METRIC SPACES

Open Journal of Applied & Theoretical Mathematics (OJATM) Vol. 2, No. 2, June 2016, pp. 65~78 ISSN: 2455-7102

SOME THEOREMS ON FIXED POINTS & COMMON FIXED POINTS OF EXPANSIVE TYPE MAPPINGS IN b- METRIC SPACES R. D. DAHERIYA (and) MOTIRAM LIKHITKER Department of Mathematics, Government J.H. Post Graduate College, Betul, P.O. Box 460001, Madhya Pradesh, India Email: [email protected]

& MANOJ UGHADE Department of Mathematics, Sarvepalli Radhakrishnan University, Bhopal, P.O. Box 462026, Madhya Pradesh, India

Article Info

ABSTRACT

Article history: Received March. 11th, 2016 Revised May. 14th, 2016 Accepted June. 10th, 2016

In this article, we present some fixed point and common fixed point theorems under various expansive conditions in b-metric spaces. Our results improve and supplement some recent

Keyword:

results in the literature. We introduce some

Fixed point, common fixed point, b-metric space.

examples the support the validity of our results.

MSC: 47H10, 54H25 Copyright © 2016

Open Journal of Applied & Theoretical Mathematics (OJATM) All rights reserved.

Corresponding Author: MOTIRAM LIKHITKER Email: [email protected]

1. INTRODUCTION Metric fixed point theory is playing an increasing role in mathematics because of its wide range of applications in applied mathematics and sciences. There have been a number of generalizations of the usual notion of a metric space. One such generalization is a b-metric space introduced and studied by Bakhtin [3] and Czerwik [4] and a lot of fixed point results for single and multi-valued mappings by many authors have been obtained in (ordered) bmetric spaces (see e.g.[2, 5, 9]. Alghamdi, et al. [11] obtained some fixed point and coupled Journal homepage: http://theojal.com/ojatm/ 65


fixed point theorems on b-metric-like spaces. The study of expansive mappings is a very interesting research area in fixed point theory. Wang et.al [18] introduced the concept of expanding mappings and proved some fixed point theorems in complete metric spaces. Daffer and Kaneko [6] defined an expanding condition for a pair of mappings and proved some common fixed point theorems for two mappings in complete metric spaces. Aage and Salunke [1] introduced several meaningful fixed point theorems for one expanding mapping. Daheriya et al. [7] proved some fixed point theorems for continuous and surjective expansive mappings in dislocated metric spaces. Jain et al. [9] established some fixed point, coincidence point and common fixed point theorems under various expansive conditions in b-metric spaces. Also, Jain et al. [9] established some fixed point, coincidence point and common fixed point theorems under various expansive conditions in parametric metric spaces and parametric b-metric Spaces. In this paper, we present some fixed point and common point theorems under various expansive conditions in b-metric spaces. These results improve and generalize some important known results in the literature. We introduce some examples the support the validity of our results.

2. PRELIMINARIES Throughout this paper ℝ and ℝ + will represents the set of real numbers and nonnegative real numbers, respectively.The following definitions are required in the sequel which can be found in [3-5]. Definition 2.1 Let X be a nonempty set, s ≥ 1 be a given real number and d ∶ X × X → ℝ + be a function. We say d is a b-metric on X 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, y) ≤ s[d(x, z) + d(z, y)]. A triplet (X, d, s) is called a b-metric space. Obviously, for s = 1, b-metric reduces to metric. Journal homepage: http://theojal.com/ojatm/ 66


Definition 2.2 Let {xn }∞ n=1 be a sequence in a b-metric space (X, d, s). 1.

{xn }∞ n=1 is said to be convergent to x ∈ X, written as limn→∞ xn = x, if for each ϵ > 0, there exists a positive integer n0 such that, for all n ≥ n0 , d(xn , x) < ϵ.

2.

{xn }∞ n=1 is said to be a Cauchy sequence in X, if for each ϵ > 0, there exists a positive integer n0 such that, for all n, m ≥ n0 , d(xn , xm ) < ϵ.

3.

(X, d) is said to be complete if every Cauchy sequence is a convergent sequence.

Definition 2.3 Let (X, d, s) be a b-metric space and T: X → X be a mapping. We say T is a continuous mapping at x in X, if for any sequence {xn }∞ n=1 in X such that limn→∞ xn = x, then limn→∞ Txn = Tx. Lemma 2.4 [11] Let (𝑋, 𝑑, 𝑠) be a b-metric space with the coefficient 𝑠 ≥ 1 and let {𝑥𝑛 }∞ 𝑛=1 ∞ be a sequence in 𝑋. if {𝑥𝑛 }∞ 𝑛=1 converges to x and also {𝑥𝑛 }𝑛=1 converges to y, then 𝑥 = 𝑦.

That is, the limit of {𝑥𝑛 }∞ 𝑛=1 is unique. Lemma 2.5 [11] Let (𝑋, 𝑑, 𝑠) be a b-metric space with the coefficient 𝑠 ≥ 1 and let {𝑥𝑘 }𝑛𝑘=0 ⊂ 𝑋. Then 𝑑(𝑥𝑛 , 𝑥0 ) ≤ 𝑠𝑑(𝑥0 , 𝑥1 ) + 𝑠 2 𝑑(𝑥2 , 𝑥3 ) + ⋯ + 𝑠 𝑛−1 𝑑(𝑥𝑛−2 , 𝑥𝑛−1 ) + 𝑠 𝑛−1 𝑑(𝑥𝑛−1 , 𝑥𝑛 ) Lemma 2.6 [11] Let(𝑋, 𝑑, 𝑠) be a b-metric space with the coefficient 𝑠 ≥ 1. Let {𝑥𝑛 }∞ 𝑛=1 be a sequence of points of 𝑋such that 𝑑(𝑥𝑛 , 𝑥𝑛+1 ) ⪯ 𝜆 𝑑(𝑥𝑛−1 , 𝑥𝑛 ) 1

where𝜆 ∈ [0 , 𝑠 ) and 𝑛 = 1, 2, . . .. Then {𝑥𝑛 }∞ 𝑛=1 is a Cauchy sequence in (𝑋, 𝑑, 𝑠). 3. FIXED POINT THEOREMS In this section, we prove some fixed point theorem for continuous and surjective mapping satisfying expansion condition in the setting of complete b-metric space. Theorem 3.1 Let (𝑋, 𝑑, 𝑠) be a b-metric space with the coefficient 𝑠 ≥ 1 and 𝑇 a continuous mapping satisfying the following condition: (3.1)

d(𝑇𝑥, 𝑇𝑦) ≥ 𝛼

d(𝑥,𝑇𝑥)[1+d(𝑦,𝑇𝑦)] 1+d(𝑥,𝑦)

+ 𝛽d(𝑥, 𝑦)

Journal homepage: http://theojal.com/ojatm/ 67


for all 𝑥, 𝑦 ∈ 𝑋, 𝑥 ≠ 𝑦, where 𝛼, 𝛽 ≥ 0 are real constants and 𝑠𝛼 + 𝛽 > 𝑠, 𝛽 > 1. Then 𝑇 has a fixed point in 𝑋. Proof Choose 𝑥0 ∈ 𝑋 be arbitrary, to define the iterative sequence {𝑥𝑛 }𝑛∈ℕ as follows: 𝑇𝑥𝑛 = 𝑥𝑛−1 for 𝑛 = 1,2,3, …. Taking 𝑥 = 𝑥𝑛+1 and 𝑦 = 𝑥𝑛+2 in (3.1), we obtain d(𝑇𝑥𝑛+1 , 𝑇𝑥𝑛+2 ) ≥ 𝛼

d(𝑥𝑛+1 ,𝑇𝑥𝑛+1 )[1+d(𝑥𝑛+2 ,𝑇𝑥𝑛+2 )] 1+d(𝑥𝑛+1 ,𝑥𝑛+2 ) d(𝑥𝑛+1 ,𝑥𝑛 )[1+d(𝑥𝑛+2 ,𝑥𝑛+1 )] 1+d(𝑥𝑛+1 ,𝑥𝑛+2 )

+ 𝛽d(𝑥𝑛+1 , 𝑥𝑛+2 )

⟹

d(𝑥𝑛 , 𝑥𝑛+1 ) ≥ 𝛼

+ 𝛽d(𝑥𝑛+1 , 𝑥𝑛+2 )

⟹

d(𝑥𝑛 , 𝑥𝑛+1 ) ≥ 𝛼d(𝑥𝑛 , 𝑥𝑛+1 ) + 𝛽d(𝑥𝑛+1 , 𝑥𝑛+2 )

⟹ (1 − 𝛼)d(𝑥𝑛 , 𝑥𝑛+1 ) ≥ 𝛽d(𝑥𝑛+1 , 𝑥𝑛+2 ) The last inequality gives d(𝑥𝑛+1 , 𝑥𝑛+2 ) ≤

(3.2)

1−𝛼 𝛽

d(𝑥𝑛 , 𝑥𝑛+1 )

= 𝜆 d(𝑥𝑛 , 𝑥𝑛+1 ) for all 𝑡 > 0, where 𝜆 =

1−𝛼 𝛽

1

< 𝑠 . Hence by induction, we obtain

d(𝑥𝑛+1 , 𝑥𝑛+2 ) ≤ 𝜆𝑛+1 d(𝑥0 , 𝑥1 )

(3.3)

By Lemma 2.6, {𝑥𝑛 }𝑛∈ℕ is a Cauchy sequence in 𝑋. But 𝑋 is a complete b-metric space; hence,{𝑥𝑛 }𝑛∈ℕ is converges. Call the limit 𝑥 ∗ ∈ 𝑋. Then, 𝑥𝑛 → 𝑥 ∗ as 𝑛 → +∞. By continuity of 𝑇 we have 𝑇𝑥 ∗ = 𝑇(𝑙𝑖𝑚𝑛→∞ 𝑥𝑛 ) = 𝑙𝑖𝑚n→∞ 𝑇𝑥𝑛 = 𝑙𝑖𝑚𝑛→∞ 𝑥𝑛−1 = 𝑥 ∗ That is, 𝑇𝑥 ∗ = 𝑥 ∗ ; thus, 𝑇 has a fixed point in 𝑋. Uniqueness Let 𝑦 ∗ be another fixed point of 𝑇 in 𝑋, then 𝑇𝑦 ∗ = 𝑦 ∗ and 𝑇𝑥 ∗ = 𝑥 ∗ . Now, d(𝑇𝑥 ∗ , 𝑇𝑦 ∗ ) ≥ 𝛼

d(𝑥 ∗ ,𝑇𝑥 ∗ )[1+𝒫d(𝑦 ∗ ,𝑇𝑦 ∗ )] 1+d(𝑥 ∗ ,𝑦 ∗ )

+ 𝛽d(𝑥 ∗ , 𝑦 ∗ )



This implies that d(𝑥 ∗ , 𝑦 ∗ ) ≥ 𝛽d(𝑥 ∗ , 𝑦 ∗ ) 1

⟹ d(𝑥 ∗ , 𝑦 ∗ ) ≤ 𝛾 d(𝑥 ∗ , 𝑦 ∗ ) This is true only when d(𝑥 ∗ , 𝑦 ∗ ) = 0. So 𝑥 ∗ = 𝑦 ∗ . Hence 𝑇 has a unique fixed point in𝑋. Next we prove Theorem 3.1 for surjective mapping. Theorem 3.2 Let (𝑋, 𝑑, 𝑠) be a b-metric space with the coefficient 𝑠 ≥ 1 and 𝑇 a surjective mapping satisfying the condition (3.1) for all 𝑥, 𝑦 ∈ 𝑋, 𝑥 ≠ 𝑦, where 𝛼, 𝛽 ≥ 0 are real constants and 𝑠𝛼 + 𝛽 > 𝑠, 𝛽 > 1. Then, 𝑇 has a fixed point in 𝑋. Proof Choose 𝑥0 ∈ 𝑋 to be arbitrary, and define the iterative sequence {𝑥𝑛 }𝑛∈ℕ as follows: 𝑇𝑥𝑛 = 𝑥𝑛−1 for 𝑛 = 1,2,3, …. Then, using (3.1), we obtain, sequence {𝑥𝑛 }𝑛∈ℕ is a Cauchy sequence in 𝑋. But 𝑋 is a complete metric space; hence {𝑥𝑛 }𝑛∈ℕ is converges. Call the limit 𝑥 ∗ ∈ 𝑋. Then, 𝑥𝑛 → 𝑥 ∗ as 𝑛 → +∞. Existence of Fixed Point Since 𝑇 is a Surjective map, so there exists a point 𝑦 in 𝑋, such that 𝑥 = 𝑇𝑦. Consider (3.4)

d(𝑥𝑛 , 𝑥) = d(𝑇𝑥𝑛+1 , 𝑇𝑦) ≥𝛼

d(𝑥𝑛+1 ,𝑇𝑥𝑛+1 )[1+d(𝑦,𝑇𝑦)] 1+d(𝑥𝑛+1 ,𝑦)

+ 𝛽d(𝑥𝑛+1 , 𝑦).

Taking 𝑛 → +∞, we get (3.5)

d(𝑥, 𝑥) ≥ 𝛼

d(𝑥,𝑥)[1+d(𝑦,𝑥)] 1+d(𝑥,𝑦)

+ 𝛽d(𝑥, 𝑦)

⟹ 0 ≥ 𝛽d(𝑦, 𝑥) ⟹ d(𝑥, 𝑦) = 0 as 𝛽 > 1. Hence 𝑥 = 𝑦 and so 𝑇𝑥 = 𝑥, that is, 𝑥 is a fixed point of 𝑇.



Uniqueness: Uniqueness follows from Theorem 3.1. Theorem 3.3 Let (𝑋, 𝑑, 𝑠) be a b-metric space with the coefficient 𝑠 ≥ 1 and 𝑇 a continuous mapping satisfying the following condition: (3.6)

d(𝑇𝑥, 𝑇𝑦) + 𝛼 𝑚𝑎𝑥{d(𝑥, 𝑇𝑦), d(𝑦, 𝑇𝑥)} ≥𝛽

d(𝑥,𝑇𝑥)[1+d(𝑦,𝑇𝑦)] 1+d(𝑥,𝑦)

+ 𝛾d(𝑥, 𝑦)

for all 𝑥, 𝑦 ∈ 𝑋, 𝑥 ≠ 𝑦, where 𝛼, 𝛽, 𝛾 ≥ 0 are real constants and 𝑠𝛽 + 𝛾 > (1 + 𝛼)𝑠 + 𝑠 2 𝛼, 𝛾 > 1 + 𝛼. Then 𝑇 has a fixed point in 𝑋. Proof Choose 𝑥0 ∈ 𝑋 be arbitrary, to define the iterative sequence {𝑥𝑛 }𝑛∈ℕ as follows: 𝑇𝑥𝑛 = 𝑥𝑛−1 for 𝑛 = 1,2,3, …. Taking 𝑥 = 𝑥𝑛+1 and 𝑦 = 𝑥𝑛+2 in (3.6), we obtain d(𝑇𝑥𝑛+1 , 𝑇𝑥𝑛+2 ) + 𝛼 𝑚𝑎𝑥{d(𝑥𝑛+1 , 𝑇𝑥𝑛+2 ), d(𝑥𝑛+2 , 𝑇𝑥𝑛+1 )} ≥𝛽

d(𝑥𝑛+1 ,𝑇𝑥𝑛+1 )[1+d(𝑥𝑛+2 ,𝑇𝑥𝑛+2 )] 1+d(𝑥𝑛+1 ,𝑥𝑛+2 )

+ 𝛾d(𝑥𝑛+1 , 𝑥𝑛+2 )

⟹ d(𝑥𝑛 , 𝑥𝑛+1 ) + 𝛼 𝑚𝑎𝑥{d(𝑥𝑛+1 , 𝑥𝑛+1 ), d(𝑥𝑛+2 , 𝑥𝑛 )} ≥𝛽

d(𝑥𝑛+1 ,𝑥𝑛 )[1+d(𝑥𝑛+2 ,𝑥𝑛+1 )] + 1+d(𝑥𝑛+1 ,𝑥𝑛+2 )

𝛾d(𝑥𝑛+1 , 𝑥𝑛+2 )

⟹ d(𝑥𝑛 , 𝑥𝑛+1 ) + 𝛼 𝑚𝑎𝑥{d(𝑥𝑛+1 , 𝑥𝑛+1 ), d(𝑥𝑛+2 , 𝑥𝑛 )} ≥𝛽

d(𝑥𝑛+1 ,𝑥𝑛 )[1+d(𝑥𝑛+1 ,𝑥𝑛+2 )] + 1+d(𝑥𝑛+1 ,𝑥𝑛+2 )

𝛾d(𝑥𝑛+1 , 𝑥𝑛+2 )

⟹ d(𝑥𝑛 , 𝑥𝑛+1 ) + 𝛼d(𝑥𝑛 , 𝑥𝑛+2 ) ≥ 𝛽d(𝑥𝑛 , 𝑥𝑛+1 ) + 𝛾d(𝑥𝑛+1 , 𝑥𝑛+2 ) ⟹ d(𝑥𝑛 , 𝑥𝑛+1 ) + 𝛼𝑠[d(𝑥𝑛 , 𝑥𝑛+1 ) + d(𝑥𝑛+1 , 𝑥𝑛+2 )] ≥ 𝛽d(𝑥𝑛 , 𝑥𝑛+1 ) + 𝛾d(𝑥𝑛+1 , 𝑥𝑛+2 ) ⟹ (1 + 𝑠𝛼 − 𝛽)d(𝑥𝑛 , 𝑥𝑛+1 ) ≥ (𝛾 − 𝑠𝛼)d(𝑥𝑛+1 , 𝑥𝑛+2 ) The last inequality gives (3.7)

d(𝑥𝑛+1 , 𝑥𝑛+2 ) ≤

1+𝑠𝛼−𝛽 𝛾−𝑠𝛼

d(𝑥𝑛 , 𝑥𝑛+1 )



= 𝜆 d(𝑥𝑛 , 𝑥𝑛+1 ) where 𝜆 =

1+𝑠𝛼−𝛽 𝛾−𝑠𝛼

1

< 𝑠 . Hence by induction, we obtain d(𝑥𝑛+1 , 𝑥𝑛+2 ) ≤ 𝜆𝑛+1 d(𝑥0 , 𝑥1 )

(3.8)

By Lemma 2.6, {𝑥𝑛 }𝑛∈ℕ is a Cauchy sequence in 𝑋. But 𝑋 is a complete b-metric space; hence,{𝑥𝑛 }𝑛∈ℕ is converges. Call the limit 𝑥 ∗ ∈ 𝑋. Then, 𝑥𝑛 → 𝑥 ∗ as 𝑛 → +∞. By continuity of 𝑇 we have 𝑇𝑥 ∗ = 𝑇(𝑙𝑖𝑚𝑛→∞ 𝑥𝑛 ) = 𝑙𝑖𝑚n→∞ 𝑇𝑥𝑛 = 𝑙𝑖𝑚𝑛→∞ 𝑥𝑛−1 = 𝑥 ∗ That is, 𝑇𝑥 ∗ = 𝑥 ∗ ; thus, 𝑇 has a fixed point in 𝑋. Uniqueness Let 𝑦 ∗ be another fixed point of 𝑇 in 𝑋, then 𝑇𝑦 ∗ = 𝑦 ∗ and 𝑇𝑥 ∗ = 𝑥 ∗ . Now, d(𝑇𝑥 ∗ , 𝑇𝑦 ∗ ) + 𝛼 𝑚𝑎𝑥{d(𝑥 ∗ , 𝑇𝑦 ∗ ), d(𝑦 ∗ , 𝑇𝑥 ∗ )} ≥𝛽

d(𝑥 ∗ ,𝑇𝑥 ∗ )[1+𝒫d(𝑦 ∗ ,𝑇𝑦 ∗ )] 1+d(𝑥 ∗ ,𝑦 ∗ )

+ 𝛾d(𝑥 ∗ , 𝑦 ∗ )

This implies that d(𝑥 ∗ , 𝑦 ∗ ) + 𝛼d(𝑥 ∗ , 𝑦 ∗ ) ≥ 𝛾d(𝑥 ∗ , 𝑦 ∗ ) That is (3.9)

d(𝑥 ∗ , 𝑦 ∗ ) ≥ (𝛾 − 𝛼)d(𝑥 ∗ , 𝑦 ∗ ) 1

⟹ d(𝑥 ∗ , 𝑦 ∗ ) ≤ 𝛾−𝛼 d(𝑥 ∗ , 𝑦 ∗ ) This is true only when d(𝑥 ∗ , 𝑦 ∗ ) = 0. So 𝑥 ∗ = 𝑦 ∗ . Hence 𝑇 has a unique fixed point in𝑋. Now we prove Theorem 3.3 for surjective mapping. Theorem 3.4 Let (𝑋, 𝑑, 𝑠) be a b-metric space with the coefficient 𝑠 ≥ 1 and 𝑇 a surjective mapping satisfying the condition (3.6) for all 𝑥, 𝑦 ∈ 𝑋, 𝑥 ≠ 𝑦, where 𝛼, 𝛽, 𝛾 ≥ 0 are real constants and 𝑠𝛽 + 𝛾 > (1 + 𝛼)𝑠 + 𝑠 2 𝛼, 𝛾 > 1 + 𝛼. Then, 𝑇 has a fixed point in 𝑋. Journal homepage: http://theojal.com/ojatm/ 71


Proof Choose 𝑥0 ∈ 𝑋 to be arbitrary, and define the iterative sequence {𝑥𝑛 }𝑛∈ℕ as follows: 𝑇𝑥𝑛 = 𝑥𝑛−1 for 𝑛 = 1,2,3, …. Then, using (3.6), we obtain, sequence {𝑥𝑛 }𝑛∈ℕ is a Cauchy sequence in 𝑋. But 𝑋 is a complete metric space; hence {𝑥𝑛 }𝑛∈ℕ is converges. Call the limit 𝑥 ∗ ∈ 𝑋. Then, 𝑥𝑛 → 𝑥 ∗ as 𝑛 → +∞. Existence of Fixed Point Since 𝑇 is a Surjective map, so there exists a point 𝑦 in 𝑋, such that 𝑥 = 𝑇𝑦. Consider d(𝑥𝑛 , 𝑥) = d(𝑇𝑥𝑛+1 , 𝑇𝑦)

(3.10)

≥ −𝛼 𝑚𝑎𝑥{d(𝑥𝑛+1 , 𝑇𝑦), d(𝑦, 𝑇𝑥𝑛+1 )} +𝛽

d(𝑥𝑛+1 ,𝑇𝑥𝑛+1 )[1+d(𝑦,𝑇𝑦)] 1+d(𝑥𝑛+1 ,𝑦)

+ 𝛾d(𝑥𝑛+1 , 𝑦).

Taking 𝑛 → +∞, we get (3.11)

d(𝑥, 𝑥) ≥ −𝛼 𝑚𝑎𝑥{d(𝑥, 𝑥), d(𝑦, 𝑥)} + 𝛽

d(𝑥,𝑥)[1+d(𝑦,𝑥)] 1+d(𝑥,𝑦)

+ 𝛾d(𝑥, 𝑦)

⟹ 0 ≥ −𝛼d(𝑥, 𝑦) + 𝛾d(𝑦, 𝑥) ⟹ (𝛾 − 𝛼)d(𝑥, 𝑦) ≤ 0 ⟹ d(𝑥, 𝑦) = 0 as 𝛾 > 𝛼. Hence 𝑥 = 𝑦 and so 𝑇𝑥 = 𝑥, that is, 𝑥 is a fixed point of 𝑇. Uniqueness: Uniqueness follows from Theorem 3.3. The proof is completed. Remark 3.5 1. If we take 𝑠 = 1, 𝛼 = 0, 𝛽 = 𝜆 in Theorem 3.1 and Theorem 3.2 then we get main results of Wang et al. [18]. 2. If we take 𝛼 = 0, 𝛽 = 𝜆 in Theorem 3.1 and Theorem 3.2, then we get Theorem of Jain et al. [9].



3. If we take 𝛼 = 0, 𝛽 = 𝛼, 𝛾 = 𝛽 in Theorem 3.3 and Theorem 3.4, we get Theorem 3.1 and Theorem 3.2 of this article. 4. If we take 𝛼 = 𝛽 = 0, 𝛾 = 𝜆 in Theorem 3.3 and Theorem 3.4, then we get Theorem of Jain et al. [9]. 5. If we take 𝑠 = 1, 𝛼 = 𝛽 = 0, 𝛾 = 𝜆 in Theorem 3.3 and Theorem 3.4, then we get main results of Wang et al. [18]. 6. If we take 𝑠 = 1 and (𝑋, 𝑑) is complete dislocated quasi-metric space in Theorem 3.3 and Theorem 3.4 then we get Theorem 1 and Theorem 2 of Daheriya et al. [7]. 7. If we take 𝑠 = 1 and (𝑋, 𝑑) is complete dislocated quasi-metric space in Theorem 3.1and Theorem 3.2 then we get Theorem 1 and Theorem 2 of Pagey et al. [15]. 8. If we take 𝑠 = 1, 𝛼 = 0, 𝛽 = 𝜆 and (𝑋, 𝑑) is complete partial-metric space in Theorem 3.1 and Theorem 3.2 then we get Corollary 2.1 of Huang et al. [19]. 9. If we take 𝛼 = 0, 𝛽 = 𝜆 and (𝑋, 𝑑) is complete cone metric space in Theorem 3.1 and Theorem 3.2 then we get Corollary 2.4 of Mahanta et al. [12].

4. COMMON FIXED POINT THEOREMS In this section, we prove some fixed point theorem for continuous and surjective mapping satisfying expansion condition, which is generalization of Theorem 4.1 of [21] in the setting of complete b-metric space. Theorem 4.1 Let (𝑋, 𝑑, 𝑠) be a b-metric space with the coefficient 𝑠 ≥ 1 and Let 𝑓 and 𝑔 be surjective self-mappings of 𝑋 satisfying the following inequalities: (4.1)

𝑑(𝑔𝑓𝑥, 𝑓𝑥) ≥ 𝑎𝑑(𝑓𝑥, 𝑥)

(4.2)

𝑑(𝑓𝑔𝑥, 𝑔𝑥) ≥ 𝑏𝑑(𝑔𝑥, 𝑥)

for all 𝑥 ∈ 𝑋, where 𝑎, 𝑏 ≥ 𝑠. If either 𝑓 or 𝑔 is continuous, then 𝑓 and 𝑔 have a common fixed point.



Proof. Let 𝑥0 be an arbitrary point in 𝑋. Since 𝑓 and 𝑔 be surjective self-mappings, there exist points 𝑥1 ∈ 𝑓 −1 (𝑥0 )

and 𝑥2 ∈ 𝑔−1 (𝑥1 ).

Continuing in this way, we obtain the

sequence {𝑥𝑛 }𝑛∈ℕ with 𝑥2𝑛+1 ∈ 𝑓 −1 (𝑥2𝑛 ) and 𝑥2𝑛+2 ∈ 𝑔−1 (𝑥2𝑛 ). Note that if 𝑥𝑛 = 𝑥𝑛+1 for some 𝑛, then 𝑥𝑛 is a fixed point of 𝑓 and 𝑔. Indeed, if 𝑥2𝑛 = 𝑥2𝑛+1 for some 𝑛 ≥ 0, then 𝑥2𝑛 is a fixed point of 𝑓. On the other hand, we have from using inequality (4.2) that 0 = 𝑑(𝑥2𝑛 , 𝑥2𝑛+1 ) = 𝑑(𝑓𝑥2𝑛+1 , 𝑔𝑥2𝑛+2 ) = 𝑑(𝑓𝑔𝑥2𝑛+2 , 𝑔𝑥2𝑛+2 ) ≥ 𝑏𝑑(𝑔𝑥2𝑛+2 , 𝑥2𝑛+2 ) = 𝑏𝑑(𝑥2𝑛+1 , 𝑥2𝑛+2 ) This is true only when 𝑑(𝑥2𝑛+1 , 𝑥2𝑛+2 ) = 0 and so 𝑥2𝑛+1 = 𝑥2𝑛+2. Thus 𝑥2𝑛 is a common fixed point of 𝑓 and 𝑔. If 𝑥2𝑛+1 = 𝑥2𝑛+2 for some 𝑛 ≥ 0, similarly, by using inequality (4.1) leads to 𝑥2𝑛+1 is a common fixed point of 𝑓 and 𝑔. Now suppose that 𝑥𝑛 ≠ 𝑥𝑛+1 for all 𝑛. Using inequality (4.1), we have 𝑑(𝑥2𝑛+1 , 𝑥2𝑛+2 ) = 𝑑(𝑔𝑥2𝑛+2 , 𝑓𝑥2𝑛+3 ) = 𝑑(𝑔𝑓𝑥2𝑛+3 , 𝑓𝑥2𝑛+3 ) ≥ 𝑎𝑑(𝑓𝑥2𝑛+3 , 𝑥2𝑛+3 ) = 𝑎𝑑(𝑥2𝑛+2 , 𝑥2𝑛+3 ) Therefore, (4.3)

𝑑(𝑥2𝑛+1 , 𝑥2𝑛+2 ) ≥ 𝑎𝑑(𝑥2𝑛+2 , 𝑥2𝑛+3 )

Similarly, using inequality (4.2), we have 𝑑(𝑥2𝑛 , 𝑥2𝑛+1 ) = 𝑑(𝑓𝑥2𝑛+1 , 𝑔𝑥2𝑛+2 ) = 𝑑(𝑓𝑔𝑥2𝑛+2 , 𝑔𝑥2𝑛+2 ) ≥ 𝑏𝑑(𝑔𝑥2𝑛+2 , 𝑥2𝑛+2 ) = 𝑏𝑑(𝑥2𝑛+1 , 𝑥2𝑛+2 ) Thus, (4.4)

𝑑(𝑥2𝑛 , 𝑥2𝑛+1 ) ≥ 𝑏𝑑(𝑥2𝑛+1 , 𝑥2𝑛+2 )

Let 𝑘 = 𝑚𝑖𝑛{𝑎, 𝑏} > 𝑠. Then from inequalities (4.3) and (4.4), we have 𝑑(𝑥2𝑛+2 , 𝑥2𝑛+3 ) ≤ 𝜆𝑑(𝑥2𝑛+1 , 𝑥2𝑛+2 ) Journal homepage: http://theojal.com/ojatm/ 74


and

𝑑(𝑥2𝑛+1 , 𝑥2𝑛+2 ) ≤ 𝜆𝑑(𝑥2𝑛 , 𝑥2𝑛+1 )

where 𝜆 = 𝑟 −1. Thus we obtain 𝑑(𝑥𝑛+1 , 𝑥𝑛+2 ) ≤ 𝜆𝑑(𝑥𝑛 , 𝑥𝑛+1 ) for all 𝑛 = 1,2,3, . .. and it follows that (4.5)

𝑑(𝑥𝑛 , 𝑥𝑛+1 ) ≤ 𝜆𝑛 𝑑(𝑥0 , 𝑥1 )

By Lemma 2.6, {𝑥𝑛 }𝑛∈ℕ is a Cauchy sequence in 𝑋. But 𝑋 is a complete b-metric space; hence,{𝑥𝑛 }𝑛∈ℕ is converges. Call the limit 𝑥 ∗ ∈ 𝑋. Then, 𝑥𝑛 → 𝑥 ∗ as 𝑛 → +∞. Now we consider that 𝑓 is continuous. Since 𝑥2𝑛 = 𝑓𝑥2𝑛+1, it follows that 𝑥 ∗ = 𝑙𝑖𝑚𝑛→∞ 𝑥2𝑛 = 𝑙𝑖𝑚n→∞ 𝑓𝑥2𝑛+1 = 𝑓(𝑙𝑖𝑚n→∞ 𝑥2𝑛+1 ) = 𝑓𝑥 ∗ and so 𝑥 ∗ is a fixed point of 𝑓. Since 𝑔 is surjective, there exists 𝑥 such that 𝑔𝑥 = 𝑥 ∗ . Thus, using inequality (4.2), we have 0 = 𝑑(𝑓𝑥 ∗ , 𝑔𝑥) = 𝑑(𝑓𝑔𝑥, 𝑔𝑥) ≥ 𝑏𝑑(𝑔𝑥, 𝑥) = 𝑏𝑑(𝑥 ∗ , 𝑥) This is true only when 𝑑(𝑥 ∗ , 𝑥) = 0 and so 𝑥 ∗ = 𝑥. Thus 𝑥 ∗ = 𝑔𝑥 ∗ = 𝑓𝑥 ∗. We have therefore proved that 𝑥 ∗ is a common fixed point of 𝑓 and 𝑔. Similarly, considering the continuity of 𝑔, it can be seen that 𝑓 and 𝑔 have a common fixed point and this completes the proof. Putting 𝑓 = 𝑔 and 𝜆 = 𝑚𝑖𝑛{𝑎, 𝑏} in Theorem 4.1, we have following: Corollary 4.2 Let (𝑋, 𝑑, 𝑠) be a b-metric space with the coefficient 𝑠 ≥ 1 and Let 𝑓 be a surjective self-mapping of 𝑋 satisfying the following inequality: (4.6)

𝑑(𝑓 2 𝑥, 𝑓𝑥) ≥ 𝜆𝑑(𝑓𝑥, 𝑥)

for all 𝑥 ∈ 𝑋, where 𝜆 > 𝑠. If 𝑓 is continuous, then 𝑓 has a fixed point. Taking 𝑠 = 1 in Corollary 4.1, we have the following results: Corollary 4.3 ([18], Theorem 4) Let (𝑋, d) be a complete metric space and Let 𝑓 be a surjective self-mapping of 𝑋 satisfying the following inequality: (4.7)

𝑑(𝑓 2 𝑥, 𝑓𝑥) ≥ 𝜆𝑑(𝑓𝑥, 𝑥) Journal homepage: http://theojal.com/ojatm/ 75


for all 𝑥 ∈ 𝑋, where 𝜆 > 1. If 𝑓 is continuous, then 𝑓 has a fixed point. Putting 𝑠 = 1 in Theorem 4.1, we have following: Corollary 4.4 Let (𝑋, d) be a complete metric space and Let 𝑓 and 𝑔 be surjective selfmappings of 𝑋 satisfying the following inequalities: (4.8)

𝑑(𝑔𝑓𝑥, 𝑓𝑥) ≥ 𝑎𝑑(𝑓𝑥, 𝑥)

(4.9)

𝑑(𝑓𝑔𝑥, 𝑔𝑥) ≥ 𝑏𝑑(𝑔𝑥, 𝑥)

for all 𝑥 ∈ 𝑋, where 𝑎, 𝑏 ≥ 1. If either 𝑓 or 𝑔 is continuous, then 𝑓 and 𝑔 have a common fixed point. Example 4.5 Let 𝑋 = [0, +∞) and let 𝑑(𝑥, 𝑦) = |𝑥 − 𝑦|2 , ∀ 𝑥, 𝑦 ∈ 𝑋. It is obvious that 𝑑 is a b-metric on 𝑋 with 𝑠 = 2 > 1 and (𝑋, 𝑑) is complete. Also, 𝑑 is not a metric on 𝑋. Define the surjective self-mappings 𝑓, 𝑔: 𝑋 → 𝑋 by 𝑓𝑥 = 2𝑥 and 𝑔𝑥 = 4𝑥, ∀ 𝑥 ∈ 𝑋. Then we have 𝑑(𝑓𝑔𝑥, 𝑔𝑥) = |8𝑥 − 4𝑥|2 = 16|𝑥|2 ≥ 15|2𝑥 − 𝑥|2 = 15 𝑑(𝑓𝑥, 𝑥) and 𝑑(𝑔𝑓𝑥, 𝑓𝑥) = |8𝑥 − 2𝑥|2 = 36|𝑥|2 ≥ 11|4𝑥 − 𝑥|2 = 11 𝑑(𝑔𝑥, 𝑥) The inequalities (4.1) and (4.2) are satisfied for all 𝑥, 𝑦 ∈ 𝑋 with 𝑎 = 15, 𝑏 = 11 > 𝑠 = 2. The conditions of Theorem are satisfied and 𝑓 and 𝑔 have a common fixed point x ⋆ = 0 ∈ X. CONFLICT OF INTEREST No conflict of interest was declared by the authors. AUTHOR’S CONTRIBUTIONS All authors contributed equally and significantly to writing this paper. All authors read and approved the final manuscript. REFERENCES [1] Aage, CT, Salunke, JN: Some fixed point theorems for expansion onto mappings on cone metric spaces. Acta Math. Sin. Engl. Ser. 27(6), 1101-1106 (2011). Journal homepage: http://theojal.com/ojatm/ 76


[2] Alghamdi, M. A., Hussain, N., Salimi, P., Fixed point and coupled fixed point theorems on b-metric-like spaces, J. Inequal. Appl., 2013 (2013), 25 pages. [3] Bakhtin, I. A., The contraction mapping principle in almost metric spaces, Funct. Anal.,Gos. Ped. Inst. Unianowsk, 30(1989), 26.37. [4] Czerwik, S., Contraction mappings in b-metric spaces, Acta Math. Inf. Univ. Ostravensis, 1 (1993), 5-11. [5] Czerwik, S., Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 263-276. [6] Daffer, P. Z. , Kaneko, H., On expansive mappings, Math. Japonica. 37 (1992), 733735. [7] Daheriya, R. D., Jain, R., Ughade, M., “Some Fixed Point Theorem for Expansive Type Mapping in Dislocated Metric Space”, ISRN Mathematical Analysis, Volume 2012, Article ID 376832, 5 pages, doi:10.5402/2012/376832. [8] Daheriya, R. D., Jain, R., Ughade, M., Unified fixed point theorems for non-surjective expansion mappings by using class 𝛹 on 2- metric spaces, Bessel J. Math., 3(3), (2013), 271-291. [9] Jain, R., Daheriya, R. D., Ughade, M., Fixed Point, Coincidence Point and Common Fixed Point Theorems under Various Expansive Conditions in b-Metric Spaces, International Journal of Scientific and Innovative Mathematical Research, vol.3, no.9, pp. 26-34, 2015. [10] Jain, R., Daheriya, R. D., Ughade, M., Fixed Point, Coincidence Point and Common Fixed Point Theorems under Various Expansive Conditions in Parametric Metric Spaces and Parametric b-Metric Spaces, Gazi University Journal of Science, Accepted, 2015. [11] Marta Demma and Pasquale Vetro, Picard Sequence and Fixed Point Results on 𝑏Metric Spaces, Journal of Function Spaces,Volume 2015, Article ID 189861, 6 pages http://dx.doi.org /10.1155/2015/189861. C3-8 [12] Mohanta, S.K., Maitra, R., Coincidence Points And Common Fixed Points For Expansive Type Mappings In Cone b-Metric Spaces, Applied Mathematics E-Notes, 14(2014), 200-208. [13] Muraliraj, A., Hussain, R.J., Coincidence and Fixed Point Theorems for Expansive Maps in d−Metric Spaces, Int. Journal of Math. Analysis, Vol. 7, 2013, no. 44, 2171 – 2179. [14] Mustafa, Z., Awawdeh, F., Shatanawi, W., Fixed Point Theorem for Expansive Mappings in G-Metric Spaces, Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 50, 2463-2472. Journal homepage: http://theojal.com/ojatm/ 77


[15] Pagey, S.S., Nighojkar, R., A fixed point theorem for expansive type mapping in dislocated Quasi-metric space, International Journal of Theoretical and Applied Science, 3(1):26-27(2011). [16] Saluja, A. S., Dhakde, A. K., Magarde, D., Some fixed point theorems for expansive type mapping in dislocated metric space, Mathematical Theorey and Modeling, 3(2013). [17] Shatanawi, W., Awawdeh, F., Some fixed and coincidence point theorems for expansive maps in cone metric spaces, Fixed Point Theory and Applications 2012, 2012:19. [18] Wang, S. Z., Li, B. Y., Gao, Z. M. , Iseki, K., Some fixed point theorems for expansion mappings, Math. Japonica. 29 (1984), 631-636. [19] Xianjiu Huang, Chuanxi Zhu, Xi Wen, Fixed point theorems for expanding mappings in partial metric spaces, An. St. Univ. Ovidius Constant Vol. 20(1), 2012, 213-224. [20] Yan Han and Shaoyuan Xu, Some new theorems of expanding mappings without continuity in cone metric spaces, Fixed Point Theory and Applications, 2013, 2013:3 [21] Ilker Sahin and Mustafa Telcia, Theorem on Common Fixed Points of Expansion Type Mappings in Cone Metric Spaces. An. S¸T. Univ. Ovidius Constanta, Vol. 18(1), 2010, 329–336.
