A new construction of the degree of maximal monotone maps
aa r X i v : . [ m a t h . A P ] A p r A NEW CONSTRUCTION OF THE DEGREE OF MAXIMALMONOTONE MAPS
MOHAMMAD NIKSIRAT
Abstract.
The inclusion equations of the type f ∈ T ( x ) where T : X → X ∗ is a maximal monotone map, are extensively studied in nonlinear analysis. Inthis paper, we present a new construction of the degree of maximal monotonemaps of the form T : Y → X ∗ , where Y is a locally uniformly convex andseparable Banach space continuously embedded in X . The advantage of thenew construction lies in the remarkable simplicity it offers for calculation ofdegree in comparison with the classical one suggested by F. Browder. We provea few classical theorems in convex analysis through the suggested degree. Introduction
Assume that X is a uniformly convex Banach space, Y is separable and reflexiveBanach spaces equipped with uniformly convex norms, and i : Y → X is thecontinuous embedding. Furthermore, assume that T : Y → X ∗ is a maximalmonotone map with the effective domain D ( T ) = Y in the following sense. A pair(˜ y, ˜ x ∗ ) is in the graph of T if the condition h x ∗ − ˜ x ∗ , i ( y − ˜ y ) i ≥ y, x ∗ ) ∈ graph( T ) where h , i denotes the continuous pairing between X, X ∗ . Inthis article, we propose a topological degree T possessing classical properties oftopological degree in certain sense. The construction generalizes the F. Browder’sclassical degree of maximal monotone maps [1]. The Browder’s degree is constructedas follows. Assume that X is a reflexive Banach space equipped with a uniformlyconvex norm and T : X → X ∗ a maximal monotone map. The map T ǫ = T + ǫJ , for ǫ > J : X → X ∗ is the duality mapping possesses the following properties:(1) T ǫ is a map satisfying the following condition: for any x n ⇀ x in X , if thereis x ∗ n ∈ T ( x n ) such thatlim sup n →∞ h x ∗ n + ǫJ ( x n ) , x n − x i ≤ , then x n → x .(2) The map T ǫ is onto X ∗ ,(3) if x = x , the sets T ( x ) , T ( x ) are disjoint, that is, T ( x ) ∩ T ( x ) = ∅ . (4) The map T − ǫ : X ∗ × (0 , ∞ ) → X is well defined and continuous.Notice that J is single valued, bijective and bi-continuous if X is uniformly convex.It is shown that the map ( T − ǫ + ǫJ − ) − : X → X ∗ is a single valued demi-continuous and ( S ) + for which a degree theory has been developed by F. Browder. Mathematics Subject Classification.
Primary 47H11, Secondary 47H07.
Key words and phrases. degree theory, finite rank approximation, maximal monotone maps,multivalued maps.
The degree of T at 0 ∈ X ∗ in an open bounded set D ⊂ X is defined by thefollowing relation(1.1) deg( T, D,
0) = lim ǫ → deg(( T − ǫ + ǫJ − ) − , D, . The degree suggested in this article generalizes the Browder’s degree in the sensethat if Y = X , the two degrees are the same. The main advantage of the suggesteddegree is the direct use of finite rank approximation we employed in our previouswork [2] for single valued mappings. It is seen that the constructed degree makescalculations much simpler than (1.1). We take note that that the suggested degreeis different from the degree of the map i ∗ ◦ T : Y → Y ∗ . The difference betweentwo formulations is discussed in [2] for single valued maps. Definition 1.1.
Assume that X and X are Banach spaces. A map A : X → X is called upper semi-continuous at x ∈ X if for every neighborhood V of A ( x ), thereexists an open neighborhood U of x such that T ( U ) ⊂ V .We have the following theorem for the upper semi-continuous multi-valued map-pings; see for example [3, 4]. Theorem 1.2. ( ǫ -continuous subgraph) Assume that X and X are Banachspaces, and the map A : X → X is upper semi-continuous. If A ( x ) is closedand convex for all x ∈ X , then for any ǫ > , there exists a continuous singlevalued function A ǫ : X → X such that for any x ∈ X , there exists z ∈ X and ˜ z ∈ A ( z ) such that k x − z k < ǫ and k A ǫ ( x ) − ˜ z k < ǫ . Proposition 1.3.
Let
X, Y be Banach spaces, i : Y → X a continuous embeddingand T : Y → X ∗ a maximal monotone map with the effective domain Y . Then T ( y ) is closed and convex for all y ∈ Y , and T is norm to weak-star upper semi-continuous in the following sense. For arbitrary y ∈ Y , and arbitrary sequence ( y n ) converges to y in norm, there is a weakly limit point x ∗ of ∪ n T ( y n ) such that x ∗ ∈ T ( y ) . The proof is completely similar to one for the map T : X → X ∗ . For a proof ofthe standard version see for example [5].If Y is a separable and reflexive Banach space equipped with a uniformly convexnorm, a theorem by Browder and Ton [6] guarantees the existence of a separa-ble Hilbert space H such that the embedding j : H ֒ → Y is dense and com-pact. Choosing an orthogonal basis { h k } ∞ k =1 for H , we obtaine the basis Y = { y , y , · · · , y n , · · · } for Y where y k = j ( h k ), and accordingly, the filtration Y = { Y n } , where Y n = span { y , . . . , y n } . The following proposition is simply verified. Proposition 1.4.
For any y ∈ Y , there is a sequence ( y n ) , y n ∈ Y n such that y n → y . The pairing in Y n is denoted by ( , ) and is defined by the relation ( y i , y j ) = δ ij for all y i , y j ∈ Y . We define the maximal monotone operato T : Y → X ∗ in thefollowing sense. Definition 1.5.
Suppose X and Y are separable and reflexive Banach spacesequipped with uniformly convex norm, and assume that T : Y → X ∗ is a maximalmonotone map. The finite rank approximation of arbitrary x ∗ ∈ T ( y ) in Y n ∈ Y , is NEW CONSTRUCTION OF THE DEGREE OF MAXIMAL MONOTONE MAPS 3 defined by ˆ x n = P nk =1 h x ∗ , i ( y k ) i y k . Accordingly, the finite rank map T n : Y → Y n is defined by the relation(1.2) T n ( y ) = [ x ∗ ∈ T ( y ) ˆ x n . For any x ∗ ∈ X ∗ and y ∈ Y n , we have the property(ˆ x n , y ) = h x ∗ , i ( y ) i , where h , i is the pairing between X ∗ , X . In fact, if x ∗ ∈ X ∗ , then for ˆ x n = P nk =1 h x ∗ , y k i y k , we have n X k =1 h x ∗ , i ( y k ) i ( y k , y ) = n X k =1 h x ∗ , i (( y k , y ) y k ) i = * x ∗ , i n X k =1 ( y k , y ) y k !+ , and thus h x ∗ , i ( y ) i = (ˆ x n , y ). Lemma 1.6.
The finite rank approximation T n is upper semi-continuous and forevery x ∈ Y , the set T n ( x ) is closed and convex.Proof. Fix n and ǫ >
0. If T n is not upper semi-continuous at x ∈ Y , there isa sequence ( δ m ) , δ m → x m ∈ B δ m ( x ) such that for some ˆ x n,m ∈ T n ( x m ),we have ˆ x n,m V ǫ ( T n ( x )). T is maximal monotone, and thus locally bounded.Therefore, there is a subsequence (shown for the sake of simplicity again by ˆ x n,m )such that ˆ x n,m → ˆ x . We show ˆ x ∈ T n ( x ). Since ˆ x n,m ∈ T n ( x m ), there is x ∗ m ∈ T ( x m ) such that ˆ x n,m are the finite rank approximation in Y n of x ∗ m , that is,ˆ x n,m = n X k =1 h x ∗ m , i ( y k ) i y k . Since T is norm to weak-star upper semi-continuous, and x m → x , we have x ∗ m ⇀ x ∗ for some x ∗ ∈ T ( x ). Thus h x ∗ m , i ( y k ) i → h x ∗ , i ( y k ) i for all 1 ≤ k ≤ n . Thereforeˆ x n,m = n X k =1 h x ∗ m , i ( y k ) i y k → n X k =1 h x ∗ , i ( y k ) i y k ∈ T n ( x ) , and therefore ˆ x ∈ T n ( x ). Now we show that T n ( x ) is closed for all x ∈ Y . Con-sider an arbitrary sequence ˆ x m ∈ T n ( x ), and ˆ x m → ˆ x . Let x ∗ m ∈ T ( y ) be thesequence such that ˆ x m = P nk =1 h x ∗ m , i ( y k ) i y k . Since T ( y ) is bounded and convex,the sequence x ∗ m converges weakly (in a subsequence) to some x ∗ ∈ T ( x ) and thusˆ x m = n X k =1 h x ∗ m , i ( y k ) i y k → n X k =1 h x ∗ , i ( y k ) i y k , and thus ˆ x = n X k =1 h x ∗ , i ( y k ) i y k ∈ T n ( x ) . That T n ( x ) is convex follows simply from the convexity of T ( x ). (cid:3) By the Lemma (1.6) and Theorem (1.2), the ǫ -continuous selection T n,ǫ of T n iswell defined. The single valued map T n,ǫ is continuous and for any x ∈ Y n , there is z ∈ Y n and ˆ z ∈ T n ( z ) such that k z − x k ∈ ǫ and k ˆ z − T n,ǫ ( x ) k < ǫ . MOHAMMAD NIKSIRAT Degree definition
Let ( ǫ n ) be a positive sequence such that ǫ n →
0. Fix ǫ >
0. Consider thefunction ˜ T n,ǫ n : Y n → Y n defined by the relation(2.1) ˜ T n,ǫ n = T n,ǫ n + ǫJ n , where T n,ǫ n is the ǫ n -continuous selection of T n and J n is the finite rank approx-imation of J ◦ i : Y → X ∗ in Y n where J : X → X ∗ is the bi-continuous dualitymap. Lemma 2.1.
Let D ⊂ Y be an open bounded set and assume that for some ǫ > ,we have cl T ǫ ( ∂D ) , where T ǫ = T + ǫJ ◦ i . Then there is N > such that ˜ T n,ǫ n ( ∂D n ) for all n ≥ N where D n = D ∩ Y n .Proof. Otherwise, there is a sequence z n ∈ ∂D n such that ˜ T n,ǫ n ( z n ) = 0 for all n ≥
1. Since ∂D is bounded, there is a subsequence (we show again by z n ) thatweakly converges to z . We first show that z n converges strongly to z . Choose asequence ζ n ∈ Y n that converges to z in norm. Since ˜ T n,ǫ n ( z n ) = 0 on Y n , we have( T n,ǫ n ( z n ) , z n − ζ n ) + ǫ ( J n ( z n ) , z n − ζ n ) = 0 , because z n − ζ n ∈ Y n . By the relation( J n ( z n ) , z n − ζ n ) = h J ◦ i ( z n ) , i ( z n − ζ n ) i , we can write(2.2) ( T n,ǫ n ( z n ) , z n − ζ n ) + ǫ h J ◦ i ( z n ) , i ( z n − ζ n ) i = 0 . On the other hand, for each z n , there is x n ∈ Y n and ˆ x n ∈ T n ( x n ) such that k x n − z n k < ǫ n , k ˆ x n − T n,ǫ n ( z n ) k < ǫ n . Therefore, we have( T n,ǫ n ( z n ) , z n − ζ n ) = ( T n,ǫ n ( z n ) − ˆ x n , z n − ζ n ) + (ˆ x n , z n − ζ n ) , and by the relation k T n,ǫ n ( z n ) − ˆ x n k < ǫ n , we obtain( T n,ǫ n ( z n ) , z n − ζ n ) ≥ − ǫ n k z n − ζ n k + (ˆ x n , z n − ζ n )By the relation k x n − z n k < ǫ n , we have(ˆ x n , z n − ζ n ) ≥ − ǫ n k ˆ x n k + (ˆ x n , x n − ζ n ) . Since ˆ x n ∈ T n ( x ), there are x ∗ n ∈ T ( x n ) such that ˆ x n are the finite rank approxi-mations of x ∗ n in Y n . Thus, we can write(ˆ x n , x n − ζ n ) = h x ∗ n , i ( x n − ζ n ) i . Also, for some
C >
0, we can write h x ∗ n , i ( x n − ζ n ) i ≥ − C k x ∗ n kk z − ζ n k + h x ∗ n , i ( x n − z ) i . Choose an arbitrary z ∗ ∈ T ( z ). We have h x ∗ n , i ( x n − z ) i ≥ h x ∗ n − z ∗ , i ( x n − z ) i + h z ∗ , i ( x n − z ) i ≥ h z ∗ , i ( x n − z ) i . Since z n ⇀ z and k x n − z n k →
0, we concludelim n →∞ ( T n,ǫ n ( z n ) , z n − ζ n ) ≥ . NEW CONSTRUCTION OF THE DEGREE OF MAXIMAL MONOTONE MAPS 5
Thus, the relation (2.2) implieslim sup n →∞ h J ◦ i ( z n ) , i ( z n − ζ n ) i ≤ . By the relation ζ n → z , we concludelim sup n →∞ h J ◦ i ( z n ) , i ( z n − ζ n ) i ≤ , and since J is a map of class ( S ) + , we obtain z n → z ∈ ∂D . Now we show0 ∈ cl T ǫ ( z ). Choose arbitrary y ∈ Y and sequence y n ∈ Y n , y n → y . We have( T n,ǫ n ( z n ) , y n ) + ǫ h J ◦ i ( z n ) , i ( y n ) i = 0 . Since J is continuous, we have h J ◦ i ( z n ) , i ( y n ) i → h J ◦ i ( z ) , y i . Choose x n ∈ Y n and ˆ x n ∈ T n ( x n ) such that k x n − z n k < ǫ n , k ˆ x n − T n,ǫ n ( z n ) k < ǫ n . We have lim | ( T n,ǫ n ( z n ) , y n ) − (ˆ x n , y n ) | = 0 . For x ∗ n ∈ T ( x n ), and by the relation y n → y , we obtainlim | ( T n,ǫ n ( z n ) , y n ) − h x ∗ n , i ( y ) i| = 0 . Since T is norm to weak-star upper semi-continuous, we conclude h x ∗ n , i ( y ) i →h x ∗ , y i for some x ∗ ∈ T ( z ). This implies that x ∗ + ǫJ ◦ i ( z )=0 and thus 0 ∈ cl T ǫ ( z )that contradicts the condition 0 cl T ǫ ( ∂D ). (cid:3) Proposition 2.2.
Assume that cl T ( ∂D ) . Then there is ǫ > such that cl T ǫ ( ∂D ) .Proof. By the assumption, there is r > , cl T ( ∂D )) = r . Let z ∈ ∂D is arbitrary. Take arbitrary z ∗ ∈ T ( z ). We have k z ∗ + ǫJ ( z ) k ≥ k z ∗ k − ǫ k z k ≥ r − ǫ k z k . Therefore 0 cl T ǫ ( ∂D ) if(2.3) 0 < ǫ < r max z ∈ ∂D k z k . The boundedness of ∂D guarantees the existence of ǫ > (cid:3) Definition 2.3.
Assume that X and Y are separable and reflexive Banach spacesequipped with uniformly convex norms, D ⊂ Y is an open bounded set and T : Y → X ∗ is a maximal monotone map such that 0 cl T ( ∂D ). Choose ǫ > .
3) and consider the map ˜ T n,ǫ n defined in (2 . T in D with respect to 0 is defined by the following formula(2.4) deg( T, D,
0) = lim n →∞ deg B ( ˜ T n,ǫ n , D n , , where deg B is the usual Brouwer’s degree of the map ˜ T n,ǫ n . Proposition 2.4.
The degree defined in the relation (2 . is stable with respect to n . MOHAMMAD NIKSIRAT
Proof.
Consider the sequence of mappings ( ˜ T k,ǫ k ) such that for sufficiently large n the condition 0 cl ˜ T k,ǫ k ( ∂D k ) is satisfied for k ≥ n − ǫ > < ǫ , ǫ < ǫ , the following relation holds(2.5) deg B ( ˜ T n,ǫ , D n ,
0) = deg B ( ˜ T n,ǫ , D n , , In fact, for any x ∈ Y n , there is z , z ∈ Y n and ˆ z ∈ T n ( z ), ˆ z ∈ T n ( z ) such that k z − x k < ǫ , k ˆ z − T n,ǫ ( x ) k < ǫ , k z − x k < ǫ , k ˆ z − T n,ǫ ( x ) k < ǫ . The continuity of T n,ǫ , T n,ǫ implies that k ˜ T n,ǫ − ˜ T n,ǫ k can be controlled and thus(2.5) holds. Let us write ˜ T n,ǫ n as˜ T n,ǫ n = ˜ T n,ǫ n + ˜ T n,ǫ n , where ˜ T n,ǫ n is the projection of ˜ T n,ǫ n into Y n − and ˜ T n,ǫ n is the projection into { y n } . Define the map S n,ǫ n : Y n → Y n as S n,ǫ n ( x ) = ˜ T n,ǫ n ( x ) + ( x, y n ) y n Obviously, we have(2.6) deg B ( S n,ǫ n , D n ,
0) = deg B ( ˜ T n,ǫ n , D n − , . First we show(2.7) deg B ( ˜ T n,ǫ n , D n − ,
0) = deg B ( ˜ T n − ,ǫ n , D n − ,
0) = deg B ( ˜ T n − ,ǫ n − , D n − , . The last equality follows from (2.5). In order to prove the first equality, we notethat if T n,ǫ n is an ǫ n -continuous selection of T n , then T n,ǫ is also an ǫ n -continuousselection of T n − ,ǫ n . In fact, let x ∈ Y n − be arbitrary, then there is z ∈ Y n andˆ z ∈ T n ( y ) such that k ˆ z − T n,ǫ n ( x ) + ˆ z − T n, ǫ n k < ǫ n , where ˆ z ∈ Y n − and ˆ z ∈ { y n } . This implies k ˆ z − T n,ǫ n ( x ) k < ǫ n , k z − y k < ǫ n . Again it follows that k T n − ,ǫ n − T n,ǫ n k can be controlled and thus the first equalityin (2.7) is proved. Now, we show(2.8) deg B ( ˜ T n,ǫ n , D n ,
0) = deg B ( S n,ǫ n , D n , . Consider the convex homotopy h n ( t ) = (1 − t ) ˜ T n,ǫ n + tS n,ǫ n . It is enough to show 0 h n ( t )( ∂D n ) for t ∈ [0 , h n ( t )( ∂D n ) for t = 0 ,
1. For t ∈ (0 ,
1) assume that there exists a sequence t n ∈ (0 ,
1) and ( z n ), z n ∈ ∂D n such that h n ( t n )( z n ) = 0. According to the construction of h n ( t ) wehave 0 = h n ( t n )( z n ) = ˜ T n,ǫ n ( z n ) + (1 − t n ) ˜ T n,ǫ n ( z n ) + t n ( z n , y n ) y n . The above relation implies ˜ T n,ǫ n ( z n ) = 0 and˜ T n,ǫ n ( z n ) = − t n − t n ( z n , y n ) y n . NEW CONSTRUCTION OF THE DEGREE OF MAXIMAL MONOTONE MAPS 7
Since ∂D is bounded then z n ⇀ z in a subsequence. Choose the sequence ( ζ n ) , ζ n ∈ Y n and ζ n → z and obtain( ˜ T n,ǫ n ( z n ) , z n − ζ n ) = − t n − t n | ( z n , y n ) | + t n − t n ( z n , y n )( ζ n , y n ) . On the other hand since ζ n → z , we have ( ζ n , y n ) →
0. Since there exists sequenceˆ z n ∈ T n ( z n ) such that k ˆ z n − T n,ǫ n ( z n ) k < ǫ n we can write for some z ∗ n ∈ T ( z n )lim sup n →∞ h z ∗ n + ǫJ ◦ i ( z n ) , i ( z n − z ) i = lim sup n →∞ ( ˜ T n,ǫ n ( z n ) , z n − ζ n ) . Therefore we obtain lim sup n →∞ h z ∗ n + ǫJ ◦ i ( z n ) , i ( z n − z ) i ≤ . Since lim n →∞ h z ∗ n , i ( z n − z ) i ≥ , we obtain lim sup n →∞ h J ◦ i ( z n ) , i ( z n − z ) i ≤ , and thus z n → z . This is impossible because 0 cl T ǫ ( ∂D ). (cid:3) Now, we show that the definition (2.4) satisfies the classical properties of atopological degree including the solvability and the homotopy invariance.
Theorem 2.5.
Let D ⊂ Y be an open bounded set and assume that T : Y → X ∗ is maximal monotone and cl T ( ∂D ) . If deg( T, D, = 0 , then there is y ∈ D such that ∈ T ( y ) .Proof. Assume deg(
T, D, = 0, then there exists a sequence z n ∈ D such that˜ T n,ǫ n ( z n ) = 0 for sufficiently small ǫ n >
0. This implies that there is the sequenceˆ z n ∈ T n ( z n ) such that k ˆ z n + ǫJ n ( z n ) k < ǫ n . Since D is bounded then z n convergesweakly (in a subsequence) to some z . Since T is monotone, we concludelim sup n →∞ h J ◦ i ( z n ) , i ( z n − z ) i ≤ , and thus z n converges strongly to z ∈ cl( D ). Let ˆ z n be the n -approximation of z ∗ n ∈ T ( z n ). Since T is norm to weak-star upper semi-continuous, the sequence z ∗ n converges weakly in a subsequence to some z ∗ ∈ T ( z ). Let v ∈ Y be arbitrary.Consider the sequence ( v n ) , v n ∈ Y n and v n → y . Then we have h z ∗ n , i ( v n ) i = − ǫ h J ◦ i ( z n ) , i ( v n ) i − ǫ ( T n,ǫ n ( z n ) − ˆ z n , v n i → , On the other hand, h z ∗ n , i ( v n ) i → h z ∗ , i ( y ) i , and thus z ∗ = 0 or equivalently 0 ∈ T ( z ). Since 0 cl T ( ∂D ), we conclude z ∈ D . (cid:3) Definition 2.6.
Let D ⊂ X be an open bounded set, and assume h : [0 , × Y → X ∗ be a continuous homotopy with respect to t such that for any t ∈ [0 , h ( t ) : Y → X ∗ is maximal monotone. Furthermore assume that 0 cl h ([0 , × ∂D ). The map h is called an admissible homotopy for maximal monotone maps. Proposition 2.7.
The degree defined in (2 . is stable under the admissible homo-topy of maximal monotone maps. MOHAMMAD NIKSIRAT
Proof.
According to the definition of the admissible homotopy, the degreedeg B (˜ h n,ǫ n ( t ) , D n , t due to the fact 0 ˜ h n,ǫ n ( t )( ∂D n ) for t ∈ [0 ,
1] and the homotopyinvariance of the Brouwer’s degree. Now, the stability of the defined degree (2 . n implies that the degree deg( h ( t ) , D,
0) is independent of t . (cid:3) Degree theoretic proofs
We give degree theoretic proofs of some theorems in convex analysis. The firsttheorem is due to D. DeFigueirdo [7].
Theorem 3.1.
Assume that X is a separable uniformly convex Banach space, T : X → X ∗ is a maximal monotone map such that ( T + λJ )( S r ) , where S r is the sphere of radius r and λ > is arbitrary. Then there exists u ∈ cl( B r ) suchthat ∈ T ( u ) , where B r is the ball of radius r in X .Proof. Assume that 0 T (cl( B r )). We show first that 0 cl T ( S r ). Otherwise thereexist a sequence u n ∈ S r and u ∗ n ∈ T ( u n ) such that u ∗ n →
0. The sequence ( u n )converges weakly in a subsequence (that we show again by u n ) to some u ∈ cl( B r ) . Claim: [ u, ∈ graph( T ). For any [ x, x ∗ ] ∈ graph( T ) we have the inequality h x ∗ , x − u i = lim h x ∗ − u ∗ n , x − u n i ≥ . Since T is maximal monotone, the above inequality implies [ x, ∈ graph( T ) orequivalently 0 ∈ T ( u ). This contradicts the assumption 0 T (cl( B r )). It is alsoapparent that 0 cl( J ( S r )). Next, we show 0 cl((1 − t ) T + tJ )( S r )) for t ∈ (0 , t n ∈ (0 , u n ∈ S r and u ∗ n ∈ T ( u n ) such that(1 − t n ) u ∗ n + t n J ( u n ) → . Again for u n ⇀ u and t n → t we obtain by the monotonicity property of T thefollowing inequality lim sup n →∞ h J ( u n ) , u n − u i ≤ , that implies u n → u ∈ S r . Claim: [ u, − t − t J ( u )] ∈ graph( T ). For any [ x, x ∗ ] ∈ graph( T ) we obtain by the fact − t − t J ( u n ) ∈ T ( u n ) the following relation h x ∗ + t − t J ( u ) , x − u i = lim h x ∗ + t − t J ( u n ) , x − u n i ≥ , that proves the claim. Now by degree theoretic argument we havedeg( T, B r ,
0) = deg((1 − t ) T + tJ, B r ,
0) = deg(
J, B r ,
0) = 1 . The above calculation guarantees the existence of u ∈ B r such that 0 ∈ T ( u )and this contradicts the assumption 0 T (cl( B r )). Therefore the assumption0 T (cl( B r )) is wrong and thus 0 ∈ T (cl( B r )). (cid:3) The next theorem is again from DeFigueirdo [7].
Proposition 3.2.
Let X be a separable uniformly convex Banach space and assumethat f : X → X ∗ is a pseudo-monotone map. Then Rang( ∂N r + f ) = X ∗ where N r is the map N r ( x ) = (cid:26) x ∈ B r x ∈ S r , and ∂N r is the set of the sub-gradients of N r . NEW CONSTRUCTION OF THE DEGREE OF MAXIMAL MONOTONE MAPS 9
Proof.
Apparently, we have(3.1) ∂N r ( x ) = (cid:26) x ∈ B r { λJ ( x ) , λ ≥ } if x ∈ S R . Claim: for every f ∈ X ∗ , we have(3.2) deg( ∂N r + f − f , B r , = 0 . First we show if 0 ( ∂N r + f − f )(cl( B r )) then(3.3) 0 cl( ∂N r + f − f )( S r ) . Otherwise, there is a sequence u n ∈ S r and u ∗ n ∈ ∂N r ( u n ) such that u ∗ n + f ( u n ) − f →
0. But u n ⇀ u ∈ cl( B r ) in a sub-sequence. We prove that [ u, f − f ( u )] ∈ graph( ∂N r ). Let f = u ∗ n + f ( u n ) + ǫ ( n ) where ǫ ( n ) ∈ X ∗ and ǫ ( n ) →
0. For anyarbitrary [ x, x ∗ ] ∈ graph( ∂N r ), we have h x ∗ − f + f ( u ) , x − u i = lim h x ∗ + f ( u ) − u ∗ n − f ( u n ) , x − u n i ≥ lim h f ( u ) − f ( u n ) , x − u i . But(3.4) 0 = lim h u ∗ n + f ( u n ) − f , u n − u i ≥ lim sup h f ( u n ) , u n − u i Since f is pseudo-monotone we obtain f ( u n ) ⇀ f ( u ) and therefore h x ∗ − f + f ( u ) , x − u i ≥ . This implies that [ u, f − f ( u )] ∈ graph( ∂N r ) and thus 0 ∈ ( ∂N r + f − f )(cl( B r ))which is impossible by the assumption. Now consider the affine homotopy(3.5) h ( t ) = (1 − t )( ∂N r + f − f ) + tJ. for t ∈ (0 , cl((1 − t )( ∂N r + f − f ) + tJ )( S r ) . Otherwise, there is a sequence u n ∈ S r , u ∗ n ∈ ∂N r ( u n ) and t n → t such that(1 − t n )( u ∗ n + f ( u n ) − w ) + t n Ju n → . But u n ⇀ u ∈ cl( B r ) in a subsequence. W show[ u, f − f ( u ) − t − t J ( u )] ∈ graph( ∂N r ) . For any [ x, x ∗ ] ∈ graph( ∂N r ) we have h x ∗ − f + f ( u ) + t − t J ( u ) , x − u i =lim h x ∗ + f ( u ) + t − t J ( u ) − u ∗ n − f ( u n ) − t n − t n J ( u n ) , x − u i ≥≥ lim sup h f ( u ) − f ( u n ) , x − u i + lim inf h t − t J ( u ) − t n − t n J ( u n ) , x − u i We conclude u n → u ∈ S r and f ( u n ) ⇀ f ( u ) because f is pseudo-monotone.Therefore we obtain again0 ∈ ( ∂N r + f + t − t J − f )(cl( B r )) . But ( ∂N r + f + t − t J − f )(cl( B r )) = ( ∂N r + f − f )(cl( B r )) and then 0 ∈ ( ∂N R + f − f )(cl( B r )) that is impossible. Finally we use the homotopy invariance propertyof degree and writedeg( ∂N R + f − f , B r ,
0) = deg( h ( t ) , B R ,
0) = deg(
J, B r ,
0) = 1 . Therefore there exist u ∈ cl( B r ) such that f ∈ ∂N r ( u ) + f ( u ). (cid:3) The following theorem is due to F. Browder [8] for the surjectivity of the mono-tone maps with locally bounded inverse.
Theorem 3.3.
Assume A : X → X ∗ is a demi-continuous monotone map suchthat A − is locally bounded, that is, for every f ∈ X ∗ there is a bounded V f ∋ f such that A − ( V f ) is bounded. Then A is onto.Proof. For any f ∈ X ∗ , we show that there is sufficiently large r = r ( f ) such that:deg( A, B r , f ) = 0 . Choose r > V f ∋ f the following condition issatisfied S r ∩ A − ( V f ) = ∅ , or equivalently f cl A ( S r ). Since there is ǫ > A, B r , f ) = deg( A + ǫJ, B r , f ) , it is enough to show(3.6) deg( A + ǫJ, B r , f ) = 0 , for sufficiently large r and sufficiently small ǫ >
0. First, we showdeg( A + ǫJ, B r , = 0 . In fact, if ( A + ǫJ )( z ) = 0 for z ∈ ∂B r , then h A ( z ) − A (0) , z i + ǫ k z k + h A (0) , z i = 0 . Since A is monotone, the inequality ǫ k z k + h A (0) , z i ≤ ǫ k z k ≤ k A (0) k ,that is impossible for sufficiently large r . Since A + ǫJ is a map of class ( S ) + , definethe homotopy h ( t ) = tA + ǫJ . It is simply seen that 0 h ( t )( ∂B r ) and thendeg( A + ǫJ, B r ,
0) = deg( h ( t ) , B r ,
0) = deg(
J, B r , = 0 . The proof of (3.6) is completely similar to one presented above. (cid:3)
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A new generalization of the Browder’s degree of (S)+ maps , appear soon.[3] A. Cellina,
Approximation of set valued functions and fixed point theorems , Ann. Mat. PuraAppl., 82:1724, 1969.[4] , N. G. Lloyd,
Degree Theory , Cambrige University Press, 1978.[5] V. P. Barbu,
Nonlinear differential equations of monotone types in Banach spaces , Springer,2010.[6] F. Browder and B. A. Ton,
Nonlinear functional equations in Banach spaces and ellipticsuper regularization , Math Z., 105:177–195, 1968.[7] D. de Figuerido,
An existence theorem for pseudo-monotone operator equation in Banachspaces , J. Math. Anal. Appl., 34:151156, 1971.[8] F. Browder.
Nonlinear operators and nonlinear equations of evolution in Banach spaces .Americam Math. Soc., 1976.
NEW CONSTRUCTION OF THE DEGREE OF MAXIMAL MONOTONE MAPS 11
Department of Mathematics, University of Alberta, Edmonton, Canada, T6G 2J5
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