International Journal of Analysis and Applications
ISSN 2291-8639
Volume 12, Number 1 (2016), 15-21
http://www.etamaths.com

SOME FIXED POINT RESULTS FOR CARISTI TYPE MAPPINGS IN

MODULAR METRIC SPACES WITH AN APPLICATION

DURAN TURKOGLU1 AND EMINE KILINÇ2,∗

Abstract. In this paper we give Caristi type fixed point theorem in complete modular metric

spaces. Moreover we give a theorem which can be derived from Caristi type. Also an application for

the bounded solution of funcional equations is investigated.

1. Introduction

Fixed point theory is one of the very popular tools in various fields. Since Banach intoruced this
theory in 1922[6], ıt has been extended and generalized by several authors. Caristi type fixed point
theorem is one of these genealizations. It is a modification of ε−variational principle of Ekeland[13].It is
crucial in nonlinear analysis, in particular, optimization, variational inequalites, differantial equations
and contol theory.

The notion of modular space was introduced by Nakano [20] and was intensively developed by Koshi,
Shimogaki, Yamamuro (see [18, 21]) and others. A lot of mathematicians are interested in fixed point
of modular space. In 2008, Chistyakov introduced the notion of modular metric space generated by
F-modular and developed the theory of this space [8], on the same idea was defined the notion of
a modular on an arbitrary set and developed the theory of metric space generated by modular such
that called the modular metric spaces in 2010 [9]. Afrah A. N. Abdou [1] studied and proved some
new fixed points theorems for pointwise and asymptotic pointwise contraction mappings in modular
metric spaces. Azadifer et. al. [3] introduced the notion of modular G-metric spaces.Azadifer et. al.
[5] proved the existence and uniqueness of a common fixed point of compatible mappings of integral
type in this space. Kılınç and Alaca [14] defined (ε,k)−uniformly locally contractive mappings and
η-chainable concept and proved a fixed point theorem for these concepts in a complete modular metric
spaces. Kılınç and Alaca [15] proved that two main fixed point theorems for commuting mappings in
modular metric spaces. Recently, many authors [4, 7, 10, 11, 12, 19] studied on different fixed point
results for modular metric spaces. In 2014 Khamsi and Abdou investigated Hausdorff modular metric
in modular metric spaces[16], and proved fixed point theorem for multivalued mappings.

In this paper we investigate Caristi type fixed point theorems for multivalued mappings in modular
metric spaces, which is more general than the results of Khojasteh, Karapinar and Khandani[17]. And
we also give an application of results for functional equations.

2. Preliminaries

In this section, we will give some basic concepts and definitions about modular metric spaces.
Definition 2.1 [[9], Definition 2.1] Let X be a nonempty set, a function w : (0,∞)×X×X → [0,∞]

is said to be a metric modular on X if satisfying, for all x,y,z ∈ X the following condition holds:
(i) wλ (x,y) = 0 for all λ > 0 ⇔ x = y;

(ii) wλ (x,y) = wλ (y,x) for all λ > 0;
(iii) wλ+µ (x,y) ≤ wλ (x,z) +wµ (z,y) for all λ,µ > 0.
If instead of (i), we have only the condition(
i
)́
wλ (x,x) = 0 for all λ > 0, then w is said to be a (metric) pseudomodular on X.

2010 Mathematics Subject Classification. 46A80, 47H10, 54E35.
Key words and phrases. modular metric spaces; fixed point; Hausdorff metric.

c©2016 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

15



16 TURKOGLU, AND KILINÇ

The main property of a metric modular [[9]] w on a set X is the following: given x,y ∈ X, the
function 0 < λ 7→ wλ (x,y) ∈ [0,∞] is nonincreasing on (0,∞). In fact, if 0 < µ < λ, then (iii),

(
i
)́

and (ii) imply

wλ (x,y) ≤ wλ−µ (x,x) + wµ (x,y) = wµ (x,y) .

It follows that at each point λ > 0 the right limit wλ+0 (x,y) = lim
µ→λ+0

wµ (x,y) and the left limit

wλ−0 (x,y) = lim
ε→+0

wλ−ε (x,y) exist in [0,∞] and the following two inequalities hold:

wλ+0 (x,y) ≤ wλ (x,y) ≤ wλ−0 (x,y) .

Theorem 2.1 [[19]] Let Xw be a complete modular metric space and T a contraction on Xw. Then,
the sequence (Tnx)n∈N converges to the unique fixed point of T in Xw for any initial x ∈ Xw.

Now we give some definitions, which are useful for our main results.
Definition 2.2 Let Xw be a modular metric space. Then following definitions exist:

(1) The sequence (xn)n∈N in Xw is said to be convergent to x ∈ Xw if w1 (xn,x) → 0, as n →∞
(2) The sequence (xn)n∈N in Xw is said to be Cauchy if w1 (xm,xn) → 0, as m,n →∞
(3) A subset C of Xw is said to be closed if the limit of a convergent sequence of C always belong

to C.
(4) A subset C of Xw is said to be complete if any Cauchy sequence in C is a convergent sequence

and its limit is in C.
(5) A subset C of Xw is said to be w−bounded if

δw(C) = sup{w1(x,y); x,y ∈ C} < ∞.

(6) A subset C of Xw is said to be w−compact if for any (xn) in C there exists a subset sequence
(xnk) and x ∈ C such that w1(xnk,x) → 0

(7) w is said to satisfy the Fatou property if and only if for any sequence (xn)n∈N in Xw w−convergent
to x, we have

w1(x,y) ≤ lim inf
n→∞

w1(xn,y),

for any y ∈ Xw.
Now we will give some basic properties and notions of multivalued mappings in modular metric

spaces which was given in [2] For a subset M of modular metric space Xw set

(i) CB(M) = {C : C is nonempty w − closed and w − bounded subset of M} ;
(ii) K(M) = {C : C is nonempty w − compact subset of M} ;

(iii) the Haussdorf modular metric is defined onCB(M) by

Hw(A,B) = max

{
sup
x∈A

w1(x,B), sup
y∈B

w1(y,A)

}
,

where w1(x,B) = inf
y∈B

w1(x,y).

Definition 2.3 Let Xw be a complete modular metric space and M be a nonempty subset of Xw.
A mapping T : M → CB(M) is called a multivalued Lipschitzian mapping, if there exists a constant
k > 0 such that

Hw(Tx,Ty) ≤ kw1(x,y),

for any x,y ∈ M.
A point x ∈ M is called fixed point of T whenever x ∈ Tx.The set of fixed points of T will be

denoted by Fix(T)
It was shown in[2] that Definition 2.4 is more general than Theorem 2.1.



SOME FIXED POINT RESULTS IN MODULAR METRIC SPACES 17

3. main results

In this section we will give a fixed point theorem for Caristi type mappings and a generalization of
the theorem in modular metrc spaces. This works are more general than the results of [17].

Theorem Let Xw be a complete modular metric space and T : Xw → CB(Xw) be a nonexpansive
mapping such that for each x ∈ Xw and for all y ∈ Tx, there exists z ∈ Ty such that

(3.1) wλ(x,y) ≤
1

λ
(ϕw(x,y) −ϕw(y,z))

specificially for λ = 1 we can write

w1(x,y) ≤ ϕw(x,y) −ϕw(y,z)
where ϕ : Xw× Xw → [0,∞] is lower semicontinuous with respect to the firs variable. Then for

w1(xn,xn+1) < ∞, T has a fixed point.

Proof. Let x0 ∈ Xw and x1 ∈ Tx0. If x1 = x0, then x0 is a fixed point and theorem is satisfied.
Otherwise, let x0 6= x1. By assumption there exists x2 ∈ Tx1 such that

w1(x0,x1) ≤ ϕw(x0,x1) −ϕw(x1,x2)
Alternatively, one can choose xn ∈ Txn−1 such that xn 6= xn−1 and find xn+1 ∈ Txn such that

(3.2) 0 < w1(xn−1,xn) ≤ ϕw(xn−1,xn) −ϕw(xn,xn+1),
which means that (ϕw(xn−1,xn))n is a nonincreasing sequence in fact

0 < w1(xn−1,xn) + ϕw(xn,xn+1) ≤ ϕw(xn−1,xn)
0 < ϕw(xn,xn+1) ≤ ϕw(xn−1,xn)

Thus it is bounded below, so it converges to some r ≥ 0. By taking the limit on both sides of (3.2) we
get

lim
n→∞

w1(xn−1,xn) ≤ lim
n→∞

(ϕw(xn−1,xn) −ϕw(xn,xn+1))

lim
n→∞

w1(xn−1,xn) ≤ lim
n→∞

ϕw(xn−1,xn) − lim
n→∞

ϕw(xn,xn+1)

lim
n→∞

w1(xn−1,xn) = r −r

lim
n→∞

w1(xn−1,xn) = 0

Now we show that w1(xn,xm−n) is Cauchy sequence . For all m,n ∈ N with m > n,

(3.3) w1(xn,xm) ≤
m∑

i=n+1

w 1
m−n

(xi−1,xi)

On the other hand from the main property of metric modular one can choose that

w1(xn−1,xn) ≤ w 1
m−n

(xn−1,xn)

w 1
m−n

(xn−1,xn) ≤
1

(m−n)
ϕw(t)

when we write this in (3.3) we get

w1(xn,xm) ≤
1

m−n
(ϕw(xn−1,xn) −···−ϕw(xm,xm+1))

w1(xn,xm) ≤
1

m−n
(ϕw(xn−1,xn) −ϕw(xm,xm+1))



18 TURKOGLU, AND KILINÇ

When we take the limsup on both sides of the inequalities above, we have

lim
n→∞

(sup{w1(xn,xm) : m > n} ≤ lim
n→∞

(sup
1

(m−n)
(ϕw(xn−1,xn) −ϕw(xm,xm+1)))

lim
n→∞

(sup{w1(xn,xm) : m > n} ≤ lim
n→∞

(sup
1

(m−n)
)(r −r)

lim
n→∞

(sup{w1(xn,xm) : m > n} = 0

Thus (xn) is a Cauchy sequence. Since Xw is complete it converges to u ∈ Xw. Now we show that
u is a fixed point of T.We have

w1(u,Tu) ≤ w1
2
(u,xn+1) + w1

2
(Tu,xn+1)

≤ w1
2
(u,xn+1) + Hw(Tu,Txn)

≤ w1
2
(u,xn+1) + w1

2
(u,xn)

When we take the limit on both sides of the inequalities above, we get

lim
n→∞

w1(u,Tu) ≤ lim
n→∞

w1
2
(u,xn+1) + lim

n→∞
w1

2
(u,xn)

lim
n→∞

w1(u,Tu) ≤ w1
2
(u,u) + w1

2
(u,u)

lim
n→∞

w1(u,Tu) = 0

Hence we get u ∈ Tu. Therefore u is fixed point of T. �

Now let us give the theorem which is a generalized version of the theorem above.
Theorem Let Xw be a complete modular metric space and T : Xw → CB(Xw) be a multivalued

mapping such that

Hw(Tx,Ty) ≤ η(w1(x,y))
for all x,y ∈ Xw; where η : [0,∞] → [0,∞] is a lower semicontunious map such that η(t) < t for all

t ∈ [0,∞] , η(t)
t

is nondecreasing. Then T has a fixed point.

Proof. Let x ∈ Xw and y ∈ Tx. If x = y then T has a fixed point and the proof is complete, so we
suppose that x 6= y. Let define that

θ(t) =
η(t) + t

2
, for all t ∈ [0,∞]

Then we have

Hw(Tx,Ty) ≤ η(w1(x,y))
Since η is nondecreasing and η(t) < t we get

η(t) < t ⇔ η(w1(x,y)) ≤ w1(x,y)

θ(w1(x,y)) =
η(w1(x,y)) + w1(x,y)

2
≤ w1(x,y)

θ(w1(x,y)) =
η(w1(x,y)) + w1(x,y)

2
≥ η(w1(x,y))

Then we get

(3.4) Hw(Tx,Ty) ≤ η(w1(x,y)) ≤ θ(w1(x,y)) ≤ w1(x,y)

Thus there exists �0 > 0 such that θ(w1(x,y) = Hw(Tx,Ty)+ �0. So there exists z ∈ Ty such that



SOME FIXED POINT RESULTS IN MODULAR METRIC SPACES 19

(3.5) w1(y,z) ≤ Hw(Tx,Ty) + �0 = θ(w1(x,y)) ≤ w1(x,y)
Again suppose that y 6= z.Then

w1(x,y) −θ(w1(x,y)) ≤ w1(x,y) −w1(y,z)
or we can rewrite this inequality as

(3.6) w1(x,y) ≤
w1(x,y)

1 − θ(w1(x,y))
w1(x,y)

−
w1(y,z)

1 − θ(w1(x,y))
w1(x,y)

Since
θ(t)
t

is nondecreasing and w1(y,z) < w1(x,y) we rewrite the (3.5) as

w1(x,y) ≤
w1(x,y)

1 − θ(w1(x,y))
w1(x,y)

−
w1(y,z)

1 − θ(w1(y,z))
w1(y,z)

Define

φw(x,y) =

{
w1(x,y)

1−θ(w1(x,y))
w1(x,y)

,x 6= y

0 ,x = y

Therefore T satisfies (3.1) of Theorem 1, so we conclude that T has a fixed point u ∈ Xw and the
proof is complete. �

4. an application to functional equations

We use fixed point theory in many fields of mathematics. One of theese fields is mathematical opti-
mization.Dynamic programing is useful for mathematical optimizationand it is related to a multistage
process reduces to solving functional equation p(x). In this section we try to give an application of
Theorem 2 to a functional equation defined as

(4.1) p(x) = sup
y∈T

{f(x,y) + =(x,y,p(η(x,y)))} , x ∈ Z,

where η : Z ×T → Z, f : Z ×T → R, and = : Z ×T ×R → R. We assume that M and N are
Banach spacesZ ⊂Mand T ⊂N

Now we study the existence of the bounded solution of the functional equation (4.1). Let
B(Z) denote the set of all bounded real-valued functions on Z, and for an arbitrary h ∈ B(Z), define
‖h‖ = sup

x∈Z
|h(x)| . Clearly, (B(Z),‖.‖) endowed with the metric modular

(4.2) wλ(h,k) =
1

1 + λ
sup
x∈Z
|h(x) −k(x)|

specificially for λ = 1 writen as

w1(h,k) =
1

2
sup
x∈Z
|h(x) −k(x)|

for all h,k ∈B(Z), is Banach space, so the convergence in this space according to w1(h,k) is uniform
which means a Cauchy sequence in B(Z) is uniformly convergent to a function say h∗, that is bounded
and so h∗ ∈B(Z)

Now we will give the theorem that gives the solution of the functional equation defined in (4.1). To
give the solution let us define an operator as follows:

(4.3) S(h)(x) = sup
y∈T

{f(x,y) + =(x,y,h(η(x,y)))}

for all h∈B(Z) and x ∈Z
Now we give the theorem for the bounded solution of functional equation given in (4.4)



20 TURKOGLU, AND KILINÇ

Theorem Let S : B(Z) → B(Z)be an upper semicontinuous operator defined by (4.4) and assume
that the following conditions are satisfied:

(i) f :Z ×T→ R, and = : Z×T ×R → R are continuous and bounded;
(ii) for all h,k ∈B(Z), if

0 < w1(h,k) < 1

implies
1

2
|=(x,y,h(η(x,y))) −=(x,y,k(η(x,y)))| ≤ w21 (h,k)

w1(h,k) ≥
implies

1

2
|=(x,y,h(η(x,y))) −=(x,y,k(η(x,y)))| ≤ w1(h,k)

.where x ∈Z and y ∈T . Then the functional equation (4.1) has a bounded solution.

Proof. ClearlyB(Z) is a complete modular metric space .Let µ > 0 be arbitrary, x ∈ Z, and h1,h2 ∈
B(Z), then there exists y1,y2 ∈T such that

S(h1)(x) < f(x,y) + =(x,y,h1(η(x,y))) + µ
S(h2)(x) < f(x,y) + =(x,y,h2(η(x,y))) + µ
S(h1)(x) ≥ f(x,y) + =(x,y,h1(η(x,y)))
S(h2)(x) ≥ f(x,y) + =(x,y,h2(η(x,y)))

Let % : [0,∞] → [0,∞] be defined as

%(t) =

{
t2, 0 < t < 1
t, t ≥ 1

Then we get

1

2
|=(x,y,h(η(x,y))) −=(x,y,k(η(x,y)))| ≤ %(w1(h,k))

for all h,k ∈B(Z).It is clear that %(t) < t for all t > 0 and %(t)
t

is nondecreasing function. Therefore
when we use the inequalities above we get

1

2
(S(h1)(x) −S(h2)(x)) <

1

2
{=(x,y,h1(η(x,y))) −=(x,y,h2(η(x,y)))} + µ

≤
1

2
|=(x,y,h1(η(x,y))) −=(x,y,h2(η(x,y)))| + µ

≤ %(w1(h1,h2)) + µ
Then inequality turns into

1

2
{S(h1)(x) −S(h2)(x)} < %(w1(h1,h2)) + µ

Analogously we get

1

2
{S(h2)(x) −S(h1)(x)} < %(w1(h1,h2)) + µ

Hence theese inequalities equal to

1

2
|S(h1)(x) −S(h2)(x)| < %(w1(h1,h2)) + µ

which means that

w1(S(h1)(x),S(h2)(x)) < %(w1(h1,h2)) + µ



SOME FIXED POINT RESULTS IN MODULAR METRIC SPACES 21

Since the above inequality turns into

w1(S(h1)(x),S(h2)(x)) ≤ %(w1(h1,h2)),
hence we get that S is a %−contraction. And S satisfies the conditions in Theorem 2 and S has a fixed
point say h∗ ∈B(Z), which means is a bounded solution of the functional equation (4.1). �

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1Department of Mathematics, Faculty of Science, Gazi University, Ankara, Turkey

2Department of Mathematics, Institute of Natural and Applied Science, Gazi University, Ankara, Turkey

∗Corresponding author: eklnc07@gmail.com