COINCIDENCES AND FIXED POINTS OF RECIPROCALLY CONTINUOUS AND COMPATIBLE

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COINCIDENCES AND FIXED POINTS OF RECIPROCALLY CONTINUOUS AND COMPATIBLE

HYBRID MAPS

S. L. SINGH and S. N. MISHRA

Received 16 February 2001 and in revised form 27 March 2001

It is proved that a pair of reciprocally continuous and nonvacuously compatible single- valued and multivalued maps on a metric space possesses a coincidence. Besides address- ing two historical problems in fixed point theory, this result is applied to obtain new general coincidence and fixed point theorems for single-valued and multivalued maps on metric spaces under tight minimal conditions.

2000 Mathematics Subject Classification: 47H10.

1. Introduction. The concept of compatible maps has proven useful for general- izing results in the context of metric fixed point theory for continuous single-valued and multivalued maps (cf. [1,2,4,11,15,16,20,21,22,23,30,31]). Recently, recip- rocal continuity for a pair of (discontinuous) single-valued maps has been introduced in [23] and promoted as a means to comprehensive results.

First, we introduce reciprocal continuity for a hybrid pair of single-valued and mul- tivalued maps, and emulate the joint merits of reciprocal continuity and compati- bility of a hybrid pair in the setting of metric spaces. We give a general principle (Theorem 2.8) stating that nonvacuously compatible and reciprocally continuous hy- brid pair on a metric space has a coincidence. This seems to be of vital interest in view of a historically significant and negatively settled problem that a pair of continuous and commuting self-maps on the closed interval[0,1]has a common fixed point (see [3,13,17]) and that continuous and commuting maps on a complete metric space need not have a coincidence even (seeRemark 2.7(v)). This principle presents another view of a significant result of Jungck [17, Theorem 3.6] on a metric space as well. We apply Theorem 2.8to obtain a coincidence and fixed point theorem for a hybrid quadruple of maps on a metric space satisfying a very general contractive type condition which includes several general conditions studied by Beg and Azam [1], ´Ciri´c [6], Das and Naik [10], Jungck [16], Kaneko [19,20], Rhoades et al. [27], Singh et al. [29], Tan and Minh [33], and others. One of our results presents another view of recent resolutions to a still open fixed point problem of Simon Reich (see [4,5,8,9,25,26]). Our final result on a compact metric space extends and generalizes fixed point theorems from [7,17,32,33].

In this paper, consistent with [22, page 620], (X, d) denotes a metric space, id the identity map onX,CL(X)(resp., CB(X)) the nonempty closed (resp., closed and bounded) subsets ofX andH for the Hausdorff (resp., generalized Hausdorff) met- ric on CB(X)(resp., CL(X)). Further,d(A, B)denotes the ordinary distance between

nonempty subsetsAandBofXwhiled(x, B)stands ford(A, B)whenA= {x}. The set of natural numbers is denoted byN.

2. Reciprocal continuity

Definition 2.1. The maps T : X→CL(X)and f :X→X are reciprocally con- tinuous on X (resp., at t∈ X) if and only if f T x∈CL(X) for each x∈ X (resp., f T t∈CL(X)) and limnf T xn=f M, limnT f xn=T twhenever{xn}is a sequence in Xsuch that

limn T xn=M∈CL(X), lim

n f xn=t∈M. (2.1)

We may use r.c. for “reciprocal continuity” or “reciprocal continuous” as the situ- ation demands. For self-mapsf , g:X→X, this definition due to Pant [23] reads:f andgare r.c. if and only if limngf xn=gtand limnf gxn=f twhenever{xn} ⊂X is such that limngxn=limnf xn=t∈X. Clearly, any continuous pair is reciprocally continuous but, as the following examples show, the converse is not true.

Example2.2(see [24, Example 2.1]). LetX=[0,∞)with the usual metric. Define g, f:X→Xby

gx=















 1

2+x if 0≤x <1 2, 1 ifx=1

2, 0 ifx >1

2,

f x=















 1

2−x if 0≤x <1 2, 1

2 ifx=1 2, 1 ifx >1

2.

(2.2)

These maps are discontinuous atx=1/2. However, they are r.c. (take a decreasing sequence{xn}converging to 0).

Example2.3. LetX=R¹,

T X=













 1

2, x+1

ifx >0, {0} ifx=0,

x−1,−1 2

ifx <0,

f x=











2x+1 ifx >0, 0 ifx=0, 2x−1 ifx <0.

(2.3)

ThenTandfare r.c. atx=0 (takexn=0, n∈N). Notice that there is a discontinuity at their common fixed point(x=0).

For continuity of multivalued maps at their fixed and common fixed points, refer to [12]. The following definition is due to Kaneko and Sessa [20] and Beg and Azam [1] whenT:X→CB(X).

Definition2.4. The mapsT :X →CL(X)and f :X→X are compatible if and only iff T x∈CL(X)for eachx∈Xand limnH(T f xn, f T xn)=0 whenever{xn}is a sequence inXsuch that limnT xn=M∈CL(X)and limnf xn=t∈M.

Evidently commuting mapsT, f (i.e., whenf T x=T f x,x∈X) are weakly com- muting (i.e., H(T f x, f T x)≤d(f x, T x), x ∈X, [19, 29]), weakly commuting T, f are compatible, and compatibleT,f are weakly compatible (i.e., whenf T x=T f x

whenever f x∈T x, [18]) but the reverse implication is not true. For an excellent discussion on the role of weak compatibility in fixed point considerations, refer to Jungck and Rhoades [18]. For self-mapsf , g:X →X, Definition 2.4 due to Jungck [15,16] reads:fandgare compatible if and only if limnd(gf xn, f gxn)=0 wherever {xn}is a sequence inXsuch that limnf xn=limngxn=t∈X. Notice that the maps g, f ofExample 2.2 are not compatible (take{xn}as in Example 2.2). So r.c. need not imply compatibility. TakingXas inExample 2.2and definingf x=3x²/(x²+8) and gx=x, ifx <2,gx=2 if x≥2, one may conclude thatf,g are r.c. but not compatible (takexn=4,n∈N). Hence we assert that:the r.c. and compatibility are independent concepts.

Following Itoh and Takahashi [14],T :X→CL(X)andf:X→Xare IT-commuting (commuting in the sense of Itoh-Takahashi [14]) at a point v ∈X if T f v ⊂f T v.

Further,Tandfare IT-commuting onXif they are IT-commuting at each pointv∈X.

We remark that the IT-commutativity of a hybrid pair(T , f )at a pointvis more general than its compatibility (cf. [31, Example 1]) and weak compatibility at the pointv.

Example2.5. LetX=[0,∞)be endowed with the usual metric and

T x=





[0, x] ifx <2,

[4,2+x] ifx≥2, f x=





x ifx <2,

4 ifx≥2. (2.4)

We see thatT andfare compatible forx <2 but not forx≥2 (e.g., takexn=2+n, n∈N). Further,T andf are not r.c. (e.g., takexn=2−1/n,n∈N).

Example2.6. LetX=[2,∞)with the usual metric and T x= {1+x}and f x= 2x+1. We see that there does not exist a sequence{xn} ⊂X such that{f xn}and {T xn}both converge to the same element inX. Thus requirements of compatibility are vacuously satisfied .

Remark2.7. (i) If a compatible pair(T , f )is such thatT t=f M, then it is evident from Definitions2.1and2.4that the continuity of one ofT orfis sufficient to ensure the r.c. of the pair(T , f ).

(ii) IfT andf are r.c. and nonvacuously compatible, thenT t=f M. See the proof ofTheorem 2.8.

(iii) The r.c. at a pointt∈Xmay be verified by considering all sequences{xn} ⊂X such that limnf xn=t∈M=limnT xn. If there does not exist such a sequence then the definition holds vacuously, and the maps are r.c. (seeExample 2.6). This observa- tion applies to the compatibility ofT andfas well. Hence nonvacuous compatibility ofT andfimplies the existence of at least a sequence{xn}inXsuch that{f xn}and {T xn}both converge as per requirements ofDefinition 2.4.

(iv) If the pair(T , f )is compatible at a pointv∈Xandvis a coincidence point ofT andf, that is,f v∈T v, thenf T v=T f v(see [20, page 260]). Indeed, commutativity, weak commutativity, and compatibility ofT and f are equivalent at a coincidence pointvofT andf.

(v) A pair of continuous and commuting selfmaps of a complete metric space need not have a coincidence; for example,gx=1+xandf x=x,x∈[0,∞). (Notice the vacuous compatibility ofgandf.)

The following is the main result of this section. In all that follows,C(T , f )stands for the collection of coincidence points ofTandf, that is,C(T , f )= {v:f v∈T v}.

Theorem2.8. Let(X, d)be a metric space andT :X→CL(X)andf :X→X. If T andf are reciprocally continuous and nonvacuously compatible onXthenC(T , f ) is nonempty. Further,T andf have a common fixed pointf t, providedf f t=f tfor somet∈C(T , f ).

Proof. SinceT andf are nonvacuously compatible, there exits a sequence{xn} inXsuch that{f xn}and{T xn}converge, respectively, tot∈XandM∈CL(X)such thatt∈M and limnH(T f xn, f T xn)=0. This, in view of the r.c. ofT andf, yields H(T t, f M)=0, andT t=f M. Now,t∈Mimpliesf t∈f M. Thereforef t∈T t, and C(T , f )is nonempty. Further,f t=f f T impliesf t∈f T t=T f t (cf.Remark 2.7(i)).

This completes the proof.

For a better appreciation ofTheorem 2.8and the relative roles of r.c. and compati- bility, consider the following result of Mizoguchi and Takahashi [21, Theorem 3].

Theorem2.9. LetKbe a closed convex subset of a uniformly convex Banach space.

LetT:K→CB(X)andf:K→Kbe such thatT xis convex for eachx∈K,H(T x, T y)≤ qx−y,x, y∈K,0≤q <1, andf x−f p ≤ x−p,x∈K,p∈F (f ), whereF (f ) denotes the set of fixed points off. IfT andf are IT-commuting onK, thenT andf have a common fixed point, that is, there existsz∈F (f )withf z∈T z.

Further, Theorem 2.8 applies to discontinuous maps and a common fixed point may be a point of discontinuity as well (seeExample 2.3). Notice that, besides several stronger conditions on the space,T ofTheorem 2.9is a multivalued contraction and fis nonexpansive about fixed points.

3. Results on metric spaces and Reich’s problem. First, we give a very general coincidence and fixed point theorem under very tight conditions. Letψdenote the family of mapsφfrom the setR⁺of nonnegative reals to itself such thatφ(t) < tfor allt >0.

Theorem3.1. Let(X, d)be a metric space andS, T:X→CL(X)andf , g:X→X such that

(1) S(X)⊂g(X) and the pair(S, f )is reciprocally continuous and nonvacuously compatible.

If there existsφ∈ψsuch that

(2) H(Sx, T y)≤φ(M(x, y))forx,yinX, where M(x, y)=max

d(f x, gy), d(f x, Sy), d(gy, T y), d(f x, T y), d(gy, Sx)

, (3.1) thenC(S, f )andC(T , g)are nonempty. Further,

(Ia) S andf have a common fixed pointf t, providedf f t=f tfor somet∈ C(S, t);

(Ib) Tandghave a common fixed pointgu, providedggu=guandT,gare IT-commuting atu∈C(T , g);

(Ic) S,T,f, andghave a common fixed point, provided (Ia) and (Ib) both are true.

Proof. ByTheorem 2.8, (1) implies thatC(S, f )is nonempty, that is,f t∈Stfor somet∈X. SinceS(X)⊂g(X), there is a pointu∈X such thatf t=gu∈St. So by (2),

d(gu, T u)≤H(St, T u)

≤φ max

d(f t, gu), d(f t, St), d(gu, T u), d(f t, T u), d(gu, St)

=φ d(gu, T u)

< d(gu, T u) ifgu∈T u.

(3.2)

Sogu∈T uandC(T , g)is nonempty.

(Ia) and (Ib) may be shown following the last part of the proof ofTheorem 2.8. Now (Ic) is immediate.

Several contractive conditions studied in [1,4,6,10,11,12,16,18,19,20,27,28, 29,33] are special cases of (2). For example, if we takeφ(t)=qt (0≤q <1),f=g, S=T, andT :X→X, then we obtain the condition studied in [10]. Notice that the contractive condition studied in [6] is a special case of that of [10].

Reich [25,26] posed the following question: let(X, d)be a complete metric space andT :X→CB(X)such thatH(T x, T y)≤k(d(x, y))d(x, y)for all distinctx, y∈ X, where k:(0 :∞)→(0,1)with limr→t⁺supk(r ) <1 for each t >0. Then, does T have a fixed point? Mizoguchi and Takahashi [21] have shown thatT has a fixed point when lim_r→t+k(r ) <1 for eacht≥0. Chang [4] has generalized this result, and Theorem 3.1presents an extension of Chang’s main result [4, Theorem 1]. However, Reich’s problem remains open and needs further resolution.

The following example shows that the nonvacuous compatibility of one of the pairs (S, f )or(T , g)is essential even iff=g=id.

Example3.2(see [28]). LetX= {1,2,3,4}with metricddefined by

d(1,2)=d(3,4)=2, d(1,3)=d(2,4)=1, d(1,4)=d(2,3)=3

2. (3.3) DefineS andT byS1=S4= {2},S2=S3= {1}andT1=T3= {4},T2=T4= {3}. Takef=g=id. ThenDefinition 2.4is satisfied withφ(t)=3t/4. All conditions of Theorem 3.1are satisfied except (1). Indeed the pairs(S,id)and(T ,id)are vacuously compatible.

The following result generalizes several main results from [1, 11, 16,18,19, 20]

and others.

Corollary3.3. Let(X, d)be a complete metric space andS, T:X→CL(X),f , g: X→Xsuch that

(3) S(X)⊂g(X), T (X)⊂f (X), and the pair(S, f )is compatible and reciprocally continuous.

If there existsq∈(0,1)such that

(4) H(Sx, T y)≤qm(x, y)forx,yinX, where

m(x, y)=max

d(f x, gy), d(f x, Sx), d(gy, T y),1 2

d(f x, T y)+d(gy, Sx) , (3.4)

thenC(S, f )andC(T , g)are nonempty. Further, conclusions (Ia), (Ib), and (Ic) are also true.

Proof. SinceS(X)⊂g(X)and T (X)⊂f (X), we construct the sequences{xn}, {yn} ⊂Xas in [11,31] such that, for eachn∈N,

y2n−1=gx2n−1∈Sx2n−2, y2n=f x2n∈T x2n−1. (3.5)

Then as in [11],

d y2n−1, y2n

≤q^−1/2d y2n−2, y2n−1 , d y2n, y2n+1

≤q^−1/2d y2n−1, y2n

. (3.6)

So{yn}converges to a pointt∈X, [11, Theorem 2]. By (4), H Sx2n, T x2n−1

≤qm x2n, x2n−1

≤qmax

d y2n, y2n−1

, d y2n, y2n+1

→0 asn→ ∞. (3.7)

Therefore, lim_k,n→∞H(Sx2n, Sx2k) ≤ lim_k,n→∞d(y_2n−1, y_2k−1) = 0, and {Sx2n} is Cauchy in CL(X). The spaceXbeing complete, the hyper space CL(X)is also complete, and{Sx2n}converges to anMin CL(X). So

d(t, M)≤d t, gx_2n−1

+H Sx_2n−2, M

→0 asn → ∞, t∈M. (3.8)

Thus, for the sequence{xn}inX, we have{Sx2n}and{gx2n−1}converging, respec- tively, toM andt∈M. Therefore the compatibility of the pair(S, g)is nonvacuous.

Hence the proof is immediate fromTheorem 3.1 by observing that (4) implies (2).

Remark3.4. (i) It is evident from the proof ofTheorem 3.1thatf t∈Standf t= gu∈T u, that is,(S, t)and(T , g)may have different coincidence points withf t=gu.

SeeExample 3.5in support of this observation, which applies toCorollary 3.3as well.

(ii) The power of Corollary 3.3is appreciated by observing that the main result of [11, Theorem 2] is obtained under condition (4) when all the mapsS,T,f,gare continuous and the pairs (S, f )and (T , g) are compatible. Moreover,Corollary 3.3 generalizes several other interesting results from [1,16,19,20] and the references therein.

(iii) Iff =id under the condition (4) ofCorollary 3.3, then the pair(S, f )is au- tomatically compatible and r.c. Thus if f = g= id then Corollary 3.3 states that S, T:X→CL(X), satisfying condition (4), have a common fixed point in completeX.

Therefore, several general results surveyed in [21] are contained inCorollary 3.3(e.g., Theorems 12, 15, and 16).

Example3.5. LetX=[0,∞)be endowed with the usual metric. Letf , g, S, T:X→X be such thatf x=8x⁴, gx=8x⁸, Sx=x⁴+7/16, andT x=x⁸+7/16. Evidently, for anyx, y∈X, d(Sx, T y)=1/8d(f x, gy), that is, condition (4) is satisfied with q= 1/8. Further, S(X)=T (X) =[7/16,∞]⊂X =f (X)=g(X), the pair (S, f ) is compatible and continuous. So all the hypotheses ofCorollary 3.3are satisfied and f (1/2)=S(1/2)andg(2^−1/2)=T (2^−1/2), that is,f,Shave a coincidence atx=1/2, andg,T have a (different) coincidence atx=(2^−1/2). Notice thatf (1/2)=g(2^−1/2) and the pair(T , g)is not compatible.

Corollary3.6. Letf,g,S,T be self-maps of a complete metric space(X, d)such that S(X)⊂g(X), T (X)⊂f (X), and the pair(S, f ) is compatible and reciprocally continuous. If there existsq∈(0,1)such that d(Sx, T y)≤qm(x, y) for allx, y in X, then

(IIa) S andf have a common fixed point;

(IIb) T andghave a coincidence atx=u∈X;

(IIc) f,g,S, andT have a common fixed point provided thatT andgare weakly compatible.

The tightness of the conditions in this result is evident from the fact that Jungck’s [16, Theorem 3.1] isCorollary 3.6(IIc) with the r.c. replaced by continuity off,g, and the weak compatibility ofT,g(atu∈X) replaced by the compatibility ofT,gonX.

4. Results on compact spaces. We applyCorollary 3.3to obtain a new coincidence theorem for a hybrid of multivalued and single-valued maps on the setting of a com- pact metric space generalizing and extending [7, Theorem 1], [16, Theorem 3.2], and relevant results of Smithson [32] and Tan and Minh [33].

Theorem4.1. Letf,gbe continuous self-maps of a compact metric space(X, d) and letS, T:X→CB(X)be continuous such thatS(X)⊂g(X),T (X)⊂f (X), and the pair(S, f )is compatible. IfH(Sx, T y) < m(x, y)(see condition (4)) whenm(x, y) >0, thenC(S, f )andC(T , g)are nonempty. Further, (Ia)–(Ic) are also true.

Proof. In view of the conclusions of Corollary 3.3, it is enough to show that C(S, t) and C(T , g) are nonempty. We claim thatm(x, y)=0 for some x, y ∈X.

Otherwise the function w(x, y)= H(Sx, T y)/m(x, y)is continuous and satisfies w(x, y) <1 for (x, y)∈X×X. SinceX×X is compact, there exist v, z∈X such thatw(x, y)≤w(v, z)=q <1 forx, y∈X. Consequently,H(Sx, T y)≤qm(x, y) forx, y∈Xand someq∈(0,1). So, byCorollary 3.3(see alsoRemark 3.4(i)), there existu, t∈X such thatf t∈St and f t=gu∈T u, and we have m(t, u)=0, con- tradictingm(t, u) >0. Thusm(x, y)=0 for somex, y∈X. Consequently,f x=gy, f x∈Sx,gy∈T y, and this completes the proof.

Corollary4.2. Letf, g, S, and T be continuous self-maps of a compact metric space(X, d)such thatS(X)⊂g(X),T (X)⊂f (X), and the pair(S, f )is compatible. If d(Sx, T y) < m(x, y)whenm(x, y) >0, then conclusions (IIa)–(IIc) are true.

Its proof may also be completed using Corollary 3.6 and following the proof of Theorem 4.1. Jungck [16, Theorem 3.2] is Corollary 3.6(IIc) when the pair(T , g) is also compatible onX.

Acknowledgments. The authors are highly indebted to the referee for his/her valuable suggestions for improving this paper. They thank Er. Yash Kumar for his work in preparing this paper. The research of the second author was supported by FRD Grant GUN: 2039007.

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[29] S. L. Singh, K. S. Ha, and Y. J. Cho,Coincidence and fixed points of nonlinear hybrid con- tractions, Int. J. Math. Math. Sci.12(1989), 247–256.

[30] S. L. Singh and S. N. Mishra, Nonlinear hybrid contractions, J. Natur. Phys. Sci. 5/8 (1991/1994), 191–206.

[31] ,On general hybrid contractions, J. Austral. Math. Soc. Ser. A66(1999), 244–254.

[32] R. E. Smithson,Fixed points for contractive multifunctions, Proc. Amer. Math. Soc.27 (1971), 192–194.

[33] D. H. Tan and N. A. Minh,Some fixed-point theorems for mappings of contractive type, Acta Math. Vietnam.3(1978), no. 1, 24–42.

S. L. Singh: Department of Mathematics, Gurukula Kangri Vishwavidyalaya, Hardwar249404, India

E-mail address:[email protected]

S. N. Mishra: Department of Mathematics, University of Transkei, Umtata 5100, South Africa

E-mail address:[email protected]

Special Issue on Space Dynamics

Call for Papers

Space dynamics is a very general title that can accommodate a long list of activities. This kind of research started with the study of the motion of the stars and the planets back to the origin of astronomy, and nowadays it has a large list of topics. It is possible to make a division in two main categories: astronomy and astrodynamics. By astronomy, we can relate topics that deal with the motion of the planets, natural satellites, comets, and so forth. Many important topics of research nowadays are related to those subjects.

By astrodynamics, we mean topics related to spaceflight dynamics.

It means topics where a satellite, a rocket, or any kind of man-made object is travelling in space governed by the grav- itational forces of celestial bodies and/or forces generated by propulsion systems that are available in those objects. Many topics are related to orbit determination, propagation, and orbital maneuvers related to those spacecrafts. Several other topics that are related to this subject are numerical methods, nonlinear dynamics, chaos, and control.

The main objective of this Special Issue is to publish topics that are under study in one of those lines. The idea is to get the most recent researches and published them in a very short time, so we can give a step in order to help scientists and engineers that work in this field to be aware of actual research. All the published papers have to be peer reviewed, but in a fast and accurate way so that the topics are not outdated by the large speed that the information flows nowadays.

Before submission authors should carefully read over the journal’s Author Guidelines, which are located at

thors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking Sy- stem at

according to the following timetable:

Manuscript Due July 1, 2009 First Round of Reviews October 1, 2009 Publication Date January 1, 2010

Lead Guest Editor

Antonio F. Bertachini A. Prado,

Instituto Nacional de Pesquisas Espaciais (INPE), São José dos Campos, 12227-010 São Paulo, Brazil;

Guest Editors

Maria Cecilia Zanardi,

São Paulo State University (UNESP), Guaratinguetá, 12516-410 São Paulo, Brazil;

[email protected]

Tadashi Yokoyama,

Universidade Estadual Paulista (UNESP), Rio Claro, 13506-900 São Paulo, Brazil;

[email protected]

Silvia Maria Giuliatti Winter,

São Paulo State University (UNESP), Guaratinguetá, 12516-410 São Paulo, Brazil;

[email protected]

Hindawi Publishing Corporation http://www.hindawi.com