### 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

*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*

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

_{∗}*(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*

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

_{∗}*(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 .*

**Proof. Suppose V is any open neighborhood of F (0). Consider open neighborhood U**

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

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

*∗*

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

**Proof. Consider x**

**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 .*

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

**Lemma 2.1. Suppose (X, τ ) is a locally convex space and x**

_{∗}*(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**Proof. Let U be an open convex neighborhood of 0 which is also balanced. From the**

*> 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}*(α*

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

^{m}### + *· · · + c*

^{n}*)α*

_{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*

*)α*

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*

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

_{∗}*(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*

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

*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*

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

*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*

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

*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.*

**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.*

**Corollary 2.2. Suppose (X, τ ) is a sequentially complete locally convex space, also suppose**

**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*

**Theorem 2.3. Let (X, τ ) be a locally convex space and f : X**

*o*

*∈ X there exists a convergence subsequence f*

^{n}

^{i}*(x*

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

**Proof. F is quasi contractive, so (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*

**Definition 2.3. A family**

*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*

**Lemma 2.2. [1] Suppose F : X**

*∗*

*(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**

**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*

**Theorem 2.4. Suppose X is a topological vector space, O(x) has the finite intersection**

*∗*

*(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 :

**Proof. Assume x /**

*(a) x = lim*

_{i}*y*

_{n}

_{i}*where n*

_{i}*≥ 2, y*

*n*

*i*

*∈ F*

^{n}

^{i}*(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}