AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/406759 406759 Research Article On Angrisani and Clavelli Synthetic Approaches to Problems of Fixed Points in Convex Metric Space Gajić Ljiljana 1 Stojaković Mila 2 Carić Biljana 2 Kumam Poom 1 Department of Mathematics Faculty of Science University of Novi Sad 21000 Novi Sad Serbia uns.ac.rs 2 Department of Mathematics Faculty of Technical Sciences University of Novi Sad 21000 Novi Sad Serbia uns.ac.rs 2014 772014 2014 30 03 2014 22 06 2014 7 7 2014 2014 Copyright © 2014 Ljiljana Gajić et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The purpose of this paper is to prove some fixed point results for mapping without continuity condition on Takahashi convex metric space as an application of synthetic approaches to fixed point problems of Angrisani and Clavelli. Our results are generalizations in Banach space of fixed point results proved by Kirk and Saliga, 2000; Ahmed and Zeyada, 2010.

1. Introduction and Preliminaries

It is well-known that continuity is an ideal property, while in some applications the mapping under consideration may not be continuous, yet at the same time it may be “not very discontinuous.”

In  Angrisani and Clavelli introduced regular-global-inf functions. Such functions satisfy a condition weaker than continuity, yet in many circumstances it is precisely the condition needed to assure either the uniqueness or compactness of the set of solutions in fixed point problems.

Definition 1.

Function F:MR, defined on topological space M, is regular-global-inf (r.g.i.) in xM if F(x)>infM(F) implies that there exist an ε>0 such that ε<F(x)-infM(F) and a neighbourhood Nx such that F(y)>F(x)-ε for each yNx. If this condition holds for each xM, then F is said to be an r.g.i. on M.

An equivalent condition to be r.g.i. on metric space for infMf- is proved by Kirk and Saliga.

Proposition 2 (see [<xref ref-type="bibr" rid="B10">2</xref>]).

Let M be a metric space and F:MR. Then F is an r.g.i. on M if and only if, for any sequence {xn}M, the conditions (1)limnF(xn)=infM(F),limnxn=x imply F(x)=infM(F).

One of the basic results in  is the following one. (Here we use μ to denote the usual Kuratowski measure of noncompactness on metric space (M,d) and Lc={xMF(x)c} for F:MR,cR.)

Theorem 3 (see [<xref ref-type="bibr" rid="B2">1</xref>]).

Let F:MR be an r.g.i. defined on a complete metric space M. If limc(infM(F))+μ(Lc)=0, then the set of global minimum points of  F is nonempty and compact.

Remark 4.

The last theorem assures that if T is a mapping of compact metric space into itself with infM(F)=0, and if F(x)=d(x,Tx),xM, is an r.g.i. on M, then the fixed point set of T is nonempty and compact even when T is discontinuous.

Example 5.

Let (X,d) be a complete metric space and T:XX a mapping such that, for some q>1 and all x,yX, (2)d(Tx,Ty)qmax{d(x,y),d(x,Tx),d(y,Ty),ggggggggd(x,Ty),d(y,Tx)} (C´iric´ quasi-contraction). Then T  is discontinuous and F(x)=d(x,Tx), xX, is r.g.i. (see ).

Let A be a bounded subset of metric space M. The Kuratowski measure of noncompactness μ(A) means the inf of numbers ε such that A can be covered by a finite number of sets with a diameter less than or equal to ε. With β(A) we are going to denote the Hausdorff measure of noncompactness, where β(A) is the infimum of numbers ε such that A can be covered by a finite number of balls of radii smaller than ε.

It is easy to prove that for α{μ,β} and bounded subsets A,BM

α(A)=0A is totally bounded;

α(A-)=α(A);

ABα(A)α(B);

α(AB)=max{α(A),α(B)}.

Moreover, these two measures of noncompactness are equivalent in the sense that β(A)μ(A)2β(A) so limnμ(An)=0 if and only if limnβ(An)=0 (for any sequence {An} of bounded subsets of M). The last property indicates that fixed point results are independent of choice of measure of noncompactness.

In Banach spaces this function has some additional properties connected with the linear structure. One of these is (3)α(convA)=α(A) (convA is a convex hull of A—the intersection of all convex sets in X containing A).

This property has a great importance in fixed point theory. In locally convex spaces this is always true, but when topological vector space is not locally convex it need not be true (see ).

In the absence of linear structure the concept of convexity can be introduced in an abstract form. In metric spaces at first it was done by Menger in 1928. In 1970 Takahashi  introduced a new concept of convexity in metric space.

Definition 6 (see [<xref ref-type="bibr" rid="B14">4</xref>]).

Let (X,d) be a metric space and I a closed unit interval. A mapping W:X×X×IX is said to be convex structure on X if for all x,y,uX,λI, (4)d(u,W(x,y,λ))λd(u,x)+(1-λ)d(u,y).X together with a convex structure is called a (Takahashi) convex metric space (X,d,W) or abbreviated TCS.

Any convex subset of a normed space is a convex metric space with W(x,y,λ)=λx+(1-λ)y.

Definition 7 (see [<xref ref-type="bibr" rid="B14">4</xref>]).

Let (X,d,W) be a TCS. A nonempty subset K of X is said to be convex if and only if W(x,y,λ)K whenever x,yK and λI.

Proposition 8 (see [<xref ref-type="bibr" rid="B14">4</xref>]).

Let (X,d,W) be a TCS. If x,yX and λI, then

W(x,y,1)=x and W(x,y,0)=y;

W(x,x,λ)=x;

d(x,W(x,y,λ))=(1-λ)d(x,y) and d(y,W(x,y,λ))=λd(x,y);

balls (either open or closed) in X are convex;

intersections of convex subsets of X are convex.

For fixed x,yX let [x,y]={W(x,y,λ)λI}.

Definition 9.

A TCS (X,d,W) has property (P) if for every x1,x2,y1,y2X,λI, (5)d(W(x1,x2,λ),W(y1,y2,λ))λd(x1,y1)+(1-λ)d(x2,y2).

Obviously in a normed space the last inequality is always satisfied.

Example 10 (see [<xref ref-type="bibr" rid="B14">4</xref>]).

Let (X,d) be a linear metric space with the following properties:

d(x,y)=d(x-y,0), for all x,yX;

d(λx+(1-λ)y,0)λd(x,0)+(1-λ)d(y,0), for all x,yX and λI.

For W(x,y,λ)=λx+(1-λ)y,x,yX,λI,(X,d,W) is a TCS with property (P).

Remark 11.

Property (P) implies that convex structure W is continuous at least in first two variables which gives that the closure of convex set is convex.

Definition 12.

A TCS (X,d,W) has property (Q) if for any finite subset AX convA is a compact set.

Example 13 (see [<xref ref-type="bibr" rid="B14">4</xref>]).

Let K be a compact convex subset of Banach space and let X be the set of all nonexpansive mappings on K into itself. Define a metric on X by d(A,B)=supxKAx-Bx,A,BX and W:X×X×IX by W(A,B,λ)(x)=λAx+(1-λ)Bx, for xK and λI. Then (X,d,W) is a compact TCS, so X is with property (Q). The property (P) is also satisfied.

Talman in  introduced a new notion of convex structure for metric space based on Takahashi notion—the so called strong convex structure (SCS for short). In SCS condition (Q) is always satisfied so it seems to be “natural.”

Any TCS satisfying (P) and (Q) has the next important property.

Proposition 14 (see [<xref ref-type="bibr" rid="B15">5</xref>]).

Let (X,d,W) be a TCS with properties (P) and (Q). Then for any bounded subset AX(6)α(convA)=α(A).

Some, among the many studies concerning the fixed point theory in convex metric spaces, can be found in .

2. Main Results

Measures of noncompactness which arise in the study of fixed point theory usually involve the study of either condensing mappings or k-set contractions. Continuity is always implicit in the definitions of these classes of mappings. Kirk and Saliga  show that in many instances it suffices to replace the continuity assumption with the weaker r.g.i. condition. We are going to follow this idea in frame of TCS.

Theorem 15.

Let (X,d,W) be a complete TCS with properties (P) and (Q), K a closed convex bounded subset of X, and T:KK a mapping satisfying the following:

infC(F)=0 for any nonempty closed convex T-invariant subset C of  K, where F(x)=d(x,Tx),xK;

α(T(A))<α(A) for all AK for which α(A)>0;

F is r.g.i. on K.

Then the fixed point set fix (T) of  T is nonempty and compact.

Proof.

Choose a point mK. Let σ denote the family of all closed convex subsets A of K for which mA and T(A)A. Since Kσ, σ. Let (7)B:=AσA,C:=conv{T(B){m}}¯. Convex structure W has property (P) so C is a convex set as a closure of convex set. We are going to prove that B=C.

Since B is a closed convex set containing T(B) and {m},CB. This implies that T(C)T(B)C so Cσ and hence BC. The last two statements clearly force B=C.

Properties (1)–(4) of measure α and Proposition 14 imply that (8)α(B)=α(conv{T(B){m}})¯=α(T(B)), so in view of (ii) B must be compact.

Now, Proposition 2 ensures that T has a fixed point on B so fix(T) is nonempty. Condition (ii) implies that fix(T) is totally bounded. Since F is r.g.i. fix(T) has to be closed. Finally, we conclude that fix(T) is compact.

The assumption infK(F)=0 is strong, especially in the absence of conditions which at the same time imply continuity. So we are going to give some sufficient conditions which are easier to check and more suitable for application.

Let us recall some well-known definitions. A mapping T:KK is called nonexpansive if d(Tx,Ty)d(x,y), for all x,yK, and directionally nonexpansive if d(Tx,Ty)d(x,y) for each xK and y[x,Tx]. If there exists α(0,1) such that this inequality holds for y=W(Tx,x,α), then we say that T is uniformly locally directionally nonexpansive.

Proposition 16.

Let (X,d,W) be a complete TCS with property (P), K a closed convex bounded subset of X, and T:KK a uniformly locally directionally nonexpansive. Let Tαx=W(Tx,x,α). For the fixed x0K, sequences {xn} and {yn} are defined as follows: (9)xn+1=Tαxn,yn=Txn,n=0,1,2,. Then for each i,nN(10)d(yi+n,xi)(1-α)-n(d(yi+n,xi+n)-d(yi,xi))+(1+nα)d(yi,xi),(11)limnd(xn,Txn)=0.

Proof.

We prove (10) by induction on n. For n=0 inequality (10) is trivial. Assume that (10) holds for given n and all i.

In order to prove that (10) holds for n+1, we proceed as follows: replacing i with i+1 in (10) yields (12)d(yi+n+1,xi+1)(1-α)-n(d(yi+n+1,xi+n+1)-d(yi+1,xi+1))+(1+nα)d(yi+1,xi+1). Also (13)d(yi+n+1,xi+1)d(yi+n+1,W(yi+n+1,xi,α))+d(W(yi+n+1,xi,α),W(Txi,xi,α))(1-α)d(yi+n+1,xi)+αd(yi+n+1,Txi)(1-α)d(yi+n+1,xi)+αk=0nd(Txi+k+1,Txi+k)(1-α)d(yi+n+1,xi)+αk=0nd(xi+k+1,xi+k) since xi+k+1=W(Txi+k,xi+k,α) and T is uniformly locally directionally nonexpansive. Combining (12) and (13) (14)d(yi+n+1,xi)(1-α)-(n+1)(d(yi+n+1,xi+n+1)-d(yi+1,xi+1))+(1-α)-1(1+nα)d(yi+1,xi+1)-α(1-α)-1k=0nd(xk+i+1,xk+i). By Proposition 8 (c), (15)d(xk+i+1,xk+i)=d(W(Txk+i,xk+i,α),xk+i)=αd(yk+i,xk+i), so (16)d(yi+n+1,xi)(1-α)-(n+1)(d(yi+n+1,xi+n+1)-d(yi+1,xi+1))+(1-α)-1(1+nα)d(yi+1,xi+1)-α2(1-α)-1k=0nd(yk+i,xk+i). On the other hand, (17)d(yn,xn)=d(Txn,W(Txn-1,xn-1,α))d(Txn,Txn-1)+d(Txn-1,W(Txn-1,xn-1,α))d(xn,xn-1)+(1-α)d(Txn-1,xn-1)=αd(yn-1,xn-1)+(1-α)d(yn-1,xn-1)=d(yn-1,xn-1) for any nN, meaning that {d(yn,xn)} is a decreasing sequence.

Now, using inequality (1+nα)-(1-α)-n0, we have that (18)d(yi+n+1,xi)(1-α)-(n+1)(d(yi+n+1,xi+n+1)-d(yi+1,xi+1))+(1-α)-1(1+nα)d(yi+1,xi+1)-α2(1-α)-1(n+1)d(yi,xi)=(1-α)-(n+1)(d(yi+n+1,xi+n+1)-d(yi,xi))+((1-α)-1(1+nα)-(1-α)-(n+1))d(yi+1,xi+1)+((1-α)-(n+1)-α2(1-α)-1(n+1))d(yi,xi)(1-α)-(n+1)(d(yi+n+1,xi+n+1)-d(yi,xi))+((1-α)-1(1+nα)-(1-α)-(n+1))d(yi,xi)+((1-α)-(n+1)-α2(1-α)-1(n+1))d(yi,xi)=(1-α)-(n+1)(d(yi+n+1,xi+n+1)-d(yi,xi))+(1+(n+1)α)d(yi,xi). Thus (10) holds for n+1, completing the proof of inequality.

Further, the sequence {d(yn,xn)} is decreasing, so there exists limnd(yn,xn)=r0. Let us suppose that r>0. Select positive integer n0d/(r·α),d=diamK, and ε>0, satisfying ε(1-α)-n0<r. Now choose positive integer k such that (19)0d(yk,xk)-d(yk+n0,xk+n0)<ε. Using (10), we obtain (20)d+rr(αn0+1)(αn0+1)d(yk,xk)d(yk+n0,xk)+ε(1-α)-n0<d+r.

By the last contradiction we conclude that r=0 and limnd(yn,xn)=limnd(Txn,xn)=0 what we had to prove.

Remark 17.

This statement is a generalization of Lemma 9.4 from .

Combining the last result with Theorem 15 we have the following consequence.

Corollary 18.

Let K be a bounded closed convex subset of complete TCS (X,d,W) with properties (P) and (Q) and let T:KK satisfy the following:

T is uniformly locally directionally nonexpansive on K;

α(T(A))<α(A), for all AK for which α(A)>0;

F is r.g.i. on K.

Then the fixed point set fix (T) of T is nonempty and compact.

Moreover, using Proposition 16 we also get generalizations of some other Kirk and Saliga  fixed point results.

Corollary 19.

Let K be a bounded closed convex subset of a complete TCS (X,d,W) with properties (P) and (Q) and let T:KK satisfy the following:

T is uniformly locally directionally nonexpansive on K;

d(Tx,Ty)θ(max{d(x,Tx),d(y,Ty)}), where θ:R+R+ is any function for which limt0+θ(t)=0.

Then T has a unique fixed point x0K if and only if  F is an r.g.i. on K.

Proof.

Proposition 16 gives infK(F)=0 and as in  one can prove that limc0+diam(Lc)=0. By Theorem 1.2 , T has a unique fixed point if and only if F is r.g.i. on K.

Theorem 20.

Let K be a bounded closed convex subset of a complete TCS (X,d,W) with properties (P) and (Q) and suppose T:KK satisfies the following:

T is directionally nonexpansive on K;

μ(T(Lc))k·μ(Lc), for some k<1 and all c>0;

F is an r.g.i. on K.

Then the fixed point set fix (T) of T is nonempty and compact. Moreover, if {xn}K satisfies limnd(xn,Txn)=0, then limnd(xn, fix (T))=0.

Proof.

By Proposition 16, infK(F)=0. Since (i) implies that (21)d(Tx,T2x)d(x,Tx),xK, the conclusion follows immediately from Theorem 2.3 .

We established that limnd(xn,Txn)=0 for every sequence {xn} defined by xn=Tαxn-1, nN, where x0K and α(0,1). Therefore limnd(xn,fix(T))=0 meaning that {xn} converges to the set fix(T), but the convergence to the specific point from fix(T) is not provided. Putting some additional assumption, we could arrange that the sequence {xn} converges to a fixed point of the mapping T.

Next, we recall the concept of weakly quasi-nonexpansive mappings with respect to sequence introduced by Ahmed and Zeyada in .

Definition 21 (see [<xref ref-type="bibr" rid="B1">15</xref>]).

Let (X,d) be a metric space and let {xn} be a sequence in DX. Assume that T:DX is a mapping with fix(T) satisfying limnd(xn,fix(T))=0. Thus, for a given ε>0 there exists n1(ε)N such that d(xn,fix(T))<ε for all nn1(ε). Mapping T is called weakly quasi-nonexpansive with respect to {xn}D if for each ε>0 there exists p(ε)fix(T) such that, for all nN with nn1(ε),d(xn,p(ε))<ε.

The next result is improvement of Theorem 20 and also a generalisation of Theorem 2.24 from .

Theorem 22.

Let K be a bounded closed convex subset of a complete TCS (X,d,W) with properties (P) and (Q) and let T:KK satisfy the following:

T is directionally nonexpansive on K;

α(T(Lc))kα(Lc) for some k<1 and all c>0;

F is r.g.i. on K;

{xn}K satisfies limlimnd(xn,Txn)=0 and T is weakly quasi-nonexpansive with respect to {xn}.

Then {xn} converges to a point in fix (T).

Proof.

Our assertion is a consequence of Theorem 20 and Theorem 2.5(b) from .

Using Proposition 16, the next corollary holds.

Corollary 23.

Let K be a bounded closed convex subset of a complete TCS (X,d,W) with properties (P) and (Q) and let T:KK satisfy the following:

T is directionally nonexpansive on K;

α(T(Lc))kα(Lc) for some k<1 and all c>0;

F is r.g.i. on K;

T is weakly quasi-nonexpansive with respect to sequence xn=Tαnx0,nN,x0K,α(0,1).

Then {xn} converges to a point in fix (T).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are very grateful to the anonymous referees for their careful reading of the paper and suggestions which have contributed to the improvement of the paper. This paper is partially supported by Ministarstvo nauke i zˇivotne sredine Republike Srbije.

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