JOURNAL OF COMPUTER AND MATHEMATICAL SCIENCES An International Open Free Access, Peer Reviewed Research Journal www.compmath-journal.org
ISSN 0976-5727 (Print) ISSN 2319-8133 (Online) Abbr:J.Comp.&Math.Sci. 2014, Vol.5(3): Pg.274-281
Approximation Fixed Point in Uniformly Convex Banach Space C. L. Dewangan and H. K. Nayak Department of Mathematics, Govt. Chhattisgarh College, Raipur, C. G. INDIA. (Received on: May 15, 2014) ABSTRACT The purpose of this paper is study the two-step iteration process to approximate the common fixed points of nonexpansive mappings and we have proved strong and weak convergence results for fixed points of nonexpansive mappings in uniformly convex Banach spaces. 2000 Mathematics Subject Classification: Primary 47H05; Secondary 47H09, 47H10. Keywords and phrases: Ishikawa type Iteration process, Nonexpansive mappings, Uniformly convex Banach space Condition (A'), Opial's condition, Common fixed point.
1. INTRODUCTION Let K be a nonempty closed convex subset of a real Banach space E. Throughout this paper, Ν denotes the set of all positive integers and F(T) ≠ φ . A mapping
T : K → K is said to be nonexpansive if || Tx - Ty || ≤ || x - y || for all x, y ∈ K . We know that a point x ∈ K is a fixed point of T if Tx = x , i.e., F(T) = {x ∈ K : Tx = x} .
Several authors have been studied iterative techniques for approximation fixed points of nonexpansive mapping (see11, 13, 1, 7 and8) using the Mann iteration methods (see15) or the Ishikawa iteration method (see10). The Picard and Mann15 iteration schemes for a mapping T : K → K are defined by
x 1 = x 0 ∈ K, x n +1 = Tx n , n ∈ Ν
Journal of Computer and Mathematical Sciences Vol. 5, Issue 3, 30 June, 2014 Pages (258-331)
(1.1)
275
C. L. Dewangan, et al., J. Comp. & Math. Sci. Vol.5 (3), 274-281 (2014)
and
x1 = x 0 ∈K, x n +1 = (1 − α n )x n + α n Tx n , n ∈ Ν
(1.2)
where {α n } is in (0, 1) . It is well-known that Picard iteration scheme converges for contractions but not converges for nonexpansive mapping whereas Mann iteration scheme converges for nonexpansive. The sequence {x n }∞n =0 defined by
x 1 = x 0 ∈K , x n +1 = (1 − α n ) x n + α n Ty n y = (1 − β ) x + β Tx , n ∈ Ν, n n n n n
mappings. If β n = 0 all n ≥ 1 , then (1.4) reduces to (1.2). In this paper, we study the iteration scheme given in (1.4) for strong and weak convergence for a nonexpansive mapping in a uniformly convex Banach space. 2. PRELIMINARIES Let X = { x ∈ E :|| x || = 1} and E * be the dual of E. The space E has: (i) Gateaux differentiable norm if
|| x + ty || − || x || , t→0 t exists for each x , y ∈ K ; lim
(1.3)
is known as the Ishikawa iteration process10, where {α n } and {β n } are sequences in [0, 1] is known as the Ishikawa iteration process3. This scheme also reduces to Mann iteration scheme (1.2) when β n = 0 . Recently, the Ishikawa type iteration process have generalized by Thianwan12 as follows:
x 1 = x 0 ∈K , x n +1 = (1 − α n ) y n + α n Ty n y = (1 − β ) x + β Tx , n = 0,1, 2, 3,.... n n n n n (1.4) where {α n } ∞n = 0 , {β n }∞n = 0 are sequences of real numbers in [0, 1] . They used this iteration scheme to find the common fixed points of two asymptotically nonexpansive nonself mappings in a Banach space. The iteration scheme (1.4) is called the projection type Ishikawa for nonexpansive
(ii) Frechet differentiable norm (see e.g.8) for each x in S, the above limit exists and is attained uniformly for y in S and in this case, it is also well-known that 1 1 h, J ( x) + || x || 2 ≤ || x + h || 2 2 2 , 1 2 ≤ h, J ( x) + || x || + b(|| h ||) 2
(2.1)
for all x , h ∈ E , where J is the Frechet
1 || ⋅ || 2 at x∈ E , 2 ⋅ , ⋅ is the dual pairing between E and E * ,
derivative of the function
and b is an increasing function defined on [ 0, ∞ ) such that
lim
t →0
b( t ) = 0; t
(iii) Opial's condition [6] if for any sequence { x n } in E, x n → x implies that
lim sup || x n − x || < lim sup || x n − y || , n →∞
n→∞
for all y ∈ E with y ≠ x .
Journal of Computer and Mathematical Sciences Vol. 5, Issue 3, 30 June, 2014 Pages (258-331)
C. L. Dewangan, et al., J. Comp. & Math. Sci. Vol.5 (3), 274-281 (2014)
Following are the definitions used to prove the results in the next section. Let E be a Banach space, K be a nonempty closed convex subset of E, and T : K → K be a nonexpansive mapping. Then I - T is said to be demiclosed at 0, if x n → x converges weakly and x n - Tx n → 0 converges strongly, then it is implies that x ∈ K and Tx = x . Suppose two mappings S, T : K → K , where K is a subset of a normed space E, said to be satisfy condition (A ′ ) if there exists a nondecreasing function
F : [0, ∞ ) → [0, ∞ ) with F(0) = 0 , f(r) > 0 for all r ∈ (0, ∞ ) such that either || x - Sx || ≥ f(d(x, F)) or || x - Tx || ≥ f(d(x, F)) for all x ∈ K where d(x, F) = inf { || x - p || : p ∈ F = F(S) ∩ F(T)} . The following lemma will be required in the sequel. Lemma 2.1. [5]: Let K be a nonempty bounded closed convex subset of a uniformly convex Banach space E and T be a nonexpansive map of K into itself. If x n → x ({x n } ⊂ K, x ∈ K) then there exists strictly increasing convex map g : [0, ∞ ) → [0, ∞ ) with g(0) = 0 such that
g (|| x − Tx ||) ≤ lim inf || x n − Tx n || n → ∞
Lemma 2.2. [3]: Let r > 0 be a fixed real number. Then a Banach space E is uniformly convex if and only if there is a continuous strictly increasing convex map
276
g : [0, ∞ ) → [0, ∞ ) with g(0) = 0 such that for all x, y ∈ B r [0] = {x ∈ E : || x || ≤ r} || λx + (1 − λ ) || 2 ≤ λ || x || 2 + (1 − λ ) || y || 2 − λ (1 − λ ) g (|| x − y ||)
for all λ ∈[ 0, 1] . Lemma 2.3. [4]: Let g : [0, ∞ ) → [0, ∞ ) with g(0) = 0 be a strictly increasing map. If a sequence {x n } in [0, 1) satisfies lim g ( x n ) = 0 , then lim x n = 0. n→∞
n→∞
Lemma 2.4. : Let E be a uniformly convex Banach space and K its nonempty closed convex subset of E. Suppose T : K → K be a nonexpansive mappings. If F(T) ≠ φ and the {x n } be the sequence defined by (1.4). Then lim || x n − p || exists for all n→∞
p ∈ F(T) . Proof. Suppose p ∈ F(T) and using (1.4), we have || x n +1 − p || = || (1 − α n ) y n + α n Ty n − p ||
= || (1 − α n )( y n − p) + α n (Ty n − p) || ≤ (1 − α n ) || y n − p || +α n || Ty n − p || ≤ (1 − α n ) || y n − p || + α n || y n − p || ≤ || y n − p ||
(2.2) 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 || ≤ (1 − β n ) || x n − p || +β n || x n − p || ≤ || x n − p || .
(2.3) Journal of Computer and Mathematical Sciences Vol. 5, Issue 3, 30 June, 2014 Pages (258-331)
277
C. L. Dewangan, et al., J. Comp. & Math. Sci. Vol.5 (3), 274-281 (2014)
Substituting (2.3)) into (2.2)), we obtain
|| x n +1 − p || ≤ || x n − p || . Since { || x n − p ||} is a non-increasing and bounded sequence. So that lim || x n − p || n→∞
is exists. This completes the proof. 3. MAIN RESULTS In this section, we have proved the approximate common fixed points of nonexpansive mapping for weak and strong convergence results, using a Ishikawa iteration type process. In the consequence, F denotes the set of common fixed point of the mapping T. Lemma 3.1. : Let E be a uniformly convex Banach space and K be a nonempty closed convex subset of E. Let T : K → K be nonexpansive mappings. Suppose {x n } be the real sequence defined by iteration (1.4) for arbitrary x 0 ∈ K , where {α n } and
{β n } are sequences of real numbers in [0, 1] ∞
with restriction ∑ β n (1 − β n ) = ∞ , n =1
lim inf β n > 0 and β n ∈[δ, 1 − δ] for n→∞
some δ ∈ ( 0, 1 / 2 ) . If F ( T ) ≠ φ , then lim || x n − Tx n || = 0 . n→∞
Proof. Suppose p ∈ F(T) . Since lim || x n − p || exists as proved in Lemma n→∞
2.4 and so {x n } is bounded. Consequently, {y n − p}, {Tx n − p} are bounded so that,
we can get a closed ball B r [0] such that
{x n − p, y n − p, Tx n − p} ⊂ B r [0] I K
and with help of Lemma 2.2 and the iteration (1.4), we get || x n + 1 − p || 2 = || α n Ty n + (1 − α n ) y n − p || 2 = || α n (Ty n − p ) + (1 − α n )( y n − p ) || 2 ≤αn || Tyn − p ||2 +(1−αn ) || yn − p ||2 −αn (1−αn )g(|| yn −Tyn ||) ≤αn || yn − p ||2 +(1−αn ) || yn − p ||2 −αn (1−αn )g(|| yn −Tyn ||) ≤|| yn − p ||2 −αn (1−αn )g(|| yn −Tyn ||) ≤|| yn − p ||2 .
(3.1) Again, using iteration process (1.4) and Lemma 2.2, we have || y n − p || 2 = || β nTxn + (1 − β n ) xn − p || 2 = || β n (Txn − p) + (1 − β n )( x n − p) || 2 ≤ β n || Tx n − p || 2 + (1 − β n ) || xn − p || 2 − β n (1 − β n ) g (|| xn − Txn || ≤ β n || x n − p || 2 +(1 − β n ) || x n − p || 2 − β n (1 − β n ) g (|| xn − Txn || ≤ | xn − p || 2 − β n (1 − β n ) g (|| xn − Txn || .
(3.2) Substituting (3.2) into (3.1), we obtain || x n +1 − p || 2 ≤ || x n − p || 2 − β n (1 − β n ) g (|| x n − Tx n ||).
(3.3)
From the above estimate, we get the following inequality, β n (1 − β n ) g (|| x n − Tx n ||) ≤ || x n − p || 2 − || x n +1 − p || 2 ,
∞
Note that ∑ β n (1 − β n ) = ∞ and n =1
Journal of Computer and Mathematical Sciences Vol. 5, Issue 3, 30 June, 2014 Pages (258-331)
(3.4)
C. L. Dewangan, et al., J. Comp. & Math. Sci. Vol.5 (3), 274-281 (2014)
lim inf β n > 0 . Suppose k ≥ 1 .
lim x n , J(p1 − p 2 )
n→∞
n →∞
Furthermore, since g : [0, ∞ ) → [0, ∞ ) is a continuous increasing convex function and {x n − Tx n } is bounded sequence in E. We assert that {g (|| x n − Tx n ||)} is bounded. Let K be any positive integer. Summing up the terms from 1 to k on both sides in the inequality (3.4), we obtain k
∑ β (1 − β n
n ) g (||
xn − Txn ||) ≤ || xn − p || 2
n =1
− || x n +1 − p || 2 < ∞
(3.5)
When k → ∞ in the above inequality, we have
278
exist, in particular,
p - q; J(p1 - p 2 ) = 0 for all p, q ∈ ωω (x n ). Proof. Take x = p1 - p 2 , with p1 ≠ p 2 and h = t(x n - p1 ) in the inequality (2.1) to get: 1 || p1 − p 2 || 2 + t xn − p, J ( p1 − p2 ) 2 1 ≤ || txn + (1 − t ) p1 − p2 || 2 2 1 ≤ || p1 − p 2 || 2 + t xn − p1, J ( p1 − p 2 ) + b(t n || xn − p1 ||). 2
As sup || x n − p1 || ≤ M ′ for some M ′ > 0 , it n ≥1
follows that 1 || p1 − p 2 || 2 + t lim sup xn − p, J ( p1 − p 2 ) 2 n →∞ 1 lim || tx n + (1 − t ) p1 − p 2 || 2 2 n →∞ 1 ≤ || p1 − p 2 || 2 + b(tM ′) + t lim inf xn − p , J ( p1 − p 2 ) . n→∞ 2 ≤
k
∑ βn (1 − βn )g(|| yn − Tyn ||) ≤ ∞. (3.6)
n =1
That is
By the properties of g, we get lim inf g (|| x n − Tx n ||) = 0. n→∞
Since lim inf β n > 0 and β n ∈[δ, 1 − δ] n→∞
for every δ ∈ ( 0, 1 / 2 ) . Then the above inequality implies that
lim || x n − Tx n || = 0.
n→∞
This completes the proof. Lemma 3.2. Let K be a nonempty closed convex subset of a Banach space E. Suppose {x n } be the sequence defined in Theorem (3.3) with F ≠ φ . Then, for any p 1 , p 2 ∈ F ,
lim sup xn − p1 , J ( p1 − p2 ) n →∞
≤ lim inf xn − p1 , J ( p1 − p 2 ) + n →∞
If t → 0 then
b(tM ′) M ′. tM ′
lim x n − p1 , J(p1 − p 2 )
n →∞
exists for all p 1 , p 2 ∈ F in particular, we get
p − q, J(p1 − p 2 ) = 0 , for all p, q ∈ ωω (x n ). Theorem 3.3. Let E be a Banach space satisfying Opial condition and K, T, S and {x n } be taken as Lemma 3.1. If
F(S) ∩ (T) ≠ φ , I - T and I - S are demiclosded at zero, then {x n } converges weakly to a common fixed point of S and T.
Journal of Computer and Mathematical Sciences Vol. 5, Issue 3, 30 June, 2014 Pages (258-331)
279
C. L. Dewangan, et al., J. Comp. & Math. Sci. Vol.5 (3), 274-281 (2014)
Proof. Let p ∈ F(S) ∩ F(T) , then as proved in Lemma 3.1 lim || x n − p || exist. Since E
{x n } converges strongly to a point of F if and only if lim inf d ( x n , F) = 0.
is Banach space. Thus there exists subsequences {x n k } ⊂ {x n } such that
Proof.
n →∞
n→∞
{x n k } converges weakly to z 1 ∈ K . From Lemma 3.1, we have lim || Tx n k − x n k || = 0 .
Necessity
is
evident,
n →∞
lim || x n − p || exists for all p ∈ F , so that
n→∞
lim d ( x n , F) exists. Since by hypothesis,
n→∞
n→∞
Since I - T is demiclosed at zero, therefore Tz 1 = z1 . Finally, we prove that {x n }
lim inf d ( x n , F) = 0. , so that, we get
converges weakly to z 1 . Let on contrary that there exists a subsequence {x n i } ⊂ {x n } and {x n j } ⊂ {x n } such that
n→∞
{x n j } converges weakly to z 2 ∈ K and z 1 ≠ z 2 . Again in the same way, we can prove that z 2 ∈ F(T). From Lemma 2.4 the limits lim || x n − z 1 || and lim || x n − z 2 || n →∞
n →∞
exists. Suppose that z1 ≠ z 2 , then by the Opial's condition, we get lim || x n − z1 || = lim || x n i − z 1 || < lim || x n i − z 2 ||
n→∞
ni → ∞
ni → ∞
= lim || x n − z 2 || = lim || x n j − z 2 || n →∞
nj →∞
< lim || x n j − z 1 || = lim || x n − z 1 || . n j →∞
let
lim inf d ( x n , F) = 0. From Lemma 2.4,
n →∞
lim d ( x n , F) = 0.
But {x n } is Cauchy sequence and therefore converges to p. We know that lim d( x n , F) = 0 , we obtained d(p, F) = 0 , n →∞
therefore p ∈ F . Using Theorem 3.2, we obtain a strong convergence theorem of the iteration scheme (1.4) under the condition (A ′) as below: Theorem 3.5. Let E be a Banach space and K, S, T, F, {x n } be as in Lemma 3.1. Let S, T satisfy the condition (A ′) and F ≠ φ . Then {x n } converges strongly to a point of F.
n →∞
This is a contradiction so z 1 = z 2 . Hence
{x n } converges weakly to a common fixed point of T. Theorem 3.4. Let E be a Banach space and K, S, T, F, {x n } be as in Lemma 3.1. Then
Proof. We proved in Lemma 3.1, i.e. lim || x n − Tx n || = 0 . n→∞
Then from the definition of condition (A ′) , we obtain lim f (d ( x n , F)) ≤ lim || x n − Tx n || = 0 . n→∞
n→∞
In above cases, we get
Journal of Computer and Mathematical Sciences Vol. 5, Issue 3, 30 June, 2014 Pages (258-331)
C. L. Dewangan, et al., J. Comp. & Math. Sci. Vol.5 (3), 274-281 (2014)
lim f ( d ( x n , F)) = 0 .
n→∞
But f : [ 0, ∞ ) → [ 0, ∞ ) is a nondecreasing function satisfying f ( 0 ) = 0 , f ( r ) > 0 for all r ∈ ( 0, ∞ ) , so that we get
lim d ( x n , F) = 0.
n→∞
All the conditions of Theorem 3.2 are satisfied, therefore by its conclusion {x n } converges to strongly to a fixed point of F. The following result is immediate sequel of our strong convergence theorem. Corollary 3.1. Let K be a nonempty closed convex subset of a uniformly convex Banach space E. Suppose T be a nonexpansive mapping of K. Let {x n } be defined by the iteration
(1.2),
where
{α n }
is
real
sequences in [0, 1] for all n ∈ N . If F(T) ≠ φ , then {x n } converges strongly to a fixed point of T. Corollary 3.2. Let K be a nonempty closed convex subset of a uniformly convex Banach space E. Suppose T be a nonexpansive mapping of K. Let {x n } be defined by the
iteration (1.3), where {α n } and {β n } are real sequences in [0, 1] for all n ∈ N . If F(T) ≠ φ , then {x n } converges strongly to a fixed point of T. Remark 3.1.
(1) Lemma 3.1 improves lemma 2.2 due to Thianwan12 in the case of one mapping in the uniformly convex Banach space.
280
(2) Theorem 3.1 improves Theorem 3.2 of Takahasi and Tamura14 and also Theorem 3.1. of Khan and Fukharuddin9 in the case of one mapping. REFERENCES 1. H. F. Senter and W.G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44, 375-380 (1974). 2. H. Fukhar-ud-din and S. H. Khan, Convergence of iterate with errors of asymptotically quasi-nonexpansive mappings and application, J. Math. Anal. Appl. 328, 821-829 (2007). 3. H.K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. TMA 16, No.2, 1127-1138 (1991). 4. H.Y. Zhou, G.T. Guo, H.J. Hwang, and Y.J. Cho, On the iterative methods for nonlinear operator equations in Banach space, Proc.Amer. Math. J., 14(4), 61-68 (2004). 5. J. Gornicki, Nonlinear ergodic theorems for asymptotically nonexpansive mappings in Banach space satisfying opial's condition, J. Math. Anal. Appl. 161, 440-446 (1991). 6. K. Goebel and W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35, 171-174 (1972). 7. K. K. Tan and H.K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178, 301-308 (1993). 8. L.C. Zeng, A note on approximating fixed points of nonexpansive mappings
Journal of Computer and Mathematical Sciences Vol. 5, Issue 3, 30 June, 2014 Pages (258-331)
281
9.
10.
11.
12.
C. L. Dewangan, et al., J. Comp. & Math. Sci. Vol.5 (3), 274-281 (2014)
by the Ishikawa iteration process, J. Math. Anal. Appl. 276, 245-250 (1981). S.H. Khan and H. Fukhar-ud-din, Weak and strong convergence of a scheme with errors for two nonexpansive mappings, Nonlinear Anal. 61, No.8, 1295-1301 (2005). S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44, 147-150 (1974). S. Ishikawa, Fixed points and iteration of nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59, 65-71 (1976). S. Thianwan, Common fixed points of the new iterations for two asymptotically nonexpansive nonself-mappings in a
13.
14.
15. 16.
Banch space, J. Omp. Appl. Math. 224, 688-695 (2009). S. Reich, Weak convergence theorems for nonexpansive mappings in Banach space, J. Math. Anal. Appl. 67, 274-276 (1979). W. Takahashi and T. Tamura, Convergence theorems for a pair of nonexpansive mappings, J. Convex Anal. 5, No.1, 45-56 (1998). W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4, 506510 (1953). Z. Opial, Weak and convergence of successive approximations for nonexpansive mapping, Bull. Amer. Math. Soc. 73, 591-597 (1967).
Journal of Computer and Mathematical Sciences Vol. 5, Issue 3, 30 June, 2014 Pages (258-331)