Nguyen Due Lang T?ip chf KHOA HOC & CONG NGHE 122(08): 179 - 183 N o t e on common fixed point for noticommuting mappings without
continuity in cone metric spaces
Nguyen Due Lang University of Science, Thainguyen University, Vietnam E-mail: [email protected] Abstract: In this work, we prove a common fixed point theorem by using the gener- alized distance in a cone metric space.
Keywords: Cone metric space, Common fixed point. Fixed point.
Mathematics Subject Classification: 47H10, 54H25.
1 Introduction
In 2007, Huang and Zhang [8] introduced the concept of the cone metric space, replacing the set of real numbers by an ordered Banach space, and proved some fixed point theorems of contractive type mappings in cone metric spaces. Afterward, several fixed and common fixed point results in cone metric spaces were introduced in [1, 2, 10, 11, 14] and the references contained therein Also, the existence of fixed and common fixed points in partially ordered cone metric spaces was studied in [3, 4, 5, 12],
The ann of this paper is to generalize and unify the common fixed point theorems of Abbas and .lungek [1]. Hardy and Rogers [7], Huang and Zhang [8], Abbas et al [2], Song ct al. [14]. Wang and Guo [15] and Cho ct a!. [6] on c-distance in a cone metric space
2 Preliminaries
Lemma 2.1. ([4. 9]) Let R be a real Banach space with a cone P in E Then, for all II.. V, w. c e E, the following hold-
(pi) Ifu :< u and v -C w, then u <Si w (p2) //O :< li <g; c for each c € intP. then u = 0.
(P:t} Ifu :< Xv where ii e P and 0 < A < 1. then n ^ Q.
(PJ) Let c £ intP, .L,, —> 0 and 0 ^ x„. Then there exists positive integer no such that x„ <S. c for each n > no
Lemma 2.2. {{6. 13. 15]) Let (X.d) be a cone metric space and let q be a c-distance on X. Also, let {x„] and {y„} be sequences in X and x.y.z ^ X. Suppose that {u^}
and {I'n} are two sequences in P converging to 0. Then the following hold:
(qPi) ^/f/(.fn, y) d: <i„ and q{x„,z) -^ v„ for ii e N , then y ^ z Specifically, if qix.y) = 0 and q[x. z) = 0, then y = z.
(qpa) //"y(jn-yn) :5 "n and q{.in. z) < v^ for u ^'^, then {yr,} converges to z
Nguyin Dlic Lang Tap chi KHOA HOC & CONG NGHE 122(08): 179 - 183 (qPa) U gi^n, ^m) i^ ^n foT TU > u, then {xn} is a Cauchy sequence in X (qp4) Ifq{y. x„) :< u„ for n eN, then {Xn} is a Cauchy sequence in X.
3 Main R e s u l t s
Our main result is the following theorem. We prove a common fixed point theorem by using c-distance and we do not require that / and g are weakly compatible. The following theorem extends and improves Theorems 2.1 and 2.3 of [1], Theorem 2,1 of [15] and Corollary 2.11 of [2] under generalized distance in a cone metric space.
T h e o r e m 3 . 1 . Let {X,d) be a cone metric space, P be a normal cone with constant K and q be a c-distance on X. Suppose that the mappings f.g : X —^ X satisfy the following two contractive conditions:
Q{fx, fy) d: ocMgx, gy) + a2qigx. fx) + a^q{gy, fy)
+o:M9-^Jy) + <^sq{gy:fx), (3.1) g{fy,fx-) ^ Oiiq{gy,gx) + a2q{fx,gx) -f o-3q{fy,gy)
+a4q{fy, gx) + a^qifx, gy) (3.2) for all X. y e X. where GJ for 7 — 1,2, • • • . 5 are nonnegative constants such that
Ql + Q2 + aa -f 2(^4 + as) < L (3.3) If the range of g contains the range of f, g{X) is a complete subspace of X, f and g
satisfy
mf{Mfx.y)\\ + Mgx.y)\\ + Mgr. fx)\\ : x e X} > 0
for all, y e X with y f fy or y ^ gy. then f and g have a common fixed point m X.
If fz ~ qz = z, then q[z.z) =• 0.
Proof Let Xo e -Y be an arbitrary point Since the range of g contains the range of / , there exists an i i 6 X such that /.TQ = J.T, By induction, a sequence {.T„} can be chosen such that / x „ = j i „ + i for n = 0,1,2, • •. Now, set x = xv,-i and y = x„ in (3-1). Thus, by (172), for any natural number n, we have
g{gxn,gxn+i) ?(/x„_i,/l„)
< Oigigx„_i.gx„) + ff2?(s.x„_,, / x „ _ , ) + a3q(gx„,fx„) + a 4 « ( s x „ _ , , / i „ ) + aiq(gx„,fxn-i)
aig(gx„-i. gx„) + n2()(,9.i„-i,,9.x„) + Oiqiqx,,. gx„+i) +aiq{gx„-i,gxn+i) + a!,q{gxn,gx„)
^ ail{gx.,-i.gx„) + a2?(gx„-i.9X„) +Q3?(,i)x„.9x„+i) + ai{q(gx„_t,gx„) + q{gx„. gx„^i)]
+a,|i;(gi„,9x„+i) + 9(,9x„+,,9x„)|. (3,4)
Nguyen Diic Ung Tap chi KHOA HQC & CONG NGHE 122(08): 179 - 183 Similarly, set i = x„-i and y — Xn in (3.2) Thus, by (92), for any natural number n.
wc have
g(gx„+i.gx„) :< aiq{gx... gx„_i) + n2q{gx„. gx.,^.)
+"39(9x„+,.(;x„) + Ut\q(gx„+i, gx„) + q{gx„. gx„-i)]
+ Q5|g(9l„,9l„+l) + q{gx„i.gx„)]. (3.5) Adding up (3.4) and (3.5), we get that
9(.93:„,.9X„+i) f g(,9x„+i,.9i„) i ( a , + Q 2 + a4)[9(.9X„-i,.»x„) + Qlgx„, gxn-i)] + (QJ + a. + 2cj5)|g((/x„, j.l„+i)
+ i!(SX„+,,9l„)]. (3.6) Now. set u„ = 5(p,Xn,,gx„+i) -f g(.9X„+i,Xn) in (3.6) Thus, we have
Pn :< ( Q I + Q2 + Cl4)iJn-l + (^3 + ^4 + 2 Q 5 ) I ; „ ,
So. i^n :< /ifn-1 for all 71 > 1 with
1 - (QI + a4 + 2a^)
since ai + tt2 + 't:i 1 2((V4 + CVS) < 1 Repealing this process, we get v„ < h^u,, for 11 = (1,1.2,'.'. Thus,
q{qx„,gx„+,) ^ [,'„ d h" lq(gXa. qxi] + q{gxi.gXo)j (3 7) for all n ^ 0,1, 2, • • • Lcl ni. > n, then it follows from (3.7) and /i < 1 that
q[gx„,gx„) < 7(s.T„,ST„+i)+f;(g:r„.,.j.T„4.2) + •- '^ q{gx„-i,g.T„)
< (A" + A"+' + • • • + A."'"')((7(9TD, 9X,) + f;((;xi. 9T|,))
s Y T T ( ' t * - ' ' " ' ' • ' • ' ' * ' ' * • ' ' ' • ' ^ ° ' ) '^*'' Lemma 2.2 implies llial {g:i„] is a Caurln- sequence in X Since g{X} is a complete
subspace of X. there exists a point x' £ 9( A') such that qx„ -> x' as >i -^ DC By (3 8) and (93)
q(gx„.x') < —^ (q(gxo, qxt) + q{qx,. gXB)j , n = 0,1.2. •••
Since P is a normal cone with normal constant h', we get
Mgx„.T')\\< K(^j!^)\\q{g.To,gx,)-l q{gT,.gxo)\\. n = 0 , l , 2 , - - (3 9) and
\Mgx„.g.r.„)\\ < A ' ( Y ^ ) 1 < / ( 9 . I » ' < / - ' i ) + «(9X,. 9.l»)||. (3 10)
Nguyjn Dire Lang Tap chf KHOA HOC & CONG NGHE 122(08): 179-183 for all m > rj > 1. It fx' ^ x' or qx^ / x', then, by the hypothesis, (3.9) and (3.10) with m — n + 1, we get
0 < inf{||9(,/x,i')ll + lk(.9x,i')ll + \\'l{gx,fx)\\ : x a X)
< inf{||9(/.T„..x')il + Mgrn.x'n + lk(9.T„,/x„)|| : r, > 1}
inf{||g(.9x„+i,x')i| + |[9(.9X„,x')|| + ||9(.9X„,,9i„4.i)|| - n > 1}
< inf | f t ' ( - - - - ) | | 9 ( 9 l o . 9 i i ) +9(.gii.9Xo)|[
+A'(~^)l[9(9.xo.9Xi) + q(.gXi.qxo)\\
+ A - ( Y 3 - ^ ) 1 J 9 ( < ) X „ . 9 . T I ) + 9(9.TI.9.X„)|| - n S l } = 0 ,
which is a contradiction. Hence x' = fx' = gx' Also, suppose that fz = gz = z. Then, by (3 1) we have
9(z. z) = 9(/z, SA
< Oiq{gz. gz) + a2q(gz, fz) + Ozq(gz, f z) + ntqigz, f z) + a^qigz, fz) (ttl + a2 + as + 04 + 05)9(2^ 2)
Since «! + nj + »3 + "4 + 05 < "1 + "2 + 03 + 2(n4 + rej) < 1 , we get that q{z. z) = 0 by Lemma 2 1(^3). This completes the proof
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