A Fixed Point Theorem for Multifunctions in Partial Metric spaces

Available online at www.ispacs.com/jnaa Volume 2013, Year 2013 Article ID jnaa-00200, 7 Pages

doi:10.5899/2013/jnaa-00200 Research Article

A Fixed Point Theorem for Multifunctions in Partial Metric

spaces

Priscilla S. Macansantos∗

University of the Philippines Baguio, Baguio City 2600, Philippines, October 2012

Copyright 2013 c⃝Priscilla S. Macansantos. 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.

Abstract

Fixed Point theorems on partial metric spaces have been the subject of recent work, with the interest generated in partial metric spaces (as a suitable structure for studies in theoretical computer science). Several approaches to fixed point theory for point-valued functions on complete metric spaces have been generalized to partial metric spaces (see, for instance, Alghamdi [1]). On the other hand, it appears that substantial work may still be done to generalize the theory (in the partial metric space context) to set-valued functions. Recently, Damjanovic et al [3] looked into pairs of multi-valued and single-valued maps in complete metric spaces, and used coincidence and common fixed points, to establish a theorem on fixed points for pairs of multivalued functions. In this paper we take off from Damjanovic and proceed to establish the same result in the setting of partial metric spaces. As a consequence of our generalization, we are able to include as special cases the theorem of Aydi et al [2] and our [9] generalization of [4]. Further, Reich’s result is also generalized to multivalued functions in partial metric spaces. Special cases include the partial metric space version of Kannan’s theorem, as well as that due to Hardy and Rogers.

Keywords:Partial metric space, Multivalued mappings; Fixed Point theorem; coincidence and common fixed point.

1 Introduction

The concept of partial metric spaces as a generalization of metric spaces was introduced in 1994 by Mathews [10], in his treatment of denotational semantics of dataflow networks. For partial metric spaces, self-distance need not be 0. Some applications of partial metrics to problems in theoretical computer science, including the use of fixed point theorems to determine program output from partially defined information, are cited in X. Huang et al [6] and references therein. Banach’s fixed point theorem for contraction mappings on complete metric spaces is widely utilized both in theoretical and applied mathematics, including that of establishing the existence of solutions to differential and integral equations, and the theorem has been generalized in many ways. Of note is the generalization of Nadler [11] to multifunctions on metric spaces satisfying a contraction condition. Some recent treatments and extensions/generalizations of fixed point theorems for multivalued functions are given in [14] and [12]. Significant research has been undertaken recently on the study of fixed point theorems in partial metric spaces (see, for instance, recent work by Alghamdi et al [1]). Nadler’s fixed point theorem for multifunctions has been generalized as well, and recently, an analogue for contractive mutifunctions on partial metric spaces, was established by H. Aydi et al [2]. In [9] we established a further generalization of the result of Aydi et al. Recently, Damjanovic et al [3] looked into pairs

of multi-valued and single-valued maps in complete metric spaces, and used coincidence and common fixed points, to establish a theorem on fixed points for pairs of multivalued functions. In this paper we take off from Damjanovic and proceed to establish the same in the setting of partial metric spaces. As a consequence of our generalization, we are able to include as special cases the theorem/s in [2] and [9]. Further, Reich’s result is also generalized to multivalued functions in partial metric spaces. Special cases include the partial metric space version of Kannan’s theorem, as well as that due to Hardy and Rogers. The paper is organized as follows. In Section 2, some basic definitions which will be used later in the paper are provided. In Section 3, we present the main theorem and some corollaries, extending known fixed point theorems of multifunctions, to partial metric spaces.

2 Preliminaries

Definition 2.1. Let X be a nonempty set. A function p:X_×X_→ℜ+_{is said to be a partial metric on X if for any}

x,y,z_∈X , the following conditions hold:

(P1) p(x,x) =p(y,y) =p(x,y)if and only if x=y; (P2) p(x,x)≤p(x,y);

(P3) p(x,y) =p(y,x);

(P4) p(x,z)≤p(x,y) +p(y,z)−p(y,y).

The pair(X,p)is then called a partial metric space.

Ifp(x,y) =0, thenx=y. But the converse does not always hold. A standard example of a partial metric space is the pair(ℜ+,p), wherep:ℜ+_×ℜ+_→ℜ+is defined asp(x,y) =max_{x,y_}.

A partial metricponX generates aT0topologyτponX which has as a base the family of openp-balls{Bp(x,ε):

x_∈X,ε>0_},whereBp(x,ε) ={y_∈X :p(x,y)<p(x,x) +ε_},for allx_∈X andε>0. Observe (from [10], p. 187) that a sequence{xn}in a partial metric space(X,p)converges to a pointx∈X, with respect toτp, if and only

if p(x,x) =limn_→∞p(x,xn). Ifp is a partial metric on X, then the function ps:X×X →ℜ+given by ps(x,y) = 2p(x,y)−p(x,x)−p(y,y)defines a metric onX. Further, a sequencexnconverges in(X,ps)to a pointx∈X if and

only iflimn,m→∞p(xn,xm) =limn→∞p(xn,x) =p(x,x).

From Matthews [10], we have the following definition and lemma:

Definition 2.2. Let (X,p) be a partial metric space. (a) A sequence xn in X is said to be a Cauchy sequence if

limn,m→∞p(xn,xm)exists and is finite. (b)(X,p)is said to be complete if every Cauchy sequence xnin X converges

with respect toτpto a point x∈X such that limn→∞p(x,xn) =p(x,x). In this case, we say that the partial metric p is

complete.

Lemma 2.1. Let(X,p)be a partial metric space. Then:

(a) A sequence xnin X is a Cauchy sequence in(X,p) if and only if it is a Cauchy sequence in the metric space (X,ps).

(b) A partial metric space(X,p)is complete if and only if the metric space(X,ps)is complete.

Towards generalizing Nadler’s Theorem to multifunctions in partial metric spaces, a partial Hausdorff metric is defined and its properties investigated by Aydi et al in [2]: In all cases, assume(X,p)is a partial metric space.

LetCBp_{(X)be the family of all nonempty, closed and bounded subsets of the partial metric space(X,p). Closedness}

is in the setting of (X,τp)whereτp is the topology induced by p, and boundedness is given as follows: Ais a

bounded subset in(X,p)if there exist x0 ∈X andM>0 such that for all a∈A, we havea∈Bp(x0,M), that is,

p(x0,a)<p(x0,x0) +M.

ForA,B_∈CBp(X)andx_∈X, definep(x,A) =in f _{p(x,a),a_∈A_},δp(A,B) =sup_{p(a,B):a_∈A_}andδp(B,A) =

sup_{p(b,A):b_∈B_}. It is noted thatp(x,A) =0 only ifps(x,A) =0 whereps(x,A) =in f_{ps(x,a),a_∈A_}.

The following properties of the mappingδp:CBp(X)×CBp(X)→[0,∞) are established in [2]: Proposition 2.1. For any A,B,C_∈CBp(X), we have the following:

(i) δp(A,A)= sup_{p(a,a):a_∈A_}; (ii) δp(A,A)≤δp(A,B);

(iii) δp(A,B) =0implies that A_⊆B;

(iv) δp(A,B)≤δp(A,C)+δp(C,B)−in fc∈Cp(c,c).

Let(X,p)be a partial metric space. ForA,B_∈CBp(X), defineHp(A,B) =max_{δp(A,B),δp(B,A)}.

Proposition 2.2. For all A,B,C_∈CBp(X),we have (h1) Hp(A,A)≤Hp(A,B);

(h2) Hp(A,B) =Hp(B,A);

(h3) Hp(A,B)≤Hp(A,C) +Hp(C,B)−in fc∈Cp(c,c).

Corollary 2.1. Let(X,p)be a partial metric space. For A,B_∈CBp(X)the following holds Hp(A,B) =0implies that A=B.

Remark 2.2. The converse of the above Corollary is not true in general.

The mappingHp:CBp(X)×CBp(X)→[0,+∞)is then called a partial Hausdorff metric induced byp. Remark 2.3. Any Hausdorff metric is a partial Hausdorff metric. The converse is not true.

We will be using pairs of multi-valued functions and single-valued maps to approach coincidence and common fixed points. The following definitions will be used.

Definition 2.3. An element x_∈X is said to be a coincidence point of T:X_→CBp_{(X)and f:X→X if f x∈T x. We}

denote C(f,T) ={x_∈X_|f x_∈T x_}, the set of coincidence points of T and f .

Definition 2.4. Maps f :X _→X and T :X _→CBp(X)are weakly compatible if they commute at their coincidence points, that is, f(T x) =T(f x)whenever f x_∈T x.

Definition 2.5. (see Kamran [7]) Let T:X_→CBp_{(X)be a multi-valued map and f :X→X be a single-valued map.}

The map f is said to be T -weakly commuting at x_∈X if f f x_∈T f x.

Definition 2.6. An element x_∈X is a common fixed point of T,S:X_→CBp(X)and f:X_→X if x= f x_∈T x_∩Sx.

3 Main Results

The following was proven in Aydi [2].

Lemma 3.1. Let(X,p)be a partial metric space, A,B_∈CBp(X)and h>1.For any a_∈A, there exists b=b(a)∈B such that p(a,b)≤hHp(A,B).

Following Damjanovic [3], we have the following theorem on partial metric spaces:

Theorem 3.1. Let(X,p)be a complete partial metric space and let T,S:X_→CBp(X)be a pair of multi-valued maps and f,g:X_→X be a pair of single-valued maps. Suppose that for all x,y_∈X,

Proof. Since 0<α+2β+2γ<1,there existsr>0 such that α+2β+2γ<√r<1.Letλ :=√α+β_r +γ

−(β+γ).Note

that 0<λ <1. Letx0∈X . Then f x0 andSx0are well-defined. SinceSX ⊆gX, there existsx1∈X withgx1∈

Sx0. According to Lemma 3.1, (withh=1/√r) there existsx2∈X,with f x2∈T x1 (sinceT X ⊆ f X) such that

p(gx1,f x2)≤√1_rHp(Sx0,T x1).From inequality 3.1, we get

p(gx1,f x2) ≤

1

√

rαp(f x0,gx1) +

1

√

rβ[p(f x0,Sx0) + p(gx1,T x1)] +

1

√

rγ[p(f x0,T x1) +p(gx1,Sx0)]. (3.2)

But observe that

p(f x0,Sx0)≤p(f x0,gx1),p(gx1,T x1)≤p(gx1,f x2),p(gx1,Sx0)≤p(gx1,gx1),

p(f x0,T x1)≤p(f x0,f x2)≤p(f x0,gx1) +p(gx1,f x2)−p(gx1,gx1).

(3.3)

Hence, from inequality 3.2 and the above inequalities, we get

p(gx1,f x2) ≤

1

√

rαp(f x0,gx1) +

1

√

rβ[p(f x0,gx1) +p(gx1,f x2)] (3.4)

+ √1

rγ[p(f x0,gx1) +p(gx1,f x2)]

= [(α+β+γ)/√r](p(f x0,gx1)) + [(β+γ)/√r](p(gx1,f x2)). (3.5)

Hence,[√r₋(β+γ)]p(gx1,f x2)≤(α+β+γ)p(f x0,gx1).

Recall thatλ:=√α+β+γ_r

−(β+γ),so we have

p(gx1,f x2)≤λp(f x0,gx1). (3.6)

Continuing in this way, with f x2∈T x1,there existsx3∈X,withgx3∈Sx2such thatp(f x2,gx3)≤√1_rHp(Sx2,T x1).

As before, we obtain

p(f x2,gx3)≤

1

√

rHp(Sx2,T x1)

≤√1

rαp(f x2,gx1) +

1

√

rβ[p(f x2,Sx2) +p(gx1,T x1)] +

1

√

rγ[p(f x2,T x1) +p(gx1,Sx2)].

Note that

p(f x2,Sx2)≤p(f x2,gx3),p(gx1,T x1)≤p(gx1,f x2),p(f x2,T x1)≤p(f x2,f x2), (3.7)

p(gx1,Sx2)≤p(gx1,gx3)≤p(gx1,f x2) +p(f(x2,gx3)−p(f x2,f x2). (3.8)

Hence,

p(f x2,gx3)≤

1

√

rαp(f x2,gx1) +

1

√

rβ[p(f x2,gx3) +p(gx1,f x2)]

+√1

rγ[p(gx1,f x2) +p(f x2,gx3)].

We obtain from above the inequality thatp(f x2,gx3)≤λp(f x2,gx1).Continuing this process, we construct a sequence

yninX, such thaty0=gx1and, for eachn,y2n=gx2n+1∈Sx2n,y2n+1= f x2n+2∈T x2n+1. It can be established that

p(yn,yn+1)≤λp(yn−1,yn).For instance, from p(f x2,gx3)≤λp(f x2,gx1),we havep(y1,y2)≤λp(y0,y1).Now, if

T X _⊆ f X) of the form f x for some xand this we now call x2j+2, satisfying the inequality p(gx2j+1,f x2j+2)≤

1

√

rHp(Sx2j,T x2j+1).Next we invoke inequality 3.1 in the hypothesis to get

Hp(Sx2j,T x2j+1)≤αp(f x2j,gx2j+1) +β[p(f x2j,Sx2j) +p(gx2j+1,T x2j+1)]

+γ[p(f x2j,T x2j+1) +p(gx2j+1,Sx2j)]. (3.9)

As above, we have

p(f x2j,Sx2j)≤p(f x2j,gx2j+1),

p(gx2j+1,T x2j+1)≤p(gx2j+1,f x2j+2),

p(gx2j+1,Sx2j)≤p(gx2j+1,gx2j+1),

(3.10)

and

p(f x2j,T x2j+1)≤p(f x2j,f x2j+2)≤p(f x2j,gx2j+1) +p(gx2j+1,f x2j+2)−p(gx2j+1,gx2j+1). Hence,

p(gx2j+1,f x2j+2)≤√1

rαp(f x2j,gx2j+1) +

1

√

rβ[p(f x2j,gx2j+1) +p(gx2j+1,f x2j+2)]

+√1

rγ[p(f x2j,gx2j+1) +p(gx2j+1,f x2j+2)−p(gx2j+1,gx2j+1) +p(gx2j+1,gx2j+1)].

Again, we get

(√r₋(β+γ))p(gx2j+1,f x2j+2)≤(α+β+γ)p(f x2j,gx2j+1) (3.11) and this yieldsp(y2j,y2j+1)≤λp(y2j,y2j−1).The same argument applies ifn=2k+1.We then have the inequality

p(yn,yn+1)≤λnp(y0,y1),for alln≥1.To show that the sequenceynis a Cauchy sequence in the complete metric

space(X,ps),we observe that

p(yn,yn+m)≤p(yn,yn+1) +p(yn+1,yn+2) +. . .+p(yn+m−1,yn+m)

≤λn_{(1+λ+. . .+λ}m−1_)p(y

0,y1) =λnp(y0,y1)/(1−λ)→0 (3.12)

asn_→∞.By the definition of the metricps,we haveps(yn,yn+m)≤2p(yn,yn+m)→0, henceynis a Cauchy sequence

in(X,ps).Lemma 2.1 above allows us to conclude that(X,ps)is complete, since the partial metric space(X,p)is complete. Hence,ynconverges to somey∈Xwith respect to the metricps,that is, limn→∞ps(yn,y) =0.From the

char-acterization of Cauchy sequence convergence earlier cited (namely, A sequencexnconverges in(X,ps)to a pointx∈X

if and only iflimn,m→∞p(xn,xm) =limn→∞p(xn,x) =p(x,x)),we havep(y,y) =limn→∞p(yn,y) =limn→∞p(yn,yn) =

0.Withy2n=gx2n+1andy2n+1=f x2n+2, we must havelimn→∞y2n=limn→∞gx2n+1=limn→∞f x2n+2=y,and since

gXand f X are closed, theny_∈f X andy_∈gX.Hence, there are elementsuandwinXsuch thaty=f u=gw.With

f x2n+2∈T x2n+1, note that

p(f u,Su)≤p(f u,f x2n+2) +p(f x2n+2,Su)≤p(f u,f x2n+2) +Hp(Su,T x2n+1)

≤p(f u,f x2n+2) +αp(f u,gx2n+1) +β[p(f u,Su) +p(gx2n+1,T x2n+1)]

+γ[p(f u,T x2n+1) +p(gx2n+1,Su)]

≤p(f u,f x2n+2) +αp(f u,gx2n+1) +β[p(f u,Su) +p(gx2n+1,f x2n+2)]

+γ[p(f u,f x2n+2) +p(gx2n+1,Su)]. (3.13)

Taking limits asn_→∞,we getp(f u,Su)≤(β+γ)p(f u,Su).Since(β+γ)<1,we havep(f u,Su) =0.SinceSu

(β+γ)p(gw,Tw),allowing us to similarly conclude thatp(gw,Tw) =0,andgw_∈Tw.Finally,

Hp(Su,Tw)≤αp(f u,gu) +β[p(f u,Su) +p(gw,Tw)] +γ[p(f u,Tw) +p(gw,Su)]

=αp(f u,gu) +β[p(f u,Su) +p(gw,Tw)] +γ[p(gw,Tw) +p(f u,Su)] =0.

Hence,Su=Tw.

Iff =gthen we obtain the following coincidence result.

Theorem 3.2. Let(X,d) be a complete partial metric space. Let T,S:X _→CBp(X) be multivalued maps and f:X_→X be a single-valued map satisfying, for each x,y_∈X,

Hp(Sx,Ty)≤αp(f x,f y) +β[p(f x,Sx) +p(f y,Ty)] +γ[p(f x,Ty) +p(f y,Sx)] (3.14)

whereα,β andγ ≥0,α+2β+2γ <1.If f X is a closed subset of X and T X_∪SX_⊆ f X , then, f,T,and S have a coincidence in X.Moreover, if f is both T -weakly commuting and S-weakly commuting at each z_∈C(f,T), and

f f z=f z, then, f,T,and S have a common fixed point in X.

Proof. If f =gin Theorem 3.1, there exist pointsuandwinXsuch that f u_∈Su, f w_∈Tw, f u= f wandSu=Tw. Asu_∈C(f,T),f isT -weakly commuting atuand f(f u) =f u. Setv= f u. Then, we have f v=vandv= f(f u)∈

T(f u) =T v. Since alsou_∈C(f,S), then f isS-weakly commuting atu, and so we obtainv=f v= f(f u)∈S(f u) =

Sv. Thus, we have proved thatv=f v_∈T v_∩Sv,that is,vis a common fixed point of f,T andS.

If f =g=IX (IX being the identity map onX) in Theorem 3.1, then, we obtain the following common fixed-point

result.

Corollary 3.1. Let(X,p)be a complete partial metric space. Let T,S:X_→CBp(X)be multi-valued maps satisfying, for each x,y_∈X,

Hp(Sx,Ty)≤αp(x,y) +β[p(x,Sx) +p(y,Ty)] +γ[p(x,Ty) +p(y,Sx)], (3.15)

whereα,β,γ≥0and0<α+2β+2γ<1. Then, there exists a point z in X , such that z_∈Sz_∩T z and Sz=T z.

Remark 3.1. If S=T in the Corollary, then, we obtain our previous result. (See [9].)

Remark 3.2. If in Theorem 3.1 (i) β =γ=0 and S=T; f =g=IX, then, we obtain the extension of Nadler’s

theorem to partial metric spaces (see also Aydi [2]); (ii) if S=T and f =g=IX,then, we obtain the extension to

partial metric spaces a result of Reich [13].

IfSandT in Corollary 3.1 above are single-valued maps, then, we obtain the following result:

Corollary 3.2. Let(X,p)be a complete partial metric space. Let T,S:X_→X be single-valued maps satisfying, for each x,y_∈X, p(Sx,Ty)≤αp(x,y) +β[p(x,Sx) +p(y,Ty)] +γ[p(x,Ty) +p(y,Sx)],whereα,β,γ≥0 and0<

α+2β+2γ<1. Then, S and T have a common fixed point in X , that is, there exists z_∈X such that z=Sz=T z.

Remark 3.3. If S=T in Corollary 3.2 , then, we obtain the partial metric space version of a result of Hardy and Rogers [5].

Acknowledgements

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