Contractive multivalued maps in terms of Q-functions on complete quasimetric spaces

Karapınar et al. Fixed Point Theory and Applications 2014, 2014:53 http://www.fixedpointtheoryandapplications.com/content/2014/1/53 R E S E A R C H Open Access Contractive multivalued maps in terms of Q-functions on complete quasimetric spaces Erdal Karapınar1,2, Salvador Romaguera3and Pedro Tirado3* *Correspondence: [email protected].es 3Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Valencia, 46022, Spain Full list of author information is available at the end of the article Abstract In this paper we prove the existence of a fixed point for multivalued maps satisfying a contraction condition in terms of Q-functions, and via Bianchini-Grandolfi gauge functions, for complete T0-quasipseudometric spaces. Our results extend, improve, and generalize some recent results in the literature. We present some examples to validate and illustrate our results. MSC: 54H25; 47H10; 54E50 Keywords: fixed point; T0-quasipseudometric; multivalued map; Q-function 1 Introduction and preliminaries The notion of metric space, introduced by Fréchet [], is one of the cornerstones of both applied and pure mathematics. The metric space is indispensable in many branches of mathematics. For example, in these days, one of the core topics in group theory is to construct a metric on a given group under the certain conditions. Due to its wide application areas in all quantitative sciences, this notion has been generalized and extended in various way, such as quasimetrics, symmetrics, b-metrics, G-metrics, fuzzy metrics, etc. Among all, we attract attention to the notion of Q-function, introduced by Al-Homidan et al. []in the framework of quasimetric space as an extension of the concept of w-distance defined by Kada et al. []. In fact, the authors of [] proved, among other results, a quasimetric version of the celebrated Nadler fixed point theorem []. Recently, Marín et al. [] generalized some results of [] by using Bianchini-Grandolfi gauge functions. Almost simultaneously, Latif and Al-Mezel [] obtained a quasimetric generalization of a well-known fixed point theorem of Mizoguchi and Takahashi [, Theorem ] (see also [, ]) for multivalued maps on complete metric spaces. In this paper we prove the existence of fixed point for a lower semicontinuous multivalued map satisfying certain contraction condition in terms of Q-functions via BianchiniGrandolfi gauge functions on a complete T-quasipseudometric space. We also prove a weaker version of that theorem by removing the lower semicontinuity assumption. We state some examples to show the validity of the conditions and to indicate our generalizations have worth, and finally give applications to the case of contractive multivalued maps on complete partial metric spaces. Our results improve, generalize, and extend several known results in this direction. Let Ndenote the set of positive integer numbers, while ωdenotes the set of nonnegative integer numbers. ©2014 Karapınar et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Karapınar et al. Fixed Point Theory and Applications 2014, 2014:53 Page 2 of 15 http://www.fixedpointtheoryandapplications.com/content/2014/1/53 For the sake of completeness of the paper, we recall several pertinent notions and fundamental results. Let Xbe nonempty set and d:X×X→[, ∞) be a function such that (qpm)d(x,y)=d(y,x)=⇔x=y,and (qpm)d(x,z)≤d(x,y)+d(y,z), for all x,y,z∈X.Thendis called a T-quasipseudometric on a set X.Thepair(X,d)is said to be a T-quasipseudometric space. If one replaces the condition (qpm) with the stronger condition (qpm)∗d(x,y)=⇔x=y, then dis called a quasimetric on X.Inthiscase,thepair(X,d)issaidtobeaquasimetric space. InthesequelwewillusetheabbreviationT-qpm (respectively, T-qpm space) instead of T-quasipseudometric (respectively, T-quasipseudometric space). Given a T-qpm don a set X, the function d– defined by d–(x,y)=d(y,x)isalsoa T-qpm, called the conjugate of d. It is clear that the function dsdefined by ds(x,y)= max{d–(x,y), d(x,y)}is a metric on X. (Note that if dis a metric on X,thend=ds.) Consequently, every T-qpm don Xinduces three topologies defined as follows. (τ)Thefirsttopology,τdwhich has as a base the family of open balls {Bd(x,ε):x∈ Xand ε>},whereBd(x,ε)={y∈X:d(x,y)<ε}for all x∈Xand ε>. (τ)Thesecondtopology,τd– which has as a base the family of open balls {Bd– (x,ε):x∈ Xand ε>},whereBd– (x,ε)={y∈X:d–(x,y)<ε}for all x∈Xand ε>. (τ) The last topology induced by the metric dsand denoted by τds. Notice that both τdand τd– are Ttopologies on X.Furthermore,ifdis a quasimetric on X,thend– is also a quasimetric on Xand hence, both τdand τd– are Ttopologies on X. It immediately follows that a sequence (xn)n∈Nin a T-qpm space (X,d)isτd-convergent to x∈Xif and only if limn→∞ d(x,xn) = . Analogously, a sequence (xn)n∈Nin a T-qpm space (X,d)isτd– -convergent to x∈Xif and only if limn→∞ d(xn,xn)=. In the literature, the notion of completeness for quasimetric spaces can be varied; see e.g. [, ,]. In the context of our paper we shall use the following very general notion: AT-qpm space (X,d) is said to be complete if every Cauchy sequence in the metric space (X,ds)isτd– -convergent. Now, we recall the definition of Q-function, as introduced by Al-Homidan-AnsariYao []. Definition  Let (X,d)beaT-qpm space and q:X×X→[, ∞) be a function which satisfies (Q)q(x,z)≤q(x,y)+q(y,z), for all x,y,z∈X, (Q)ifx∈X,M>and (yn)n∈Nis a sequence in Xthat τd– -converges to a point y∈X, and satisfies q(x,yn)≤M, for all n∈N,thenq(x,y)≤M, (Q)foreachε>there exists δ>such that q(x,y)≤δand q(x,z)≤δimply d(y,z)≤ε. Then qis called a Q-function on (X,d). Karapınar et al. Fixed Point Theory and Applications 2014, 2014:53 Page 3 of 15 http://www.fixedpointtheoryandapplications.com/content/2014/1/53 If qsatisfies conditions (Q)and(Q ), and (Q )foreachx,y∈Xthe function q(x,·):X→[,∞)is τd– -lower semicontinuous on (X,d), then qis called a w-distance on (X,d). Note that every w-distance is a Q-function. Remark  It is evident that dis a w-distance on (X,d)ifdis a metric on X.Notealsothat if (X,d)isaT-qpm space then dis not necessarily a Q-function on (X,d)[, Example .] (see also [, Proposition .]). We conclude this section with the following simple fact which will be useful in the rest of the paper. Lemma  [] Let q be a Q-function on a T-qpm space (X,d), let ε>and let δ=δ(ε)> for which condition (Q)holds.If q(x,y)≤δand q(x,z)≤δthen ds(y,z)≤ε. 2 Main results Let (X,d)beaT-qpm space. The collection of all nonempty subsets (respectively, τdsclosed subsets) of Xwill be denoted by X(respectively, Clds(X)). Let be the family of functions ϕ:[,∞)→[,∞) satisfying the following conditions: (ϕ)ϕis nondecreasing; (ϕ)+∞ n= ϕn(t)<∞for all t>,whereϕnis the nth iterate of ϕ. These functions are known in the literature as Bianchini-Grandolfi gauge functions in some sources (see e.g. [–]) and as (c)-comparison functions in some other sources (see e.g. []). It is easily proved that if ϕ∈,thenϕ(t)<tfor any t>(seee.g. []). The following lemma will be crucial to prove our first theorem. Lemma  Let (X,d)be a T-qpm space,q a Q-function on (X,d), ϕ:[,∞)→[,∞)a Bianchini-Grandolfi gauge function and T :X→Xa multivalued map such that for each x,y∈Xandu∈Tx;there is v ∈Ty satisfying q(u,v)≤ϕmaxq(x,y),q(x,u),q(y,v).() Then,for each x∈Xthereisasequence(xn)n∈ωsatisfying the following three conditions: (a) xn+ ∈Txnfor all n∈ω. (b) For each δ>there exists nδ∈Nsuch that q(xn,xm)<δwhenever m>n≥nδ. (c) (xn)n∈ωis a Cauchy sequence in the metric space (X,ds). Proof Fix x∈X.Letx∈Tx. By hypothesis, there exists x∈Txsuch that q(x,x)≤ϕmaxq(x,x),q(x,x). Similarly, there exists x∈Txsuch that q(x,x)≤ϕmaxq(x,x),q(x,x). Karapınar et al. Fixed Point Theory and Applications 2014, 2014:53 Page 4 of 15 http://www.fixedpointtheoryandapplications.com/content/2014/1/53 Following this process we construct a sequence (xn)n∈ωin Xsuch that xn+ ∈Txnand q(xn+,xn+)≤ϕmaxq(xn,xn+), q(xn+,xn+),() for all n∈ω. Now we distinguish two cases. Case . There exists k∈ωsuch that q(xk,xk+) = . Then, by condition ()andthefact that ϕ(t)<tfor all t>,wededucethatq(xk+,xk+)=.Repeatingthisargument,weobtain q(xk+j,xk+j+) =  for all j∈ω, so, by condition (Q), q(xn,xm)=wheneverm>n≥k. It follows from Lemma that for each ε>,ds(xn,xm)≤εwhenever n,m>k,andthus (xn)n∈ωis a Cauchy sequence in (X,ds). Thus we have shown that conditions (a), (b), and (c) are satisfied. Case . q(xn,xn+) >  for all n∈ω. Then, by condition ()andthefactthatϕ(t)<tfor all t>,wededucethatq(xn,xn+)>q(xn+,xn+) for all n∈ω,so q(xn+,xn+)≤ϕq(xn,xn+)<q(xn,qn+), for all n∈ω. Therefore q(xn,xn+)≤ϕnq(x,x),() for all n∈ω. Now choose an arbitrary ε>.Letδ=δ(ε)∈(, ε) for which condition (Q) holds. We shall show that conditions (b) and (c) hold. Indeed, since q(x,x)>, ∞ n= ϕn(q(x,x)) < ∞,sothereisnδ∈ωsuch that ∞  n=nδ ϕnq(x,x)<δ.() Then, for m>n≥nδ,weobtain q(xn,xm)≤q(xn,xn+)+q(xn+,xn+)+···+q(xm– ,xm) ≤ϕnq(x,x)+ϕn+q(x,x)+···+ϕm–q(x,x) ≤ ∞  j=nδ ϕjq(x,x)<δ.() In particular, q(xnδ,xn)≤δand q(xnδ,xm)≤δwhenever n,m>nδ. Thus, by Lemma , ds(xn,xm)≤εwhenever n,m>nδ.Hence(xn)n∈ωis a Cauchy sequence in (X,ds). This concludes the proof.  We also need the following notion. Definition  Let qbe a Q-functiononaT-qpm space (X,d). We say that a multivalued map T:X→Xis q-lower semicontinuous (q-l.s.c. in short) if the function x→ q(x,Tx) is lower semicontinuous on the metric space (X,ds), where q(x,Tx)=inf{q(x,y):y∈Tx}. Remark  An antecedent of the above concept can be found in Theorem . of the paper by Daffer and Kaneko [], where they proved a generalization of Nadler’s fixed point Karapınar et al. Fixed Point Theory and Applications 2014, 2014:53 Page 5 of 15 http://www.fixedpointtheoryandapplications.com/content/2014/1/53 theorem for a multivalued map Ton a complete metric space (X,d) by assuming that the function x→ d(x,Tx) is lower semicontinuous on (X,d). Before establishing our first fixed point result we recall that a point z∈Xis said to be a fixed point of a multivalued map T:X→Xif z∈Tz. Theorem  Let (X,d)be a complete T-qpm space,q a Q-function on (X,d), ϕ:[,∞)→ [,∞)a Bianchini-Grandolfi gauge function and T :X→Clds(X)aq-l.s.c.multivalued map such that for each x,y∈Xandu∈Tx,there is v ∈Ty satisfying q(u,v)≤ϕmaxq(x,y),q(x,u),q(y,v).() Then T has a fixed point z ∈Xsuchthatq(z,z)=. Proof Fix x∈X. Then there is a sequence (xn)n∈ωsatisfying the three conditions (a), (b) and (c) of Lemma .Since(X,d) is complete, there exists z∈Xsuch that limn→∞ d(xn, z)=. We shall prove that z∈Tz.Tothisend,firstweprovethatlimn→∞ q(xn,z) = . Indeed, given ε>takeδ=δ(ε)<εfor which condition (Q)holds.Fixn≥nδ. By condition (b), we have q(xn,xm)≤δwhenever m>n, so from condition (Q)wededucethatq(xn,z)≤δ<ε whenever n≥nδ. Next we show that limn→∞ ds(xn,z) = . Indeed, given ε>takeδ=δ(ε)<εfor which condition (Q)holds.Sinceq(xnδ,z)≤δand q(xnδ,xn)≤δwhenever n>nδ, it follows from Lemma that ds(z,xn)≤εwhenever n>nδ. Now we prove that there is a sequence (zk)k∈Nin Tz such that limk→∞ q(z,zk) = . Indeed, since the sequence (xn)n∈Nsatisfies (b) and, by assumption, Tis q-l.s.c., we deduce that there exists a subsequence (xnk)k∈Nof (xn)n∈Nsuch that q(xnk,xnk+)<  kand q(z,Tz)<q(xnk,Txnk)+  k, for all k∈N. Therefore, there exists a sequence (zk)k∈Nin Tz satisfying q(z,zk)<q(z,Tz)+  k<q(xnk,Txnk)+ k≤q(xnk,xnk+)+ k, for all k∈N.Hence lim k→∞ q(z,zk)=. () Then, by (Q)andthefactthatlimn→∞ q(xnk,z) = , we deduce that limk→∞ q(xnk,zk)=. So, by (Q) and Lemma ,weobtain lim n→∞ ds(z,zk)=. () Consequently z∈CldsTz =Tz. Finally, q(z,z)=by(), (), and condition (Q).  Next we give an example which shows that q-lower semicontinuity of Tcannot be omitted in Theorem not even when (X,d)isacompletemetricspace. Karapınar et al. Fixed Point Theory and Applications 2014, 2014:53 Page 6 of 15 http://www.fixedpointtheoryandapplications.com/content/2014/1/53 Example  Let X={}∪N∪A,whereA={/n:n∈N\{}},andletdbe the restriction to Xof the usual metric on the set of real numbers. It is clear that (X,d)isacompletemetric space. Now let q:X×X→[, ∞)begivenby q(x,x)=for all x∈X, q(,x)=for all x∈N∪A, q(x,y)=q(y,x)=for all x∈N,y∈A, q(x,y)=for all x,y∈N, q(x,)=for all x∈N, q(x,y)=|x–y|for all x,y∈A,and q(x,)=xfor all x∈A. It is easy to check that qis a Q-function (actually it is a w-distance) on (X,d). Define T:X→Cld(X)as T=N, Tx = /xfor all x∈N,and Tx =x/ for all x∈A. Since q(,T) =  and q(x,Tx)=x/ for all x∈A,wededucethatTis not q-l.s.c. Moreover, it is obvious that Thas no fixed point. However, we shall show that the contraction condition () is satisfied for the Bianchini-Grandolfi gauge function ϕdefined as ϕ(t)=t/ for all t≥. To this end, we first note that for x=,y∈N∪A,andu∈Tx,wehaveu∈N,Ty ={v} with v∈A, and hence q(u,v)==ϕ() = ϕq(x,y). Similarly, if x∈N,y=andu∈Tx,wetakev=x∈Ty,andthus q(u,v)==ϕ() = ϕq(x,y). If x∈A,y=,andu∈Tx,wehaveu=x/ and taking v=∈Ty,wededuce q(u,v)==ϕ() = ϕq(y,v). Now, if x,y∈Aand u∈Tx,wehaveu=x/ and Ty ={v}where v=y/, so that q(u,v)=  |x–y|= q(x,y)=ϕq(x,y). Similarly, if x,y∈N,withx=y,andu∈Tx,wehaveu= /x∈Aand Ty ={v}where v= /y,sothat q(u,v)=    x– y < =ϕ() = ϕq(x,y). Finally, for x∈N,y∈A,andu∈Tx,wehaveu∈Aand Ty ={v},withv∈A,sothat q(u,v)=|u–v|< =ϕ() = ϕq(x,y). Thecasethatx∈Aand y∈Nis similar, and hence it is omitted. Karapınar et al. Fixed Point Theory and Applications 2014, 2014:53 Page 7 of 15 http://www.fixedpointtheoryandapplications.com/content/2014/1/53 Our next fixed point result shows that q-lower semicontinuity of Tcan be removed if the contraction condition ()isreplacedwithq(u,v)≤ϕ(max{q(x,y),q(x,u)}). Theorem  Let (X,d)be a complete T-qpm space,q a Q-function on (X,d), ϕ:[,∞)→ [,∞)a Bianchini-Grandolfi gauge function and T :X→Clds(X)amultivaluedmapsuch that for each x,y∈Xandu∈Tx,there is v ∈Ty satisfying q(u,v)≤ϕmaxq(x,y),q(x,u).() Then T has a fixed point. Proof Fix x∈X. Then there is a sequence (xn)n∈ωsatisfying the three conditions (a), (b), and (c) of Lemma .Since(X,d) is complete, there exists z∈Xsuch that limn→∞ d(xn, z)=. Now, as in the proof of Theorem ,weobtainlimn→∞ q(xn,z)=. For each n∈ω,takezn∈Tz such that q(xn,zn)≤ϕmaxq(xn–,z), q(xn–,xn).() We show that limn→∞ q(xn,zn) = . Indeed, given ε>thereexistsn∈Nsuch that q(xn–,z)<εand q(xn–,xn)<εfor all n>n.Takeanyn>n.Ifq(xn–,z)=q(xn–,xn)=, then q(xn,zn) = . Otherwise, we have <maxq(xn–,z), q(xn–,xn)<ε, so, by ()andthefactthatϕ(t)<tfor all t>,wededucethatq(xn,zn)<ε. Consequently lim n→∞ ds(z,zn)=, by Lemma .Weconcludethatz∈Tz. The following consequences of Theorem , which are also illustrated by Example below, improve and generalize in several directions the Banach contraction principle. Corollary  Let (X,d)be a complete T-qpm space,q a Q-function on (X,d), ϕ:[,∞)→ [,∞)a Bianchini-Grandolfi gauge function and T :X→Clds(X)amultivaluedmapsuch that for each x,y∈Xandu∈Tx,there is v ∈Ty satisfying q(u,v)≤ϕq(x,y). Then T has a fixed point. If we take ϕ(t)=rt where r∈[,)wegetoneofthemainresultsin[]. Corollary  Let (X,d)be a complete T-qpm space,q a Q-function on (X,d), T:X→ Clds(X)amultivaluedmapandr∈[,) such that for each x,y∈Xandu∈Tx,there is Karapınar et al. Fixed Point Theory and Applications 2014, 2014:53 Page 8 of 15 http://www.fixedpointtheoryandapplications.com/content/2014/1/53 v∈Ty satisfying q(u,v)≤rq(x,y). Then T has a fixed point. Corollary was proved in [, Theorem .]. In fact, it was showed that there is a fixed point z∈Xof Tsuch that q(z,z) = . This suggests the following question that remains open: Under the conditions of Theorem ,isthereisafixedpointz∈Xof Tsuch that q(z,z)=? Remark  Example shows that Theorem  is not true when the contraction condition ()isreplacedwithq(u,v)≤ϕ(max{q(x,y),q(y,v)}). Indeed, take in that example, x∈A, y=andu∈Tx.Thenwehaveu=x/, and hence q(u,v)=>x=max{q(x,y), q(x,u)}. Theorems and are independent from each other. The following two examples show this fact. Example  Let X=ω,i.e.,X={}∪N,andletdbe the quasimetric on Xdefined as d(x,x)=for all x∈X, d(x,y)=xif x>y,and d(x,y)=x+yif x<y. Clearly (X,d) is a complete quasimetric space and τdis the discrete topology on X,so τd=τds. Furthermore, it is almost obvious that dis a w-distance on (X,d). Now let T:X→Clds(X)givenas T=, T={x∈N:x>},and Tx ={}∪{y∈N:y>x}for all x∈N\{}. Since τdsis the discrete topology on Xit immediately follows that Tis d-l.s.c. Consider the Bianchini-Grandolfi gauge function ϕgiven by ϕ(t)=t/ if ≤t<,and ϕ(t)=nif t∈[n+,n+),n∈N. An easy computation of the different cases shows that the contraction condition () is satisfied. Indeed, let x,y∈Xand u∈Tx. In the cases where for u=wecanchoose v=∈Ty, the conclusion is obvious. Therefore, we briefly discuss the rest of the cases. If x=,y=,wehaveu= , and taking v=∈Ty we deduce d(u,v)==ϕ() = ϕd(y,v). If x=,y=andu∈Tx,wehavev=andthus d(u,v)=u=ϕ(u+ )=ϕd(x,u). If x∈N\{},y=andu∈Tx,withu=,wededuce d(u,v)=u=ϕ(u+)≤ϕd(x,u). Karapınar et al. Fixed Point Theory and Applications 2014, 2014:53 Page 9 of 15 http://www.fixedpointtheoryandapplications.com/content/2014/1/53 If x=,y∈N\{}and u∈Tx,takev=∈Ty,and,asintheprecedingcase, d(u,v)=u=ϕ(u+)≤ϕd(x,u). If x∈N\{},y=andu∈Tx,takev=∈Ty and thus (recall that u=oru>x) d(u,v)=max{u,v}≤max{u+x–,v}=ϕmaxd(x,u),d(y,v). Finally, if x,y∈N\{}and u∈Tx,withu=,takev=∈Ty and thus d(u,v)=u<u+x–=ϕ(u+x)=ϕd(x,u). Hence, all conditions of Theorem are satisfied. However, we cannot apply Theorem  because for x=,y=,u=andanyv∈Ty,wehave d(u,v)=v>=maxd(x,y),d(x,u)>ϕ() = ϕmaxd(x,y),d(x,u). Example  Let X={,}∪A,whereA={–/n:n∈N\{}},andletdbe the restriction to Xof the usual metric on the set of real numbers. It is clear that (X,d)isacomplete metric space. Now let q:X×X→[, ∞)begivenby q(x,x)=for all x∈X\{}, q(,) = , q(,x)=q(x, ) = / for all x∈X\{}, q(,x)=xfor all x∈A, q(x,)=–xfor all x∈A,and q(x,y)=|x–y|for all x,y∈A. It is not difficult to check that qis a w-distance on (X,d). Define T:X→Cld(X)as T=, T={,},and Tx ={,( + x)/}for all x∈A, and let ϕbe the Bianchini-Grandolfi gauge function given by ϕ(t)=t/ for all t≥. Notice that Tis not q-l.s.c. because q(, T) = q(,) = /, but for each x∈A, q(x,Tx)=qx,+x =–x ≤ . Hence, we cannot apply Theorem to this example. We show that, nevertheless, the contraction condition () is satisfied and consequently the conditions of Theorem hold. Indeed, let x,y∈Xand u∈Tx. In the cases where for u=wecanchoosev=∈Ty, the conclusion is obvious. Therefore we discuss the rest of the cases. If x=y=andu=,takev=,andthus q(u,v)=  =ϕ() = ϕq(x,u). If x=,y=andu=,wehavev=,and,asintheprecedingcase,q(u,v)=ϕ(q(x,u)).