The Theory of Reich's Fixed Point Theorem for Multivalued Operators

Abstract

The purpose of this paper is to present a theory of Reich's fixed point theorem for multivalued operators in terms of fixed points, strict fixed points, multivalued weakly Picard operators, multivalued Picard operators, data dependence of the fixed point set, sequence of multivalued operators and fixed points, Ulam-Hyers stability of a multivalued fixed point equation, well-posedness of the fixed point problem, and the generated fractal operator.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Tania Lazar 1 and Ghiocel Mot 2 and Gabriela Petrusel 3 and Silviu Szentesi 4

Recommended by S. Reich

1, Commercial Academy of Satu Mare, Mihai Eminescu Street No. 5, Satu Mare, Romania
2, Aurel Vlaicu University of Arad, Elena Dragoi Street, No. 2, 310330 Arad, Romania
3, Department of Business, Babes-Bolyai University, Cluj-Napoca, Horea Street No. 7, 400174 Cluj-Napoca, Romania
4, Aurel Vlaicu University of Arad, Revoultiei Bd., No. 77, 310130 Arad, Romania

Received 12 April 2010; Revised 12 July 2010; Accepted 18 July 2010

1. Introduction

Let (X,d) be a metric space and consider the following family of subsets _Pcl (X):={Y⊆X|"Y is nonempty and closed}. We also consider the following (generalized) functionals: [figure omitted; refer to PDF] D is called the gap functional between A and B . In particular, if _x0 ∈X, then D(_x0 ,B):=D({_x0 },B) : [figure omitted; refer to PDF] ρ is called the (generalized) excess functional: [figure omitted; refer to PDF] H is the (generalized) Pompeiu-Hausdorff functional.

It is well known that if (X,d) is a complete metric space, then the pair (_Pcl (X),H) is a complete generalized metric space. (See [1, 2]).

Definition 1.1.

If (X,d) is a metric space, then a multivalued operator T:X[arrow right]_Pcl (X) is said to be a Reich-type multivalued (a,b,c) -contraction if and only if there exist a,b,c∈_...+ with a+b+c<1 such that [figure omitted; refer to PDF]

Reich proved that any Reich-type multivalued (a,b,c) -contraction on a complete metric space has at least one fixed point (see [3]).

In a recent paper Petrusel and Rus introduced the concept of "theory of a metric fixed point theorem" and used this theory for the case of multivalued contraction (see [4]). For the singlevalued case, see [5].

The purpose of this paper is to extend this approach to the case of Reich-type multivalued (a,b,c) -contraction. We will discuss Reich's fixed point theorem in terms of

(i) fixed points and strict fixed points,

(ii) multivalued weakly Picard operators,

(iii): multivalued Picard operators,

(iv) data dependence of the fixed point set,

(v) sequence of multivalued operators and fixed points,

(vi) Ulam-Hyers stability of a multivaled fixed point equation,

(vii): well-posedness of the fixed point problem;

(viii): fractal operators.

Notice also that the theory of fixed points and strict fixed points for multivalued operators is closely related to some important models in mathematical economics, such as optimal preferences, game theory, and equilibrium of an abstract economy. See [6] for a nice survey.

2. Notations and Basic Concepts

Throughout this paper, the standard notations and terminologies in nonlinear analysis are used (see the papers by Kirk and Sims [7], Granas and Dugundji [8], Hu and Papageorgiou [2], Rus et al. [9], Petrusel [10], and Rus [11]).

Let X be a nonempty set. Then we denote. [figure omitted; refer to PDF]

Let (X,d) be a metric space. Then δ(Y)=sup {d(a,b)|"a,b∈Y} and [figure omitted; refer to PDF]

Let T:X[arrow right]P(X) be a multivalued operator. Then the operator T...:P(X)[arrow right]P(X) , which is defined by [figure omitted; refer to PDF] is called the fractal operator generated by T . For a well-written introduction on the theory of fractals see the papers of Barnsley [12], Hutchinson [13], Yamaguti et al. [14].

It is known that if (X,d) is a metric space and T:X[arrow right]_Pcp (X) , then the following statements hold:

(a) if T is upper semicontinuous, then T(Y)∈_Pcp (X) , for every Y∈_Pcp (X) ;

(b) the continuity of T implies the continuity of T...:_Pcp (X)[arrow right]_Pcp (X) .

The set of all nonempty invariant subsets of T is denoted by I(T) , that is, [figure omitted; refer to PDF]

A sequence of successive approximations of T starting from x∈X is a sequence (_xn)n∈... of elements in X with _x0 =x, _xn+1 ∈T(_xn ) , for n∈... .

If T:Y⊆X[arrow right]P(X), then _FT :={x∈Y|"x∈T(x)} denotes the fixed point set of T and (SF_)T :={x∈Y|"{x}=T(x)} denotes the strict fixed point set of T . By [figure omitted; refer to PDF] we denote the graph of the multivalued operator T .

If T:X[arrow right]P(X) , then ^T0 :=_1X , ^T1 :=T,...,^Tn+1 =T[composite function]^Tn , n∈..., denote the iterate operators of T .

Definition 2.1 (see [15]).

Let (X,d) be a metric space. Then, T:X[arrow right]P(X) is called a multivalued weakly Picard operator (briefly MWP operator) if for each x∈X and each y∈T(x) there exists a sequence (_xn)n∈... in X such that

(i) _x0 =x and _x1 =y ;

(ii) _xn+1 ∈T(_xn ) for all n∈... ;

(iii): the sequence (_xn)n∈... is convergent and its limit is a fixed point of T .

For the following concepts see the papers by Rus et al. [15], Petrusel [10], Petrusel and Rus [16], and Rus et al. [9].

Definition 2.2.

Let (X,d) be a metric space, and let T:X[arrow right]P(X) be an MWP operator. The multivalued operator ^T∞ :Graph(T)[arrow right]P(_FT ) is defined by the formula ^T∞ (x,y)={z∈_FT |" there exists a sequence of successive approximations of T starting from (x,y) that converges to z} .

Definition 2.3.

Let (X,d) be a metric space and T:X[arrow right]P(X) an MWP operator. Then T is said to be a c -multivalued weakly Picard operator (briefly c -MWP operator) if and only if there exists a selection ^t∞ of ^T∞ such that d(x,^t∞ (x,y))≤cd(x,y) for all (x,y)∈Graph(T) .

We recall now the notion of multivalued Picard operator.

Definition 2.4.

Let (X,d) be a metric space and T:X[arrow right]P(X) . By definition, T is called a multivalued Picard operator (briefly MP operator) if and only if

(i) (SF_)T =_FT ={^x* } ;

(ii) ^Tn (x)[arrow right]H{^x* } as n[arrow right]∞ , for each x∈X .

In [10] other results on MWP operators are presented. For related concepts and results see, for example, [1, 17-23].

3. A Theory of Reich's Fixed Point Principle

We recall the fixed point theorem for a single-valued Reich-type operator, which is needed for the proof of our first main result.

Theorem 3.1 (see [3]).

Let (X,d) be a complete metric space, and let f:X[arrow right]X be a Reich-type single-valued (a,b,c) -contraction, that is, there exist a,b,c∈_...+ with a+b+c<1 such that [figure omitted; refer to PDF] Then f is a Picard operator, that is, we have:

(i) _Ff ={^x* } ;

(ii) for each x∈X the sequence (^fn (x)_)n∈... converges in (X,d) to ^x* .

Our main result concerning Reich's fixed point theorem is the following.

Theorem 3.2.

Let (X,d) be a complete metric space, and let T:X[arrow right]_Pcl (X) be a Reich-type multivalued (a,b,c) -contraction. Let α:=(a+b)/(1-c) . Then one has the following

(i) _FT ≠∅ ;

(ii) T is a 1/(1-α) -multivalued weakly Picard operator;

(iii): let S:X[arrow right]_Pcl (X) be a Reich-type multivalued (a,b,c) -contraction and η>0 such that H(S(x),T(x))≤η for each x∈X , then H(_FS ,_FT )≤η/(1-α);

(iv) let _Tn :X[arrow right]_Pcl (X) (n∈... ) be a sequence of Reich-type multivalued (a,b,c) -contraction, such that _Tn (x)[arrow right]HT(x) uniformly as n[arrow right]+∞ . Then, _FTn [arrow right]H_FT as n[arrow right]+∞ .

If, moreover T(x)∈_Pcp (X) for each x∈X , then one additionally has:

(v) (Ulam-Hyers stability of the inclusion x∈T(x) ) Let ...>0 and x∈X be such that D(x,T(x))≤..., then there exists ^x* ∈_FT such that d(x,^x* )≤.../(1-α) ;

(vi) T...:(_Pcp (X),H)[arrow right](_Pcp (X),H) , T...(Y):=_...x∈Y T(x) is a set-to-set (a,b,c) -contraction and (thus) _FT... ={^AT* } ;

(vii): ^Tn (x)[arrow right]H^AT* as n[arrow right]+∞ , for each x∈X ;

(viii): _FT ⊂^AT* and _FT are compact;

(ix) ^AT* =_{...n∈...\{0}}^Tn (x) for each x∈_FT .

Proof.

(i) Let _x0 ∈X and _x1 ∈T(_x0 ) be arbitrarily chosen. Then, for each arbitrary q>1 there exists _x2 ∈T(_x1 ) such that d(_x1 ,_x2 )≤qH(T(_x0 ),T(_x1 )) . Hence [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] Denote β:=q(a+b)/(1-qc). By an inductive procedure, we obtain a sequence of successive approximations for T starting from (_x0 ,_x1 )∈Graph(T) such that, for each n∈... , we have d(_xn ,_xn+1 )≤^βn d(_x0 ,_x1 ). Then [figure omitted; refer to PDF] If we choose 1<q<1/(a+b+c) , then by (3.4) we get that the sequence (_xn)n∈... is Cauchy and hence convergent in (X,d) to some ^x* ∈X

Notice that, by D(^x* ,T(^x* ))≤ d(^x* ,_xn+1 )+D(_xn+1 ,T(^x* )) ≤d(_xn+1 ,^x* )+H(T(_xn ),T(^x* )) ≤d(_xn+1 ,^x* )+ad(_xn ,^x* )+bD(_xn ,T(_xn ))+cD(^x* ,T(^x* )) ≤d(_xn+1 ,^x* )+ad(_xn ,^x* )+bd(_xn ,_xn+1 ))+cD(^x* ,T(^x* )) , we obtain that [figure omitted; refer to PDF] Hence ^x* ∈_FT .

(ii) Let p[arrow right]+∞ in (3.4). Then we get that [figure omitted; refer to PDF] For n=1, we get [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] Let q...1 in (3.8), then [figure omitted; refer to PDF] Hence T is a 1/(1-α) -multivalued weakly Picard operator.

(iii) Let _x0 ∈_FS be arbitrarily chosen. Then, by (ii), we have that [figure omitted; refer to PDF] Let q>1 be an arbitrary. Then, there exists _x1 ∈T(_x0 ) such that [figure omitted; refer to PDF] In a similar way, we can prove that for each _y0 ∈_FT there exists _y1 ∈S(_y0 ) such that [figure omitted; refer to PDF] Thus, (3.11) and (3.12) together imply that H(_FS ,_FT )≤qη/(1-α) for every q>1 . Let q...1 and we get the desired conclusion.

(iv) follows immediately from (iii).

(v) Let ...>0 and x∈X be such that D(x,T(x))≤... . Then, since T(x) is compact, there exists y∈T(x) such that d(x,y)≤... . From the proof of (i), we have that [figure omitted; refer to PDF] Since ^x* :=^t∞ (x,y)∈_FT , we get that d(x,^x* )≤.../(1-α) .

(vi) We will prove for any A,B∈_Pcp (X) that [figure omitted; refer to PDF] For this purpose, let A,B∈_Pcp (X) and let u∈T(A) . Then, there exists x∈A such that u∈T(x) . Since the sets A,B are compact, there exists y∈B such that [figure omitted; refer to PDF] From (3.15) we get that D(u,T(B)) ≤D(u,T(y)) ≤H(T(x),T(y)) ≤ad(x,y)+bD(x,T(x))+cD(y,T(y)) ≤ad(x,y)+bρ(A,T(x))+cρ(B,T(y)) ≤aH(A,B)+bρ(A,T(A))+cρ(B,T(B)) ≤aH(A,B)+bH(A,T(A))+cH(B,T(B)) . Hence [figure omitted; refer to PDF] In a similar way we obtain that [figure omitted; refer to PDF] Thus, (3.16) and (3.17) together imply that [figure omitted; refer to PDF] Hence, T... is a Reich-type single-valued (a,b,c) -contraction on the complete metric space (_Pcp (X),H) . From Theorem 3.1 we obtain that

(a) _FT... ={^AT* } and

(b) ^T...n (A)[arrow right]H^AT* as n[arrow right]+∞ , for each A∈_Pcp (X) .

(vii) From (vi)-(b) we get that ^Tn ({x})=^T...n ({x})[arrow right]H^AT* as n[arrow right]+∞ , for each x∈X .

(viii)-(ix) Let x∈FT be an arbitrary. Then x∈T(x)⊂T2 (x)⊂...⊂Tn (x)⊂.... Hence x∈Tn (x) , for each n∈...* . Moreover, lim n[arrow right]+∞Tn (x)=...n∈...*Tn (x) . From (vii), we immediately get that AT* =...n∈...*Tn (x) . Hence x∈...n∈...*Tn (x)=AT* . The proof is complete.

A second result for Reich-type multivalued (a,b,c) -contractions formulates as follows.

Theorem 3.3.

Let (X,d) be a complete metric space and T:X[arrow right]_Pcl (X) a Reich-type multivalued (a,b,c) -contraction with (SF_)T ≠∅ . Then, the following assertions hold:

(x) _FT =(SF_)T ={^x* } ;

(xi) (Well-posedness of the fixed point problem with respect to D [24]) If (_xn)n∈... is a sequence in X such that D(_xn ,T(_xn ))[arrow right]0 as n[arrow right]∞ , then _xn [arrow right]d^x* as n[arrow right]∞ ;

(xii) (Well-posedness of the fixed point problem with respect to H [24]) If (_xn)n∈... is a sequence in X such that H(_xn ,T(_xn ))[arrow right]0 as n[arrow right]∞ , then _xn [arrow right]d^x* as n[arrow right]∞ .

Proof.

(x) Let ^x* ∈(SF_)T . Note that (SF_)T ={^x* } . Indeed, if y∈(SF_)T , then d(^x* ,y)=H(T(^x* ),T(y))≤ad(^x* ,y)+bD(^x* ,T(^x* ))+cD(y,T(y))=ad(^x* ,y) . Thus y=^x* .

Let us show now that _FT ={^x* } . Suppose that y∈_FT . Then, d(^x* ,y)=D(T(^x* ),y)≤H(T(^x* ),T(y))≤ad(^x* ,y)+bD(^x* ,T(^x* ))+cD(y,T(y))=ad(^x* ,y) . Thus y=^x* . Hence _FT ⊂(SF_)T ={^x* } . Since (SF_)T ⊂_FT , we get that (SF_)T =_FT ={^x* } .

(xi) Let (_xn)n∈... be a sequence in X such that D(_xn ,T(_xn ))[arrow right]0 as n[arrow right]∞ . Then, d(_xn ,^x* )≤D(_xn ,T(_xn )) +H(T(_xn ),T(^x* ))≤D(_xn ,T(_xn )) +ad(_xn ,^x* ) +bD(_xn ,T(_xn )) +cD(^x* ,T(^x* ))=(1 +b)D(_xn ,T(_xn )) +ad(_xn ,^x* ) . Then d(_xn ,^x* )≤((1+b)/(1-a))D(_xn ,T(_xn ))[arrow right]0 as n[arrow right]+∞ .

(xii) follows by (xi) since D(_xn ,T(_xn ))≤H(_xn ,T(_xn ))[arrow right]0 as n[arrow right]+∞ .

A third result for the case of (a,b,c) -contraction is the following.

Theorem 3.4.

Let (X,d) be a complete metric space, and let T:X[arrow right]_Pcp (X) be a Reich-type multivalued (a,b,c) -contraction such that T(_FT )=_FT . Then one has

(xiii)^Tn (x)[arrow right]H_FT as n[arrow right]+∞ , for each x∈X ;

(xiv)T(x)=_FT for each x∈_FT ;

(xv)If (_xn)n∈... ⊂X is a sequence such that _xn [arrow right]d^x* ∈_FT as n[arrow right]∞ and T is H -continuous, then T(_xn )[arrow right]H_FT as n[arrow right]+∞ .

Proof.

(xiii) From the fact that T(_FT )=_FT and Theorem 3.2 (vi) we have that _FT =^AT* . The conclusion follows by Theorem 3.2 (vii).

(xiv) Let x∈_FT be an arbitrary. Then x∈T(x) , and thus _FT ⊂T(x) . On the other hand T(x)⊂T(_FT )⊂_FT . Thus T(x)=_FT , for each x∈_FT .

(xv) Let (_xn)n∈... ⊂X be a sequence such that _xn [arrow right]d^x* ∈_FT as n[arrow right]∞ . Then, we have T(_xn )[arrow right]HT(^x* )=_FT as n[arrow right]∞ . The proof is complete.

For compact metric spaces we have the following result.

Theorem 3.5.

Let (X,d) be a compact metric space, and let T:X[arrow right]_Pcl (X) be a H -continuous Reich-type multivalued (a,b,c) -contraction. Then

(xvi) if (_xn)n∈... is such that D(_xn ,T(_xn ))[arrow right]0 as n[arrow right]∞ , then there exists a subsequence (_xni)i∈... of (_xn)n∈... such that _xni [arrow right]d^x* ∈_FT as i[arrow right]∞ (generalized well-posedness of the fixed point problem with respect to D [24, 25]).

Proof.

(xvi) Let (_xn)n∈... be a sequence in X such that D(_xn ,T(_xn ))[arrow right]0 as n[arrow right]∞ . Let (_xni)i∈... be a subsequence of (_xn)n∈... such that _xni [arrow right]d^x* as i[arrow right]∞ . Then, there exists _yni ∈T(_xni ) , such that _yni [arrow right]d^x* as i[arrow right]∞ . Then D(^x* ,T(^x* ))≤d(^x* ,_yni )+D(_yni ,T(_xni ))+H(T(_xni ),T(^x* ))≤d(^x* ,_yni )+ad(^x* ,_xni )+bD(_xni ,T(_xni ))+cD(^x* ,T(^x* )) . Hence [figure omitted; refer to PDF] as n[arrow right]+∞. Hence ^x* ∈_FT .

Remark 3.6.

For b=c=0 we obtain the results given in [4]. On the other hand, our results unify and generalize some results given in [12, 13, 17, 26-34]. Notice that, if the operator T is singlevalued, then we obtain the well-posedness concept introduced in [35].

Remark 3.7.

An open question is to present a theory of the Ciric-type multivalued contraction theorem (see [36]). For some problems for other classes of generalized contractions, see for example, [17, 21, 27, 34, 37].

Acknowledgments

The second and the forth authors wish to thank National Council of Research of Higher Education in Romania (CNCSIS) by "Planul National, PN II (2007-2013)--Programul IDEI-1239" for the provided financial support. The authors are grateful for the reviewer(s) for the careful reading of the paper and for the suggestions which improved the quality of this work.

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