Some Fixed Point and Common Fixed Point Theorems in 2-Banach Spaces

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1 Americn Journl of Engineering eserch (AJE) e-issn : -7 p-issn : -9 Volume- Issue-5 pp--7 eserch Pper Open Access Some Fied Point nd Common Fied Point heorems in -Bnch Spces A.S.Sluj Alkesh Kumr Dhkde Govt. J.H. P.G. College Betul (M.P.) Indi IES College of echnolog Bhopl (M.P.) Indi Abstrct: In the present pper we prove some fied point nd common fied point theorems in -Bnch spces for new rtionl epression. Which generlie the well known results. AMS: 7H 5H5. Kewords: Bnch Spce -Bnch Spces Fied point Common Fied point. I. INODUCION he stud of non-contrction mpping concerning the eistence of fied points drws ttention of vrious uthors in non-liner nlsis. It is well known tht the differentil nd integrl equtions tht rise in phsicl problems re generll non-liner therefore the fied point methods specill Bnch s contrction principle provides powerful tool for obtining the solutions of these equtions which were ver difficult to solve b n other methods. ecentl Verm [9] described bout the ppliction of Bnch s contrction principle []. Ghlr [5] introduced the concept of -Bnch spces. ecentl Bdshh nd Gupt [] Ydv jput nd Bhrdwj [] Ydv jput Choudhr nd Bhrdwj [] lso worked for Bnch nd -Bnch spces for non contrction mppings. In present pper we prove some fied point nd common fied point theorems for non-contrction mppings in -Bnch spces motivted b bove before strting the min result first we write some definitions. Definition (.A) -Bnch Spces: In pper Ghler [5] define liner -normed spce to be pir non negtive rel vlued function defined on L such tht b cl (i) b if nd onl if nd b re linerl dependent (ii) (iii) (iv) b b b b is rel b c b c Hence.. is clled -norm. w w w. j e r. u s Pge L. where L is liner spce nd.. is Definition (.B): A sequence in liner -normed spce L is clled convergent sequence if there is L such tht n lim for ll L. n n Definition (.C): A sequence in liner -normed spce L is clled Cuch sequence if there eists L such tht n nd re linerl independent nd

2 w w w. j e r. u s Pge lim n m n m Definition (.D): A liner -normed spce in which ever Cuch sequence is convergent is clled -Bnch spces. II. MAIN ESULS heorem.: Let be mpping of -Bnch spces into itself. If stisfies the following conditions: I where I is identit mpping (.) (.) Where is rel with.hen hs unique fied point. Proof: Suppose is n point in -Bnch spce X. king I

3 w w w. j e r. u s Pge Now for u Now u u (.) On the other hnd

4 w w w. j e r. u s Pge 5 u (.) So s k Where k Let I then k B the definition of we clim tht n is Cuch sequence in X n is converges to so element in X. So lim n n. So. Hence So is fied point of. Uniqueness: If possible let is nother fied point of.hen Which is contrdiction so.hence fied point in unique. heorem.: Let nd G be two epnsion mppings of -Bnch spce X into itself. nd G stisf the following conditions: (.) nd G commute (.) I nd I G where I is identit mpping.

5 (.) G G G G G G G G G G G G G G G G G X. with nd G G nd. For ever hen there eists unique common fied of nd G such tht nd G. Proof: - Suppose is point in -Bnch spce X it is cler tht G I GG. GG G GG G GG. GG GG G GG G GG G GG GG GG GG G GG G GG G GG G G G G G G G GG G G G G G G G G G G G GG G GG G GG G GG G GG G GG GGG G G king G p G q where p q p Gp q Gq q Gq q Gp p Gq p q p G king G we get p q p p q p q Gq p Gq q Gp p p q p q p q q q q q p p q p q p q p p q q p It is cler b theorem (.); tht G And so. G Or G G Now p q p q G hs t lest one fied point s in K tht is G G p q w w w. j e r. u s Pge

6 . G G G G G G G G G G G G G G G G G So ht is is the fied point of. G G. But so Hence is the fied point of nd G. Uniqueness: If possible let is nother common fied point of nd G. hen G G G G G G G G G G G G G G G G G But So. So common fied point in unique. EFEENCES: []. Ahmd nd Shkil M. Some fied point theorems in Bnch spces Nonliner Funct.Anl. & Appl. () -9. []. Bnch S. Surles opertion dns les ensembles bstrits et leur ppliction u equtions integrls Fund. Mth. (9) -. []. Bdshh V.H. nd Gupt O.P. Fied point theorems in Bnch nd -Bnch spces Jnnbh 5(5) 7-7. []. Goebel K. nd Zlotkiewics E. Some fied point theorems in Bnch spces Colloq Mth (97) -. [5]. Ghlr S. -metrche rume nd ihre topologiscche structure Mth.Ndh. (9-) 5-. []. Iseke K. fied point theorem in Bnch spce Mth.Sem.Notes Kobe Universit (97) -5. [7]. Jong S.J.Viscosit pproimtion methods for fmil of finite non epnsive in Bnch spces nonliner Anlsis () []. Shrm P.L. nd jput S.S. Fied point theorem in Bnch spce Vikrm Mthemticl Journl (9) 5-. [9]. Verm B.P. Appliction of Bnch fied point theorem to solve non liner equtions nd its generlition Jnnbh () -. []. Ydv.N. jput S.S. nd Bhrdwj.K. Some fied point nd common fied point theorems in Bnch spces Act Cienci Indic No (7) 5-. []. Ydv.N. jput S.S. Choudhr S. nd Bhrdwj.K. Some fied point nd common fied point theorems for noncontrction mpping on -Bnch spces Act Cienci Indic No (7) w w w. j e r. u s Pge 7