# Regularization of ill-posed mixed variational inequalities with non-monotone perturbations

- Nguyen TT Thuy
^{1}Email author

**2011**:25

https://doi.org/10.1186/1029-242X-2011-25

© Thuy; licensee Springer. 2011

**Received: **10 February 2011

**Accepted: **21 July 2011

**Published: **21 July 2011

## Abstract

In this paper, we study a regularization method for ill-posed mixed variational inequalities with non-monotone perturbations in Banach spaces. The convergence and convergence rates of regularized solutions are established by using a priori and a posteriori regularization parameter choice that is based upon the generalized discrepancy principle.

## Keywords

## 1 Introduction

*f*∈

*X**, find an element

*x*

_{0}∈

*X*such that

where *A* : *X* → *X** is a monotone-bounded hemicontinuous operator with domain *D*(*A*) = *X*, *φ* : *X* → ℝ is a proper convex lower semicontinuous functional and *X* is a real reflexive Banach space with its dual space *X**. For the sake of simplicity, the norms of *X* and *X** are denoted by the same symbol || · ||. We write 〈*x**, *x*〉 instead of *x**(*x*) for *x** ∈ *X** and *x* ∈ *X*.

By *S*_{0} we denote the solution set of the problem (1). It is easy to see that *S*_{0} is closed and convex whenever it is not empty. For the existence of a solution to (1), we have the following well-known result (see [4]):

*then* (1) *has at least one solution*.

*φ*by the indicator function of a closed convex set

*K*in

*X*,

*K*is the whole space

*X*, the later variational inequality is of the form of the following operator equation:

*A*is the Gâteaux derivative of a finite-valued convex function

*F*defined on

*X*, the problem (1) becomes the nondifferentiable convex optimization problem (see [4]):

*A*,

*f*,

*φ*), we used stable methods for solving it. A widely used and efficient method is the regularization method introduced by Liskovets [7] using the perturbative mixed variational inequality:

where *A*_{
h
} is a monotone operator, *α* is a regularization parameter, *U* is the duality mapping of *X*, *x*_{
*
}∈ *X* and (*A*_{
h
} , *f*_{
δ
} , *φ*_{
ε
} ) are approximations of (*A*, *f*, *φ*), *τ* = (*h*, *δ*, *ε*). The convergence rates of the regularized solutions defined by (6) are considered by Buong and Thuy [8].

*A*

_{ h }: to be monotone. In this case, the regularized variational inequality (6) may be unsolvable. In order to avoid this fact, we introduce the regularized problem of finding such that

where *μ* is positive small enough, *U*^{
s
} is the generalized duality mapping of *X* (see Definition 1.3) and
is in *X* which plays the role of a criterion of selection, *g* is defined below.

Assume that the solution set *S*_{0} of the inequality (1) is non-empty, and its data *A*, *f*, *φ* are given by *A*_{
h
} , *f*_{
δ
} , *φ*_{
ε
} satisfying the conditions:

**(1)** || *f* - *f*_{
δ
} || ≤ *δ*, *δ* → 0;

**(3)**

*φ*

_{ ε }:

*X*→ ℝ is a proper convex lower semicontinuous functional for which there exist positive numbers

*c*

_{ ε }and

*r*

_{ ε }such that

where *C*_{0} is some positive constant, *d*(*t*) has the same properties as *g*(*t*).

*α >*0. Then, we show that the regularized solutions converge to

*x*

_{0}∈

*S*

_{0}, the -minimal norm solution defined by

The convergence rate of the regularized solutions
to *x*_{0} will be established under the condition of inverse-strongly monotonicity for *A* and the regularization parameter choice based on the generalized discrepancy principle.

We now recall some known definitions (see [9–11]).

**Definition 1.1**. An operator

*A*:

*D*(

*A*) =

*X*→

*X** is said to be

- (a)
hemicontinuous if

*A*(*x*+*t*_{ n }*y*) ⇀*Ax*as*t*_{ n }→ 0^{+},*x*,*y*∈*X*, and demicontinuous if*x*_{ n }→*x*implies*Ax*_{ n }⇀*Ax*; - (b)
monotone if 〈

*Ax*-*Ay*,*x*-*y*〉 ≥ 0, ∀*x*,*y*∈*X*; - (c)

where *m*_{
A
} is a positive constant.

It is well-known that a monotone and hemicontinuous operator is demicontinuous and a convex and lower semicontinuous functional is weakly lower semicontinuous (see [9]). And an inverse-strongly monotone operator is not strongly monotone (see [10]).

**Definition 1.2**. It is said that an operator *A* : *X* → *X** has *S*-property if the weak convergence *x*_{
n
} ⇀ *x* and 〈*Ax*_{
n
} - *Ax*, *x*_{
n
} - *x*〉 → 0 imply the strong convergence *x*_{
n
} → *x* as *n* → ∞.

**Definition 1.3**. The operator

*U*

^{ s }:

*X*→

*X** is called the generalized duality mapping of

*X*if

When *s* = 2, we have the duality mapping *U*. If *X* and *X** are strictly convex spaces, *U*^{
s
} is single-valued, strictly monotone, coercive, and demicontinuous (see [9]).

where *m*_{
s
} is a positive constant. It is well-known that when *X* is a Hilbert space, then *U*^{
s
} = *I*, *s* = 2 and *m*_{
s
} = 1, where *I* denotes the identity operator in the setting space (see [12]).

## 2 Main result

**Lemma 2.1**. *Let X** *be a strictly convex Banach space. Assume that A is a monotone-bounded hemicontinuous operator with D*(*A*) = *X and conditions* **(2)** *and* **(3)** *are satisfied. Then, the inequality* (7) *has a non-empty solution set S*_{
ε
} *for each α >* 0 *and f*_{
δ
} ∈ *X**.

**Proof**. Let

*x*

_{ ε }∈ dom

*φ*

_{ ε }. The monotonicity of

*A*and assumption (

**3)**imply the following inequality:

*x*||

*> r*

_{ ε }. Consequently, (2) is fulfilled for the pair (

*A*+

*αU*

^{ s },

*φ*

_{ ε }). Thus, for each

*α >*0 and

*f*

_{ δ }∈

*X**, there exists a solution of the following inequality:

*A*and the strict monotonicity of

*U*

^{ s }. Indeed, let

*x*

_{1}and

*x*

_{2}be two different solutions of (14). Then,

Due to the monotonicity of *A* and the strict monotonicity of *U*^{
s
} , the last inequality occurs only if *x*_{1} = *x*_{2}.

Since *μ* ≥ *h*, we can conclude that each
is a solution of (7).

□

Let be a solution of (7). We have the following result.

**Theorem 2.1**.

*Let X and X**

*be strictly convex Banach spaces and A be a monotone-bounded hemicontinuous operator with D*(

*A*) =

*X. Assume that conditions*

**(1)**-

**(3)**

*are satisfied, the operator U*

^{ s }

*satisfies condition*(13)

*and, in addition, the operator A has the S-property. Let*

*Then*
*converges strongly to the*
-*minimal norm solution x*_{0} ∈ *S*_{0}.

*A*, assumption

**(1)**, and the inequalities (8), (9), (13) and (20) yield the relation

Since *μ*/*α* → 0 as *α* → 0 (and consequently, *h/α* → 0), it follows from (19) and the last inequality that the set
are bounded. Therefore, there exists a subsequence of which we denote by the same
weakly converges to
.

Finally, the *S* property of *A* implies the strong convergence of
to
.

*α*→ 0 in the inequality (7), provided that

*A*is demicontinuous, from (8), (9), (28), (29) and condition

**(1)**imply that

*U*

^{ s }and the inequalities (8), (9) and (13), we can rewrite (17) as

*x*by ,

*t*∈ (0, 1) in the last inequality, dividing by (1 -

*t*) and then letting

*t*to 1, we get

Using the property of *U*^{
s
} , we have that
, ∀*x* ∈ *S*_{0}. Because of the convexity and the closedness of *S*_{0}, and the strictly convexity of *X*, we can conclude that
. The proof is complete.

□

with
, where
is the solution of (7) with
, *c* is some positive constant.

**Lemma 2.2**. *Let X and X** *be strictly convex Banach spaces and A* : *X* → *X** *be a monotone-bounded hemicontinuous operator with D*(*A*) = *X. Assume that conditions* **(1)**, **(2)** *are satisfied, the operator U*^{
s
} *satisfies condition* (13). *Then, the function*
*is single-valued and continuous for α*≥ *α*_{0}*>* 0, *where*
*is the solution of* (7).

**Proof**. Single-valued solvability of the inequality (7) implies the continuity property of the function

*ρ*(

*α*). Let

*α*

_{1},

*α*

_{2}≥

*α*

_{0}be arbitrary (

*α*

_{0}

*>*0). It follows from (7) that

*α*=

*α*

_{1}and

*α*=

*α*

_{2}. Using the condition

**(2)**and the monotonicity of

*A*, we have

Obviously,
as *μ* → 0 and *α*_{1} → *α*_{2}. It means that the function
is continuous on [*α*_{0}; +∞). Therefore, *ρ*(*α*) is also continuous on [*α*_{0}; +∞).

**Theorem 2.2**. *Let X and X** *be strictly convex Banach spaces and A* : *X* → *X** *be a monotone-bounded hemicontinuous operator with D*(*A*) = *X. Assume that conditions* **(1)**-**(3)** *are satisfied, the operator U*^{
s
} *satisfies condition* (13). *Then*

*(i) there exists at least a solution*
*of the equation* (30),

*(ii) let μ*, *δ*, *ε* → 0. *Then*

*(2) if*0

*< p < q then*,

*with*-

*minimal norm and there exist constants C*

_{1},

*C*

_{2}

*>*0

*such that for sufficiently small μ*,

*δ*,

*ε >*0

*the relation*

*holds*.

**Proof**.

**(1)**, the monotonicity of

*A*, (8), (10), (12), and the last inequality to deduce that

Therefore, lim_{α→+0}*α*^{
q
}*ρ*(*α*) = 0.

Since *ρ*(*α*) is continuous, there exists at leat one
which satisfies (30).

By Theorem 2.1 the sequence
converges to *x*_{0} ∈ *S*_{0} with
-minimal norm as *μ*, *δ*, *ε* → 0.

therefore, there exists a positive constant *C*_{2} such that (32). On the other hand, because *c >* 0 so there exists a positive constant *C*_{1} satisfied (32). This finishes the proof.

□

**Theorem 2.3**. *Let X be a strictly convex Banach space and A be a monotone-bounded hemicontinuous operator with D*(*A*) = *X. Suppose that*

*(i) for each h, δ, ε >* 0 *conditions* **(1)**-**(3)** *are satisfied*;

*(ii) U*^{
s
} *satisfies condition* (13);

*(iii) A is an inverse-strongly monotone operator from X into X**,

*Fréchet differentiable at some neighborhood of x*

_{0}∈

*S*

_{0}

*and satisfies*

**Proof**. By an argument analogous to that used for the proof of the first part of Theorem 2.1, we have (21). The boundedness of the sequence follows from (21) and the properties of

*g*(

*t*),

*d*(

*t*) and

*α*. On the other hand, based on (20), the property of

*U*

^{ s }and the inverse-strongly monotone property of

*A*we get that

*Remark 2.1*If

*α*is chosen a priori such that

*α*~ (

*μ*+

*δ*+

*ε*)

^{ η }, 0

*< η <*1, it follows from (35) that

*Remark 2.2* Condition (34) was proposed in [13] for studying convergence analysis of the Landweber iteration method for a class of nonlinear operators. This condition is used to estimate convergence rates of regularized solutions of ill-posed variational inequalities in [14].

*Remark 2.3* The generalized discrepancy principle for regularization parameter choice is presented in [15] for the ill-posed operator equation (4) when *A* is a linear and bounded operator in Hilbert space. It is considered and applied to estimating convergence rates of the regularized solution for equation (4) involving an accretive operator in [16].

## Declarations

## Authors’ Affiliations

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