1 ) + d(xi+1 , xi+2 ), and then 1 d(xi , xi+2 ) < max{d(xi−1 , xi+1 ), d(xi−1 , xi ), d(xi+1 , xi+2 ), (d(xi−1 , xi+2 ) 2 + d(xi , xi+1 ))} 1 ≤ max{d(xi−1 , xi+1 ), d(xi−1 , xi ), d(xi+1 , xi+2 ), (d(xi−1 , xi ) 2 + 2d(xi , xi+1 ) + d(xi+1 , xi+2 ))} = max{d(xi−1 , xi+1 ), d(xi−1 , xi ), d(xi+1 , xi+2 ), d(xi , xi+1 ) 1 + (d(xi−1 , xi ) + d(xi+1 , xi+2 ))} 2 ≤ max{d(xi−1 , xi+1 ), 2d(xi−1 , xi ), 2d(xi , xi+1 ), 2d(xi+1 , xi+2 )}. Let ai = 2d(xi , xi+1 ) and bi = d(xi , xi+2 ) for i = 1, 2, . . .. By the above discussions, we have ai < max{ai−1 , bi−1 },

i = 1, 2, . . .

and bi < max{bi−1 , ai−1 , ai , ai+1 },

i = 1, 2, . . .

On Common Fixed Points in the Context of Brianciari Metric Spaces

Take A =

max ai and B =

max bi . By the easy facts an+k = an and

0≤i≤k−1

0≤i≤k−1

bn+k = bn for any n, we derive that A = max ai < max max{ai−1 , bi−1 } 1≤i≤k

1≤i≤k

= max{ max ai−1 , max bi−1 } = max{A, B} 1≤i≤k

1≤i≤k

and B = max bi < max max{bi−1 , ai−1 , ai , ai+1 } ≤ max{B, A}. 1≤i≤k

1≤i≤k

So, we get that max{A, B} < max{A, B}, which is a contradiction. Therefore, we deduce that xn = xm ,

for all n = m.

Step 2: lim d(xn , xn+1 ) = 0 and lim d(xn , xn+2 ) = 0. n→+∞

n→+∞

Note that ai+1 < max{ai , bi }. We have bi < max{bi−1 , ai−1 , ai , ai+1 } ≤ max{bi−1 , ai−1 , ai , ai , bi } = max{bi−1 , ai−1 , ai , bi }. If max{bi−1 , ai−1 , ai } ≤ bi , then bi < max{bi−1 , ai−1 , ai , bi } ≤ bi , which is a contradiction. Thus, max{bi−1 , ai−1 , ai , bi } > bi and then bi < max{bi−1 , ai−1 , ai , bi } = max{bi−1 , ai−1 , ai }. Since ai < max{ai−1 , bi−1 }, so bi < max{bi−1 , ai−1 , ai } = max{ai−1 , bi−1 }. By the above discussions, we have max{ai , bi } < max{ai−1 , bi−1 },

i = 1, 2, . . . .

Let ci = max{ai , bi }, i = 0, 1, 2, . . ., then lim ci exists and we denote it c. i→+∞

Assume that c > 0. By the above discussions, we can easily get M (xi , xi+1 ) = max{ a2i , b2i } = c2i and M (xi , xi+2 ) = max{ai , bi } = ci , i = 0, 1, 2, . . .. Hence, we have c ai i−1 = d(xi , xi+1 ) ≤ ψ(M (xi−1 , xi )) = ψ 2 2 and bi = d(xi , xi+2 ) ≤ ψ(M (xi−1 , xi+1 )) = ψ(ci−1 ). We deduce that

a  

c i i−1 ci = max 2 , bi ≤ max 2ψ , ψ(ci−1 ) . 2 2

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Thus,

c  i−1 c = lim ci ≤ lim sup max 2ψ , ψ(ci−1 ) i→+∞ 2 i→+∞  

ci−1 ≤ max 2ψ lim , ψ( lim ci−1 ) i→+∞ 2 i→+∞

c  = max 2ψ , ψ(c)

c 2 < max 2 , c = c, 2 which is a contradiction. So c = 0, i.e., lim d(xn , xn+1 ) = 0 and lim n→+∞

n→+∞

d(xn , xn+2 ) = 0. Step 3: {xn } is a Cauchy sequence. Since 2ψ( 2t ) ≤ ψ(t) for t ≥ 0, so we have cn ≤ ψ(cn−1 ) ≤ · · · ≤ ψ n (c0 ), and then d(xn , xn+1 ) ≤ cn ≤ ψ n (c0 ). Case 3.1: d(xn , xn+2p+1 ) ≤

2p+1 

d(xn+i−1 , xn+i )

i=1

≤

2p+1 

ψ n+i−1 (c0 )

i=1

=

+∞ 

ψ k (c0 ) → 0,

n → +∞, ∀p ≥ 0.

k=n

Case 3.2: d(xn , xn+2p ) ≤

2p−2 

d(xn+i−1 , xn+i ) + d(xn+2p−2 , xn+2p )

i=1

≤ =

2p−2  i=1 +∞ 

ψ n+i−1 (c0 ) + d(xn+2p−2 , xn+2p )

ψ k (c0 ) + d(xn+2p−2 , xn+2p ) → 0,

n → +∞, ∀p ≥ 2.

k=n

Hence, we conclude that {xn } is a BMS Cauchy sequence in the complete Brianciari metric space (X, d). Thus, there exists u ∈ X such that lim d(xn , u) = 0.

n→+∞

Since T is continuous, u = lim xn+1 = lim T xn = T u, n→+∞

n→+∞

On Common Fixed Points in the Context of Brianciari Metric Spaces

that is, u is a fixed point of T . This completes the proof.



Remark 3. We can not remove the condition α(x0 , T 2 x0 ) ≥ 1 in (ii) in Theorem 4. A counter-example is given in Example 1. Theorem 4 is an analog result of Theorem 15 in [17]. We also state the following result. Theorem 5. Let (X, d) be a complete Brianciari metric space and T : X → X be a generalized-(α, ψ)-Meir–Keeler contractive mapping, where ψ ∈ Ψ. Suppose that (i) T is strong α-admissible; (ii) there exists x0 ∈ X such that α(x0 , T x0 ) ≥ 1 and α(x0 , T 2 x0 ) ≥ 1; (iii) T is continuous. Then, there exists u ∈ X such that T u = u. Proof. By the conditions of α, we can easily get that α(xn , xm ) ≥ 1. The rest of proof is the same as the proof of Theorem 4, except the proof of that {xn } is a BMS Cauchy sequence. Suppose to the contrary that there exists ε, for all n there exists rn > qn ≥ n such that d(xrn , xqn ) ≥ ε. Note rn is the least one, that is d(xrn −1 , xqn ) < ε, d(xrn −2 , xqn ) < ε, . . .. Since lim d(xqn +1 , xqn ) = 0 and lim d(xqn +2 , xqn ) = 0, so we have n→+∞

n→+∞

rn ≥ qn + 3 for sufficiently large n. Then,

ε ≤ d(xrn , xqn ) ≤ d(xrn , xrn −1 ) + d(xrn −1 , xrn −2 ) + d(xrn −2 , xqn ) < d(xrn , xrn −1 ) + d(xrn −2 , xqn ) + ε. Letting n → +∞, we can deduce that d(xrn , xqn ) → ε. Having in mind that |d(xrn , xqn ) − d(xrn −1 , xqn )| ≤ d(xrn , xrn −2 ) + d(xrn −2 , xrn −1 ) → 0, we can get that d(xrn −1 , xqn ) → ε as n → +∞. Since |d(xrn , xqn ) − d(xrn , xqn −1 )| ≤ d(xqn , xqn −2 ) + d(xqn −1 , xqn −2 ) → 0, we find d(xrn , xqn −1 ) → ε, n → +∞. By |d(xrn , xqn ) − d(xrn −1 , xqn −1 )| ≤ d(xqn , xqn −1 ) + d(xrn −1 , xrn ) → 0,

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we can obtain that d(xrn −1 , xqn −1 ) → ε as n → +∞. Therefore ε=

lim d(xrn , xqn )  1 ≤ lim sup ψ max d(xrn −1 , xqn −1 ), d(xrn −1 , xrn ), d(xqn −1 , xqn ), (d(xrn −1 , xqn ) 2 n→+∞ n→+∞

+ d(xrn , xqn −1 ))})  

1 ≤ψ lim max d(xrn −1 , xqn −1 ), (d(xrn −1 , xqn ) + d(xrn , xqn −1 )) n→+∞ 2 = ψ(ε) < ε,

which is a contradiction.



Acknowledgements The authors are grateful to the reviewers for their careful reviews and useful comments. Conflict of interest The authors declare that they have no competing interests.

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On Common Fixed Points in the Context of Brianciari Metric Spaces [11] Bilgili, N., Karapınar, E.: A note on “common fixed points for (ψ, α, β)weakly contractive mappings in generalized metric spaces”. Fixed Point Theory Appl. 2013, 287 (2013) [12] Banach, S.: Sur les op´erations dans les ensembles abstraits et leur application aux ´equations int´egrales. Fundam. Math. 3, 133–181 (1922) [13] Branciari, A.: A fixed point theorem of Banach–Caccioppoli type on a class of generalized metric spaces. Publ. Math. Debrecen 57, 31–37 (2000) [14] Jleli, M., Karapınar, E., Samet, B.: Fixed point results for α − ψλ contractions on gauge spaces and applications. Abstr. Appl. Anal., Article ID 730825 (2013) [15] Di Bari, C., Vetro, P.: Common fixed points in generalized metric spaces. Appl. Math. Comput. 218(13), 7322–7325 (2012) [16] Kirk, W.A., Shahzad, N.: Generalized metrics and Caristi’s theorem. Fixed Point Theory Appl. 2013, 129 (2013) [17] Latif, A., Eshaghi Gordji, M., Karapinar, E., Sintunavarat, W.: Fixed point results for generalized (α, ψ)-Meir–Keeler contractive mappings and applications. J. Inequal. Appl. 2014, 68 (2014) [18] La Rosa, V., Vetro, P.: Common fixed points for α–ψ–ϕ-contractions in generalized metric spaces. Nonlinear Anal. Model. Control 19(1), 43–54 (2014) [19] Samet, B.: Discussion on: a fixed point theorem of Banach–Caccioppoli type on a class of generalized metric spaces by A. Branciari. Publ. Math. Debrecen 76(4), 493–494 (2010) [20] Samet, B., Vetro, C., Vetro, P.: Fixed point theorems for α–ψ-contractive type mappings. Nonlinear Anal. 75, 2154–2165 (2012) Hassen Aydi Department of Medical Research China Medical University Hospital China Medical University Taichung Taiwan Hassen Aydi Department of Mathematics College of Education of Jubail University of Dammam P.O: 12020 Industrial Jubail 31961 Saudi Arabia e-mail: [email protected] Erdal Karapınar Department of Mathematics Atilim University 06836 ˙ Incek Ankara Turkey

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Results. Math.

Erdal Karapınar Nonlinear Analysis and Applied Mathematics Research Group (NAAM) King Abdulaziz University Jeddah Saudi Arabia e-mail: [email protected] Dong Zhang School of Mathematical Sciences Peking University 100871 Beijing China e-mail: [email protected]

Received: December 3, 2014. Accepted: November 24, 2015.