A continuation method for weakly Kannan maps

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 321594, 12 pages doi:10.1155/2010/321594 Research Article A Continuation Method for Weakly Kannan Maps David Ariza-Ruiz and Antonio Jim´ enez-Melado Departamento de An´ alisis Matem´ atico, Facultad de Ciencias, Universidad de M´ alaga, 29071 M´ alaga, Spain Correspondence should be addressed to Antonio Jim´ enez-Melado, [email protected] Received 25 September 2009; Revised 4 December 2009; Accepted 6 December 2009 Academic Editor: Mohamed A. Khamsi Copyright q2010 D. Ariza-Ruiz and A. Jim´ enez-Melado. 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. The first continuation method for contractive maps in the setting of a metric space was given by Granas. Later, Frigon extended Granas theorem to the class of weakly contractive maps, and recently Agarwal and O’Regan have given the corresponding result for a certain type of quasicontractions which includes maps of Kannan type. In this note we introduce the concept of weakly Kannan maps and give a fixed point theorem, and then a continuation method, for this class of maps. 1. Introduction Suppose that X, dis a metric space and that f:D⊂X→Xis a map. We say that f is contractive if there exists α∈0,1such that dfx,fy ≤αdx,yfor all x, y ∈D. The well-known Banach fixed point theorem states that fhas a fixed point if DXand X, dis complete. In 1962, Rakotch 1obtained an extension of Banach theorem replacing the constant αby a function of dx, y,ααdx, y, provided that αis nonincreasing and 0 ≤αt<1 for all t>0for a recent refinement of this result see 2. A similar generalization of the contractive condition was considered by Dugundji and Granas 3, who extended Banach theorem to the class of weakly contractive mappings i.e., ααx, y,with sup{αx, y:a≤dx,y≤b}<1 for all 0 <a≤b. Another focus of attention in Fixed Point Theory is to establish fixed point theorems for non-self mappings. In the setting of a Banach space, Gatica and Kirk 4proved that if f:U→Xis contractive, with Uan open neighborhood of the origin, then fhas a fixed point if it satisfies the well-known Leray-Schauder condition: fx/ λx, for x∈∂U, λ > 1.L-S Recently, Kirk 5has extended this result to the abstract setting of a certain class of metric spaces: the CAT0spaces. In the proof, the author uses a homotopy result due to 2 Fixed Point Theory and Applications Granas 6, which is known as continuation method for contractive maps. In fact, the jump from a Banach space setting to the metric space setting was given by Granas himself in 6 for more information on this topic see, for instance, 7–9. After Granas, Frigon 8gave a similar result for weakly contractive maps. A variant of the Banach contraction principle was given by Kannan 10, who proved that a map f:X→X, where X, dis a complete metric space, has a unique fixed point if f is what we call a Kannan map, that is, there exists α∈0,1such that, for all x, y ∈X, dfx,fy≤α 2dx,fxdy,fy.1.1 In this note, following the pattern of Dugundji and Granas 3, we extend Kannan theorem to the class of weakly Kannan maps i.e., ααx, y,withsup{αx, y:a≤dx, y≤ b}<1 for all 0 <a≤b.ThisisdoneinSection 2.InSection 3 we use a local version of the previous result to obtain a continuation method for weakly Kannan maps. 2. Weakly Kannan Maps In this section we follow the pattern of Dugundji and Granas 3to introduce the concept of weakly Kannan maps. Definition 2.1. Let X, dbe a metric space, D⊂X,andf:D→X. Therefore fis a weakly Kannan map if there exists α:D×D→0,1,withθa, b:sup{αx,y:a≤dx, y≤ b}<1 for every 0 <a≤bsuch that, for all x, y ∈D, dfx,fy≤αx, y 2dx,fxdy,fy.2.1 Remark 2.2. Clearly, any weakly Kannan map fhas at most one fixed point: if xfxand yfy, then dx,ydfx,fy≤1 2dx,fxdy,fy0.2.2 Remark 2.3. Notice that if f:D⊂X→Xis a weakly Kannan map and we define αfx, yon D×Das αfx,y⎧ ⎪ ⎨ ⎪ ⎩ 2dfx,fy dx,fxdy,fy if dx, fxdy,fy/ 0, 0 otherwise, 2.3 then αfis well defined, takes values in 0,1, satisfies sup{αfx, y:a≤dx, y≤b}<1 for all 0 <a≤bfor αfis smaller than any αassociated to f, and also satisfies 2.1,withα replaced by αf, for all x, y ∈D. Conversely, if αfis defined as in 2.3and satisfies the above set of conditions, then fis a weakly Kannan map, establishing in this way an equivalent definition for Kannan maps. Fixed Point Theory and Applications 3 Remark 2.4. Although Kannan showed that the concept of Kannan map is independent of the concept of contractive map, Janos 11observed that any contractive map f:D⊂X→X whose Lipschitz constant defined by Lfsupdfx,fy dx,y:x,y ∈X, x / y2.4 is less than 1/3 is a Kannan map. Next, we exhibit an example of a weakly Kannan map f,withLf1/3, which is not a Kannan map, thus showing that the constant 1/3inthe aforementioned result by Janos is sharp. Example 2.5. Consider the metric space X0,∞with the usual metric dx, y|x−y|,and let f:X→Xbe the function defined as fx1/3log1ex. Then, Lf1/3andfis a weakly Kannan map, but not a Kannan map. The equality Lf1/3 follows from the fact that |fx|<1/3 for all x∈0,∞ together with lim x→∞ dfx,f0 dx,01 3.2.5 We also have that fis not a Kannan map because lim x→∞ 2dfx,f0 dx,fxd0,f01.2.6 To check that fis a weakly Kannan map, consider the function α:X×X→0,∞ given by 2.3. This function is well defined and also takes values in 0,1since Lf1/3. Next, assume that 0 <a≤band let us see that θa, bsup{αx, y:a≤dx,y≤b}<1. To see this, observe that |u−fu|→∞as u→∞, so there is M>0 such that |u−fu|>bfor all u>M. Observe also that fM, the restriction of fto 0,M, is a Kannan map with constant αM∈0,1, due to the fact that LfM<1/3, for fMis continuously differentiable on 0,M and |fu|<1/3 for all u∈0,M. We will see θa, b≤max{2/3,α M}.Todoit,suppose that x,y ∈0,∞with a≤|x−y|≤band 0 ≤x<y. Then, if y>M,use|y−fy|>band that Lf≤1/3toobtainαx,y≤2/3. Otherwise, we would have 0 ≤x<y≤Mand then αx,y≤αM. Although the way we have introduced the concept of weakly Kannan map has been by analogy with the work done by Dugundji and Granas in 3, we would like to mention that this extension may be done in some different ways. For instance, Pathak et al. 12, Theorem 3.1have proved the following result. Theorem A. Let X, dbe a complete metric space and suppose that f:X→Xis a map such that dfx,fy≤α1dx, fxdx, fxα2dy,fydy,fy,2.7 for all x,y ∈X,whereαi:R→0,1. If, in addition, there exists a sequence {xn}in Xwith dxn,fxn →0,thenfhas a fixed point in X. 4 Fixed Point Theory and Applications Observe that relation 2.7can be written in the following more general form: dfx,fy≤A1xdx, fxA2ydy,fy,2.8 for all x, y ∈X, where Ai:X→0,1,i1,2, and notice that any map satisfying 2.8also satisfies the relation 2.1with αx, y2 max{A1x,A 2y}. In fact, the arguments used by the authors in the proof of Theorem A are also valid for this class of maps. Next, we state this slightly more general result and include the proof for the sake of completeness. Then, we obtain, as a consequence, a fixed point theorem for weakly Kannan maps. Theorem 2.6. Let X, dbe a complete metric space and assume that A:X×X→0,∞is a bounded function satisfying the following condition: for any sequence {xn}in Xand u∈X, xn−→ u⇒lim sup Axn,u <1.∗ Assume also that f:X→Xis a map such that dfx,fy≤Ax, ydx,fxdy,fy,2.9 for all x,y ∈X. If there exists a sequence {xn}in Xwith dxn,fxn →0,thenfhas a unique fixed point uin X, and xn→u. Proof. Since Ais bounded, there exists M>0 such that |Ax, y|≤Mfor all x, y ∈X. Suppose that {xn}is a sequence in Xwith dxn,fxn →0anduse2.9to obtain that, for all n, m ∈N, dfxn,fxm≤Mdxn,fxndxm,fxm.2.10 This implies that {fxn}is a Cauchy sequence. Since X, dis complete, the sequence {fxn}is convergent, say to u∈X. Then xn→ubecause dxn,fxn →0. Thus, by ∗, lim sup Axn,u<1. That ufuis a consequence of the following relation and the fact that lim sup Axn,u<1, then du, fulim dfxn,fu≤lim sup Axn,u dxn,fxndu, fu du, fulim sup Axn,u . 2.11 Finally, uis the unique fixed point of fbecause if zfz: du, zdfu,fz≤Au, zdu, fudz, fz0.2.12 Corollary 2.7. Let X, dbe a complete metric space and suppose that f:X→Xis a weakly Kannan map. Then, fhas a unique fixed point u∈Xand, for any x0∈X, the sequence of iterates {fnx0}converges to u. Fixed Point Theory and Applications 5 Proof. Since fis a weakly Kannan map, there exists a function α:X×X→0,1with θa, b:sup{αx, y:a≤dx,y≤b}<1 for all 0 <a≤b, satisfying 2.1for all x, y ∈X. Hence, the function A:X×X→0,1/2given as Ax, y1/2αx,yis bounded and satisfies the conditions ∗and 2.9. Consider any x0∈Xand define xnfxn−1,n1,2,.... We may assume that dx0,x 1>0 because otherwise we have finished. We will prove that dxn,fxn →0and hence, by Theorem 2.6,{xn}will converge to a point uwhich is the unique fixed point of f. First of all, observe that the inequality dxn1,x n≤αxn,x n−1dxn,x n−12.13 holds for all n≥1. In fact, it is a consequence of the following one, which is true by 2.1: dxn1,x n≤αxn,x n−1 2dxn1,x ndxn,x n−1.2.14 From 2.13we obtain that the sequence {dxn,x n−1}is nonincreasing, for 0 ≤ αxn,x n−1≤1, and then it is convergent to the real number dinf{dxn,x n−1:n1,2,... }.2.15 To prove that d0, suppose that d>0 and arrive to a contradiction as follows: use 0<d≤dxn,x n−1≤dx1,x 02.16 and the definition of θθd, dx1,x 0 to obtain αxn,x n−1≤θfor all n1,2,.... This, together with 2.13,givesthat d≤dxn1,x n≤θndx1,x 0,2.17 for all n1,2,..., which is impossible since d>0and0≤θ<1. Remark 2.8. We do not know whether Theorem A is, or not, a particular case of Theorem 2.6, although that is the case if the functions α1,α 2satisfy the additional assumption sup{α1t α2t:t≥0}<2. To see this, suppose that the map f:X→Xis in the conditions of Theorem A, that is, fsatisfies relation 2.7for some given functions αi:R→0,1,i1,2, and suppose also that the functions α1,α 2satisfy in addition sup{α1tα2t:t≥0}<2. Define A:X×X→0,∞as Ax,ymax{ax,ay}, where a:X→0,1is given by az1 2α1dz, fzα2dz, fz.2.18 6 Fixed Point Theory and Applications Let us see that, with this function A,fsatisfies the hypotheses of Theorem 2.6. Indeed, Ais clearly bounded and also satisfies ∗;if{xn}is a sequence in Xand u∈X,withxn→u, then sup{axn:n1,2,... }≤supα1tα2t 2:t≥0<1.2.19 Since we also have that au<1, we obtain that sup{Axn,u:n1,2,...}<1. Finally, to see that fsatisfies relation 2.9,userelation2.7with x, y ∈X, together with the same relation interchanging the roles of xand y, and the fact that dfx,fy  dfy,fx,toobtainthat dfx,fy≤axdx, fxaydy,fy,2.20 from which the result follows. To prove the homotopy result of the next section, we will need the following local version of Corollary 2.7. Corollary 2.9. Assume that X, dis a complete metric space, x0∈X, r > 0, and f:Bx0,r→X is a weakly Kannan map with associated function αsatisfying 2.1.Ifθis defined as usual, and dx0,fx0<1 3minr 2,r1−θr 2,r,2.21 then fhas a fixed point. Proof. In view of Corollary 2.7,itsuffices to show that the closed ball Bx0,ris invariant under f. To prove it, consider any x∈Bx0,rand obtain the relation dx0,fx≤dx0,fx0dfx0,fx ≤dx0,fx0αx0,x 2dx0,fx0dx, fx ≤dx0,fx0αx0,x 2dx0,fx0dx, x0dx0,fx, 2.22 from which, having in mind that αx0,x≤1, dx0,fx≤3dx0,fx0αx0,xdx0,x.2.23 To end the proof, obtain that dx0,fx ≤rthrough the above inequality by considering two cases: if dx0,x≤r/2, then dx0,fx ≤rbecause dx0,fx0 ≤r/6. Otherwise, we would have r/2≤dx0,x≤r, and consequently αx0,x≤θr/2,r,from Fixed Point Theory and Applications 7 which dx0,fx≤r1−θr 2,rrθr 2,rr. 2.24 3. A Homotopy Result In 1974 ´ Ciri´ c13introduced the concept of quasicontractions and proved the following fixed point theorem: suppose that X, dis a complete metric space and that f:X→Xis a quasicontraction, that is, there exists q∈0,1such that, for all x,y ∈X, dfx,fy≤qmaxdx, y,dx,fx,dy,fy,dx, fy,dy,fx.3.1 Then, fhas a fixed point in X. Observe that any contractive map, as well as any Kannan map, is a quasicontraction; thus, the theorem by ´ Ciri´ c generalizes the well known fixed point theorems by Banach and Kannan. On the other hand, Agarwal and O’Regan 14considered a certain class of quasicontractions: those maps f:X→X, where X, dis a metric space, for which there exists q∈0,1such that, for all x, y ∈X, dfx,fy ≤qmaxdx, y,dx, fx,dy,fy,1 2dx,fydy,fx, Q and gave the following homotopy result. Theorem B. Let X, dbe a complete metric space, Uan open subset of X, and H:U×0,1→X satisfying the following properties: iHx,λ/ xfor all x∈∂U and all λ∈0,1, iithere exists q∈0,1such that for all x, y ∈Uand λ∈0,1we have dHx,λ,Hy,λ ≤qmaxdx, y,dx, Hx,λ,dy,Hy,λ,1 2dx,Hy,λ,dy,Hx, λ, 3.2 iiiHx,λis continuous in λ, uniformly for x∈U. If H·,0has a fixed point in U,thenH·,λalso has a fixed point in Ufor all λ∈0,1. The above homotopy result includes the corresponding one for the class of Kannan maps, and in the following theorem we show that an analogous result is true for the wider class of weakly Kannan maps. 8 Fixed Point Theory and Applications Theorem 3.1. Let X, dbe a complete metric space, Uan open subset of X, and H:U×0,1→X satisfying the following properties: P1Hx,λ/ xfor all x∈∂U and all λ∈0,1, P2there exists α:U×U→0,1such that for all x, y ∈Uand λ∈0,1one has dHx,λ,Hy, λ≤αx,y 2dx,Hx, λ dy,Hy, λ,3.3 and θa, bsup{αx, y:a≤dx,y≤b}<1for all 0<a≤b, P3there exists a continuous function φ:0,1→Rsuch that, for every x∈Uand t, s ∈ 0,1,dHx,t,Hx, s ≤|φt−φs|. If H·,0has a fixed point in U,thenH·,λalso has a fixed point in Ufor all λ∈0,1. Proof. Consider the nonempty set A{λ∈0,1:Hx,λxfor some x∈U}.3.4 We will prove that A0,1, and for this it suffices to show that Ais both closed and open in 0,1. We start showing that Ais closed in 0,1: suppose that {λn}is a sequence in A converging to λ∈0,1and let us show that λ∈A. By definition of A, there exists a sequence {xn}in Uwith xnHxn,λ n. We will prove that {xn}converges to a point x0∈Uwith Hx0,λx0, thus showing that λ∈A. That {xn}is a Cauchy sequence is a consequence of the following relation, where we have used P2,P3, and the fact that xmHxm,λ m: dxn,x mdHxn,λ n,Hxm,λ m ≤dHxn,λ n,Hxn,λ m dHxn,λ m,Hxm,λ m ≤φλn−φλmαxn,x m 2dxn,Hxn,λ m dxm,Hxm,λ m φλn−φλmαxn,x m 2dHxn,λ n,Hxn,λ m ≤3 2φλn−φλm. 3.5 Fixed Point Theory and Applications 9 Write x0lim xnand let us see that x0∈Uand also that x0Hx0,λ.Thatx0Hx0,λis a consequence of the following relation: dx0,Hx0,λ  ≤dx0,x ndxn,Hx0,λ  ≤dx0,x ndHxn,λ n,Hxn,λ  dHxn,λ ,Hx0,λ  ≤dx0,x nφλn−φλ1 2dxn,Hxn,λ  dx0,Hx0,λ  ≤dx0,x n3 2φλn−φλ1 2dx0,Hx0,λ , 3.6 and that x0∈Uis straightforward from P1. Next we prove that Ais open in 0,1: suppose that λ0∈Aand let us show that λ0−δ, λ0δ∩0,1⊂A, for some δ>0. Since λ0∈A, there exists x0∈Uwith x0Hx0,λ 0. Consider r>0withBx0,r⊂Uand use the continuity of φto obtain δ>0 such that φλ−φλ0<minr 2,r1−θr 2,r,3.7 for all λ∈λ0−δ, λ0δ∩0,1. To show now that any λ∈λ0−δ, λ0δ∩0,1is also in A,itsuffices to prove that the map H·,λ:Bx0,r→Xhas a fixed point. And this is true by Corollary 2.9,since dx0,Hx0,λ  dHx0,λ 0,Hx0,λ  ≤φλ0−φλ <minr 2,r1−θr 2,r. 3.8 Remark 3.2. A careful reading of the proof shows that hypothesis P3in Theorem 3.1 can be easily replaced by the weaker hypothesis iiiin Theorem B. Remark 3.3. The counterpart to Theorem 3.1 for weakly contractive maps was proved by Frigon 8. In that result, it was assumed, in place of our 3.3, an equivalent formulation of the following condition H’: dHx,λ,Hy,λ≤αx,ydx,y.H’ Observe that condition H’means that all the maps H·,λ:U→X,λ∈0,1are weakly contractive, and with the same function α. Our condition 3.3is no surprise then. It also means that all the maps H·,λare of weakly Kannan type, and with the same function α. We end the section with an example of a homotopy Hsatisfying P1,P2,andP3 but not the hypotheses of Theorem B. In fact, the function fH·,1will be of weakly Kannan type, but will not satisfy the quasicontractivity condition Qhence, it will not be