About convergence rates in regularization for ill-posed operator equations of hammerstein type

Ta. p ch´ı Tin ho. c va` Die`ˆu khieˆ’n ho. c, T.23, S.1 (2007), 50—58
ABOUT CONVERGENCE RATES IN REGULARIZATION
FOR ILL-POSED OPERATOR EQUATIONS OF HAMMERSTEIN TYPE
NGUYEN BUONG1, DANG THI HAI HA2
1Vietnamse Academy of Science and Technology, Institute of Information Technology
2Vietnamese Forestry University, Xuan Mai, Ha Tay
Abstract. The aim of this paper is to study convergence rates of the regularized solutions in
connection with the finite-dimensional approximations for the operator equation of Hammerstein
type x+F2F1(x) = f in reflexive Banach spaces under the perturbations for not only the operators
Fi, i = 1, 2, but also f . The conditions of convergence and convergence rates given in this paper for
a class of inverse-strongly monotone operators Fi, i = 1, 2, are much simpler than those in the past
papers.
To´m ta˘´t. Mu. c d´ıch cu’a ba`i ba´o na`y la` nghieˆn cu´
.u toˆ´c doˆ. hoˆ. i tu. cu’a nghieˆ.m hieˆ.u chı’nh da˜ du
.o.. c
xaˆ´p xı’ hu˜.u ha.n chie`ˆu cho phu
.o.ng tr`ınh toa´n tu.’ loa. i Hammerstein x + F2F1(x) = f trong khoˆng
gian Banach pha’n xa. vo´
.i nhie˜ˆu khoˆng chı’ co´ o.’ ca´c toa´n tu.’ Fi, i = 1, 2 ma` ca’ o.’ f . Die`ˆu kieˆ.n hoˆ. i tu.
va` toˆ´c doˆ. hoˆ. i tu. trong ba`i ba´o na`y cho toa´n tu
.’ ngu.o.. c do
.n dieˆ.u ma.nh Fi, i = 1, 2 la` yeˆ´u ho
.n nhie`ˆu
so vo´.i ca´c keˆ´t qua’ tru.´o.c.
1. INTRODUCTION
Let X be a reflexive real Banach space, and X∗ be its dual which both are strictly convex.
For the sake of simplicity the norms of X and X∗ are denoted by the symbol ‖.‖. We write〈
x∗, x
〉
or
〈
x, x∗
〉
instead of x∗(x) for x∗ ∈ X∗ and x ∈ X . Concerning the space X , in
addition assume that it possesses the property: the weak convergence and convergence of
norms for any sequence follows its strong convergence. Let F1 : X → X∗ and F2 : X∗ → X
be monotone, in general nonlinear, bounded (i.e. image of any bounded subset is bounded)
and continuous operators.
Our main aim of this paper is to study a stable method of finding an approximative solution
for the equation of Hammerstein type
x+ F2F1(x) = f, f ∈ X. (1.1)
Usually instead of Fi, i = 1, 2, and f we know their monotone continuous approximations F
h
i
and fδ, such that
‖F h1 (x)− F1(x)‖ 
 hg(‖x‖) ∀x ∈ X,
‖F h2 (x∗)− F2(x∗)‖ 
 hg(‖x∗‖) ∀x∗ ∈ X∗,
g(t) 
Mt+N, M,N  0,
ABOUT CONVERGENCE RATES IN REGULARIZATION FOR ILL-POSED 51
where g(t) is a real nonegative, non-decreasing, bounded function (the image of a bounded set
is bounded) with g(0) = 0, and ‖fδ−f‖ 
 δ. Without additional conditions for the operators
Fi such as the strongly monotone property, equation (1.1) is ill-posed (see the example at the
end of the paper). To solve (1.1) we need to use stable methods. One of them is the operator
equation
x+ F h2,αF
h
1,α(x) = fδ (1.2)
(see [1], [2]), where F hi,α = F
h
i +αUi, Ui, i = 1, 2, are the normalized dual mappings of X and
X∗, respectively (see [9]), and α > 0 is the small parameter of regularization. For every α > 0
equation (1.2) has a unique solution xh,δα , and the sequence {xh,δα } converges to a solution x0
of (1.1) as (h + δ)/α, α → 0. Moreover, this solution xh,δα , for every fixed α > 0, depends
continuously on F hi , i = 1, 2 and fδ, the finite-dimensional problems
x+ F h2,α,nF
h
1,α,n(x) = fδ,n, x ∈ Xn, (1.3)
where F h2,α,n = PnF
h
2,αP
∗
n , F
h
1,α,n = P
∗
nF
h
1,αPn, fδ,n = Pnfδ, Pn is a linear projection from X
onto its finite-dimensional subspace Xn such that Xn ⊂ Xn+1, Pnx→ x, as n→∞ for every
x ∈ X , and P ∗n is the dual of Pn with ‖Pn‖ 
 c˜ = constant, for all n, have a unique solution
xh,δα,n, and the sequence {xh,δα,n} converges to xh,δα , as n→∞, without additional conditions on
Fi, i = 1, 2. In the case of linearity for F2 and fδ = f for all δ > 0, the convergence rates for
the sequences {xh,δα } and {xh,δα,n} are given in the paper [3] provided the existence of bounded
inversion (I + F2F
′
1(x0))
−1, where I denotes the identity operator in X. It is not difficult to
verify that this condition can be replaced by the bounded inversion of (I + F ′2(x
∗
0)F
′
1(x0))
−1,
when F2 also is nonlinear, where x
∗
0 = F1(x0). The last requirement is equivalent to that
-1 is not an eigenvalue of the operator F ′2(x
∗
0)F
′
1(x0) and is used in studying a method of
collocation-type for nonlinear integral equations of Hammerstein type (see [6]). In general
case, i.e., when both the operators Fi, i = 1, 2, are nonlinear, it means that R, the range of
the operator I + F ′2(x
∗
0)F
′
1(x0), is the whole space X . It is natural to ask if we can estimate
the convergence rates for the sequences {xh,δα }, {xh,δα,n}, when R is not the whole space X . For
this purpose, only demanding that R contains a necessary element of X, the convergence rates
of {xh,δα } and {xh,δα,n} are estimated in [4], [5] on the base of the zero value of the derivatives
of higher order for F1 and F2 at x0 and x
∗
0, respectively. This result is formulated in the
following theorem.
Theorem 1.1. (see [4] or [5]). Let the following conditions hold:
(i) F1 is Fre´chet differentiable at some neighbourhood U0 of x0 s1−1-times if s1 = [s1], the
integer part of s1, [s1]-times if s1 = [s1], and F2 is Fre´chet differentiable at some neighbourhood
V0 of x∗0 s2 − 1-times, if s2 = [s2], [s2]-times if s2 = [s2],
(ii) there exists a constant L > 0 such that
‖F (k)1 (x0)− F (k)1 (y)‖ 
 L‖x0 − y‖, ∀ y ∈ U0,
‖F (k)2 (x∗0)− F (k)2 (y∗)‖ 
 L‖x∗0 − y∗‖, ∀ y∗ ∈ V0,
for F
(k)
i : k = si − 1 if si = [si], k = [si] if si = [si], and if [si]  3, then F (2)1 (x0) = ... =
F
(k)
1 (x0) = 0, and F
(2)
2 (x
∗
0) = ... = F
(k)
1 (x
∗
0) = 0,
52 NGUYEN BUONG, DANG THI HAI HA
(iii) there exists an element x1 ∈ X such that(
I + F ′2(x
∗
0)
∗F ′1(x0)
∗
)
x1 = F ′2(x
∗
0)
∗U1(x0)− U2(x∗0),
if s1 = [s1] then L‖x1‖ < m1s1!, and if s2 = [s2] then L‖F ′1(x0)∗x1 − U1(x0)‖ < m2s2!
Then, if α is chosen such that α ∼ (h+ ε)ρ, 0 < ρ < 1, we have
‖xω − x0‖ = O((h+ ε)θ),
θ = min {θ1, 1− ρ+ θ2
s1 − 1 },
θi = min {1− ρ
si
,
ρ
si
}, i = 1, 2.
In this paper, the convergence rates of {xh,δα } and {xh,δα,n} are established under much weaker
conditions on Fi, i = 1, 2. These are the assumptions that R contains some element of X , and
Fi, i = 1, 2, are inverse-strongly monotone, i.e.
〈F1(x)− F1(y), x− y〉  m˜1‖F1(x)− F1(y)‖2, x, y ∈ X,
〈F2(x∗)− F2(y∗), x∗ − y∗〉  m˜2‖F2(x∗)− F2(y∗)‖2, x∗, y∗ ∈ X∗,
(1.4)
where m˜i, i = 1, 2, are some positive constants. Note that in [7] the inverse-strongly monotone
property was used to estimate the convergence rates of the regularized solutions for ill-posed
variational inequalities.
Below, by “a ∼ b” we mean “a = O(b) and b = O(a)”.
2. MAIN RESULTS
Assume that the normalized dual mappings Ui, i = 1, 2, of the spaces X and X∗ satisfy
the following conditions (see [8])〈
Ui(y
i
1)− Ui(yi2), yi1 − yi2
〉
 mi‖yi1 − yi2‖si , mi > 0, si  2, (2.1)
‖Ui(yi1)− Ui(yi2)‖ 
 ci(Ri)‖yi1 − yi2‖νi , 0 < νi 
 1, (2.2)
where yi1, y
i
2 ∈ X or X∗ on dependence of i = 1 or 2, respectively, and ci(Ri), Ri > 0, are
the positive increasing functions on Ri = max {‖yi1‖, ‖yi2‖}.
The following theorem answers the question on convergence rates for {xh,δα }.
Theorem 2.1. Assume that the following conditions hold:
(i) Fi, i = 1, 2, are inverse-strongly monotone and continuously Fre´chet differentiable at
some neighbourhoods U of x0 and V of x∗0 , respectively, and
‖F1(x)− F1(x0)− F ′1(x0)(x− x0)‖ 
 τ1‖F1(x)− F1(x0)‖, ∀x ∈ U ,
‖F2(x∗)− F2(x∗0)− F ′2(x∗0)(x∗ − x∗0)‖ 
 τ2‖F2(x∗)− F2(x∗0)‖, ∀x∗ ∈ V,
where τi, i = 1, 2, are some positive constants,
ABOUT CONVERGENCE RATES IN REGULARIZATION FOR ILL-POSED 53
(ii) there exists an element x1 ∈ X such that(
I + F ′2(x
∗
0)
∗F ′1(x0)
∗
)
x1 = F ′2(x
∗
0)
∗U1(x0)− U2(x∗0).
Then, if α is chosen such that α ∼ (h+ δ)ρ, 0 < ρ < 1, we have
‖xh,δα − x0‖ = O
(
(h+ δ)θ/s1), θ = min {ρ/2, 1− ρ}.
Proof. Set
A = m1‖xh,δα − x0‖s1 +m2‖xh,δ,∗α − x∗0‖s2 , xh,δ,∗α = F h1,α(xh,δα ).
It is easy to see that x0 is a solution of (1.1) iff z0 = [x0, x
∗
0] is a solution of the system of
following operator equations
F1(x)− x∗ = 0,
F2(x
∗) + x− f = 0.
Similarily, xh,δα is a regularized solution of the operator equation (1.2) iff z
h,δ
α = [x
h,δ
α , x
h,δ,∗
α ] is
a solution of the system of following equations
F h1 (x) + αU1(x)− x∗ = 0,
F h2 (x
∗) + αU2(x
∗) + x− fδ = 0.
Consider the space Z = X ×X∗ with the norm ‖z‖2 = ‖x‖2 + ‖x∗‖2, z = [x, x∗], x ∈ X, and
x∗ ∈ X∗. Then, the two above systems of equations can be written, respectively, in form of
equations
A(z) = f,
Ahα(z) ≡ Ah(z) + αJ(z) = f δ,
(2.3)
where
A(z) = [F1(x), F2(x∗)] + [−x∗, x],
Ah(z) = [F h1 (x), F h2 (x∗)] + [−x∗, x],
J(z) = [U1(x), U2(x
∗)],
f = [0, f ], f δ = [0, fδ].
(2.4)
It is easy to verify that A and Ah are the monotone operators from Z to Z∗ = X∗ ×X, and
the operator J is the normalized duality mapping of the space Z. Hence, from (2.1), (2.3),
(2.4) and the monotone property of Ah it implies that
A 
〈J(zh,δα )− J(z0), zh,δα − z0〉 
 〈J(z0), z0 − zh,δα 〉
+
1
α
[〈f δ − f, zh,δα − z0〉+ 〈A(z0)−Ah(z0), zh,δα − z0〉].
(2.5)
It is not difficult to verify that
‖Ah(z)−A(z)‖ 

√
2hg(‖z‖). (2.6)
54 NGUYEN BUONG, DANG THI HAI HA
Further, from (1.4) it follows
〈A(zh,δα )−A(z0), zh,δα − z0〉 = 〈F1(xh,δα )− xh,δ,∗α − (F1(x0)− x∗0), xh,δα − x0〉
+ 〈F2(xh,δ,∗α ) + xh,δα − (F2(x∗0) + x0), xh,δ,∗α − x∗0〉
= 〈F1(xh,δα )− F1(x0), xh,δα − x0〉+ 〈F2(xh,δ,∗α )− F2(x∗0), xh,δ,∗α − x∗0〉
 m˜1‖F1(xh,δα )− F1(x0)‖2 + m˜2‖F2(xh,δ,∗α )− F2(x∗0)‖2
 min{m˜1, m˜2}C2, C2 = ‖F1(xh,δα )− F1(x0)‖2 + ‖F2(xh,δ,∗α )− F2(x∗0)‖2.
On the other hand, from (2.3), (2.4)-(2.6) and the properties of A,Ah, J, g we have
〈A(zh,δα )−A(z0), zh,δα − z0〉 
 〈f δ − f, zh,δα − z0〉
+ α〈J(z0), z0 − zh,δα 〉+ 〈A(zh,δα )−Ah(zh,δα ), zh,δα − z0〉,
and {zh,δα } is bounded, as (h+ δ)/α→ 0. Therefore,
C2 

1
min{m˜1, m˜2} [δ + α‖J(z0)‖+
√
2hg(‖zh,δα ‖)]‖zh,δα − z0‖.
Consequently, C 
 O(
√
h+ δ + α). Hence,
‖F1(xh,δα )− F1(x0)‖ 
 O(
√
h+ δ + α),
‖F2(xh,δ,∗α )− F2(x∗0)‖ 
 O(
√
h+ δ + α).
(2.7)
Now, we shall estimate the value 〈J(z0), z0 − zh,δα 〉. For this purpose, set x2 = U1(x0) −
F ′1(x0)
∗x1. From condition (ii) of the theorem it follows that x1 and x2 (∈ X∗) satisfy the
system of following equalities
F ′1(x0)
∗x1 + x2 = U1(x0),
F ′2(x
∗
0)
∗x2 − x1 = U2(x∗0).
By virtue of
〈J(z0), z0 − zh,δα 〉 = 〈U1(x0), x0 − xh,δα 〉+ 〈U2(x∗0), x∗0 − xh,δ,∗α 〉
=〈F ′1(x0)∗x1 + x2, x0 − xh,δα 〉+ 〈F ′2(x∗0)∗x2 − x1, x∗0 − xh,δ,∗α 〉
=〈xh,δ,∗α − x∗0 − F ′1(x0)(xh,δα − x0), x1〉
+ 〈x0 − xh,δα − F ′2(x∗0)(xh,δ,∗α − x∗0), x2〉
=〈F1(xh,δα )− F1(x0)− F ′1(x0)(xh,δα − x0), x1〉
+ α〈U1(xh,δα ), x1〉+ 〈F h1 (xh,δα )− F1(xh,δα ), x1〉
+ 〈F2(xh,δ,∗α )− F2(x∗0)− F ′2(x∗0)(xh,δ,∗α − x∗0), x2〉
+ 〈αU2(xh,δ,∗α ) + f − fδ, x2〉+ 〈F h2 (xh,δ,∗α )− F(xh,δ,∗α ), x2〉,
we have
〈J(z0), z0 − zh,δα 〉 
 max{τ1‖x1‖, τ2‖x2‖} ×
(‖F1(xh,δα )− F1(x0)‖+ ‖F2(xh,δ,∗α )− F2(x∗0)‖) +O(h+ δ + α).
ABOUT CONVERGENCE RATES IN REGULARIZATION FOR ILL-POSED 55
Thus, for sufficiently small h, δ, α (h+ δ + α < 1) from (2.5)-(2.7) we have got
A 
 O((h+ δ)1−ρ) +O(
√
h+ δ + α),
It means that
‖xh,δα − x0‖ = O
(
(h+ δ)θ/s1).
Theorem is proved. 

Theorem 2.2. Assume that the conditions of Theorem 2.1 hold, and α is chosen such that
α ∼ (h+ δ + γn)ρ, 0 < ρ < 1, where
γn = max{‖(I − Pn)x0‖, ‖(I − Pn)f‖, ‖(I − Pn)x1‖, ‖(I∗ − P ∗n)x∗0‖, ‖(I∗ − P ∗n)x2‖},
and I∗ denotes the identity operator in X∗. Then,
‖xh,δα,n − x0‖ = O
(
(h+ δ)η + γµn
)
,
η = min {1− ρ
s1
,
ρ
2s1
},
µ = min {η, ν1
s1
,
ν2
s1
}.
Proof. Set
B = m1‖xh,δα,n − x0,n‖s1 +m2‖xh,δ,∗α,n − x∗0,n‖s2 ,
with x0,n = Pnx0, x
h,δ,∗
α,n = F h1,α,n(x
h,δ
α,n), and x∗0,n = P
∗
nx
∗
0. It is easy to see that x
h,δ
α,n is a
solution of (1.3) iff xh,δα,n and x
h,δ,∗
α,n are the solutions of the system of following equations
F h1,n(x) + αU
n
1 (x)− x∗ = 0,
F h2n(x
∗) + αUn2 (x
∗) + x− fδ,n = 0,
with Un1 = P
∗
nU1Pn, U
n
2 = PnU2P
∗
n , F
h
1,n = P
∗
nF
h
1 Pn, F
h
2,n = PnF
h
2 P
∗
n , and fδ,n = Pnfδ. As in
the proof of theorem 2.1, zh,δα,n := [x
h,δ
α,n, x
h,δ,∗
α,n ] is the solution of the following operator equation
Ahα,n(z) ≡ Ahn(z) + αJn(z) = f δ,n, (2.8)
where
Ahn(z) = [F h1,n(x), F h2,n(x∗)] + [−x∗, x],
Jn(z) = [Un1 (x), U
n
2 (x
∗)], f δ,n = [0, fδ,n].
(2.9)
The operatorsAhn andAn, defined byAn(z) = [F1,n(x), F2,n(x∗)]+[−x∗, x], F1,n = P ∗nF1Pn, F2,n =
PnF2P
∗
n , are the monotone operators, and act from Zn := Xn × X∗n into Z∗n, and Jn is the
normalized duality mapping of the space Zn.
From (2.8) we obtain
An(zh,δα,n)−An(z0,n) + α[Jn(zh,δα,n)− Jn(z0,n)] = f δ,n +
An(zh,δα,n)−Ahn(zh,δα,n)−An(z0,n)− αJn(z0,n).
(2.10)
56 NGUYEN BUONG, DANG THI HAI HA
Therefore, from (1.4) and the properties of the projections Pn, P
∗
n it implies that
〈An(zh,δα,n)−An(z0,n), zh,δα,n − z0,n〉 = 〈F1(xh,δα,n)− F1(x0,n), xh,δα,n − x0,n〉
+ 〈F2(xh,δ,∗α,n )− F2(x∗0,n), xh,δ,∗α,n − x∗0,n〉
 m˜1‖F1(xh,δα,n)− F1(x0,n)‖2 + m˜2‖F2(xh,δ,∗α,n )− F2(x∗0,n)‖2
 min{m˜1, m˜2}C2n, C2n = ‖F1(xh,δα,n)− F1(x0,n)‖2 + ‖F2(xh,δ,∗α,n )− F2(x∗0,n)‖2.
On the other hand, from (2.8), (2.9) we also obtain
Ahn(zh,δα,n)−Ahn(z0,n) + α[Jn(zh,δα,n)− Jn(z0,n)] = f δ,n
−Ahn(z0,n)− αJn(z0,n).
(2.11)
Hence, on the base of the property of J and (2.11) we can write
B 

1
α
〈f δ − f − αJ(z0,n), zh,δα,n − z0,n〉
+
1
α
〈A(z0)−Ah(z0,n), zh,δα,n − z0,n〉


1
α
[δ + ‖A(z0)−A(z0,n)‖+ hg(‖z0,n‖)]‖zh,δα,n − z0,n‖
+ 〈Jn(z0,n), z0,n − zh,δα,n〉.
(2.12)
Moreover, using the continously Fre´chet differentiable property of F1, F2 and the definition of
γn we can also write
‖A(z0,n)−A(z0)‖ 
 (‖F1(x0,n)− F1(x0)‖2
+ ‖F2(x∗0,n)− F2(x∗0)‖2)1/2 +
√
2γn

 (max{c˜1, c˜2}+
√
2)γn,
where c˜1 = max0t1 ‖F ′1(x0 + t(x0,n − x0))‖ and c˜2 = max0t1 ‖F ′2(x∗0 + t(x∗0,n − x∗0))‖.
Consequently, {zh,δα,n} is bounded, when (h+ δ + γn)/α→ 0. By virtue of (2.10) we have
〈An(zh,δα,n)−An(z0,n), zh,δα,n − z0,n〉 
 〈f δ,n −An(z0,n), zh,δα,n − z0,n〉
+ 〈An(zh,δα,n)−Ahn(zh,δα,n)− αJn(z0,n), zh,δα,n − z0,n〉

 〈f δ − f +A(z0)−A(z0,n), zh,δα,n − z0,n〉
+ 〈An(zh,δα,n)−Ahn(zh,δα,n)− αJ(z0,n), zh,δα,n − z0,n〉

 O(h+ δ + α+ γn)‖z0,n − zh,δα,n〉‖.
Therefore, C˜n 
 O(
√
h+ δ + α+ γn). Hence,
‖F1(xh,δα,n)− F1(x0,n)‖ 
 O(
√
h+ δ + α+ γn),
‖F2(xh,δ,∗α,n )− F2(x∗0,n)‖ 
 O(
√
h+ δ + α+ γn).
ABOUT CONVERGENCE RATES IN REGULARIZATION FOR ILL-POSED 57
Now, we obtain the esimation for 〈Jn(z0,n), z0,n − zh,δα,n〉. From (2.2), (2.8) and the condition
of the theorem we have got
〈Jn(z0,n), z0,n − zh,δα,n〉 = 〈J(z0,n), z0,n − zh,δα,n〉
= 〈J(z0,n)− J(z0), z0,n − zh,δα,n〉+ 〈J(z0), z0,n − zh,δα,n〉

 Cγνn‖zh,δα,n − z0,n‖+ 〈F ′1(x0)∗x1 + x2, x0,n − xh,δα,n〉
+ 〈F ′2(x∗0)∗x2 − x1, x∗0,n − xh,δ,∗α,n 〉

 Cγνn‖zh,δα,n − z0,n‖+ 〈x1, xh,δ,∗α,n − x∗0,n − F ′1(x0)(xh,δα,n − x0,n)〉
+ 〈x2, x0,n − xh,δα,n − F ′2(x∗0)(xh,δ,∗α,n − x∗0,n)〉,
where C is some positive constant, and ν = min{ν1, ν2}. Obviously,
〈x1, xh,δ,∗α,n − x∗0,n − F ′1(x0)(xh,δα,n − x0,n)〉 = 〈x1, F h1,n(xh,δα,n) + αUn1 (xh,δα,n)− x∗0,n〉
+ 〈x1,−F ′1(x0)(xh,δα,n − x0) + F ′1(x0)(x0,n − x0)〉
= 〈x1n,F1(xh,δα,n)− F1(x0)− F ′1(x0)(xh,δα,n − x0)〉
+ α〈x1, Un1 (xh,δα,n)〉+ 〈x1, F ′1(x0)(x0,n − x0)〉
+ 〈(I − Pn)x1,−F ′1(x0)(xh,δα,n − x0)〉+ 〈x1n, F h1 (xh,δα,n)− F1(xh,δα,n)〉

 τ1‖x1n‖‖F1(xh,δα,n)− F1(x0)‖+O(h+ α+ γn),
where x1n = Pnx
1. By the argument, we also obtain the estimate
〈x2, x0,n − xh,δα,n − F ′2(x∗0)(xh,δ,∗α,n − x∗0,n)〉 
 τ2‖x2n‖‖F2(xh,δ,∗α,n )− F2(x∗0)‖
+O(h+ δ + α+ γn).
Therefore,
〈Jn(z0), z0 − zh,δα 〉 
 O(γνn) +O(
√
h+ δ + γn + α).
Thus, from (2.12) and the properties of Ah, J it follows
B 
 O((h+ δ + γn)
1−ρ + γνn +O((h+ δ + γn)
ρ/2).
Consequently,
‖xh,δα,n − x0‖ = O
(
(h+ δ)η + γµn).
Theorem is proved. 

Example 1. Consider the simple example, when X ≡ X∗ = E2, the Euclid space, and
F1 =
[
1 −1
1 0
]
, F2 =
[
0 −1
1 1
]
, x = (x1, x2).
It is easy to verify that 〈F1x, x〉 = x21  0, and 〈F2x, x〉 = x22  0∀x ∈ E2. It means that
Fi, i = 1, 2, are monotone. Equation (1.1) has the form 0x1 = f1, 2x1 = f2 with f = (f1, f2).
Obviously, this system of equations has a unique solution when f = (0, f2) for arbitrary f2.
When fδ = (f
δ
1 , f2) with f
δ
1 = 0 equation (1.1) in this case there isn’t a solution. So, equation
58 NGUYEN BUONG, DANG THI HAI HA
(1.1) with the monotone operators F1, i = 1, 2, in general is ill-posed. On the other hand,
equation A(z) = f for z = (x1, x2, x∗1, x∗2) is the system of 4 linear equations with the matrix
A =

1 −1 −1 0
1 0 0 −1
1 0 0 −1
0 1 1 1
 .
having det A = 0. Consequently, the system of equations is also ill-posed.
REFERENCES
[1] N. Buong, On solutions of the equations of Hammerstein type in Banach spaces, Zh.
Vychisl. Matematiki i Matem. Fiziki 25 (8) (1985) 1256—1280 (in Russian).
[2] N. Buong, On solution of Hammerstein’s equation with monotone perturbations, Viet-
namese Math. Journal 3 (1985) 28—32.
[3] N. Buong, On approximate solution for operator equations of Hammerstein type, J. of
Comput. and Applied Math. 75 (1996) 77—86.
[4] N. Buong, Convergence rates in regularization for Hammerstein equations, Zh. Vychisl.
Matematiki i Matem. Fiziki 39 (4) (1999) 3—7.
[5] N. Buong, Convergence rates in regularization for the case of monotone perturbations,
Ukrainian Math. Zh. 52 (2) (2000) 285—293.
[6] S. Kumar, Superconvergence of a Collocation - type Method for Hammerstein Equations,
IMA Journal of Numerical Analysis 7 (1987) 313—325.
[7] F. Liu and M. Z. Nashed, Regularization of nonlinear ill-posed variational inequalities and
convergence rates, Set-Valued Analysis 6 (1998) 313—344.
[8] I. P. Ryazantseva, On an algorithm for solving nonlinear monotone equations with unknow
estimate input errors, Zh. Vychisl. Matematiki i Matem. Fiziki 29 (1989) 1572—1576 (in
Russian).
[9] M. M. Vainberg, Variational method and method of monotone operators, Nauka, Moscow
1972 (in Russian).
Received on April 6 - 2006
Revised on November 14 - 2006