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@ Appl. Gen. Topol. 17, no. 2(2016), 185-198doi:10.4995/agt.2016.5660
c© AGT, UPV, 2016

Best proximity point for Z-contraction and

Suzuki type Z-contraction mappings with an

application to fractional calculus

Somayya Komal a, Poom Kumam a,b,∗ and Dhananjay Gopal b,c

a
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology

Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thai-

land. (komal.musab@gmail.com, poom.kum@mail.kmutt.ac.th)
b
Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of

Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd.,

Bang Mod, Thung Khru, Bangkok 10140, Thailand. (poom.kum@mail.kmutt.ac.th)
c

Department of applied Mathematics and Humanities, SV National Institute of Technology,

Surat, Gujarat, India. (gopal.dhananjay@rediffmail.com)

Abstract

In this article, we introduced the best proximity point theorems for Z-

contraction and Suzuki type Z-contraction in the setting of complete

metric spaces. Also by the help of weak P-property and P-property,

we proved existence and uniqueness of best proximity point. There is a

simple example to show the validity of our results. Our results extended

and unify many existing results in the literature. Moreover, an appli-

cation to fractional order functional differential equation is discussed.

2010 MSC: 54H25; 34A08.

Keywords: best proximity point; weak P-property; Suzuki type Z-
contraction; functional differential equation.

∗Corresponding author: poom.kumam@mail.kmutt.ac.th and poom.kum@kmutt.ac.th

Received 26 April 2016 – Accepted 23 June 2016

http://dx.doi.org/10.4995/agt.2016.5660


S. Komal, P. Kumam and D. Gopal

1. Introduction

When we study about fixed points of different mappings satisfying certain
conditions, then it is observed that this theory has enormous applications in
various branches of mathematics and mathematical sciences and hence become
the source of inspiration for many researchers and mathematicians working in
the metric fixed point theory (see for instant [5, 16, 12, 26]). When a self
mapping in a metric space has no fixed points, then it could be interesting to
study the existence and uniqueness of some points that minimize the distance
between the point and its corresponding image. These points are known as best
proximity points. Best proximity points theorems for several types of non-self
mappings have been derived in [1], [2], [3], [6], [7], [8], [10], [9] and [24]. The best
proximity points were introduced by [13] and modified by Sadiq Basha in [7].
The results about best proximity point theory have been found very briefly in
the work of [6] to [9]. Now, after the new generalization of Banach contraction
principle given by khoj. et al. in [15] by defining a notion of Z-contraction,
after that Kumam et. al. in [16] introduced Suzuki type Z-contraction and
unified many fixed point results. Some recent contribution in this field can be
found in ([18, 17, 4, 20, 21, 22, 23]). Because of its importance in nonlinear
analysis, we extend these generalizations and contractions to find out the unique
best proximity point in metric spaces and introduced these notions for non self
mappings in the light of Yaq. et al. [25] by using some suitable properties.
Some examples and an application to fractional order functional differential
equation is given to illustrate the usability of new theory.

2. Preliminaries

In this section, we collect some notations and notions which will be used
throughout the rest of this work.
Let A and B be two nonempty subsets of a metric space (X, d). We will use
the following notations:

d(A, B) := inf{d(a, b) : a ∈ A, b ∈ B};

A0 := {a ∈ A : d(a, b) = d(A, B) for some b ∈ B};

B0 := {b ∈ B : d(a, b) = d(A, B) for some a ∈ A}.

Definition 2.1. An element x∗ ∈ A is said to be a best proximity point of
the non-self-mapping T : A → B if it satisfies the condition that d(x∗, T x∗) =
d(A, B).

Remark 2.2. It can be observed that a best proximity reduces to a fixed point
if the underlying mapping is a self-mapping.

Definition 2.3 ([15]). Let ζ : [0, ∞) × [0, ∞) → R be a mapping, then ζ is
called a simulation function if it satisfies the following conditions:

(1) ζ(0, 0) = 0;
(2) ζ(t, s) < s − t for t, s > 0;

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 186



Best proximity point for Z-contraction and Suzuki type Z-contraction mappings

(3) if {tn}, {sn} are sequences in (0, ∞) such that

lim
n→∞

tn = lim
n→∞

sn > 0,

then

lim sup
n→∞

ζ(tn, sn) < 0.

We denote the set of all simulation functions by Z.

Definition 2.4 ([15]). Let (X, d) be a metric space, F : X → X is a mapping
and ζ ∈ Z. Then F is called a Z-contraction with respect to ζ if the following
condition holds:

(2.1) ζ(d(Fx, Fy), d(x, y)) ≥ 0

where x, y ∈ X, with x 6= y.

Definition 2.5 ([16]). Let (X, d) be a metric space, F : X → X is a mapping
and ζ ∈ Z. Then F is called a Suzuki type Z-contraction with respect to ζ if
the following condition holds:

(2.2)
1

2
d(x, Fx) < d(x, y) ⇒ ζ(d(Fx, Fy), d(x, y)) ≥ 0

where x, y ∈ X, with x 6= y.

Definition 2.6 ([19]). Let (A, B) be a pair of nonempty subsets of a metric
space (X, d) with A0 6= φ. Then the pair (A, B) is said to have the P-property
if and only if

d(x1, y1) = d(A, B) and d(x2, y2) = d(A, B) ⇒ d(x1, x2) = d(y1, y2),

where x1, x2 ∈ A0 and y1, y2 ∈ B0.

Definition 2.7 ([27]). Let (A,B) be a pair of nonempty subsets of a metric
space (X,d) with A0 6= ∅. Then the pair (A,B) is said to have weak P-property
if and only if for any x1, x2 ∈ A0 and y1, y2 ∈ B0

d(x1, y1) = d(A, B)
d(x2, y2) = d(A, B)

}

⇒ d(x1, x2) ≤ d(y1, y2).

Theorem 2.8 ([16]). Let (X, d) be a complete metric space. Define a mapping
F : X → X satisfying the following conditions:

(1) F is Suzuki type Z-contraction with respect to ζ;
(2) for every bounded Picard sequence there exists a natural number k such

that 1
2
d(xmk , xmk+1) < d(xmk , xnk ) for mk > nk ≥ k.

Then there exists unique fixed point in X and the Picard iteration sequence
{xn} defined by

xn = Fxn−1, n = 1, 2, ...

converges to a fixed point of F,

Remark 2.9 ([15]). Every Z-contraction is contractive and hence Banach con-
traction.

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 187



S. Komal, P. Kumam and D. Gopal

Theorem 2.10 ([5]). Let (X, d) be a complete metric space. Then every con-
traction mapping has a unique fixed point. It is known as Banach contraction
principle.

3. Main Results

In this section, we will introduced the notion of generalized contraction prin-
ciple for non self mappings by combining Suzuki and Z- contraction mappings
and will find the unique best proximity point.

Definition 3.1. Let (X, d) be a metric space, F : A → B is a mapping and
ζ ∈ Z. Then F is called a Z-contraction with respect to ζ if the following
condition holds:

(3.1) ζ(d(Fx, Fy), d(x, y)) ≥ 0

where A, B ⊆ X and x, y ∈ A, with x 6= y.

Definition 3.2. Let (X, d) be a metric space, F : A → B is a mapping and
ζ ∈ Z. Then F is called a Suzuki type Z-contraction with respect to ζ if the
following condition holds:

(3.2)
1

2
d(x, Fx) < d(x, y) ⇒ ζ(d(Fx, Fy), d(x, y)) ≥ 0

where A, B ⊆ X and x, y ∈ A, with x 6= y.

Remark 3.3. Since the definition of simulation function implies that ζ(t, s) < 0
for all t ≥ s > 0. Therefore F is Suzuki type Z contraction with respect to ζ,
then

1

2
d(x, Fx) < d(x, y) ⇒ d(Fx, Fy) < d(x, y)

for any distinct x, y ∈ A.

Remark 3.4. Every Suzuki type Z-contraction is also a Z-contraction.

Now, we are in a position to prove best proximity point theorems for Z and
Suzuki type Z-contractions in metric spaces.

Theorem 3.5. Let (A, B) be the pair of nonempty closed subsets of a complete
metric space (X, d) such that A0 is nonempty. Define a mapping F : A → B
satisfying the following conditions:

(1) F is Z-contraction with F(A0) ⊆ B0;
(2) the pair (A, B) has weak P-property.

Then there exists unique best proximity point in A and the iteration sequence
{x2n} defined by

x2n+1 = Fx2n, d(x2n+2, x2n+1) = d(A, B), n = 0, 1, 2, ...

converges, to x∗, for every x0 ∈ A0.

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 188



Best proximity point for Z-contraction and Suzuki type Z-contraction mappings

Proof. First of all, we have to show that B0 is closed. For this, let us take
{yn} ⊆ B0 a sequence such that yn → t ∈ B. Since the pair (A, B) has weak
P-property, it follows from the weak P-property that

d(yn, ym) → 0 ⇒ d(xn, xm) → 0,

as m, n → ∞, and xn, xm ∈ A0 and d(xn, yn) = d(xm, ym) = d(A, B).
Thus {xn} is a Cauchy sequence and converges strongly to a point s ∈ A. By
the continuity of the metric d, we have d(s, t) = d(A, B), that is t ∈ B0 and
hence B0 is closed.
Let A0 be the closure of A0; now we have to prove that F(A0) ⊆ B0. If we take
x ∈ A0 \ A0, then there exists a sequence {xn} ⊆ A0 such that xn → x. By the
continuity of F and the closeness of B0, we get as Fx = limn→∞ Fxn ∈ B0.
That is, F(A0) ⊆ B0.
Since F is Z-contraction,which implies that

0 ≤ ζ(d(Fx1, Fx2), d(x1, x2))

< d(x1, x2) − d(Fx1, Fx2),

implies that

(3.3) d(Fx1, Fx2) < d(x1, x2).

Define an operator PA0 : F(A0) → A0, by PA0y = {x ∈ A0 : d(x, y) = d(A, B)}.

Since the pair (A, B) has weak P-property and using (5), we have

d(PA0Fx1, PA0Fx2) ≤ d(Fx1, Fx2)

< d(x1, x2)

for any x1, x2 ∈ Ao. Hence ζ(d(PA0 Fx1, PA0Fx2), d(x1, x2)) ≥ 0. So, PA0F :
A0 → A0 is a Z-contraction from complete metric subspace A0 into itself.
Since by using Remark (2.1), every Z-contraction is a contraction and hence
a Banach contraction. Thus by using theorem (2.2), PA0F has unique fixed
point, that is PA0Fx

∗ = x∗ ∈ A0, which implies that

d(x∗, Fx∗) = d(A, B).

Therefore, x∗ is unique in A0 such that d(x
∗, Fx∗) = d(A, B). It is easily seen

that x∗ is unique one in A such that d(x∗, Fx∗) = d(A, B). The Picard Iterative
sequence

xn+1 = PA0Fxn, n = 0, 1, 2, ...

converges, for every x0 ∈ A0, to x
∗. The iteration sequence {x2n}, for n =

0, 1, 2, ... defined by,

x2n+1 = Fx2n, d(x2n+2, x2n+1) = d(A, B), n = 0, 1, 2, ...

is exactly a subsequence of {xn}, so that it converges to x
∗, for every x0 ∈

A0. �

Theorem 3.6. Let (A, B) be the pair of nonempty closed subsets of a complete
metric space (X, d) such that A0 is nonempty. Define a mapping F : A → B
satisfying the following conditions:

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 189



S. Komal, P. Kumam and D. Gopal

(1) F is Suzuki type Z-contraction with F(A0) ⊆ B0;
(2) the pair (A, B) has the weak P-property.

Then there exists unique x∗ in A such that d(x∗, Fx∗) = d(A, B) and the
iteration sequence {x2n} defined by

x2n+1 = Fx2n, d(x2n+2, x2n+1) = d(A, B), n = 0, 1, 2, ...

converges, for every x0 ∈ A0 to x
∗.

Proof. First of all, we have to show that B0 is closed. For this, let us take
{yn} ⊆ B0 a sequence such that yn → g ∈ B. Since the pair (A, B) has weak
P-property, it follows from weak P-property that

d(yn, ym) → 0 ⇒ d(xn, xm) → 0,

as m, n → ∞, and xn, xm ∈ A0 and d(xn, yn) = d(xm, ym) = d(A, B).
Thus {xn} is a Cauchy sequence and converges strongly to a point f ∈ A. By
the continuity of the metric d, we have d(f, g) = d(A, B), that is g ∈ B0 and
hence B0 is closed.
Let A0 be the closure of A0; now we have to prove that F(A0) ⊆ B0. If we take
x ∈ A0 \ A0, then there exists a sequence {xn} ⊆ A0 such that xn → x. By the
continuity of F and the closeness of B0, we get as Fx = limn→∞ Fxn ∈ B0.
That is, F(A0) ⊆ B0.
Define an operator PA0 : F(A0) → A0, by PA0y = {x ∈ A0 : d(x, y) = d(A, B)}.

Since F is Suzuki type Z-contraction, such that for 1
2
d(x1, Fx1) < d(x1, y1),

we have

ζ(d(Fx1, Fy1), d(x1, y1)) ≥ 0.

Now, we claim that PA0F is Suzuki type Z-contraction. For this, we have to
prove that 1

2
d(x1, PA0Fx1) < d(x1, y1), for all x, y ∈ A. Since F is Suzuki type

Z-contraction, that is d(Fx, Fy) < d(x, y). By using P-property, P
A0

y = {x ∈

A0 : d(x, y) = d(A, B)} and triangular inequality, we obtain

1

2
d(x1, PA0Fx1) ≤

1

2
[d(x1, y1) + d(y1, PA0Fx1)]

=
1

2
[d(x1, y1) + d(y1, x1)]

= d(x1, y1)

= d(Fx1, Fy1)

< d(x1, y1)

Hence,

(3.4)
1

2
d(x1, PA0Fx1) < d(x1, y1).

for any x1, y1 ∈ A0. Which shows that ζ(d(PA0 Fx1, PA0Fy1), d(x1, y1)) ≥ 0,
where x1, y1 ∈ A0. Thus, PA0F : A0 → A0 is a Suzuki type Z-contraction from

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 190



Best proximity point for Z-contraction and Suzuki type Z-contraction mappings

complete metric subspace A0 into itself. Consequently, one may write by using
the fact that PA0F is a Suzuki type Z-contraction and remark (3.1) as

⇒ d(PA0Fx1, PA0Fy1) < d(x1, y1).

Then by using Theorem (2.1), PA0F has unique fixed point, that is PA0Fx
∗ =

x∗ ∈ A0, which implies that

d(x∗, Fx∗) = d(A, B).

Therefore, x∗ is unique in A0 such that d(x
∗, Fx∗) = d(A, B). It is easily seen

that x∗ is unique one in A such that d(x∗, Fx∗) = d(A, B). The Picard Iterative
sequence

xn+1 = PA0Fxn, n = 0, 1, 2, ...

converges, for every x0 ∈ A0, to x
∗. The iteration sequence {x2n}, for n =

0, 1, 2, ... defined by,

x2n+1 = Fx2n, d(x2n+2, x2n+1) = d(A, B), n = 0, 1, 2, ...

is exactly a subsequence of {xn}, so that it converges to x
∗, for every x0 ∈

A0. �

Corollary 3.7. Let (X, d) be a complete metric space. Define a mapping
F : X → X satisfying the following conditions:

(1) F is Z-contraction.

Then there exists unique fixed point in X and the iteration sequence {x2n}
defined by

x2n+1 = Fx2n, d(x2n+2, x2n+1) = d(A, B), n = 0, 1, 2, ...

converges to x∗, for every x0 ∈ A0.

Proof. Taking self mapping A = B = X in Theorem (3.1), then we get desired
result. �

Remark 3.8. By Taking self mapping in Theorem (3.2), we obtain Theorem
(2.1).

There is an example to justify our results and remarks.

Example 3.9. Consider X = R2, with the usual metric d. Define the sets
A = {(x, 1) : x ≥ 0} and B = {(x, 0) : x ≥ 0}. Let A0 = A and B0 = B
and clearly, the pair (A, B) has the P-property, also satisfies weak P-property.
Also define f : A → B as:

f(x, 1) = (
x2

x + 1
, 0),

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 191



S. Komal, P. Kumam and D. Gopal

we take A0 = A 6= ∅, B0 = B, f(A0) ⊆ B0. Then,

d(f(x1, 1), f(x2, 1)) = |
x21

x1 + 1
−

x22
x2 + 1

|

=
|x21(x2 + 1) − x

2
2(x1 + 1)

(x1 + 1)(x2 + 1)
=

|(x1x2 + x1 + x2)(x2 − x1)|

|(x1 + 1)(x2 + 1)|

=
x1x2 + x1 + x2
(x1 + 1)(x2 + 1)

|x1 − x2|

=
x1x2 + x1 + x2

x1x2 + x1 + x2 + 1
|x1 − x2|

< |x1 − x2| = d((x1, 1), (x2, 1)).

i.e. d(f(x1, 1), f(x2, 1)) < d((x1, 1), (x2, 1)),
which implies that ζ(d(f(x1, 1), f(x2, 1)), d((x1, 1), (x2, 1))) ≥ 0, i.e. f is ζ-
contraction. Thus, all the conditions of the Theorem (3.1) are satisfied, and the
conclusion of that theorem is also correct, that is, f has a unique best proximity
point z∗ = (0, 1) ∈ A0 such that d(z

∗, fz∗) = d((0, 1), (0, 0)) = d(A, B) = 1.
On the other hand, it is clear that the iteration sequence {z2k}, k = 0, 1, 2, ...
defined by

z2k+1 = f{z2k}, d(z2k+2, z2k+1) = d(A, B) = 1, k = 0, 1, 2, ...,

converges for every z0 ∈ A0, to z
∗, since

z2(k+1) = (x2(k+1), 1) = (
x22k

x2k + 1
, 1) → (0, 1).

In fact, from x2(k+1) =
x
2
2k

x2k+1
, we know that x2k+1 ≤ x2k, so there exists a

number x∗ such that x2k → x
∗. Furthermore, x∗ =

(x∗)2

x∗+1
and hence x∗ = 0.

Example 3.10. Consider X = R2, with the usual metric d. Define the sets
A = {(x, 1) : x ≥ 0} and B = {(x, 0) : x ≥ 0}. Let A0 = A and B0 = B
and clearly, the pair (A, B) has the P-property, also satisfies weak P-property.
Also define f : A → B as:

f(x, 1) = (
x2

x + 1
, 0),

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 192



Best proximity point for Z-contraction and Suzuki type Z-contraction mappings

we take A0 = A 6= ∅, B0 = B, f(A0) ⊆ B0. Then,

1

2
d((x1, 1), f(x1, 1)) =

1

2
d((x1, 1), (

x21
x1 + 1

, 0))

=
1

2
|1 + (x1 −

x21
x1 + 1

)|

=
1

2
|1 +

1

1 + x1
|

=
1

2

|1 + 1
1+x1

|

|x1 − x2|
|x1 − x2|

=
1

2

|x1 + 2|

|(x1 − x2)(x1 + 1)|
|x1 − x2|

< |x1 − x2| = d((x1, 1), (x2, 1)).

Thus, d((x1, 1), f(x1, 1)) < d((x1, 1), (x2, 1)), which implies that ζ(d(f(x1, 1),
f(x2, 1)), d((x1, 1), (x2, 1))) ≥ 0, and f is Suzuki type Z-contraction with re-
spect to ζ. Thus, all the conditions of the Theorem (3.2) are satisfied, and the
conclusion of that theorem is also correct, that is, f has a unique best proxim-
ity point z∗ = (0, 1) ∈ A0 such that d(z

∗, fz∗) = d((0, 1), (0, 0)) = d(A, B) = 1
On the other hand, it is clear that the iteration sequence {z2k}, k = 0, 1, 2, ...
defined by

z2k+1 = f{z2k} d(z2k+2, z2k+1) = d(A, B) = 1, k = 0, 1, 2, ...,

converges for every z0 ∈ A0, to z
∗, since

z2(k+1) = (x2(k+1), 1) = (
x22k

x2k + 1
, 1) → (0, 1).

In fact, from x2(k+1) =
x
2
2k

x2k+1
, we know that x2k+1 ≤ x2k, so there exists a

number x∗ such that x2k → x
∗. Furthermore, x∗ =

(x∗)2

x∗+1
and hence x∗ = 0.

Example 3.11. If we change the defined mapping on same conditions of above
example and on little change on given sets like for A = {(1, y) : y ≥ 0} and
B = {(0, y) : y ≥ 0} and A0 = A and B0 = B. Define f : A → B as:

f(1, y) = (0,
y2

y + 1
),

as given in [25], then also with this defined mapping there exists a best prox-
imity point for both Z and Suzuki type Z-contractions, also after such change
in the conditions, examples (3.1) and (3.2), theorems (3.1) and (3.2) ver-
ified, and that best proximity point is (1, 0) for both, that is, d(x, fx) =
d((1, 0), (0, 0)) = d(A, B) = 1. If there are two best proximity points for
same sets, (1, 0) and (0, 1), then their uniqueness can be proved easily as
d((1, 0), (0, 0)) = d(A, B) and d((0, 1), (0, 0)) = d(A, B) = 1, then one can write
as: d((1, 0), (0, 0)) = d((0, 1), (0, 0)) = d(A, B), this implies that (1, 0) = (0, 1).
Hence, existence and uniqueness of best proximity point in the metric space
has proved.

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S. Komal, P. Kumam and D. Gopal

4. Application

In this section, we present an application of our fixed point results derived
in previous section to establish the existence of solution of fractional order
functional differential equation.
Consider the following initial value problem (IVP for short) of the form

(4.1) Dαy(t) = f(t, yt), for each t ∈ J = [0, b], 0 < α < 1,

(4.2) y(t) = φ(t), t ∈ (−∞, 0]

where Dα is the standard Riemann-Liouville fractional derivative, f : J × B →
R, φ ∈ B, φ(0) = 0 and B is called a phase space or state space satisfying
some fundamental axioms (H-1, H-2, H-3) given below which were introduced
by Hale and Kato in [14].
For any function y defined on (−∞, b] and any t ∈ J , we denote by yt the
element of B defined by

yt(θ) = y(t + θ), θ ∈ (−∞, 0].

Here yt(·) represents the history of the state from −∞ up to present time t.

By C(J, R) we denote the Banach space of all continuous functions from J
into R with the norm

||y||∞ := sup{|y(t)| : t ∈ J}

where | · | denotes a suitable complete norm on R.

(H-1) If y : (−∞, b] → R, and y0 ∈ B, then for every t ∈ [0, b] the following
conditions hold:
(i) yt is in B,
(ii) ||yt||B ≤ K(t) sup{|y(s)| : 0 ≤ s ≤ t} + M(t)||y0||B,
(iii) |y(t)| ≤ H||yt||B,

where H ≥ 0 is a constant, K : [0, b] → [0, ∞) is continuous, M :
[0, ∞) → [0, ∞) is locally bounded and H, K, M are independent
of y(·).

(H-2) For the function y(·) in (H-1), yt is a B-valued continuous function on
[0, b].

(H-3) The space B is complete.

By a solution of problem (4.1)-(4.2), we mean a space Ω = {y : (−∞, b] →
R : y|(−∞,0] ∈ B and y|[0,b] is continuous}. Thus a function y ∈ Ω is said to be
a solution of (4.1)-(4.2) if y satisfies the equation Dαy(t) = f(t, yt) on J, and
the condition y(t) = φ(t) on (−∞, 0].

The following lemma is crucial to prove our existence theorem for the pro-
blem (4.1)-(4.2).

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Best proximity point for Z-contraction and Suzuki type Z-contraction mappings

Lemma 4.1 (see [11]). Let 0 < α < 1 and let h : (0, b] → R be continuous and
lim
t→0+

h(t) = h(0+) ∈ R. Then y is a solution of the fractional integral equation

y(t) =
1

Γ(α)

∫ t

0

(t − s)α−1h(s)ds,

if and only if y is a solution of the initial value problem for the fractional
differential equation

Dαy(t) = h(t), t ∈ (0, b],

y(0) = 0.

Now we are ready to prove following existence theorem.

Theorem 4.2. Let f : J × B → R. Assume
(H) there exists q > 0 such that

|f(t, u) − f(t, v)| ≤ q||u − v||B, for t ∈ J and every u, v ∈ B.

If b
α
Kbq

Γ(α+1)
= λ < 1 where

Kb = sup{|K(t)| : t ∈ [0, b]},

then there exists a unique solution for the IVP (4.1)-(4.2) on the interval
(−∞, b].

Proof. To prove the existence of solution for the IVP (4.1)-(4.2), we transform
it into a fixed point problem. For this, consider the operator N : Ω → Ω defined
by

N(y)(t) =

{

φ(t) t ∈ (−∞, 0],
1

Γ(α)

∫ t

0
(t − s)α−1f(s, ys)ds t ∈ [0, b].

Let x(·) : (−∞, b] → R be the function defined by

x(t) =

{

φ(t) t ∈ (−∞, 0],
0 t ∈ [0, b].

Then x0 = φ. For each z ∈ C([0, b], R) with z(0) = 0, we denote by z̄ the
function defined by

z̄(t) =

{

0 if t ∈ (−∞, 0],
z(t) if t ∈ [0, b].

If y(·) satisfies the integral equation

y(t) =
1

Γ(α)

∫ t

0

(t − s)α−1f(s, ys)ds,

we can decompose y(·) as y(t) = z̄(t)+x(t), 0 ≤ t ≤ b, which implies yt = z̄t+xt,
for every 0 ≤ t ≤ b, and the function z(·) satisfies

z(t) =
1

Γ(α)

∫ t

0

(t − s)α−1f(s, z̄s + xs)ds

Set
C0 = {z ∈ C([0, b], R) : z0 = 0},

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 195



S. Komal, P. Kumam and D. Gopal

and let || · ||b be the seminorm in C0 defined by

||z||b = ||z0||B + sup{|z(t)|; 0 ≤ t ≤ b} = sup{|z(t)|; 0 ≤ t ≤ b}, z ∈ C0.

C0 is a Banach space with norm ||·||b. Let the operator P : C0 → C0 be defined
by

(4.3) (Pz)(t) =
1

Γ(α)

∫ t

0

(t − s)α−1f(s, z̄s + xs)ds, t ∈ [0, b].

That the operator N has a fixed point is equivalent to P has a fixed point, and
so we turn to proving that P has a fixed point. Indeed, consider z, z∗ ∈ C0.
Then we have for each t ∈ [0, b]

|P(z)(t) − P(z∗)(t)| ≤
1

Γ(α)

∫ t

0

(t − s)α−1|f(s, z̄s + xs) − f(s, z̄
∗
s + xs)| ds

≤
1

Γ(α)

∫ t

0

(t − s)α−1q||z̄s − z̄
∗
s ||B ds

≤
1

Γ(α)

∫ t

0

(t − s)α−1qKb sup
s∈[o,t]

||z(s) − z∗(s)|| ds

≤
Kb

Γ(α)

∫ t

0

(t − s)α−1q ds ||z − z∗||b.

Therefore

||P(z) − P(z∗)||b ≤
qbαKb

Γ(α + 1)
||z − z∗||b,

i.e.

d(P(z), P(z∗)) ≤ λd(z, z∗).

Now we observe that the function ζ : [0, ∞) × [0, ∞) → R defined by ζ(t, s) =
λs−t for all t, s ∈ [0, ∞), is in Z and so we deduce that the operator P satisfies
all the hypothesis of corollary (3.7). Thus P has unique fixed point. �

Acknowledgements. The authors thank Editor-in-Chief and Referee(s) for
their valuable comments and suggestions, which were very useful to improve
the paper significantly. The authors would like to thank the Petchra Pra Jom
Klao Ph.d. Research Scholarship for financial support. Also, Somayya Komal
was supported by the Petchra Pra Jom Klao Doctoral Scholarship Academic
for Ph.D. Program at KMUTT. This work was completed while the third au-
thor (Dr. Gopal) was visiting Theoretical and Computational Science Cen-
ter (TaCS), Science Laboratory Building, Faculty of Science, King Mongkut’s
University of Technology Thonburi (KMUTT), Bangkok, Thailand. He thanks
Professor Poom Kumam and the University for their hospitality and support.
Moreover, this project was supported by the Theoretical and Computational
Science (TaCS) Center under Computational and Applied Science for Smart
Innovation Research Cluster (CLASSIC), Faculty of Science, KMUTT.

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 196



Best proximity point for Z-contraction and Suzuki type Z-contraction mappings

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