A NEW TOPOLOGICAL DEGREE THEORY FOR DENSELY DEFINED QUASIBOUNDED (S̃+)-PERTURBATIONS OF MULTIVALUED MAXIMAL MONOTONE OPERATORS IN REFLEXIVE BANACH SPACES

Let X be an infinite-dimensional real reflexive Banach space with dual space X∗ and G⊂ X open and bounded. Assume that X and X∗ are locally uniformly convex. Let T : X ⊃ D(T)→ 2X be maximal monotone and C : X ⊃ D(C)→ X∗ quasibounded and of type (S̃+). Assume that L ⊂ D(C), where L is a dense subspace of X , and 0 ∈ T(0). A new topological degree theory is introduced for the sum T +C. Browder’s degree theory has thus been extended to densely defined perturbations of maximal monotone operators while results of Browder and Hess have been extended to various classes of single-valued densely defined generalized pseudomonotone perturbationsC. Although the main results are of theoretical nature, possible applications of the new degree theory are given for several other theoretical problems in nonlinear functional analysis.


Introduction and preliminaries
In what follows, the symbol X stands for an infinite-dimensional real reflexive Banach space which has been renormed so that it and its dual X * are locally uniformly convex.The symbol • stands for the norm of X, X * and J : X → X * is the normalized duality mapping.In what follows, "continuous" means "strongly continuous" and the symbol "→" (" ") means strong (weak) convergence.
The symbol R(R + ) stands for the set (−∞,∞)([0,∞)) and the symbols ∂D, D denote the strong boundary and closure of the set D, respectively.We denote by B r (0) the open ball of X or X * with center at zero and radius r > 0.
For an operator T : X → 2 X * , we denote by D(T) the effective domain of T, that is, D(T) = {x ∈ X : Tx = ∅}.We denote by G(T) the graph of T, that is, G(T) = {(x, y) : x ∈ D(T), y ∈ Tx}.An operator T : X ⊃ D(T) → 2 X * is called "monotone" if for every x, y ∈ D(T) and every u ∈ Tx, v ∈ T y, we have u − v,x − y ≥ 0. (1.1) and only if R(T + λJ) = X * for all λ ∈ (0,∞).If T is maximal monotone, then the operator T t ≡ (T −1 + tJ −1 ) −1 : X → X * is bounded, continuous (see Lemma 3.1 below), maximal monotone and such that T t x T {0} x as t → 0 + for every x ∈ D(T), where T {0} x denotes the element y * ∈ Tx of minimum norm, that is, T {0} x = inf{ y * : y * ∈ Tx}.
In our setting, this infimum is always attained and D(T {0} ) = D(T).Also, T t x ∈ TJ t x, where J t ≡ I − tJ −1 T t : X → X and satisfies lim t→0 J t x = x for all x ∈ co D(T), where co A denotes the convex hull of the set A. The operators T t , J t were introduced by Brézis et al. in [2].For their basic properties, we refer the reader to [2] as well as Pascali and Sburlan [22, pages 128-130].In our setting, the duality mapping J is single-valued and bicontinuous.
An operator T : X ⊃ D(T) → Y , with Y another real Banach space, is "bounded" if it maps bounded subsets of D(T) onto bounded sets.It is "compact" if it is continuous and maps bounded subsets of D(T) onto relatively compact subsets of Y .It is "demicontinuous" ("completely continuous") if it is strong-weak (weak-strong) continuous on D(T).
Lemma 1.1.Let T : X ⊃ D(T) → 2 X * be maximal monotone.Then the following are true: (i) {x n } ⊂ D(T), x n → x 0 and Tx n y n y 0 imply x 0 ∈ D(T) and y 0 ∈ Tx 0 ; (ii) {x n } ⊂ D(T), x n x 0 and Tx n y n → y 0 imply x 0 ∈ D(T) and y 0 ∈ Tx 0 .
From Lemma 1.1, we see that either one of (i) and (ii) implies that the graph G(T) of the operator T is closed, that is, G(T) is a closed subset of X × X * .
Recent related eigenvalue problems can be found in the paper of the authors [14] and the paper of Li and Huang [20].
In Section 2, we summarize the construction of our recent degree theory for two densely defined mappings T, C, where T is at least single-valued and maximal monotone, and C satisfies an L-related quasiboundedness condition and an L-related generalized (S + )-condition with respect to T. Here, L is a dense subspace of X.
Section 3 contains the construction of the new degree.We only assume that 0 ∈ D(T), 0 ∈ T(0), and T : X → 2 X * is maximal monotone.Unlike [15], we do not assume that T is densely defined and conditions like (t 2 )-(t 4 ) (see Section 2), which make T stronger than just maximal monotone.The operator C : X ⊃ D(C) → X * , with L ⊂ D(C), is quasibounded, finitely continuous on subspaces of L (see (c 3 ) below) and satisfies condition ( S + ).In [15], we assumed that C satisfies a quasiboundedness condition and a condition of type (S + ), which involve the operator T and the space L.
The basic characteristic of the new degree is that the maximal monotone operator T may be multivalued but not necessarily densely defined.We should note here that the new degree theory does not contain the theory developed in [15] as a special case.Although the two degree theories overlap for certain combinations of operators T, C, and the degree in [15] is used for the construction herein, they are generally different even in the important case of a single-valued maximal monotone operator T. The new degree theory is also a substantial extension of Browder's degree theory in [5].Browder's perturbation term is defined on the closure of an open and bounded set in the space X.However, we should mention here that our degree definition uses the degree of the mapping T t + C, which is constant for all small values of t.Such an approach was first used by Browder in [5].Naturally, we have to show here that this homotopy function T t + C is admissible for our degree in [15].This is the content of Theorem 3.3.
Naturally, every new degree theory is useful provided that it carries appropriate homotopies that can be used for the calculation of the degree.Theorem 4.3 contains a basic homotopy result.This result is used in Theorem 4.4(iii), of Section 4, in order to obtain a rather important homotopy that we have actually used in all the applications of the new mapping theorems of Section 6.Again, unlike the main homotopy that has been used for the degree of [15], the main feature of the above homotopy is that we no longer assume that the dense linear space L lies in both domains D(T), D(C).Such an assumption must be made for the degree in [15], and it precludes us from considering many simple affine homotopies of the type H(t,•) ≡ t(T + C 1 ) + (1 − t)C 2 for general maximal monotone operators T. Simply, under this assumption, the degree d(H(t,•),G,0) might not be well defined, although 0 ∈ H(t,•)(∂G), t ∈ [0,1], and both degrees d(C 1 ,G,0), d(C 2 ,G,0) are well defined.This, at times, is due to the fact that the mappings T, C 1 have domains that contain different dense subspaces L.
An index theory for densely defined operators and the degree developed in [15] can be found in the authors' paper [17].
Section 4 contains some basic properties of the new degree including two basic homotopies.
In Section 5, we extend some results of Browder and Hess [6] about generalized pseudomonotone operators to pairs of operators T, C covered by the new degree theory.
Further mapping theorems for the new degree are given in Section 6.

The degree for densely defined mappings T, C
We exhibit below, in a summary, the degree theory that was recently developed by the authors in [15].In this degree theory, both operators T, C are densely defined, and d(T + C,G,0) comes from approximation by finite-dimensional Brouwer degrees.We note that the operator T is single-valued and the operator C satisfies two basic conditions (quasiboundedness and generalized (S + )) involving the dense subspace L ⊂ D(T) ∩ D(C) of the space X as well as the operator T itself.This introduction is instructive in view of the degree theory that we are going to develop later in this paper.
Let L be a subspace of X and let Ᏺ(L) be the set of all finite-dimensional subspaces of L. Consider a single-valued operator T : X ⊃ D(T) → X * satisfying the following conditions: (t 1 ) T is monotone, that is, for every u,v ∈ D(T).Moreover, we have u 0 ∈ D(T) and Tu 0 = h * 0 ; (t 3 ) for any u 0 ∈ D(T), we have (2.4) (t 4 ) for every F ∈ Ᏺ(L), v ∈ L, the mapping σ(F,v) : F → , defined by σ(F,v)u = Tu,v is continuous.Note that the conditions (t 2 ), (t 3 ) are automatically satisfied by a maximal monotone operator T whose domain D(T) = L.
We also consider a second operator C : X ⊃ D(C) → X * satisfying the following conditions: (c 1 ) and C is quasibounded with respect to T, that is, for every number S > 0, there exists a number K(S) > 0 such that from the inequalities we have Cu ≤ K(S); (c 2 ) the operator C satisfies the following generalized (S + ) condition with respect to T: for every sequence {u n } ⊂ L such that u n u 0 , Cu n h 0 and for some u 0 ∈ X, h 0 ∈ X * , we have u n → u 0 , u 0 ∈ D(C) and Cu 0 = h 0 ; A. G. Kartsatos and I. V. Skrypnik 125 where Then there exists a space F 0 ∈ Ᏺ(L) such that for every space (2.9) Let F ∈ Ᏺ(L) and let v 1 ,...,v k be a basis for F. We define a finite-dimensional mapping (2.10) be the space defined in Lemma 2.1.Then for every space F ∈ Ᏺ(L) with F 0 ⊂ F, the following relation holds: where (T + C) F is the finite-dimensional mapping defined by (2.10), and deg denotes the Brouwer degree.
Definition 2.3 (degree for densely defined T, C).Assume that the operators T, C and the set G satisfy the conditions of Theorem 2.2.Then the degree d(T + C,G,0) is defined by where the operator (T + C) F is defined by (2.10), and F 0 is the finite-dimensional subspace of L determined by Lemma 2.1.
The basic properties of our degree can be found in [15].We do need to exhibit the basic homotopy invariance property of this degree.It is contained in Theorem 2.5.Before we state it, we need certain facts and a definition.
Remark 2.6.It is important to mention here that our degree theory above was actually developed in [15] with S in place of 0 in the first inequality in (2.6) and (2.13).It can be seen that the present situation is sufficient for the development of our degree after a careful study of the construction in [15].

The construction of the new degree
We are now ready to state the hypotheses needed for our new degree.As above, L is a dense subspace of X carrying the family Ᏺ(L).For the operator T, we assume the following: (t1) T : X ⊃ D(T) → 2 X * is maximal monotone with 0 ∈ D(T) and 0 ∈ T(0).
For the operator C, we assume that (c1) C : X ⊃ D(C) → X * , with L ⊂ D(C), is quasibounded, that is, for every S > 0, there exists K(S) > 0 such that u ∈ D(C) with implies Cu ≤ K(S); (c2) the operator C satisfies condition ( S + ); (c3) for every Condition (c3) here is the same as condition (c 3 ).It is included with a new symbol for convenience.
The following lemma is a new result that shows the continuity of the operator (t,x) → T t x on (0, ∞) × X.For D(T) in place of X, this was shown differently in the paper [30].
Proof.Fix δ > 0. Let {x n } ⊂ X, {t n } ⊂ [δ,∞) be such that x n → x 0 and t n → t 0 .Let y * n = T tn x n .Then, for some z n ∈ D(T) with y * n ∈ Tz n , Using the monotonicity of the operator T and the condition 0 ∈ T(0), we get which gives us the boundedness of the sequence {y * n } and hence the boundedness of {z n } by (3.2).Since X, X * are reflexive, we may assume that y * n y * 0 , z n z 0 and J −1 y * n y 0 .Using this and the monotonicity of the duality mapping The second inequality of (3.4) follows from which implies Using the (S + )-property of the operator J −1 , we obtain y * n → y * 0 .Passing to the limit in (3.2) and taking into consideration that y * 0 ∈ Tz 0 , we get Thus, y * 0 = (T −1 + t 0 J −1 ) −1 x 0 = T t0 x 0 , and the proof is complete.The following theorem will allow us to define the degree

Assume that the operator T satisfies condition (t1) and the operator C satisfies conditions (c1)-(c3). Assume that G ⊂ X is open and bounded and that
(3.17) Then there exists Proof.Assume that (3.17) is true and that the conclusion is false.Then there exists a sequence {t n } such that t n ↓ 0, and a sequence Since G is bounded, we may assume that x n x 0 ∈ X.Since {x n } is bounded and Cx n ,x n ≤ 0, because T tn x n ,x n ≥ 0, we have by the quasiboundedness of C that { Cx n } is also bounded.We may thus assume that Cx n h * ∈ X * .We claim that (3.18) implies (1.2).Assume that this is not true.Then there exists a subsequence of {x n }, denoted again by {x n }, such that This implies We also have T tn x n −h * .Consequently, along with we obtain Thus, by (3.22), or Using the fact that the operator C satisfies condition ( S + ), we conclude that x n → x 0 , x 0 ∈ D(C) ∩ ∂G, and Cx 0 = h * .This says T tn x n −h * and and implies, as in the argument starting with (3.20) above, that x 0 ∈ D(T) and Tx 0 −h * .Consequently, Tx 0 + Cx 0 0 with The proof is complete.
Theorem 3.3 below contains the fact that the degree d(T t + C,G,0) is well defined and constant.The latter follows from the fact that the operator T t + C defines a homotopy which is admissible for all small t > 0, according to Definition 2.4.

Theorem 3.3. Assume that the operator T satisfies condition (t1) and the operator C satisfies conditions (c1)-(c3). Assume that G ⊂ X is open and bounded and that
is well defined and constant for every t ∈ (0,t 1 ]. Proof.We first note that 0 ∈ T(0) implies T t (0) = 0, t > 0. In order to define the degree d(T t + C,G,0), we need to show that the operators T t , C satisfy the conditions (t 1 )-(t 4 ) and (c 1 )-(c 3 ), respectively.We know that T t is maximal monotone and continuous.This takes care of (t 1 ) and (t 4 ).
To show (t 2 ), fix t > 0 and let (u 0 ,h * 0 ) ∈ X × X * be such that Since L is dense in X and T t is continuous, it follows easily that this inequality holds for all u ∈ X.Since T t is maximal monotone, this says that u 0 ∈ D(T t ) = X and T t u 0 = h * 0 .Thus, (t 2 ) is true.
To show (t 3 ), we fix t > 0 and note that u 0 ∈ X and imply, by the continuity of T t and the density of L in X, that the same inequality is true for v ∈ X.This however is false because one such v is the element u 0 .We now show that C satisfies (c 1 ), (c 2 ), and (c 3 ) is identical to (c3) with T t in place of T. To see that (c 1 ) is satisfied, it suffices to observe that, in view of c1, the first inequality of (2.6) is true without the term Tu because Tu,u ≥ 0. To see that (c 2 ) is satisfied, it suffices to observe that (c2) is stronger than (c 2 ).
It follows that the degree d(T t + C,G,0), t ∈ (0,t 1 ], is well defined.Now, fix the point t 0 ∈ (0,t 1 ) and let λ(t . Since t 0 is picked arbitrarily in (0,t 1 ), in order to show that this degree is constant on (0, t 1 ], it suffices to show that {T λ(t) + C}, t ∈ [0,1], is an admissible homotopy in the sense of Definition 2.4 with M t = T λ(t) and A t = C.
To this end, we observe first that (m (1)  t ) is satisfied by what we saw above.
To see that (m (2)  t ) is true, we observe that for every v ∈ L, actually for every v ∈ X ⊃ L, and every t ∈ [0,1], we have by Lemma 3.1 above, which shows our assertion.
As far as A t = C is concerned, we have already checked the validity of (a (1)  t ) and (a (3)  t ).To show (a (2)  t ), we observe that A tj = C and that the assumptions on C in it are stronger than those of ( S + ).
Definition 3.4 (degree for ( S + )-perturbations C).Assume that the operators T, C and the set G satisfy the conditions of Theorem 3.3.Assume that 0 Then the new degree d(T + C,G,0) is defined by where t 1 is as in the conclusion of Theorem 3.2.We also set Remark 3.5.We note that in the above definition the operator C − p * satisfies all the assumptions (c1)-(c3).Thus, the degree d(T + C − p * ,G,0) is well defined.To see, in particular, that C − p * is quasibounded, let u ≤ S and Cu − p * ,u ≤ S, where S is a positive constant.Then where Thus, the quasiboundedness of C implies Cu ≤ K(S 1 ) and Cu − p * ≤ K(S 1 ) + p * ≡ K(S), where K(S) is now the quasiboundedness constant for C − p * .We should also point out that the degree d(J,G,0) is well defined if 0 ∈ J(∂G), which is equivalent to 0 ∈ ∂G.We are allowed to take C = εJ, ε > 0 (or C = εJ ψ , with J ψ defined in Section 5), in Definition 3.4.However, we are not allowed to have C = 0 there.This is due to the fact that C = 0 does not satisfy the ( S + )-condition.

Basic properties of the new degree
We are now going to establish (see Theorem 4.3 below) a homotopy property of the new degree.This property is used in Theorem 4.4(iii) in order to establish a more concrete and useful homotopy.
Lemma 4.1.Assume that the family of operators {T τ } satisfies the condition (t (1)  τ ).Then the condition (t (2)  τ ) is equivalent to the following condition: (t (3)  τ ) let {τ n } ⊂ [0,1] be such that τ n → τ 0 and let x ∈ D(T τ0 ), x * ∈ T τ0 x, then there exist sequences x n ∈ D(T τn ), x * n ∈ T τn x n such that x n → x and , be a second one-parameter family of operators satisfying the following conditions: (c (1)  τ ) the family {C τ } is "uniformly quasibounded", that is, for every S > 0, there exists K(S) > 0 such that imply the estimate C τ u ≤ K(S); (c (2)  τ ) for every pair of sequences for some c1)-(c3) of Section 3, respectively, with the space L independent of i.We say that the operators T (0) + C (0) , T (1) + C (1) are "homotopic" with respect to the open bounded set G ⊂ X if there exist one-parameter families of operators T τ : , satisfying conditions (t (1)  τ ), (t (2)  τ ) and (c (1)  τ )-(c (3) τ ), respectively, and such that ) When the operators T τ , C τ are as above, we also say that the mapping H(τ,x) ≡ (T τ + C τ )x is an "admissible homotopy." Theorem 4.3.Assume that the operators of Section 3, respectively.Assume that the operators T (0) + C (0) , T (1) + C (1) are homotopic with respect to the bounded open set G ⊂ X.Then if T τ , C τ are as in Definition 4.2, it holds that where the degrees are well defined according to Definition 3.4.
Proof.We let We assume that the contrary is true.Then there exist sequences Since T τn tn u n ,u n ≥ 0, we have C τn u n ,u n ≤ 0 and from the uniform quasiboundedness of {C τ } follows the boundedness of { C τn u n }.We may assume that C τn u n h * ∈ X * .We are going to show that Using the condition (c (2)  τ ), we conclude that u n → u 0 , u 0 ∈ D(C τ0 ) ∩ ∂G, and Repeating the argument that we carried out above starting with (4.12), we obtain from (4.16) (with ">" replaced by "≥") u 0 ∈ D(T τ0 ), −h * ∈ T τ0 u 0 .Thus, 0 ∈ T τ0 u 0 + C τ0 u 0 with u 0 ∈ D(T τ0 ) ∩ D(C τ0 ) ∩ ∂G, and we have a contradiction with (4.6).The proof of (4.8) is complete.
We fix t 0 ∈ (0, t1 ] and introduce the operator M τ = T τ t0 .We need to check that conditions (m (1)  t )-(m (3)  t ) are satisfied for the operator M τ and conditions (a (1)  t )-(a (3)  t ) are satisfied for the operator C τ .Then the assertion of the theorem will follow immediately from Theorem 2.5.
Condition (m (1)  t ).Conditions (t 1 )-(t 4 ) have already been checked in the proof of Theorem 3.3 for the operator satisfying the condition t1.
Condition (m (2)  t ).We have to show that for every u ∈ X, the mapping τ → M τ u is continuous.Consider the sequence {τ n } ⊂ [0,1] such that τ n → τ 0 .Let v * n = M τn u.Then there exists w n ∈ D(T τn ) such that Using the monotonicity of the operator T τn and the condition 0 ∈ T τn (0), we get A. G. Kartsatos and I. V. Skrypnik 135 which yields the boundedness of the sequence {v * n } and hence the sequence {w n }.Thus, we may assume that for some v 0 ,w 0 ∈ X, v * 0 ∈ X * .From (4.18), (4.20), we obtain Using the condition (t (2)  τ ), we get from (4.21) v * 0 ∈ T τ0 w 0 , v * n ,w n → v * 0 ,w 0 .From this and (4.18), we have This implies v * n → v * 0 by the (S + )-property of the operator J −1 .Passing to the limit in (4.18), we get u = t 0 J −1 v * 0 + w 0 , which gives v * 0 = T τ0 t0 u = M τ0 u, and the proof of the condition (m (2)  t ) is complete.
. The proof of this condition goes as in the case of condition (m (2)  t ).It is therefore omitted.
An important homotopy is included in the statement of Theorem 4.4(iii) below.Let the mapping φ : + → + be such that φ(0) = 0 and if r n > 0, n = 1,2,..., satisfies lim n→∞ φ r n = 0, (4.24) then r n → 0 + .We say that the operator C : X → X * belongs to the class Γ φ if there exists a function φ, as above, such that Cx,x ≥ φ( x ), x ∈ X. ) Here, C 1 satisfies c1-c3 and C 2 : X → X * is bounded, demicontinuous, of type (S + ), and belongs to the class Γ φ , for some function where Proof.Property (i) is a well-known property of the degree mapping which goes back to Skrypnik in 1973 (see [27]).In fact, (4.25) follows from the fact that because the operator T s + λJ is demicontinuous, bounded, strictly monotone (and thus one-to-one), and satisfies T s x + λJx,x ≥ 0, x ∈ ∂G (cf.Browder [4, Theorem 3(iv)]).Equation (4.26) follows from the same Browder reference as well.
We first consider the case τ 0 = 0. Assume that τ n = 0 for all large n.Then C τn u n = C 2 u n h * 0 and the first inequality of (4.36), along with the (S + )-property of C 2 , implies u n → u 0 and C τn u n C 0 u 0 = C 2 u 0 = h * 0 .Now, assume that there exists a subsequence {n k } of {n} such that τ nk = 0 for all k.Then C τn k u nk = C 2 u nk h * 0 and the first inequality of (4.36), which holds for {n k } instead of {n}, along with the (S + )-property of C 2 , implies u nk → u 0 , C τn k u nk C 0 u 0 = C 2 u 0 = h * 0 .Let, for another subsequence {n j } of {n}, τ nj > 0 for all j.Then the second inequality of (4.36) says that and implies easily the boundedness of { C 1 u nj } and |τ nj C 1 u nj ,u nj | → 0. Then the first inequality of (4.36), along with the (S + )-property of C 2 , implies again u nj → u 0 and It is evident from the above analysis that u n → u 0 , We are thus done with the case τ 0 = 0.
For the case τ 0 > 0, we set We assume first that τ 0 ∈ (0,1).We also assume that the first inequality of (4.41) is true.Then The assumption ( S + ) for C 1 implies that u nk → u 0 , u 0 ∈ D(C 1 ), and and the (S + )-property of C 2 imply u nk → u 0 ∈ X.Since C 2 is demicontinuous, C 2 u 0 = h * 2 .Since C 1 satisfies condition ( S + ) and the first inequality in (4.41) is true again, we have u 0 ∈ D(C 1 ) and C 1 u 0 = h * 1 .The rest of the proof for this case follows exactly as above.It is therefore omitted.
We now assume that τ 0 = 1.Then since Again, the ( S + )-property of C 1 says that u n → u 0 ∈ D(C 1 ) and C 1 u 0 = h * 1 .Thus, C 2 u 0 = h * 2 , and the situation repeats itself as above.Property c (3)  τ follows immediately from the corresponding property of C 1 and the demicontinuity of C 2 .
We will now prove that d(H(t,•),G,0) does not depend on t ∈ (0,1].We fix t 0 ∈ (0,1) and consider the homotopy We set T τ t = (T τ −1 + tJ −1 ) −1 .We will establish the existence of a number δ > 0 such that Assume that the contrary is true.Then there exist sequences τ n ∈ (0,1), t n ∈ (0,∞), x n ∈ ∂G such that τ n → 0, t n → 0, and The other possible case of τ n = 0, for some n, can be easily discarded.Using the monotonicity of the operator T τ t , the equality T τ t 0 = 0 and the inequality C 2 x,x ≥ 0, we obtain from (4.52) The quasiboundedness of the operator C 1 implies the boundedness of the sequence {C 1 x n }.Now, we have from (4.52) which contradicts the fact that the operator C 2 belongs to the class Γ φ .It follows that Consider the homotopy We need to prove that this homotopy satisfies all the conditions of Theorem 2.5.We will check the conditions (m (2)  t ), (m (3) t ).It is sufficient to establish the continuity of T δt δ x with respect to t, x.Define the mapping where J : X → X * is the duality mapping.Then x n ∈ X be such that t n → t 0 , x n → x 0 .From (4.57), we obtain the existence of y * n ∈ TJ tn x n , y * 0 ∈ TJ t0 x 0 such that Using this, the monotonicity of the operator T and the assumptions 0 ∈ D(T), 0 ∈ T(0), we have which implies the boundedness of the sequence {J tn x n }.
From (4.58) and the monotonicity of the operator T, we get From this inequality, the boundedness of the sequence {J tn x n }, t n → t 0 , and x n → x 0 , we obtain where z n = x n − J tn x n , z 0 = x 0 − J t0 x 0 .This and a well-known property of the duality mapping imply z n → z 0 and, consequently, J tn x n → J t0 x 0 .Then T δtn δ x n → T δt0 δ x 0 by virtue of (4.57), and the desired continuity of the mapping T δt δ has been established.Using Definition 3.4, the properties of the operator C τ established above and Theorem 2.5, we obtain the equality (4.50) and the proof of part (iii) is complete.
The proof of part (iv) follows simply from Theorem 4.3 and it is therefore omitted.
(v) Let T t : X → X * be as in Theorem 3.2.As in the proof of Theorem 3.2, we establish the existence of t 1 ∈ (0,∞) such that Using the additivity property of the degree of the operator T t + C, which follows simply from our construction in [15], we have Assertion (v) follows from this and Definition 3.4.

Extending results of Browder and Hess
We denote by J ψ the duality mapping with gauge function ψ.The function ψ : + → + is continuous, strictly increasing and such that ψ(0) = 0 and ψ(r) → ∞ at r → ∞.This mapping J ψ is continuous, bounded, surjective, strictly and maximal monotone, and satisfies condition (S + ).Also, J ψ x,x = ψ( x ) x and J ψ x = ψ( x ), x ∈ X.Thus, J ψ ∈ Γ φ , where φ(r) = ψ(r)r.For these facts, we refer to Petryshyn [23, pages 32-33 and 132].Petryshyn used in [23, Lemma 2.5] the separability of X in order to get a convergent subsequence of a bounded sequence {J ψ x j } there.However, the separability of X is not needed in our setting because of the Eberlein-Smulyan theorem about reflexive spaces.For the property d(J ψ ,G,0) = 1, for any bounded open set G containing zero, see Lemma 5.10 below.
The following proposition shows how we can solve an important approximate problem for the operator T + C.This approximate problem, inclusion (5.3) below, can be used in a variety of problems in nonlinear analysis, that is, problems of solvability, existence of eigenvalues, ranges of sums, invariance of domain, bifurcation, and so forth. where p * ∈ X * is fixed, and ε is a positive constant.Then the degree d(H(t,•),G,0) is well defined and In particular, the inclusion Proof.The conclusion of this proposition follows from (i)-(iii) of Theorem 4.4.In fact, one may take here C 1 = C − p * + εJ ψ and C 2 = J ψ .Then the homotopy invariance in (iii) of Theorem 4.4 says that (5.2) is true.This says that by Theorem 4.4(i), because 0 ∈ G. Finally, Theorem 4.4(ii) implies (5.3).
We need the following definition from Browder and Hess [6].

maximal monotone, and has effective domain D(T) = X.
A generalized pseudomonotone operator C : The operator T + C in our degree theory (as well as in Proposition 5.1 and Theorem 5.4 below) is generalized pseudomonotone.This is included in the following lemma.Lemma 5.3.Let T satisfy condition (t1).Let C : X ⊃ D(C) → X * be generalized pseudomonotone and satisfy (c1), (c3).Then the operator T + C is generalized pseudomonotone.Proof.Our assertion follows from Theorem 1 of Browder and Hess [6, page 260].In fact, it suffices to notice that T is generalized pseudomonotone (see [6, Proposition 2, page 257]) and such that u,x ≥ 0 for all (x,u) ∈ G(T), while C is generalized pseudomonotone and quasibounded ("strongly quasibounded" according to Browder and Hess [6]).However, we cannot replace, within our methodology, the operator T + C by a single multivalued generalized pseudomonotone operator, because we have no degree theory, as yet, for such mappings.
In order to demonstrate the applicability of our new degree theory, we give below an existence theorem concerning single-valued and densely defined generalized pseudomonotone perturbations.This result uses the homotopy function of Proposition 5.1, where the condition ( S + ) for the operator C is actually replaced by the weaker assumption of generalized pseudomonotonicity and does not follow from any of the results of Browder and Hess [6].A related result is in [10,Theorem 2.1].
A. G. Kartsatos and I. V. Skrypnik 143 Theorem 5.4 (existence).Let T satisfy (t1).Let C : X ⊃ D(C) → X * satisfy (c1), (c3) and be generalized pseudomonotone.Assume that there exist a constant Q > 0 and β : [Q,∞) → + , with β(r) → 0 as r → ∞, such that: for every x ∈ D(T) ∩ D(C) with x ≥ Q and every u ∈ Tx, it holds that where ψ is a gauge function.Then, for every ε > 0, R(T ) Proof.We fix p * ∈ X * , ε > 0, and consider the problem As in Proposition 5.1, we consider the homotopy inclusion and apply Theorem 4.4(iii).To this end, we need to show first that the operator U = C + εJ ψ − p * satisfies the conditions (c1)-(c3), and the operator J ψ satisfies the conditions on C 2 in Theorem 4.4(iii).The latter is obviously true.Also, it is evident that the operator U satisfies c1, c3.To show that U satisfies c2, assume that x n x 0 , Ux n h * , and for some x 0 ∈ X, h * ∈ X * .Since {J ψ x n } is bounded, we may assume that Cx n h * 1 .We show that x n → x 0 , x 0 ∈ D(U) and Ux 0 = h * .Since p * ,x n − x 0 → 0, (5.11) implies (5.12) Using the monotonicity of J ψ , we get which, along with (5.12), gives Since C is generalized pseudomonotone, we obtain x 0 ∈ D(C), Cx 0 = h * 1 and Cx n ,x n → Cx 0 ,x 0 .Thus, (5.15) Using this in (5.12), we get Since J ψ is of type (S + ), we have x n → x 0 and J ψ x n → J ψ x 0 .Consequently, It follows that c2 is satisfied.We now show that all possible solutions of the inclusion (5.10) are bounded by a constant which is independent of t ∈ [0,1].To this end, assume that there exists a sequence If there exists a subsequence {t mk } of {t m } such that t mk = 0, k = 1,2,..., then x mk = 0 for all k, which contradicts x mk → ∞ as k → ∞.We may thus assume that or, for some u m ∈ Tx m , By our hypothesis, assuming that x m ≥ Q for all m, we find where β( x m ) → 0 as m → ∞.Using this along with (5.18), we obtain (5.20) This says that ε ≤ β( x n ) → 0 as m → ∞, that is, a contradiction.Thus, there exists a number r > 0 such that all possible solutions of (5.10) lie in the ball B r (0).Consequently, no solution of (5.10) lies in ∂B r (0), and the degree mapping d(H(t,•),B r (0),0) is well A. G. Kartsatos and I. V. Skrypnik 145 defined.By the homotopy invariance property of this degree (Theorem 4.4(iii)), we obtain By (ii) of Theorem 4.4, the inclusion (5.9) is solvable for every ε > 0. Let x n be a solution of However, for some u n ∈ Tx n , we have that is, a contradiction.Since As Kartsatos has noted in [11, page 1673], neither one of the conditions (5.7), (5.8) is sufficient for the surjectivity of the operator T + C under the rest of the assumptions of Theorem 5.4.The simple counterexamples of [11] hold true here as well.
From the proof of the above theorem, we have the following useful lemma.As a special case of the above theorem, we obtain the following single-valued extension of Theorem 5 of Browder and Hess [6, page 273].In [6], it was assumed that the operator C is coercive.
Corollary 5.7.Assume that T satisfies (t1) and C : X ⊃ D(C) → X * satisfies (c1), (c3) and is generalized pseudomonotone.Assume that (a) there exist constants k > 0, Q > 0 such that Proof.We observe first that (5.29) implies where β(r) = k/r.Thus, (5.7) is true with ψ(r) = r.Consequently, Theorem 5.4 implies R(T + C + εJ) = X * , that is, given any p * ∈ X * , the inclusion is solvable for every ε > 0. Here, J ψ = J, that is, J ψ is the normalized duality mapping.We fix p * ∈ X * and consider a solution x n of the inclusion To show that {x n } is bounded, we assume that the contrary is true.Then, without any loss of generality, we may also assume that In that theorem, the operator C is multivalued and coercive.If C is coercive in Corollary 5.7, then both conditions (a), (b) in it are trivially satisfied because T + C is also coercive.When C is coercive in Corollary 5.7, then this corollary is also related to [6,Theorem 5].In that theorem, T is the zero operator and C is multivalued, "weakly quasibounded" (i.e., for every S > 0, there exists K(S) > 0 such that: (x, y * ) ∈ G(C) with x ≤ S and y * ,x ≤ S x imply y * ≤ K(S)) generalized pseudomonotone, and such that L ⊂ D(C) and a condition like c3 is satisfied.However, unlike our simple degree-theoretic argument, the proof of Theorem 5 in [6] is about 5 pages long (cf.[6, pages 273-279]).
We now consider the solvability of a Leray-Schauder type of problem.Proof.We consider again the homotopy equation It is obvious, by our assumption, that (5.39) has no solution x ∈ ∂G for t = 1.This is also true for t = 0 because Jx = 0 implies x = 0 ∈ ∂G.We now assume that for some t ∈ (0,1), the inclusion (5.39) has a solution x ∈ ∂G.Then The problem in Theorem 5.8 was solved first by de Figueiredo [7] and then by Browder and Hess [6] for single multivalued pseudomonotone operators C with D(C) = X and regular generalized pseudomonotone operators C, respectively.The set G in these references was B r (0).It was also assumed in [6] that the operator C satisfies u,x ≥ −k x , for every x ∈ D(C), u ∈ Cx, where k is a fixed positive constant.The authors of [6,7] used Rockafellar's mapping from [24]: which is maximal monotone and quasibounded because intD(T r ) = ∅ (cf.[6, Proposition 14]).Thus, in [6], the operator T r + C is regular and generalized pseudomonotone.This allows the solvability of the problem T r x + Cx + λJx 0 in B r (0) and, eventually, the solvability of T r x + Cx 0. Also, in [7] the operator T r + C is shown to be surjective via a different method of proof.Kenmochi extended this result in [19,Theorem 22] by considering a more general boundary condition on a closed convex subset of X instead of the ball B r (0).The reader is also referred to Kenmochi [19] for other results involving the class of operators of type (M), which is more general than the class of pseudomonotone mappings.
Corollary 5.9.Let T : It is actually possible to replace J ψ x by J ψ (x − x 0 ) in various homotopies provided that x 0 ∈ G.In this case, we do not need 0 ∈ G.In fact, our assertion will become obvious from the following lemma.Lemma 5.10.Let G be a bounded open subset of X and fix x 0 ∈ G and constants µ for a gauge function ψ : + → + .Then the degree mapping d(H(t,•),G,0) is well defined and constant on [0,1].In particular, (5.47) Proof.Since the mappings µ 1 J ψ (• − x 0 ), µ 2 (J ψ − J ψ x 0 ) are continuous, bounded and satisfy (S + ), they also satisfy ( S + ) and the degree d(H(t,•),G,0) is well defined, provided that 0 ∈ H(t,∂G), for t ∈ [0,1].Assume that this last assertion is not true.Then, for some x ∈ ∂G, H(t, x) = 0.If t = 0 or t = 1, we obtain x = x 0 , which contradicts ∂G ∩ intG = ∅.Let t ∈ (0,1).Then which says t = 1 and x = x 0 , that is, a contradiction again.It follows that the mapping H(t,x) is an admissible homotopy for our degree.Thus, d(H(t,•),G,0) is well defined and constant for all t ∈ [0,1].In particular, (5.49) In fact, to show that d(µ(J ψ − J ψ x 0 ),G,0) = 1, we first observe that we can consider instead the translated mapping J ψ x = µ(J ψ (x + x 0 ) − J ψ x 0 ) on the translated set G = G − x 0 .We do this because we now have 0 ∈ G and J ψ (0) = 0. Another way of saying this is to consider the mapping g(x) = x − x 0 and the degree d(J ψ g −1 − J ψ x 0 ,g(G),0), where g −1 (x) = x + x 0 .Since g is a homeomorphism on X with all the desirable properties, the mapping d( f g −1 ,g(G),0) is another degree mapping on the demicontinuous, bounded, and (S + )mappings f : G → X * .Since this degree is unique (cf.Browder [5]), we must have ), and µ 2 (J ψ g −1 − J ψ x 0 ) is continuous and one-to-one on g(G) = G − x 0 , and satisfies Proof.(i) Without loss of generality, we may assume that C(0) = 0, 0 ∈ M, and 0 ∈ G. Fix a point y * ∈ M such that y * = 0 and let s(t), t ∈ [0,1] be a path in M such that s(0) = 0 and s(1) = y * .We consider the two homotopy inclusions: ) where C n = C + (1/n)J.We first show that (6.24) has no solution x ∈ D(T + C) ∩ ∂G for any t ∈ [0,1].Assume that this is not true.Then we may also assume that there exist sequences and y * m + C n x m → s(t 0 ) and the operator C n satisfies ( S + ) and is quasibounded, we can repeat the proof of Theorem 7 in [13] to obtain x n → x 0 ∈ D(T + C) ∩ ∂G and s(t 0 ) ∈ Tx 0 + C n x 0 .This, however, is a contradiction to our assumption that the set M does not intersect the set (T + C)(D(T + C) ∩ ∂G).The quoted proof is for single-valued operators T, but it goes through for multivalued ones as well.
To show that (6.25) has no solutions x ∈ ∂G, we assume again that this is not the case and that {t m } ⊂ [0,1], {x m } ⊂ ∂G are such that t m → t 0 , x m x 0 ∈ X.If t m = 1, then we get a contradiction again by the injectivity of T + C n and the fact that 0 ∈ (T + C n )(0).Thus, t m ∈ (0,1) for all m.Again, we can now repeat the relevant part of the proof of Theorem 7 in [13] in order to obtain a contradiction.
It follows that H 1 (t,x), H 2 (t,x) are admissible homotopies for our degree.As such, they have constant degrees.Thus, we have If (b) is true, we can use the fact that the operator C satisfies ( S + ) along with the proof of Theorem 7 in [13] to arrive at the same conclusion.
(ii) We fix y * 0 ∈ (T + C)(D(T + C) ∩ G) with y * 0 = (T + C)x 0 .By our assumption, there exists a ball B q (x 0 ) ⊂ X such that B q (x 0 ) ⊂ G and the operator T + C is injective on D(T + C) ∩ B q (x 0 ).We show that there exists r > 0 such that (T + C) D(T + C) ∩ ∂B q x 0 ∩ B r y * 0 = ∅, (6.34)where B r (y * 0 ) ⊂ X * .Assume the contrary and let r n ↓ 0, p * n ∈ B rn (y * 0 ) ⊂ X * , {x n } ∈ D(T + C) ∩ ∂B q (x 0 ) be such that we can use the generalized pseudomonotonicity of the operator T + C to obtain x ∈ D(T + C) and y * 0 ∈ T x + C x. Naturally, x ∈ ∂B q (x 0 ) because we already have y * 0 ∈ Tx 0 + Cx 0 and the operator T + C is injective on D(T + C) ∩ B q (x 0 ).Thus, (6.34) is true.

Discussion
"Ranges of sums" problems can also be handled with our new degree theory in the spirit of the results of the paper [10].However, Theorem 2.1 in that paper is a very general result for densely defined, (weakly) quasibounded, finitely continuous, and generalized pseudomonotone perturbations C of maximal monotone operators T. That result uses an approximation involving a duality mapping J φ , where the gauge function φ is produced by the weak quasiboundedness property of the operator C.
In the proof of Theorem 6.2, we made use of the following lemma that can be found in [6, page 263].Since the operator T + C in our new degree theory is generalized pseudomonotone, it is useful to state explicitly this lemma here for future use.Lemma 7.1.Let T : X ⊃ D(T) → 2 X * be generalized pseudomonotone.Let M be a bounded weakly closed subset of X.Then T(D(T) ∩ M) is closed.In particular, T(D(T) ∩ B r (x 0 )) is closed for every x 0 ∈ X and every r > 0.

. 25 )
Since (x,x * ) are arbitrary in the graph G(T) and T is maximal monotone, we have x 0 ∈ D(T) and Tx 0 −h * .Taking x = x 0 and x * = −h * in (3.25), we obtain a contradiction.Consequently, (1.2) is true.

Proposition 5 . 1 .
Assume that the operator T satisfies (t1) and the operator C satisfies (c1)-(c3).Let G be an open and bounded subset of X with 0 {x n } bounded, there exists a sequence u n ∈ Tx n such that lim n→∞ u n + Cx n = p * .(5.25) Now, we may assume that x n x 0 .Since limsup n→∞ u n + Cx n ,x n − x 0 = lim n→∞ p * ,x n − x 0 = 0 (5.26) and the operator T + C is generalized pseudomonotone (see Lemma 5.3), we have x 0 ∈ D(T + C) and Tx 0 + Cx 0 p * .The proof is complete.
) and be generalized pseudomonotone.Assume, further, that there exists an open, bounded and convex set G ⊂ X containing zero and such that for every x ∈ D(T + C) ∩ ∂G and every u * ∈ Tx we haveu * + Cx ≥ 0.(5.45)Then the inclusion Tx + Cx 0 has a solution x ∈ D(T + C) ∩ G.Proof.It suffices to note that (5.38) is impossible for λ ≤ 0.A. G. Kartsatos and I. V. Skrypnik 149 {t m } ⊂ [0,1], {x n } ⊂ D(T + C) ∩ ∂G such that t m → t 0 ∈ [0,1], x m x 0 ∈ X and Tx m + C n x m s t m ,(6.26)ory * m + C n x m = s t m , (6.27) for some y * m ∈ Tx m .Since lim m→∞ y * m + C n x m ,x m − x 0 = lim m→∞ s t m ,x m − x 0 = 0 (6.28) t m y * m + C n x m + 1 − t m Jx m 0,(6.29)forsome y * m ∈ Tx n .From y * m + (1/n)Jx m ,x m ≥ 0 and the quasiboundedness of C, we can now obtain the boundedness of the sequence {C n x m }, which implies the boundedness of {y * m } as well.Consequently, we may also assume that y * m y * 0 ∈ X * and C n x m h * ∈ X * .If t m = 0 for some m, then (6.29) says that 0 ∈ ∂G, that is, a contradiction. A. G. Kartsatos and I. V.Skrypnik 155

Corollary 6 . 4 .
156 A topological degree theory Using part (i) of the proof with M = B r (y * 0 ) and the convex open set G = B q (x 0 ), we obtainB r y * 0 ⊂ (T + C) D(T + C) ∩ B q x 0 ⊂ (T + C) D(T + C) ∩ G .(6.37)It follows that the set (T + C)(D(T + C) ∩ G) is open, and the proof is finished.Since every open set G is the union of bounded open subsets of it (i.e., open balls about its points lying in it), part (ii) of Theorem 6.3 is actually true for any open set G. We state this fact in the following corollary.Assume that T satisfies (t1), while C : X ⊃ D(C) → X * satisfies (c1), (c3) and is generalized pseudomonotone.Then(i) if T + C is locally injective on an open set G ⊂ X, the set (T + C)(D(T + C) ∩ G) is open; (ii) if T + C is locally injective and R(T + C) is closed, then R(T + C) = X * .Proof.(ii) If T + C is locally injective, then (i) implies that R(T + C) = (T + C)(D(T + C) ∩ X) is open.If R(T + C) is also closed, then it must equal X * because the only open and closed sets in a Banach space are the empty set and the space itself.
.10)Assume that this is not true.Then we may also assume that {τ n }, {u n } are such that ) ∈ G(T τ0 ).Then, by Lemma 4.1, there exist x n ∈ D(T τn ), x * n ∈ T τn x n such that x n → x, x * n → x * .By the monotonicity of T τn , we obtain0 ≤ g * n − x * n ,u n − t n J −1 g * n −x n .∈ T τ0 x are otherwise arbitrary and T τ0 is maximal monotone, we have from (4.16) u 0 ∈ D(T τ0 ) and −h * ∈ T τ0 u 0 .Taking x = u 0 in (4.16), we obtain a contradiction.Consequently, (4.10) is true.
n ,u n − u 0 < 0. (4.12) Denote g * n = T τn tn u n .Then there exists w n ∈ D(T τn ) such that u n = t n J −1 g * n + w n , g * n ∈ T τn w n .(4.13) Now, let (x,x * − h * − x * ,u 0 − x > 0. (4.16)Since x ∈ D(T τ0 ), x * t 1 ].This implies that for each t ∈ (0,t 1 ], there exists x t ∈ D(C) ∩ G such that T t x t + Cx t = p * .Kartsatos and I. V. Skrypnik 137 we can use the quasiboundedness of C − p * to conclude, without any loss of generality, that Cx n − p * h * ∈ X * .We can now show, as in the proof of Theorem 3.2, that limsup ∩ G such that Tt n x n + Cx n = p * .(4.32)Since {x n } is bounded, we may assume that x n x 0 ∈ X.SinceCx n − p * ,x n = − Tt n x n ,x n ≤ 0, (4.33) A. G. n→∞ Cx n − p * ,x n − x 0 ≤ 0. (4.34)Using the condition ( S + ), we see that x n → x 0 , x 0 ∈ D(C) and Cx 0 − p * = h * {a n }, {b n } bounded.We also know that C 2 is bounded and that (4.36) and the quasiboundedness of C 1 imply the boundedness of {C 1 u n } as well.Thus, we may assume that C 1 u n h * 1 and C 2 u n h * 2 .We also observe that there exists a subsequence {n k } of {n} such that one of the inequalities 22))We assume that (5.8) holds and show that the sequence {x n } is bounded.To this end, assume that there exists a subsequence of {x n }, denoted again by {x n }, such that x n → ∞.Then there exists α > 0 such that Since the operator T + C is generalized pseudomonotone (see Lemma 5.3), we can conclude that x 0 ∈ D(T + C) and p * ∈ (T + C)x 0 .It follows that p * ∈ R(T + C) and the proof is complete.Corollary 5.7 is related to Theorem 7 of Browder and Hess [6, page 282].
Using again the fact that T + C is generalized pseudomonotone (see Lemma 5.3), we obtain x 0 ∈ D(T + C) and 0 ∈ Tx 0 + Cx 0 .Obviously, x 0 ∈ co G = G, but x 0 ∈ ∂G because of our assumption on (5.38).The proof is complete.