SQU Journal for Science, 2017, 22(1), 48-52                                   DOI: http://dx.doi.org/10.24200/squjs.vol22iss1pp48-52 

2017 Sultan Qaboos University  

48 

 

Theorems on Fixed Points for 
Asymptotically Regular Sequences and 
Maps in 𝒃-Metric Space 

Mohammad S. Khan1* and Pankaj K. Jhade2 

1Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, PC 123,          
Al-Khoud, Muscat, Sultanate of Oman; 2Department of applied Science, Sagar Institute of 
Science, Technology and Engineering, Bhopal, India. *Email: mohammad@squ.edu.om. 
 

ABSTRACT: In this paper, we present some fixed point theorems for asymptotically regular sequences and 

asymptotically regular maps in complete 𝑏- metric spaces. Our results extend and generalize the well-known fixed 
point theorems of Hardy- Roger [1] and Reich [2].  

 

Keywords: b-Metric space; Fixed point; Asymptotically regular sequence; Asymptotically regular maps.      

 سقاط في فضاء متريإنقاط ثابتة لمتتالية عادية متقاربة وحول نظريات  

 ادهبونكاج كومار جو محمد سعيد خان 

تعتبر نتائجنا . بعض نظريات النقطة الثابتة لمتتاليات عادية متقاربة واسقاط عادي متقارب في فضاءات مترية متكاملةنقدم في هذه الورقة  ملخص:ال

 [.1]وريخ [ 2]روجر -توسيع وتعميم لنظريات النقطة الثابتة المعروفة لكل من هاردي 

 

 .سقاط عادي متقاربإالفضاء المتري، النقطة الثابتة، متتالية عادية متقاربة، : مفتاحيةالكلمات ال

1.  Introduction 

etric fixed point theory was born with the well-known Banach contraction principle that was initially 

published in 1922. This principle states that on a complete metric space(𝑋, 𝑑) , a self mapping 𝑇 for which 
𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝑘𝑑(𝑥, 𝑦), for all 𝑥, 𝑦 ∈ 𝑋, 0 < 𝑘 < 1, has a unique fixed point. Several generalizations and 
extensions of this celebrated result have been appeared in the last few decades. The fixed point theorem in metric 

spaces plays a significant role to construct methods to solve the problems in mathematics and sciences. Metric 

fixed point theory is a vast field of study and is capable of solving many equations. To overcome the problem of 

measurable functions with respect to a measure and their convergence, [3] needed an extension of metric space.  

Using this idea, he introduced the concept of 𝑏-metric space and presented the contraction mapping in 𝑏-metric 
spaces that is generalization of the Banach contraction principle in metric spaces [4-7]. After that, several papers 

have dealt with fixed point theory or the variational principle for single-valued and multi-valued operators in 𝑏-
metric spaces [8-14]. In this paper our aim is to show the validity of some fixed point theorems for asymptotically 

regular sequences. We also present results on fixed points of asymptotically regular mappings. 

 

2. Preliminaries: Consistent with [3] and [4,15], we recall some definitions and properties for 𝑏-metric space. 
 

Definition 2.1. Let 𝑀 be a nonempty set and the mapping 𝜌: 𝑀 × 𝑀 → ℝ+   (ℝ+ stands for nonnegative reals) 
satisfies: 

(i) 𝜌(𝑥, 𝑦) = 0 if and only if  𝑥 = 𝑦 for  all 𝑥, 𝑦 ∈ 𝑀; 
(ii) 𝜌(𝑥, 𝑦) = 𝜌(𝑦, 𝑥) for all 𝑥, 𝑦 ∈ 𝑀; 
(iii) there exists a real number 𝑠 ≥ 1 such that  𝜌(𝑥, 𝑦) ≤ 𝑠[𝜌(𝑥, 𝑧) + 𝜌(𝑧, 𝑦)] for all 𝑥, 𝑦, 𝑧 ∈ 𝑀. 
Then 𝜌 is called a b-metric on 𝑀 and the pair (𝑀, 𝜌) is called a 𝑏- metric space with coefficient 𝑠. 
 

Remark 2.1. The class of 𝑏-metric spaces is larger than the class of metric spaces since any metric space is a 𝑏-

metric space 𝑠 = 1. Therefore, it is obvious that 𝑏-metric spaces generalizes metric spaces. We present an 
example which shows that introducing a 𝑏-metric space instead of a metric space is meaningful since there exists 
𝑏-metric space instead of a metric space which are not metric spaces. 

M 

mailto:mohammad@squ.edu.om


THEOREMS ON FIXED POINTS FOR ASYMPTOTICALLY REGULAR SEQUENCE  

 

49 

 

Example 2.1. Let 𝑀 = [0, ∞) and 𝜌: 𝑀 × 𝑀 → ℝ+ defined by 𝜌(𝑥, 𝑦) = |𝑥 − 𝑦|𝑝, where 𝑝 is a real number such that 
𝑝 > 1. Let 𝑥, 𝑦, 𝑧 ∈ 𝑀. By taking 𝑢 = 𝑥 − 𝑧 and 𝑣 = 𝑧 − 𝑦 we have  

|𝑥 − 𝑦|𝑝 = |𝑢 + 𝑣|𝑝 ≤ (|𝑢| + |𝑣|)𝑝 
                                                                                          ≤ (2max{|𝑢|, |𝑣|})𝑝 
                                                                                          ≤ 2𝑝(|𝑥 − 𝑧|𝑝 + |𝑧 − 𝑦|𝑝), 
which implies that 𝜌(𝑥, 𝑦) ≤ 2𝑝[𝜌(𝑥, 𝑧) + 𝜌(𝑧, 𝑦)]. Therefore (𝑀, 𝜌) is a 𝑏-metric space with coefficient 2𝑝. On the 
other hand, for 𝑥 > 𝑧 > 𝑦, we have 
|𝑥 − 𝑦|𝑝 = |𝑢 + 𝑣|𝑝 = (𝑢 + 𝑣)𝑝 > 𝑢𝑝 + 𝑣 𝑝 = (𝑥 − 𝑧)𝑝 + (𝑧 − 𝑦)𝑝 = |𝑥 − 𝑧|𝑝 + |𝑧 − 𝑦|𝑝, 
which implies that 𝜌(𝑥, 𝑦) > 𝜌(𝑥, 𝑧) + 𝜌(𝑧, 𝑦). Therefore (𝑀, 𝜌) is not a metric space. 
 

Definition 2.2. Let (𝑀, 𝜌) be a 𝑏-metric space. Then {𝑥𝑛} in 𝑀 is called 
1. a Cauchy sequence if and only if for all 𝜖 > 0 there exists 𝑛(𝜖) ∈ ℕ such that for each 𝑛. 𝑚 ≥ 𝑛(𝜖), we have 

𝜌(𝑥𝑛 , 𝑥𝑚 ) < 𝜖. 
2. a convergent sequence if and only if there exists 𝑥 ∈ 𝑀 such that for all 𝜖 > 0 there exists 𝑛(𝜖) ∈ ℕ such that for 

every 𝑛 ≥ 𝑛(𝜖), we have 𝜌(𝑥𝑛 , 𝑥) < 𝜖. 
 

Definition 2.3  If (𝑀, 𝜌) is a 𝑏-metric space then a subset 𝐿 ⊂ 𝑀 is called  
(i) compact if and only if for every sequence of elements of 𝐿 there exists a subsequence that converges to an 

element of 𝐿. 
(ii) closed if and only if for each sequence  {𝑥𝑛 } in 𝐿 which converges to an element 𝑥, we have 𝑥 ∈ 𝐿. 
(iii)  The 𝑏-metric space is complete if every Cauchy sequence in 𝑀 converges in M. 

 

Definition 2.4 [13] Let (𝑀, 𝜌) be a 𝑏-metric space. A sequence {𝑥𝑛} in 𝑀 is said to be asymptotically T-regular if  
lim
𝑛→∞

𝜌( 𝑥𝑛, 𝑇𝑥𝑛 ) = 0. 

 

Example 2.2. Let 𝑀 = [0, ∞) and 𝜌: 𝑋 × 𝑋 → ℝ+  defined by 𝜌(𝑥, 𝑦) = |𝑥 − 𝑦|𝑝, 𝑝 ≥ 1 then clearly (𝑀, 𝜌) is a 𝑏-

metric space with coefficient 2𝑝. Now let 𝑇 be a self map of 𝑀 such that 𝑇𝑥 =
𝑥

2
 and choose a sequence {𝑥𝑛},  𝑥𝑛 ≠ 0 

for any positive integer 𝑛, which converges to zero in metric in 𝑀 . We deduce that   
 

lim
𝑛→∞

𝜌( 𝑥𝑛 , 𝑇𝑥𝑛 ) = lim
𝑛→∞

|𝑥𝑛 − 𝑇𝑥𝑛 |
𝑝  

                                                                                   = lim
𝑛→∞

|𝑥𝑛 −
𝑥𝑛

2
|

𝑝

= lim
𝑛→∞

|
𝑥𝑛

2
|

𝑝

= 0. 

 

Hence {𝑥𝑛 } is an asymptotically T-regular sequence in(𝑀, 𝜌).  
 

Definition 2.5 [17] Let (𝑀, 𝜌) be a 𝑏-metric space. A mappings 𝑇 of  𝑀 into itself is said to be asymptotically regular 
at a point 𝑥 in 𝑀 if  

lim
𝑛→∞

𝜌(𝑇𝑛𝑥, 𝑇𝑛+1𝑥) = 0. 

 

Example 2.3. Let (𝑀, 𝜌) be a 𝑏-metric space as defined in Example 2.2 and let 𝑇: 𝑀 → 𝑀 be such that 𝑇𝑥 =
𝑥

4
 where 

𝑥 ∈ 𝑀. Then we have  
lim
𝑛→∞

𝜌( 𝑇𝑛𝑥, 𝑇 𝑛+1𝑥) = lim
𝑛→∞

|𝑇𝑛𝑥 − 𝑇𝑛+1𝑥|𝑝 

                                                                                   = lim
𝑛→∞

|
𝑥

4𝑛
−

𝑥

4𝑛+1
|

𝑝

 

                                                                                   = lim
𝑛→∞

|
3𝑥

4𝑛+1
|

𝑝

= 0. 

Hence T is an asymptotically regular map at all points in M. 

3. Main Results 

Theorems 3.1. Let (𝑀, 𝜌) be a complete 𝑏-metric space with the coefficient 𝑠 ≥ 1 and 𝑇 be a self mapping of 𝑀 
satisfying the following inequality 

 

𝜌(𝑇𝑥, 𝑇𝑦) ≤ 𝑎1𝜌(𝑥, 𝑇𝑥) + 𝑎2𝜌(𝑦, 𝑇𝑦) + 𝑎3𝜌(𝑥, 𝑇𝑦) + 𝑎4 𝜌(𝑦, 𝑇𝑥) + 𝑎5𝜌(𝑥, 𝑦), 
 

for all 𝑥, 𝑦 ∈ 𝑀, where 𝑎𝑖  ( 𝑖 = 1,2,3,4,5) are non-negative real numbers with 𝑚𝑎𝑥{(𝑎1𝑠 + 𝑎4𝑠
2), (𝑎3𝑠

3 + 𝑎4𝑠
2 +

𝑎5)} < 1 for 𝑠 ≥ 1. If there exists an asymptotically 𝑇-regular sequence in 𝑀, then 𝑇 has a unique fixed point. 
 

Proof. Let {𝑥𝑛 } be an asymptotically 𝑇-regular sequence in 𝑀. Then for  𝑛, 𝑚 ∈ ℕ 𝑤𝑖𝑡ℎ 𝑚 ≥ 𝑛 , we have  
    𝜌(𝑥𝑛 , 𝑥𝑚 ) ≤ 𝑠[𝜌(𝑥𝑛 , 𝑇𝑥𝑛 ) + 𝜌(𝑇𝑥𝑛 , 𝑥𝑚 )]  



MOHAMMAD S. KHAN and PANKAJ K. JHADE 

 

50 

 

                     ≤ 𝑠[𝜌(𝑥𝑛 , 𝑇𝑥𝑛 ) + 𝑠𝜌(𝑇𝑥𝑛 , 𝑇𝑥𝑚 ) + 𝑠𝜌(𝑇𝑥𝑚 , 𝑥𝑚 )] 
                     ≤ 𝑠 𝜌(𝑥𝑛 , 𝑇𝑥𝑛 ) + 𝑠

2𝜌(𝑇𝑥𝑛 , 𝑇𝑥𝑚 ) + 𝑠
2𝜌(𝑇𝑥𝑚 , 𝑥𝑚 ) 

                     ≤ 𝑠 𝜌(𝑥𝑛 , 𝑇𝑥𝑛 ) + 𝑠
2𝜌(𝑇𝑥𝑚 , 𝑥𝑚 ) + 𝑠

2[𝑎1𝜌(𝑥𝑛, 𝑇𝑥𝑛 ) + 𝑎2𝜌(𝑥𝑚 , 𝑇𝑥𝑚 ) 
                          +𝑎3 𝜌(𝑥𝑛, 𝑇𝑥𝑚 )+𝑎4𝜌(𝑥𝑚 , 𝑇𝑥𝑛 ) + 𝑎5𝜌(𝑥𝑛 , 𝑥𝑚 )] 
                     ≤ (𝑠 + 𝑎1𝑠

2 + 𝑎4𝑠
3)𝜌(𝑥𝑛 , 𝑇𝑥𝑛 ) + (𝑠

2 + 𝑎2𝑠
2 + 𝑎3𝑠

3)𝜌(𝑥𝑚 , 𝑇𝑥𝑚 ) 
                         +(𝑎3𝑠

3 + 𝑎4 𝑠
3 + 𝑎5)𝜌(𝑥𝑛 , 𝑥𝑚 ). 

Thus, we get  

                𝜌(𝑥𝑛 , 𝑥𝑚 ) ≤ {
(𝑠+𝑎1𝑠

2+𝑎4𝑠
3)

1−(𝑎3𝑠
3+𝑎4𝑠

3+𝑎5)
}  𝜌(𝑥𝑛 , 𝑇𝑥𝑛 ) + {

(𝑠2+𝑎2𝑠
2+𝑎3𝑠

3)

1−(𝑎3𝑠
3+𝑎4𝑠

3+𝑎5)
} 𝜌(𝑥𝑚 , 𝑇𝑥𝑚 ). 

 

Taking the limit as 𝑛 → ∞, we get  
   lim

𝑛→∞
𝜌( 𝑥𝑛 , 𝑥𝑚 ) = 0,  

 

which shows that {𝑥𝑛 } is a Cauchy sequence. Since 𝑀 is a complete 𝑏-metric space, the sequence {𝑥𝑛 }  converges 
in 𝑀. So let   lim

𝑛→∞
𝑥𝑛 = 𝑧 for some 𝑧 ∈ 𝑀. 

Now we show that 𝑧 is a fixed point of T.  
Consider, 

 𝜌(𝑇𝑧, 𝑧) ≤ 𝑠[𝜌(𝑇𝑧, 𝑇𝑥𝑛 ) + 𝜌(𝑇𝑥𝑛 , 𝑧)] 
                ≤ 𝑠𝜌(𝑇𝑧, 𝑇𝑥𝑛 ) + 𝑠

2𝜌(𝑇𝑥𝑛 , 𝑥𝑛 ) + 𝑠
2𝜌(𝑥𝑛 , 𝑧). 

                  ≤ 𝑠[𝑎1𝜌(𝑧, 𝑇𝑧) + 𝑎2𝜌(𝑥𝑛 , 𝑇𝑥𝑛 ) + 𝑎3𝜌(𝑧, 𝑇𝑥𝑛 ) + 𝑎4𝜌(𝑥𝑛 , 𝑇𝑧) + 𝑎5𝜌(𝑧, 𝑥𝑛 )] 
                     + 𝑠2𝜌(𝑇𝑥𝑛 , 𝑥𝑛) + 𝑠

2𝜌(𝑥𝑛 , 𝑧) 
               ≤ 𝑎1𝑠𝜌(𝑧, 𝑇𝑧) + 𝑎2 𝑠𝜌(𝑥𝑛 , 𝑇𝑥𝑛 ) + 𝑎3𝑠

2𝜌(𝑧, 𝑥𝑛 ) 
                       +𝑎3𝑠

2𝜌(𝑥𝑛 , 𝑇𝑥𝑛 ) + 𝑠
2𝑎4𝜌(𝑥𝑛 , 𝑧) + 𝑠

2𝑎4𝜌(𝑧, 𝑇𝑧) 
                    +𝑎5𝑠𝜌(𝑧, 𝑥𝑛 ) + 𝑠

2𝜌(𝑇𝑥𝑛 , 𝑥𝑛 ) + 𝑠
2𝜌(𝑥𝑛 , 𝑧). 

Therefore,  

(1 − 𝑎1𝑠 − 𝑠
2𝑎4)𝜌(𝑇𝑧, 𝑧) ≤  (𝑠

2 + 𝑎2𝑠 + 𝑎3𝑠
2)𝜌(𝑥𝑛 , 𝑇𝑥𝑛 ) 

                                                 +(𝑠2 + 𝑎3𝑠
2 + 𝑎4𝑠

2 + 𝑎5𝑠)𝜌(𝑥𝑛 , 𝑧),  
which implies that  

𝜌(𝑇𝑧, 𝑧) ≤ {
(𝑠2+𝑎2𝑠+𝑎3𝑠

2)

(1−𝑎1𝑠−𝑠
2𝑎4)

}  𝜌(𝑥𝑛 , 𝑇𝑥𝑛 )  + {
(𝑠2+𝑎3𝑠

2+𝑎4𝑠
2+𝑎5𝑠)

(1−𝑎1𝑠−𝑠
2𝑎4)

} 𝜌(𝑥𝑛 , 𝑧). 

 

Since T is asymptotically T-regular, letting limit 𝑛 → ∞ we get 𝜌(𝑇𝑧, 𝑧) = 0 𝑖. 𝑒. 𝑇𝑧 = 𝑧. Hence z is a fixed point of T. 
Uniqueness:  Let 𝑢 be another fixed point such that 𝑧 ≠ 𝑢. Then, 
 

                                 𝜌(𝑧, 𝑢) = 𝜌(𝑇𝑧, 𝑇𝑢) 
                                              ≤ 𝑎1𝜌(𝑧, 𝑇𝑧) + 𝑎2𝜌(𝑢, 𝑇𝑢) + 𝑎3𝜌(𝑧, 𝑇𝑢) + 𝑎4𝜌(𝑢, 𝑇𝑧) + 𝑎5𝜌(𝑧, 𝑢). 
From the last inequality, we have   (1 − 𝑎3 − 𝑎4 − 𝑎5) 𝜌(𝑧, 𝑢) = 0. Since  𝑎3 + 𝑎4 + 𝑎5 < 1, therefore   𝑧 = 𝑢,. 
Next, we discuss the problem of the existence of a fixed point of an operator without using any contractive condition. 

We shall first consider the situation in a metric space. 

 

Theorem 3.2. Let (𝑀, 𝜌) be a metric space, and  𝑇 be a continuous self-mapping on 𝑀. If there exists an 
asymptotically 𝑇-regular sequence {𝑥𝑛 } such that  lim

𝑛→∞
𝑥𝑛 = 𝑧, then 𝑧 is a fixed point of  𝑇. 

 

Proof. Let us consider the inequality  

          𝜌(𝑇𝑧, 𝑧) ≤ [𝜌(𝑇𝑧, 𝑇𝑥𝑛 ) + 𝜌(𝑇𝑥𝑛 , 𝑧)] 
                        ≤ 𝜌(𝑇𝑧, 𝑇𝑥𝑛 ) + 𝜌(𝑇𝑥𝑛 , 𝑥𝑛) + 𝜌(𝑥𝑛 , 𝑧). 
Taking the limit as 𝑛 → ∞, we obtain   𝜌(𝑇𝑧, 𝑧) ≤ 0 giving thereby  𝑇𝑧 = 𝑧.  
Now, let us see the similar situation in a b-metric space. 

 

Theorem 3.3. Let (𝑀, 𝜌) be a 𝑏-metric space with the coefficient  𝑠 ≥ 1 , and 𝑇 be a self-mapping of  𝑀. If there exists 
an asymptotically 𝑇-regular sequence {𝑥𝑛 } with lim

𝑛→∞
𝑥𝑛 = 𝑧, then 𝑧 is not necessarily a fixed point. 

 

Proof. Using the triangle inequality twice in a b-metric space, we obtain  

          𝜌(𝑇𝑧, 𝑧) ≤ 𝑠[𝜌(𝑇𝑧, 𝑇𝑥𝑛 ) + 𝜌(𝑇𝑥𝑛 , 𝑧)] 
                        ≤ 𝑠 𝜌(𝑇𝑧, 𝑇𝑥𝑛 ) + 𝑠

2𝜌(𝑇𝑥𝑛 , 𝑥𝑛 ) + 𝑠
2𝜌(𝑥𝑛 , 𝑧). 

Taking the limit as 𝑛 → ∞, we obtain 
                                                         (1 − 𝑠2)𝜌(𝑇𝑧, 𝑧) ≤ 0 ,  
which is equivalent to the inequality  

                                                   (𝑠2 − 1)𝜌(𝑇𝑧, 𝑧) ≥ 0,  
which provides no definite information. Thus z may or may not be a fixed point of  T.  



THEOREMS ON FIXED POINTS FOR ASYMPTOTICALLY REGULAR SEQUENCE  

 

51 

 

 

Remarks 3.1.  

1. Even if T is continuous in Theorem 3.3, we cannot get any information from it about the existence of a/ the 
fixed point of  T.  

2. From Theorem 3.2 and Theorem 3.3, we notice yet another difference between the behavior of the two types 
of metrics defined on a set. 

 

Theorem 3.4. Let (𝑀, 𝜌) be a complete 𝑏-metric space with the coefficient 𝑠 ≥ 1 and 𝑇 be a self mapping of 𝑀 
satisfying the inequality 

                       𝜌(𝑇𝑥, 𝑇𝑦) ≤ 𝑎1𝜌(𝑥, 𝑇𝑥) + 𝑎2𝜌(𝑦, 𝑇𝑦) + 𝑎3𝜌(𝑥, 𝑇𝑦) + 𝑎4𝜌(𝑦, 𝑇𝑥) + 𝑎5𝜌(𝑥, 𝑦),          
for all 𝑥, 𝑦 ∈ 𝑀, where 𝑎𝑖  ( 𝑖 = 1,2,3,4,5) are non-negative real numbers with 𝑚𝑎𝑥{(𝑎1𝑠 + 𝑎4𝑠

2), (𝑎3 + 𝑎4 + 𝑎5)} <
1 for 𝑠 ≥ 1. If 𝑇 is asymptotically regular at some point   𝑥0 ∈ 𝑀, then there exists a unique fixed point of 𝑇. 
 

Proof. Let 𝑇 be an asymptotically regular mapping at 𝑥0 ∈ 𝑀. Considering the sequence {𝑇
𝑛 𝑥0}  and   

 𝑚, 𝑛 ∈ ℕ, we have 
      𝜌(𝑇𝑚 𝑥0, 𝑇

𝑛 𝑥0) ≤ 𝑎1𝜌(𝑇
𝑚−1𝑥0, 𝑇

𝑚 𝑥0) + 𝑎2𝜌(𝑇
𝑛−1𝑥0, 𝑇

𝑛𝑥0) + 𝑎3𝜌(𝑇
𝑚−1𝑥0, 𝑇

𝑛𝑥0) 
                                  +𝑎4𝜌(𝑇

𝑛−1𝑥0, 𝑇
𝑚 𝑥0) + 𝑎5𝜌(𝑇

𝑚−1𝑥0, 𝑇
𝑛−1𝑥0) 

                                ≤ 𝑎1𝜌(𝑇
𝑚−1𝑥0, 𝑇

𝑚 𝑥0) + 𝑎2𝜌(𝑇
𝑛−1𝑥0, 𝑇

𝑛𝑥0) 
                                  +𝑎3𝑠𝜌(𝑇

𝑚−1𝑥0, 𝑇
𝑚 𝑥0) + 𝑎3𝑠𝜌(𝑇

𝑚 𝑥0, 𝑇
𝑛𝑥0) 

                                  +𝑎4𝑠𝜌(𝑇
𝑛−1𝑥0, 𝑇

𝑛𝑥0) + +𝑎4𝜌𝑠(𝑇
𝑛 𝑥0, 𝑇

𝑚 𝑥0) 
                                  +𝑎5𝑠𝜌(𝑇

𝑚−1𝑥0, 𝑇
𝑚 𝑥0) + 𝑎5𝑠

2𝜌(𝑇𝑚 𝑥0, 𝑇
𝑛𝑥0) + 𝑎5𝑠

2𝜌(𝑇𝑛𝑥0, 𝑇
𝑛−1𝑥0), 

which implies that 

    𝜌(𝑇𝑚 𝑥0, 𝑇
𝑛𝑥0) ≤ {

𝑎1+𝑎3𝑠+𝑎5𝑠

1−(𝑎3𝑠+𝑎4𝑠+𝑎5𝑠
2)

} 𝜌(𝑇𝑚−1𝑥0, 𝑇
𝑚 𝑥0) + {

𝑎2+𝑎4𝑠+𝑎5𝑠
2

1−(𝑎3𝑠+𝑎4𝑠+𝑎5𝑠
2)

} 𝜌(𝑇𝑛−1𝑥0, 𝑇
𝑛𝑥0) .    

Since 𝑇 is asymptotically regular, as 𝑚, 𝑛 → ∞, the above inequality yields lim
𝑚,𝑛→∞

𝜌(𝑇𝑚 𝑥0, 𝑇
𝑛 𝑥0) = 0 . This shows 

that {𝑇𝑛𝑥0} is a Cauchy sequence. Since 𝑀 is complete,   lim
𝑛→∞

𝑇𝑛 𝑥0  = 0 = 𝑧 for some 𝑧 ∈ 𝑀. 

Next we will show that 𝑧 is a fixed point of  𝑇. 
Consider, 

     𝜌(𝑇𝑧, 𝑧) ≤ 𝑠𝜌(𝑇𝑧, 𝑇 𝑛𝑥0) + 𝑠𝜌(𝑇
𝑛𝑥0, 𝑧) 

                   ≤ 𝑎1𝑠𝜌(𝑧, 𝑇𝑧) + 𝑎2𝑠𝜌(𝑇
𝑛−1𝑥0, 𝑇

𝑛 𝑥0) + 𝑎3𝑠𝜌(𝑧, 𝑇
𝑛𝑥0) 

                      +𝑎4𝑠𝜌(𝑇
𝑛−1𝑥0, 𝑇𝑧) + 𝑎5𝑠𝜌(𝑧, 𝑇

𝑛−1𝑥0) + 𝑠𝜌(𝑇
𝑛𝑥0, 𝑧) 

                   ≤ 𝑎1𝑠𝜌(𝑧, 𝑇𝑧) + 𝑎2𝑠𝜌(𝑇
𝑛−1𝑥0, 𝑇

𝑛 𝑥0) + 𝑎3𝑠𝜌(𝑧, 𝑇
𝑛𝑥0) 

                      +𝑎4𝑠
2𝜌(𝑇 𝑛−1𝑥0, 𝑇

𝑛𝑥0) + 𝑎4𝑠
2𝜌(𝑇𝑛𝑥0, 𝑇𝑧) 

                      +𝑎5𝑠
2𝜌(𝑧, 𝑇𝑛 𝑥0) + 𝑎5𝑠

2𝜌(𝑇𝑛 𝑥0, 𝑇
𝑛−1𝑥0) + 𝑠𝜌(𝑇

𝑛𝑥0, 𝑧). 
Letting the limit be  𝑛 → ∞ and since {𝑇𝑛−1𝑥0} is a subsequence of   {𝑇

𝑛 𝑥0} , we obtain 
𝜌(𝑇𝑧, 𝑧) ≤ 𝑎1𝑠𝜌(𝑧, 𝑇𝑧) + 𝑎4𝑠

2𝜌(𝑧, 𝑇𝑧) 
which gives 𝜌(𝑇𝑧, 𝑧) = 0 i.e. 𝑇𝑧 = 𝑧. Hence 𝑧 is a fixed point of 𝑇. The uniqueness of the fixed point follows from 
Theorem 3.1. 

 

Theorem 3.5. Let (𝑀, 𝜌) be a complete 𝑏-metric space with coefficient 𝑠 ≥ 1 and 𝑇 be a self mapping of 𝑀 satisfying 
the inequality 

                       𝜌(𝑇𝑥, 𝑇𝑦) ≤ 𝑎1𝜌(𝑥, 𝑇𝑥) + 𝑎2𝜌(𝑦, 𝑇𝑦) + 𝑎3𝜌(𝑥, 𝑇𝑦) + 𝑎4𝜌(𝑦, 𝑇𝑥) + 𝑎5𝜌(𝑥, 𝑦),          
for all 𝑥, 𝑦 ∈ 𝑀, where 𝑎𝑖  ( 𝑖 = 1,2,3,4,5) are non-negative real numbers with 𝑚𝑎𝑥{(𝑎2 + 𝑎3)𝑠

2, (𝑎3 + 𝑎4𝑠 + 𝑎5𝑠)} <
1 for 𝑠 ≥ 1. If 𝑇 is asymptotically regular at some point 𝑥 ∈ 𝑀 and the sequence {𝑇𝑛𝑥} of iterates has a subsequence 
{𝑇𝑛𝑘 𝑥} converging to a point 𝑧 of 𝑀, then 𝑧 is a unique fixed point of 𝑇 and {𝑇 𝑛𝑥} also converges to 𝑧. 
Proof. Let  lim

𝑘→∞
𝑇𝑛𝑘 𝑥 = 𝑧, then 

     𝜌(𝑧, 𝑇𝑧) ≤ 𝑠𝜌(𝑧, 𝑇𝑛𝑘 𝑥) + 𝑠2𝜌(𝑇𝑛𝑘 𝑥, 𝑇𝑛𝑘+1𝑥) + 𝑠2𝜌(𝑇𝑛𝑘+1𝑥, 𝑇𝑧) 
                   ≤  𝑠𝜌(𝑧, 𝑇𝑛𝑘 𝑥) + 𝑠2𝜌(𝑇𝑛𝑘 𝑥, 𝑇𝑛𝑘+1𝑥) + 𝑎1𝑠

2𝜌(𝑇𝑛𝑘 𝑥, 𝑇𝑛𝑘+1𝑥) + 𝑎2𝑠
2𝜌(𝑧, 𝑇𝑧) 

                     +𝑎3𝑠
2𝜌(𝑇 𝑛𝑘 𝑥, 𝑇𝑧) + 𝑎4𝑠

2𝜌(𝑧, 𝑇𝑛𝑘+1𝑥) + 𝑎5𝑠
2𝜌(𝑇𝑛𝑘 𝑥, 𝑧). 

Letting 𝑘 → ∞, we get 
𝜌(𝑧, 𝑇𝑧) ≤ (𝑎2𝑠

2 + 𝑎3𝑠
2)𝜌(𝑧, 𝑇𝑧) 

which gives 𝑧 is a fixed point of 𝑇. 
Now, 

      𝜌(𝑧, 𝑇𝑛 𝑥) = 𝜌(𝑇𝑧, 𝑇 𝑛𝑥) 
                       ≤ 𝑎1𝜌(𝑧, 𝑇𝑧) + 𝑎2𝜌(𝑇

𝑛−1𝑥, 𝑇𝑛𝑥) + 𝑎3𝜌(𝑧, 𝑇
𝑛𝑥) + 𝑎4𝜌(𝑇

𝑛−1𝑥, 𝑇𝑧) + 𝑎5𝜌(𝑧, 𝑇
𝑛−1𝑥) 

                       ≤ 𝑎1𝜌(𝑧, 𝑇𝑧) + 𝑎2𝜌(𝑇
𝑛−1𝑥, 𝑇𝑛𝑥) + 𝑎3𝜌(𝑧, 𝑇

𝑛𝑥) 
                         +𝑎4𝑠𝜌(𝑇

𝑛−1𝑥, 𝑇𝑛 𝑥) + 𝑎4𝑠𝜌(𝑇
𝑛 𝑥, 𝑇𝑧) 

                         +𝑎5𝑠𝜌(𝑧, 𝑇
𝑛𝑥) + 𝑎5𝑠𝜌(𝑇

𝑛 𝑥, 𝑇𝑛−1𝑥) 
which implies that 

     (1 − 𝑎3 − 𝑎4𝑠 − 𝑎5𝑠)𝜌(𝑧, 𝑇
𝑛𝑥) ≤ (𝑎2 + 𝑎4𝑠 + 𝑎5𝑠)𝜌(𝑇

𝑛−1𝑥, 𝑇𝑛𝑥). 



MOHAMMAD S. KHAN and PANKAJ K. JHADE 

 

52 

 

Since 𝑇 is asymptotically regular at 𝑥 ∈ 𝑀 and using the fact that 𝑚𝑎𝑥{(𝑎2 + 𝑎3)𝑠
2, (𝑎3 + 𝑎4𝑠 + 𝑎5𝑠)} < 1 for 𝑠 ≥ 1 

implies that the sequence {𝑇𝑛 𝑥} converges to 𝑧 in 𝑀. This completes the proof. 
 

Example 3.1. Let 𝑀 = ℝ and 𝜌: 𝑀 × 𝑀 → ℝ + be defined by 𝜌(𝑥, 𝑦) = |𝑥 − 𝑦|𝑝, where 𝑝 > 1. Then (𝑀, 𝜌) is a 𝑏-

metric space. Define a self map 𝑇 on 𝑀 as follows 𝑇𝑥 =
𝑥

2
 for all 𝑥 ∈ 𝑀. Clearly 𝑇 is asymptotically regular for all 𝑥 ∈

𝑀. If we take 𝑎1 = 𝑎2 = 𝑎3 = 𝑎4 = 0 and 𝑎5 =
1

2𝑝
 , then the contractive condition used here holds and 0 is the unique 

fixed point of  𝑇. 
 

Remark 3.2. The asymptotic regularity of the mapping 𝑇 satisfies the Hardy-Roger’s contraction condition. It is 
actually a consequence of ∑ 𝑎𝑖 < 1

5
𝑖=1 . Thus Theorem 3.3 and Theorem 3.4 extend results due to Hardy-Roger [1] in 𝑏-

metric space. It is also worth mentioning that our condition on control constants says that ∑ 𝑎𝑖
5
𝑖=1  may exceed 1. 

4. Conclusion  

It has been demonstrated that one can use asymptotically regular sequences rather than sequences of iterates to 

obtain interesting results related to fixed point theorems in b-metric spaces.  

5. Acknowledgements 

The authors would like to thank the referees for their valuable comments. 

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Received   9 November 2015            



THEOREMS ON FIXED POINTS FOR ASYMPTOTICALLY REGULAR SEQUENCE  

 

53 

 

Accepted   5 January 2017