A continuation method for weakly contractive mappings under the interior condition

Hindawi Publishing Co po a ion Fixed Poin Theo y and Applica ions Volume 2009, A icle ID 809315, 8pages doi:10.1155/2009/809315 Resea ch A icle A Con inua ion Me hod o Weakly Con ac i e Mappings unde he In e io Condi ion Da id A iza-Ruiz and An onio Jim´ enez-Melado Depa amen o de An´ alisis Ma em´ a ico, Facul ad de Ciencias, Uni e sidad de M´ alaga, 29071 M´ alaga, Spain Co espondence should be add essed o An onio Jim´ enez-Melado, [email p o ec ed] Recei ed 29 July 2009; Accep ed 8 Oc obe 2009 Recommended by Ma lene F igon Recen ly, F igon p o ed ha , o weakly con ac i e maps, he p ope y o ha ing a ixed poin is in a ian by a ce ain class o homo opies, ob aining as a consequence a Le ay-Schaude al e na i e o his class o maps in a Banach space. We p o e he e ha he Le ay-Schaude condi ion in he a o emen ioned esul can be eplaced by a modi ica ion o i , he in e io condi ion. We also show ha ou a gumen s wo k o a ce ain class o gene alized con ac ions, hus complemen ing a esul o Aga wal and O’Regan. Copy igh q2009 D. A iza-Ruiz and A. Jim´ enez-Melado. This is an open access a icle dis ibu ed unde he C ea i e Commons A ibu ion License, which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed. 1. In oduc ion Suppose ha Xis a Banach space, ha U⊂Xis an open bounded subse o X, con aining he o igin, and ha :U→Xis a mapping. I is well known ha i sa is ies he Le ay- Schaude condi ion de ined as x/ λx, o x∈∂U, λ > 1L-S and is a s ic se -con ac ion o , mo e gene ally, condensing, hen has a ixed poin in U see, e.g., 1o 2. The i s con inua ion me hod in he se ing o a comple e me ic space o con ac i e maps comes om he hands o G anas 3, in 1994, who ga e a homo opy esul o con ac i e maps  o mo e in o ma ion on his opic see, e.g., 4,5o 6. On he o he hand, i has been ecen ly shown in 7 ha , o condensing mappings, he condi ion L-Scan be eplaced by a modi ica ion o i which we call he in e io condi ion, 2 Fixed Poin Theo y and Applica ions and is de ined as ollows: a mapping :U→Xsa is ies he In e io Condi ion I-C, i he e exis s δ>0 such ha x/ λx, o x∈Uδ,λ>1, x/ ∈U, I-C whe e Uδ{x∈U:dis x, ∂U<δ}some gene aliza ions o his esul can be ound in 8,9. We ema k ha he condi ion I-Cby i sel canno be a subs i u e o he condi ion L-S, and an addi ional assump ion on he domain o needs o be made in o de o gua an ee he exis ence o a ixed poin o . The class o se s ha we need is de ined as ollows: suppose ha U⊂Xis an open neighbo hood o he o igin. We say ha Uis s ic ly s a shaped i o any x∈∂U we ha e ha {λx :λ>0}∩∂U {x}.I was shown in 7 ha i Uis bounded and s ic ly s a shaped and :U→Xis a condensing mapping sa is ying he condi ion I-C, hen has a ixed poin . O cou se, his esul includes he case o a con ac i e map i.e., a map o which he e exis s k∈0,1such ha d x, y ≤kdx, y o all x, y ∈U, bu ou aim in his no e is, ollowing he pa e n o G anas 3and F igon e al. 10, o gi e a con inua ion me hod o weakly con ac i e mappings, in he se ing o a comple e me ic space, unde some condi ions on he homo opy which a e he coun e pa o he condi ion I-Cand he no ion o a s ic ly s a shaped se in a space wi hou a ec o s uc u e. Finally, in he las sec ion we show ha ou a gumen s also wo k o a class o gene alized con ac ions, hus complemen ing a esul o Aga wal and O’Regan 11. 2. Weakly Con ac i e Maps In his chap e we deal wi h he concep o weakly con ac i e maps, as i was in oduced by Dugundji and G anas in 12. De ini ion 2.1. Le X, dbe a comple e me ic space and Uan open subse o X.A unc ion :U→Xis said o be weakly con ac i e i he e exis s ψ:X×X→0,∞compac ly posi i e i.e., in {ψx, y:a≤dx, y≤b}θa, b>0 o e e y 0 <a≤bsuch ha d x, y≤dx, y−ψx, y.2.1 I ψis a compac ly posi i e unc ion, we de ine o 0 <a≤b γa, bmin{a, θa, b}.2.2 I was shown in 12 ha any weakly con ac i e map :X→Xde ined on a comple e me ic space Xhas a unique ixed poin . Some yea s la e , F igon 5p o ed ha , o weakly con ac i e maps, he p ope y o ha ing a ixed poin is in a ian by a ce ain class o homo opies, ob aining as a consequence a Le ay-Schaude al e na i e o weakly con ac i e maps in he se ing o a Banach space. We p o e he e ha he Le ay-Schaude condi ion in he a o emen ioned esul can be eplaced by he condi ion I-C, and i will also be ob ained as a consequence o a con inua ion me hod. The de ini ion o homo opy ha we need o ou pu poses is he ollowing. Fixed Poin Theo y and Applica ions 3 De ini ion 2.2. Le X, dbe a comple e me ic space, and Uan open subse o X.Le ,g : U→Xbe wo weakly con ac i e maps. We say ha is I-C-homo opic o gi he e exis s H:U×0,1→Xwi h he ollowing p ope ies: P1Hx,1 xand Hx, 0gx o e e y x∈U; P2 he e exis s δ>0 such ha x/ Hx,  o e e y x∈Uδ,wi h x/ ∈U,and ∈0,1, whe e Uδ{x∈U:dis x, ∂U<δ}; P3 he e exis s a compac ly posi i e unc ion ψ:X×X→0,∞such ha dHx, ,Hy,  ≤dx, y−ψx, y o e e y x, y ∈U,and ∈0,1; P4 he e exis s a con inuous unc ion φ:0,1→Rsuch ha , o e e y x∈Uand , s ∈0,1,dHx, ,Hx, s ≤|φ −φs|; P5i x∈∂U and 0 ≤λ<1, wi h Hx, λ∈∂U, hen Hx, 1/ ∈U. In he p oo o he main esul o his chap e we shall make use o he ollowing lemma see F igon 5. Lemma 2.3. Le x0∈X, >0, and h:Bx0, →Xweakly con ac i e. I dx0,hx0 < γ /2, , henhhas a ixed poin . Theo em 2.4. Le ,g :U→Xbe wo weakly con ac i e maps. Suppose ha is homo opic o g and gUis bounded. I ghas a ixed poin in U, hen has a ixed poin in U. P oo . We a gue by con adic ion. Suppose ha does no ha e any ixed poin in U,andle Hbe a homo opy be ween and g, in he sense o De ini ion 2.1. Conside he se A{λ∈0,1:xHx,λ o some x∈U},2.3 and no ice ha Ais nonemp y since ghas a ixed poin in U, ha is,0∈A. We will show ha Ais bo h open and closed in 0,1, and hence, by connec edness, we will ha e ha A0,1. As a esul , will ha e a ixed poin in U, which es ablishes a con adic ion. To show ha Ais closed, suppose ha {λn}is a sequence in Acon e ging o λ∈0,1 and le us show ha λ∈A. Since λn∈A, he e exis s xn∈Uwi h xnHxn,λ n.Fix ε>0. Using ha gUis bounded and ha φis con inuous on he compac in e al 0,1, i is easy o show ha he e exis s M>εsuch ha diam HU×0,1 ≤M, and hence dxn,x m≤M o all n, m ∈N. De ine μθε, Mand le n0∈Nbe such ha o all n, m ≥n0,|φλn−φλm|<μ. Then dxn,x m<ε o all n, m ≥n0because, o he wise, we would ha e dxn,x m≥ε o some n, m ≥n0, and hen 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 Poin Theo y and Applica ions which is a con adic ion. Then {xn}is a Cauchy sequence and, since X, dis comple e, he e exis s x0∈Usuch ha xn→x0as n→∞. In addi ion, x0Hx0,λsince o all n∈Nwe ha e ha 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 Obse e ha 0 ≤λ<1, because i λ1, hen x0Hx0,1 x0, which con adic s he ac ha does no ha e any ixed poin in U.No ice ha x0∈U, because, o he wise, we would ha e x0∈∂U, ha is,Hx0,λ∈∂U, and since 0 ≤λ<1, by P5, we ha e ha Hx0,1/ ∈U. Howe e , since x0∈∂U,{xn}→x0and xn∈U o all n∈N, he e exis s n0∈Nsuch ha xn∈Uδ o all n≥n0. Hence, since xnHxn,λ n o all n≥n0, applying P2, we ha e ha xn∈U o all n≥n0, ha is,Hxn,1∈U o all n≥n0. Taking limi s, we a i e o he con adic ion Hx0,1∈U. The e o e, x0∈Uand, consequen ly, λ∈A. Nex we show ha Ais open in 0,1.Le λ0∈A. Then he e exis s x0∈Uwi h x0 Hx0,λ 0.Le >0 be such ha Bx0, ⊂U,andle δ>0 such ha |φλ−φλ0|<γ /2,  o e e y λ∈0,1wi h |λ0−λ|<δ. Then, i 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  2, . 2.6 Using Lemma 2.3,weob ain ha H·,λhas a ixed poin in U o e e y λ∈0,1such ha |λ0−λ|<δ.Thusλ∈A o any λ∈λ0−δ, λ0δ∩0,1, and he e o e Ais open in 0,1. As an immedia e consequence o he p e ious heo em, we ob ain he ollowing ixed poin esul o he Le ay-Schaude ype o weakly con ac i e maps unde he condi ion I-C. Theo em 2.5. Suppose ha Uis an open and s ic ly s a shaped subse o a Banach space X, ·, wi h 0∈U, and ha :U→Xis a weakly con ac i e map wi h Ubeing bounded. I sa is ies he condi ion I-C, hen has a ixed poin in U. P oo . Since sa is ies he condi ion I-C, he e exis s δ>0 such ha x/ λx o λ>1 and x∈Uδwi h x/ ∈U. We may assume ha x/  x o e e y x∈Uδ, because o he wise we a e inished. De ine H:U×0,1→Xas Hx,  x,andle gbe he ze o map. No ice ha ghas a ixed poin in U, ha is,0g0and also ha and ga e wo weakly con ac i e mappings. So, he esul will ollow om Theo em 2.4 once we p o e ha is I-C-homo opic o g. Le us check i . Fixed Poin Theo y and Applica ions 5 P1Fo all x∈U,Hx,00· x0gxand Hx,11· x x. P2Since sa is ies he condi ion I-C, we ha e ha x/ λx o x∈Uδwi h x/ ∈U and λ>1. Hence, x/ Hx,  o e e y x∈Uδ,wi h x/ ∈U,and ∈0,1. P3Since is weakly con ac i e, he e exis s a compac ly posi i e unc ion ψ:X×X→ 0,∞such ha d x, y ≤dx,y−ψx, y o e e y x, y ∈U. Then, i x,y ∈U and ∈0,1, dHx, ,Hy,    x− y  ≤d x, y ≤dx,y−ψx, y. 2.7 P4Since Uis bounded, he e exis s M≥0 such ha  x≤M o all x∈U. Hence, dHx, ,Hx, s   x | −s| ≤M| −s| φ −φs, 2.8 whe e φ:0,1→Ris he con inuous unc ion de ined as φ M . P5Suppose ha o some x∈∂U and λ<1 we ha e ha Hx, λ∈∂U. Then, x/ 0 since Hx,λλ x,0∈Uand Uis open. Le us see ha Hx, 1/ ∈U: suppose, on he con a y, ha Hx, 1∈U, ha is, x∈Uand de ine  λ:sup ≥1: x∈U.2.9 Then, i is easy o see ha  λ x∈∂U, which con adic s ha Uis s ic ly s a shaped, since we also ha e ha λ x∈∂U. 3. A Class o Gene alized Con ac ions A mul i ude o gene aliza ions and a ian s o Banach’s con ac i e condi ion ha e been gi en a e Banach’s heo em see, e.g., Rhoades 13 and, ecen ly, Aga wal and O’Regan 11ha e gi en a homo opy esul  hus gene alizing a ixed poin heo em o Ha dy and Roge s 14 unde he ollowing gene alized con ac i e condi ion: he e exis s a∈0,1 such ha o all x, y ∈X d x, y≤amaxdx, y,dx, x,dy, y,1 2dx, ydy, x.3.1 6 Fixed Poin Theo y and Applica ions In his sec ion we gi e a homo opy esul o his class o mappings unde he condi ion I-C. In he p oo o ou heo em we shall use he ollowing esul 11. Lemma 3.1. Le X, dbe a comple e me ic space, x0∈X, >0, and h:Bx0, →X. Suppose ha he e exis s a∈0,1such ha o x,y ∈Bx0, 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 . 3.2 Then he e exis s x∈Bx0, wi h xhx. The p oo o he ollowing heo em is e y simila o he p oo o Theo em 2.4, and we gi e a ske ch o i . Theo em 3.2. Le X, dbe a comple e me ic space, and Uan open subse o X.Le ,g :U→X be wo maps such ha he e exis s H:U×0,1→Xwi h he ollowing p ope ies: P1Hx,1 xand Hx, 0gx o e e y x∈U; P2 he e exis s δ>0such ha x/ Hx,  o e e y x∈Uδ,wi h x/ ∈U, and ∈0,1, whe e Uδ{x∈U:dis x, ∂U<δ}; P3 he e exis s a∈0,1such ha o 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 he e exis s a con inuos unc ion φ:0,1→Rsuch ha , o e e y x∈Uand , s ∈0,1, dHx, ,Hx, s ≤|φ −φs|; P5i x∈∂U and 0≤λ<1,wi hHx, λ∈∂U, henHx, 1/ ∈U. I ghas a ixed poin in U, hen has a ixed poin in U. P oo . Suppose ha does no ha e any ixed poin in Uand conside he nonemp y se A{λ∈0,1:Hx,λx o some x∈U}.3.4 We will a i e o a con adic ion by showing ha A0,1, and o his we only need p o e ha Ais closed and open in 0,1. To show ha Ais closed in 0,1, conside a sequence {λn}in A,wi hλn→λ∈0,1 as n→∞, and show ha λ∈A; ha is, ha he e exis s x0∈Uwi h Hx0,λx0. To p o e ha x0exis s, ake any sequence {xn}in Uwi h xnHxn,λ n, p o e ha {xn}is Cauchy, and de ine x0as he limi o {xn},asn→∞. Tha {xn}is a Cauchy sequence, as well as x0Hx0,λ, ollows om s anda d a gumen s which can be seen in 11, Theo em 3.1. I emains o show ha x0∈U. Fixed Poin Theo y and Applica ions 7 To p o e his, suppose ha i is no ue and a i e o a con adic ion as ollows: we ha e ha Hx0,λx0∈U U∂U, and also ha 0 ≤λ<1, because does no ha e any ixed poin in U. Then, by P5 x0/ ∈∂U. On he o he hand, x0lim xn∈Ubecause xn∈U o nla ge enough. To be con inced o i , jus apply P2:sincex0∈∂U,{xn}→x0 and xn∈U o all n∈N, he e exis s n0∈Nsuch ha xn∈Uδ o all n≥n0. Then, xn∈U o all n≥n0since xnHxn,λ n. To p o e ha Ais open a gue as in Theo em 2.4,useLemma 3.1 ins ead o Lemma 2.3. As an immedia e consequence, we ob ain he ollowing esul , whose p oo is omi ed because i is analogous o he p oo o Theo em 2.5. Theo em 3.3. Suppose ha Uis an open and s ic ly s a shaped subse o a Banach space X, ·, wi h 0∈U, and ha :U→Xis map wi h Ubeing bounded. Assume also ha he e exis s a∈0,1such ha o all x, y ∈Uand λ∈0,1one has dλ x,λ y ≤amaxdx, y,dx, λ x,dy,λ y,1 2dx,λ ydy,λ x. 3.5 I sa is ies he condi ion I-C, hen has a ixed poin in U. Acknowledgmen This esea ch is pa ially suppo ed by he Spanish G an MTM2007-60854and egional Andalusian G an s FQM210, FQM1504Go e nmen s. Re e ences 1W. V. 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