COMMON FIXED POINT THEOREMS FOR g ...

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x = gx = fx, then x is called a common fixed point of f and g. In 1993, S. Czerwik [20], with a view to improve the idea of a metric space, introduced the notion of a ...
COMMON FIXED POINT THEOREMS FOR g-GENERALIZED CONTRACTIVE MAPPINGS IN b-METRIC SPACES MOHAMMAD IMDAD1 , MOHAMMAD ASIM2 AND RQEEB GUBRAN3

Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India. Email addresses: [email protected] , [email protected] and [email protected] Abstract. In this paper, we establish some common fixed point results for a pair of self-mappings satisfying g-generalized weakly contractive conditions (governed by an implicit function) in a b-metric space endowed with an amorphous binary relation. Our results generalize relevant core results of the existing literature, which include several rational contractions as well as some weakly contractive conditions.

Keywords: Common fixed point; binary relation; b-metric space, g-generalized contractive mapping. 2000 AMS classification: 47H10, 54H25.

1. Introduction Fixed point theory is a very vast topic of nonlinear functional analysis which is relatively old but still a hot domain of vigorous research activity. The strength of metric fixed point theory lies in the applications of fixed point theoretic results which are scattered in mathematics as well as outside of mathematics. The origin of metric fixed point theory is often traced back to Banach contraction principle which was proved by S. Banach [11] in 1922 which continues to be the most celebrated result of fixed point theory. In 1975, Dass et al. [22] and Jaggi [28] proved an extension of Banach contraction principle employing rational contractions. Alber and Guerre-Delabriere [6] generalized contraction condition by introducing the idea of weak contractions. Thereafter, this idea was used by Rhoades [38], Dutta and Choudhury [23] and several others. For the work of this kind one can be referred to Harjani et al. [25], Luong [33], Gubran et al. [24] and references therein. Recently, Alam and Imdad [4, 5] extended the classical Banach contraction principle to a complete metric space endowed with a binary relation. On the other hand, the concept of b-metric space was introduced by Czerwik [20]. In recent years, Mustafa [34], Suzuki [41], Wong [42], Piri-Afshari [35] and others proved some fixed point results in b-metric space. Amini [9] extended the main results contained in [22, 28] and proved common fixed point results for rational contractions. In 1997, Popa [36] initiated the idea of implicit function which is general enough to unify several contraction conditions in one go. In the recent past, several authors improved their results utilising this idea. For more details on implicit function one can 1

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consult [7, 12, 13, 26, 27, 37]. Inspired by foregoing observations, we prove some existence and uniqueness common fixed point results in b-metric space endowed with an arbitrary binary relation satisfying a suitably chosen implicit function. 2. Preliminaries In what follows, we collect relevant definitions and auxiliary results needed in our subsequent discussions. Recall that for a nonempty set X, a subset R of X 2 is called a binary relation on X, i.e., R = {(x, y) ∈ X 2 : x, y ∈ X}. We say that x is comparable to y in the relation R and denote it by: xRy. Write R−1 = {(x, y) ∈ X 2 : (y, x) ∈ R}. Put S = R ∪ R−1 which is indeed a subset of X 2 and hence defines another relation on X. Two elements x, y ∈ X are said to be comparable in relation S if (x, y) ∈ S and denote it by: xSy. An element x ∈ X is said to be a coincidence point of a pair (f, g) of self-mappings on X if gx = f x and x∗ ∈ X is said to be a point of coincidence if x∗ = gx = f x. If x = gx = f x, then x is called a common fixed point of f and g. In 1993, S. Czerwik [20], with a view to improve the idea of a metric space, introduced the notion of a b-metric space which runs as follows: Definition 2.1. [20] Let X be a non-empty set. The pair (X, σ) is called a b-metric space with coefficient s ≥ 1, if a mapping σ : X ×X → R+ satisfies the following axioms for all x, y, z ∈ X (1) σ(x, y) = 0 if and only if x = y, (2) σ(x, y) = σ(y, x) (3) σ(x, y) ≤ s{σ(x, z) + σ(z, y)}. Naturally, with s = 1, the class of b-metric spaces coincides with class of metric spaces. Now, we present an example which shows that a b-metric need not be a metric in general (see also [ [21] p. 264]): Example 2.1. Let (X, d) be a metric space and σ(x, y) = (d(x, y))p where p > 1. Then σ is a b-metric space for s = 2p−1 . Definition 2.2. [29,39,40] Let (f, g) be a pair of a self-mappings defined on a b-metric space (X, σ). Then (i) f is said to be a g-comparative if gxSgy ⇒ f xSf y, for all x, y ∈ X, (ii) the pair (f, g) is said to be weakly compatible if g(f x) = f (gx), for every coincidence point x ∈ X of the pair (f, g), (iii) f is said to be g-continuous at x ∈ X if for all sequences {xn } ⊂ X, σ

σ

gxn −→ gx ⇒ f xn −→ f x. Moreover, f is called g-continuous if it is g-continuous at each point of X.

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Definition 2.3. [39] Let (X, σ, S) be a metric spaces with a symmetric relation S and f a self mapping on X. Then (X, σ, S) is said to be regular if f xn Sf x, whenever {xn } is a sequence in X and x ∈ X, such that lim f xn = f x.

n→∞

Definition 2.4. [39] Let X be endowed with a symmetric relation S and f a selfmapping on X. Then a subset Y of X is said to be an S-f -directed if for all x, y ∈ Y, there exists z ∈ X such that f xRf z and f zRf y. Lemma 2.1. [3] Let (f, g) be a pair of weakly compatible self-mappings defined on a non-empty set X. Then every point of coincidence of the pair (f, g) remains a coincidence point. Now, we present a suitable implicit function, besides giving some examples, which include most of the well known contractions of the existing literature besides deducing several new ones. Definition 2.5. Let Λ be the family of the lower semi-continuous functions λ : R5+ → R+ which are non-decreasing in the 4th component and also satisfy the following conditions (∀ s ≥ 1): (i) λ(u, u, v, u + v, 0) ≤ sv, ∀ u, v ≥ 0 (ii) λ(v, 0, 0, v, v) ≤ sv, ∀ v ≥ 0. Usually, such λ function is called implicit function. In the following examples, we present some members of Λ. Example 2.2. λ(u1 , u2 , u3 , u4 , u5 ) = s max{u3 , u5 }. (i) λ(u, u, v, u + v, 0) = sv, ∀ u, v ≥ 0 (ii) λ(v, 0, 0, v, v) = sv, ∀ v ≥ 0. Example 2.3. λ(u1 , u2 , u3 , u4 , u5 ) = s(u3 + u5 ). (i) λ(u, u, v, u + v, 0) = sv, ∀ u, v ≥ 0 (ii) λ(v, 0, 0, v, v) = sv, ∀ v ≥ 0. √ √ Example 2.4. λ(u1 , u2 , u3 , u4 , u5 ) = s max u3 + u5 , u4 − u1 , u2 u5 . (i) λ(u, u, v, u + v, 0) = sv, ∀ u, v ≥ 1 (ii) λ(v, 0, 0, v, v) ≤ sv, ∀ v ≥ 1. u2 5 , u5 1+u } Example 2.5. λ(u1 , u2 , u3 , u4 , u5 ) = s max{ u3 +u 2 2

(i) λ(u, u, v, u + v, 0) = s v2 , ∀ u, v ≥ 0 (ii) λ(v, 0, 0, v, v) = s v2 , ∀ v ≥ 0.  1+u4 5 Example 2.6. λ(u1 , u2 , u3 , u4 , u5 ) = s max u3 +u . , u 3 2 1+u2 +u3 (i) λ(u, u, v, u + v, 0) = sv, ∀ u, v ≥ 0 (ii) λ(v, 0, 0, v, v) = s v2 , ∀ v ≥ 0. Example 2.7. λ(u1 , u2 , u3 , u4 , u5 ) = s max{u3 , u5 , u22 u5 , u2 u25 }. (i) λ(u, u, v, u + v, 0) = sv, ∀ u, v ≥ 0 (ii) λ(v, 0, 0, v, v) = sv, ∀ v ≥ 0.

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Example 2.8. λ(u1 , u2 , u3 , u4 , u5 ) = s max{u3 , u5 } + a

√





√

u2 u5 u3 +u5

, where a1 ∈ R

(i) λ(u, u, v, u + v, 0) = sv, ∀ u, v ≥ 0 (ii) λ(v, 0, 0, v, v) = sv, ∀ v ≥ 0. Example 2.9. λ(u1 , u2 , u3 , u4 , u5 ) = s{u3 +u5 }+a



u22 u25 1+u4

+b

u2 u5 u3 +u5



, where a1 , a2 ∈ R

(i) λ(u, u, v, u + v, 0) = sv, ∀ u, v ≥ 0 (ii) λ(v, 0, 0, v, v) = sv, ∀ v ≥ 0.  2   2  u u2 u5 u u4 u5 Example 2.10. λ(u1 , u2 , u3 , u4 , u5 ) = s{u3 +u5 }+a u13 +u +b u21 +u , where a1 , a2 ∈ 4 3 R (i) λ(u, u, v, u + v, 0) = sv, ∀ u, v ≥ 0 (ii) λ(v, 0, 0, v, v) = sv, ∀ v ≥ 0.  u2 u3 s u1 , if u1 > 0, Example 2.11. λ(u1 , u2 , u3 , u4 , u5 ) = 0, if u1 = 0. (i) λ(u, u, v, u + v, 0) = sv, ∀ u, v ≥ 0 (ii) λ(v, 0, 0, v, v) = 0, ∀ v ≥ 0. (1+u2 ) Example 2.12. λ(u1 , u2 , u3 , u4 , u5 ) = s u31+u . 1 (i) λ(u, u, v, u + v, 0) = sv, ∀ u, v ≥ 0 (ii) λ(v, 0, 0, v, v) = 0, ∀ v ≥ 0. 2 )(1+u5 ) Example 2.13. λ(u1 , u2 , u3 , u4 , u5 ) = s u3 (1+u . 1+u1 (i) λ(u, u, v, u + v, 0) = sv, ∀ u, v ≥ 0 (ii) λ(v, 0, 0, v, v) = 0, ∀ v ≥ 0.

3. Main results The following definition is used in our subsequent discussions. Definition 3.1. Let (f, g) be a pair of self-mappings on a b-metric space (X, σ) for some (s ≥ 1). Then, f is said to be a g-generalized contractive mapping if for all x, y ∈ X with x 6= y and gxSgy, we have σ(f x, f y) ≤ α(σ(gx, gy))Ng (x, y) + β(σ(gx, gy))Mg (x, y),

(3.1)

where,   1 Ng (x, y) = λ σ(gx, gy), σ(gx, f x), σ(gy, f y), σ(gx, f y), σ(gy, f x) , s o n 1 Mg (x, y) = max σ(gx, gy), σ(gx, f x), σ(gy, f y), [σ(gx, f y) + σ(gy, f x)] , 2s wherein, λ ∈ Λ and α, β : [0, ∞) → [0, 1) are mappings such that α is continuous and sα(t) + s lim supq→t+ β(q) < 1, for each t ≥ 0. Our main result runs as follows: Theorem 3.1. Let (X, σ) be a b-metric space (for some (s ≥ 1)) equipped with a binary relation R, Y a complete subspace of X and f, g : X → X such that f is a g-comparative mapping. Suppose that the following conditions hold:

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(i) (ii) (iii) (iv)

there exists an x0 ∈ X such that gx0 Sf x0 , f is a g-generalized contractive mapping, f (X) ⊆ Y ⊆ g(X), either (a) f is g-continuous or (b) (X, σ, S) is regular.

Then the pair (f, g) has a coincidence point. Proof. Take x0 ∈ X such as in (i). Due to (iii), we can choose x1 ∈ X so that gx1 = f x0 and x2 ∈ X so that gx2 = f x1 . Continuing this process, we can construct a sequence {xn } such that, for all n ∈ N0 := N ∪ {0} gxn+1 = f xn . Observe that, if σ(gxm , gxm+1 ) = 0 for some m ∈ N0 , then xm is a coincidence point and we are done. Otherwise, assume that σ(gxn , gxn+1 ) > 0 for all n ∈ N0 . Now, we proceed to show that lim σ(gxn , gxn+1 ) = 0. To accomplished this consider n→∞

Ng (xn−1 , xn ) =

=

≤ ≤

 λ σ(gxn−1 , gxn ), σ(gxn−1 , gxn ), σ(gxn , gxn+1 ),  1 σ(gxn−1 , gxn+1 ), σ(gxn , gxn ) s λ σ(gxn−1 , gxn ), σ(gxn−1 , gxn ), σ(gxn , gxn+1 ),  1 σ(gxn−1 , gxn+1 ), 0 s λ σ(gxn−1 , gxn ), σ(gxn−1 , gxn ), σ(gxn , gxn+1 ),  σ(gxn−1 , gxn ) + σ(gxn , gxn+1 ), 0 sσ(gxn , gxn+1 )

and n Mg (xn−1 , xn ) = max σ(gxn−1 , gxn ), σ(gxn−1 , gxn ), σ(gxn , gxn+1 ), o 1 [σ(gxn−1 , gxn+1 ) + σ(gxn , gxn )] 2s n o 1 = max σ(gxn−1 , gxn ), σ(gxn , gxn+1 ), [σ(gxn−1 , gxn+1 )] , 2s by the triangular inequality in b-metric space, we have Mg (xn−1 , xn ) ≤ max{σ(gxn−1 , gxn ), σ(gxn , gxn+1 )}. Now, on setting x = xn−1 and y = xn in (3.1), we get σ(gxn , gxn+1 ) ≤ α(σ(gxn−1 , gxn ))N (xn−1 , xn ) + β(σ(gxn−1 , gxn ))M (xn−1 , xn ) ≤ sα(σ(gxn−1 , gxn ))σ(gxn , gxn+1 ) + β(σ(gxn−1 , gxn )) max{σ(gxn−1 , gxn ), σ(gxn , gxn+1 )}

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which is equivalent to (1 − sα(σ(gxn−1 , gxn ))σ(gxn , gxn+1 )) ≤ β(σ(gxn−1 , gxn )) max{σ(gxn−1 , gxn ), σ(gxn , gxn+1 )}. (3.2) Assume that max{σ(gxn−1 , gxn ), σ(gxn , gxn+1 )} = σ(gxn , gxn+1 ), then, from (3.2), we have 1 ≤ sα(σ(gxn−1 , gxn )) + β(σ(gxn−1 , gxn )) a contradiction. Thus, max{σ(gxn−1 , gxn ), σ(gxn , gxn+1 )} = σ(gxn−1 , gxn ). Therefore, (3.2) gives rise (1 − sα(σ(gxn−1 , gxn ))σ(gxn , gxn+1 )) ≤ β(σ(gxn−1 , gxn ))σ(gxn−1 , gxn ) so that β(σ(gxn−1 , gxn )) σ(gxn−1 , gxn ). 1 − sα(σ(gxn−1 , gxn )

σ(gxn , gxn+1 ) ≤

(3.3)

sβ β Observe that, sα + sβ < 1 ⇒ 1−sα < 1 ⇒ 1−sα < 1 (for all s ≥ 1). Let γ(t) = for each t ∈ R+ . Then for each t ≥ 0, we have

β(t) 1−sα(t)

lim sup β(δ)

lim sup γ(δ) =

δ→t+

δ→t+

1 − sα(t)

< 1.

(3.4)

By (3.3), we have (for n ≥ 1) σ(gxn , gxn+1 ) ≤ γ(σ(gxn−1 , gxn ))σ(gxn−1 , gxn ).

(3.5)

Hence, {σ(gxn , gxn+1 )} is a decreasing sequence of positive real numbers so that lim σ(gxn , gxn+1 ) = r ≥ 0.

n→∞

Assume that r > 0. Taking limit superior as n → ∞ in (3.5), we have lim sup γ(δ) = lim sup γ(σ(gxn , gxn+1 )) ≥ lim sup

δ→r

n→∞

n→∞

σ(gxn , gxn+1 ) = 1, σ(gxn−1 , gxn )

a contradiction, so that lim σ(gxn , gxn+1 ) = 0.

(3.6)

n→∞

Next, we assert that {gxn } is a Cauchy sequence. In view of (3.4), let c be such that lim+ sup γ(δ) < sc < 1. Owing to (3.5) and as lim sup γ(σ(gxn , gxn+1 )) = lim+ sup γ(r) < n→∞

δ→t

r→0

1, there exists N ∈ N with σ(gxn , gxn+1 ) ≤ scσ(gxn−1 , gxn ), ∀ n ≥ N. Hence, ∞ X

σ(gxn , gxn+1 ) < ∞.

n=1

Then, for all m, n ∈ N0 with m ≥ n σ(gxn , gxm ) ≤ scσ(gxn , gxn+1 ) + (sc)2 σ(gxn+1 , gxn+2 ) + ... + (sc)m−n σ(gxm−1 , gxm ) implying thereby lim σ(gxn , gxm ) = 0

n,m→∞

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proving that {gxn } is a Cauchy sequence in Y. In view of the completeness of Y, there exists z ∈ X such that gxn → z as n → ∞. On the other hand, Y ⊆ g(X), so that there exists x ∈ X such that gx = z. Thus, we have lim σ(gxn , gx) = 0.

(3.7)

n→∞

Firstly assume that f is g-continuous, then σ

σ

gxn −→ gx ⇒ f xn −→ f x. As gxn+1 = f xn , owing to the uniqueness of the limit, we conclude that f x = gx. Alternatively, owing to the regularity of X, contraction condition (3.1) gives rise σ(gxn+1 , f x) = σ(f xn , f x) ≤ α(σ(gxn , gx))Ng (xn , x) + β(σ(gxn , gx))Mg (xn , x), so that σ(gx, f x) = ≤

lim σ(gxn+1 , f x)

n→∞

lim sup[α(σ(gxn , gx))Ng (xn , x)] + lim sup[β(σ(gxn , gx))Mg (xn , x)]

n→∞

n→∞

≤ α(0)λ(0, 0, σ(gx, f x), σ(gx, f x), 0) + σ(gx, f x) lim+ sup β(t) t→0

≤ σ(gx, f x)[(sα(0) + lim+ sup β(t))].

(3.8)

t→0

Owing to the fact that sα(0) + lim+ sup β(t) < 1, whenever σ(gx, f x) > 0, (3.8) gives t→0

rise σ(gx, f x) < σ(gx, f x), a contradiction. Then σ(gx, f x) = 0 i.e., x ∈ X is a coincidence point of f and g. This completes the proof of the theorem.  Corollary 3.1. Theorem 2.1 of Amini-Harandi [9] is immediate from Theorem 3.1. Corollary 3.2. Theorem 3.1 remains true if contractive condition (3.1) is replaced by any one of the following, σ(f x, f y) ≤ sα(σ(gx, gy))

 σ(gx, f x)σ(gy, f y) + β σ(gx, gy) Mg (x, y). σ(gx, gy)

(3.9)

σ(gy, f y)(1 + σ(gx, f x)) σ(f x, f y) ≤ sα(σ(gx, gy)) 1 + σ(gx, gy)  +β σ(gx, gy) Mg (x, y).

(3.10)

σ(gy, f y)(1 + σ(gx, f x))(1 + σ(gy, f x)) σ(f x, f y) ≤ sα(σ(gx, gy)) 1 + σ(gx, gy)  +β σ(gx, gy) Mg (x, y).

(3.11)

Proof. In view of Theorem 3.1, the above results are immediate from Examples 2.11-2.13 respectively. 

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Remark 3.1. Corollary 3.2 corresponding to contraction conditions (3.9), (3.10) and (3.11) respectively generalize Theorem 1 in [28] together with Theorem 2.2 in [25], main results in [22] together with Theorem 2 in [19] and a generalized form of Theorem 2.1 utilized in [9]. The following results are possibly new to the literature as we never come across such results in the existing literature. Corollary 3.3. The conclusions of Theorem 3.1 remain true if contractive condition (3.1) is replaced by any one of the following,  σ(f x, f y) ≤ sα(σ(gx, gy)) max{σ(gy, f y), σ(gy, f x)} + β σ(gx, gy) Mg (x, y). (3.12)  σ(f x, f y) ≤ sα(σ(gx, gy)){σ(gy, f y) + σ(gy, f x)} + β σ(gx, gy) Mg (x, y). (3.13) p σ(f x, f y) ≤ sα(σ(gx, gy)) σ(gy, f y) + σ(gy, f x), σ(gx, f y) − σ(gx, gy), p  σ(gx, f x)σ(gy, f x) + β σ(gx, gy) Mg (x, y). (3.14)   σ(gy, f y) + σ(gy, f x) σ(gx, f x) σ(f x, f y) ≤ sα(σ(gx, gy)) , σ(gy, f x) 2 1 + σ(gx, f x)  +β σ(gx, gy) Mg (x, y). (3.15)   σ(gy, f y) + σ(gy, f x) 1 + σ(gx, f y) σ(f x, f y) ≤ sα(σ(gx, gy)) , σ(gy, f y) 2 1 + σ(gx, f x) + σ(gy, f y)  +β σ(gx, gy) Mg (x, y). (3.16)  σ(f x, f y) ≤ sα(σ(gx, gy)) max σ(gy, f y), σ(gy, f x), σ(gx, f x)2 σ(gy, f x),  σ(gx, f x)σ(gy, f x)2 + β σ(gx, gy) Mg (x, y). (3.17) p  σ(gx, f x)σ(gy, f x) σ(f x, f y) ≤ sα(σ(gx, gy)) max{σ(gy, f y), σ(gy, f x)} + a1 σ(gy, f y) + σ(gy, f x)  +β σ(gx, gy) Mg (x, y), (3.18) where a1 ∈ R 

σ(gx, f x)2 σ(gy, f x)2 1 + σ(gx, f y)

σ(f x, f y) ≤ sα(σ(gx, gy)){σ(gy, f y) + σ(gy, f x)} + a1 p   σ(gx, f x)σ(gy, f x) a2 + β σ(gx, gy) Mg (x, y), σ(gy, f y) + σ(gy, f x)



(3.19)

where a1 , a2 ∈ R  σ(gx, gy)2 σ(gx, f x)σ(gy, f x) σ(f x, f y) ≤ sα(σ(gx, gy)){σ(gy, f y) + σ(gy, f x)} + a1 σ(gy, f y) + σ(gx, f y)   2  σ(gx, f x)σ(gx, f y) σ(gy, f x) a2 + β σ(gx, gy) Mg (x, y). (3.20) σ(gx, f y) + σ(gy, f y) 

where a1 , a2 ∈ R

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Proof. In view of Theorem 3.1, results are immediate from examples 2.2-2.10 respectively.    Example 3.1. Let X = 21 , 32 ∪ [200, 300]  be equipped with a binary relation R = (1, 1), (n, n + 1) : n = 200, 201, 202, ...299. . Let (X, σ) be a b-metric space defined as in Example 2.1 (with p = 2). Define f, g : X → X by: f x = cos ln x and gx = cos ln |2 − x|, ∀ x ∈ X. In order to verify the contraction condition (3.12), for α(t) = 21 (∀ t ∈ [0, ∞)) and β = 0, the two cases arise: if x = y = 1, then condition (3.12) holds trivially. Otherwise, with x = n and y = n + 1, we have 2 (3.21) σ(f x, f y) = cos ln n − cos ln(n + 1) Further,  sα(σ(gx, gy)) max{σ(gy, f y), σ(gy, f x)}+β σ(gx, gy) Mg (x, y) 2  1 =2 × max cos ln(n − 1) − cos ln(n + 1) 2 2 , cos ln(n − 1) − cos ln n + 0 2 = cos ln(n − 1) − cos ln n . (3.22) Now, owing to the fact that cos ln n − cos ln(n + 1) 2 ≤ 1, cos ln(n − 1) − cos ln n 2

(3.23)

for all n = 200, ..., 299 (the verification of (3.23) is carried out by MATLAB), contraction condition (3.12) holds for all x, y ∈ X with xRy. Thus all the conditions of Corollary 3.3 are satisfied ensuring the existence of a coincidence point f and g (namely x = 1). Here, it can be pointed out that contraction condition (3.12) does not hold if we choose p = 1, demonstrating the utility of Theorem 3.1 over corresponding results in metric spaces.  1 1 Example 3.2. Let X = [0, 1] be equipped with a binary relation R = (0, 0), ( 2n , 2n+1 ): x 2x n ∈ N . Let f, g : X → X be defined by: f x = 3 and gx = 3 , ∀ x ∈ X. Define, b-metric σ as σ(x, y) = |x − y|2 on X. In order to verify inequality (3.10) for β = 0, 1 1 observe that it is holds trivially for x = y = 0. Now, on setting x = 2n and y = 2n+1 in (3.10), we have 2 2  2  1 2 1 2 2 1 3(2n) − 3(2n + 1) ≤ 2α 3(2n) − 3(2n + 1) 3(2n + 1) − 3(2n + 1) 2   2 1 1 + 3(2n) − 3(2n) ×  2  + 0, 2 2 1 + 3(2n) − 3(2n+1)

10

which can be written as 2   2 9n2 (2n + 1)2 + 1 α ≥ 3(2n)(2n + 1) 2n2 (36n2 + 1)(2n + 1)2 1 9 + = . 2(36n2 + 1) 2n2 (36n2 + 1)(2n + 1)2

(3.24)

Observe that the maximum of right hand side of inequality (3.24) is at n = 1. Therefore, (3.10) is satisfied on setting α(t) = 81 for all t ∈ [0, ∞). Consequently, Theorem 3.1 is applicable and the pair (f, g) has a coincidence point (namely x = 0). However, this example can not be covered by any corresponding metrical fixed point theorem as b-metric considered in this example is not a metric. 4. Uniqueness Results Theorem 4.1. The pair (f, g) in the hypothesis of Theorem 3.1 has a unique point of coincidence provided that f (X) is an S-g-directed. Proof. If the pair (f, g) has a unique coincidence point, then we are done. Otherwise, suppose that x and y are two coincidence points of the pair f and g. Now, we show that gx = gy. By the hypothesis, there exists z ∈ X such that gz is comparable to both f x and f y. Without loss of generality, we may assume that f xSgz and f ySgz. Put z = z0 . Since f (X) ⊆ g(X) and f is g-comparative mapping, one can define a sequence zn ⊂ X such that gzn+1 = f zn and gxSgzn for all n ∈ N0 . We assert that lim σ(gx, gzn ) = 0. n→∞

As in (3.6), we have shown lim σ(gzn , gzn+1 ) = 0.

n→∞

(4.1)

Making use of (3.1), we can have σ(gx, gzn+1 ) = σ(f x, f zn ) = α(σ(gx, gzn ))Ng (x, zn ) + β(σ(gx, gzn ))Mg (x, zn ),

(4.2)

where,   1 Ng (x, zn ) ≤ λ σ(gx, gzn ), σ(gx, f x), σ(gzn , f zn ), σ(gx, f zn ), σ(gzn , f x) s and o n 1 Mg (x, zn ) = max σ(gx, gzn ), σ(gx, f x), σ(gzn , f zn ), [σ(gx, f zn ) + σ(gzn , f x)] , 2s On using (4.1) and triangular inequality in b-metric 1 σ(gx, gzn+1 ) ≤ σ(gx, gzn ) + σ(gzn , gzn+1 ) s which implies 1 lim sup σ(gx, gzn+1 ) ≤ σ(gx, gzn ) n→∞ s

11

Thus, we have, 

lim sup Ng (x, zn ) = lim sup λ σ(gx, gzn ), 0, σ(gzn , gzn+1 ), n→∞ n→∞  1 σ(gx, gzn+1 ), σ(gzn , gx) s   ≤ lim sup λ σ(gx, gzn ), 0, 0, σ(gx, gzn ), σ(gzn , gx) n→∞

≤ s lim sup σ(gx, gzn ). n→∞

Or, n Mg (x, zn ) = max σ(gx, gzn ), σ(gx, f x), σ(gzn , gzn+1 ), o 1 [σ(gx, gzn+1 ) + σ(gzn , gx)] . 2s Using triangular inequality, we have, 1 1 lim sup [σ(gx, gzn+1 ) + σ(gzn , gx)] ≤ lim sup [sσ(gx, gzn ) + sσ(gzn , gzn+1 ) n→∞ 2s n→∞ 2s +sσ(gzn , gzn ) + sσ(gzn , gx)] = lim sup σ(gzn , gx). n→∞

Therefore, (4.1) gives rise lim sup Mg (x, zn ) ≤ lim sup σ(gx, gzn ). n→∞

n→∞

Taking limit superior as n → ∞ in (4.2), we have lim sup σ(gx, gzn+1 ) ≤ lim sup σ(gx, gzn ){sα(σ(gx, gzn )) + β(σ(gx, gzn ))}, n→∞

n→∞

so that, lim supn→∞ sα(σ(gx, gzn )) + β(σ(gx, gzn )) ≥ 1, a contradiction. Therefore, lim sup σ(gx, gzn ) = 0.

(4.3)

n→∞

Similarly, one can show that lim sup σ(gy, gzn ) = 0.

(4.4)

n→∞

Using (4.3), (4.4) and triangular inequality (for s ≥ 1), we have σ(gx, gy) ≤ lim sup s[σ(gx, gzn ) + σ(gy, gzn )] = 0, n→∞

which shows that the pair (f, g) has a unique point of coincidence.



Theorem 4.2. The pair (f, g) in the hypothesis of Theorem 4.1 has a unique common fixed point provided that the pair is weakly compatible. Proof. Let x ∈ X be an arbitrary coincidence point of the pair (f, g). Appealing, Theorem 4.1, there exists a unique point of coincidence x∗ ∈ X (say) such that f x = gx = x∗ . In view of Lemma 2.1, x∗ is itself a coincidence point, i.e., f x∗ = gx∗ . Again appealing, Theorem 4.1, x∗ is a unique common fixed point of the pair f and g. 

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