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ISSN 0976-5727 (Print) ISSN 2319-8133 (Online) Abbr:J.Comp.&Math.Sci. 2014, Vol.5(2): Pg.251-257
Common Fixed Point for Contractive Operators by a Faster Iterative Process in Real Banach Spaces M. R. Yadav School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur, Chhattisgarh, INDIA. (Received on: April 21, 2014) ABSTRACT In this article, we introduced a new two step iteration scheme for the class of Zamfirescu operators in real Banach spaces and proved strong convergence theorem to approximate common fixed points of Zamfirescu operators. We showed that our iteration process is faster than generalized Mann iteration scheme is defined by Yao and Chen9 and studied by3, 5, 7, 8. We also give the comparison table for iteration schemes. An example is given to support our main results and satisfied Zamfirescu operators. 2000 Mathematics Subject Classification: 47H05, 47H09, 47H10. Keywords and phrases: Two-step iteration scheme, Zamfirescu Operator, real Banach spaces, Strong convergence, Common foxed point.
1. INTRODUCTION Throughout this paper, N denotes the set of all positive integers. Let K be nonempty subset of real Banach space X and T : K → K a given operator. Let x 0 ∈ K be arbitrary and {α n }∈ [0, 1] a sequence or real numbers. The Mann iteration process starting at x 0 is defined by
x 1 = x 0 ∈ K; (1.1) x n +1 = (1 - α n )x n + α n Tx n ; n ∈ N is called the Mann6 iteration schemes or Mann iteration procedure and the generalized Mann iteration scheme is defined as follows: x1 = x0 ∈K; (1.2) x = α x + β Tx + γ Sx ; n ∈ N n +1 n n n n n n
Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)
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M. R. Yadav, J. Comp. & Math. Sci. Vol.5 (2), 251-257 (2014)
where {α n } , {β n }
and {γ n } are real
sequence in [0, 1] with α n + β n + γ n = 1. Above iteration scheme studied by Yao and Chen9 for common fixed points of two mappings. The following Ishikawa type iteration scheme has been studied by various authors for common fixed points of two mappings, see for example3, 5, 7, 8:
x1 = x 0 ∈ K; x n +1 = (1- α n )x n + α n Syn y = (1- β )x + βα Tx ; n ∈ N , n n n n n
(1.3)
where {α n } and {β n } are sequences in [0; 1]. This scheme also reduces to Mann iteration scheme (1.1) when T = I or S = I. The purpose of this paper is to define a new type of two-step iteration scheme for Zemfrarescu operators and showed strong convergence theorem to approximate fixed point in the framework of real Banach space. Let K be a nonempty, closed and convex subset of a real Banach space X. S; T : K → K be two Suppose ∞
nonlinear operators and {x n }n =0 be the sequence in [0, 1]: x 1 = x 0 ∈ K; (1.4) x n +1 = α n x n + β n Tx n + γ n Sy n y = (1 - β )x + β Tx , n ∈ N , n n n n n
where {α n } , {β n }
and {γ n } are real
sequence in [0, 1] with α n + β n + γ n = 1. This scheme also reduces to Mann iteration scheme (1.1) when T = I or β n = 0.
We recall the following definitions in a metric space (X, d) . A mapping T : X → X is called a-contraction if d(Tx, Ty) ≤ ad(x, y) , (1.5) for all x, y ∈ X, where a ∈[0, 1) . The map T is called Kannan mapping [4] if there exists b∈ (0, 1/2) such that d(Tx, Ty) ≤ b[d(x, Tx) + d(y, Ty)] , (1.6) for all x, y ∈ X, . The map T is called Chatterjea2 if there exists c∈ (0, 1/2) such that d(Tx, Ty) ≤ c[d(x, Ty) + d(y, Tx)] , (1.7) for all. An operator T which satisfies at least one of the contractive condition (1.5), (1.6) and (1.7) is called Zamfirescu Operator or a Z-operator10. In 1972 Zamfirescu10 obtained very interesting fixed point Theorem, by combining (1.5), (1.6) and (1.7) given below: Theorem 1.1. Let (X, d) be complete metric space and T : X → X a map for which there exist the real number a, b, and c satisfying 0 < a < 1, 0 < b, c < 0.5 such that for each pair x, y ∈ X, , at least one the following is true: (i) d(Tx, Ty) ≤ ad(x, y) , (ii) d(Tx, Ty) ≤ b[d(x, Tx) + d(y, Ty)] , (iii) d(Tx, Ty) ≤ c[d(x, Ty) + d(y, Tx)] Then T is a Picard operator x n +1 = Tx n . 2. MAIN RESULTS In this section, we have proved strong convergence theorem and find approximate common fixed points of two self mapping S and T.
Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)
M. R. Yadav, J. Comp. & Math. Sci. Vol.5 (2), 251-257 (2014)
Theorem 2.1. Let K be a nonempty closed convex subset of real Banach space E and T : X → X be a Zemfrarescu operator. Suppose {x n } be the sequence defined by iteration (1.4) for arbitrary x 0 ∈ K with ∞
∑ β n = ∞ , where {α n } , {β n } and {γ n }
n =0
253
α n + β n + γ n = 1 . Then {x n } converges strongly to a common fixed point of T and S. Proof. Suppose F(T) ≠ φ and p∈ F(T) . Since T is Zamfirescu operator, then T satisfies at least one of the conditions (1.5), (1.6) and (1.7). If (1.6) holds, then for any x, y ∈ K, , we obtain
are sequences in [0, 1] with
|| Tx - Ty || ≤ b[|| x − Tx || + || y − Ty ||] ≤ b[|| x − Tx || + || y − x || + || x − Tx || + || Tx − Ty ||] which yield that
(1 - b) || Tx - Ty || ≤ b[ || x - y || + 2b || x - Tx || , b 2b since 0 ≤ b < 1/2 , we have || Tx − Ty || ≤ || x − y || + || x − Tx || . 1− b 1− b Similarly, if (1.7) holds, then we obtain for any x, y ∈ K, || Tx - Ty || ≤ c[|| x − Ty || + || y − Tx ||]
(2.1)
≤ b[|| x − Tx || + || Tx − Ty || + || y − x || + || x − Tx ||] which yield that
(1 - c) || Tx - Ty || ≤ c[ || x - y || + 2c || x - Tx || , since 0 ≤ c < 1/2 , we have
|| Tx − Ty || ≤
c 2c || x − y || + || x − Tx || . 1− c 1− c
(2.2)
Suppose,
c b δ = max a , , 1− b 1− c Then we have 0 ≤ δ < 1 and from (2.1), (2.2) and (2.3), we get || Tx − Ty || ≤ δ || x − y || + 2δ || x − Tx ||
(2.3)
(2.4)
Similarly, since S is a Zamfirescu operator, we obtain
|| Sx − Sy || ≤ δ || x − y || + 2δ || x − Sx || holds for all x, y ∈ K, . Let p is a fixed point of T, then if x = p and y = x n
(2.5) in (2.4), we
obtain
|| Tx n − p || ≤ δ || x n − p || Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)
(2.6)
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M. R. Yadav, J. Comp. & Math. Sci. Vol.5 (2), 251-257 (2014)
and put x = p and y = y n in (2.5), we get
|| Sy n − p || ≤ δ || y n − p || ∞ n n =0
Suppose {x }
(2.7)
be the sequences defined by (1.4). Then we have
|| x n +1 − p || = α n x n + β n Tx n + γ n Sy n − p || ≤ || α n ( x n − p) + β n (Tx n − p) + γ n (Sy n − p) ||
(2.8)
≤ α n || x n − p || + β n || Tx n − p || + γ n || Sy n − p || From (2.6), (2.7) and (2.8), we obtain
|| x n +1 − p || ≤ α n || x n − p || + β n δ || x n − p || + γ n δ || y n − p || ≤ (α n + β n δ) || x n − p || + γ n δ || y n − p ||
(2.9)
and,
|| y n − p || = (1 − β n ) x n + β n Tx n − p || ≤ || (1 − β n )( x n − p) + β n (Tx n − p) || ≤ (1 − β n ) || x n − p || + β n || Tx n − p ||
(2.10)
≤ (1 − β n ) || x n − p || + β n δ || x n − p || Combining (2.9) and (2.10), we get
|| x n +1 − p || = (α n + βn δ) || x n − p || + γ n δ{(1 − βn ) || x n − p || + βn δ || x n − p ||} ≤ (α n + βn δ) || x n − p || + γ n δ(1 − βn ) || x n − p || + γ n βn δ 2 || x n − p || ≤ (α n + βn δ + γ n δ − γ n βn δ + γ n βn δ 2 ) || x n − p || ≤ [α n + (1 − α n )δ − βn γ n δ(1 − δ)] || x n − p || ≤ [δ + α n − α n δ + βn γ n δ(1 − δ)] || x n − p || ≤ [1 − 1 + δ + α n (1 − δ) − βn γ n δ(1 − δ)] || x n − p || ≤ [1 − 1 + δ + α n (1 − δ) − βn γ n δ(1 − δ)] || x n − p || ≤ [1 − (1 − δ) + α n (1 − δ) − βn γ n δ(1 − δ)] || x n − p || ≤ [1 − (1 − δ)(βn + γ n + +βn γ n δ] || x n − p || . Now,
[1 − (1 − δ){β n + γ n (1 + β n δ)}] = [1 − (1 − δ)β n − (1 − δ) γ n (1 + β n δ)] ≤ [1 − (1 − δ)β n ] By the induction method, we have Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)
M. R. Yadav, J. Comp. & Math. Sci. Vol.5 (2), 251-257 (2014) n
|| x n +1 − p || ∏ [1 − (1 − δ)β n ]. || x 0 − p ||, n = 0,1,2,...
255
(2.11)
m =0
Now, since
0 ≤ δ < 1 , β m ∈ [0, 1] and
∞
∑ β m = ∞ , then it follows that
When T = I then Berinde result ([1], Theorem 2.1) is a corollary to our result. Corollary 2.2. Let K be a nonempty closed convex subset of real Banach space E and S : K → K be a Zemfrarescu operator. Suppose {x n } be the sequence defined by
m =0
n
lim ∏ [1 − (1 − δ)β n ] = 0 ,
m → ∞ m =0
which implies by (2.11), we obtain lim || x n +1 − p || = 0 .
iteration (1.1) for arbitrary x 0 ∈ K with
m →∞
∞
∞
Therefore, {x n }n =0 converges strongly to p, which is the common fixed point of T and S. This complete the proof.
∑ γ n = ∞ , where {γ n } is sequence in
n =0
[0, 1] . Then {x n } converges strongly to a common fixed point of S.
Following examples illustrating our main results. Example : Consider the mappings define by S, T : R → R as follows
Tx =
(2 − x ) (2 y + 1) and Sy = for all x, y ∈ R 3 4
It is clear that, both S and T are satisfied the condition of Zamfirescu operators with the
1 . 2 Furthermore, since 0 ≤ δ < 1 and from (2.4) and (2.5), we have || Tx - Ty | ≤ δ || x - y || + 2δ || x - Tx || , and || Sx - Sy | ≤ δ || x - y || + 2δ || x - Sx || , for all x, y ∈ R . In fact, common fixed point
|| Tx − Ty || =
(2 − x ) (2 − y ) x − y , and − = 3 3 3
δ || x − y || + 2δ || x − Tx || = δ x − y + 2δ x − = δ x − y + 2δ =δ
2−x 3x − 2 + x = δ x − y + 2δ 3 3
4x − 2 3
= δ x − y + 4δ
2x − 1 3
3x − 3y + 8x − 4 11x − 3y − 4 =δ . 3 3
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M. R. Yadav, J. Comp. & Math. Sci. Vol.5 (2), 251-257 (2014)
Similarly, we conclude that
x−y , 2
|| Sx − Sy || = and
δ || x − y || + 2δ || x − Sx || =
1 δ [ 4x − 2 y − 1 ]. 2
Put x = 0.25, y = 0.14 and δ = 0.5, we have
|| Tx − Ty || =
x−y = 0.036 , 3
and
δ || x − y || + 2δ || x − Tx || = δ
11x − 3y − 4 = 0.388 . 3
Table 1. A comparison table of our process with other processes Steps
Iter.1
Steps
Iter.1
Steps
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.2500 9.5100 4.6846 2.4649 1.4439 0.9742 0.7581 0.6587 0.6130 0.5920 0.5823 0.5779 0.5758 0.5749 0.5744
16 17 18 19 20 21 22 23 24 25 26 17 28 29 30
0.5742 0.5742 0.5741 0.5741 0.5741 0.5741 0.5741 0.5741 0.5741 0.5741 0.5741 0.5741 0.5741 0.5741 0.5741
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Iter.2 0.2500 10.5825 5.7325 3. 2347 1.9484 1.2859 0.9447 0.7690 0.6786 0.6320 0.6080 0.5956 0.5892 0.5860 0.5843
Hence, for x = 0.25, y = 0.14 and δ = 0.5 the condition (2.4) is satisfied. Similarly we show that the condition (2.5) is also satisfied. It is not difficult to show that T and S are contraction. Now, choose α = 0.5, β = 0.33 and γ = 0.25 for all n with initial value
Steps
Iter.2
Steps
16 17 18 19 20 21 22 23 24 25 26 17 28 29 30
0.5834 0.5829 0.5827 0.5826 0.5825 0.5825 0.5825 0.5825 0.5825 0.5825 0.5825 0.5825 0.5825 0.5825 0.5825
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Iter.3 0.2500 11.4900 6.6393 3.8744 2.2984 1.4001 0.8881 0.5962 0.4298 0.3350 0.2810 0.2501 0.232 0.2226 0.2169
Steps
Iter.4
16 17 18 19 20 21 22 23 24 25 26 17 28 29 30
0.2136 0.2118 0.2107 0.2101 0.2098 0.2096 0.2095 0.2094 0.2094 0.2093 0.2093 0.2093 0.2093 0.2093 0.2093
x1 = 20 . The comparison table given in the following table shows that our iterative process (1.4) converges faster than all Yao et al.9 and Das et al.3 iterative processes up to the accuracy of forth decimal places. Here we observe that the values the above calculations have been repeated by
Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)
M. R. Yadav, J. Comp. & Math. Sci. Vol.5 (2), 251-257 (2014)
taking different values of parameters α , β and γ . It has been verified every time that our iterative process Iter1 converges faster than all studied by Yao and Chen9 and3, 5, 7, 8 iterative processes. REFERENCES 1. Berinde, V., "On the convergence theorem for Mann iteration in the calss of Zamfirescu operators,"Analele Universitatii de Vest. Timisoara Seria Mathematica-informatic, XLV.1, pp.3341 (2007). 2. Chatterjea, S.K.,"Fixed point theorems," C.R.A. Acad. Bulgare. Sci., 25, pp.727730 (1986). 3. Das, G., and Debta, J.P., "Fixed points of quasi-nonexpansive mapping," Inda J. Pure Appl. Math., 17, pp.1263-1269 (1972). 4. Kannan, R., "Some results on _xed points," Bull. Call. Math. Soc., 10, pp.71-76 (1968).
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5. Khan, S. H., and Takahashi, W., "Approximation common _xed points of two asymptotically nonexpansive mappings," Sci. Math. Jpn., 53(10), pp.143-148 (2001). 6. Mann, W.R., "Mean value methods in iteration," Proc. Amer. Math. Soc. 4, pp.506-510 (1953). 7. Shahzad, N., "Approximating xed points of non-self nonexpansive mappings in Banach spaces", Nonlinear Anal. 61, No. 6, pp.1031-1039 (2005). 8. Takahashi, W. and Tamura, T., "Convergence theorems for a pair of nonexpansive map- pings", J. Convex Anal. 5, No. 1, pp.45-56 (1998). 9. Yao, Y., and Chen, R., "Weak and strong convergence of a modi_ed Mann iteration for asymptotically nonexpansive mappings," Nonlinear Funct. Anal. Appl., 12(2), pp.307-325 (2007). 10. Zamirescu, T., "Fix point theorems in matric spaces," Arch. Math., 23, pp.292298 (1972).
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