Some Approximate Fixed Point Theorems

Transcription

1 It. Joural of Math. Aalysis, Vol. 3, 009, o. 5, 03-0 Some Approximate Fixed Poit Theorems Bhagwati Prasad, Bai Sigh ad Ritu Sahi Departmet of Mathematics Jaypee Istitute of Iformatio Techology Uiversity A-0, Sector-6, Noida 0307, Idia Abstract The purpose of this paper is to obtai some basic approximate fixed poit theorems i geeral settigs. Few special cases are also obtaied. Mathematics Subject Classificatios: 54H5, 47H0, 47H5 Keywords: Fixed poit, Approximate fixed poit, b-metric space.. Itroductio Let T be a self map of a metric space ( X, d). Let us look for a approximate solutio of Tx = x. If there exists a poit z X such that d( Tz, z) ε, where ε is a positive umber, the z is called a approximate solutio of the equatio Tx = or equivaletl z X is a approximate fixed poit (or ε -fixed poit ) of T. I may situatios of practical utilit the mappig uder cosideratio may ot have a exact fixed poit due to some tight restrictio o the space or the map, or a approximate fixed poit is more tha eough, a approximate solutio plays a importat role i such situatios. The theory of fixed poits ad cosequetly of approximate fixed poits fids applicatio i mathematical ecoomics, ocooperative game theor dyamic programmig, oliear aalysis, variatioal calculus, theory of itegro-differetial equatios ad several other areas of applicable aalysis (see, for istace, [5], [9], [0], [4], [5] ad several refereces thereof). Cromme ad Dieer [7] have foud approximate fixed poits by geeralizig Brouwer s fixed poit theorem to a discotiuous map, Hou ad Che [] have exteded their results to set valued maps. Espiola ad Kirk [0] obtaied

2 04 B. Prasad, B. Sigh ad R. Sahi iterestig results i product spaces. Tijs et al [5] have discussed approximate fixed poit theorems for cotractive ad o-expasive maps by weakeig the coditios o the spaces. R. Brazei et al [5] further exteded these results to multifuctios i Baach spaces. Recetly M. Beride [4] obtaied approximate fixed poit theorems for operators satisfyig Kaa, Chatterjea ad Zamfirescu type of coditios o metric spaces. I this paper we study some basic approximate fixed poit results i geeralized metric spaces.. Prelimiaries Defiitio. [8]. Let X be a o empty set ad s be a give real umber. A fuctio d : X X R + (set of oegative real umbers) is said to be a b-metric iff for all z X the followig coditios are satisfied: (i) d ( = 0 iff x = (ii) d ( = d( x), (iii) d ( z) s[ d( + d( z)]. A pair ( X, d) is called a b-metric space. The class of b-metric spaces is effectively larger tha that of metric spaces, sice a b-metric space is a metric space whe s = i the above coditio (iii). The followig example shows that a b-metric o X eed ot be a metric o X (see also [8, p. 64]). Example. [3]. Let X = { x, x, x3, x4} d( x, x) = k ad, d ( x, x3 ) = d( x, x4 ) = d ( x3) = d( x4) = d ( x3, x4 ) =, d ( xi, x j ) = d( x j, xi ) fo r all i, j =,,3, 4 ad d( x, x ) = 0, i =,,3,4. The ad if k >, i i [ d( x, x ) d( x, x )] k d ( xi, x j ) i + j for, i, j =,,3, 4 the ordiary triagle iequality does ot hold. Defiitio.. Let T : X X, ε > 0 ad x0 X. The a elemet x 0 X is a approximate fixed poit (orε -fixed poit) of T if d ( Tx0, x0 ) < ε. T is said to satisfy approximate fixed poit property (AFPP) if for every ε > 0, Fix ( T) φ. ε Remark.. Throughout this paper, for give ε > 0, we shall deote the set of all approximate fixed poits of T by Fix ε (T). Followig otios of asymptotically regular mappigs is essetially due to Browder ad Petryshy [6]:

3 Some approximate fixed poit theorems 05 Defiitio.3 [6]. A mapt : X X is said to be asymptotically regular if for + ay x X, lim d( T T x) = 0. Cosider the followig coditios for all y X ad some a (0, ), b, c (0, ): d( T ad( (.) [ d( Tx) d( )] d ( T b + Ty (.) [ d( d( )] d ( T c + Tx (.3) As oted i Beride [], it is well kow that the coditios (.) ad (.), (.) ad (.3), as well as (.) ad (.3), respectivel are idepedet (see also Rhoades []). Zamfirescu [6] obtaied some iterestig results by combiig coditios (.), (.) ad (.3) i metric spaces. 3. Mai Results Followig is the slightly exteded versio of the Lemma. of Beride [4]: Lemma 3.. Let ( X, d) be a b-metric space ad T : X X. If T is a asymptotically regular map, the T has AFPP. Proof. It may be completed followig Beride [4]. Theorem 3.. Let ( X, d) be a b-metric space ad T : X X satisfies (.), the T has AFPP. Proof. This follows immediately from Beride [4]. Theorem 3.. Let ( X, d) be a b-metric space ad T : X X satisfies (.). sε ( + s) The for eachε > 0, the diameter of Fix ε (T ) is ot larger tha. as Proof: We kow that T has the approximate fixed poit propert so we ca take x ad y ay two ε -fixed poit of T, the d( s[ d( Tx) + d( T ] + s [ d( T + d( T ] + s d( T + s ε + s ε + as d( sε ( + s) d(, for each ε > 0. as This completes the proof.

4 06 B. Prasad, B. Sigh ad R. Sahi If we put s = i Theorem 3. we obtai the result of Beride [4]. Corollary 3. [4]. Let ( X, d) be a metric space ad T : X X satisfies (.). ε The for each ε > 0, the diameter of Fix ε (T ) is ot larger tha. a Theorem 3.3. Let ( X, d) be a b-metric space ad T : X X satisfies (.). The T has AFPP. Proof. It follows o the same lies as i [4] for metric spaces. Theorem 3.4. Let ( X, d) be a b-metric space ad T : X X satisfies (.). The for eachε > 0, the diameter of Fix ε (T ) is ot larger tha s ε ( + s + bs). Proof: We kow that T has the approximate fixed poit propert so we ca take x ad y ay twoε -fixed poit of T, the d( s[ d( Tx) + d( T ] + s [ d( T + d( T ] + s d( T + s ε + s ε + s b d( Tx) + d( [ ] + s ε + s bε (+ s+ bs) d( (+ s+ bs), for each ε > 0. This completes the proof. If we put s = i Theorem 3.3 we obtai the result of Beride [4]. Corollary 3. [4]. Let ( X, d) be a metric space ad T : X X a Kaa operator. The for each ε > 0, the diameter of Fix ε (T ) is ot larger tha ε ( + b). Remark 3.. A map T is called Kaa operator if T satisfies (.) ad Chatterjea operator if T satisfies (.3) (see [4], []). Theorem 3.5. Let ( X, d) be a b-metric space ad T : X X satisfies (.3) with cs <. The T has AFPP. Proof: Let ε > 0 ad x X. The + dt ( xt, x) = dtt ( ( x), TT ( x)) cdt [ ( xttx, ( )) + dtxtt (, ( x))]

5 Some approximate fixed poit theorems 07 = cdt xt x + dt xt x = cdt xt x + + [ (, ) (, )] (, ) cs d T x T x + d T x T x + [ (, ) (, )] + cs cs (, ) (, )... (, ) dt xt x dt xt x dxtx cs cs This implies d( T T + x) 0, as, x X. Now by Lemma 3. it follows that ε > 0, Fix ε ( T ) φ. Theorem 3.6. Let ( X, d) be a b-metric space ad T : X X satisfies (.3). sε ( + s + cs) The for eachε > 0, the diameter of Fix ε (T ) is ot larger tha. s c Proof: We kow that T has the approximate fixed poit propert so we ca take x ad y ay two ε -fixed poit of T, the d( s[ d( Tx) + d( T ] + s [ d( T + d( T ] + s d( T + s ε + s ε + s c[ d( + d( Tx) ] + s ε + s c[ d( + d( ] + s c[ d( x) + d( Tx)] + s ε + s cd( + s cε sε (+ s+ cs) d( s c This completes the proof., for each ε > 0. If we put s = i Theorem 3.6 we obtai the result of Beride [4]. Corollary 3.3 [4]. Let ( X, d) be a metric space ad T : X X satisfies (.3). ε ( + c) The for eachε > 0, the diameter of Fix ε (T ) is ot larger tha. c Remark 3.. A mappig T : X X is a Zamfirescu operator if it satisfies at least oe of the coditios (.), (.) ad (.3) (cf. [], [3], [5]). Theorem 3.7. Let ( X, d) be a b-metric space ad T : X X a Zamfirescu operator o X. The, forbs < /, cs < / T has AFPP. Proof: First we will try to cocetrate the three idepedet coditios ito a sigle oe they all imply. Let y X. Suppose (.) holds, the we have

6 08 B. Prasad, B. Sigh ad R. Sahi [ ] dtxty (, ) bdxtx (, ) + d( yty, ) bd(, x Tx) + bs[(,) d y x + d(, x ] bd(, x Tx) + bsd (,) y x + bs [(, d x Tx) + d( T ] b( + s) dxtx (, ) + bsdxy (, ) + bsdtxty (, ) b( + s ) bs dxtx (, ) + dxy (, ) () bs bs Suppose (.3) holds, the we obtai d( T c[ d( + d( Tx) ] csd [ ( + d( yty, )] + csd [ ( yty, ) + dtytx (, )] csd( + csd( + csd( T cs cs dtxty (, ) d( yty, ) + dxy (, ) ( a) cs cs Similarl we have dtxty (, ) cdxty [ (, ) + d( ytx, )] cs[ d( Tx) + d( T ] + cs[ d( x) + d( Tx)] csd( + csd( Tx) + csd( T cs cs dtxty (, ) d( xtx, ) + d( ( b) cs cs bs b( + s ) cs Let δ = max{ a,,, }. It is easy to see that δ [0, ). bs ( bs ) cs If T satisfies at least oe of the coditios (.), (.) ad (.3), the ad d( T δ d( Tx) + δ d( (3a) d( T δ d( + δ d( (3b) hold. Usig these coditios implied by (.)-(.3), we obtai + dt ( xt, x) = dtt ( ( x), TT ( x)) δ dt ( xtt, ( x)) + δ dt ( xt, x) + = (3 δ) dt ( xt, x) dt ( xt, x)...(3 δ) dxtx (, ) + dt ( xt, x) 0 as, x X. Now by Lemma 3. it follows that ε > 0, Fix ε ( T ) φ. Theorem 3.8. Let ( X, d) be a b-metric space ad T : X X a Zamfirescu operator. The for eachε > 0, the diameter of Fix ε (T ) is ot larger

7 Some approximate fixed poit theorems 09 tha sε ( + s + δ s), where,. δ max bs b( + s ) cs δ = a,, bs ( bs ) cs Proof: I the proof of Theorem 3.7 we have already show that if T satisfies at least oe of the coditios (.), (.) or (.3), the d( T δ d( Tx) + δ d( ad d( T δ d( + δ d( hold. We kow that T has the approximate fixed poit propert so we ca take x ad y ay twoε -fixed poit of T, the d ( y ) s[ d ( Tx ) + d ( Tx, y )] + s + s [ d ( Tx, Ty ) + + s d ( Tx, Ty ) + s ε ε + d ( Ty, y )] δ d ( Tx ) + δ d ( y ) + s ε + s δε + δ d ( y ) sε ( + s + δ s) d (, for each ε > 0. δ This completes the proof. If we put s = i Theorem 3.3 we obtai the result of Beride [4]. Corollary 3.4 [4]. Let ( X, d) be a metric space ad T : X X a Zamfirescu operator. The for each ε > 0, the diameter of Fix ε (T ) is ot larger ε ( + δ ) b c tha, where δ = max{ a,, }. δ b c s Refereces [] V. Beride, O the approximatio of fixed poits of weak cotractive mappigs, Carpathia J. Math. 9 (003), o., 7-. [] V. Beride, Approximatig fixed poits of weak cotractios usig Picard iteratio, Noliear Aal. Forum, 9 (004), o., [3] V. Beride, A covergece theorem for Ma iteratio i the class of Zamfirescu operators. A. Uiv. Vest Timi\ c s. Ser. Mat.-Iform. 45 (007), o., [4] M. Beride, Approximate Fixed Poit Theorems, Stud. Uiv. Babes Bolyai, Math. 5 (006), o., -5.

8 0 B. Prasad, B. Sigh ad R. Sahi [5] R. Brazei, J. Morga, V. Scalzo ad S. Tijs, Approximate fixed poit theorems i Baach spaces with applicatio i game theor J. Math. Aal. Appl. 85 (003) [6] F. E. Browder ad W. V. Petryshy, The solutio by iteratio of oliear fuctioal equatios i Baach spaces, Bull. Amer. Math.Soc. 7 (966), [7] L. J. Cromme ad I Dieer, Fixed poit theorems for discotiuous mappig, Math. Programmig 5(99) o., (Ser. A), [8] S. Czerwik, Noliear set-valued cotractio mappigs i b-metric spaces, Atti Sem. Mat. Fis. Uiv. Modea 46 (998), No., [9] J. Dugudji ad A. Graas, Fixed poit Theor PWN-Polish Scietific Publishers, Warsaw (98). [0] R. Espiola ad W. A. Kirk, Fixed poits ad approximate fixed poit i product spaces, Taiwaese J. Math. 5() (00), [] S. H. Hou, G. Ya Che, Approximate fixed poits for discotiuous set-valued mappigs, Math. Meth. Oper. Res. (998) 48: [] B. E. Rhoades, A comparisio of various defiitios of cotractive mappigs, Proc. Amer. Math. Soc. 6 (977), [3] S. L. Sigh, B. Prasad, Some coicidece theorems ad stability of iterative procedures, Comput. Math. Appl. 55 (008), [4] S. L. Sigh, B. P. Chamola, Quasi-Cotractios ad Approximate Fixed Poits, J. Natur. Phys. Sci. Vol. 6 (-) (00) [5] S. Tijs, A. Torre ad R. Brazei, Approximate fixed poit theorems, Libertas Math. 3 (003), [6] T. Zamfirescu, Fix poit theorems i metric spaces, Arch. Math. (Basel), 3 (97), Received: Jul 008