IJERD (www.ijerd.com) International Journal of Engineering Research and Development IJERD : hard copy of journal, Call for Papers 2012, publishing of journal, journal of science and technology, research paper publishing, where to publish research paper,
IJERD (www.ijerd.com) International Journal of Engineering Research and Development IJERD : hard copy of journal, Call for Papers 2012, publishing of journal, journal of science and technology, research paper publishing, where to publish research paper,
IJERD (www.ijerd.com) International Journal of Engineering Research and Development IJERD : hard copy of journal, Call for Papers 2012, publishing of journal, journal of science and technology, research paper publishing, where to publish research paper,
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International Journal of Engineering Research and Developmente-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.comVolume 3, Issue 2 (August 2012), PP. 35-40 Fixed Point Theorems for Mappings under General Contractive Condition of Integral Type Shailesh T. Patel, Sunil Garg, Dr.Ramakant Bhardwaj, The student of Singhania University Senior Scientist,MPCST Bhopal Truba Institute of Engineering & Information Technology,Bhopal(M.P.)Abstract––In the present paper, we establish a fixed point theorem for a mapping and a common fixed point theorem fora pair of mappings for rational expression. The mapping involved here generalizes various type of contractive mappingsin integral setting.Keywords––Fixed point, Common Fixed point, complete metric space, Continuous Mapping, Compatible Mapping I. INTRODUCTION AND PRELIMINARIESImpact of fixed point theory in different branches of mathematics and its applications is immense.The first important resulton fixed point for contractive type mapping was the much celebrated Banach’s contraction principle by S.Banach [1] in1922. In the general setting of complete metric space,this theorem runs as follows(see Theorem 2.1,[4] or,Theorem1.2.2,[10]) In the present paper we will find some fixed point and common fixed point theorems for rational expression,which will satisfy the well known results. I have developed new result in above theorem with rational expression, by usingR.Bhardwaj,Some common fixed point theorem in Metric space using integral type mappings[12].Theorem 1.1 (Banach’s contraction principal)Let (X,d) be a complete matric space,cє(0,1) and f:X→X be a mapping such that for each x,yєX,d(fx,fy) ≤ c d(x,y) …………………(1.1)then f has a unique fixed point aєX,such that for each xєX, lim f n x a. n after this classical result , Kannaan [5] gave a substantially new contractive mapping to prove fixed point theorem.Since thena number of mathematicians have been working on fixed point theory dealing with mappings satisfying various type ofcontractive conditions(see[3],[5],[7],[8],[9] and [11] for details).In 2002,A.Branciari [2] analyzed the existence of fixed point for mapping f defined on a complete matric space (X,d)satisfying a general contractive condition of integral type.Theorem 1.2 (Branciari)Let (X,d) be a complete matric space,cє(0,1) and f:X→X be a mapping such that for each x,yєX,d ( fx , fy ) d ( x, y ) (t )dt c (t )dt 0 0 …………………(1.2)Where :[0,+∞)→[0,+∞) is a Lesbesgue-integrable mapping which is summable(i.e.with finite integral) on each compact subset of [0,+∞),nonnegative,and such that for each є>0, (t )dt ,then f has a unique fixed point aєX,such that for each 0xєX, lim f x a. n n After the paper of Branciari, a lot of research works have been carried out on generalizing contractive conditions of integraltype for different contractive mappings satisfying various known prpperties. A fine work has been done by Rhoades [6]extending the result of Branciari by replacing the condition (1.2) by the following [ d ( x , fy ) d ( y , fx )] max{d ( x , y ),d ( x , fx ),d ( y , fy ), }d ( fx , fy ) 2 (t )dt c 0 (t )dt 0 ………………(1.3)The aim of this paper is to generalize some mixed type of contractive conditions to the mapping and then a pair of mappingssatisfying a general contractive condition of integral type,which includes several known contractive mappings such asKannan type [5],Chatterjea type [3],Zamfirescu type [11],etc. 35
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Fixed Point Theorems for Mappings under General Contractive Condition of Integral Type II. MAIN RESULTSLet f beaself mapping of a complete matric space (x,d) satisfying the following condition:d ( fx , fy ) [ d ( x , fx ) d ( y , fy )] d ( x, y ) max{d ( x , fy ),d ( y , fx )} (t )dt 0 (t )dt (t )dt 0 0 (t )dt 0 d ( x , fy ) d ( y , fx ) d ( x , fx ) 1 d ( x , fy ).d ( y , fx ).d ( x , fx ) (t )dt 0 ……………….(2.1)For each x,yєX with non-negative reals , , , , such that 0< 2 2 3 <1,where : R R is a Lesbesgue-integrable mapping which is summable(i.e.with finite integral) on each compact (t )dt …………….(2.2) subset of R ,nonnegative,and such that for each є>0, 0Then f has a unique fixed point zєX,such that for each xєX, lim f n x z. n Proof: Let x0єX and, for brevity,definexn=fxn-1.For each integer n≥1,from (2.1) we get,d ( Xn, Xn1) d ( fXn 1, fXn ) (t )dt 0 (t )dt 0 [ d ( Xn1, Xn ) d ( Xn, Xn 1)] d ( Xn1, Xn ) max{d ( Xn1, Xn1),d ( Xn, Xn )} (t )dt 0 (t )dt 0 (t )dt 0 d ( Xn 1, Xn1) d ( Xn, Xn ) d ( Xn 1, Xn ) 1 d ( Xn 1, Xn 1).d ( Xn, Xn ).d ( Xn 1, Xn ) (t )dt 0 [ d ( Xn1, Xn ) d ( Xn, Xn 1)] d ( Xn1, Xn ) d ( Xn1, Xn1) (t )dt 0 (t )dt 0 (t )dt 0 d ( Xn1, Xn1) d ( Xn1, Xn ) (t )dt 0 d ( Xn1, Xn ) d ( Xn, Xn1) ( 2 ) (t )dt ( ) (t )dt 0 0Which implies thatd ( Xn, Xn1) d ( Xn1, Xn ) (t )dt ( (t )dt 2 1 ) 0 0 d ( Xn, Xn1) d ( Xn1, Xn) (t )dt h (t )dt 2And so, where h= 1 <1 …………..(2.3) 0 0Thus by routine calculation, d ( Xn, Xn 1) d ( X 0 , X1 ) (t )dt h (t )dt n 0 0 ……………(2.4) d ( Xn, Xn1)Taking limit of (2.4) as n→∞, we get lim n (t )dt 0 oWhich,from (2.2) implies that lim d ( Xn, Xn 1) 0 ……………(2.5) n 36
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Fixed Point Theorems for Mappings under General Contractive Condition of Integral TypeWe now show that {Xn} is a Cauchy sequence. Suppose that it is not. Then there exists an ϵ˃0 and subsequences {m(p)} and{n(p)} such that m(p)˂n(p)˂m(p+1) with d(Xm(p),Xn(P))≥є, d(Xm(p),Xn(P)-1)<є ……………(2.6) d(Xm(p)-1,Xn(P)-1)≤ d(Xm(p)-1,Xm(P))+ d(Xm(p),Xn(P)-1) <d(Xm(p)-1,Xm(P))+є …………..(2.7) d ( Xm ( p ) 1,, Xn ( p ) 1) Hence lim p (t )dt (t)dt o o ……………(2.8)Using (2.3),(2.6) and (2.8) we get d ( Xm( p ),, Xn( p )) d ( Xm( p ) 1,, Xn ( p ) 1) (t)dt o (t )dt h o (t )dt h (t )dt o oWhich is a contradiction, since hє(0,1). Therefore {Xn} is Cauchy, hence convergent. Call the limit z.From (2.1) we getd ( fz , Xn1) [ d ( z , fz ) d ( Xn, Xn1)] d ( z , Xn) max{d ( z , Xn1),d ( Xn, fz )} (t )dt o (t )dt 0 (t )dt 0 (t )dt 0 d ( z , Xn1) d ( Xn, fz ) d ( z , fz ) 1 d ( z , Xn1).d ( Xn, fz ).d ( z , fz ) (t )dt 0Taking limit as n→∞, we get d ( fz , z ) d ( z , fz ) (t )dt o ( 2 ) (t )dt 0As d ( fz , z ) 2 2 3 <1, (t )dt 0 oWhich, from (2.2), implies that d(fz,z)=0 or fz=z.Next suppose that w (≠ z) be another fixed point of f. Then from (2.1) we haved ( z , w) d ( fz , fw ) (t )dt o (t )dt od ( z , w) [ d ( z , fz ) d ( w, fw )] d ( z , w) max{d ( z , fw ),d ( w, fz )} (t )dt o (t )dt (t )dt 0 0 (t )dt 0 d ( z , fw ) d ( w, fz ) d ( z , fz ) 1 d ( z , fw ).d ( w, fz ).d ( z , fz ) (t )dt 0 d ( z , w) ( 2 ) (t )dt 0Since, d ( z , w) 2 <1 this implies that, , (t )dt 0 oWhich from (2.2), implies that d(z,w)=0 or, z=w and so the fixed point is unique. 37
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Fixed Point Theorems for Mappings under General Contractive Condition of Integral TypeRemark. On setting (t ) 1 over R+, the contractive condition of integral type transforms into a general contractivecondition not involving integrals.Remark. From condition (2.1) of integraltype,several contractive mappings of integral type can be obtained.1. 0 є(0, 1 ) gives Kannanof integral type. 2 and2. 0 and (0, 1 ) gives Chatterjea [3] map of integral type. 2Theorem 2.3. Let f and g be self mappings of a complete metric space (X,d) satisfying the following condition:d ( fx , gy ) [ d ( x , fx ) d ( y , gy )] d ( x, y ) max{d ( x , gy ),d ( y , fx )} (t )dt o (t )dt 0 (t )dt 0 (t )dt 0 d ( x , gy ) d ( y , fx ) d ( x , fx ) 1 d ( x , gy ).d ( y , fx ).d ( x , fx ) (t )dt 0 ………………(2.9)For each x,yєX with nonnegative reals , , , such that 2 2 3 <1, where : R R is a Lesbesgue-integrable mapping which is summable(i.e.with finite integral) on each compact subset of R ,nonnegative,and such that for each є>0, (t )dt …………….(2.10) 0Then f and g have a unique common fixed point zєX.Proof. Let x0єX and, for brevity,define x2n+1=fx2n and x2n+2=gx2n+1. For each integer n≥0, from (2.9) we get,d ( X 2 n 1, X 2 n 2 ) d ( fX 2 n , gX 2 n 1) (t )dt 0 (t )dt 0 [ d ( X 2 n , X 2 n 1) d ( X 2 n 1, X 2 n 2 )] d ( X 2 n , X 2 n 1) max{d ( X 2 n , X 2 n 2 ),d ( X 2 n 1, X 2 n 1)} (t )dt 0 (t )dt 0 (t )dt 0 d ( X 2 n , X 2 n 2 ) d ( X 2 n 1, X 2 n 1) d ( X 2 n , X 2 n 1) 1 d ( X 2 n , X 2 n 2 ).d ( X 2 n 1, X 2 n 1).d ( X 2 n , X 2 n 1) (t )dt 0 d ( X 2 n , X 2 n 1) d ( X 2 n 1, X 2 n 2 ) ( 2 ) (t )dt ( ) 0 (t )dt 0Which implies thatd ( X 2 n 1, X 2 n 2 ) d ( X 2 n , X 2 n 1) (t )dt ( (t )dt 2 1 ) 0 0 d ( X 2 n 1, X 2 n 2 ) d ( X 2 n , X 2 n 1) (t )dt h (t )dt where 2 And so, h= 1 <1 ………….(2.11) 0 0 d ( X 2 n , X 2 n 1) d ( X 2 n 1, X 2 n )Similarly (t )dt h (t )dt 0 0 ………….(2.12)Thus in general, for all n=1,2,…………… d ( Xn, Xn1) d ( Xn1, Xn) (t )dt h (t )dt 0 0 ………….(2.13) 38
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Fixed Point Theorems for Mappings under General Contractive Condition of Integral TypeThus by routine calculation, we have d ( Xn, Xn 1) d ( X 0 , X1 ) (t )dt h (t )dt n 0 0 d ( Xn, Xn1)Taking limit as n→∞, we get lim n (t )dt 0 oWhich,from (2.10) implies that lim d ( Xn, Xn 1) 0 ……………(2.14) nWe now show that {Xn} is a Cauchy sequence. Suppose that it is not. Then there exists an є>0 and subsequences {2m(p)}and {2n(p)} such that p<2m(p)<2n(p) with d(X2m(p),X2n(P))≥є, d(X2m(p),X2n(P)-2)<є ……………(2.15)Now d(X2m(p),X2n(P))≤ d(X2m(p),X2n(P)-2)+ d(X2n(p)-2,X2n(P)-1)+ d(X2n(p)-1,X2n(P)) <є+d(X2n(p)-2,X2n(P)-1)+ d(X2n(p)-1,X2n(P)) ……………(2.16) …………..(2.7) d ( X 2 m ( p ), X 2 n ( p )) Hence lim p (t )dt (t)dt o o ……………(2.17)Then by (2.13) we get d ( X 2 m ( p ),, X 2 n ( p )) d ( X 2 m ( p ) 1,, X 2 n ( p ) 1) (t )dt h o (t )dt o d ( X 2 m ( p ) 1,, X 2 m ( p )) d ( X 2 m ( p ),, X 2 n ( p )) d ( X 2 n ( p ) 1, X 2 n ( p )) h[ (t )dt o (t )dt o (t )dt ] oTaking limit as p→∞ we get (t )dt h (t )dt 0 0Which is a contradiction, since hє(0,1). Therefore {Xn} is Cauchy, hence convergent. Call the limit z.From (2.9) we getd ( fz , X 2 n 2 ) d ( fz , gX 2 n 1) (t )dt o (t )dt 0 [ d ( z , fz ) d ( X 2 n 1, X 2 n 2 )] d ( z , X 2 n 1) max{d ( z , X 2 n 2 ),d ( X 2 n 1, fz )} (t )dt 0 (t )dt 0 (t )dt 0 d ( z , X 2 n 2 ) d ( X 2 n 1, fz ) d ( z , fz ) 1 d ( z , X 2 n 2 ).d ( X 2 n 1, fz ).d ( z , fz ) (t )dt 0Taking limit as n→∞, we get d ( fz , z ) d ( z , fz ) (t )dt o ( 2 ) (t )dt 0As d ( fz , z ) 2 2 3 <1, (t )dt 0 o 39
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Fixed Point Theorems for Mappings under General Contractive Condition of Integral TypeWhich, from (2.10), implies that d(fz,z)=0 or fz=z. Similarly it can be shown that gz=z. So f and g have a common fixedpoint zєX. We now show that z is the unique common fixed point of f and g.Ifnot,then letw (≠ z) be another fixed point of f and g. Then from (2.9) we haved ( z , w) d ( fz , gw ) (t )dt (t )dt o od ( z , w) [ d ( z , fz ) d ( w, gw)] d ( z , w) max{d ( z , gw),d ( w, fz )} (t )dt o (t )dt (t )dt 0 0 (t )dt 0 d ( z , gw ) d ( w, fz ) d ( z , fz ) 1 d ( z , gw ).d ( w, fz ).d ( z , fz ) (t )dt 0 d ( z , w) ( 2 ) (t )dt 0Since, d ( z , w) 2 <1 this implies that, , (t )dt 0 oWhich from (2.10), implies that d(z,w)=0 or, z=w and so the fixed point is unique. ACKNOWLEDGMENTSThe authors would like to thank the referee for his comments that helped us improve this article. REFERENCES [1]. S.Banach, Sur les oprationsdans les ensembles abstraitsetleur application aux quationsintgrales, Fund. Math.3,(1922)133181 (French). [2]. A.Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, Int.J.Math.Math.Sci, 29(2002), no.9, 531-536. [3]. S.K.Chatterjea, Fixed point theorems, C.R.Acad.Bulgare Sci. 25(1972),727-730. [4]. K.Goebel and W.A.Kirk ,Topiqs in Metric fixed point theory, Combridge University Press, New York, 1990. [5]. R.Kannan, Some results on fixed points, Bull. Calcutta Math. Soc.,60(1968), 71-76. [6]. B.E.Rhoades, Two fixed point theorems for mappings satisfying a general contractive condition of integral type, International Journal of Mathematics and Mathematical Sciences, 63,(2003),4007-4013. [7]. B.E.Rhoades, A Comparison of Various Definitions of Contractive Mappings, Trans. Amer.Math. Soc. 226(1977),257-290. [8]. B.E.Rhoades, Contractive definitions revisited, Topological Methods in Nonlinear Functional Analysis (Toronto, Ont.,1982), Contemp. Math.21, American Mathematical Society, Rhode Island, (1983), 189-203. [9]. B.E.Rhoades, Contractive Definitions, NonlinearAnalysis,World Science Publishing, Singapore, 1987, 513-526. [10]. O.R.Smart, Fixed Point Theorems, Cambridge University Press, London, 1974. [11]. T.Zamfirescu,, Fixed Point Theorems in metric spaces, Arch.Math.(Basel) 23 (1972), 292-298. [12]. R.Bhardwaj,Some common fixed point theorem in Metric space using integral type mappings,Iosr Journal 187(2012),187-190. 40
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