# On Angrisani and Clavelli Synthetic Approaches to  Problems of Fixed Points in Convex Metric Space

Authors: Ljiljana Gajić; Mila Stojaković; Biljana Carić
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

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.


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## Body

## 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 [1] 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.
FunctionF:M→R, defined on topological space M, is regular-global-inf (r.g.i.) in x∈M if F(x)>inf⁡M(F) implies that there exist an ε>0 such that ε<F(x)-inf⁡M(F) and a neighbourhood Nx such that F(y)>F(x)-ε for each y∈Nx. If this condition holds for each x∈M, then F is said to be an r.g.i. on M.An equivalent condition to be r.g.i. on metric space forinf⁡Mf≠-∞ is proved by Kirk and Saliga.Proposition 2 (see [2]).
LetM be a metric space and F:M→R. Then F is an r.g.i. on M if and only if, for any sequence {xn}⊂M, the conditions
(1)lim⁡n→∞F(xn)=inf⁡M(F),lim⁡n→∞xn=x
imply F(x)=inf⁡M(F).One of the basic results in [1] is the following one. (Here we use μ to denote the usual Kuratowski measure of noncompactness on metric space (M,d) and Lc∶={x∈M∣F(x)≤c} for F:M→R,c∈R.)Theorem 3 (see [1]).
LetF:M→R be an r.g.i. defined on a complete metric space M. If lim⁡c→(inf⁡M(F))+μ(Lc)=0, then the set of global minimum points of  F is nonempty and compact.Remark 4.
The last theorem assures that ifT is a mapping of compact metric space into itself with inf⁡M(F)=0, and if F(x)∶=d(x,Tx),x∈M, 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:X→X a mapping such that, for some q>1 and all x,y∈X,
(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), x∈X, is r.g.i. (see [1]).LetA be a bounded subset of metric space M. TheKuratowski measure of noncompactnessμ(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 theHausdorff 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 forα∈{μ,β} and bounded subsets A,B⊆M(1)
α(A)=0⇔A is totally bounded;(2)
α(A-)=α(A);(3)
A⊂B⇒α(A)≤α(B);(4)
α(A∪B)=max⁡{α(A),α(B)}.Moreover, these two measures of noncompactness are equivalent in the sense thatβ(A)≤μ(A)≤2β(A) so lim⁡nμ(An)=0 if and only if lim⁡nβ(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 [3]).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 [4] introduced a new concept of convexity in metric space.Definition 6 (see [4]).
Let(X,d) be a metric space and I a closed unit interval. A mapping W:X×X×I→X is said to beconvex structure on X if for all x,y,u∈X,λ∈I,
(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 withW(x,y,λ)=λx+(1-λ)y.Definition 7 (see [4]).
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,y∈K and λ∈I.Proposition 8 (see [4]).
Let(X,d,W) be a TCS. If x,y∈X and λ∈I, then (a)
W(x,y,1)=x and W(x,y,0)=y;(b)
W(x,x,λ)=x;(c)
d(x,W(x,y,λ))=(1-λ)d(x,y) and d(y,W(x,y,λ))=λd(x,y);(d)
balls (either open or closed) inX are convex;(e)
intersections of convex subsets ofX are convex.For fixedx,y∈X let [x,y]={W(x,y,λ)∣λ∈I}.Definition 9.
A TCS(X,d,W) hasproperty(P) if for every x1,x2,y1,y2∈X,λ∈I,
(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 [4]).
Let(X,d) be a linear metric space with the following properties: (1)
d(x,y)=d(x-y,0), for all x,y∈X;(2)
d(λx+(1-λ)y,0)≤λd(x,0)+(1-λ)d(y,0), for all x,y∈X and λ∈I.
ForW(x,y,λ)=λx+(1-λ)y,x,y∈X,λ∈I,(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) hasproperty(Q) if for any finite subset A⊆X convA is a compact set.Example 13 (see [4]).
LetK 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)=sup⁡x∈K∥Ax-Bx∥,A,B∈X and W:X×X×I→X by W(A,B,λ)(x)=λAx+(1-λ)Bx, for x∈K and λ∈I. Then (X,d,W) is a compact TCS, so X is with property (Q). The property (P) is also satisfied.Talman in [5] introduced a new notion of convex structure for metric space based on Takahashi notion—the so calledstrong 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 [5]).
Let(X,d,W) be a TCS with properties (P) and (Q). Then for any bounded subset A⊆X(6)α(convA)=α(A).Some, among the many studies concerning the fixed point theory in convex metric spaces, can be found in [6–13].


## 2. Main Results

Measures of noncompactness which arise in the study of fixed point theory usually involve the study of either condensing mappings ork-set contractions. Continuity is always implicit in the definitions of these classes of mappings. Kirk and Saliga [2] 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:K→K a mapping satisfying the following:(i)
inf⁡C(F)=0 for any nonempty closed convex T-invariant subset C of  K, where F(x)=d(x,Tx),x∈K;(ii)
α(T(A))<α(A) for all A⊆K for which α(A)>0;(iii)
F is r.g.i. on K. Then the fixed point set fix(T) of  T is nonempty and compact.Proof.
Choose a pointm∈K. Let σ denote the family of all closed convex subsets A of K for which m∈A and T(A)⊆A. Since K∈σ, σ≠∅. Let
(7)B:=⋂A∈σA,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.
SinceB is a closed convex set containing T(B) and {m},C⊆B. This implies that T(C)⊆T(B)⊆C so C∈σ and hence B⊆C. The last two statements clearly force B=C.
Properties (1)–(4) of measureα 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, Proposition2 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 assumptioninf⁡K(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 mappingT:K→K is callednonexpansive if d(Tx,Ty)≤d(x,y), for all x,y∈K, anddirectionally nonexpansive if d(Tx,Ty)≤d(x,y) for each x∈K 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 isuniformly 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:K→K a uniformly locally directionally nonexpansive. Let Tαx=W(Tx,x,α). For the fixed x0∈K, sequences {xn} and {yn} are defined as follows:
(9)xn+1=Tαxn,yn=Txn,n=0,1,2,….
Then for each i,n∈N(10)d(yi+n,xi)≥(1-α)-n(d(yi+n,xi+n)-d(yi,xi))+(1+nα)d(yi,xi),(11)lim⁡n→∞d(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)+α∑k=0n‍d(Txi+k+1,Txi+k)≤(1-α)d(yi+n+1,xi)+α∑k=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)-1∑k=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)-1∑k=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 n∈N, meaning that {d(yn,xn)} is a decreasing sequence.
Now, using inequality(1+nα)-(1-α)-n≤0, 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 lim⁡n→∞d(yn,xn)=r≥0. Let us suppose that r>0. Select positive integer n0≥d/(r·α),d=diam⁡⁡K, and ε>0, satisfying ε(1-α)-n0<r. Now choose positive integer k such that
(19)0≤d(yk,xk)-d(yk+n0,xk+n0)<ε.
Using (10), we obtain
(20)d+r≤r(αn0+1)≤(αn0+1)d(yk,xk)≤d(yk+n0,xk)+ε(1-α)-n0<d+r.
By the last contradiction we conclude thatr=0 and lim⁡n→∞d(yn,xn)=lim⁡n→∞d(Txn,xn)=0 what we had to prove.Remark 17.
This statement is a generalization of Lemma 9.4 from [14].Combining the last result with Theorem15 we have the following consequence.Corollary 18.
LetK be a bounded closed convex subset of complete TCS (X,d,W) with properties (P) and (Q) and let T:K→K satisfy the following: (i)
T is uniformly locally directionally nonexpansive on K;(ii)
α(T(A))<α(A), for all A⊆K for which α(A)>0;(iii)
F is r.g.i. on K. Then the fixed point set fix(T) of T is nonempty and compact.Moreover, using Proposition16 we also get generalizations of some other Kirk and Saliga [2] fixed point results.Corollary 19.
LetK be a bounded closed convex subset of a complete TCS (X,d,W) with properties (P) and (Q) and let T:K→K satisfy the following: (i)
T is uniformly locally directionally nonexpansive on K;(ii)
d(Tx,Ty)≤θ(max⁡{d(x,Tx),d(y,Ty)}), where θ:R+→R+ is any function for which lim⁡t→0+θ(t)=0. Then T has a unique fixed point x0∈K if and only if  F is an r.g.i. on K.Proof.
Proposition16 gives inf⁡K(F)=0 and as in [2] one can prove that lim⁡c→0+diam⁡(Lc)=0. By Theorem 1.2 [1], T has a unique fixed point if and only if F is r.g.i. on K.Theorem 20.
LetK be a bounded closed convex subset of a complete TCS (X,d,W) with properties (P) and (Q) and suppose T:K→K satisfies the following: (i)
T is directionally nonexpansive on K;(ii)
μ(T(Lc))≤k·μ(Lc), for some k<1 and all c>0;(iii)
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 lim⁡n→∞d(xn,Txn)=0, then lim⁡n→∞d(xn,fix(T))=0.Proof.
By Proposition16, inf⁡K(F)=0. Since (i) implies that
(21)d(Tx,T2x)≤d(x,Tx),∀x∈K,
the conclusion follows immediately from Theorem 2.3 [2].We established thatlim⁡n→∞d(xn,Txn)=0 for every sequence {xn} defined by xn=Tαxn-1, n∈N, where x0∈K and α∈(0,1). Therefore lim⁡n→∞d(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 ofweakly quasi-nonexpansive mappings with respect to sequence introduced by Ahmed and Zeyada in [15].Definition 21 (see [15]).
Let(X,d) be a metric space and let {xn} be a sequence in D⊆X. Assume that T:D→X is a mapping with fix(T)≠∅ satisfying lim⁡n→∞d(xn,fix(T))=0. Thus, for a given ε>0 there exists n1(ε)∈N such that d(xn,fix(T))<ε for all n≥n1(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 n∈N with n≥n1(ε),d(xn,p(ε))<ε.The next result is improvement of Theorem20 and also a generalisation of Theorem 2.24 from [15].Theorem 22.
LetK be a bounded closed convex subset of a complete TCS (X,d,W) with properties (P) and (Q) and let T:K→K satisfy the following: (i)
T is directionally nonexpansive on K;(ii)
α(T(Lc))≤kα(Lc) for some k<1 and all c>0;(iii)
F is r.g.i. on K;(iv)
{xn}⊆K satisfies lim⁡lim⁡n→∞d(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 Theorem20 and Theorem 2.5(b) from [15].Using Proposition16, the next corollary holds.Corollary 23.
LetK be a bounded closed convex subset of a complete TCS (X,d,W) with properties (P) and (Q) and let T:K→K satisfy the following: (i)
T is directionally nonexpansive on K;(ii)
α(T(Lc))≤kα(Lc) for some k<1 and all c>0;(iii)
F is r.g.i. on K;(iv)
T is weakly quasi-nonexpansive with respect to sequence xn=Tαnx0,n∈N,x0∈K,α∈(0,1). Then {xn} converges to a point in fix(T).



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