Journal of Convex Analysis Volume 1 (1994), No. 2, 121–133

Topological Degree for Maximal Monotone Operators and Application to Parametric Optimization Problems^{1}

Hassan Riahi

Université CADI AYYAD, Faculté des Sciences Semlalia, Département de Mathématiques, B.P. S 15, 40 000 Marrakech, Marocco.

Received 16 November 1993 Revised manuscript received 17 October 1994

The generalized topological degree theory is based on the Brouwer and Leray-Schauder degrees. It can be defined for general classes of mappings. The purpose of this article is two-fold. One goal is to define the topological degree for maximal monotone operators. Particular attention is paid to the continuation methods for this kind of operators and real functions of convex type. This allows us to extend some recent results (see [5], [6]) by withdrawing the compactness assumptions.

Keywords : Topological degree, Pseudo-monotone operator, Operator of class (S+), Maximal monotone operator, Generalized Yosida approximation, Graph-continuity.

1. Introduction

The question of stability in optimization deals with what happens to an optimization problem when the elements of the problem are in some way deformed. As being expressed by Felix E. Browder, the concept of degree of a mapping, in all its different forms, is one of the most effective tools for studying the properties of the existence and multiplicity of solutions of nonlinear equations. Historically, the well known topological degree is a useful tool in applied mathematics, for example to prove that some nonlinear equations have solutions and to investigate the stability by using the continuation method. The notion of the degree was first introduced explicitly by Brouwer in 1912 in the case of finite dimensional spaces. Leray and Schauder extended this theme in 1934 to the context of Banach spaces and mappings of the form f = I − g, with I the identity and g a compact mapping (we refer to [15], [27] and [39] for a wide bibliography on the subject.) Afterwards many authors defined and developed the topological degree theory for various classes of non-compact nonlinear mappings between Banach spaces. For references on these notions see [1], [2], [3], [15], [16], [22]–[24], [26], [27], [29], [31], [33]–[38], [41], [43],

122 H. Riahi / Topological degree for maximal monotone operators

[47] and [50]. In a series of articles [15]–[18] in 1983 Browder has defined and extended this concept of the classical topological degree for operators of monotone type (class (S_{+}) and pseudomonotone operators). His method is based on Galerkin approximations for which the classical Brouwer degree is defined. Recently, Berkovits and Mustonen [11]–[13] introduced a new construction of the Browder degree which is based on the Leray-Schauder degree. The degree theory obtained can be used by relaying on the continuation methods to deduce existence theorems for nonlinear inclusion differential equations (see the references mentioned above), fixed point theory ([14], [16], [25], [36], [44], [52], ...) and optimisation ([5], [6], [45]). In this paper we show how Browder’s degree, given for operators of class (S_{+}), can be naturally extended to the case of maximal monotone operators by relying on generalized Yosida approximates. Particular attention is paid to the normalization and invariance under homotopies for the topological degree we define. Homotopy methods are used to prove several theorems on the existence of solutions. This allows us to extend some recent results (see [5], [6]) of Attouch, Penot and Riahi about the continuation method for solutions of parametrized monotone nonlinear equations (Theorem 4.5). It is also possible, by relying on subdifferentials, that our definition could be used to define topological degrees for real convex functions and convex-concave saddle functions. Here is the summary of the paper. In section 2, some basic properties of Browder’s degree are set out. In section 3, we demonstrate an auxiliary continuation theorem (Theorem 3.3). It concerns to pass from graph continuity of maximal monotone operators family to class (S_{+}) property of the associated generalized Yosida approximation. Section 4 is devoted to define the topological degree of a maximal monotone operator and to give familiar properties (Theorem 4.2). Afterwards we give various results on parametrized nonlinear monotone equations (Theorem 4.5, Propositions 4.6, 4.7). Finally, in section 5 we apply the results of the previous sections to real functions of convex type (Propositions 5.2, 5.3).

1 A preliminary version of this paper was presented at the Congress on “Set-convergence in Nonlinear

Analysis and Optimization” that was held at Marseille-Luminy on June 22-26, 1992.

ISSN 0944-6532 / $ 2.50 ◯^{c} Heldermann Verlag

2. Notation and preliminaries

Let be given a real reflexive Banach space X with the topological dual X^{∗}. Without loss of generality we will always assume (X, ∥.∥) and (X^{∗}, ∥.∥) to be locally unifomly convex, by virtue of the powerful renorming theorem of Asplund, Lindenstrauss, Trojanski and Zizler (see [27], p.185 or [49]). In particular this implies that the duality mapping J of X into X^{∗} given by

J(x) = {x^{∗} ∈ X^{∗}; ⟨x^{∗}, x⟩ = ∥x∥.∥x^{∗}∥ = ∥x∥^{2}}

is a homeomorphism between X and X^{∗}. The strong and the weak convergences in each of the spaces X and X^{∗} are denoted by “→^{s}” and “→^{w}”, respectively.

Let us recall some definitions and results that will be needed in the sequel.

Definition 2.1. Let Ω be an open subset of the reflexive Banach space X and {f_{t} : \overline{Ω} ⟶ X^{∗}; t ∈ T \} be a family of demi-continuous operators from \overline{Ω} in X^{∗}. Then the family (f_{t})_{t∈T} is called pseudomonotone (resp. of class (S_{+})) if for any net (t_{i})_{i∈I} converging to t in T and (x_{i})_{i∈I} in \overline{Ω}, the relations

x_{i} →^{w} x and lim sup_{i∈I}⟨f_{t}_{i}(x_{i}), x_{i} − x⟩ ≤ 0

imply

lim_{i∈I}⟨f_{t}_{i}(x_{i}), x_{i} − x⟩ = 0 (resp. x_{i} →^{s} x),

if moreover x ∈ \overline{Ω} the net (f_{t}_{i}(x_{i}))_{i∈I} does weakly converge to f_{t}(x).

Theorem 2.2. (Browder’s degree) Let X be a reflexive Banach space, Ω an open bounded subset of X, F(Ω, y) be the family of all operators of class (S_{+}) such that y ∉ f(∂Ω), and A(Ω, y) the family of homotopies in F(Ω, y) of class (S_{+}). On these admissible triplets (f, Ω, y), i.e. (f, Ω, y) ∈ F(Ω, y), one can define a unique Z -valued function d that satisfies the three basic conditions corresponding to the ones of Brouwer’s topological degree, namely : (d_{1}) d(J, Ω, y) = 1 for y ∈ Ω , and d(f, Ω, y) ≠ 0 implies y ∈ f(Ω) ; (d_{2}) d(f, Ω, y) = d(f, Ω_{1}, y) + d(f, Ω_{2}, y) whenever Ω_{1} and Ω_{2} are disjoint open subsets

of Ω such that y ∉ f(\overline{Ω} \ Ω_{1} ∪ Ω_{2}) ; (d_{3}) d(f_{t}, Ω, y(t)) is independent of t ∈ T whenever y is continuous on T, the homotopy

(f_{t})_{t∈T} is of class (S_{+}) and y(t) ∉ f_{t}(∂Ω) on T.

The reader is refered to Browder [15]–[20] for more details.

3. Generalized Yosida approximation

Let X be a reflexive Banach space. In the sequel we assume that X and X^{∗} are locally uniformly convex, and we will identify a multi-valued mapping (or operator) A : X ⟼ X^{∗} with its graph in X × X^{∗} i.e. A = {(x, y) ∈ X × X^{∗}; y ∈ A(x)}. The domain of A is denoted by dom(A) = {x ∈ X; A(x) ≠ ∅}. A multi-valued operator A ⊂ X × X^{∗} is said to be monotone if for any (x_{i}, y_{i}) ∈ A, with i = 1,2, one has ⟨y_{1} − y_{2} , x_{1} − x_{2}⟩ ≥ 0. A is said to be maximal monotone if it is maximal in the family of monotone operators in X × X^{∗}, ordered by inclusion. Let now A ⊂ X × X^{∗} be a maximal monotone operator. Then the resolvent J_{λ}^{A}(x) for λ > 0 and A is defined as the unique solution of the equation

0 ∈ J(J_{λ}^{A}(x) − x) + λA(J_{λ}^{A}(x)).

The Yosida approximation is given by A_{λ}(x) = \frac{1}{λ}J(x − J_{λ}(x)) = (A^{−1} + λJ^{−1})^{−1}(x). For more details see [4], [10], [15], [27]. We will use an extension of the concept of Yosida approximation that is given by A^{λ} = A_{λ} + λJ. This notion will be called generalized Yosida approximation.

Definition 3.1. a) A net {A_{i} ⊂ X × X^{∗}; i ∈ I} is said to be graph convergent to A ⊂ X × X^{∗} if it converges in X × X^{∗} in the Kuratowski-Painlevé sense. In other words A = graph − lim A_{i} if the following inclusions hold :

lim sup_{i∈I} A_{i} ⊂ A ⊂ lim inf_{i∈I} A_{i}

with

lim inf_{i∈I} A_{i} = {(x, y) ∈ X × X^{∗}; ∃(x_{i}, y_{i}) ∈ A_{i} → (x, y)}

and

lim sup_{i∈I} A_{i} = {(x, y) ∈ X × X^{∗}; ∃K ⊂ I such that ∀ j ∈ K ∃(x_{j}, y_{j}) ∈ A_{j} → (x, y)}.

For maximal monotone operators the graph-convergence of (A_{i})_{i∈I} to A is equivalent to

A ⊂ lim inf_{i∈I} A_{i}.

b) A family {A_{t} ⊂ X × X^{∗}; t ∈ T } is graph-continuous if whenever t_{i} → t in T one has

A_{t} = graph − lim A_{t_{i}}.

Proposition 3.2. ([4], Prop. 3.60) Let (A_{i})_{i∈I} and A be a family of maximal monotone operators in X × X^{∗}. The following statements are equivalent :

(a) (A_{i})_{i∈I} is graph-convergent to A ;

(b) A_{i,λ}(x) converges strongly to A_{λ}(x) for every x ∈ X and every λ > 0.

The following theorem summarizes some properties which will be needed in the sequel.

Theorem 3.3. Let X be a reflexive Banach space such that both X and X^{∗} are locally uniformly convex. Let {A_{t} ⊂ X × X^{∗}; t ∈ T } be a graph-continuous family of maximal monotone operators. Then for any open bounded Ω ⊂ X the family \{A_{t}^{λ} : \overline{Ω} ⟶ X^{∗}; t ∈ T, λ > 0} is of class (S_{+}).

Proof. Let Ω be an open bounded subset of X. Suppose that t_{i} → t in T, λ_{i} → λ > 0 and x_{i} →^{w} x in X satisfy

lim sup_{i∈I}⟨A_{t_{i}}^{λ_{i}}(x_{i}), x_{i} − x⟩ ≤ 0.

First of all we obtain

lim sup_{i∈I}⟨A_{t_{i},λ_{i}}(x_{i}), x_{i} − x⟩ ≤ −λ lim inf_{i∈I}⟨J(x_{i}), x_{i} − x⟩ ≤ lim inf_{i∈I}⟨J(x), x_{i} − x⟩ ≤ 0. (1)

If we set y_{i} = A_{t_{i},λ_{i}}(x_{i}), y = A_{t,λ}(x) and z_{i} = A_{t_{i},λ}(x) for every i ∈ I we obtain that the points (x_{i} − λ_{i}J^{−1}y_{i}, y_{i}) and (x − λJ^{−1}z_{i}, z_{i}) are in A_{t_{i}}. From the monotonicity of A_{t_{i}} and J^{−1} it follows that

0 ≤ ⟨y_{i} − z_{i}, J^{−1}y_{i} − J^{−1}z_{i}⟩ ≤ \frac{1}{λ}⟨y_{i} − z_{i}, x_{i} − x⟩ + \frac{λ − λ_{i}}{λ} ⟨y_{i} − z_{i}, J^{−1}y_{i}⟩.  (2)

Because of Proposition 3.2, the net (z_{i})_{i∈I} converges strongly to y. Now, let (u_{0}, v_{0}) ∈ A_{t}. Since A_{t_{i}} graph-converges to A_{t} , there exists (u_{i}, v_{i}) ∈ A_{t_{i}} strongly convergent to (u_{0}, v_{0}). The monotonicity of A_{t_{i}} implies

0 ≤ ⟨y_{i} − v_{i} , x_{i} − λ_{i}J^{−1}y_{i} − u_{i}⟩ ≤ −λ_{i}∥y_{i}∥^{2} + (∥v_{i}∥ + \frac{1}{λ_{i}} ∥u_{i} − x_{i}∥)∥y_{i}∥ + ⟨v_{i}, u_{i} − x_{i}⟩,

and ∥y_{i}∥^{2} + α∥y_{i}∥ + β ≤ 0 for some real numbers α and β. Hence (J^{−1}y_{i})_{i∈I} is bounded and by using (2) we obtain

lim inf_{i∈I}⟨y_{i}, x_{i} − x⟩ ≥ lim inf_{i∈I}(⟨z_{i}, x_{i} − x⟩ + (λ_{i} − λ)⟨y_{i} − z_{i}, J^{−1}y_{i}⟩) ≥ 0.

Taking into account (1), we conclude that ⟨A_{t_{i},λ_{i}}(x_{i}) , x_{i} − x⟩ converges to zero . Because

of lim sup_{i∈I}⟨A_{t_{i}}^{λ_{i}}(x_{i}), x_{i} − x⟩ ≤ 0 we deduce

lim sup_{i∈I}⟨J(x_{i}), x_{i} − x⟩ ≤ −\frac{1}{λ} lim sup_{i∈I}⟨A_{t_{i}}^{λ_{i}}(x_{i}), x_{i} − x⟩ ≤ 0.

As a consequence of the local uniform convexity of X and the weak convergence of (x_{i})_{i∈I} to x, we conclude that (x_{i})_{i∈I} converges strongly to x.

4. Degree for maximal monotone operator

Now we are in a position to define the topological degree for maximal monotone operators. Let X be a reflexive Banach space, A ⊂ X × X^{∗} a maximal monotone operator and Ω a bounded open subset of X. We approximate the operator A, for λ > 0, by the generalized Yosida approximation A^{λ} = A_{λ} + λJ. Observe that A^{λ} is demicontinuous and of class (S_{+}).

Definition 4.1. We define the topological degree of A over Ω at 0 by the formula :

deg(A, Ω, 0) = lim_{λ↘0} d(A^{λ}, Ω, 0).  (3)

We first verify that in the definition above, the degree function deg is independent of λ > 0 for λ sufficiently small. On any closed subinterval of [0, λ], the family {A^{λ}; λ ∈ I} is of class (S_{+}). Therefore, by invariance of the Browder’s degree function under homotopies of class (S_{+}), the function d(A^{λ}, Ω, 0) will be independent of λ ∈ I , provided 0 ∉ A^{λ}(∂Ω) for every λ ∈ I. Otherwise, one can find a decreasing sequence (λ_{n}) converging to zero such that 0 ∈ A^{λ_{n}}(∂Ω). Since ∂Ω is bounded and the Banach space X is reflexive, one can find a weakly convergent sequence x_{n} ∈ ∂Ω to x in X such that

0 = A^{λ_{n}}(x_{n}) = A_{λ_{n}}(x_{n}) + λ_{n}Jx_{n}.

Thus

lim sup_{n→∞}⟨A^{λ_{n}}(x_{n}) , x_{n} − x⟩ ≤ 0.

By Theorem 3.3 it follows that the sequence (x_{n}) converges strongly to x and so x ∈ ∂Ω. On the other hand 0 = A^{λ_{n}}(x_{n}) is equivalent to v_{n} ∈ Au_{n} with v_{n} = −λ_{n}Jx_{n} and u_{n} = x_{n} − λ_{n}^{2}J^{∗}(−Jx_{n}). Since A is maximal we deduce that 0 ∈ Ax, for v_{n} →^{s} 0 and u_{n} →^{w} x. Thus we reach a contradiction with the assumption that 0 ∉ A(∂Ω). This common value permits us to define the extended degree function (3). With the above definition one can state the familiar properties of degree theory for maximal monotone mappings.

Theorem 4.2. Let Ω be an open bounded subset of a reflexive Banach space X which is, with its dual X^{∗}, locally uniformly convex . Let A ⊂ X × X^{∗} be a maximal monotone operator. Then we have (i) deg(J, Ω, 0) = 1, provided 0 ∈ Ω ; (ii) deg(A, Ω, 0) = 0 whenever 0 ∉ A(\overline{Ω}); (iii) if the homotopy of the maximal monotone operators {A_{t} ⊂ X × X^{∗}; t ∈ T } is graph

continuous and satisfies 0 ∉ ⋃{A_{t}(∂Ω); t ∈ T }, then deg(A_{t}, Ω, 0) is independent of t in T;

(iv) if Ω_{1} and Ω_{2} are two disjoint open subsets of Ω such that 0 ∉ A(\overline{Ω} \ Ω_{1} ⋃ Ω_{2}) , then

deg(A, Ω, 0) = deg(A, Ω_{1}, 0) + deg(A, Ω_{2}, 0).

Proof. Part (i) is obvious, since J^{λ} = J_{λ} + λJ = \frac{1 + λ + λ^{2}}{1 + λ} J and

deg(J, Ω, 0) = lim_{λ→0^{+}} d(J^{λ}, Ω, 0) = d(J, Ω, 0).

To prove (ii), suppose deg(A, Ω, 0) ≠ 0. Let λ_{0} > 0 be such that d(A^{λ}, Ω, 0) ≠ 0 for each 0 < λ ≤ λ_{0}. By Browder’s degree theorem (d_{1}) there exists x_{λ} ∈ \overline{Ω} such that 0 = A^{λ}(x_{λ}) for each λ ∈]0, λ_{0}]. Let (λ_{i})_{i∈I} be a net of positive constants which converge to zero. \overline{Ω} is bounded, therefore we can find a subnet, also denoted by (x_{i} = x_{λ_{i}})_{i∈I} and elements of \overline{Ω}, such that (x_{i})_{i∈I} converges weakly to x in X and ((1 + λ_{i}^{2})x_{i}, −λ_{i}Jx_{i}) ∈ A for every i ∈ I. Since A is maximal monotone we conclude that 0 ∈ A(x). On the other hand {A^{λ}; λ > 0} is of class (S_{+}), as pointed out in Theorem 3.3. We deduce that (x_{i})_{i∈I} converges strongly to x and x ∈ \overline{Ω}. Thus 0 ∈ A(\overline{Ω}), as desired.

(iii) By relying on Browder’s degree theorem (d_{3}) and Theorem 3.3 it suffices to show that for each t ∈ T there exists λ(t) > 0 such that for every λ ∈]0, λ(t)] one has 0 ∉ A_{t}^{λ}(∂Ω) and inf_{t∈T} λ(t) > 0. Assuming the contrary, since \overline{Ω} is bounded, we can find t_{0} in T and

nets (λ_{i})_{i∈I} in ]0, +∞[ and (x_{i})_{i∈I} in ∂Ω such that λ_{i} → 0, x_{i} →^{w} x in X and 0 ∈ A_{t_{0}}^{λ_{i}}(x_{i}). As in (ii) one deduces 0 ∈ A_{t_{0}}(\overline{x}). From this and the fact that A_{t_{0}} is monotone, we infer that

⟨λ_{i}Jx_{i} , \overline{x} − (1 + λ_{i}^{2})x_{i}⟩ ≤ 0

and

lim sup_{i∈I} ∥x_{i}∥ ≤ lim sup_{i∈I} \frac{1}{1 + λ_{i}^{2}} ∥\overline{x}∥ ≤ ∥\overline{x}∥.

Hence (x_{i})_{i∈I} converges strongly to \overline{x} since X is locally uniformly convex. As ∂Ω is closed we conclude that x ∈ ∂Ω, a contradiction to 0 ∉ A_{t_{0}}(∂Ω). For (iv), let λ_{0} > 0 be such that for any 0 < λ ≤ λ_{0} we have 0 ∉ A^{λ}(\overline{Ω} \ (Ω_{1} ∪ Ω_{2})). Otherwise there are λ_{i} → 0 and x_{i} →^{w} \overline{x} ∈ X such that (x_{i})_{i∈I} ⊂ (\overline{Ω}\(Ω_{1} ∪ Ω_{2})) . From this and the maximality of A we infer, by virtue of (iii), that \overline{x} ∈ (\overline{Ω}\(Ω_{1} ∪ Ω_{2}) ∩ A^{−1}(0), which is a contradiction. According to the property (d_{2}) of Browder’s degree theorem we obtain

d(A^{λ}, Ω, 0) = d(A^{λ}, Ω_{1}, 0) + d(A^{λ}, Ω_{2}, 0),

and so, by letting λ → 0,

deg(A, Ω, 0) = deg(A, Ω_{1}, 0) + deg(A, Ω_{2}, 0).

Theorem 4.3. On the family of maximal monotone operators there exists one and only one degree function with invariance under graph-continuous homotopies.

Proof. Let A be a maximal monotone operator and Ω a bounded open subset of X. Suppose that 0 ∉ A(∂Ω). As stated in Definition 4.1, for λ > 0 sufficiently small we have

deg(A, Ω, 0) = d(A^{λ}, Ω, 0)

and

0 ∉ A^{λ}(∂Ω).

Let us consider now a degree function d_{0} satisfying the properties (i), (ii), (iii) and (iv) of Theorem 4.2, and let M_{λ} be the family of the maximal monotone operators A_{t} = tA^{λ} + (1 − t)A , where λ > 0 is sufficiently small and t ∈ T = [0, 1]. It is clear that M_{λ} is graph-continuous. So that, if we suppose that 0 ∉ ⋃{A_{t}(∂Ω); t ∈ T }, we deduce by Theorem 4.2, (iii) that d_{0}(A, Ω, 0) = d_{0}(A^{λ}, Ω, 0); it suffices to take t = 0, 1. Using the unicity of the Browder’s degree for operators of class (S_{+}), one has d_{0}(A^{λ}, Ω, 0) = d(A^{λ}, Ω, 0). As the degree deg is independent of λ > 0 sufficiently small, it follows that d_{0}(A, Ω, 0) = deg(A, Ω, 0). Hence, if we suppose that the assertion of the degree function were false, there would exist u_{n} ∈ ∂Ω such that u_{n} →^{w} u in X, λ_{n} → 0 and t_{n} → t with 0 ∈ A_{t_{n}}(u_{n}). By taking some v_{n} ∈ A(u_{n}) and w_{n} = A_{λ_{n}}(u_{n}) one has −t_{n}λ_{n}Ju_{n} = t_{n}w_{n} + (1 − t_{n})v_{n}. Now for (x, y) in A, we have by monotonicity that

⟨w_{n} − y , u_{n} − λ_{n}J^{−1}w_{n} − x⟩ ≥ 0

and

⟨v_{n} − y , u_{n} − x⟩ ≥ 0.

Thus, by multiplying these two relations respectively by t_{n} and (1 − t_{n}), then adding, we obtain

⟨t_{n}w_{n} + (1 − t_{n})y , u_{n} − x⟩ = ⟨t_{n}λ_{n}Ju_{n} − y , x − u_{n}⟩ ≤ t_{n}λ_{n}⟨w_{n} − y , J^{−1}w_{n}⟩. (4)

A straightforward calculation shows that

t_{n}λ_{n}∥w_{n}∥^{2} ≤ (t_{n}λ_{n})^{1/2}∥y∥ + (∥t_{n}λ_{n}Ju_{n} − y∥.∥u_{n} − u∥)^{1/2}.

Thus the sequence (t_{n}λ_{n}∥w_{n}∥^{2}) is bounded. This permits us to deduce that (t_{n}λ_{n}w_{n}) converges strongly to zero. Returning to (4), we have

⟨y , u_{n} − x⟩ ≤ t_{n}λ_{n}∥w_{n}∥.∥y∥ + t_{n}λ_{n}∥u_{n}∥.∥u_{n} − x∥.

Letting n → ∞, it follows that ⟨y, x − u⟩ ≥ 0 for all (x, y) ∈ A. Hence 0 ∈ A(u) by maximility. Let us take now x = u and y = 0 in (4), then we have

lim sup_{n→∞}⟨Ju_{n} , u_{n} − u⟩ ≤ 0.

Since J is of class (S_{+}) it follows that u_{n} converges strongly to u. This implies that 0 ∈ A(∂Ω) and gives a contradiction.

Remark 4.4. Other properties of the topological degree given by Definition 4.1 can be proved by using the arguments of the preceding proposition. For example :

a) Up to a translation one can define the degree for maximal operators relative to Ω at a point y ∈ X, i.e. deg(A, Ω, y) = deg(A′, Ω′, 0) where A′(x) = A(x) − y and Ω′ = Ω − y.

b) (Homotopy) Suppose that all the conditions of Theorem 4.2 (iii) hold except that condition 0 ∉ ⋃{A_{t}(∂Ω) ; t ∈ T } is replaced by: t ⟼ y(t) is continuous and y(t) ∉ ⋃{A_{t}(∂Ω) ; t ∈ T } for each t in T. Then deg(A_{t}, Ω, y(t)) is independent of t. In fact, it is not hard to show that {A_{t} = A_{t} − y(t) ; t ∈ T } is a graph-continuous homotopy of maximal operators.

Comment. Let us now indicate how the above concepts and results for maximal monotone operators can be extended to m-accretive (hyperaccretive) mappings. An operator A ⊂ X × X is called accretive if for each (x_{i}, y_{i}) ∈ A , i = 1, 2, one has

⟨J(x_{1} − x_{2}), y_{1} − y_{2}⟩ ≥ 0.

The operator A is m-accretive iff A + λI is onto X for each λ > 0. To construct the topological degree for A it is sufficient to replace in the generalized Yosida approximation the duality mapping J by the identity I of X :

A^{λ} = (A^{−1} + λI^{−1})^{−1} + λI.

Note. Recently, Attouch, Penot and Riahi (see [5], Thm. 2.7) have investigated via the continuation methods the existence of solutions for parametrized nonlinear monotone problems. The approch in [5], [6] was based upon a connectedness argument. As a consequence of Theorem 4.2 we shall sharpen and extend these results from Hilbert to reflexive spaces, and also rule out the compactness conditions.

Theorem 4.5. Suppose that assumptions (iii) of Theorem 4.2 hold and the following condition is satisfied: for some t_{0} ∈ T one has deg(A_{t_{0}}, Ω, 0) ≠ 0. Then for each t ∈ T, A_{t}^{−1}(0) is nonempty and contained in Ω.

The proof relies only on the independence of d(A_{t}, Ω, 0) of t in T and the convexity (connectedness) of A_{t}^{−1}(0) since A_{t} is maximal monotone.

Proposition 4.6. Let A and B be two maximal monotone operators such that dom(A)∩ int(dom(B)) ≠ ∅ and A(x) = B(x) for each x ∈ ∂Ω, and moreover 0 ∉ A(∂Ω). Then deg(A, Ω, 0) = deg(B, Ω, 0).

Proof. The homotopy {A_{t} = tA + (1 − t)B; t ∈ T = [0, 1]} is a graph-continuous family of maximal monotone operators and satisfies all assumptions of Theorem 4.2 (iii). Then deg(A_{t}, Ω, 0) is independent of t in T. Take t = 0 and t = 1 and the claim follows.

Proposition 4.7. Suppose that A is maximal monotone and −μJu ∉ A(u) for each u in ∂Ω and μ > 1 (resp. μ ≥ 0). Then deg(A + J, Ω, 0) = 1 if 0 ∈ Ω and deg(A + J, Ω, 0) = 0 if 0 ∉ Ω (resp. deg(A, Ω, 0) = 1 if 0 ∈ Ω and deg(A, Ω, 0) = 0 if 0 ∉ Ω).

Proof. Considering the homotopy of maximal monotone operators A_{t} = tA + J (resp. A_{t} = tA + (1 − t)J), we have 0 ∉ A_{t}(∂Ω) for each t ∈ T = [0, 1]. Thus by Theorem 4.2 (iii) the statement follows, i.e. deg(A + J, Ω, 0) = deg(J, Ω, 0) (resp. deg(A, Ω, 0) = deg(J, Ω, 0)).

Corollary 4.8. Let A be a maximal monotone operator. Suppose that for some r > 0 and for each λ ≥ 0 ∥(A^{−1})_{λ}∥ ≠ r. Then A^{−1}(0) is nonempty and contained in B_{r}, the ball with radius r.

Proof. We apply Theorem 4.5 to {A_{t} = tA + (1 − t)J; t ∈ [0, 1]}. Since ∥(A^{−1})_{λ}∥ ≠ r for each λ ≥ 0, one has 0 ∉ ⋃{A_{t}(∂B_{r}); t ∈ [0, 1]}. We obtain the desired conclusion.

5. Application to real functions of convex type

In this section we apply the results of section 4 to convex functions and convex-concave saddle bifunctions. A - Before we state further consequences, let us first introduce some definitions. For further details see [4], [10], [27] and [52].

Definition 5.1. 1) Let X be a reflexive space. A function f : X ⟼ IR ∪ {+∞} is said to be convex lower semicontinuous (lsc) whenever its epigraph, epi(f) = {(x, t) ∈ X × IR ; f(x) ≤ t}, is convex and closed. Here X × IR is endowed with the product topology.

2) Let F = {f_{t} : X ⟼ IR ∪ {+∞}; t ∈ T } be a family of convex lsc functions. F is said to be Mosco-epicontinuous provided the homotopy {epi(f_{t}) ⊂ X × IR ; t ∈ T } is graph-continuous with respect to the strong topology and the weak convergence of X.

3) Let f be a proper convex lsc function. Then the subdifferential of f at x in X and the minimum set are given by

∂f(x) = {x^{∗} ∈ X^{∗}; f(x) ≤ f(u) + ⟨x^{∗}, x − u⟩ ∀u ∈ X}

and

M(f) = {x ∈ X; f(x) ≤ f(u) ∀u ∈ X}.

Since the subdifferential of f is a maximal monotone operator, one can define the degree of a proper convex lsc function at y relatively to an open bounded subset Ω of X by:

deg(f, Ω, y) = deg(∂f, Ω, y).  (5)

Remark 5.2. This definition permits us to extend the next result from Hilbert (see [45]) to reflexive spaces.

Proposition 5.3.

a) For a proper convex lsc function f such that M(f)∩∂Ω = ∅ , deg(f, Ω, 0) ≠ 0 implies ∅ ≠ M(f) ⊂ Ω. b) Let {f_{t}; t ∈ T } be a Mosco-epicontinuous family of proper convex lsc functions such

that M(f_{t}) ∩ ∂Ω = ∅ ∀t ∈ T and deg(f_{t_{0}}, Ω, 0) ≠ 0 for some t_{0} ∈ T. Then for each t in T one has ∅ ≠ M(f_{t}) ⊂ Ω.

Proof. a) This follows immediately from Theorem 4.2 (ii) and the relation (5).

b) Since the Mosco-epicontinuity of {f_{t}; t ∈ T } implies that the family {∂f_{t}; t ∈ T } is graph-continuous and M(f_{t}) = (∂f_{t})^{−1}(0), the conclusion follows from Theorem 4.5.

B - For bivariate functions (bi-functions), let X and Y be two reflexive Banach spaces which are in separate duality with X^{∗} and Y^{∗} via pairings denoted by ⟨, ⟩.

Let us consider a closed convex-concave bifunction F : X × Y ⟼ \overline{IR}, which is convex lower semicontinuous (resp. concave upper semicontinuous) with respect to the variable x in X (resp. y in Y). In [46] R. T. Rockafellar introduced the operator A = ∂_{1}F × −∂_{2}F, where ∂_{1}F and ∂_{2}F denote the partial subdifferentials of the convex functions F (., y) and −F (x, .). and proved that A is maximal monotone in X × Y. It is well known that the set S(F) = {(x, y) ∈ X × Y; inf_{u∈X} F (u, y) = sup_{v∈Y} F (x, v)} of

saddle points of F is exactly A^{−1}(0, 0). Let us consider a family {F_{t}; t ∈ T } of closed convex-concave bifunctions, which is Moscoepi/hypocontinuous whenever the homotopy

{C_{t}(x, y^{∗}) = sup_{y∈Y}(F_{t}(x, y) + ⟨y^{∗}, y⟩) ; t ∈ T }

of convex parents is Mosco-epicontinuous. Then the homotopy {∂C_{t}; t ∈ T } is graphcontinuous. Hence {A_{t} = ∂_{1}F_{t} × ∂_{2}F_{t}; t ∈ T } is graph-continuous. Further properties of these notions can be found in [7]–[9], [30], [46] and [52]. Let F : X × Y ⟼ \overline{IR} be a closed convex concave bifunction, Ω an open bounded subset of X^{∗} × Y^{∗} and (x, y) ∈ X × Y. The degree of F at (x, y) is defined by

deg(F, Ω, (x, y)) = deg(A, Ω, (x, y)).

With the above definition and properties one can easily state the analogue of Proposition 5.2 for bifunctions :

Proposition 5.4.

a) deg(F, Ω, 0) ≠ 0 and S(F) ∩ ∂Ω = ∅ imply that there exists a saddle point of F at Ω.

b) Let {F_{t}; t ∈ T } be a Mosco-epi/hypocontinuous homotopy such that deg(F_{t_{0}}, Ω, 0) ≠ 0 for some t_{0} ∈ T and S(F_{t}) ∩ ∂Ω = ∅ for each t in T. Then for every t ∈ T one has

∅ ≠ S(F_{t}) ⊂ Ω.

Acknowledgment. The author would like to thank the anonymous referees for their careful reading and useful remarks.

H. Riahi / Topological degree for maximal monotone operators 131

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