Quasi Contraction and Fixed Points

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doi:10.5899/2012/jnaa-00168 Research Article

Quasi Contraction and Fixed Points

Mehdi Roohi

1

, Mohsen Alimohammady

2

(1) Department of Mathematics, Faculty of Sciences, Golestan University, P.O.Box. 155, Gorgan, Iran.

(2) Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran.

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Copyright 2012 c ⃝ Mehdi Roohi and Mohsen Alimohammady. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and re- production in any medium, provided the original work is properly cited.

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Abstract

In this note, we establish and improve some results on fixed point theory in topological vector spaces. As a generalization of contraction maps, the concept of quasi contraction multivalued maps on a topological vector space will be defined. Further, it is shown that a quasi contraction and closed multivalued map on a topological vector space has a unique fixed point if it is bounded value.

Keywords: : Topological vector space, Quasi contraction, Fixed point, Multi-valued map.

1 Introduction

Fixed point theory has many applications in almost all branches of mathematics. The Banach contraction principles occupies a central position in fixed point theory. Banach’s original theorem is expressed in metric spaces and some authors have extended this re- sult in some other versions [1], [4], [6] and [7]. Kirk [5] and Edelstien [3] studied and achieved some basic results in fixed point theory. Here, we would improve their results for multivalued maps in topological vector spaces.

For two sets X and Y and each element x of X we associate a nonempty subset F (x) of Y and this correspondence x 7→ F (x) is called a multivalued mapping or a multifunction from X into Y ; i.e., F is a function from X to P

(Y ), where P

(Y ) is the set of all nonempty subsets of Y . The lower inverse of a multivalued mapping F is the multi-valued mapping F

l

of Y into X defined by

F

l

(y) = {x ∈ X : y ∈ F (x)},

Corresponding author. Email address: m.roohi@gu.ac.ir

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also for any nonempty subset B of Y we have,

F

l

(B) = {x ∈ X : F (x) ∩ B ̸= ∅},

finally it is understood that F

l

( ∅) = ∅. The set {x ∈ X : F (x) ⊆ B} is the upper inverse of B and is denoted by F

u

(B). F is lower semicontinuous (upper semicontinuous), if for every open set U ⊆ Y , F

l

(U )(F

u

(U )) is open in X. It is well known [2] that F is upper semi-continuous at x

0

if for each open set V containing F (x

0

) there exists a neighborhood U of x

0

such that x ∈ U implies that F (x) ⊆ V.

For two multivalued maps F : X → P

(Y ) and G : Y → P

(Z) the composition GoF : X → P

(Z) is defined by

GoF (x) =

y∈F (x)

G(y).

2 Main Results

The following definition of quasi contraction is an extended definition of contraction in metric spaces and we achieve some results which they extend some results in invariant metric spaces.

Definition 2.1. Suppose (X, τ ) is a topological vector space. A multi-valued map F : X → P

(X) is said to be :

(a) quasi contraction map if for all x, y ∈ X and any open neighborhood U of 0 there is a constant 0 ≤ c < 1 such that x − y ∈ U implies that F (x) − F (y) ⊆ cU.

(b) closed map if x

n

→ x, y

n

→ y and y

n

∈ F (x

n

) imply that y ∈ F (x).

(c) bounded valued map if F (x) is a bounded set in X for all x ∈ X.

(d) upper semi-continuous at x

0

if for each open set V containing F (x

0

) there exists a neighborhood U (x

0

) such that x ∈ U(x

0

) implies that F (x) ⊆ V.

It is well known that any contraction multivalued map between metric spaces is con- tinuous. Extending this fact is our next aim.

Theorem 2.1. Suppose (X, τ ) is a topological vector space and F : X → P

(X) is quasi contraction. If 0 ∈ F (0) then F is upper semicontinuous at 0.

Proof. Suppose V is any open neighborhood of F (0). Consider open neighborhood U of 0 for which cU ⊆ V . Then U = U − {0} ⊆ U. Therefore, F (U) ⊆ F (U) − F (0) ⊆ cU ⊆ V .

Corollary 2.1. Suppose (X, τ ) is a topological vector space and F : X → P

(X) is quasi contraction. Then F is upper semicontinuous.

Proof. Consider x

0

∈ X, y

0

∈ F (x

0

) and V an open neighborhood of F (x

0

). Then V − y

0

= U is an open neighborhood of 0. Define G(x) = F (x + x

0

) − y

0

. We claim that g is contraction, too. To see this, if W is an open neighborhood of 0 and x − y ∈ W , then

G(x) − G(y) = (F (x + x

0

) − y

0

) − (F (y + x

0

) − y

0

) ⊆ cW.

Also, G(0) = F (x

0

) − y

0

contains 0, so from Theorem 2.1 for V − y

0

, there is open

neighborhood W of 0 such that G(W ) ⊆ V − y

0

, so F (W + x

0

) ⊆ V .

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Lemma 2.1. Suppose (X, τ ) is a locally convex space and x ∈ X. If F : X → P

(X) is a contraction bounded valued map, then any sequence {y

n

} is a Cauchy sequence, where y

n

∈ F

n

(x) for all n ∈ N.

Proof. Let U be an open convex neighborhood of 0 which is also balanced. From the assumption F (x) and so F (x) − x are bounded sets. Since U is absorbent, there is α

0

> 0 such that

F (x) − x ∈ αU, for all α with |α| ≥ α

0

,

so F (x) − x ∈ α

0

U . Then F

n+1

(x) − F

n

(x) ⊆ c

n

0

)U . Since 0 ≤ c < 1, there is N ∈ N such that (c

m

+ · · · + c

n

0

< 1 for all m, n ≥ N. Therefore, if m > n ≥ N we have

F

m+1

(x) − F

n+1

(x) ⊆ c

m

α

0

U + · · · + c

n

α

0

U

= (c

m

+ · · · + c

n

0

U

⊆ U.

This completes the proof.

The following result is other version of Banach contraction Theorem. First we need to the next lemma.

Theorem 2.2. Suppose (X, τ ) is a sequentially complete locally convex space and F : X → P

(X) is a quasi contraction bounded valued map. If F is closed multi-valued map, then F has a unique fixed point.

Proof. Suppose U is any open neighborhood of 0 and x is any element in X. Make a sequence {y

n

} in Y by induction, where y

1

∈ F (x) and y

n+1

∈ F (y

n

) for all n ∈ N. Ap- plying Lemma 2.1, {y

n

} is a Cauchy sequence. Since X is sequentially complete, so {y

n

} converges to an element y ∈ X. That y ∈ F (y) follows from closedness of F , y

n

∈ F (y

n−1

) and y

n−1

→ y. For the uniqueness, suppose x, y ∈ X are two distinct fixed points of F . Then there is a convex open neighborhood U of 0 such that x − y /∈ U. Since U is absorbent so there is α

0

> 0 such that x − y ∈ α

0

U . Therefore, F (x) − F (y) ⊆ cα

0

U and so x − y ∈ c

n

α

0

U for each n ∈ N. Since 0 ≤ c < 1, we can assume that c

n

α

0

< 1 for some n ∈ N. On the other hand, U is convex so c

n

α

0

U ⊆ U. Consequently, x − y ∈ U which is a contradiction.

Remark 2.1. It should be noticed that closedness in Theorem 2.2 could be reduced to the following condition

y

n

−→ y and y

n

∈ F (y

n−1

) = ⇒ y ∈ F (y).

As a special case of quasi contraction multi-valued maps, we introduce the quasi con- traction maps.

Definition 2.2. Suppose (X, τ ) is a topological vector space. A function f : X → X is

said to be quasi contraction if for all x, y ∈ X and any open neighborhood U of 0 there is

a constant 0 ≤ c < 1 such that x − y ∈ U implies that f(x) − f(y) ∈ cU.

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Corollary 2.2. Suppose (X, τ ) is a sequentially complete locally convex space, also suppose f : X → X is quasi contraction. Then f has a unique fixed point.

Proof. It is a direct result of Theorem 2.2.

Theorem 2.3. Let (X, τ ) be a locally convex space and f : X → X be a quasi contraction map. If for some x

o

∈ X there exists a convergence subsequence f

ni

(x

0

) to an element u ∈ X, then u is a fixed point for f.

Proof. F is quasi contractive, so (f

n

(x

0

))

n

is a Cauchy sequence from the proof of Lemma 2.1. Hence from the assumption f

n

(x

0

) −→ u. From Theorem 2.1, f is continu- ous, so f (u) = f (lim f

n

(x

0

)) = lim f

n+1

(x

0

) = u.

Definition 2.3. A family {A

j

: j ∈ J} of sets in X has finite intersection property if each finite subfamily of it, has nonempty intersection.

For a multi-valued map F : X → P

(X), set O(F

n

(x)) =

m≥n

F

m

(x), where it is understood that F

0

(x) = {x}. The following result would improve a result of Ciric [4].

First we need to the following lemma.

Lemma 2.2. [1] Suppose F : X → P

(X) is a multi-valued map and there is x

0

∈ X such that O(x

0

) has finite intersection property. Then F has a fixed point if O(F

2

(x)) ⊆ F (x) for all x ∈ X.

Proof. It is easy to see that F (O(x

0

)) ⊆ O(x

0

). Set K = {A ⊆ O(x

0

) : A ̸=

∅, F (A) ⊆ A}. Then partially ordered K by inclusion. Since O(x

0

) has finite intersection property, so from Zorns lemma K has minimal element, say C. Then F (C) ⊆ C, but F (F (C)) ⊆ F (C) implies that F (C) = C. Now, if u /∈ F (u) for each u ∈ C, then from as- sumption u / ∈ O(F

2

(u)). Since u ∈ C, so F (u) ⊆ F (C) = C, therefore F

k

(u) ⊆ C for any nonnegative integer k. Now O(F

2

(u)) = C follows from minimality of C. Consequently, u ∈ O(F

2

(u)) which is a contradiction.

Theorem 2.4. Suppose X is a topological vector space, O(x) has the finite intersection property for each x ∈ X and F : X → P

(X) is a multi-valued map. Then F has a fixed point if x / ∈ F (x) implies that x /∈ O(F

m

(x)) for all m ≥ 2.

Proof. Assume x / ∈ F (x). We claim that x /∈ O(F 2(x)). On the contrary there are two cases :

(a) x = lim

i

y

ni

where n

i

≥ 2, y

ni

∈ F

ni

(x) (b) x ∈ F

m

(x) for some m ≥ 2.

If (a) satisfies, then x ∈ (O(F m(x)))

⊆ O(F

m

(x)) hence, from the assumption

x ∈ F (x), which is a contradiction. Assume (b) satisfies, then x ∈ O(F

m

(x)) which

is impossible again. Therefore, x / ∈ O(F

2

(x)) and so O(F

2

(x)) ⊆ F (x). That F has a

fixed point follows from Lemma 2.2.

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