CZASOPISMO TECHNICZNE TECHNICAL TRANSACTIONS FUNDAMENTAL SCIENCES NAUKI PODSTAWOWE 1-NP/2015 KRZYSZTOF WESOŁOWSKI*

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1 TECHNICAL TRANSACTIONS FUNDAMENTAL SCIENCES CZASOPISMO TECHNICZNE NAUKI PODSTAWOWE -NP/05 KRZYSZTOF WESOŁOWSKI* FIXED POINTS OF α-nonexpansive MAPPINGS PUNKTY STAŁE ODWZOROWAŃ α-nieoddalających Astract Ths paper s coected th the theor of a-oexpasve mappgs, hch ere troduced K. Goeel ad M. A. J. Peda 007. These mappgs are a atural geeralsato of oexpasve mappgs from the pot of ve of the fxed pot theor. I partcular, the proved that Baach spaces all α = ( α,, α) -oexpasve mappgs th a g eough, amel α ³, have mmal dsplacemet equal to zero. Ths paper troduces some e results coected th ths prolem. Keords: α-oexpasve mappgs, mmal dsplacemet, fxed pot Streszczee Nesz artuł est ząza z odzoroaam a-eoddalaącm, tóre został proadzoe przez K. Goela M. A. J. Pedę 007 r. Odzoroaa te są aturalm uogóleem odzoroań eoddalaącch z putu dzea teor putu stałego. Wże spoma autorz azal, że przestrzeach Baacha odzoroaa α = ( α,, α) -eoddalaące, maące odpoedo duże a, a dołade α ³, posadaą mmale przesuęce róe zeru. W artule przedstaoo pee oe z zązae z tm prolemem. Słoa luczoe: odzoroaa α-eoddalaące, mmale przesuęce, put stał DOI: /353737XCT * Isttute of Mathematcs, Facult of Phscs, Mathematcs ad Computer Scece, Craco Uverst of Techolog; rzsztof.esolos@p.edu.pl.

2 6. Itroducto ad prelmares Let (X, d) e a metrc space, ad let α = ( α,, α ) e a mult-dex satsfg α > 0, α > 0, α 0, =,, ad. I [], the follog otos ere = troduced: The mappg T : X X s sad to e a-lpschtza th costat ³ 0, f = α dtxt (, ) d( x, ) for all x, X. The mappg T : X X s sad to e a-oexpasve (a-cotracto), f T s a-lpschtza th costat = ( < resp.). Deote the Lpschtz costat th (T) ad the a-lpschtz costat of T th (a, T). Defe also dt ( ): = f{ d( x, Tx), xî X}, hch e ll call the mmal dsplacemet of T. Sometmes t s also called the approxmate fxed pot of T. These otos are atural geeralsatos of Lpschtza mappgs, oexpasve mappgs ad cotractos from the pot of ve of the fxed pot theor. For more formato cocerg a-oexpasve mappgs ad other Lpschtza mappgs coected th the fxed pot theor, e refer to [4]. I [], the authors proved the follog: Theorem.. (see also [4], chapter 3) Let X e a Baach space, let C e a oempt, closed, covex ad ouded suset of X. Let T : C Ceaα = ( α,, α ) - -oexpasve mappg here α ³. The dt ( ) = 0. Notce that the prolem of determg the set of mult-dces α for hch each a-oexpasve mappg T has d (T ) = 0 s stll ope. The am of ths paper s to prove to results (Theorem., Theorem.) hch gve a partal aser to the aove ope prolem (see [4]). Before proceedg further, let us recall the geeralsed Baach cotracto prcple (arevated to GBCP), hch s formulated as follos: Theorem.. ([], [5]) I complete metrc space X f for some N ³ ad 0 < M < the mappg T : X X satsfes m{ d( T xt, ), N} Md( x, ) for a x, Î X, the T has the uque fxed pot. I author s PhD thess [6] the more geeral verso of the aove theorem as preseted. Let us recall t thout proof. Theorem.3. Let (X, d) e a complete metrc space, N ³. Assume that φ :[ 0, ) [,] 0 s a cotuous, o-creasg fucto satsfg φ( t ) = f, ad ol f, t = 0. Let T : X X e such that m{ d( T xt, ), N} φ( d( x, )) d( x, ) for all x, Î X. The T has the uque fxed pot.

3 7. Ma results Frstl, let us ote a smple fact, there exst some a-lpschtza mappgs hch are ot α-oexpasve; hoever, ther mmal dsplacemet s equal to zero; moreover, the ma have the uque fxed pot. Ths s llustrated : Example.. Let T : l { x l : x 0, N} l { x l : x 0, N } x x 3 e defed the follog a: T : x = ( x, x, ) Tx : =,,, + x3 + x x x4 5,,. The T s ot a-oexpasve for a a; hoever, for properl chose + x5 +x4 α = ( α, α ) the mappg T s a-lpschtza th costat artrarl close to. Moreover, T has the uque fxed pot. Ovousl, the mappg T has the uque fxed pot (, 0, 0,...). Also, e have Tx T x ad Tx T x, for a x, l { x l : x 0, N }. O the other had, tag x = 00,,, 00,, ad Tx T = (,, 00 ) e have = =, ; x + therefore, (T ) =. + Smlarl, T ( ) = ad T ( ) ³, ³ 3. I l, t s ot possle to choose a such that a > 0 ad T s a-oexpasve; hoever, Tx T + T x T x + ; therefore, assumg to e g eough, the mappg T s α=, -Lpschtza th costat ( α, T) artrarl close to. It s orth metog that the exstece ad uqueess of the fxed pot x x3 of T also follos from Theorem.3. Ideed, e have T x =, 3,, + x + 3x3

4 8 x x 4 x 5,, ad + 3x5 x 3 + x + 3 x x ( x x x + ) x x t, N. The latter equalt follos from the fact that t + x + + x + t s a creasg fucto o [ 0, ). Smlarl, T x T φ( t): =. + t x x ; therefore, + 3x x x, so T satsfes the assumptos of Theorem.3 th + x No, let us exchage the codto α ³ th the other regulart codto of a mappg T. Theorem.. Let X e a Baach space, let 0 Î C X e oempt, closed, covex ad ouded. Let T : C C e a α = ( α,, α )-oexpasve mappg such that T ( µ x) T ( µ ) T ( λx) T ( λ) for a x, C, 0 µ λ, {,, }. The dt ( ) = 0. Proof. Fx ³. Defe S : = T. Ovousl, S x = 0 + Tx Î C. The Sx S = Tx T. Next, e have: Sx S Tx = T = Tx ; therefore, assumptos: Sx S T Tx = T T = T Tx T T T x T.

5 9 Smlarl, for ³ 3 e have: S x T Tx = = T T Tx ; therefore: S x S T = Tx T T T = T Tx T T T T T Tx T T T T x T. B assumptos, Tx Tx x for some α = ( α,, α ) satsfg α 0 α ad α =. Therefore: = { } = m S x S, S x S α x, B Theorem., S has the uque fxed pot. Deote ths fxed pot x. We get: x Tx = Sx Tx = Tx Tx = Tx 0,, ths completes the proof. For ³ 3, there exsts a mappg T hch does ot satsf the assumptos of Theorem.; hoever, for ³, t satsfes the assumptos of Theorem.. Ths ll e llustrated the follog example: Example.. Fx ³. Let τ :[,] [, ] e a o-decreasg fucto, havg the Lpschtz costat () τ =, cocave o [,0], covex o [0,] ad such that t(0) = 0. No defe T : Bl x = ( x, x, ) Tx : (( x ), x, x, x, ) Bl. = τ Î We ll sho that the assumptos of Theorem. are ot satsfed for a mult- -dex a of legth. Notce, that:

6 0 Tx T = τ( x) τ( ) + x + x 3 3 = 4 x + x x, = 4 T x T = x x x τ τ + = 5 x + x + x = 5. Therefore, T ( ) = ad ( T ) = >,. It s eas to see that for ³ 3, the assumptos of Theorem. are ot satsfed for a mult-dex a of legth. Ideed, e ould eed to have α 3 = > ad thus for a such α = ( α,, α ), the mappg T ould ot e a-oexpasve. We ll o sho that T satsfes the assumptos of Theorem.; therefore, dt ( ) = 0. It s eough to tae =. It s eas to chec that Tx T + T x T x ; therefore, T s, -oexpasve. We ol have to sho that T( µ x) T( µ ) T( λx) T( λ ) for a x, B l, 0 µ λ. If τµ ( x ) τµ ( ) τλ ( x ) τλ ( ), the ovousl, T( µ x) T( µ ) = τµ ( x) τµ ( ) + µ x µ + µ x µ 3 3 = 4 τ( λx) τ( λ) + λx λ λx λ = 4 = T( λx) T( λ) It s eough to prove that τµ ( v) τµ ( ) τλ ( v) τλ ( ) for v ad 0 < µ < λ. Frstl, assume that v, > 0. Wthout the loss of geeralt, e ca assume that < v. Therefore, 0 < < µ v, λ< λv.

7 Assume that 0 < < µ v λ< λv. Choose aî ( λ, λ v] such that λ = µ v. µ Of course, such a a exsts sce λ( v ) ³ uv ( ). Therefore, µ v = v + µ v + a ad λ λ µ µ λ µ = a + a. Due to the covext of t o [0,] µ v τµ τ µ µ µ µ ( v) a v a a = + µ v µ τµ µ v ( ) + τ( a) λ τλ τ µ µ λ µ ( ) a a a = + λ µ τµ λ ( ) + τ( a) = µ v µ τµ a µ v ( ) + µ τ ( ) a Addg the aove estmates sde--sde ad tag to cosderato the fact that t s o-decreasg, e get: µ v µ v µ v a µ v τµ ( v) + τλ ( ) + τµ ( ) + + τ( a) a = τµ ( ) + τ( a) τ ( ) + τ( λv), ths mples that τµ ( v) τµ ( ) τλ ( v) τλ ( ). O the other had, f 0 < < λ µ v< λv, the let us choose aî ( µ v, λ v] such that µ v = λ. Of course, such a a exsts sce ( λ µ ) v³ ( λ µ ). The λ λ µ µ λ µ µ µ µ µ µ µ µ = a a a v v a v + ad = + a. Due to the covext of t o [0,], e have: λ τλ τ µ µ λ µ ( ) a a a = + λ µ τµ λ ( ) + τ( a)

8 µ v τµ τ µ µ µ µ ( v) a v a a = + µ v µ τµ µ v ( ) + τ( a) = λ a λ τµ ( ) + µ µ τ ( ) a Aga, addg the aove estmatos sde--sde, e get: λ λ λ a λ τλ ( ) + τµ ( v) + τµ ( ) + + τ( a) a = τµ ( ) + τ( a) τ ( ) + τ( λv), ths leads to τµ ( v) τµ ( ) τλ ( v) τλ ( ). Smlarl, t s eas to chec that the estmato τµ ( v) τµ ( ) τλ ( v) τλ ( ) remas true for v, < 0 ad for other cases. Ths shos that T satsfes the assumptos of Theorem.. A set satsfg λx0 + ( λ) ÎC for all ÎC, λ Î[ 0,] e call star-le set C th respect to x 0. Theorem.. Let X e a Baach space, x0 Î X, N ÎN, let C X e a ouded, star-le set th respect to x 0. Let T : C C e such that { }. m T x T, N x for all x, C,. there exsts 0 such that for all 0, N, x, C TT ( x) T( T ) ( + ) T x T, here T x= ( Tx ) + x0. The dt ( ) = 0. Proof. Fx artrar x, Î C ad tae Î{,, N} such that T x T x. Let us ote that: T x T = T ( T x) T ( T ) = ( TT ) ( x) + x ( T ) ( T ) x = ( ) T( T x) TT ( ) ( )( + ) T x T ) ( x 0 0 0

9 3 Therefore, for a x, Î C there exsts Î{,, N} such that T x T ( ) x. Theorem. esures, that T has the uque fxed pot. No, fx a artrar ε > 0 ad choose 0 such that Tz Tz 0 = = ( Tz ) + x Tz = x Tz ε for a z Î C. 0 0 Let z Î C e such that T z = z. Therefore, z Tz z Tz + Tz Tz 0 + ε = ε, ths proves that dt ( ) = 0. Let us llustrate the possle applcato of Theorem.. Example.3. Let T e the same as Example.. The T satsfes Theorem. (e have alread sho that T does ot satsf Theorem. for ³ 3). Ideed, let us calculate TTx ( ) = τ ( ) x, ( x ),( x ),( x ), ad T x = τ x x x x 3, 4, 5, 6, We have: TTx ( ) T( T ) = ( ) x ( ) τ τ ( x ) 4 ( ) 4 + ( x ) 5 ( ) 5 + x 3 τ τ x x = T x T ( + ) T x T. We have alread tae to accout the fact, that τµ ( s) τµ ( t) τλ ( s) τλ ( t) for a 0 µ λ, st, [, ]. We proved ths fact Example.. The estmate m Tx T, T x T shos that T satsfes Theorem.. { } + Tx T T x T x

10 4 Refereces [] Arvatas A.D., A proof of the geeralzed Baach cotracto coecture, Proc. of the Amer. Math. Soc. 3, 003, : , MR [] Goeel K., Japo Peda M.A., O a tpe of geeralzed oexpasveess, Proceedgs of the 8th Iteratoal Coferece of Fxed Pot Theor ad ts Applcatos, 007, 7-8. [3] Goeel K., Sms B., Mea Lpschtza Mappgs, Cotemporar Mathematcs, 53, 00, [4] Pasec Ł., Classfcato of Lpschtz mappgs, Pure ad appled mathematcs (307), CRC Press, Boca Rato, FL, 04. [5] Merrfeld J., Ste J.D. Jr., A geeralzato of the Baach cotracto prcple, J. Math. Aal. Appl. 73, 00, -0, MR (003g:5400). [6] Put stałe odzoroań przestrzeach metrczch SF-przestrzeach, Thess, Jagelloa Uverst, Kraó 05.