A Fixed Point Result Using a Function of 5-Variables

Transcription

1 Joural of Physical Scieces, Vol., 2007, 57-6 Fixed Poit Result Usig a Fuctio of 5-Variables P. N. Dutta ad Biayak S. Choudhury Departmet of Mathematics Begal Egieerig ad Sciece Uiversity, Shibpur P.O.: B. Garde, Shibpur, Howrah - 703, West Begal, INDI Received December 2, 2007; accepted December 2, 2007 BSTRCT Here we prove two fixed poit theorems i metric spaces. The results are obtaied for mappig which satisfy certai iequalities through a fuctio of five variables.. Itroductio The Baach cotractio mappig theorem is a pivotal result i mathematical aalysis []. fter that efforts have bee made by mathematicias to obtai fixed poit theorems of mappigs satisfyig several cotractive iequalities [2],[3]. s a result, a large umber of research articles o the existece of fixed poits of several types of mappigs satisfyig differet cotractive iequalities have appeared i literature. Some of such works deal with oe or more mappigs which are assumed to satisfy iequalities through give fuctios. I the preset work we assume such a iequality with the help of a five variable fuctio. We have two theorems i oe of which we prove a commo fixed poit result i a metric space which is ot ecessarily complete. I the other theorem we prove the existece of a uique fixed poit of a self-mappig defied o a complete metric space. We support our result by a example. The followig is the defiitio of a class of fuctios which we call G-fuctios. Defiitio.. [ 0, [ 5 0, 5 g : is said to be a G-type mappig if (i g is cotiuous (ii g is odecreasig i each variable (iii if h(r = g (r, r, r, r, r. the r r h (r is strictly icreasig ad positive i (0,. Examples of G-type mappig are the followig 57

2 58 P. N. Dutta ad Biayak S. Choudhury g r, r = α r + α r + α r + α r + α r r, r, r, (i ( where α are o-egative ad < α <. s i 5 0 i (ii g ( r, r 2, r 3, r 4, r 5 λ max{ r, r 2, r 3, r 4, r 5 } (iii g ( r, r, r, r, r = l[ + max{ r, r, r, r r }] = where 0 < λ < , 2. Mai Results Theorem 2.. Let ( X,d be a metric space ad, B : X X be two self-mappigs satisfyig the iequality d By y, By, y, y, By, x where g belogs to the ( class of G-type fuctios. (2. Let { x } be ay sequece i X which satisfies d ( x, x 0 as (2.2 If x coverges to a poit x the ay other sequece (, 0 5 y havig the property that d y By as will also coverge to x ad x is a commo fixed poit of ad B. Proof. We assume that x x as (2.3 Let ε > 0 be give. We choose > 0 δ as δ ( ε h ( ε = (2.4 This choice of δ is possible i view of defiitio.. s (2.2 is true, correspodig to the choice of δ > 0 which by (2.4, depeds also o ε > 0, a positive iteger 0 ca be foud out such that for > 0. d (, < δ ad ( < δ x x Cosequetly for > 0 3 d y, By (2.5 ( x, y x, x + d ( x, By + d ( By y 2 δ + g ( x, y, x, x, By, y, x, y, By, x d, (usig (2. ad (2.5 s g is o-decreasig i each variable, we have d x y 2δ + g d x, y, d x, x, (, ( ( ( By, y, d ( x, x + x, y, By, y + y, x δ + g ( x, y, δ, δ, δ + x, y, δ + x, y, 2

3 Fixed Poit Result Usig a Fuctio of 5-Variables 59 2 δ + h ( δ + d ( x, y or, d ( x, y + δ 3δ + h( δ + x, y or, ( x, y + δ h( δ + x, y 3δ = ε h( ε ( by( 2.4 Sice r h(r is strictly icreasig ad positive i (,, x, y + δ < ε for 0 > Cosequetly ( < ε 0 the above iequality implies d x, y for > 0 (2.6 (2.2, (2.3 ad (2.6 joitly imply that if x x the y x By x as (2.7 gai, usig (2. By < g( y,by, y, y,b y,x Makig ad otig that g is cotiuous, we obtai from (2.7 that which implies d ( x h( x 0 ( x g( 0, x,0, x,0 h ( x d usig the property of h i defiitio., it follows that d x = or equivaletly x = x Similarly it ca be proved that This completes the proof. ( 0 Bx = x Theorem 2.2. Let : X X be a self mappig i a complete metric space (X, d satisfyig ( x y g( y, x, y, y, y, k y x d,, (2.8 Where g belogs to the class G ad 0 < k (2.9 2 The for ay x X, the sequece { x} is such that d + ( x 0 as Further if { x} is coverget the it coverges to the uique fixed poit of. lso i that case ay other sequece { } coverge to the uique fixed poit. y satisfyig (, 0 d y y as will also

4 60 P. N. Dutta ad Biayak S. Choudhury Proof. We costruct α = =,2, If α = 0 for some, the has a fixed poit at y= x We assume that α 0 for all =,2, (2.0 Replacig y with x respectively i (2.8 ad usig defiitio. We obtai + + x,, k + x + x, 0,k( + x + x which implies α g α, α, α,0, K( α α (2. + ( If possible, letα + α Sice 0 < k, it follows that 2 α h α 0 or α 0 which cotradicts ( = Cosequetly, 0 < α + < α for =,2, which implies { α },beig a decreasig sequece,is coverget. Let α α as s α + < α, usig (2.9 ad (2., we obtai α h( α + Makig, it follows that α h ( α or α = 0 (2.2 Let x = =,2, The x = x = α+ 0 as Let x z as. We observe that whe the assume =B, we obtai (2.8 from (2. as a special case. The by the applicatio of theorem - the result of the preset theorem follows except for the uiqueess of fixed poit. To prove the uiqueess, we suppose x ad y as two fixed poits of.

5 Fixed Poit Result Usig a Fuctio of 5-Variables 6 The from (2.8 y y,0,0, y, y h( y or y h( y 0 or y = 0 or x = y This completes the proof. Example 2.. x + Let X = R, d ( y = x y ad x = 4 If g ( α, α2, α3, α4, α5 = α 3 + α α α α 8 5 The satisfies the coditio of theorem 2.2. It is see that x = is the uique fixed 3 poit of. REFERENCES [] W.. Kirk ad B. Sims, Hadbook of metric fixed poit theory, Kluwer cademic Publishers, Netherlads (2002 [2] J. Meszaros, compariso of various defiitios of cotractive type mappigs, Bull. Cal. Math. Soc. 90 (992, [3] B. E. Rhoades, compariso of various defiitios of cotractive mappigs, Tra. mer. Math. Soc, 226 (977,