Paper24-Acta26-2011. 139KB Jun 04 2011 12:03:16 AM
Acta Universitatis Apulensis
ISSN: 1582-5329
No. 26/2011
pp. 225-236
CONVERGENCE AND STABILITY RESULTS FOR SOME
ITERATIVE SCHEMES
Memudu Olaposi Olatinwo
Abstract. We obtain strong convergence results for quasi-contractive operators
in arbitrary Banach space via some recently introduced iterative schemes and also
establish stablity theorems for Kirk’s iterative process. Our results generalize and
extend some well-known results in the literature.
2000 Mathematics Subject Classification: 47H06, 47H10.
Keywords: strong convergence results; quasi-contractive operators, arbitrary Banach space.
1. Introduction
Our purpose in this paper is to obtain some strong convergence results for quasicontractive operators in arbitrary Banach space via the iterative schemes introduced
in [13, 14]. We also establish stablity theorems for Kirk’s iterative process. The
convergence results obtained are generalizations and extensions of those of [3, 8, 9,
18, 19] while our stability results generalize some of the results of Rhoades [20, 21]
and the results of the author [12].
Let (E, ||.||) be a normed linear space and T : E → E a selfmap of E. Suppose
that FT = { p ∈ E | T p = p } is the set of fixed points of T.
In a normed linear space or a Banach space setting, we have several iterative
processes that have been defined by many researchers to approximate the fixed
points of different operators. Some of them are the following:
For x0 ∈ E, define the sequence {xn }∞
n=0 by
xn+1 =
k
X
αi T i xn , x0 ∈ E, n = 0, 1, 2, · · · ,
i=0
k
X
i=0
225
αi = 1,
(1)
M.O. Olatinwo - Convergence and stability results for some iterative schemes
αi ≥ 0, α0 6= 0, αi ∈ [0, 1], where k is a fixed integer. See Kirk [11] for this iterative
process.
Another iterative process of interest is the Ishikawa scheme defined as follows:
For x0 ∈ E, define the sequence {xn }∞
n=0 by
xn+1 = (1 − αn )xn + αn T zn ,
n = 0, 1, · · · ,
(2)
zn = (1 − βn )xn + βn T xn
For the iterative process in (2), see Ishikawa [7].
The following two iterative processes have been recently introduced in [13, 14]:
(I) For x0 ∈ E, define the sequence {xn }∞
n=0 by
P
Pk
xn+1 P
= αn,0 xn + ki=1P
αn,i T i zn ,
i=0 αn,i = 1, n = 0, 1, 2, · · · ,
s
zn = sj=0 βn,j T j xn ,
j=0 βn,j = 1,
(3)
k ≥ s, αn,i ≥ 0, αn,0 6= 0, βn,j ≥ 0, βn,0 6= 0, αn,i , βn,j ∈ [0, 1], where k and s are
fixed integers.
(II) For x0 ∈ E, define the sequence {xn }∞
n=0 by
P
Pk
xn+1 P
= αn,0 xn + ki=1
αn,i Ti zn ,
i=0 αn,i = 1, n = 0, 1, 2, · · · ,
P
s
zn = sj=0 βn,j Tj xn ,
j=0 βn,j = 1,
(4)
k ≥ s, αn,i ≥ 0, αn,0 6= 0, βn,j ≥ 0, βn,0 6= 0, αn,i , βn,j ∈ [0, 1], where k and s are
fixed integers and S0 is an identity operator.
Remark 1.1 It has been shown in [13, 14] that the iterative algorithms defined
in (3) and (4) generalize many well-known schemes in the literature.
Definition 1.1 [6].Let (E, d) be a complete metric space, T : E → E a selfmap
of E. Suppose that FT 6= φ (i.e. nonempty) is the set of fixed points of T. Let
{xn }∞
n=0 ⊂ E be the sequence generated by an iterative procedure involving T which
is defined by
xn+1 = f (T, xn ), n = 0, 1, · · · ,
(⋆)
where x0 ∈ E is the initial approximation and f is some function. Suppose {xn }∞
n=0
converges to a fixed point p of T . Let {yn }∞
⊂
E
and
set
n=0
ǫn = d(yn+1 , f (T, yn )), (n = 0, 1, · · · ). Then, the iterative procedure (⋆) is said to be
T -stable or stable with respect to T if and only if lim ǫn = 0 implies lim yn = p.
n→∞
n→∞
Remark 1.2. (i) Since the metric is induced by the norm, we have ǫn = ||yn+1 −
f (T, yn )||, n = 0, 1, · · · , in place of ǫn = d(yn+1 , f (T, yn )), n = 0, 1, · · · , in the
definitions of stability stated above whenever we are working in normed linear space
or Banach space.
226
M.O. Olatinwo - Convergence and stability results for some iterative schemes
P
(ii) we obtain the Kirk’s iterative process from (⋆) if f (T, xn ) = ki=0 αi T i xn , n =
Pk
0, 1, 2, · · · ,
i=0 αi = 1, where αi ≥ 0, α0 6= 0, αi ∈ [0, 1] and k is a fixed integer.
Any other iterative process can be obtained in a similar manner from (⋆).
Several stability results established in metric spaces and normed linear spaces
are available in the literature. Some of the various authors whose contributions are
of colossal value in the study of stability of the fixed point iterative procedures are
Ostrowski [17], Harder and Hicks [6], Rhoades [18, 20], Osilike and Udomene [16],
and Berinde [3, 4]. The first stability result on T − stable mappings was due to
Ostrowski [17].
Definition 1.2 [4, 22] (a) A function ψ : R+ → R+ is called a comparison function if it satisfies the following conditions:
(i) ψ is monotone increasing;
(ii) lim ψ n (t) = 0, ∀ t ≥ 0.
n→∞
Remark 1.3. (i) Every comparison function satisfies ψ(0) = 0.
(ii) ψ n (t) is the n−th iterate of ψ(t).
We shall employ the following contractive definitions: Let E be an arbitrary Banach
space,
(i) for an operator T : E → E, there exist a ∈ [0, 1) and a monotone increasing
function ϕ : R+ → R+ , with ϕ(0) = 0, such that
||T x − T y|| ≤ ϕ(||x − T x||) + a||x − y||, ∀ x, y ∈ E.
(5)
(ii) also, for operators Ti : E → E, i = 0, 1, 2, · · · , k there exist ai ∈ [0, 1), i =
0, 1, 2, · · · , k, and a monotone increasing function ϕ : R+ → R+ , with ϕ(0) = 0,
such that
||Ti x − Ti y|| ≤ ϕ(||x − Ti x||) + ai ||x − y||, ∀ x, y ∈ E,
(6)
where T0 =identity operator.
Other forms of the contractive conditions which shall be used are given in two
of our Lemmas in the sequel:
Lemma 1.1 [12]. Let (E, || · ||) be a normed linear space and let T : E → E
be a selfmap of E satisfying (5), where ϕ : R+ → R+ is a sublinear, monotone
increasing function such thatPϕ(0) =0, ϕ(Lu) ≤ Lϕ(u), L ≥ 0. Then, ∀ i ∈ N, and
∀ x, y ∈ E, ||T i x − T i y|| ≤ ij=1 ji bi−j ϕj (||x − T x||) + bi ||x − y||, ∀ x, y ∈ E.
n
Lemma 1.2 [15]. Let {ψ k (t)}P
k=0 be a sequence of comparison functions. Then,
n
j
any convex linear combination
Pn j=0 cj ψ (t) of the comparison functions is also a
comparison function, where j=0 cj = 1 and co , c1 , · · · , cn are positive constants.
Lemma 1.3 [15]. If ψ : R+ → R+ is a subadditive comparison function and {ǫn }∞
n=0
227
M.O. Olatinwo - Convergence and stability results for some iterative schemes
is a sequence of positive numbers such that lim ǫn = 0, then for any sequence of
n→∞
positive numbers {un }∞
n=0 satisfying
un+1 ≤
m
X
δk ψ k (un ) + ǫn , n = 0, 1, · · · ,
(7)
k=0
where δ0 , δ1 , · · · , δm ∈ [0, 1] with 0 ≤
Pm
k=0 δk
≤ 1, we have lim un = 0.
n→∞
Lemma 1.4. Let (E, ||·||) be a normed linear space and let T : E → E be a selfmap
of E satisfying
||T x − T y|| ≤ ϕ(||x − T x||) + ψ(||x − y||), ∀ x, y ∈ E,
(8)
where ψ : R+ → R+ is a sublinear comparison function and ϕ : R+ → R+ , a sublinear monotone increasing function such that ϕ(0) = 0 and ψ s (ϕr (x)) ≤ ϕr (ψ s (x)),
∀ x ∈ R+ , r, s ∈ N. Then, ∀ i ∈ N, we have
i
i
||T x − T y|| ≤
i
X
i
j=1
j
ϕj (ψ i−j (||x − T x||)) + ψ i (||x − y||), ∀ x, y ∈ E.
(9)
Proof: We first proof the sublinearity of both ϕ and ψ as follows: In order to show
that ψ i (i.e. iterate of ψ) is sublinear, we have to show that ψ i is both subadditive
and positively homogeneous. We first establish that ψ subadditive implies that each
iterate ψ i of ψ is also subadditive: Since ψ is subadditive, we have ψ(x + y) ≤
ψ(x) + ψ(y), ∀ x, y ∈ R+ . Therefore,using subadditivity of ψ in ψ 2 yields
ψ 2 (x + y) = ψ(ψ(x + y)) ≤ ψ(ψ(x) + ψ(y)) ≤ ψ(ψ(x)) + ψ(ψ(y)) = ψ 2 (x) + ψ 2 (y),
which implies that ψ 2 is subadditive. Similarly, applying subadditivity of ψ 2 in ψ 3 ,
we get
ψ 3 (x+y) = ψ(ψ 2 (x+y)) ≤ ψ(ψ 2 (x) + ψ 2 (y)) ≤ ψ(ψ 2 (x))+ψ(ψ 2 (y)) = ψ 3 (x) + ψ 3 (y),
which implies that ψ 3 is also subadditive. Hence, in general, each ψ n , n = 1, 2, · · · ,
is subadditive. We now prove that ψ positively homogeneous implies that each
iterate ψ i of ψ is also positively homogeneous: Therefore, we have that ψ(αx) =
αψ(x), ∀ x ∈ R+ , α > 0. Using positive homogeneity of ψ in ψ 2 , we have
ψ 2 (αx) = ψ(ψ(αx)) = ψ(αψ(x)) = αψ(ψ(x)) = αψ 2 (x), ∀ x ∈ R+ , α > 0,
which implies that ψ 2 is positively homogeneous. Hence, in general, each ψ n , n =
1, 2, · · · , is positively homogeneous. Thus, we have that ψ n , n = 1, 2, · · · , is sublinear. In a similar manner, we can prove that ϕj , j = 1, 2, · · · , is sublinear.
228
M.O. Olatinwo - Convergence and stability results for some iterative schemes
The second part of the proof of this lemma is by induction on i as follows: If
i = 1, then (9)Pbecomes
||T x−T y|| ≤ 1j=1 1j ϕj (ψ 1−j (||x−T x||))+ψ(||x−y||) = ϕ(||x−T x||)+ψ(||x−y||),
that is, (9) reduces to (8) when i = 1 and hence the result holds. Assume as an
inductive hypothesis that (9) holds for i = m, m ∈ N, i.e.
m
m
||T x − T y|| ≤
m
X
m
j
j=1
ϕj (ψ m−j (||x − T x||)) + ψ m (||x − y||), ∀ x, y ∈ E.
We then show that the statement is true for i = m + 1;
||T m+1 x − T m+1 y|| = P
||T m (T x) − T m (T y)||
m
j
m−j (||T x − T 2 x||)) + ψ m (||T x − T y||)
≤ m
j=1 j ϕ (ψ
Pm
j
m−j (ϕ(||x − T x||) + ψ(||x − T x||)))
≤ j=1 m
j ϕ (ψ
+ ψ m (ϕ(||x
− T x||) + ψ(||x − y||))
j m−j
Pm
≤ j=1 m
ϕ
[ψ
(ϕ(||x − T x||)) + ψ m+1−j (||x − T x||)]
j
+ ψ m (ϕ(||x
T x||)) + ψ m+1 (||x − y||)
j − m−j
Pm
m
≤ j=1 j ϕ [ϕ(ψ
(||x − T x||)) + ψ m+1−j (||x − T x||)]
+ ϕ(ψ m(||x − T x||)) + ψ m+1 (||x − P
y||)
j m+1−j
P
m
m
m
j+1 (ψ m−j (||x − T x||)) +
≤ m
ϕ
(||x
j=1 j
j=1 j ϕ (ψ
m
m+1
−T x||)) + ϕ(ψ (||x − T x||))
(||x − y||)
+ ψm m
m+1 (||x − T x||) + [ m
= m
ϕ
+
]ϕ
(ψ(||x − T x||))
m
m−1
m
m−1
m
m
2
+ [ m−2 +
m−1
]ϕ m (ψ (||x − T x||)) m+1
m
+ ·· · + [ m
+
(||x − y||)
1
0 ]ϕ(ψ (||x − T x||)) + ψ
m+1
m+1
m+1
m
= m+1 ϕ
(||x − T x||) + m ϕ (ψ(||x − T x||))
m−1 2
m+1
+ m−1 ϕ
(ψ (||x − T x||)) + · · ·
+ m+1
ϕ2 (ψ m−1 (||x − T x||)) + m+1
ϕ(ψ m (||x − T x||))
2
1
+ ψ m+1 (||x − y||)
Pm+1 m+1 j m+1−j
= j=1
ϕ (ψ
(||x − T x||)) + ψ m+1 (||x − y||).
j
Lemma 1.5. Let (E, || · ||) be a normed linear space and let T : E → E be a
selfmap of E satisfying
||T x − T y|| ≤ L||x − T x|| + ψ(||x − y||), ∀ x, y ∈ E, L ≥ 0,
(10)
where ψ : R+ → R+ is a sublinear comparison function. Then, ∀ i ∈ N, we have
i
i
||T x − T y|| ≤
i
X
i
j=1
j
Lj ψ i−j (||x − T x||) + ψ i (||x − y||), ∀ x, y ∈ E.
229
(11)
M.O. Olatinwo - Convergence and stability results for some iterative schemes
Proof: The proof of sublinearity of ψ i (i.e. iterate of ψ) for each i = 1, 2, · · · , is
the same as that of Lemma 1.4. The second part of the proof of this lemma is by
induction on i:PIf i = 1,
(11) becomes
then
1
1
j
1−j
||T x − T y|| ≤ j=1 j L ψ (||x − T x||) + ψ(||x − y||) = L||x − T x|| + ψ(||x − y||),
that is, (11) reduces to (10) when i = 1 and hence the result holds. Assume as an
inductive hypothesis that (11) holds for i = m, m ∈ N, i.e.
m
m
||T x − T y|| ≤
m
X
m
j=1
j
Lj ψ m−j (||x − T x||) + ψ m (||x − y||), ∀ x, y ∈ E.
We then show that the statement is true for i = m + 1;
m
||T m+1 x − T m+1 y|| = ||T
x) − T m (T y)||
Pm (T m
≤ j=1 j Lj ψ m−j (||T x − T 2 x||) + ψ m (||T x − T y||)
j m−j
P
m
≤ m
(L||x − T x|| + ψ(||x − T x||))
j=1 j L ψ
m
− T x|| + ψ(||x − y||))
P + ψm(L||x
Pm m j m+1−j
j+1 ψ m−j (||x − T x||) +
≤ m
L
(||x
j=1 j
j=1 j L ψ
m
m+1
− T x||) + Lψ (||x − T x||)
− y||)
+ ψm (||x
m
m
m+1
m
= m L
||x
m−1 + m ]L ψ(||x − T x||)
− T x||
+ [ m−1
m
m
+ [ m−2
+
]L
ψ 2 (||x − T x||) + · · ·
m−1
m
m
m
m
+ [ 1 + 2 ]L2 ψ m−1 (||x − T x||) + [ m
1 + 0 ]Lψ (||x
m+1
− Tx||) + ψ
(||x − y||)
m+1
m+1
m
= m+1 L
||x − T x|| + m+1
m L ψ(||x − T x||)
m+1
m−1
2
+ m−1 L
ψ (||x − T x||) + · · ·
2 m−1
m
m+1
+ 2 L ψ
(||x − T x||) + m+1
Lψ (||x − T x||)
1
+ ψ m+1(||x − y||)
P
m+1
= m+1
Lj ψ m+1−j (||x − T x||) + ψ m+1 (||x − y||).
j=1
j
Remark 1.4. In addition to both (5) and (6), the contractive conditions (8)
and (10) shall also be used to prove some of our results.
2. Main results
We now establish some convergence results:
Theorem 2.1.Let E be an arbitrary Banach space, K a closed convex subset of E
and T : K → K an operator satisfying (5), where ϕ : R+ → R+ is a subadditive
monotone increasing function such that ϕ(0) = 0, ϕ(Lu) ≤ Lϕ(u), u ∈ R+ . Let
∞
x0 ∈ K, {xn }∞
n=0 be the iterative process defined by (3). Then, {xn }n=0 converges
strongly to the fixed point p of T.
Proof: We shall employ Lemma 1.1 and the triangle inequality to establish that
230
M.O. Olatinwo - Convergence and stability results for some iterative schemes
lim xn = p. However, we shall first establish that T satisfying the given contractive
condition has a unique fixed point. Suppose that there exist p1 , p2 ∈ FT , p1 6= p2 ,
with ||p1 − p2 || > 0. Therefore, we have
P
0 < ||p1 − p2 || = ||T i p1 − T i p2 || ≤ ij=1 ji ai−j ϕj (||p1 − T p1 ||) + ai ||p1 − p2 ||
P
= ij=1 ji ai−j ϕj (0) + ai ||p1 − p2 ||,
n→∞
from which we have that (1 − ai )||p1 − p2 || ≤ 0. Since a ∈ [0, 1), then, 1 − ai > 0 and
||p1 − p2 || ≤ 0.
Also, since norm is nonnegative we have ||p1 − p2 || = 0. That is, p1 = p2 .
Thus, T has a unique fixed point.
We now prove that {xn }∞
n=0 converges strongly to a fixed point of T : Then, we
have
P
i
i
||xn+1 − p|| ≤ ki=1 αn,i ||T
o
nPp − T zn || + αn,0 ||xn − p||
Pk
i
i i−j j
i ||p − z || + α
a
ϕ
(||p
−
T
p||)
+
a
≤ i=1 αn,i
n
n,0 ||xn − p||
P
j=1 j
k
=
αn,i ai ||p − zn || + αn,0 ||xn − p||
Pi=1
P
Ps
s
k
r
r
i ||
α
a
=
r=0 βn,r T xn || + αn,0 ||xn − p||
r=0 βn,r T p −
i=1 n,i
P
P
k
s
i
r
r
≤
i=1 αn,i a { r=1 βn,r ||T p − T xn || + βn,0 ||xn − p||}
αn,0 ||xn −p||
P
+
Pk
Ps
k
i β
i
r p − T r x || +
α
a
α
a
β
||T
=
n
n,0 ||xn − p||
i=1 n,i
i=1 n,i
r=1 n,r
p||
P+ αn,0 ||xn−
Pr
Ps
k
r r−j j
i
ϕ (||p − T p||) + ar ||p − xn || ]
≤
j=1 j a
r=1 βn,r [
i=1 αn,i a
P
k
i
+
i=1 αn,i a βn,0 ||xn − p|| + αn,0 ||xn − p||
P
Pk
= [( i=1 αn,i ai )( sr=0 βn,r ar ) + αn,0 ]||xn − p||
P
P
≤ Πnν=0 [ ( ki=1 αν,i ai )( sr=0 βν,r ar ) + αν,0 ]||x0 − p|| → 0 as n → ∞. (12)
P
P
We claim that 0 ≤ ( ki=1 αν,i ai )( sr=0 βν,r ar ) + αν,0 < 1 as follows:
Pk
Pk
i ≤
i
2
k
i=1 |αν,i a | = |αν,1 a| + |αν,2 a | + · · · + |αν,k a |
i=1 αν,i a
= |αν,1 ||a| + |αν,2 ||a|2 + · · · + |αν,k ||a|k
< |αν,1 | + |αν,2 | + · · · + |αν,k | = αν,1 + αν,2 + · · · + αν,k
= 1 − αν,0 , αν,0 ∈ (0, 1).
Similarly, we have that
Ps
Ps
r ≤
r
2
s
r=0 βν,r a
r=0 |βν,r a | = |βν,0 | + |βν,1 a| + |βν,2 a | + · · · + |βν,k a |
2
s
= |βν,0 | + |βν,1 ||a| + |βν,2 ||a| + · · · + |βν,s ||a|
< |βν,0 | + |βν,1 | + · · · + |βν,s | = βν,0 + βν,1 + · · · + βν,s = 1.
231
M.O. Olatinwo - Convergence and stability results for some iterative schemes
P
P
Therefore, ( ki=1 αν,i ai )( sr=0 βν,r ar ) + αν,0 < 1 − αν,0 + αν,0 = 1.
Hence, we obtain from (12) that ||xn+1 − p|| → 0 as n → ∞,
i.e. {xn }∞
n=0 converges strongly to p.
Theorem 2.2.Let E be an arbitrary Banach space, K a closed convex subset of E
and Ti : K → K, (i = 0, 1, · · · , k), selfoperators satisfying (6), where ϕ : R+ → R+
is a monotone increasing function such that ϕ(0) = 0. Let x0 ∈ K, {xn }∞
n=0 be
the iterative process defined by (4). Then, {xn }∞
converges
strongly
to
the
unique
n=0
common fixed point p of Ti (for each i).
Proof. Let FTi be the set of common fixed points of Ti (i = 0, 1, · · · , k). Suppose
that there exist p1 , p2 ∈ FTi , p1 6= p2 , with ||p1 − p2 || > 0. Therefore, we have
0 < ||p1 − p2 || = ||Ti p1 − Ti p2 || ≤ ϕ(||p1 − Ti p1 ||) + ai ||p1 − p2 ||
= ai ||p1 − p2 ||,
from which it follows that we have that p1 = p2 . That is, Ti (i = 0, 1, · · · , k) have a
unique common fixed point.
We now prove that {xn }∞
n=0 converges strongly to a fixed point of Ti .
Then, we have
P
||xn+1 − p|| ≤ ki=1 αn,i ||Ti p − Ti zn || + αn,0 ||xn − p||
P
≤ ki=1 αn,i {ϕ(||p
− Ti p||) + ai ||p − zn ||} + αn,0 ||xn − p||
Pk
αn,i ai ||p − zn || + αn,0 ||xn − p||
=
Pi=1
P
P
k
s
=
βn,r Tr p − sr=0 βn,r Tr xn || + αn,0 ||xn − p||
i=1 αn,i ai ||
P r=0
P
k
αn,i ai { sr=0 βn,r ||Tr p − Tr xn ||} + αn,0 ||xn − p||
≤
Pi=1
P
k
s
α
a
≤
i=1 n,i i
r=0 βn,r [ϕ(||p − Tr p||) + ar ||p − xn || ] + αn,0 ||xn − p||
Pk
Ps
= [ ( i=1 αn,i ai )( r=0 βn,r ar ) + αn,0 ]||xn − p||
P
P
≤ Πnν=0 [ ( ki=1 αν,i ai )( sr=0 βν,r ar ) + αν,0 ]||x0 − p|| → 0 as n → ∞, (13)
P
P
where as in Theorem 2.1, we can show that 0 ≤ ( ki=1 αν,i ai )( sr=0 βν,r ar ) + αν,0 <
1.
Hence, we obtain from (13) that ||xn+1 − p|| → 0 as n → ∞,
i.e. {xn }∞
n=0 converges strongly to p.
Theorem 2.3.Let E be an arbitrary Banach space, K a closed convex subset of
E, and T : K → K an operator satisfying
||T x − T y|| ≤
ϕ(||x − T x||) + a||x − y||
, ∀ x, y ∈ E, a ∈ [0, 1), M ≥ 0,
1 + M ||x − T x||
232
(14)
M.O. Olatinwo - Convergence and stability results for some iterative schemes
where ϕ : R+ → R+ is a monotone increasing function such that ϕ(0) = 0. Let
∞
x0 ∈ K, {x
Pn∞}n=0 defined by (2) be the Ishikawa iterative process with αn , βn ∈
[0, 1] and
k=0 αk = ∞. Then, the Ishikawa iterative process converges strongly to
the fixed point of T.
Proof. We shall first establish that T has a unique fixed point by using condition
(14): Suppose not. Then, there exist x∗ , y ∗ ∈ FT , x∗ 6= y ∗ and ||x∗ − y∗ || > 0.
Therefore, we have
−T x ||)+a||x −y
0 < ||x∗ − y ∗ || = ||T x∗ − T y ∗ || ≤ ϕ(||x 1+M
||x∗ −T x∗ ||
∗
= a||x − y ∗ ||
∗
∗
∗
∗ ||
from which it follows that (1 − a)||x∗ − y ∗ || ≤ 0, which leads to 1 − a > 0 ( since a ∈
[0, 1) ), but ||x∗ − y ∗ || ≤ 0 (which is a contradiction).
Therefore, since norm is nonnegative, ||x∗ − y ∗ || = 0 i.e. x∗ = y ∗ = p,
thus proving the uniqueness of the fixed point for T. Hence, FT = {p} .
We now prove that {xn }∞
n=0 converges strongly to the fixed point p using condition (14). Therefore, we have
||xn+1 − p|| ≤ ||(1 − αn )xn + αn T zn − (1 − αn + αn )p||
≤ (1 − αn )||xn − p|| + aαn ||p − zn ||
≤ [1 − αn (1 − a) − aαn βn (1 − a)]||xn − p||
≤ [1 − (1 − a)αn ]||xn − p||
≤ Πnk=0 [1 − (1 − a)αk ]||x0 − p||
k ||x − p||
≤ Πnk=0 e−(1−a)α
0
Pn
−[(1−a)
α
k=0 k ] ||x − p|| → 0 as n → ∞,
=e
0
P∞
since k=0 αk = ∞ and a ∈ [0, 1). Hence, we obtain from (15) that
||xn+1 − p|| → 0 as n → ∞, i.e. {xn }∞
n=0 converges strongly to p.
(15)
Remark 2.1. If in each of Theorem 2.1 and Theorem 2.2, the iterative processes
defined by
k
k
X
X
i
xn+1 =
αn,i T xn ,
αn,i = 1, n = 0, 1, 2, · · · ,
(16)
i=0
i=0
αn,i ≥ 0, αn,0 6= 0, αn,i ∈ [0, 1], where k is a fixed integer;
and
k
k
X
X
xn+1 =
αn,i Ti xn ,
αn,i = 1, n = 0, 1, 2, · · · ,
i=0
(17)
i=0
αn,i ≥ 0, αn,0 6= 0, αn,i ∈ [0, 1], where k is a fixed integer and T0 =identity operator;
are employed, then we obtain corresponding results for the one-step processes defined
in (16) and (17). Again, we refer to [13, 14] for these iterative processes too.
233
M.O. Olatinwo - Convergence and stability results for some iterative schemes
Remark 2.2. Theorem 2.1, Theorem 2.2 and Theorem 2.3 are generalizations
and extensions of both Theorem 1 and Theorem 2 of Berinde [3], Theorem 2 and
Theorem 3 of Kannan [8], Theorem 3 of Kannan [9], Theorem 4 of Rhoades [18] as
well as Theorem 8 of Rhoades [19]. See also Berinde [4] for the results of Rhoades
[18, 19].
We prove the following stability results:
Theorem 2.4.Let (E, ||.||) is a normed linear space and T : E → E a selfmap
of E satisfying (8). Let x0 ∈ E and {xn }∞
n=0 defined by (1) be the Kirk’s iterative
process. Suppose that T has a fixed point p. Let ψ : R+ → R+ be a continuous
sublinear comparison function and ϕ : R+ → R+ a sublinear monotone increasing
function such that ϕ(0) = 0 and ψ s (ϕr (x)) ≤ ϕr (ψ s (x)), ∀ x ∈ R+ , r, s ∈ N. Then,
the Kirk iteration process is T -stable.
Pk
i
Proof. Let {yn }∞
lim ǫn = 0.
n=0 ⊂ E and ǫn = ||yn+1 −
i=0 αi T yn ||. Let n→∞
Then, we shall prove that lim yn = p. By using both Lemma 1.4 and the triangle
n→∞
inequality we have that:
P
P
||yn+1 − p|| ≤ ||yn+1 − ki=0 αi T i yn || + || ki=0 αi T i yn − p||
P
P
= ǫn + || ki=0 αi T i yn − ki=0 αi T i p||
Pk
= ǫn + || i=0 αi (T i yn − T i p)||
P
≤ ki=0 αi ||T i p − T i yn || + ǫn
P
= α0 ||p − yn || + ki=1 αi ||T i p − T i yn || + ǫn
Pk
≤ i=0 αi ψ i (||yn − p||) + ǫn ,
(18)
where ψ i−j (0) = ψ(0) = 0 and ϕj (0)
ϕ(0) = 0.
P=
k
i
We have by Lemma 1.2 that
i=0 αi ψ (||yn − p||) is a comparison function.
Therefore, using Lemma 1.3 in (18) yields lim ||yn − p|| = 0, that is, lim yn = p.
n→∞
n→∞
Conversely, let lim yn = p. Then, by Lemma 1.4 and the triangle inequality, we
n→∞
have
P
ǫn = ||yn+1 − ki=0 αi T i yn ||
P
≤ ||yn+1 − p|| + ||p − ki=0 αi T i yn ||
P
P
= ||yn+1 − p|| + || ki=0 αi T i p − ki=0 αi T i yn ||
Pk
= ||yn+1 − p|| + || i=0 αi (T i p − T i yn )||
P
≤ ||yn+1 − p|| + ki=0 αi ||T i p − T i yn ||
P
= ||yn+1 − p|| + α0 ||p − yn || + ki=1 αi ||T i p − T i yn ||
Pk
≤ ||yn+1 − p|| + i=0 αi ψ i (||yn − p||) → 0 as n → ∞.
Theorem 2.5.Let (E, ||.||) is a normed linear space and T : E → E a selfmap
of E satisfying (10). Let x0 ∈ E and {xn }∞
n=0 defined by (1) be the Kirk’s iterative
234
M.O. Olatinwo - Convergence and stability results for some iterative schemes
process. Suppose that T has a fixed point p. Let ψ : R+ → R+ be a continuous
sublinear comparison function and ϕ : R+ → R+ a sublinear monotone increasing
function such that ϕ(0) = 0. Then, the Kirk iteration process is T -stable.
Proof. The proof of this result is similar to that of Theorem 2.4 except for the
application of Lemma 1.2, Lemma 1.3 and Lemma 1.5.
Remark 2.3. Our stability results generalize some of the results of Rhoades
[20, 21]. In particular, both Theorem 2.4 and Theorem 2.5 are generalizations of
the results of Olatinwo [12].
References
[1] R. P. Agarwal, D. O’Regan and D. R. Sahu; Fixed point theory for Lipschitziantype mappings with applications - Topological fixed point theory 6, Springer Science+Bussiness Media (www.springer.com) (2009).
[2] V. Berinde; On The Stability of Some Fixed Point Procedures, Bul. Stiint.
Univ. Baia Mare, Ser. B, Matematica-Informatica, Vol. XVIII (1) (2002), 7-14.
[3] V. Berinde; On the convergence of the Ishikawa iteration in the class of quasicontractive operators, Acta Math. Univ. Comenianae Vol. LXXIII (1) (2004),
119-126.
[4] V. Berinde; Iterative approximation of fixed points, Springer-Verlag Berlin
Heidelberg (2007).
[5] Lj. B. Ciric; Some recent results in metrical fixed point theory, C-PRINT,
Beograd (2003).
[6] A. M. Harder and T. L. Hicks; Stability Results for Fixed Point Iteration
Procedures, Math. Japonica 33 (5) (1988), 693-706.
[7] S. Ishikawa; Fixed Point by a New Iteration Method, Proc. Amer. Math. Soc.
44 (1) (1974), 147-150.
[8] R. Kannan (1971); Some results on fixed points III, Fund. Math. 70 (2),
169-177.
[9] R. Kannan; (1973); Construction of fixed points of a class of nonlinear mappings, J. Math. Anal. Appl. 41, 430-438.
[10] M. A. Khamsi and W. A. Kirk; An introduction to metric spaces and fixed
point theory, John Wiley & Sons, Inc. (2001).
[11] W. A. Kirk; On successive approximations for nonexpansive mappings in
Banach spaces, Glasgow Math. J. 12 (1971), 6-9.
[12] M. O. Olatinwo, O. O. Owojori and C. O. Imoru; Some stability results for
fixed point iteration processes, Aus. J. Math. Anal. Appl. 3 (2) (2006), Article 8,
1-7.
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M.O. Olatinwo - Convergence and stability results for some iterative schemes
[13] M. O. Olatinwo; Some stability results for two hybrid fixed points iterative
algorithms of Kirk-Ishikawa and Kirk-Mann type, J. Adv. Math. Studies 1 (2008),
No. 1-2, 87-96.
[14] M. O. Olatinwo; Some stability results for two hybrid fixed points iterative
algorithms in normed linear space, Math. Vesnik 61 (4) (2009), 247-256.
[15] M. O. Olatinwo; Some Unifying Results on Stability and Strong Convergence for Some New Iteration Processes, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis 25 (1) (2009), 105-118.
[16] M. O. Osilike and A. Udomene; Short Proofs of Stability Results for Fixed
Point Iteration Procedures for a Class of Contractive-type Mappings, Indian J.Pure
Appl. Math. 30 (12) (1999), 1229-1234.
[17] A. M. Ostrowski; The Round-off Stability of Iterations, Z. Angew. Math.
Mech. 47 (1967), 77-81.
[18] B. E. Rhoades; Fixed point iteration using infinite matrices, Trans. Amer.
Math. Soc. 196 (1974), 161-176.
[19] B. E. Rhoades; Comments on two fixed point iteration methods, J. Math.
Anal. Appl. 56 (2) (1976), 741-750.
[20] B. E. Rhoades; Fixed Point Theorems and Stability Results for Fixed Point
Iteration Procedures, Indian J. Pure Appl. Math. 21 (1) (1990), 1-9.
[21] B. E. Rhoades; Fixed Point Theorems and Stability Results for Fixed Point
Iteration Procedures II, Indian J. Pure Appl. Math. 24 (11) (1993), 691-703.
[22] I. A. Rus; Generalized Contractions and Applications, Cluj Univ. Press,
Cluj Napoca, (2001).
Memudu Olaposi Olatinwo
Department of Mathematics
Obafemi Awolowo University
Ile-Ife, Nigeria.
E-mail: [email protected]
236
ISSN: 1582-5329
No. 26/2011
pp. 225-236
CONVERGENCE AND STABILITY RESULTS FOR SOME
ITERATIVE SCHEMES
Memudu Olaposi Olatinwo
Abstract. We obtain strong convergence results for quasi-contractive operators
in arbitrary Banach space via some recently introduced iterative schemes and also
establish stablity theorems for Kirk’s iterative process. Our results generalize and
extend some well-known results in the literature.
2000 Mathematics Subject Classification: 47H06, 47H10.
Keywords: strong convergence results; quasi-contractive operators, arbitrary Banach space.
1. Introduction
Our purpose in this paper is to obtain some strong convergence results for quasicontractive operators in arbitrary Banach space via the iterative schemes introduced
in [13, 14]. We also establish stablity theorems for Kirk’s iterative process. The
convergence results obtained are generalizations and extensions of those of [3, 8, 9,
18, 19] while our stability results generalize some of the results of Rhoades [20, 21]
and the results of the author [12].
Let (E, ||.||) be a normed linear space and T : E → E a selfmap of E. Suppose
that FT = { p ∈ E | T p = p } is the set of fixed points of T.
In a normed linear space or a Banach space setting, we have several iterative
processes that have been defined by many researchers to approximate the fixed
points of different operators. Some of them are the following:
For x0 ∈ E, define the sequence {xn }∞
n=0 by
xn+1 =
k
X
αi T i xn , x0 ∈ E, n = 0, 1, 2, · · · ,
i=0
k
X
i=0
225
αi = 1,
(1)
M.O. Olatinwo - Convergence and stability results for some iterative schemes
αi ≥ 0, α0 6= 0, αi ∈ [0, 1], where k is a fixed integer. See Kirk [11] for this iterative
process.
Another iterative process of interest is the Ishikawa scheme defined as follows:
For x0 ∈ E, define the sequence {xn }∞
n=0 by
xn+1 = (1 − αn )xn + αn T zn ,
n = 0, 1, · · · ,
(2)
zn = (1 − βn )xn + βn T xn
For the iterative process in (2), see Ishikawa [7].
The following two iterative processes have been recently introduced in [13, 14]:
(I) For x0 ∈ E, define the sequence {xn }∞
n=0 by
P
Pk
xn+1 P
= αn,0 xn + ki=1P
αn,i T i zn ,
i=0 αn,i = 1, n = 0, 1, 2, · · · ,
s
zn = sj=0 βn,j T j xn ,
j=0 βn,j = 1,
(3)
k ≥ s, αn,i ≥ 0, αn,0 6= 0, βn,j ≥ 0, βn,0 6= 0, αn,i , βn,j ∈ [0, 1], where k and s are
fixed integers.
(II) For x0 ∈ E, define the sequence {xn }∞
n=0 by
P
Pk
xn+1 P
= αn,0 xn + ki=1
αn,i Ti zn ,
i=0 αn,i = 1, n = 0, 1, 2, · · · ,
P
s
zn = sj=0 βn,j Tj xn ,
j=0 βn,j = 1,
(4)
k ≥ s, αn,i ≥ 0, αn,0 6= 0, βn,j ≥ 0, βn,0 6= 0, αn,i , βn,j ∈ [0, 1], where k and s are
fixed integers and S0 is an identity operator.
Remark 1.1 It has been shown in [13, 14] that the iterative algorithms defined
in (3) and (4) generalize many well-known schemes in the literature.
Definition 1.1 [6].Let (E, d) be a complete metric space, T : E → E a selfmap
of E. Suppose that FT 6= φ (i.e. nonempty) is the set of fixed points of T. Let
{xn }∞
n=0 ⊂ E be the sequence generated by an iterative procedure involving T which
is defined by
xn+1 = f (T, xn ), n = 0, 1, · · · ,
(⋆)
where x0 ∈ E is the initial approximation and f is some function. Suppose {xn }∞
n=0
converges to a fixed point p of T . Let {yn }∞
⊂
E
and
set
n=0
ǫn = d(yn+1 , f (T, yn )), (n = 0, 1, · · · ). Then, the iterative procedure (⋆) is said to be
T -stable or stable with respect to T if and only if lim ǫn = 0 implies lim yn = p.
n→∞
n→∞
Remark 1.2. (i) Since the metric is induced by the norm, we have ǫn = ||yn+1 −
f (T, yn )||, n = 0, 1, · · · , in place of ǫn = d(yn+1 , f (T, yn )), n = 0, 1, · · · , in the
definitions of stability stated above whenever we are working in normed linear space
or Banach space.
226
M.O. Olatinwo - Convergence and stability results for some iterative schemes
P
(ii) we obtain the Kirk’s iterative process from (⋆) if f (T, xn ) = ki=0 αi T i xn , n =
Pk
0, 1, 2, · · · ,
i=0 αi = 1, where αi ≥ 0, α0 6= 0, αi ∈ [0, 1] and k is a fixed integer.
Any other iterative process can be obtained in a similar manner from (⋆).
Several stability results established in metric spaces and normed linear spaces
are available in the literature. Some of the various authors whose contributions are
of colossal value in the study of stability of the fixed point iterative procedures are
Ostrowski [17], Harder and Hicks [6], Rhoades [18, 20], Osilike and Udomene [16],
and Berinde [3, 4]. The first stability result on T − stable mappings was due to
Ostrowski [17].
Definition 1.2 [4, 22] (a) A function ψ : R+ → R+ is called a comparison function if it satisfies the following conditions:
(i) ψ is monotone increasing;
(ii) lim ψ n (t) = 0, ∀ t ≥ 0.
n→∞
Remark 1.3. (i) Every comparison function satisfies ψ(0) = 0.
(ii) ψ n (t) is the n−th iterate of ψ(t).
We shall employ the following contractive definitions: Let E be an arbitrary Banach
space,
(i) for an operator T : E → E, there exist a ∈ [0, 1) and a monotone increasing
function ϕ : R+ → R+ , with ϕ(0) = 0, such that
||T x − T y|| ≤ ϕ(||x − T x||) + a||x − y||, ∀ x, y ∈ E.
(5)
(ii) also, for operators Ti : E → E, i = 0, 1, 2, · · · , k there exist ai ∈ [0, 1), i =
0, 1, 2, · · · , k, and a monotone increasing function ϕ : R+ → R+ , with ϕ(0) = 0,
such that
||Ti x − Ti y|| ≤ ϕ(||x − Ti x||) + ai ||x − y||, ∀ x, y ∈ E,
(6)
where T0 =identity operator.
Other forms of the contractive conditions which shall be used are given in two
of our Lemmas in the sequel:
Lemma 1.1 [12]. Let (E, || · ||) be a normed linear space and let T : E → E
be a selfmap of E satisfying (5), where ϕ : R+ → R+ is a sublinear, monotone
increasing function such thatPϕ(0) =0, ϕ(Lu) ≤ Lϕ(u), L ≥ 0. Then, ∀ i ∈ N, and
∀ x, y ∈ E, ||T i x − T i y|| ≤ ij=1 ji bi−j ϕj (||x − T x||) + bi ||x − y||, ∀ x, y ∈ E.
n
Lemma 1.2 [15]. Let {ψ k (t)}P
k=0 be a sequence of comparison functions. Then,
n
j
any convex linear combination
Pn j=0 cj ψ (t) of the comparison functions is also a
comparison function, where j=0 cj = 1 and co , c1 , · · · , cn are positive constants.
Lemma 1.3 [15]. If ψ : R+ → R+ is a subadditive comparison function and {ǫn }∞
n=0
227
M.O. Olatinwo - Convergence and stability results for some iterative schemes
is a sequence of positive numbers such that lim ǫn = 0, then for any sequence of
n→∞
positive numbers {un }∞
n=0 satisfying
un+1 ≤
m
X
δk ψ k (un ) + ǫn , n = 0, 1, · · · ,
(7)
k=0
where δ0 , δ1 , · · · , δm ∈ [0, 1] with 0 ≤
Pm
k=0 δk
≤ 1, we have lim un = 0.
n→∞
Lemma 1.4. Let (E, ||·||) be a normed linear space and let T : E → E be a selfmap
of E satisfying
||T x − T y|| ≤ ϕ(||x − T x||) + ψ(||x − y||), ∀ x, y ∈ E,
(8)
where ψ : R+ → R+ is a sublinear comparison function and ϕ : R+ → R+ , a sublinear monotone increasing function such that ϕ(0) = 0 and ψ s (ϕr (x)) ≤ ϕr (ψ s (x)),
∀ x ∈ R+ , r, s ∈ N. Then, ∀ i ∈ N, we have
i
i
||T x − T y|| ≤
i
X
i
j=1
j
ϕj (ψ i−j (||x − T x||)) + ψ i (||x − y||), ∀ x, y ∈ E.
(9)
Proof: We first proof the sublinearity of both ϕ and ψ as follows: In order to show
that ψ i (i.e. iterate of ψ) is sublinear, we have to show that ψ i is both subadditive
and positively homogeneous. We first establish that ψ subadditive implies that each
iterate ψ i of ψ is also subadditive: Since ψ is subadditive, we have ψ(x + y) ≤
ψ(x) + ψ(y), ∀ x, y ∈ R+ . Therefore,using subadditivity of ψ in ψ 2 yields
ψ 2 (x + y) = ψ(ψ(x + y)) ≤ ψ(ψ(x) + ψ(y)) ≤ ψ(ψ(x)) + ψ(ψ(y)) = ψ 2 (x) + ψ 2 (y),
which implies that ψ 2 is subadditive. Similarly, applying subadditivity of ψ 2 in ψ 3 ,
we get
ψ 3 (x+y) = ψ(ψ 2 (x+y)) ≤ ψ(ψ 2 (x) + ψ 2 (y)) ≤ ψ(ψ 2 (x))+ψ(ψ 2 (y)) = ψ 3 (x) + ψ 3 (y),
which implies that ψ 3 is also subadditive. Hence, in general, each ψ n , n = 1, 2, · · · ,
is subadditive. We now prove that ψ positively homogeneous implies that each
iterate ψ i of ψ is also positively homogeneous: Therefore, we have that ψ(αx) =
αψ(x), ∀ x ∈ R+ , α > 0. Using positive homogeneity of ψ in ψ 2 , we have
ψ 2 (αx) = ψ(ψ(αx)) = ψ(αψ(x)) = αψ(ψ(x)) = αψ 2 (x), ∀ x ∈ R+ , α > 0,
which implies that ψ 2 is positively homogeneous. Hence, in general, each ψ n , n =
1, 2, · · · , is positively homogeneous. Thus, we have that ψ n , n = 1, 2, · · · , is sublinear. In a similar manner, we can prove that ϕj , j = 1, 2, · · · , is sublinear.
228
M.O. Olatinwo - Convergence and stability results for some iterative schemes
The second part of the proof of this lemma is by induction on i as follows: If
i = 1, then (9)Pbecomes
||T x−T y|| ≤ 1j=1 1j ϕj (ψ 1−j (||x−T x||))+ψ(||x−y||) = ϕ(||x−T x||)+ψ(||x−y||),
that is, (9) reduces to (8) when i = 1 and hence the result holds. Assume as an
inductive hypothesis that (9) holds for i = m, m ∈ N, i.e.
m
m
||T x − T y|| ≤
m
X
m
j
j=1
ϕj (ψ m−j (||x − T x||)) + ψ m (||x − y||), ∀ x, y ∈ E.
We then show that the statement is true for i = m + 1;
||T m+1 x − T m+1 y|| = P
||T m (T x) − T m (T y)||
m
j
m−j (||T x − T 2 x||)) + ψ m (||T x − T y||)
≤ m
j=1 j ϕ (ψ
Pm
j
m−j (ϕ(||x − T x||) + ψ(||x − T x||)))
≤ j=1 m
j ϕ (ψ
+ ψ m (ϕ(||x
− T x||) + ψ(||x − y||))
j m−j
Pm
≤ j=1 m
ϕ
[ψ
(ϕ(||x − T x||)) + ψ m+1−j (||x − T x||)]
j
+ ψ m (ϕ(||x
T x||)) + ψ m+1 (||x − y||)
j − m−j
Pm
m
≤ j=1 j ϕ [ϕ(ψ
(||x − T x||)) + ψ m+1−j (||x − T x||)]
+ ϕ(ψ m(||x − T x||)) + ψ m+1 (||x − P
y||)
j m+1−j
P
m
m
m
j+1 (ψ m−j (||x − T x||)) +
≤ m
ϕ
(||x
j=1 j
j=1 j ϕ (ψ
m
m+1
−T x||)) + ϕ(ψ (||x − T x||))
(||x − y||)
+ ψm m
m+1 (||x − T x||) + [ m
= m
ϕ
+
]ϕ
(ψ(||x − T x||))
m
m−1
m
m−1
m
m
2
+ [ m−2 +
m−1
]ϕ m (ψ (||x − T x||)) m+1
m
+ ·· · + [ m
+
(||x − y||)
1
0 ]ϕ(ψ (||x − T x||)) + ψ
m+1
m+1
m+1
m
= m+1 ϕ
(||x − T x||) + m ϕ (ψ(||x − T x||))
m−1 2
m+1
+ m−1 ϕ
(ψ (||x − T x||)) + · · ·
+ m+1
ϕ2 (ψ m−1 (||x − T x||)) + m+1
ϕ(ψ m (||x − T x||))
2
1
+ ψ m+1 (||x − y||)
Pm+1 m+1 j m+1−j
= j=1
ϕ (ψ
(||x − T x||)) + ψ m+1 (||x − y||).
j
Lemma 1.5. Let (E, || · ||) be a normed linear space and let T : E → E be a
selfmap of E satisfying
||T x − T y|| ≤ L||x − T x|| + ψ(||x − y||), ∀ x, y ∈ E, L ≥ 0,
(10)
where ψ : R+ → R+ is a sublinear comparison function. Then, ∀ i ∈ N, we have
i
i
||T x − T y|| ≤
i
X
i
j=1
j
Lj ψ i−j (||x − T x||) + ψ i (||x − y||), ∀ x, y ∈ E.
229
(11)
M.O. Olatinwo - Convergence and stability results for some iterative schemes
Proof: The proof of sublinearity of ψ i (i.e. iterate of ψ) for each i = 1, 2, · · · , is
the same as that of Lemma 1.4. The second part of the proof of this lemma is by
induction on i:PIf i = 1,
(11) becomes
then
1
1
j
1−j
||T x − T y|| ≤ j=1 j L ψ (||x − T x||) + ψ(||x − y||) = L||x − T x|| + ψ(||x − y||),
that is, (11) reduces to (10) when i = 1 and hence the result holds. Assume as an
inductive hypothesis that (11) holds for i = m, m ∈ N, i.e.
m
m
||T x − T y|| ≤
m
X
m
j=1
j
Lj ψ m−j (||x − T x||) + ψ m (||x − y||), ∀ x, y ∈ E.
We then show that the statement is true for i = m + 1;
m
||T m+1 x − T m+1 y|| = ||T
x) − T m (T y)||
Pm (T m
≤ j=1 j Lj ψ m−j (||T x − T 2 x||) + ψ m (||T x − T y||)
j m−j
P
m
≤ m
(L||x − T x|| + ψ(||x − T x||))
j=1 j L ψ
m
− T x|| + ψ(||x − y||))
P + ψm(L||x
Pm m j m+1−j
j+1 ψ m−j (||x − T x||) +
≤ m
L
(||x
j=1 j
j=1 j L ψ
m
m+1
− T x||) + Lψ (||x − T x||)
− y||)
+ ψm (||x
m
m
m+1
m
= m L
||x
m−1 + m ]L ψ(||x − T x||)
− T x||
+ [ m−1
m
m
+ [ m−2
+
]L
ψ 2 (||x − T x||) + · · ·
m−1
m
m
m
m
+ [ 1 + 2 ]L2 ψ m−1 (||x − T x||) + [ m
1 + 0 ]Lψ (||x
m+1
− Tx||) + ψ
(||x − y||)
m+1
m+1
m
= m+1 L
||x − T x|| + m+1
m L ψ(||x − T x||)
m+1
m−1
2
+ m−1 L
ψ (||x − T x||) + · · ·
2 m−1
m
m+1
+ 2 L ψ
(||x − T x||) + m+1
Lψ (||x − T x||)
1
+ ψ m+1(||x − y||)
P
m+1
= m+1
Lj ψ m+1−j (||x − T x||) + ψ m+1 (||x − y||).
j=1
j
Remark 1.4. In addition to both (5) and (6), the contractive conditions (8)
and (10) shall also be used to prove some of our results.
2. Main results
We now establish some convergence results:
Theorem 2.1.Let E be an arbitrary Banach space, K a closed convex subset of E
and T : K → K an operator satisfying (5), where ϕ : R+ → R+ is a subadditive
monotone increasing function such that ϕ(0) = 0, ϕ(Lu) ≤ Lϕ(u), u ∈ R+ . Let
∞
x0 ∈ K, {xn }∞
n=0 be the iterative process defined by (3). Then, {xn }n=0 converges
strongly to the fixed point p of T.
Proof: We shall employ Lemma 1.1 and the triangle inequality to establish that
230
M.O. Olatinwo - Convergence and stability results for some iterative schemes
lim xn = p. However, we shall first establish that T satisfying the given contractive
condition has a unique fixed point. Suppose that there exist p1 , p2 ∈ FT , p1 6= p2 ,
with ||p1 − p2 || > 0. Therefore, we have
P
0 < ||p1 − p2 || = ||T i p1 − T i p2 || ≤ ij=1 ji ai−j ϕj (||p1 − T p1 ||) + ai ||p1 − p2 ||
P
= ij=1 ji ai−j ϕj (0) + ai ||p1 − p2 ||,
n→∞
from which we have that (1 − ai )||p1 − p2 || ≤ 0. Since a ∈ [0, 1), then, 1 − ai > 0 and
||p1 − p2 || ≤ 0.
Also, since norm is nonnegative we have ||p1 − p2 || = 0. That is, p1 = p2 .
Thus, T has a unique fixed point.
We now prove that {xn }∞
n=0 converges strongly to a fixed point of T : Then, we
have
P
i
i
||xn+1 − p|| ≤ ki=1 αn,i ||T
o
nPp − T zn || + αn,0 ||xn − p||
Pk
i
i i−j j
i ||p − z || + α
a
ϕ
(||p
−
T
p||)
+
a
≤ i=1 αn,i
n
n,0 ||xn − p||
P
j=1 j
k
=
αn,i ai ||p − zn || + αn,0 ||xn − p||
Pi=1
P
Ps
s
k
r
r
i ||
α
a
=
r=0 βn,r T xn || + αn,0 ||xn − p||
r=0 βn,r T p −
i=1 n,i
P
P
k
s
i
r
r
≤
i=1 αn,i a { r=1 βn,r ||T p − T xn || + βn,0 ||xn − p||}
αn,0 ||xn −p||
P
+
Pk
Ps
k
i β
i
r p − T r x || +
α
a
α
a
β
||T
=
n
n,0 ||xn − p||
i=1 n,i
i=1 n,i
r=1 n,r
p||
P+ αn,0 ||xn−
Pr
Ps
k
r r−j j
i
ϕ (||p − T p||) + ar ||p − xn || ]
≤
j=1 j a
r=1 βn,r [
i=1 αn,i a
P
k
i
+
i=1 αn,i a βn,0 ||xn − p|| + αn,0 ||xn − p||
P
Pk
= [( i=1 αn,i ai )( sr=0 βn,r ar ) + αn,0 ]||xn − p||
P
P
≤ Πnν=0 [ ( ki=1 αν,i ai )( sr=0 βν,r ar ) + αν,0 ]||x0 − p|| → 0 as n → ∞. (12)
P
P
We claim that 0 ≤ ( ki=1 αν,i ai )( sr=0 βν,r ar ) + αν,0 < 1 as follows:
Pk
Pk
i ≤
i
2
k
i=1 |αν,i a | = |αν,1 a| + |αν,2 a | + · · · + |αν,k a |
i=1 αν,i a
= |αν,1 ||a| + |αν,2 ||a|2 + · · · + |αν,k ||a|k
< |αν,1 | + |αν,2 | + · · · + |αν,k | = αν,1 + αν,2 + · · · + αν,k
= 1 − αν,0 , αν,0 ∈ (0, 1).
Similarly, we have that
Ps
Ps
r ≤
r
2
s
r=0 βν,r a
r=0 |βν,r a | = |βν,0 | + |βν,1 a| + |βν,2 a | + · · · + |βν,k a |
2
s
= |βν,0 | + |βν,1 ||a| + |βν,2 ||a| + · · · + |βν,s ||a|
< |βν,0 | + |βν,1 | + · · · + |βν,s | = βν,0 + βν,1 + · · · + βν,s = 1.
231
M.O. Olatinwo - Convergence and stability results for some iterative schemes
P
P
Therefore, ( ki=1 αν,i ai )( sr=0 βν,r ar ) + αν,0 < 1 − αν,0 + αν,0 = 1.
Hence, we obtain from (12) that ||xn+1 − p|| → 0 as n → ∞,
i.e. {xn }∞
n=0 converges strongly to p.
Theorem 2.2.Let E be an arbitrary Banach space, K a closed convex subset of E
and Ti : K → K, (i = 0, 1, · · · , k), selfoperators satisfying (6), where ϕ : R+ → R+
is a monotone increasing function such that ϕ(0) = 0. Let x0 ∈ K, {xn }∞
n=0 be
the iterative process defined by (4). Then, {xn }∞
converges
strongly
to
the
unique
n=0
common fixed point p of Ti (for each i).
Proof. Let FTi be the set of common fixed points of Ti (i = 0, 1, · · · , k). Suppose
that there exist p1 , p2 ∈ FTi , p1 6= p2 , with ||p1 − p2 || > 0. Therefore, we have
0 < ||p1 − p2 || = ||Ti p1 − Ti p2 || ≤ ϕ(||p1 − Ti p1 ||) + ai ||p1 − p2 ||
= ai ||p1 − p2 ||,
from which it follows that we have that p1 = p2 . That is, Ti (i = 0, 1, · · · , k) have a
unique common fixed point.
We now prove that {xn }∞
n=0 converges strongly to a fixed point of Ti .
Then, we have
P
||xn+1 − p|| ≤ ki=1 αn,i ||Ti p − Ti zn || + αn,0 ||xn − p||
P
≤ ki=1 αn,i {ϕ(||p
− Ti p||) + ai ||p − zn ||} + αn,0 ||xn − p||
Pk
αn,i ai ||p − zn || + αn,0 ||xn − p||
=
Pi=1
P
P
k
s
=
βn,r Tr p − sr=0 βn,r Tr xn || + αn,0 ||xn − p||
i=1 αn,i ai ||
P r=0
P
k
αn,i ai { sr=0 βn,r ||Tr p − Tr xn ||} + αn,0 ||xn − p||
≤
Pi=1
P
k
s
α
a
≤
i=1 n,i i
r=0 βn,r [ϕ(||p − Tr p||) + ar ||p − xn || ] + αn,0 ||xn − p||
Pk
Ps
= [ ( i=1 αn,i ai )( r=0 βn,r ar ) + αn,0 ]||xn − p||
P
P
≤ Πnν=0 [ ( ki=1 αν,i ai )( sr=0 βν,r ar ) + αν,0 ]||x0 − p|| → 0 as n → ∞, (13)
P
P
where as in Theorem 2.1, we can show that 0 ≤ ( ki=1 αν,i ai )( sr=0 βν,r ar ) + αν,0 <
1.
Hence, we obtain from (13) that ||xn+1 − p|| → 0 as n → ∞,
i.e. {xn }∞
n=0 converges strongly to p.
Theorem 2.3.Let E be an arbitrary Banach space, K a closed convex subset of
E, and T : K → K an operator satisfying
||T x − T y|| ≤
ϕ(||x − T x||) + a||x − y||
, ∀ x, y ∈ E, a ∈ [0, 1), M ≥ 0,
1 + M ||x − T x||
232
(14)
M.O. Olatinwo - Convergence and stability results for some iterative schemes
where ϕ : R+ → R+ is a monotone increasing function such that ϕ(0) = 0. Let
∞
x0 ∈ K, {x
Pn∞}n=0 defined by (2) be the Ishikawa iterative process with αn , βn ∈
[0, 1] and
k=0 αk = ∞. Then, the Ishikawa iterative process converges strongly to
the fixed point of T.
Proof. We shall first establish that T has a unique fixed point by using condition
(14): Suppose not. Then, there exist x∗ , y ∗ ∈ FT , x∗ 6= y ∗ and ||x∗ − y∗ || > 0.
Therefore, we have
−T x ||)+a||x −y
0 < ||x∗ − y ∗ || = ||T x∗ − T y ∗ || ≤ ϕ(||x 1+M
||x∗ −T x∗ ||
∗
= a||x − y ∗ ||
∗
∗
∗
∗ ||
from which it follows that (1 − a)||x∗ − y ∗ || ≤ 0, which leads to 1 − a > 0 ( since a ∈
[0, 1) ), but ||x∗ − y ∗ || ≤ 0 (which is a contradiction).
Therefore, since norm is nonnegative, ||x∗ − y ∗ || = 0 i.e. x∗ = y ∗ = p,
thus proving the uniqueness of the fixed point for T. Hence, FT = {p} .
We now prove that {xn }∞
n=0 converges strongly to the fixed point p using condition (14). Therefore, we have
||xn+1 − p|| ≤ ||(1 − αn )xn + αn T zn − (1 − αn + αn )p||
≤ (1 − αn )||xn − p|| + aαn ||p − zn ||
≤ [1 − αn (1 − a) − aαn βn (1 − a)]||xn − p||
≤ [1 − (1 − a)αn ]||xn − p||
≤ Πnk=0 [1 − (1 − a)αk ]||x0 − p||
k ||x − p||
≤ Πnk=0 e−(1−a)α
0
Pn
−[(1−a)
α
k=0 k ] ||x − p|| → 0 as n → ∞,
=e
0
P∞
since k=0 αk = ∞ and a ∈ [0, 1). Hence, we obtain from (15) that
||xn+1 − p|| → 0 as n → ∞, i.e. {xn }∞
n=0 converges strongly to p.
(15)
Remark 2.1. If in each of Theorem 2.1 and Theorem 2.2, the iterative processes
defined by
k
k
X
X
i
xn+1 =
αn,i T xn ,
αn,i = 1, n = 0, 1, 2, · · · ,
(16)
i=0
i=0
αn,i ≥ 0, αn,0 6= 0, αn,i ∈ [0, 1], where k is a fixed integer;
and
k
k
X
X
xn+1 =
αn,i Ti xn ,
αn,i = 1, n = 0, 1, 2, · · · ,
i=0
(17)
i=0
αn,i ≥ 0, αn,0 6= 0, αn,i ∈ [0, 1], where k is a fixed integer and T0 =identity operator;
are employed, then we obtain corresponding results for the one-step processes defined
in (16) and (17). Again, we refer to [13, 14] for these iterative processes too.
233
M.O. Olatinwo - Convergence and stability results for some iterative schemes
Remark 2.2. Theorem 2.1, Theorem 2.2 and Theorem 2.3 are generalizations
and extensions of both Theorem 1 and Theorem 2 of Berinde [3], Theorem 2 and
Theorem 3 of Kannan [8], Theorem 3 of Kannan [9], Theorem 4 of Rhoades [18] as
well as Theorem 8 of Rhoades [19]. See also Berinde [4] for the results of Rhoades
[18, 19].
We prove the following stability results:
Theorem 2.4.Let (E, ||.||) is a normed linear space and T : E → E a selfmap
of E satisfying (8). Let x0 ∈ E and {xn }∞
n=0 defined by (1) be the Kirk’s iterative
process. Suppose that T has a fixed point p. Let ψ : R+ → R+ be a continuous
sublinear comparison function and ϕ : R+ → R+ a sublinear monotone increasing
function such that ϕ(0) = 0 and ψ s (ϕr (x)) ≤ ϕr (ψ s (x)), ∀ x ∈ R+ , r, s ∈ N. Then,
the Kirk iteration process is T -stable.
Pk
i
Proof. Let {yn }∞
lim ǫn = 0.
n=0 ⊂ E and ǫn = ||yn+1 −
i=0 αi T yn ||. Let n→∞
Then, we shall prove that lim yn = p. By using both Lemma 1.4 and the triangle
n→∞
inequality we have that:
P
P
||yn+1 − p|| ≤ ||yn+1 − ki=0 αi T i yn || + || ki=0 αi T i yn − p||
P
P
= ǫn + || ki=0 αi T i yn − ki=0 αi T i p||
Pk
= ǫn + || i=0 αi (T i yn − T i p)||
P
≤ ki=0 αi ||T i p − T i yn || + ǫn
P
= α0 ||p − yn || + ki=1 αi ||T i p − T i yn || + ǫn
Pk
≤ i=0 αi ψ i (||yn − p||) + ǫn ,
(18)
where ψ i−j (0) = ψ(0) = 0 and ϕj (0)
ϕ(0) = 0.
P=
k
i
We have by Lemma 1.2 that
i=0 αi ψ (||yn − p||) is a comparison function.
Therefore, using Lemma 1.3 in (18) yields lim ||yn − p|| = 0, that is, lim yn = p.
n→∞
n→∞
Conversely, let lim yn = p. Then, by Lemma 1.4 and the triangle inequality, we
n→∞
have
P
ǫn = ||yn+1 − ki=0 αi T i yn ||
P
≤ ||yn+1 − p|| + ||p − ki=0 αi T i yn ||
P
P
= ||yn+1 − p|| + || ki=0 αi T i p − ki=0 αi T i yn ||
Pk
= ||yn+1 − p|| + || i=0 αi (T i p − T i yn )||
P
≤ ||yn+1 − p|| + ki=0 αi ||T i p − T i yn ||
P
= ||yn+1 − p|| + α0 ||p − yn || + ki=1 αi ||T i p − T i yn ||
Pk
≤ ||yn+1 − p|| + i=0 αi ψ i (||yn − p||) → 0 as n → ∞.
Theorem 2.5.Let (E, ||.||) is a normed linear space and T : E → E a selfmap
of E satisfying (10). Let x0 ∈ E and {xn }∞
n=0 defined by (1) be the Kirk’s iterative
234
M.O. Olatinwo - Convergence and stability results for some iterative schemes
process. Suppose that T has a fixed point p. Let ψ : R+ → R+ be a continuous
sublinear comparison function and ϕ : R+ → R+ a sublinear monotone increasing
function such that ϕ(0) = 0. Then, the Kirk iteration process is T -stable.
Proof. The proof of this result is similar to that of Theorem 2.4 except for the
application of Lemma 1.2, Lemma 1.3 and Lemma 1.5.
Remark 2.3. Our stability results generalize some of the results of Rhoades
[20, 21]. In particular, both Theorem 2.4 and Theorem 2.5 are generalizations of
the results of Olatinwo [12].
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Memudu Olaposi Olatinwo
Department of Mathematics
Obafemi Awolowo University
Ile-Ife, Nigeria.
E-mail: [email protected]
236