CHAPTER 1 INTRODUCTION

Transcription

1 CHAPTER INTRODUCTION The work reported i this thesis is based o fixed poits, commo fixed poits, approximate fixed poits, ed poits ad approximate best proximity poits of a variety of maps satisfyig differet cotractive coditios. The stability of differet iterative procedures is ivestigated i a ew settig. Several existig results cocerig sigle valued ad multi valued maps are exteded to the theory of iterated fuctio systems (IFS) ad iterated multifuctio systems (IMS). Further, some miimax ad saddle poit theorems are also established. We first provide a brief historical developmet of the fixed poit theory.. HISTORICAL OUTLINE The origi of fixed poit theory lies i the method of successive approximatios used for provig existece of solutios of differetial equatios itroduced idepedetly by Joseph Liouville [] i 837 ad Charles Emile Picard [2] i 890. But formally it was started i the begiig of twetieth cetury as a importat part of aalysis. The abstractio of this classical theory is the pioeerig work of the great Polish mathematicia Stefa Baach [3] published i 922 which provides a costructive method to fid the fixed poits of a map. However, o historical poit of view, the major classical result i fixed poit theory is due to

2 L. E. J. Brouwer [4] give i 92 (also see Zeidler [5], Kirk ad Sims [6] ad Graas ad Dugudji [7]). The celebrated Baach cotractio priciple (BCP) states that a cotractio mappig o a complete metric space has a uique fixed poit. Baach used the idea of shrikig map to obtai this fudametal result. The Brouwer fixed poit theorem is of great importace i the umerical treatmet of equatios. It exactly states that a cotiuous map o a closed uit ball i R has a fixed poit. A importat extesio of this is the Schauder s fixed poit theorem [8] of 930 statig a cotiuous map o a covex compact subspace of a Baach space has a fixed poit. These celebrated results have bee used, geeralized ad exteded i various ways by several mathematicias, scietists, ecoomists for sigle valued ad multivalued mappigs uder differet cotractive coditios i various spaces. Kaa [9] proved a fixed poit theorem for the maps ot ecessarily cotiuous. This was aother importat developmet i fixed poit theory (see also Chatterjea [0]). Thus various results pertaiig to fixed poits, commo fixed poits, coicidece poits, etc. have bee ivestigated for maps satisfyig differet cotractive coditios i differet settigs, see amog others, Tychooff [], Lefschetz [2], Kakutai [3], Tarski [4], Edelstei [5], Zamfirescu [6], Ćirić [7]-[8], Mishra [9], Sigh et al [20], Sigh ad Mishra [2], Browder [22], Göhde [23], Suzuki [24]-[25], Suzuki ad Kikkawa [26] ad refereces therei. For a fudametal compariso ad developmet of various cotractive coditios, oe may refer Rhoades [27]-[30], Collaco ad Silva [3], Murthy [32], Sigh ad Tomar [33] ad Pat et al [34]. The study of fixed poits of multivalued mappigs was iitiated by Nadler [35]-[37] ad oexpasive mappig by Marki [38]-[39] usig the cocept of Hausdorff metric. Ideed, Nadler [37] proved that a multivalued cotractio has a fixed poit i a complete metric space. Ćirić [40] geeralized Nadler s result to multivalued quasi-cotractio maps. Subsequetly, it received great attetio i applicable mathematics ad was exteded ad geeralized o various settigs. Further, hybrid fixed poit theory for oliear sigle-valued ad multivalued maps is a ew developmet i the domai of multivalued aalysis (see, for istace, Krasoselskii [4], Corley [42], Mishra et al [43], Sigh ad Arora [44], Sigh ad Prasad [45], Dhage [46], Sigh ad Mishra [47]-[50], Sigh et al [5]-[52] ad refereces thereof). For a historical developmet of the hybrid fixed poit theory, oe may refer to Sigh ad Mishra [47] ad Sigh et al [5]-[52]. 2

3 Thus fixed poit theory has bee extesively studied, geeralized ad eriched i differet approaches such as, metric, topological ad order-theoretic (see Goebel ad Kirk [53], Kirk ad Sims [54], Brow [55], Carl [56], Adres ad Goriewicz [57], Wegrzyk [58] ad Smart [59]). This advacemet i fixed poit theory diversified the applicatios of various fixed poit results i various areas such as the existece theory of differetial ad itegral equatios, dyamic programmig, fractal ad chaos theory, discrete dyamics, populatio dyamics, differetial iclusios, system aalysis, iterval arithmetic, optimizatio ad game theory, variatioal iequalities ad cotrol theory, elasticity ad plasticity theory ad other diverse disciplies of mathematical scieces (see, for istace, Zeidler [5], Corley [42], Brow [55], Adres ad Góriewicz [57], Wegrzyk [58], Smart [59] ad Agarwal et al [60]). I the followig, we give a brief sketch of the developmet of the topics studied i this thesis.... I may situatios of practical utility, the mappig uder cosideratio may ot have a exact fixed poit due to some tight restrictios o the space or the map. Further, there may arrise some practical situatios where the existece of a fixed poit is ot strictly required but a approximate fixed poit is more tha eough. The theory of approximate fixed poits plays a importat role i such situatios. Approximate fixed poit property for various types of mappigs has bee a promiet area of research for the last few decades. A classical best approximatio theorem was itroduced by Fa [6] i 969. Afterward, several authors, icludig Reich [62], Prolla [63], Sehgal ad Sigh [64]-[65], have derived extesios of Fa s theorem i may directios. Approximate fixed poits of cotiuous maps have bee studied by Gajek et al [66], Rafi ad Salami [67], Hadzic [68]-[69] ad may others. Tijs et al [70] studied approximate fixed poit theorems for cotractio ad o-expasive maps by weakeig the coditios o the spaces. Brazei et al [7] further exteded these results to multifuctios i Baach spaces. Sigh ad Prasad [72] proved a approximate fixed poit theorem for quasi cotractio i metric spaces. Recetly Beride [73] obtaied some approximate fixed poit theorems for operators satisfyig Kaa, Chatterjea ad Zamfirescu type of coditios o metric spaces. P acurar ad P acurar [74] obtaied approximate fixed poit results for almost or weak cotractio maps. 3

4 If T is a o-self-mappig, it is probable that the fixed poit equatio Tx = x has o solutio. I such a case best approximatio theorems explore the existece of a approximate solutio whereas best proximity poit theorems aalyze the existece of a optimal approximate solutio. I 2003, Kirk et al [75] itroduced 2-cyclic cotractio, which becomes a relevat research area i search of best proximity poits. Eldred ad Veeramai [76] defied a ew cotractio which is more abstract formulatio tha 2-cyclic cotractio. This defiitio is more geeral tha the otio of cyclical maps, i the sese that if the sets itersect, the every poit is a best proximity poit. It turs out that may of the cotractive-type coditios which have bee ivestigated for fixed poits esure the existece of best proximity poits. Some results of this kid are obtaied i Bari et al [77], Al-Thagafi ad Shahzad [78], Karpagam ad Agrawal [79], Petric [80], Mishra ad Pat [8], etc. Recetly, Mohsealhosseii et al [82] also defied a stroger cocept of approximate best proximity poits ad obtaied some iterestig existece results...2. It is a well kow that the celebrated Baach cotractio priciple (BCP) is oe of the mai tools for both the theoretical ad the computatioal aspects i mathematical scieces. This theorem has witessed umerous geeralizatios ad extesios i the literature due to its simplicity ad costructive approach. Kaa [9] by takig a etirely differet coditio, proved a fixed poit theorem for operators, which eed ot be cotractio or cotractive. Ideed, he was the first to propose a fixed poit theorem for a discotiuous map. Jugck [83] obtaied a importat geeralizatio of (BCP) i the form of commo fixed poit theorem for commutig pair of maps. Sessa [84] itroduced the cocept of weakly commutig maps. This cocept was further improved by Jugck ad Rhoades [85] with the otio of weakly compatible mappigs. Pat [86] studied the commo fixed poit theorem of ocommutig maps. Sigh ad Mishra [49] studied coicidece ad fixed poits of reciprocally cotiuous ad compatible hybrid maps. Aamri ad Moutawakil [87] defied property E. A., Liu et al [88] geeralized it as commo property E. A. ad obtaied iterestig results. Braciari [89] studied cotractive coditios of itegral type ad obtaied a itegral versio of the Baach cotractio priciple. A umber of papers appeared for the maps satisfyig the itegral type coditios i differet settigs, see for example, Rhoades [90], Vijayaraju et al [9], Aliouche [92], Djoudi ad Aliouche [93], Pathak et al [94], Pathak ad Verma [95]-[96], Day et al [97] ad may others. 4

5 After the iitiatio of fuzzy sets by Zadeh [98], fuzzy metric spaces are itroduced by may authors such as Kramosil ad Michalek [99], George ad Veeramai [00], Gregori ad Sapea [0], Kaleva ad Seikkala [02], Park [03], Adibi et al [04], Saadati ad Park [05]-[06] i differet ways. Grabiec [07] first exteded BCP ad Edelstei fixed poit theorem to fuzzy metric space Thereafter, may authors worked o this lie (see, George ad Veeramai [00], Gregori ad Sapea [0], Schweizer et al [08], Sog [09], Mihet [0], Mishra et al []-[3], Jha [4] etc). Gogeu [5] geeralized fuzzy set to L fuzzy set i 967. Aother extesio icludes ituitioistic fuzzy sets proposed by Ataassov [6] i 986. Saadati et al [7], Saadati [8] geeralized the otios of fuzzy metric spaces due to George ad Veeramai [00] ad ituitioistic fuzzy metric spaces due to Saadati ad Park [05] by itroducig the otio of L fuzzy metric spaces with the help of cotiuous t- orms. These developmets ethused the fuzzyficatio of fixed poit theory ad authors cotributed profusely i this lie, see for example, [00]-[0], [07], [9]-[28] ad refereces thereof...3. Fixed poit theory has always bee excitig i itself ad its applicatios i ew areas. Curretly, it has foud ew ad hot areas of activity. The iitiatio of fixed poit theory i computer sciece by Tarski [4] ad Scarf [29] ehaces its applicability i differet domais. Due to the advet of speedy ad fast computatioal tools, a ew horizo has bee provided to fixed poit theory. The fixed poit equatios are solved by meas of some iterative procedures. I view of their cocrete applicatios, it is of great iterest to kow whether these iterative procedures are umerically stable or ot. The study of stability of iterative procedures ejoys a celebrated place i applicable mathematics due to chaotic behavior of fuctios i discrete dyamics, fractal graphics ad various other umerical computatios where computer programmig is ivolved. This kid of problem for real-valued fuctios was first discussed by M. Urabe [30] i 956. The first result o the stability of iterative procedures o metric spaces is due to Alexader M. Ostrowski [3]. This result was exteded to multivalued operators by Sigh ad Chadha [32]. Czerwik et al [33]-[34] exteded this to the settig of geeralized metric spaces. The stability of iterative schemes i various settigs has thus bee studied durig the last three decades by a umber of authors, see for istace, Agarwal et al [60], Czerwik et al 5

6 [33]-[34], Ahmed [35], Beride [36]-[37], Ćirić ad Ume [38], Chidume et al [39], Goebel ad Kirk [40], Harder ad Hicks [4]-[42], Osilike [43]-[44], Rhoades [45]- [46], Sigh et al [47]-[49], Olatiwo ad Imoru [50], Osilike ad Udomee [5], Sigh ad Prasad [52], Mishra ad Kalide [53], Mishra et al [54] ad several refereces thereof. Recetly Timis ad Beride [55] ad Timis [56] respectively studied weak stability ad weak 2 w stability of a operator i metric spaces...4. The study of fixed poit problems for multivalued maps was iitiated by Kakutai [3] i the year 94 i fiite dimesioal spaces by geeralizig the Brouwer s fixed poit theorem [4]. This was the begiig of the fixed poit theory of multimaps havig a vital coectio with the miimax theory i games. Kakutai used his results to obtai a simple proof of vo Neuma s miimax theorem [57]. This was exteded to ifiite dimesioal Baach spaces by Boheblust ad Karli [58]. Further, Sio [59] geeralized vo Neuma results o the basis of Kaster, Kuratowski ad Mazurkiewicz (KKM) theorem [60]. At the begiig, the results of the KKM theory were established for covex subsets of topological vector spaces maily by Ky Fa [6]-[64]. Horvath [65]-[67] exteded most of the Fa s results i the KKM theory to H-spaces by replacig the covexity coditio by cotractibility. Park [68] i 992 established ew versios of KKM theorems ad miimax iequalities o H-spaces which were exteded by Park ad Kim [69]-[73]. Thereafter more geeral spaces such as abstract covex space ad KKM spaces were itroduced by Park i [74]-[78]. I this way, KKM theory has bee cotiuously upgraded ad eriched to ehace its applicability to the wider rage of problems, such as miimax theorems, saddle poit theorems i game theory, ecoomics ad optimizatio theory, variatioal iequalities etc., (see for istace [5], [3], [58]-[59], [62]-[63], [65], [68]-[95])...5. Oe of the recet advacemet of the fixed pit theory of sigle-valued ad multivalued mappigs is i the areas of iterated fuctio systems (IFS), iterated multifuctio systems (IMS) ad fractals. Fractal theory is a ew domai iitiated by Madelbrot [96] i which Baach [3] ad Nadler [37] fixed poit theorems ad their variats are of vital importace. 6

7 Michael Barsley ad Steve Demko [97] popularized the theory of IFS after Hutchiso [98] gave a formal defiitio of it i 98. A IFS usually cosists of a complete metric space together with a fiite set of cotractio mappigs. It was bor as a applicatio of the theory of discrete dyamical systems ad is a useful tool to build fractal ad other similar sets (see, for istace, [96], [97]-[229] ad refereces thereof). Ideed, self similar sets are special class of fractals ad there are o objects i ature which have exact structures of self similar sets. These sets are perhaps the simplest ad the most basic structures i the theory of fractals which should give us much iformatio o what would happe i the geeral case of fractals (cf. Kigami [209]-[2]). This basic otio of IFS has bee exteded ad eriched to more geeral settigs by chagig the coditio o mappigs or the space by various authors (see for istace, [97]-[99], [22]-[229] ). I Jai ad Fiser [22] ad Fiser [25] cotractio maps are replaced by weakly cotractive or o-expasive maps. Mate [29] ad Rus ad Triff [229] replaced cotractio costat by a compariso fuctio to obtai their results. I [230]-[232] the formulatios of the cotractio due to Meir ad Keeler have bee used to geeralize the IFS theory. Recetly Mihail ad Miculescu [22] itroduced the otio of geeralized iterated fuctio system (GIFS), which is a family of fuctios i a complete metric space ad obtaied GIFS to be a atural geeralizatio of the otio of IFS (see [220]-[224]). Further, Llores-Fuster et al [28] defied mixed iterated fuctio system by takig more geeral coditios ad obtaied a mixed iterated fuctio system theory for cotractio ad Meir Keeler cotractio maps. Thus there has bee a huge developmet i the metric as well as topological fixed poit theory. The recet advacemet of the computatioal tools has tremedously ehaced the scope of the applicatios of it. The etire theory of cotractios ad multivalued cotractios is expected to revolutioize ot oly various braches of mathematics but also diverse dimesios of kowledge i egieerig ad scieces. I the followig sectio, we recall the basic cocepts required for our study i the subsequet chapters regardig fixed poit theory for sigle valued, multivalued ad hybrid maps. 7

8 .2 BASIC CONCEPTS We start with the basic cocepts related to metric spaces, fixed poits ad differet cotractio coditios. Defiitio.2. [233]. Let X be a o-empty set together with a distace fuctio d : X X R. + The fuctio d is said to be a metric iff for all y, z X, the followig coditios are satisfied (i) d( y) 0ad d ( y) = 0 iff x = y, (ii) d ( y) = d( y, x), (iii) d ( z) d( y) + d( y, z). The pair ( X, d) is called a metric space. Defiitio.2.2 [6]. A sequece i a metric space is a Cauchy sequece if for every ε > 0, there exists 0 N such that d ( x, x m ) < ε, for all m >., 0 Defiitio.2.3 [6]. A metric space ( X, d) is called complete, if every Cauchy sequece coverges i it. Defiitio.2.4 [6]. The diameter of a set A deoted by δ (A) is defied as δ ( A) = sup{ d( a, b) : a, b A}. It meas that diameter of the set is the least upper boud of the distaces betwee the poits of the set A. If the diameter is fiite, i.e., δ ( A) <, the A is bouded. Defiitio.2.5 [6]. Let ( X, d) be a metric space ad T : X X. The T has a fixed poit if there is a x X such that Tx = x. The poit x is called a fixed poit of T. Defiitio.2.6 [6]. Let ( X, d) be a metric space ad T, S : X X. The x is called a coicidece (respectively, commo fixed) poit of T ad S, if (respectively, x = Tx = Sx). x X such that Tx = Sx 8

9 I some situatios a approximate solutio of the problem is more tha eough, so we have to fid approximate fixed poit istead of fixed poit. Defiitio.2.7 [73]. Let ( X, d) be a metric space ad T : X X. A elemet x0 X is called a approximate fixed poit (orε fixed poit) of T if dtx ( 0, x0) < ε, whe ε > 0. Approximate fixed poit of the fuctio exists if the defied mappig has approximate fixed poit property. Defiitio.2.8 [73]. A map T is said to satisfy approximate fixed poit property (AFPP) if for every ε > 0, Fix ( T) φ, where Fix ε (T) is the set of all approximate fixed poit of T. ε The followig coditio guaratees the existece of approximate fixed poits. Defiitio.2.9 ([234]). A map T : X X is said to be asymptotically regular if for ay x X, limd( T T + x) = 0. Defiitio.2.0 [37]. Let ( X, d) be a metric space. A mappig T : X X is called (i) Lipschitzia (or d( T L d( y), (ii) Baach cotractio if T is L Lipschitzia) if there exists L > 0 such that for all y X, α Lipschitzia, with α [0, ) ad for all y X, (lm) dtxty (, ) α dx (, y), (bc) (iii) oexpasive if T is -Lipschitzia ad for all y X, dtxty (, ) dx (, y), (m) (iv) cotractive if d ( T < d( y), for all y X, x y, (cm) (v) isometry if d ( T = d( y), for all y X. (im) Examples.2. [37]. (i) Let T :[/ 2, 2] [/ 2, 2], be defied as T ( x) = / the T is 4-Lipschitzia with Fix (T) = {}, where Fix( T ) deotes fixed poit of the mappig T. (ii) T : R R, T ( x) = x / 2 + 3, x R. Obviously T is a Baach cotractio ad Fix( T ) = {6}. 9

10 (iii) Tx = x R is oexpasive ad Fix( T ) = {/2}. (iv) T :[, ] [, ], T ( x) = x + / is cotractive ad Fix( T ) = φ. (v) T ( x) = x + 2, the Fix( T ) = φ is isometry. The celebrated Baach cotractio priciple (BCP) is the simplest ad oe of the most versatile ad fudametal results i the fixed poit theory. It ot oly produces approximatio of ay desired accuracy but also determies a-priori ad a-posteriori error estimates. Theorem.2. [3]. Let ( X, d) be a complete metric space ad T : X X satisfies (bc). The T has a uique fixed poit p, i.e., Tp = p ad lim T x= p. Moreover, we have a-priori error estimate α dt ( p) dtx (, x) α ad the a-posteriori error estimate α dt x p dt xt x α (, ) (, ). d( T α ( x), p) d( T α ( x), T ( x)). There are various geeralizatios of the celebrated cotractio mappig priciple. These geeralizatios are obtaied either by weakeig the cotractio coditio of the map by givig a sufficietly rich structure to the space i order to compesate the relaxatio of the cotractio coditio or by extedig the structure of the space or sometime combiig both the approaches. Kaa [9] was the first to propose a fixed poit theorem for a discotiuous map. He exactly proved the followig. Theorem.2.2 [9] (Kaa Cotractio Theorem). Let T be a self map o a complete metric space (X, d) satisfyig the followig coditio (usually called Kaa cotractio), for all y X ad some β [0,/ 2), 0

11 d ( T β [ d( Tx) + d( y, ]. (kc) The T has a uique fixed poit. Kaa s theorem motivated umerous extesios ad geeralizatios of the BCP ad his ow fixed poit theorem o various settigs. Chatterjea [0] proved a fixed poit theorem for discotiuous mappig satisfyig a coditio which is actually a kid of dual of Kaa mappig. Defiitio.2. [0]. Let ( X, d) be a metric space. A mappig T : X X is called Chatterjea cotractio if for all y X ad some γ [0,/ 2), d ( T γ [ d( + d( y, Tx) ]. (cc) Zamfirescu [6] obtaied some iterestig results by combiig Baach, Kaa ad Chatterjea cotractio coditios. Defiitio.2.2 [6]. Let ( X, d) be a metric space. A mappig T : X X is called Zamfirescu cotractio if for all least oe of the followig coditios. (i) d( T αd( y) (ii) d ( T [ d( Tx) + d( y, ] y X ad some α [0, ), β, γ [0, ), satisfies at 2 β (zc) (iii) d ( T γ [ d( + d( y, Tx) ]. Amog various geeralizatios of the Baach cotractio, the Ćirić [7]-[8] cotractio (also called as quasi-cotractio) is cosidered to be the most geeral oe. Defiitio.2.3. Let ( X, d) be a metric space. A mappig T : X X is called quasi- cotractio (Ćirić [7]-[8]) if d( T k.max{ d( y), d( Tx), d( y,, d(, d( y, Tx)}, (qc) for some 0 k < ad all y i X.

12 Defiitio.2.4 [235]. A mappig T : X X is called weak or almost cotractio if there exist α (0, ) ad L 0 such that for all y X, d ( T α d( y) + Ld( y, Tx). (ac) It is importat to ote that ay mappig satisfyig Baach, Kaa, Chatterjea, Zamfirescu, or Ciric (with costat k i ( 0, / 2) ) type coditios are a weak or almost cotractio, see [7]. Jugck [83] exteded Baach cotractio i the followig maer. Defiitio.2.5 [83]. Let ( X, d) be a metric space ad T, S: X X. The followig coditio is kow as Jugck cotractio. d( T α d( S Sy), (jc) for some α [0, ) ad all y X. Although this coditio was kow to Sigh ad Kulshrestha [236] but Jugck was credited for presetig costructive proof regardig the existece of a commo fixed poit of commutig maps. Further various fixed poit ad commo fixed poit results are ivestigated i the literature for the maps satisfyig differet coditios. We recall some of them as follows. Defiitio.2.6. Let ( X, d) be a metric space ad T, S : X X. The mappigs T ad S are (i) commutig [83], if TSx = STx for all x X, (ii) weakly commutig [84], if d( TS STx) d( T Sx) for all x X, (iii) compatible [237, 238], if lim d( TSx, STx ) = 0, wheever x } is a sequece i X such that limtx = lim Sx = t, for some t X, (iv) weakly compatible [85], if they commute at their coicidece poits; i.e., if Tu = Su for some u i X, the TSu = STu, (v) satisfyig the property (E. A.) [87], if there exists a sequece { x } such that { limtx = lim Sx = t, for some t X. 2

13 Notice that weakly commutig mappigs are compatible ad compatible mappigs are weakly compatible but the coverse eed ot be true (see [87], [237]-[239]). It is easy to see that two ocompatible mappigs satisfy the property (E. A.). Fixed poit theory for multivalued mappigs is a atural geeralizatio of the theory of sigle valued mappigs. Most of the results icludig the well kow fixed poit theorems of Baach ad others have bee extesively studied for the multivalued cases, see, for istace [35]-[4], [55], [57]-[59], [63], [85] ad several refereces therei. These multivalued fixed poit theorems have applicatios i cotrol theory, covex optimizatio, differetial equatios, ecoomics, etc. (see also [78], [80]). I the followig, we discuss the basic cocepts ad prelimiaries regardig multivalued fixed poit theory. Defiitio.2.7 [59]. If T is a multivalued map, i.e., from X to the collectio of oempty subsets of X. The a poit p i X is called a fixed poit of T if p Tp. Defiitio.2.8 [59]. Let S : X X ad T : X CB( X ). A poit p X is a coicidece (respectively, commo fixed) poit of S ad T if Sp Tp (respectively, p = Sp Tp ). Example.2.2. Cosider X = [ 0, ) with the usual metric. Defie S : X X ad T : X CB( X ) as 0 if x [0, ) { x} Sx = ad Tx = 3x if x [, ) [, + 3x] if x [0, ) if x [, ) We have S = 3 [, 4] = T, that is x = is a coicidece poit of S ad T ad 0 is the commo fixed poit of S ad T. Defiitio.2.9 [240]. Let A ad B be the oempty compact subsets of X, the the distace betwee a poit x ad set A is defied as d ( A) = mi d( y). The distace from set A to set B is give as d( A, B) = max d( B). x A y A 3

14 Note that d is ot a metric, sice d is ot symmetric. That is, d( A, B) d( B, A) i geeral. The Hausdorff distace h ( A, B), betwee Aad B is hab (, ) = max{ dab (, ), db (, A)}. Defiitio.2.20 [240]. Let ( X, d) be a metric space ad K(X) be the oempty compact subsets of X together with a distace fuctio h. The for all A, BC, K( X), habsatisfies (, ) followig properties. (i) hab (, ) 0 ad h( A, B) = 0 iff A = B, (iii) hab (, ) = hb (, A), (iv) hab (, ) hac (, ) + hc (, B). The pair ( K, h ) is called a Hausdorff metric space. Defiitio.2.2 [37]. Let ( X, d ) be a metric space ad P( X ) is the family of oempty subsets of X. Let T : X P( X) satisfies the followig coditio. htx (, a d( y), (mc) for some a [0, ) ad all y X. The T is called a multivalued cotractio. Further, T is called (i) oexpasive, if it is -Lipschitzia, (ii) cotractive if htx (, < d( y), for all y X, x y, (iii) isometry if htx (, = d( y), for all y X. Defiitio Let T : X X ad S: X CB( X). The mappigs T ad S are (i) weakly commutig [84], if TSx CB( X ) for all x X ad h( TS STx) d( T Sx), (iii) compatible [237, 238], if lim h( TSx, STx ) = 0, wheever x } is a sequece i X such that lim Sx = A CB( X ) ad lim Tx = t A, (iv) weakly compatible [85], if they commute at their coicidece poits; i.e., if Tu Su for some u i X, the TSu = STu, (v) R-weakly commutativity [86], if TSx CB( X ) for all x X ad there exists a real umber R > 0 such that hstxtsx (, ) RdTx (, Sx), { 4

15 (v) property (E. A.) [87], if there exists a sequece { x } such that lim Sx = A CB( X ) ad limtx = t A. Liu et al [86] defied a more geeral coditio the property (E. A.) as follows. Defiitio.2.23 [88]. Let f, g : X X ad F, G : X CB( X ). The pairs ( f, F) ad ( g, G) are said to satisfy the commo property (E. A.) if there exists two sequeces { x}, { y} i X, some t i X, ad A, B i CB(X) such that lim Fx = A, lim Gy = B, lim fx = lim gx = t A I B. Nadler [37] exteded the BCP to multivalued maps i the followig maer. Theorem.2.3 [37]. Let ( X, d) be a complete metric space ad CB( X ) deotes the collectio of all oempty closed ad bouded subsets of X with the Hausdorff metric h. Suppose T :( X, d) ( CB( X), h) satisfies (mc). The T has a fixed poit i X. For approximatig fixed poit of the correspodig cotractio operator of a give oliear equatio, we eed some iterative methods. There are may iterative procedures give i the literature which ca be used i differet coditios (see, [2], [47], [24]-[244], etc.), i.e. if oe iterative scheme fails i a give situatio other may be applied or some time we may choose them accordig to their speed of covergece. The most popular iterative procedure, called Picard iteratio is defied as follows. Defiitio.2.24 [2]. Let ( X, d) be a metric space ad T : X X. Choose x0 X ad defie x = Tx0, x2 = T... ad obtai a relatio Here x is the x + = Tx, = 0,, 2,... (PI) th Picard iterate of x0 i X. 5

16 Baach fixed poit theorem uses Picard iteratio for the covergece towards a fixed poit. But i may cases Picard iteratios may ot coverge to the fixed poit of the map, some other iterative scheme may be used i such cases. I 953, Ma [242] defied a iteratio scheme i the followig maer. Defiitio.2.25 [242]. Choose a iitial poit x 0 i a give set ad let α } with 0 α <, be a sequece of real umbers. The Ma iteratio is defied by x = ( α ) x + α Tx, 0,, 2,... (MI) + = { If T is a cotiuous map ad the Ma iterative process coverges, the it coverges to a fixed poit of T. But if T is ot cotiuous, the there is o guaratee that it will coverge to a fixed poit of T, eve if the Ma process coverges. This is show by the followig example. Example.2.3 [37]. Let T :[0, ] [0, ] be give by T 0 = T = 0 ad Tx =, 0 < x <. The Fix( T ) = {0}, where Fix( T ) is the set of all fixed poits of T ad the Ma iteratio with α =, coverges to, which is ot a fixed poit of T. Some other scheme may be used i such a case. Ishikawa [243] defied the followig iterative scheme. Defiitio.2.26 [243]. Choose a iitial poit x 0 i a give set ad let α }, { β } be the sequeces of real umbers with 0 α β <. The { x + y = ( α ) x = ( β ) x + α Ty + β Tx, = 0,, 2,... (II) I the similar way, Noor [244] defied a three-step iteratio scheme as follows. 6

17 Defiitio.2.27 [244]. Choose a iitial poit x 0 i a give set ad let { α }, { β },{ γ } [0, ) be the sequeces of real umbers. The x y z + = ( α ) x = ( β ) x = ( γ ) x + α Ty + β Tz + γ Tx, = 0,, 2,... (NI) For a detailed developmet of differet iteratio scheme, oe may refer Beride [37]. Jugck [237, 238] geeralized BCP, by replacig the idetity map with two cotiuous maps ad obtaied a commo fixed poit theorem. He defied a ew iterative scheme usig two mappigs i the followig way. Defiitio.2.28 [238]. Let S, T : Y X ad T ( Y ) S( Y ). For ay x, cosider 0 Y Sx = f ( T, x), = + 0,,... This procedure was essetially itroduced by Jugck, ad it becomes the Picard iterative procedure whe Y = X ad S = idetity map. Notice that, If we take f T, x ) = Tx, the the method is called Jugck-Picard iteratio ( (JPI) ad if f T, x ) = ( α ) Sx + α Tx, where 0 α <, ( iteratio (JMI) (see [47] & [245]). the it is called Jugck-Ma Defiitio.2.29 [50]. Let S : X X ad T ( X ) S( X ). Defie Sx+ = ( α ) Sx + α Tz Sz = ( β ) Sx + β Tx where { α } ad { β } satisfies, = 0,,... (JII) ( i) α =, ( iii) 0 α =, ( ii)0 α, β, 0, ( iv) j= 0 i= j+ { α + aα } coverges. It is called Jugck-Ishikawa iteratio. i i Notice that whe S = idetity map, it is called Ishikawa iteratio. 7

18 These fixed poit iterative procedures have tremedous applicatios i the problems of solvig oliear equatios. A iterative scheme is said to be stable if small perturbatios durig computatios, produce small chages i approximate value of the fixed poit computed by meas of these itearios. Formally, Harder ad Hicks [42] defied the stability of a iteratio procedure as follows. Defiitio.2.30 [42]. Let ( X, d) be a complete metric space ad T : X X. Let { x } X be the sequece geerated by a iteratio procedure ivolvig T which is =0 defied by x+ = f ( T, x ), = 0,,..., where x 0 X is the iitial approximatio ad f is some fuctio. Suppose x coverges to a fixed poit p of T. Let { y } X ad set { } =0 =0 ε = d( y +, f ( T, y )), = 0,, 2,... The, the iteratio procedure is said to be T stable or stable with respect to T if ad oly if lim ε = 0 implies lim = p. y Sigh et al [47] first defied stability for Jugck type iterative procedures i the followig maer. Defiitio.2.3 [47]. Let S, T : Y X, T ( Y ) S( Y ) ad z be a coicidece poit of T ad S, that is, Sz = Tz = p, say. For ay x, let the sequece Sx }, geerated by the 0 Y iterative procedure Sx + = f ( T, x), = 0,,..., coverges to p. Let { Sy } X be a arbitrary sequece, ad set ε = d( Sy +, f ( T, y )), = 0,, 2,.... The the iterative procedure f ( T, x ) will be called ( S, T ) stable if ad oly if lim ε = 0 implies that lim Sy = p. { Timis ad Beride [55] defied weak stability usig the cocept of approximate sequece, which is defied as follows. Defiitio.2.32 [37]. Let ( X, d) be a metric space ad { x } X be a give sequece. We shall say that { y } X is a approximate sequece of x } if, for ay k N, there =0 exists η = η(k) such that d ( x, y ) η, for all k. = { 8

19 Defiitio.2.33 [37]. Let ( X, d) be a metric space ad T : X X. Let x } be a iteratio procedure defied by x 0 X ad x + = f ( T, x ), 0. Suppose that { x } coverges to a fixed poit p of T. If for ay approximate sequece { } X of x }, lim d ( y, f ( T, y )) = 0 implies lim = p, y { iterative procedure is weakly + y T stable or weakly stable with respect to T. { the we shall say that Further Timis [56] defied weak sequeces Defiitio.2.34 [56]. Two sequeces { x } =0 ad { } =0 2 w d ( x, y ) 0 as. stability usig the followig cocept of equivalet y are equivalet sequeces if Remark.2.. Ay equivalet sequece is a approximate sequece but the coverse may ot be true. The followig example illustrates it. Example.2.4. Let of x to be { } =0 { } =0 { } =0 2 x be a sequece with x =. First, we take a equivalet sequece 2 y where y = +. I this case, we have d( y, x ) = 0, Now, take a approximate sequece of d( y, x) = > 0, { } =0. x to be { } 2 y = 0, where y = +. The, 2 + Defiitio.2.35 [56]. Let ( X, d) be a metric space ad T : X X be a map. Let x } be a iteratio procedure defied by x 0 X ad x + = f ( T, x ), 0. { 9

20 Suppose that { x } coverges to a fixed poit p of T. If for ay equivalet sequece { y } of { x }, lim d ( y, f ( T, y )) = 0 implies lim = p, + y the we shall say that iterative scheme is weak 2 w stable with respect to T..3 SOME GENERAL SPACES I this sectio we defie some geeral spaces used i the subsequet work reported i this thesis..3. b - METRIC SPACE First we defie a b- metric space itroduced by Czerwik [246]. The basic cocepts of covergece, compactess, closedess ad completeess are also defied i such space. Defiitio.3.. [246]. Let X be a o empty set ad b be a give real umber. A fuctio d : X X R + is said to be a b metric iff for all y, z X, the followig coditios are satisfied (i) d ( y) = 0 iff x = y, (ii) d ( y) = d( y, x), (iii) d ( z) b[ d( y) + d( y, z)]. The pair ( X, d) is called a b-metric space. As oted i [246], mathematical problems such as the problem of metrizatio of covergece with respect to measure, lead to the geeralizatio of metric, so called b-metric. 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 of Sigh ad Prasad [52] shows that a b-metric o X eed ot be a metric o X (see also [246, p. 264]). Example.3.. [52]. Let X = x, x, x, } ad d x, x ) = k 2, { 2 3 x4 ( 2 d x, x ) = d( x, ) = d x, x ) = d( x, ) = d x, x ), d x, x ) = d( x, x ) for all ( 3 x4 ( x4 i, j =, 2, 3, 4 ad d( x, x ) = 0, i =, 2, 3, 4. The i i ( 3 4 = ( i j j i 20

21 [ d( x, x ) d( x, x )] k d ( xi, x j ) i + j for, i, j =, 2, 3, 4 2 ad if k > 2, the ordiary triagle iequality does ot hold. Defiitio.3..2 [24]. Let ( X, d) be a b metric space. The a sequece { x} N i X is called (a) coverget if ad oly if there exists x X such that d( x, x) 0 as. I this case, we write lim = x (b) Cauchy if ad oly if d ( x, x m ) 0 as m,. Remark.3.. [24]. I a b-metric space ( X, d) the followig assertios hold (i) a coverget sequece has a uique limit, (ii) each coverget sequece is Cauchy, (iii) i geeral, a b metric is ot cotiuous. Defiitio.3..3 [24]. Let ( X, d) be a b metric space. If Y is a oempty subset of X. The the closure Y of Y is the set of limits of all coverget sequeces of poits i Y, i.e., Y = { x X :there exists a sequece{ x } such that lim x x}. N = Defiitio.3..4 [24]. Let ( X, d) be a b metric space. The a subset Y X is called: (a) closed if ad oly if for each sequece { x } i Y which coverges to a elemet x, we have x Y, (b) compact if ad oly if for every sequece of elemets of Y there exists a subsequece that coverges to a elemet of Y, (c) bouded if ad oly if δ ( Y ) = sup{ d( a, b) : a, b Y} <. N Defiitio.3..5 [24]. The b metric space ( X, d) is complete iff every Cauchy sequece i X coverges. 2

22 I the followig sectio we recall some defiitios ad prelimiaries regardig fuzzy metric spaces required for our results. We follow R. P. Pat [86], George ad Veeramai [00], Grabiec [07], Mishra et al [], Schweizer ad Sklar [20], Subramayam [247], Vasuki [248], V. Pat [249], Sigh ad Jai [250] ad Pathak et al [25] for otatios ad prelimiaries..3.2 FUZZY METRIC SPACE Defiitio.3.2. [20]. A biary operatio T :[0, ] [0, ] [0, ] is a t orm if T satisfies followig coditios for all a, b, c, d [0, ]. (i) T ( a, b) = T ( b, a), (ii) T (( a, b), c) = T ( a, ( b, c)), (iii) T ( a, b) T ( c, d) wheever a c ad b d, for all a, b, c, d [0, ], (iv) T ( a, ) = a, for all a [0, ]. Defiitio [00]. The triplet ( X, M, T ) is said to be fuzzy metric space, if X is a 2 arbitrary set, T is a cotiuous t orm ad M is a fuzzy set o X (0, ) satisfyig the followig coditios. (i) M ( y, t ) > 0, (ii) M ( y, t) = iff x = y, (iii) T ( M( y, t), M( y, z, s) M ( z, t+ s), t, s> 0, y, z X, (iv) M ( y,.) : (0, ) [0, ] is cotiuous. Defiitio [07]. Let ( X, M, T ) be a fuzzy metric space. A sequece x } i X is said to be (i) coverget to a poit x X if lim M (, t) =, for all t > 0, (ii) a Cauchy sequece if lim M ( x, x, t) =, for all t > 0, p > 0, + p x (iii) a complete fuzzy metric space i which every Cauchy sequece coverges to a poit i it. { 22

23 The cocept of commutig mappigs i the settig of fuzzy metric space is give by Subramayam [247] as follows. Defiitio [247]. Mappigs A ad S of a fuzzy metric space ( X, M, T ) ito itself are said to be commutig if M ( AS SA t ) = for all x X. After this several other weaker coditios of commutativity are defied i fuzzy metric space, some of them are described as: weakly commutig mappigs, compatible mappigs, R-weakly commutig maps, R-weakly commutativity of type ( A g ), R-weakly commutativity of type A ), etc., (see [], [247]-[248]). ( f Defiitio [86]. Mappigs A ad S of a fuzzy metric space ( X, M, T ) ito itself is said to be weakly commutig if M( AS SA t) M ( A S t) for each x X ad t > 0. I 994, Mishra et al [] itroduced the cocept of compatible mappig as follows: Defiitio []. Mappigs A ad S of a fuzzy metric space ( X, M, T ) ito itself is said to be compatible if ( ASx, SAx, t) = X such that lim Ax = lim Sx = u for some u X. for all > 0, t wheever x } is a sequece i { Vasuki [248] exteded the otio of R-weakly commutig maps itroduced by Pat [86] i metric space to fuzzy metric spaces. Defiitio [248]. Mappigs A ad S of a fuzzy metric space ( X, M, T ) ito itself are R weakly commutig provided there exists some positive real umber R such that M ( AS SA t) M ( A S t / R) for each x X ad t > 0. Pathak et al [25] improved the otio of R-weakly commutig mappigs i metric spaces to the otios of R-weakly commutativity of type ( Ag ) ad type ( A f ). V. Pat [249] exteded it to fuzzy metric space. 23

24 Defiitio [249]. Mappigs A ad S of a fuzzy metric space ( X, M, T ) ito itself are R weakly commutig of type A ) (or of type A ) ) provided there exists some positive real umber R, for each ( f x X ad t > 0 such that M( SS AS t) M ( S A t/ R) (or M( AA SA t) M ( A S t / R) ). ( g Jugck ad Rhoades [85] itroduced the cocept of weakly compatible maps which were foud to be more geeralized tha compatible maps. Weak compatibility i fuzzy metric space is give by Sigh ad Jai [250]. Defiitio [250]. Mappigs A ad S of a fuzzy metric space ( X, M, T ) ito itself are said to be weakly compatible if they commute at the coicidece poits, i.e., if Au = Su for some u X, the ASu = SAu. May authors defied fuzzy metric spaces i differet ways (see for istace [99]-[06], [252]-[253]). Usig the idea of L -fuzzy set [2], Saadati et al [5] itroduced the otio of L -fuzzy metric spaces with the help of cotiuous t-orms as a geeralizatio of fuzzy metric space due to George ad Veeramai [00] ad ituitioistic fuzzy metric space due to Saadati ad Park [05]. The followig sectio devoted to basic cocepts related to L -fuzzy metric spaces..3.3 L -FUZZY METRIC SPACE Defiitio.3.3. [5]. Let L = ( L, ) be a complete lattice, ad U be a oempty set L called a uiverse. A L -fuzzy set o U is defied as a mappig : U L. For each u i U, (u) represets the degree ( i L ) to which u satisfies. Lemma.3.3. [254, 255]. Cosider the set * L ad the operatio * L defied by * 2 L = {( x2 ) : ( x2 ) [0, ] ad x + x2 }, 24

25 * * ( x x y ad x, for every ( x, x ), ( y, y ). The L, ), x2 ) * ( y, y2 ) L 2 y L ( L * is a complete lattice. Classically, a triagular orm o ([ 0, ], ) is defied as a icreasig, commutative, associative mappig :[0, ] 2 [0, ] satisfyig (, x ) = for all x [0, ]. These defiitios ca be straightforwardly exteded to ay lattice L = ( L, L). Also 0L = ifl ad L = supl Defiitio [20]. A triagular orm ( t satisfyig the followig coditios. (i) ( x L)( T ( ) = x); L orm) o L is a mappig 2 T : L L (ii) T T 2 ( ( y) L )( ( y) = ( y, x)); 3 (iii) ( ( y, z) L ) ( T ( T ( y, z)) = T ( T ( y), z); (iv) x x y y L x x ad y y x y x y 4 ( (,,, ) ) ( L L T (, ) LT (, )). A t orm T o L is said to be cotiuous if for ay y L ad ay sequeces x } ad { y } which coverge to x ad y, we have lim T ( x, y ) = T ( y). { Defiitio [254]. A egatio o L is ay decreasig mappig N : L L satisfyig N (0 ) = ad N ( ) = 0. If NN ( ( x)) = for all x L, the N is called a ivolutive L egatio. L L L Defiitio [8]. The 3-triplet ( X, M, T ) is said to be a L fuzzy metric space, if X is a arbitrary (o-empty) set, is a cotiuous t-orm o L ad M is a L fuzzy set 2 o X (0, ) satisfyig the followig coditios for every x, y, z i X ad t, s i ( 0, ). (i) M ( y, t ) > L 0, L (ii) M ( y, t) = L for all t > 0, iff x = y, (iii) M( y, t) = M ( y, t), (iv) T ( M( y, t), M( y, z, s) M ( z, t+ s), L 25

26 (v) M ( y,.):(0, ) L is cotiuous ad lim ( y, t) =. M L t I this case M is called a L fuzzy metric. If M = M, is a ituitioistic fuzzy set, the the 3-tuple ( X, M, N, T ) is said to be a ituitioistic fuzzy metric space, defied by Park [03] i 2004, usig the cocept of ituitioistic fuzzy set [6]. M N Defiitio A sequece { x } i a L fuzzy metric space ( X, M, T ) is called a N Cauchy sequece, if for each ε L \{0 L } ad t > 0, there exists 0 N such that for all m 0, M( x, x, t) > N ( ε). m L The sequece { x } is said to be coverget to x X i L fuzzy metric space N ( X, M, T ), if M( x, t) = M( x, t) L, wheever, for every t > 0. L fuzzy metric space is said to be complete if ad oly if every Cauchy sequece coverges to a poit i it..4 THESIS OUTLINE This thesis is orgaized as follows. I Chapter 2, we obtai some basic approximate fixed poit results i geeralized metric spaces. Further some existece results cocerig approximate fixed poits, edpoits ad approximate edpoits of multivalued cotractios are also derived. Some quatitative estimates of the sets of approximate fixed poits ad approximate edpoits for set-valued cotractios i geeralized metric space are developed. These results exted some recet results i the literature. Whe the mappig uder cosideratio is a oself mappig, the fixed poit equatio i such cases may ot have a solutio. Best proximity pair theorems deal with these problems ad provide ot oly approximate solutio but also a optimal approximate solutio. Some approximate best proximity pair theorems are also obtaied with applicatio to Hammerstei itegral equatio. The itet of Chapter 3 is to obtai some commo fixed poit theorems i geeral settigs. We study the commo fixed poits of hybrid pair of maps cosistig of sigle ad 26

27 multivalued maps satisfyig a itegral type cotractive coditio i b-metric spaces. A applicatio of the results to fuctioal equatio of dyamic programmig problem is also give. Further we defie the otio of θ L fuzzy metric space ad obtai some commo fixed poit theorems for maps satisfyig itegral type coditios i such spaces. I Chapter 4, we study the stability results of various iterative schemes of maps 2 satisfyig some geeral coditio. Weak ad w weak stability results are also provided for these iterative schemes. Our results exted ad improve several recet results. A comparative study of the differet iterative schemes is also preseted for some examples reported i the literature. A geeralized versio of the KKM theorems by usig the cocept of Chag ad Zhag [82] is studied i Chapter 5. Our results geeralize some of the recet result of Chag ad Zhag [82] ad Asari et al [79]. As applicatios of the results to game theory, miimax theorem ad saddle poit theorem for two-perso-zero-sum game are established. Cosequetly, a saddle poit theorem for two perso zero sum parametric game is also proved. I Chapter 6, we first defie some cotractio coditios of more geeral ature ad establish a geeralized iterated fuctio ad multi fuctio system theory usig those cotractio coditios. Correspodigly some existece ad uiqueess results are also obtaied. This theory exteds several recet results ad ehaces the scope of IFS ad IMS. 27

28 ONE SHOULD STUDY MATHEMATICS SIMPLY BECAUSE IT HELPS TO ARRANGE ONE S IDEAS M. W. Lomoossow 28