245
Acta Math. Univ. Comenianae
Vol. LXXVII, 2(2008), pp. 245–253
ITERATIVE SOLUTIONS OF NONLINEAR EQUATIONS WITH
φ-STRONGLY ACCRETIVE OPERATORS
SHIN MIN KANG, CHI FENG and ZEQING LIU
Abstract. Suppose that X is an arbitrary real Banach space and T : X → X
is a Lipschitz continuous φ-strongly accretive operator or uniformly continuous
φ-strongly accretive operator. We prove that under different conditions the three-step iteration methods with errors converge strongly to the solution of the equation
T x = f for a given f ∈ X.
1. Introduction
Let X be a real Banach space with norm k · k and dual X ∗ , and J denote the
∗
normalized duality mapping from X into 2X given by
J(x) = {f ∈ X ∗ : kf k2 = kxk2 = hx, f i},
x ∈ X,
where h·, ·i is the generalized duality pairing. In this paper, I denotes the identity
operator on X, R+ and δ(K) denote the set of nonnegative real numbers and the
diameter of K for any K ⊆ X, respectively. An operator T with domain D(T ) and
range R(T ) in X is called φ-strongly accretive if there exists a strictly increasing
function φ : R+ → R+ with φ(0) = 0 such that for any x, y ∈ D(T ) there exists
j(x − y) ∈ J(x − y) such that
(1.1)
hT x − T y, j(x − y)i ≥ φ(kx − yk)kx − yk.
If there exists a positive constant k > 0 such that (1.1) holds with φ(kx − yk)
replaced by kkx − yk, then T is called strongly accretive. The accretive operators
were introduced independently in 1967 by Browder [1] and Kato [8]. An early
fundamental result in the theory of accretive operator, due to Browder, states the
initial value problem
du
+ T u = 0, u(0) = u0
dt
is solvable if T is locally Lipschitz and accretive on X. Martin [11] proved that if
T : X → X is strongly accretive and continuous, then T is subjective so that the
(1.2)
Received April 9, 2007.
2000 Mathematics Subject Classification. Primary 47H05, 47H10, 47H15.
Key words and phrases. φ-strongly accretive operators; three-step iteration method with
errors; Banach spaces.
246
SHIN MIN KANG, CHI FENG and ZEQING LIU
equation
Tx = f
(1.3)
has a solution for any given f ∈ X. Using the Mann and Ishikawa iteration
methods with errors, Chang [3], Chidume [4], [5], Ding [7], Liu and Kang [10]
and Osilike [12], [13] obtained a few convergence theorems for Lipschitz φ-strongly
accretive operators. Chang [2] and Yin, Liu and Lee [16] also got some convergence
theorems for uniformly continuous φ-strongly accretive operators.
The purpose of this paper is to study the three-step iterative approximation of
solution to equation (1.3) in the case when T is a Lipschitz φ-strongly accretive
operator and X is a real Banach space. We also show that if T : X → X is a
uniformly continuous φ-strongly accretive operator, then the three-step iteration
method with errors converges strongly to the solution of equation (1.3). Our
results generalize, improve the known results in [2]–[7], [10], [12], [13] and [15].
2. Preliminaries
The following Lemmas play a crucial role in the proofs of our main results.
Lemma 2.1 ([7]). Suppose that φ : R+ → R+ is a strictly increasing function
∞
∞
∞
with φ(0) = 0. Assume that {rn }∞
n=0 , {sn }n=0 , {kn }n=0 and {tn }n=0 are sequences
of nonnegative numbers satisfying the following conditions:
(2.1)
∞
X
kn < ∞,
n=0
∞
X
n=0
tn < ∞,
∞
X
sn = ∞
n=0
and
(2.2) rn+1 ≤ (1 + kn )rn − sn rn
φ(rn+1 )
+ tn
1 + rn+1 + φ(rn+1 )
for n ≥ 0.
Then limn→∞ rn = 0.
Lemma 2.2 ([10]). Suppose that X is an arbitrary Banach space and T : X →
X is a continuous φ-strongly accretive operator. Then the equation T x = f has a
unique solution for any f ∈ X.
∞
∞
Lemma 2.3 ([9]). Let {αn }∞
n=0 , {βn }n=0 and {γn }n=0 be three nonnegative
real sequences satisfying the inequality
αn+1 ≤ (1 − ωn )αn + ωn βn + γn
for n ≥ 0,
P
P∞
∞
where {ωn }∞
n=0 ⊂ [0, 1],
n=0 γn < ∞. Then
n=0 ωn = ∞, limn→∞ βn = 0 and
limn→∞ αn = 0.
3. Main Results
Theorem 3.1. Suppose that X is an arbitrary real Banach space and T : X →
∞
X is a Lipschitz φ-strongly accretive operator. Assume that {un }∞
n=0 , {vn }n=0 ,
ITERATIVE SOLUTIONS OF NONLINEAR EQUATIONS
247
∞
∞
∞
{wn }∞
n=0 are sequences in X and {an }n=0 , {bn }n=0 and {cn }n=0 are sequences in
∞
[0, 1] such that {kwn k}n=0 is bounded and
(3.1)
∞
X
a2n < ∞,
n=0
∞
X
an bn < ∞,
n=0
kun k < ∞,
n=0
∞
X
(3.2)
∞
X
∞
X
kvn k < ∞,
n=0
an = ∞.
n=0
For any given f ∈ X, define S : X → X by Sx = f + x − T x for all x ∈ X. Then
the three-step iteration sequence with errors {xn }∞
n=0 defined for arbitrary x0 ∈ X
by
zn = (1 − cn )xn + cn Sxn + wn ,
yn = (1 − bn )xn + bn Szn + vn ,
(3.3)
xn+1 = (1 − an )xn + an Syn + un ,
n≥0
converges strongly to the unique solution q of the equation T x = f . Moreover
kxn+1 − qk ≤ [1 + (3 + 3L3 + L4 )a2n + L(1 + L2 )an bn ]kxn − qk
− A(xn+1 , q)an kxn − qk + an bn L2 (3 + L)kwn k
(3.4)
+ an L(3 + L)kvn k + (3 + L)kun k
for n ≥ 0, where A(x, y) =
φ(kx−yk)
1+kx−yk+φ(kx−yk)
∈ [0, 1) for x, y ∈ X.
Proof. It follows from Lemma 2.2 that the equation T x = f has a unique
solution q ∈ X. Let L0 denote the Lipschitz constant of T . From the definition
of S we know that q is a fixed point of S and S is also Lipschitz with constant
L = 1 + L0 . Thus for any x, y ∈ X, there exists j(x − y) ∈ J(x − y) such that
h(I − S)x − (I − S)y, j(x − y)i ≥ A(x, y)kx − yk2 .
This implies that
h(I − S − A(x, y))x − (I − S − A(x, y))y, j(x − y)i ≥ 0
and it follows from Lemma 1.1 of Kato [8] that
(3.5) kx − yk ≤ kx − y + r[(I − S − A(x, y))x − (I − S − A(x, y))y]k
for x, y ∈ X and r > 0. From (3.3) we conclude that for each n ≥ 0
xn = xn+1 + an xn − an Syn − un
(3.6)
= (1 + an )xn+1 + an (I − S − A(xn+1 , q))xn+1 − (I − A(xn+1 , q))an xn
+ an (Sxn+1 − Syn ) + (2 − A(xn+1 , q))a2n (xn − Syn )
− [1 + (2 − A(xn+1 , q))an ]un
and
(3.7) q = (1 + an )q + an (I − S − A(xn+1 , q))q − (I − A(xn+1 , q))an q.
248
SHIN MIN KANG, CHI FENG and ZEQING LIU
It follows from (3.5)–(3.7) that
kxn − qk
= k(1 + an )xn+1 + an (I − S − A(xn+1 , q))xn+1 − (I − A(xn+1 , q))an xn
+ an (Sxn+1 − Syn ) + (2 − A(xn+1 , q))a2n (xn − Syn )
− [1 + (2 − A(xn+1 , q))an ]un − (1 + an )q − an (I − S − A(xn+1 , q))q
+ (I − A(xn+1 , q))an qk
an
≥ (1 + an )
xn+1 − q + 1 + an [(I − S − A(xn+1 , q))xn+1
− (I − S − A(xn+1 , q))q − an (1 − A(xn+1 , q))kxn − qk
− (2 − A(xn+1 , q))a2n kxn − Syn k − an kSxn+1 − Syn k
− [1 + (2 − A(xn+1 , q))an ]kun k
≥ (1 + an )kxn+1 − qk − an (1 − A(xn+1 , q))kxn − qk
− (2 − A(xn+1 , q))a2n kxn − Syn k − an kSxn+1 − Syn k
− [1 + (2 − A(xn+1 , q))an ]kun k,
which implies that
kxn+1 − qk
≤
(3.8)
1 + (1 − A(xn+1 , q))an
kxn − qk + (2 − A(xn+1 , q))a2n kxn − Syn k
1 + an
+ an kSxn+1 − Syn k + [1 + (2 − A(xn+1 , q))an ]kun k
≤ (1 − A(xn+1 , q)an + a2n )kxn − qk + 2a2n kxn − Syn k
+ an kSxn+1 − Syn k + (1 + 2an )kun k
for n ≥ 0. By (3.3) we get that
kzn − qk ≤ (1 − cn )kxn − qk + cn kSxn − qk + kwn k
(3.9)
≤ (1 − cn )kxn − qk + Lcn kxn − qk + kwn k
≤ Lkxn − qk + kwn k,
(3.10)
kyn − qk ≤ (1 − bn )kxn − qk + bn kSzn − qk + kvn k
≤ (1 − bn )kxn − qk + Lbn kzn − qk + kvn k,
(3.11) kxn − Szn k ≤ kxn − qk + kSzn − qk ≤ kxn − qk + Lkzn − qk,
(3.12)
kxn − yn k ≤ bn kxn − Szn k + kvn k
and
(3.13)
kSyn − yn k ≤ kSyn − qk + kyn − qk ≤ (1 + L)kyn − qk
for n ≥ 0. From (3.9)–(3.13) we obtain that
(3.14)
kxn − Syn k ≤ (1 + L3 )kxn − qk + L2 bn kwn k + Lkvn k
ITERATIVE SOLUTIONS OF NONLINEAR EQUATIONS
249
and
kSxn+1 − Syn k ≤ (Lbn + L3 bn − Lan bn − L3 an bn + L3 an + L4 an )kxn − qk
+ (L2 bn + L3 an bn )kwn k + (L + L2 an )kvn k + Lkun k
(3.15)
for n ≥ 0. It follows from (3.8), (3.14) and (3.15) that
kxn+1 − qk ≤ [1 + (3 + 3L3 + L4 )a2n + L(1 + L2 )an bn ]kxn − qk
− A(xn+1 , q)an kxn − qk + an bn L2 (3 + L)kwn k
(3.16)
+ (3 + L)an kvn k + (3 + L)kun k
for n ≥ 0. Set
rn = kxn − qk,
kn = (3 + 3L3 + L4 )a2n + L(1 + L2 )an bn ,
sn = an ,
2
tn = an bn L (3 + L)kwn k + an L(3 + L)kvn k + (3 + L)kun k for n ≥ 0.
Then (3.16) yields that
(3.17) rn+1 ≤ (1 + kn )rn − sn rn
φ(rn+1 )
+ tn
1 + rn+1 + φ(rn+1 )
for n ≥ 0.
It follows from (3.1), (3.2), (3.17) and Lemma 2.1 that rn → 0 as n → ∞. That is
xn → q as n → ∞. This completes the proof.
Remark 3.2. Theorem 3.1 extends Theorem 5.2 of [3], Theorem 1 of [4],
Theorem 2 of [5], Theorem 1 of [6], Theorem 3.1 of [10], Theorem 1 of [12],
Theorem 1 of [13] and Theorem 4.1 of [15].
∞
∞
∞
∞
Theorem 3.3. Let X, {un }∞
n=0 , {vn }n=0 , {wn }n=0 , {an }n=0 , {bn }n=0 and
∞
{cn }n=0 be as in Theorem 3.1 and T : D(T ) ⊂ X → X be a Lipschitz φ-strongly
accretive operator. Suppose that the equation T x = f has a solution q ∈ D(T )
∞
∞
for some f ∈ X. Assume that the sequences {xn }∞
n=0 , {yn }n=0 and {zn }n=0
generated from an arbitrary x0 ∈ D(T ) by (3.3) are contained in D(T ). Then
∞
∞
{xn }∞
n=0 , {yn }n=0 and {zn }n=0 converge strongly to q and satisfied (3.4).
The proof of Theorem 3.3 uses the same idea as that of Theorem 3.1. So we
omit it.
Remark 3.4. Theorem 3.1 in [7] and Theorem 3.2 in [10] are special cases of
our Theorem 3.3.
Theorem 3.5. Suppose that X is an arbitrary real Banach space and T : X → X
is a uniformly continuous φ-strongly accretive operator, and the range of either
(I − T ) or T is bounded. For any f ∈ X, define S : X → X by Sx = f + x − T x
for all x ∈ X and the three-step iteration sequence with errors {xn }∞
n=0 by
x0 , u0 , v0 , w0 ∈ X,
(3.18)
zn = a00n xn + b00n Sxn + c00n wn ,
yn = a0n xn + b0n Szn + c0n vn ,
xn+1 = an xn + bn Syn + cn un ,
n ≥ 0,
250
SHIN MIN KANG, CHI FENG and ZEQING LIU
∞
∞
where {un }∞
n=0 , {vn }n=0 and {wn }n=0 are arbitrary bounded sequences in X and
∞
∞
∞
0 ∞
0 ∞
00 ∞
00 ∞
{an }n=0 , {bn }n=0 , {cn }n=0 , {an }n=0 , {b0n }∞
n=0 , {cn }n=0 , {an }n=0 , {bn }n=0 and
00 ∞
{cn }n=0 are real sequences in [0, 1] satisfying the following conditions
(3.19)
(3.20)
∞
X
an + bn + cn = 1,
a0n + b0n + c0n = 1,
a00n + b00n + c00n = 1,
bn + cn ∈ (0, 1),
bn = +∞,
n=0
n ≥ 0,
lim bn = lim b0n = lim c0n = lim
n→∞
n→∞
n→∞
n→∞
cn
= 0.
bn + cn
Then the sequence {xn }∞
n=0 converges strongly to the unique solution of the equation T x = f .
Proof. It follows from Lemma 2.2 that the equation T x = f has a unique
solution q ∈ X. By (1.2) we have
hT x − T y, j(x − y)i = h(I − S)x − (I − S)y, j(x − y)i ≥ A(x, y)kx − yk2 ,
where A(x, y) =
φ(kx−yk)
1+kx−yk+φ(kx−yk)
∈ [0, 1) for x, y ∈ X. This implies that
h(I − S − A(x, y))x − (I − S − A(x, y))y, j(x − y)i ≥ 0
for x, y ∈ X. It follows from Lemma 1.1 of Kato [8] that
(3.21)
kx − yk ≤ kx − y + r[(I − S − A(x, y))x − (I − S − A(x, y))y]k
for x, y ∈ X and r > 0. Now we show that R(S) is bounded. If R(I − T ) is
bounded, then
kSx − Syk = k(I − T )x − (I − T )yk ≤ δ(R(I − T ))
for x, y ∈ X. If R(T ) is bounded, we get that
kSx − Syk = k(x − y) − (T x − T y)k
≤ φ−1 (kT x − T yk) + kT x − T yk
≤ φ−1 (δ(R(T ))) + δ(R(T ))
for x, y ∈ X. Hence R(S) is bounded. Put
dn = bn + cn ,
d0n = b0n + c0n ,
d00n = b00n + c00n
for n ≥ 0
and
(3.22)
D = max{kx0 − qk,
sup{kx − qk : x ∈ {un , vn , wn , Sxn , Syn , Szn : n ≥ 0}}}.
By (3.18) and (3.22) we conclude that
(3.23)
max{kxn − qk, kyn − qk, kzn − qk} ≤ D
for n ≥ 0.
Using (3.18) we obtain that
(1 − dn )xn = xn+1 − dn Syn − cn (un − Syn )
(3.24)
= [1 − (1 − A(xn+1 , q))dn ]xn+1 + dn (I − S − A(xn+1 , q))xn+1
+ dn (Sxn+1 − Syn ) − cn (un − Syn ).
251
ITERATIVE SOLUTIONS OF NONLINEAR EQUATIONS
Note that
(1 − dn )q = [1 − (1 − A(xn+1 , q))dn ]q + dn (I − S − A(xn+1 , q))q.
(3.25)
It follows from (3.21) and (3.23)–(3.25) that
(1 − dn )kxn − qk
≥ [1 − (1 − A(xn+1 , q))dn ]kxn+1 − q
dn
[(I − S − A(xn+1 , q))xn+1
1 − (1 − A(xn+1 , q))dn
− (I − S − A(xn+1 , q))q]k − dn kSxn+1 − Syn k − cn kun − Syn k
+
≥ [1 − (1 − A(xn+1 , q))dn ]kxn+1 − qk − dn kSxn+1 − Syn k − 2Dcn .
That is
kxn+1 − qk
1 − dn
kxn − qk
1 − (1 − A(xn+1 , q))dn
(3.26)
2Dcn
dn
kSxn+1 − Syn k +
+
1 − (1 − A(xn+1 , q))dn
1 − (1 − A(xn+1 , q))dn
≤ [1 − (1 − A(xn+1 , q))dn ]kxn − qk + M dn kSxn+1 − Syn k + M cn
≤
for n ≥ 0, where M is some constant. In view of (3.18)–(3.20) we infer that
kxn+1 − yn k ≤ kxn+1 − xn k + kyn − xn k
≤ bn kSyn − xn k + cn kun − xn k + b0n kSzn − xn k + c0n kvn − xn k
≤ bn kSyn − xn k + cn kun − xn k + b0n kSzn − zn k + c0n kvn − xn k
+ b0n (b00n kSxn − xn k + c00n kwn − xn k)
≤ 2D(dn + d0n + b0n d00n ) → 0
as n → ∞. Since S is uniformly continuous, we have
(3.27)
kSxn+1 − Syn k → 0
as n → ∞.
Set inf{A(xn+1 , q) : n ≥ 0} = r. We claim that r = 0. If not, then r > 0. It is
easy to check that
kxn+1 − qk ≤ (1 − rdn )kxn − qk + M dn kSxn+1 − Syn k + M cn
for n ≥ 0.
Put
αn = kxn − qk,
cn = tn dn ,
βn = M r
−1
ωn = rdn ,
(kSxn+1 − Syn k + tn ),
γn = 0 for n ≥ 0.
(3.2) ensures that tn → 0Pas n → ∞. It follows from (3.20), (3.27)
P∞ and Lemma 2.3
∞
that ωn ∈ (0, 1] with
ω
=
∞,
lim
β
=
0,
n→∞ n
n=0 n
n=0 γn < ∞. So
kxn − qk → 0 as n → ∞, which means that r = 0. This is a contradiction.
∞
Thus r = 0 and there exists a subsequence {kxni +1 − qk}∞
i=0 of {kxn+1 − qk}n=0
satisfying
(3.28)
kxni +1 − qk → 0
as i → ∞.
252
SHIN MIN KANG, CHI FENG and ZEQING LIU
From (3.28) and (3.29) we conclude that for given ε > 0 there exists a positive
integer m such that for n ≥ m,
kxnm +1 − qk < ε
(3.29)
and
M kSxn+1 − Syn k + M
(3.30)
cn
< min
dn
1
φ(ε)ε
ε,
.
2 1 + φ( 23 ε) + 32 ε
Now we claim that
kxnm +j − qk < ε
(3.31)
for j ≥ 1.
In fact (3.29) means that (3.31) holds for j = 1. Assume that (3.31) holds for
j = k. If kxnm +k+1 − qk > ε, we get that
kxnm +k+1 − qk
≤ kxnm +k − qk + M dnm +k kSxnm +k+1 − Synm +k k + M cnm +k
φ(ε)ε
(3.32) ≤ ε + min 1 ε,
dnm +k
2 1 + φ( 23 ε) + 32 ε
3
≤ ε.
2
Note that φ(kxnm +k+1 − qk) > φ(ε). From (3.32) we get that
A(xnm +k+1 , q) ≥
(3.33)
φ(ε)
.
1 + φ( 23 ε) + 32 ε
By virtue of (3.26) (3.30) and (3.33) we obtain that
kxnm +k+1 − qk
φ(ε)ε
≤ 1−
dn +k kxnm +k − qk
1 + φ( 23 ε) + 23 ε m
+ M dnm +k kSxnm +k+1 − Synm +k k + M cnm +k
1
φ(ε)ε
φ(ε)ε
dn +k ε + min
ε,
dnm +k
≤ 1−
2 1 + φ( 32 ε) + 32 ε
1 + φ( 23 ε) + 23 ε m
≤ ε.
That is
ε < kxnm +k+1 − qk ≤ ε,
which is a contradiction. Hence kxnm +k+1 − qk ≤ ε. By induction (3.29) holds for
j ≥ 1. Thus (3.31) yields that xn → q as n → ∞. This completes the proof.
Remark 3.6. Theorem 3.5 extends and improves Theorem 3.4 in [2] and Theorem 3.1 in [16].
Acknowledgement. This work was supported by the Science Research Foundation of Educational Department of Liaoning Province (20060467).
ITERATIVE SOLUTIONS OF NONLINEAR EQUATIONS
253
References
1. Browder F. E., Nonlinear mappings of nonexpansive and accretive type in Banach spaces,
Bull. Amer. Math. Soc. 73 (1967), 875–882.
2. Chang S. S., Some problems and results in the study of nonlinear analysis, Nonlinear Anal.
30 (1997), 4197–4208.
3. Chang S. S., ChoY. J., Lee B. S. and S. M. Kang, Iterative approximations of fixed points
and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach
spaces, J. Math. Anal. Appl. 224 (1998) 149–165.
4. Chidume C. E., An iterative process for nonlinear Lipschitzian strongly accretive mapping
in Lp spaces, J. Math. Anal. Appl. 151 (1990), 453–461.
5.
, Iterative solution of nonlinear equations with strongly accretive operators, J. Math.
Anal. Appl. 192 (1995), 502–518.
6. Deng L., On Chidume’s open questions, J. Math. Anal. Appl. 174 (1993), 441–449.
7. Ding X. P., Iterative process with errors to nonlinear φ-strongly accretive operator equations
in arbitrary Banach spaces, Computers Math. Applic. 33 (1997), 75–82.
8. Kato T., Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967),
508–520.
9. Liu L. S., Ishikawa and Mann iterative process with errors for nonlinear strongly accretive
mapping in Banach spaces, J. Math. Anal. Appl. 194 (1995), 114–125.
10. Liu Z. and Kang S. M. Convergence theorems for φ-strongly accretive and φ-hemicontractive
operators, J. Math. Anal. Appl. 253 (2001), 35–49.
11. Martin R. H., Jr. A global existence theorem for autonomous differential equations in Banach
spaces, Proc. Amer. Math. Soc. 26 (1970), 307–314.
12. Osilike M. O., Iterative solution of nonlinear equations of the φ-strongly accretive type,
J. Math. Anal. Appl. 200 (1996), 259–271.
13.
, Ishikawa and Mann iteration methods with errors for nonlinear equations of the
accretive type, J. Math. Anal. Appl. 213 (1997), 91–105.
14.
, Iterative solution of nonlinear φ-strongly accretive operator equations in arbitrary
Banach spaces, Nonlinear Anal. TMA 36 (1999), 1–9.
15. Tan K. K. and Xu H. K., Iterative solutions to nonlinear equations of strongly accretive
operators in Banach spaces, J. Math. Anal. Appl. 178 (1993), 9–21.
16. Yin Q., Liu Z. and Lee B. S., Iterative solutions of nonlinear equations with φ-strongly
accretive operators, Nonlinear Anal. Forum 5 (2000), 87–89.
Shin Min Kang, Department of Mathematics and the Research Institute of Natural Science,
Gyeongsang National University, Jinju 660-701, Korea, e-mail: smkang@nongae.gsnu.ac.kr
Chi Feng, Department of Science, Dalian Fisheries College, Dalian, Liaoning, 116023, People’s
Republic of China, e-mail: windmill-1129@163.com
Zeqing Liu, Department of Mathematics, Liaoning Normal University, P.O. Box 200, Dalian,
Liaoning, 116029, People’s Republic of China, e-mail: zeqingliu@sina.com.cn