International Journal of Analysis and Applications
ISSN 2291-8639
Volume 1, Number 2 (2013), 123-127
http://www.etamaths.com

FIXED POINT THEOREM ON MULTI-VALUED MAPPINGS

J.MARIA JOSEPH1,∗, E. RAMGANESH2

Abstract. In this paper, we prove a common fixed point theorem for two
multivalued self-mappings in complete metric spaces.

1. Introduction and preliminaries:

The study of fixed points for set valued contractions and nonexpansive maps
using the Hausdorff metric was initiated by Markin. Later, an interesting and rich
fixed point theory for such maps has been developed. The theory of set valued
maps has applications in control theory, convex optimization, differential inclusions
and economics.

Following the Banach contraction principle Nadler introduced the concept of set
valued contractions and established that a set valued contraction possesses a fixed
point in a complete metric space. Subsequently many authors generalized Nadlers
fixed point theorem in different ways[[1],[2]].

Definition 1.1. Let X and Y be nonempty sets. T is said to be a multi-valued
mapping from X to Y if T is a function from X to the power set of Y . We denote
a multi-valued map by T : X → 2Y .

Definition 1.2. A point x0 ∈ X is said to be a fixed point of the multi-valued
mapping T if x0 ∈ Tx0.

Example 1.3. Every single valued mapping can be viewed as a multi-valued map-
ping. Let f : X → Y be a single valued mapping.
Define T : X → 2Y by Tx = {f(x)}.
Note that T is multi-valued mapping iff for each x ∈ X,Tx ⊆ Y. Unless otherwise
stated we always assume Tx is non-empty for each x ∈ X.

Definition 1.4. Let (X,d) be a metric space. A map T : X → X is called con-
traction if there exists 0 ≤ λ < 1 such that d(Tx,Ty) ≤ λd(x,y), for all x,y ∈ X.

Definition 1.5. Let (X,d) be a metric space. We define the Hausdorff metric on
CB(X) induced by d. That is

H(A,B) = max

{
sup
x∈A

d(x,B), sup
y∈B

d(y,A)

}
for all A,B ∈ CB(X), where CB(X) denotes the family of all nonempty closed and
bounded subsets of X and d(x,B) = inf{d(x,b) : b ∈ B}, for all x ∈ X.

2010 Mathematics Subject Classification. 54H25,47H10.

Key words and phrases. Fixed point, Multivalued map, Hausdorff metric.

c©2013 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

123



124 JOSEPH AND RAMGANESH

Definition 1.6. Let (X,d) be a metric space. A map T : X → CB(X) is said to be
multi valued contraction if there exists 0 ≤ λ < 1 such that H(Tx,Ty) ≤ λd(x,y),
for all x,y ∈ X.

Lemma 1.7. [3] If A,B ∈ CB(X) and a ∈ A, then for each ε > 0, there exists
b ∈ B such that

d(a,b) ≤ H(A,B) + ε.

2. Main Results

Theorem 2.1. Let(X,d) be complete metric space and let S,T : X → CB(X)
be multivalued maps satisfying H(Tx,Sy) ≤ ad(x,Ty) + b(d(x,Sy) + d(Ty,Tx)),
where 0 < a + 2b < 1, a,b ≥ 0, for all x,y ∈ X. Then F(T) = F(S) 6= ∅ and
Tx = Sx = F(T), for all x ∈ F(T).

Proof. Fix any x ∈ X. Define x0 = x and let x1 ∈ Tx0. By lemma(1.7), we may
choose x2 ∈ Sx0 such that d(x1,x2) ≤ H(Tx0,Sx0) + (a + b). Now

d(x1,x2) ≤ H(Tx0,Sx0) + (a + b)
≤ ad(x0,x1) + b(d(x0,x1) + d(x1,x2)) + (a + b)

≤
a + b

1 − b
d(x0,x1) +

a + b

1 − b
.

By lemma(1.7), there exists x3 ∈ Tx2 such that

d(x3,x2) ≤ H(Tx2,Sx0) +
(a + b)2

1 − b
.

Now

d(x3,x2) ≤ H(Tx2,Sx0) +
(a + b)2

1 − b

≤ ad(x2,x1) + b(d(x2,x2) + d(x1,x3)) +
(a + b)2

1 − b

≤
a + b

1 − b
d(x2,x1) + (

a + b

1 − b
)2

≤ (
a + b

1 − b
)2d(x0,x1) + 2(

a + b

1 − b
)2

Continuing this process, we obtain by induction a sequence {xn} such that x2n ∈
Sx2n−2, x2n+1 ∈ Tx2n, such that

d(x2n+1,x2n+2) ≤ H(Tx2n,Sx2n) +
(a + b)2n+1

(1 − b)2n
,

d(x2n,x2n+1) ≤ H(Sx2n−2,Tx2n) +
(a + b)2n

(1 − b)2n−1
,



FIXED POINT THEOREM ON MULTI-VALUED MAPPINGS 125

Now,

d(x2n,x2n+1) ≤ H(Sx2n−2,Tx2n) +
(a + b)2n

(1 − b)2n−1
≤ ad(x2n,Tx2n−2) + b(d(x2n,Sx2n−2) + d(Tx2n−2,Tx2n)

+
(a + b)2n

(1 − b)2n−1
≤ ad(x2n,x2n−1) + b(d(x2n−1,x2n) + d(x2n,x2n+1)

+
(a + b)2n

(1 − b)2n−1

≤
(a + b)

(1 − b)
d(x2n−1,x2n) +

(a + b)2n

(1 − b)2n

Also,

d(x2n+1,x2n+2) ≤ H(Tx2n,Sx2n) +
(a + b)2n+1

(1 − b)2n
≤ ad(x2n,Tx2n) + b(d(x2n,Sx2n) + d(Tx2n,Tx2n)

+
(a + b)2n+1

(1 − b)2n
≤ ad(x2n,x2n+1) + b(d(x2n,x2n+1) + d(x2n+1,x2n+2)

+
(a + b)2n+1

(1 − b)2n

≤
(a + b)

(1 − b)
d(x2n,x2n+1) +

(a + b)2n+1

(1 − b)2n+1

Therefore,

d(xn,xn+1) ≤
(a + b)

(1 − b)
d(xn−1,xn) +

(a + b)n

(1 − b)n

for all n ∈ N and let k = (a+b)
(1−b)

d(xn,xn+1) ≤ kd(xn−1,xn) + kn

≤ k(kd(xn−2,xn−1) + kn−1) + kn

= k2(d(xn−2,xn−1)) + kk
n−1 + kn

≤ ...
≤ knd(x0,x1) + nkn.

Since k < 1 ,
∑
kn and

∑
nkn have same radius of convergence, {xn} is a Cauchy

sequence. Since (X,d) is complete, there exists z ∈ X such that xn → z.
d(Tz,z) ≤ d(z,x2n+2) + d(x2n+2,Tz)

≤ d(z,x2n+2) + H(Tz,Sx2n)
≤ d(z,x2n+2) + [ad(z,Tx2n) + b(d(z,Sx2n) + d(Tx2n,Tz))]
≤ d(z,x2n+2) + [ad(z,x2n+1) + b(d(z,x2n+2) + d(x2n+1,Tz))]
→ ad(z,z) + b[d(z,z) + d(z,Tz)] as n →∞.

Therefore d(Tz,z)(1 − b) ≤ 0. Hence
d(Tz,z) = 0



126 JOSEPH AND RAMGANESH

H(Tz,Sz) ≤ ad(z,Tz) + b(d(z,Sz) + d(Tz,Tz))]
= ad(z,Tz) + bd(z,Sz)

≤ ad(z,Tz) + bd(z,Tz) + bd(Tz,Sz)
≤ (a + b)d(z,Tz) + bH(Tz,Sz)

H(Tz,Sz) ≤ (
a + b

1 − b
)d(z,Tz)

Hence, H(Tz,Sz) = 0, z ∈ Tz = Sz and therefore z ∈ F(T) 6= ∅, z ∈ F(S) 6= ∅, To
complete the proof, it is enough to show following four cases:
(i) F(T) ⊆ Tz and Sx = Tx for all x ∈ F(T).
(ii) Tz ⊆ F(T)
(iii) Tx = Tz for all x ∈ F(T)
(iv) F(S) ⊆ Tz
For any x ∈ F(T),

d(x,Tz) ≤ H(Tx,Sz)

≤ (
a + b

1 − b
)d(x,Tz)

This shows that d(x,Tz) = 0 and x ∈ Tz.Further

H(Sx,Tx) ≤ (
a + b

1 − b
)d(x,Tx) = 0

and x ∈ Sx = Tx.For any x ∈ Tz

d(x,Tx) ≤ H(Sz,Tx) ≤ (
a + b

1 − b
)d(Tz,x) = 0

This shows that x ∈ Tx. Now, we see thatTz = F(T) ⊆ F(S)and Sx = Tx for all
x ∈ F(T). For any x ∈ F(T),

H(Tx,Sz) ≤ (
a + b

1 − b
)d(x,Tz)

= (
a + b

1 − b
)d(x,F(T)) = 0

Hence, Tx = Sz = Tz. It remains to show that F(S) ⊆ Tz = F(T). For any
x ∈ F(S),

d(x,Tz) ≤ H(Tx,Sz)

≤ (
a + b

1 − b
)d(Tx,z)

≤ (
a + b

1 − b
)H(Tx,Sz)

≤ (
a + b

1 − b
)2d(x,Tz)

Hence, d(x,Tz) = 0. Then x ∈ Tz and F(S) ⊆ Tz. �

In what follows, let ( denote multimap.

Corollary 2.1. Let T : X ( X be a multivalued map with nonempty compact
values and r ∈ [0, 1) such that

H(Tx,T2y) ≤ rd(x,Ty),



FIXED POINT THEOREM ON MULTI-VALUED MAPPINGS 127

for all x,y ∈ X. Then, F(T) 6= ∅ and Tx = F(T) for all x ∈ F(T).

Remark 2.2. Let S be a self mapping (multi valued or single valued) defined on
X, we denote F(S) the collection of all fixed points of S.
If one of S and T in Theorem 2.1 is single valued, then the set F(T) = F(S) is
singleton and the maps S and T have a unique common fixed point in X.

3. Acknowledgments

The authors would like to thank the editor of the paper and the referees for their
precise remarks to improve the presentation of the paper.

References

[1] Lai-Jiu Lin and Sung-Yu Wang,Common Fixed Point Theorems for a Finite Family of Dis-
continuous and Noncommutative Maps,Fixed Point Theory and Applications,vol.2011,Article

ID 847170,19 pages.

[2] J. Maria Joseph, M.Marudai, Common Fixed Point Theorem for Set-Valued Maps and a
Stationary Point Theorem, Int. Journal of Math. Analysis, 6, (2012), no. 33, 1615 - 1621.

[3] S.B. Nadler, Multi-valued contraction mappings, Pacific Journal of Mathematics, 30 (1969),
475–488.

1Department of Mathematics, St.Joseph’s college,Tiruchirappalli,Tamil Nadu,India

2Department of Educational Technology, Bharathidasan University,Tiruchirappalli,Tamil
Nadu,India

∗Corresponding author