A continuation method for weakly contractive mappings under the interior condition

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009, Article ID 809315, 8pages doi:10.1155/2009/809315 Research Article A Continuation Method for Weakly Contractive Mappings under the Interior Condition 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 29 July 2009; Accepted 8 October 2009 Recommended by Marlene Frigon Recently, Frigon proved that, for weakly contractive maps, the property of having a fixed point is invariant by a certain class of homotopies, obtaining as a consequence a Leray-Schauder alternative for this class of maps in a Banach space. We prove here that the Leray-Schauder condition in the aforementioned result can be replaced by a modification of it, the interior condition. We also show that our arguments work for a certain class of generalized contractions, thus complementing a result of Agarwal and O’Regan. Copyright q2009 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. 1. Introduction Suppose that Xis a Banach space, that U⊂Xis an open bounded subset of X, containing the origin, and that f:U→Xis a mapping. It is well known that if fsatisfies the LeraySchauder condition defined as fx/ λx, for x∈∂U, λ > 1L-S and fis a strict set-contraction or, more generally, condensing, then fhas a fixed point in U see, e.g., 1or 2. The first continuation method in the setting of a complete metric space for contractive maps comes from the hands of Granas 3, in 1994, who gave a homotopy result for contractive maps for more information on this topic see, e.g., 4,5or 6. On the other hand, it has been recently shown in 7that, for condensing mappings, the condition L-Scan be replaced by a modification of it which we call the interior condition, 2 Fixed Point Theory and Applications and is defined as follows: a mapping f:U→Xsatisfies the Interior Condition I-C, if there exists δ>0 such that fx/ λx, for x∈Uδ,λ>1,f x/ ∈U, I-C where Uδ{x∈U:distx, ∂U<δ}some generalizations of this result can be found in 8,9. We remark that the condition I-Cby itself cannot be a substitute for the condition L-S, and an additional assumption on the domain of fneeds to be made in order to guarantee the existence of a fixed point for f. The class of sets that we need is defined as follows: suppose that U⊂Xis an open neighborhood of the origin. We say that Uis strictly star shaped if for any x∈∂U we have that {λx :λ>0}∩∂U {x}.Itwas shown in 7that if Uis bounded and strictly star shaped and f:U→Xis a condensing mapping satisfying the condition I-C, then fhas a fixed point. Of course, this result includes the case of a contractive map i.e., a map ffor which there exists k∈0,1such that dfx,fy ≤kdx, yfor all x, y ∈U, but our aim in this note is, following the pattern of Granas 3and Frigon et al. 10, to give a continuation method for weakly contractive mappings, in the setting of a complete metric space, under some conditions on the homotopy which are the counterpart of the condition I-Cand the notion of a strictly star shaped set in a space without a vector structure. Finally, in the last section we show that our arguments also work for a class of generalized contractions, thus complementing a result of Agarwal and O’Regan 11. 2. Weakly Contractive Maps In this chapter we deal with the concept of weakly contractive maps, as it was introduced by Dugundji and Granas in 12. Definition 2.1. Let X, dbe a complete metric space and Uan open subset of X.Afunction f:U→Xis said to be weakly contractive if there exists ψ:X×X→0,∞compactly positive i.e., inf{ψx, y:a≤dx, y≤b}θa, b>0 for every 0 <a≤bsuch that dfx,fy≤dx, y−ψx, y.2.1 If ψis a compactly positive function, we define for 0 <a≤b γa, bmin{a, θa, b}.2.2 It was shown in 12that any weakly contractive map f:X→Xdefined on a complete metric space Xhas a unique fixed point. Some years later, Frigon 5proved that, for weakly contractive maps, the property of having a fixed point is invariant by a certain class of homotopies, obtaining as a consequence a Leray-Schauder alternative for weakly contractive maps in the setting of a Banach space. We prove here that the Leray-Schauder condition in the aforementioned result can be replaced by the condition I-C, and it will also be obtained as a consequence of a continuation method. The definition of homotopy that we need for our purposes is the following. Fixed Point Theory and Applications 3 Definition 2.2. Let X, dbe a complete metric space, and Uan open subset of X.Letf,g : U→Xbe two weakly contractive maps. We say that fis I-C-homotopic to gif there exists H:U×0,1→Xwith the following properties: P1Hx,1fxand Hx, 0gxfor every x∈U; P2there exists δ>0 such that x/ Hx,tfor every x∈Uδ,withfx/ ∈U,andt∈0,1, where Uδ{x∈U:distx, ∂U<δ}; P3there exists a compactly positive function ψ:X×X→0,∞such that dHx,t,Hy,t ≤dx, y−ψx, yfor every x, y ∈U,andt∈0,1; P4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|; P5if x∈∂U and 0 ≤λ<1, with Hx, λ∈∂U, then Hx, 1/ ∈U. In the proof of the main result of this chapter we shall make use of the following lemma see Frigon 5. Lemma 2.3. Let x0∈X,r>0, and h:Bx0,r→Xweakly contractive. If dx0,hx0 < γr/2,r,thenhhas a fixed point. Theorem 2.4. Let f,g :U→Xbe two weakly contractive maps. Suppose that fis homotopic to g and gUis bounded. If ghas a fixed point in U,thenfhas a fixed point in U. Proof. We argue by contradiction. Suppose that fdoes not have any fixed point in U,andlet Hbe a homotopy between fand g, in the sense of Definition 2.1. Consider the set A{λ∈0,1:xHx,λfor some x∈U},2.3 and notice that Ais nonempty since ghas a fixed point in U,thatis,0∈A. We will show that Ais both open and closed in 0,1, and hence, by connectedness, we will have that A0,1. As a result, fwill have a fixed point in U, which establishes a contradiction. To show that Ais closed, suppose that {λn}is a sequence in Aconverging to λ∈0,1 and let us show that λ∈A. Since λn∈A, there exists xn∈Uwith xnHxn,λ n.Fix ε>0. Using that gUis bounded and that φis continuous on the compact interval 0,1, it is easy to show that there exists M>εsuch that diam HU×0,1 ≤M, and hence dxn,x m≤Mfor all n, m ∈N. Define μθε, Mand let n0∈Nbe such that for all n, m ≥n0,|φλn−φλm|<μ. Then dxn,x m<εfor all n, m ≥n0because, otherwise, we would have dxn,x m≥εfor some n, m ≥n0, and then 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dxn,x m−ψxn,x m <μdxn,x m−ψxn,x m ≤dxn,x m, 2.4 4 Fixed Point Theory and Applications which is a contradiction. Then {xn}is a Cauchy sequence and, since X, dis complete, there exists x0∈Usuch that xn→x0as n→∞. In addition, x0Hx0,λsince for all n∈Nwe have that dxn,Hx0,λ  dHxn,λ n,Hx0,λ  ≤dHxn,λ n,Hxn,λ  dHxn,λ ,Hx0,λ  ≤φλn−φλdxn,x 0−ψxn,x 0 ≤φλn−φλdxn,x 0. 2.5 Observe that 0 ≤λ<1, because if λ1,then x0Hx0,1fx0, which contradicts the fact that fdoes not have any fixed point in U.Noticethatx0∈U, because, otherwise, we would have x0∈∂U,thatis,Hx0,λ∈∂U, and since 0 ≤λ<1, by P5, we have that Hx0,1/ ∈U. However, since x0∈∂U,{xn}→x0and xn∈Ufor all n∈N, there exists n0∈Nsuch that xn∈Uδfor all n≥n0. Hence, since xnHxn,λ nfor all n≥n0, applying P2, we have that fxn∈Ufor all n≥n0,thatis,Hxn,1∈Ufor all n≥n0. Taking limits, we arrive to the contradiction Hx0,1∈U. Therefore, x0∈Uand, consequently, λ∈A. Next we show that Ais open in 0,1.Letλ0∈A. Then there exists x0∈Uwith x0 Hx0,λ 0.Letr>0 be such that Bx0,r⊂U,andletδ>0 such that |φλ−φλ0|<γr/2,r for every λ∈0,1with |λ0−λ|<δ. Then, if λ∈λ0−δ, λ0δ∩0,1, dx0,Hx0,λ  dHx0,λ 0,Hx0,λ  ≤φλ0−φλ <γ r 2,r. 2.6 Using Lemma 2.3,weobtainthatH·,λhas a fixed point in Ufor every λ∈0,1such that |λ0−λ|<δ.Thusλ∈Afor any λ∈λ0−δ, λ0δ∩0,1, and therefore Ais open in 0,1. As an immediate consequence of the previous theorem, we obtain the following fixed point result of the Leray-Schauder type for weakly contractive maps under the condition I-C. Theorem 2.5. Suppose that Uis an open and strictly star shaped subset of a Banach space X, ·, with 0∈U, and that f:U→Xis a weakly contractive map with fUbeing bounded. If fsatisfies the condition I-C,thenfhas a fixed point in U. Proof. Since fsatisfies the condition I-C, there exists δ>0 such that fx/ λx for λ>1 and x∈Uδwith fx/ ∈U. We may assume that x/ fxfor every x∈Uδ, because otherwise we are finished. Define H:U×0,1→Xas Hx, ttfx,andletgbe the zero map. Notice that ghas a fixed point in U,thatis,0g0and also that fand gare two weakly contractive mappings. So, the result will follow from Theorem 2.4 once we prove that fis I-C-homotopic to g. Let us check it. Fixed Point Theory and Applications 5 P1For all x∈U,Hx,00·fx0gxand Hx,11·fxfx. P2Since fsatisfies the condition I-C, we have that fx/ λx for x∈Uδwith fx/ ∈U and λ>1. Hence, x/ Hx,tfor every x∈Uδ,withfx/ ∈U,andt∈0,1. P3Since fis weakly contractive, there exists a compactly positive function ψ:X×X→ 0,∞such that dfx,fy ≤dx,y−ψx, yfor every x, y ∈U. Then, if x,y ∈U and t∈0,1, dHx,t,Hy,tt fx−fy  ≤dfx,fy ≤dx,y−ψx, y. 2.7 P4Since fUis bounded, there exists M≥0 such that fx≤Mfor all x∈U. Hence, dHx,t,Hx, s  fx |t−s| ≤M|t−s| φt−φs, 2.8 where φ:0,1→Ris the continuous function defined as φtMt. P5Suppose that for some x∈∂U and λ<1 we have that Hx, λ∈∂U. Then, fx/ 0 since Hx,λλfx,0∈Uand Uis open. Let us see that Hx, 1/ ∈U: suppose, on the contrary, that Hx, 1∈U,thatis,fx∈Uand define  λ:supt≥1:tfx∈U.2.9 Then, it is easy to see that  λfx∈∂U, which contradicts that Uis strictly star shaped, since we also have that λfx∈∂U. 3. A Class of Generalized Contractions A multitude of generalizations and variants of Banach’s contractive condition have been given after Banach’s theorem see, e.g., Rhoades 13 and, recently, Agarwal and O’Regan 11have given a homotopy result thus generalizing a fixed point theorem of Hardy and Rogers 14 under the following generalized contractive condition: there exists a∈0,1 such that for all x, y ∈X dfx,fy≤amaxdx, y,dx, fx,dy,fy,1 2dx,fydy,fx.3.1 6 Fixed Point Theory and Applications In this section we give a homotopy result for this class of mappings under the condition I-C. In the proof of our theorem we shall use the following result 11. Lemma 3.1. Let X, dbe a complete metric space, x0∈X,r>0, and h:Bx0,r→X. Suppose that there exists a∈0,1such that for x,y ∈Bx0,rone has dhx,h y≤amaxdx, y,dx, hx,dy, hy,1 2dx,hydy,hx, dx0,h x0 <1−ar. 3.2 Then there exists x∈Bx0,rwith xhx. The proof of the following theorem is very similar to the proof of Theorem 2.4, and we give a sketch of it. Theorem 3.2. Let X, dbe a complete metric space, and Uan open subset of X.Letf,g :U→X be two maps such that there exists H:U×0,1→Xwith the following properties: P1Hx,1fxand Hx, 0gxfor every x∈U; P2there exists δ>0such that x/ Hx,tfor every x∈Uδ,withfx/ ∈U, and t∈0,1, where Uδ{x∈U:distx, ∂U<δ}; P3there exists a∈0,1such that for all x, y ∈Uand λ∈0,1one has dHx,λ,Hy,λ ≤amax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; 3.3 P4there exists a continuos 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|; P5if x∈∂U and 0≤λ<1,withHx, λ∈∂U,thenHx, 1/ ∈U. If ghas a fixed point in U,thenfhas a fixed point in U. Proof. Suppose that fdoes not have any fixed point in Uand consider the nonempty set A{λ∈0,1:Hx,λxfor some x∈U}.3.4 We will arrive to a contradiction by showing that A0,1, and for this we only need prove that Ais closed and open in 0,1. To show that Ais closed in 0,1, consider a sequence {λn}in A,withλn→λ∈0,1 as n→∞, and show that λ∈A; that is, that there exists x0∈Uwith Hx0,λx0. To prove that x0exists, take any sequence {xn}in Uwith xnHxn,λ n, prove that {xn}is Cauchy, and define x0as the limit of {xn},asn→∞. That {xn}is a Cauchy sequence, as well as x0Hx0,λ, follows from standard arguments which can be seen in 11, Theorem 3.1. It remains to show that x0∈U. Fixed Point Theory and Applications 7 To prove this, suppose that it is not true and arrive to a contradiction as follows: we have that Hx0,λx0∈U\U∂U, and also that 0 ≤λ<1, because fdoes not have any fixed point in U. Then, by P5fx0/ ∈∂U. On the other hand, fx0lim fxn∈Ubecause fxn∈Ufor nlarge enough. To be convinced of it, just apply P2:sincex0∈∂U,{xn}→x0 and xn∈Ufor all n∈N, there exists n0∈Nsuch that xn∈Uδfor all n≥n0. Then, fxn∈U for all n≥n0since xnHxn,λ n. To prove that Ais open argue as in Theorem 2.4,useLemma 3.1 instead of Lemma 2.3. As an immediate consequence, we obtain the following result, whose proof is omitted because it is analogous to the proof of Theorem 2.5. Theorem 3.3. Suppose that Uis an open and strictly star shaped subset of a Banach space X, ·, with 0∈U, and that f:U→Xis map with fUbeing bounded. Assume also that there exists a∈0,1such that for all x, y ∈Uand λ∈0,1one has dλfx,λfy ≤amaxdx, y,dx, λfx,dy,λfy,1 2dx,λfydy,λfx. 3.5 If fsatisfies the condition I-C,thenfhas a fixed point in U. Acknowledgment This research is partially supported by the Spanish Grant MTM2007-60854and regional Andalusian Grants FQM210, FQM1504Governments. 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