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

## 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

monotone mixed variational inequality non-monotone perturbations regularization convergence rate## 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]):

**Theorem 1.1**.

*If there exists u*∈ dom

*φ satisfying the coercive condition*

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

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

*K*in

*X*,

*x*

_{0}∈

*K*such that

*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 }: Open image in new window 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 Open image in new window such that

where *μ* is positive small enough, *U*^{ s } is the generalized duality mapping of *X* (see Definition 1.3) and Open image in new window 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;

**(2)**

*A*

_{ h }:

*X*→

*X** is not necessarily monotone,

*D*(

*A*

_{ h }) =

*D*(

*A*) =

*X*, and

**(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 Open image in new window converge to

*x*

_{0}∈

*S*

_{0}, the Open image in new window -minimal norm solution defined by

The convergence rate of the regularized solutions Open image in new window 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, 10, 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)inverse-strongly monotone if

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]).

*U*

^{ s }satisfies the following condition:

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,

*z*=

*x*

_{2}in (15) and

*z*=

*x*

_{1}in (16) and add the obtained inequalities, we obtain

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

*h >*0, making use of (8), from (17) one gets

Since *μ* ≥ *h*, we can conclude that each Open image in new window is a solution of (7).

□

Let Open image in new window 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* Open image in new window *converges strongly to the* Open image in new window -*minimal norm solution x*_{0} ∈ *S*_{0}.

**Proof**. By (1) and (7), we obtain

*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 Open image in new window are bounded. Therefore, there exists a subsequence of which we denote by the same Open image in new window weakly converges to Open image in new window .

*A*and

*U*

^{ s }implies that

*φ*

_{ ε }is proper convex weakly lower semicontinuous, we have from (25) that

Finally, the *S* property of *A* implies the strong convergence of Open image in new window to Open image in new window .

*φ*is weakly lower semicontinuous,

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

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

**(1)**imply that

This means that Open image in new window .

*U*

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

*x*by Open image in new window ,

*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 Open image in new window , ∀*x* ∈ *S*_{0}. Because of the convexity and the closedness of *S*_{0}, and the strictly convexity of *X*, we can conclude that Open image in new window . The proof is complete.

□

with Open image in new window , where Open image in new window is the solution of (7) with Open image in new window , *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* Open image in new window *is single-valued and continuous for α*≥ *α*_{0}*>* 0, *where* Open image in new window *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, Open image in new window as *μ* → 0 and *α*_{1} → *α*_{2}. It means that the function Open image in new window 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* Open image in new window *of the equation* (30),

*(ii) let μ*, *δ*, *ε* → 0. *Then*

*(1)* Open image in new window ;

*(2) if*0

*< p < q then*Open image in new window , Open image in new window

*with*Open image in new window -

*minimal norm and there exist constants C*

_{1},

*C*

_{2}

*>*0

*such that for sufficiently small μ*,

*δ*,

*ε >*0

*the relation*

*holds*.

**Proof**.

*< α <*1, it follows from (7) that

**(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 Open image in new window which satisfies (30).

Therefore, Open image in new window as *μ*, *δ*, *ε* → 0.

*< p < q*, it follows from (30) and (32) that

By Theorem 2.1 the sequence Open image in new window converges to *x*_{0} ∈ *S*_{0} with Open image in new window -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*

*then, if the parameter α*=

*α*(

*μ*,

*δ*,

*ε*)

*is chosen by*(30)

*with*0

*< p < q, we have*

**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 Open image in new window 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

*α*is chosen by (30), it follows from Theorem 2.1 that

*i*= 1, 2, 3, are the positive constants. Using the implication

*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].

## Notes

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