International Journal of Analysis and Applications

Volume 17, Number 2 (2019), 275-281

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-17-2019-275

FIXED POINT THEOREM OF ĆIRIĆ-PATA TYPE

AO-LEI SIMA, FEI HE∗ AND NING LU

School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China

∗Corresponding author: hefei@imu.edu.cn

Abstract. In this article, we proved a fixed point theorem of Ćirić-Pata type in metric space. This result

extends several results in the existing literature. Moreover, an example is given in the support of our result.

In particular, the main result provides a complete solution to an open problem raised by Kadelburg and

Radenović (J. Egypt. Math. Soc. 24 (2016) 77-82).

1. Introduction

Throughout this paper, (X,d) will be a complete metric space. Fix an arbitrary point x0 ∈ X and

denote ‖x‖ = d(x,x0), for each x ∈ X. Also, ψ : [0, 1] → [0,∞) is an increasing function, continuous at zero,

with ψ(0) = 0. Given a function f : X → X.

In 2011, Pata [1] obtained the following result which is a generalization of the classical Banach contraction

principle.

Theorem 1.1. [1] Let Λ ≥ 0, α ≥ 1 and β ∈ [0,α] be fixed constants. If the inequality

d(fx,fy) ≤ (1 −ε)d(x,y) + Λεαψ(ε)[1 + ‖x‖ + ‖y‖]β (1.1)

is satisfied for every ε ∈ [0, 1] and all x,y ∈ X, then f has a unique fixed point z ∈ X.

Received 2018-11-06; accepted 2018-12-14; published 2019-03-01.

2010 Mathematics Subject Classification. 47H10, 54H25.

Key words and phrases. fixed point theorem; Pata type contraction; Ćirić type quasi-contraction; metric space.
c©2019 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

275

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-275


Int. J. Anal. Appl. 17 (2) (2019) 276

Afterward many pata type fixed point theorems have been established by various authors; see ( [2], [3],

[4], [5], [6], [7], [8], [9]). Particularly, Kadelburg and Radenović [7] proved some fixed point theorems of Pata

type and raised the following open question on Pata-version of Ćirić contraction principle (see [10]).

Problem 1.1. [7] Prove or disprove the following. Let f : X → X and let Λ ≥ 0, α ≥ 1 and β ∈ [0,α] be

fixed constants. If the inequality

d(fx,fy) ≤ (1 −ε) max{d(x,y),d(x,fx),d(y,fy),d(x,fy),d(y,fx)}

+ Λεαψ(ε)[1 + ‖x‖ + ‖y‖]β
(1.2)

is satisfied for every ε ∈ [0, 1] and all x,y ∈ X, then f has a unique fixed point z ∈ X. Furthermore, the

sequence {fnx0} converges to z.

Very recently, Jacobe et al. give the following result.

Theorem 1.2. [5] Let f : X → X and let Λ ≥ 0, α ≥ 1 and β ∈ [0,α] be fixed constants. If the inequality

d(fx,fy) ≤ (1 −ε) max
{
d(x,y),

d(x,fx) + d(y,fy)

2
,
d(x,fy) + d(y,fx)

2
}

+ Λεαψ(ε)[1 + ‖x‖ + ‖y‖ + ‖fx‖ + ‖fy‖]β
(1.3)

is satisfied for every ε ∈ [0, 1] and all x,y ∈ X, then f has a unique fixed point in X.

In this paper, we give a fixed point theorem of Ćirić-Pata type in metric space. This theorem extends the

main results in ( [1], [5], [7]) and provides a complete solution to the above Problem 1.1. Finally, an example

is given to illustrate the superiority of the main results.

2. Main results

Our result of this paper are stated as follows.

Theorem 2.1. Let Λ ≥ 0, α ≥ 1 be fixed constants. For x,y ∈ X, we denote

M(x,y) = max{d(x,y),d(x,fx),d(y,fy),d(x,fy),d(y,fx)}.

If the inequality

d(fx,fy) ≤ (1 −ε)M(x,y) + Λεαψ(ε)[1 + ‖x‖ + ‖y‖ + ‖fx‖ + ‖fy‖]α (2.1)

is satisfied for every ε ∈ [0, 1] and all x,y ∈ X, then f has a unique fixed point z ∈ X. Furthermore, the

sequence {fnx0} converges to z.



Int. J. Anal. Appl. 17 (2) (2019) 277

Proof. Starting from x0, construct a sequence {xn} such that xn = fxn−1 = fnx0. If xn0 = xn0+1 for

some n0, then xn0 is a fixed point of f. Thus, we always assume that xn 6= xn+1 for all n ∈ N.

We prove that xn 6= xm for all m,n ∈ N and n 6= m. Assume that there exist n0,m0 ∈ N such that n0 < m0

and xn0 = xm0. Denote A = max{d(xi,xj) : n0 ≤ i < j ≤ m0} and B = max{‖xi‖ : n0 ≤ i ≤ m0 + 1}.

It is obvious that A = max{d(xi,xj) : n0 + 1 ≤ i < j ≤ m0} and A > 0. For each i,j ∈ N such

that n0 + 1 ≤ i < j ≤ m0, we have

d(xi,xj) ≤ (1 −ε)M(xi−1,yj−1) + Λεαψ(ε)[1 + ‖xi−1‖ + ‖xj−1‖ + ‖xi‖ + ‖xj‖]α

≤ (1 −ε)A + Λεαψ(ε)(1 + 4B)α.

It follows that

A ≤ (1 −ε)A + Λεαψ(ε)(1 + 4B)α

and

A ≤ Λεα−1ψ(ε)(1 + 4B)α.

Letting ε → 0, we can see A ≤ 0. This is a contradiction with A > 0.

Denote Dn = max{d(xi,xj) : 0 ≤ i < j ≤ n} and δn = sup{d(xi,xj) : n ≤ i < j}. In order to prove {xn}

is a Cauchy sequence, we divide into the following three steps.

Step 1. We show that d(fx,fy) < M(x,y) for all x,y ∈ X and x 6= y. Let ε = 0 in (2.1), we

have d(fx,fy) ≤ M(x,y) for all x,y ∈ X. Assume that there exist x0,y0 ∈ X and x0 6= y0 such

that d(fx0,fy0) = M(x0,y0). Using (2.1), we get

M(x0,y0) = d(fx0,fy0) ≤ (1 −ε)M(x0,y0) + Λεαψ(ε)[1 + ‖x0‖ + ‖y0‖ + ‖fx0‖ + ‖fy0‖]α.

It follows that

M(x0,y0) ≤ Λεα−1ψ(ε)[1 + ‖x0‖ + ‖y0‖ + ‖fx0‖ + ‖fy0‖]α.

Passing to the limit as ε → 0, we see M(x0,y0) ≤ 0, a contradiction.

Step 2. We prove that {Dn} is bounded. By step 1, we see that

d(xi,xj) = d(fxi−1,fxj−1) < M(xi−1,xj−1) ≤ Dn,

for all i,j ∈ N such that 0 < i < j ≤ n. Thus there exists `n ∈ N such that 1 ≤ `n ≤ n and Dn = d(x0,x`n ).

Using (2.1), we have

Dn = d(x0,x`n )

≤ d(x0,x1) + d(x1,x`n )

≤ d(x0,x1) + (1 −ε)M(x0,x`n−1) + Λε
αψ(ε)[1 + ‖x0‖ + ‖x`n−1‖ + ‖x1‖ + ‖x`n‖]

α

≤ (1 −ε)Dn + Λεαψ(ε)(1 + 3Dn)α + d(x0,x1).



Int. J. Anal. Appl. 17 (2) (2019) 278

This implies that

εDn ≤ Λεαψ(ε)(1 + 3Dn)α + d(x0,x1).

Suppose that {Dn} is unbounded. Then there exists a subsequence {Dnk} with {Dn} such that Dnk →

∞ (k →∞) and Dnk ≥ 1 + d(x0,x1). Let ε = εk =
1+d(x0,x1)

Dnk
. Then we get

1 + d(x0,x1)

Dnk
·Dnk ≤ Λ[

1 + d(x0,x1)

Dnk
]αψ(εk)(1 + 3Dnk )

α + d(x0,x1)

and

1 ≤ Λ(
1

Dnk
+ 3)αψ(εk)[1 + d(x0,x1)]

α.

Letting k →∞, we have εk → 0 and

Λ(
1

Dnk
+ 3)αψ(εk)[1 + d(x0,x1)]

α → 0.

This is a contradiction. Thus {Dn} is bounded and there exists a constant M > 0 such that Dn ≤ M.

Step 3. We show that δn → 0. Observe that

d(xi,xj) = d(fxi−1,xj−1) < M(xi−1,xj−1) ≤ δn

for every i,j ∈ N with n + 1 ≤ i < j. Thus we get δn+1 ≤ δn ≤ ··· ≤ δ0 ≤ M. It is easy to see that {δn}

is decreasing and bounded sequence. It follows that lim
n→∞

δn = δ for some δ ≥ 0. Assume that δ > 0. From

(2.1), it holds for each i,j ∈ N with n + 1 ≤ i < j,

d(xi,xj) ≤ (1 −ε)M(xi−1,xj−1) + Λεαψ(ε)(1 + 4M)α.

This implies that

δn+1 ≤ (1 −ε)δn + Λεαψ(ε)(1 + 4M)α. (2.2)

Letting n →∞ in (2.2), we get

δ ≤ (1 −ε)δ + Λεαψ(ε)(1 + 4M)α

and

δ ≤ Λεα−1ψ(ε)(1 + 4M)α.

From

Λεα−1ψ(ε)(1 + 4M)α → 0 (ε → 0),

we see δ ≤ 0, a contradiction. For each p ∈ N, we get d(xn,xn+p) ≤ δn → 0 (n →∞). Hence, {xn} is Cauchy

sequence. Since X is complete, there exists z ∈ X such that xn → z (n →∞).

Now, we show that fz = z. Using (2.1), we get

d(fz,xn+1) ≤ (1 −ε) max{d(z,xn),d(z,fz),d(xn,xn+1),d(z,xn+1),d(xn,fz)}

+ Λεαψ(ε)(1 + 4M)α.



Int. J. Anal. Appl. 17 (2) (2019) 279

By taking limits on both sides when n →∞, we obtain

d(fz,z) ≤ (1 −ε)d(fz,z) + Λεαψ(ε)(1 + 4M)α.

Then

d(fz,z) ≤ Λεα−1ψ(ε)(1 + 4M)α → 0 (ε → 0).

This implies that d(fz,z) = 0 and fz = z.

Finally, we prove the uniqueness of z. If fu = u, fv = v for any two fixed u,v ∈ X, then we can write

(2.1) in the form

d(u,v) = d(fu,fv)

≤ (1 −ε) max{d(u,v),d(u,fu),d(v,fv),d(u,fv),d(v,fu)}

+ Λ�αψ(ε)[1 + ‖u‖ + ‖v‖ + ‖fu‖ + ‖fv‖]α

≤ (1 −ε)d(u,v) + Λεαψ(ε)[1 + 2‖u‖ + 2‖v‖]α.

Therefore

d(u,v) ≤ Λεα−1ψ(ε)[1 + 2‖u‖ + 2‖v‖]α → 0 (ε → 0),

which implies that d(u,v) = 0 and u = v. Hence, f has a unique fixed point z ∈ X. �

Remark 2.1. It is easy to see that the condition (2.1) is weaker than the condition (1.2). Hence, Theorem

2.1 provides a solution to Problem 1.1.

From Theorem 2.1 we get the following Corollaries.

Corollary 2.1. Let f : X → X and Let Λ ≥ 0, α ≥ 1 be fixed constants. If the inequality

d(fx,fy) ≤ (1 −ε)d(x,y)

+ Λεαψ(ε)[1 + ‖x‖ + ‖y‖ + ‖fx‖ + ‖fy‖]α
(2.3)

is satisfied for every ε ∈ [0, 1] and all x,y ∈ X, then f has a unique fixed point z ∈ X. Furthermore, the

sequence {fnx0} converges to z.

Remark 2.2. It is easy to see that the condition (2.3) is weaker than the condition (1.1). Thus Corollary

2.1 is an extension of Theorem 1.1.

Corollary 2.2. Let f : X → X and Let Λ ≥ 0, α ≥ 1 be fixed constants. If the inequality

d(fx,fy) ≤
1 −ε

2
(d(x,fy) + d(y,fx))

+ Λεαψ(ε)[1 + ‖x‖ + ‖y‖ + ‖fx‖ + ‖fy‖]α
(2.4)

is satisfied for every ε ∈ [0, 1] and all x,y ∈ X, then f has a unique fixed point z ∈ X. Furthermore, the

sequence {fnx0} converges to z.



Int. J. Anal. Appl. 17 (2) (2019) 280

Remark 2.3. It is easy to see that the condition 2.4 is weaker than the condition 2.1 in [7]. Thus Corollary

2.2 is an extension of Theorem 2.1 in [7].

Corollary 2.3. Let f : X → X and Let Λ ≥ 0, α ≥ 1 be fixed constants. If the inequality

d(fx,fy) ≤ (1 −ε) max
{
d(x,y),d(x,fx),d(y,fy),

d(x,fy) + d(y,fx)

2
}

+ Λεαψ(ε)[1 + ‖x‖ + ‖y‖ + ‖fx‖ + ‖fy‖]β
(2.5)

is satisfied for every ε ∈ [0, 1] and all x,y ∈ X, then f has a unique fixed point z ∈ X. Furthermore, the

sequence {fnx0} converges to z.

Remark 2.4. It is easy to see that the condition (2.5) is weaker than the condition (1.3). Thus Corollary

2.3 is an extension of Theorem 1.2.

The following is an example which can apply to Theorem 2.1 but not Corollary 2.3 or Theorem 1.2.

Example 2.1. Let X = {0, 1, 2, 4, 8, 9}
⋃
{

1

2n
: n = 1, 2, · · ·} with the usual metric. It is easily to check that

X is a complete metric space. Define f : X → X by

fx =


 8 x = 9,1

2
x others .

Then mapping f satisfies the condition (2.1) with Λ =
8

9
, β = 1 and ψ(ε) = ε

1
8 (for all ε ∈ [0, 1] ). Moreover,

it is worth mentioning that

8

9
− 1 + ε ≤

8

9
[1 +

9

8
(ε− 1)] ≤

8

9
ε

9
8 ≤

8

9
εε

1
8 .

Thus we have the following two cases.

(1) If x = 9 and y 6= 9, then

d(fx,fy) = d(8,
1

2
y) = 8 −

1

2
y

≤
8

9
(9 −

1

2
y) =

8

9
d(9,fy)

≤
8

9
M(9,y)

= (1 −ε)M(9,y) + (
8

9
− 1 + ε)M(9,y)

≤ (1 −ε)M(9,y) + (
8

9
− 1 + ε)[1 + ‖9‖ + ‖y‖ + ‖f9‖ + ‖fy‖]

≤ (1 −ε)M(9,y) +
8

9
εε

1
8 [1 + ‖9‖ + ‖y‖ + ‖f9‖ + ‖fy‖]



Int. J. Anal. Appl. 17 (2) (2019) 281

(2) If x 6= 9 and y 6= 9, then

d(fx,fy) =
1

2
(x−y) ≤

8

9
(x−y)

=
8

9
d(x,y) ≤

8

9
M(x,y)

= (1 −ε)M(x,y) + (
8

9
− 1 + ε)M(x,y)

≤ (1 −ε)M(x,y) + (
8

9
− 1 + ε)[1 + ‖x‖ + ‖y‖ + ‖fx‖ + ‖fy‖]

≤ (1 −ε)M(x,y) +
8

9
εε

1
8 [1 + ‖x‖ + ‖y‖ + ‖fx‖ + ‖fy‖]

Hence, f satisfies all conditions of Theorem 2.1. This leads to f has a unique fixed point. Indeed, 0 is the

fixed point for the mapping f.

Now, let ε = 0, x = 9 and y = 4, we have

d(f9,f4) = 6 >
11

2
= max{5, 1, 2,

11

2
}

= max{d(9, 4),d(9,f9),d(4,f4),
d(9,f4) + d(4,f9)

2
}

It is easy to see that f does not satisfy the condition (2.5) of Corollary 2.3. Also, f does not satisfy the

condition (1.3) of Theorem 1.2.

Acknowledgements. This work was supported by the National Natural Science Foundation of China

(No. 11561049, 11471236 )

References

[1] V. Pata, A fixed point theorems in metric spaces, J. Fixed Point Theory Appl. 10 (2011), 299–305.

[2] M. A. Alghamdi, A. Petrusel and N. Shahzad, Correction: A fixed point theorem for cyclic generalized contractions in

metric spaces, Fixed Point Theory Appl. 2012 (2012), 122.

[3] S. Balasubramanian, A Pata-type fixed point theorem, Math. Sci. 8 (2014), 65–69.

[4] M. Eshaghi, S. Mohseni, M. R. Delavar, M. De La Sen, G. H. Kim and A. Arian, Pata contractions and coupled type fixed

points, Fixed Point Theory Appl. 2014 (2014), 130.

[5] G. K. Jacob, M. S. Khan, C. Park and S. Jun, On generalized Pata type contractions, Mathmatics. 6 (2018), 25.

[6] Z. Kadelburg and S. Radenović, Fixed point and tripled fixed point theorems under Pata-type conditions in ordered metric

spaces, International Journal of Analysis and Applications. 6 (2014), 113–122.

[7] Z. Kadelburg and S. Radenović, Fixed points theorems under Pata-type conditions in metric spaces, J. Egypt. Math. Soc.

24 (2016), 77–82.

[8] S. M. Kolagar, M. Ramezani and M. Eshaghi, Pata type fixed point theorems of multivalued operators in ordered metric

spaces with applications to hyperbolic differential inclusions, Proc. Amer. Math. Soc. 6 (2016), 21–34.

[9] M. Paknazar, M. Eshaghi, Y. J. Cho and S. M. Vaezpour, A Pata-type fixed point theorem in modular spaces with

application, Fixed Point Theory and Appl. 2013 (2013), 239.

[10] L. J. Ćirić, A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc. 45 (1974), 267–273.


	1. Introduction
	2. Main results
	References