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Eigenvalues of quasibounded maximal monotone operators

Abstract

Let X be a real reflexive separable Banach space with dual space X and let L be a dense subspace of X. We study a nonlinear eigenvalue problem of the type

0Tx+λCx,

where T:D(T)X 2 X is a strongly quasibounded maximal monotone operator and C:D(C)X X satisfies the condition ( S + ) D ( C ) with LD(C). The method of approach is to use a topological degree theory for ( S + ) L -perturbations of strongly quasibounded maximal monotone operators, recently developed by Kartsatos and Quarcoo. Moreover, applying degree theory, a variant of the Fredholm alternative on the surjectivity of the operator λT+C is discussed, where we assume that λ is not an eigenvalue for the pair (T,C), T and C are positively homogeneous, and C satisfies the condition ( S + ) L .

1 Introduction and preliminaries

A systematic theory of compact operators emerged from the theory of integral equations of the form

Tx+λx=y,where Tx(t)= a b k ( t , s , x ( s ) ) ds.

Here, λR is a parameter, y and k are given functions, and x is the unknown function. Such equations play a role in the theory of differential equations. The study goes back to Krasnosel’skii [1]. Moreover, the eigenvalue problem of the form

Tx+λCx=0

could be solved with the Galerkin method, where C is continuous, bounded, and of type (S); see, e.g., [2].

From now on, we concentrate on the class of maximal monotone operators, as a generalization of linear self-adjoint operators. The theory of nonlinear maximal monotone operators started with a pioneer work of Minty [3] and has been extensively developed, with applications to evolution equations and to variational inequalities of elliptic and parabolic type; see [4, 5]. The eigenvalue problem for various types of nonlinear operators was investigated in [610]. As a key tool, topological degree theory was made frequent use of; for instance, the Leray-Schauder degree and the Kartsatos-Skrypnik degree; see [1115].

Let X be a real reflexive Banach space with dual space X . We consider a nonlinear eigenvalue problem of the form

0Tx+λCx,
(E)

where T:D(T)X 2 X is a maximal monotone multi-valued operator and C:D(C)X X is a single-valued operator. In the case where the operator C or the resolvents of T are compact, it was studied in [6, 7, 10] by using the Leray-Schauder degree for compact operators. When the operator C is densely defined and quasibounded and satisfies the condition ( S ˜ + ), Kartsatos and Skrypnik [9] solved the above problem (E) via the topological degree for these operators given in [13].

We are now focused on the quasiboundedness of the operator T instead of that of the operator C. Actually, a strongly quasibounded operator due to Browder and Hess [16] may not necessarily be bounded. One more thing to be considered is the condition ( S + ) L , where L is a dense subspace of X with LD(C). In fact, the condition ( S + ) 0 , L was first introduced in [12] and the structure of the class ( S + ) L or ( S + ) D ( C ) was discussed in [17], as a natural extension of the class ( S + ); see [4, 14].

In the present paper, the first goal is to study the above eigenvalue problem (E) for strongly quasibounded maximal monotone operators, provided that the operator C satisfies the condition ( S + ) D ( C ) . In addition, we assume the following property (P): For ϵ>0, there exists a λ>0 such that the inclusion

0Tx+λCx+εJx

has no solution in D(T)D(C)Ω, where Ω is a bounded open set in X and J is a normalized duality operator. This property is closely related to the use of a topological tool for finding the eigensolution on the boundary of Ω; see [9, 10]. To solve the above problem (E), we thus use the degree theory for densely defined ( S + ) L -perturbations of maximal monotone operators introduced by Kartsatos and Quarcoo in [18]. Roughly speaking, the degree function is based on the Kartsatos-Skrypnik degree [8] of the densely defined operators T t +C, which is constant for all small values of t, where T t is the approximant introduced by Brézis et al. [19]. Such an approach was first used by Browder in [11]. The second goal is to establish a variant of a Fredholm alternative result on the surjectivity for the operator λT+C, where λ1 is not an eigenvalue for the pair (T,C) and the operator C satisfies the condition ( S + ) L ; see [9, 20].

This paper is organized as follows: In Section 2, we give some eigenvalue results for strongly quasibounded maximal monotone operators by applying the Kartsatos-Quarcoo degree theory. Section 3 contains a version of the Fredholm alternative for positively homogeneous operators, with a regularization method by means of a duality operator J φ .

Let X be a real Banach space, X its dual space with the usual dual pairing ,, and Ω a nonempty subset of X. Let Ω ¯ , intΩ, and Ω denote the closure, the interior, and the boundary of Ω in X, respectively. The symbol → () stands for strong (weak) convergence. An operator A:Ω X is said to be bounded if A maps bounded subsets of Ω into bounded subsets of X . A is said to be demicontinuous if, for every x 0 Ω and for every sequence { x n } in Ω with x n x 0 , we have A x n A x 0 .

An operator T:D(T)X 2 X is said to be monotone if

u v , x y 0for every x,yD(T) and every  u Tx, v Ty,

where D(T)={xX:Tx} denotes the effective domain of T.

The operator T is said to be maximal monotone if it is monotone and it follows from (x, u )X× X and

u v , x y 0for every yD(T) and every  v Ty

that xD(T) and u Tx.

An operator T:D(T)X 2 X is said to be strongly quasibounded if for every S>0 there exists a constant K(S)>0 such that for all xD(T) with

xSand u , x S,

where u Tx, we have u K(S).

We say that T:D(T)X 2 X satisfies the condition ( S q ) on a set MD(T) if for every sequence { x n } in M with x n x 0 and every sequence { u n } with u n u where u n T x n , we have x n x 0 .

We say that T:D(T)X X satisfies the condition ( S + ) on a set MD(T) if for every sequence { x n } in M with

x n x 0 and lim sup n T x n , x n x 0 0,

we have x n x 0 .

Throughout this paper, X will always be an infinite-dimensional real reflexive separable Banach space which has been renormed so that X and its dual X are locally uniformly convex.

An operator J φ :X X is said to be a duality operator if

J φ x,x=φ ( x ) xand J φ x=φ ( x ) for xX,

where φ:[0,)[0,) is continuous, strictly increasing, φ(0)=0 and φ(t) as t. When φ is the identity map I, J:= J I is called a normalized duality operator.

It is described in [21] that J φ is continuous, bounded, surjective, strictly monotone, maximal monotone, and that it satisfies the condition ( S + ) on X.

The following properties as regards maximal monotone operators will often be used, taken from [[19], Lemma 1.3], [[13], Lemma 3.1], [[22], Lemma 1], and [[18], Lemma D] in this order.

Lemma 1.1 Let T:D(T)X 2 X be a maximal monotone operator. Then the following statements hold:

  1. (a)

    For each t(0,), the operator T t ( T 1 + t J 1 ) 1 :X X is bounded, demicontinuous, and maximal monotone.

  2. (b)

    If, in addition, 0D(T) and 0T(0), then the operator (0,)×X X , (t,x) T t x is continuous on (0,)×X.

Lemma 1.2 Let T:D(T)X 2 X and S:D(S)X X be two maximal monotone operators with 0D(T)D(S) and 0T(0)S(0) such that T+S is maximal monotone. Assume that there is a sequence { t n } in (0,) with t n 0 and a sequence { x n } in D(S) such that x n x 0 X and T t n x n + w n y 0 X , where w n S x n . Then the following hold:

  1. (a)

    The inequality lim inf n T t n x n + w n , x n x 0 0 is true.

  2. (b)

    If lim n T t n x n + w n , x n x 0 =0, then x 0 D(T+S) and y 0 (T+S) x 0 .

Lemma 1.3 Let T:D(T)X 2 X be a strongly quasibounded maximal monotone operator such that 0D(T) and 0T(0). If { t n } is a sequence in (0,) and { x n } is a sequence in X such that

x n Sand T t n x n , x n S 1 ,

where S, S 1 are positive constants, then the sequence { T t n x n } is bounded in X .

Let L be a dense subspace of X and let F(L) denote the class of all finite-dimensional subspaces of L. Let { F n } be a sequence in the class F(L) such that for each nN

F n F n + 1 ,dim F n =n,and n N F n ¯ =X.
(1.1)

Set L{ F n }:= n N F n .

Definition 1.4 Let C:D(C)X X be a single-valued operator with LD(C). We say that C satisfies the condition ( S + ) 0 , L if for every sequence { F n } in F(L) satisfying equation (1.1) and for every sequence { x n } in L with

x n x 0 , lim sup n C x n , x n 0,and lim n C x n ,y=0

for every yL{ F n }, we have x n x 0 , x 0 D(C), and C x 0 =0.

We say that C satisfies the condition ( S + ) L if the operator C h :D(C) X , defined by C h x:=Cxh, satisfies the condition ( S + ) 0 , L for every h X .

We say that the operator C satisfies the condition ( S + ) 0 , D ( C ) if it satisfies the condition ( S + ) 0 , L with ‘{ x n }L’ replaced by ‘{ x n }D(C)’. We say that C satisfies the condition ( S + ) D ( C ) if the operator C h satisfies the condition ( S + ) 0 , D ( C ) for every h X .

It is obvious from Definition 1.4 that if the operator C satisfies the condition ( S + ) D ( C ) , then C satisfies the condition ( S + ) L . However, the converse is not true in general, as we see in Example 3.2 of [17].

2 The existence of eigenvalues

In this section, we deal with some eigenvalue results for strongly quasibounded maximal monotone operators in reflexive separable Banach spaces, based on a topological degree theory for ( S + ) L -perturbations of maximal monotone operators due to Kartsatos and Quarcoo [18].

We establish the existence of an eigenvalue concerning ( S + ) D ( C ) -perturbations of strongly quasibounded maximal monotone operators.

Theorem 2.1 Let Ω be a bounded open set in X with 0Ω and let L be a dense subspace of X. Suppose that T:D(T)X 2 X is a multi-valued operator and C:D(C)X X is a single-valued operator with LD(C) such that

(t1) T is maximal monotone and strongly quasibounded with 0D(T) and 0T(0),

(c1) C satisfies the condition ( S + ) D ( C ) ,

(c2) for every FF(L) and vL, the function c(F,v):FR, defined by c(F,v)(x)=Cx,v, is continuous on F, and

(c3) there exists a nondecreasing function ψ:[0,)[0,) such that

Cx,xψ ( x ) for all xD(C).

Let Λ and ε 0 be two given positive numbers.

  1. (a)

    For a given ε>0, assume the following property (P):

There exists a λ(0,Λ] such that the inclusion

0Tx+λCx+εJx

has no solution in D(T+C)Ω.

Then there exists a ( λ ε , x ε )(0,Λ]×(D(T+C)Ω) such that

0T x ε + λ ε C x ε +εJ x ε .

Here, D(T+C) denotes the intersection of D(T) and D(C).

  1. (b)

    If property (P) is fulfilled for every ε(0, ε 0 ], T satisfies the condition ( S q ) on D(T)Ω, 0T(D(T)Ω), and the set C(D(C)Ω) is bounded, then the inclusion

    0Tx+λCx

has a solution ( λ 0 , x 0 ) in (0,Λ]×(D(T+C)Ω).

Proof (a) Assume that the conclusion of (a) is not true. Then for every λ(0,Λ], the following boundary condition holds:

0Tx+λCx+εJxfor all xD(T+C)Ω.
(2.1)

Considering a multi-valued map H given by

H(s,x):=Tx+sΛCx+εJxfor s[0,1],

the inclusion 0H(s,x) has no solution x in D(T+C)Ω for all s[0,1]. Actually, this holds for s=0, in view of the injectivity of the operator T+εJ with 0(T+εJ)(D(T)Ω).

Now we consider a single-valued map H 1 given by

H 1 (t,s,x):= T t x+sΛCx+εJxfor t(0,) and s[0,1].

We will first show that there exists a positive number t 0 such that the equation

H 1 (t,s,x)=0
(2.2)

has no solution x in D(C)Ω for all t(0, t 0 ] and all s[0,1]. For s=0, assertion (2.2) is obvious because ( T t +εJ)x=0 implies x=0. Assume that assertion (2.2) does not hold for any s(0,1]. Then there exist sequences { t n } in (0,), { s n } in (0,1], and { x n } in D(C)Ω such that t n 0, s n s 0 , x n x 0 , J x n j , and

T t n x n + s n ΛC x n +εJ x n =0,
(2.3)

where s 0 [0,1], x 0 X, and j X . Let S be a positive upper bound for the bounded sequence { x n }. Note that s 0 (0,1]. Indeed, if s 0 =0, then we have by the monotonicity of T t n with T t n (0)=0, equation (2.3), and (c3)

ε x n 2 ε J x n , x n + T t n x n , x n = s n Λ C x n , x n s n Λ ψ ( x n ) s n Λ ψ ( S )

and so x n 0Ω; but x n Ω, which is a contradiction. Since we have the inequality

T t n x n , x n = s n ΛC x n , x n εJ x n , x n Λψ(S),

Lemma 1.3 implies that the sequence { T t n x n } is bounded in the reflexive Banach space X . Passing to a subsequence, if necessary, we may suppose that T t n x n v for some v X . Set

u := 1 s 0 Λ ( v + ε j ) .

By equation (2.3), we have C x n u and hence

lim n C x n + u , y =0for every yL{ F n }.
(2.4)

Recall that if two operators A 1 :D( A 1 )X 2 X and A 2 :D( A 2 )X 2 X are maximal monotone and D( A 1 )intD( A 2 ), then the sum A 1 + A 2 :D( A 1 )D( A 2 ) 2 X is also maximal monotone; see [[5], Theorem 32.I]. Since T+εJ is thus maximal monotone and T t n x n +εJ x n v +ε j , Lemma 1.2(a) says that

lim inf n T t n x n +εJ x n , x n x 0 0.
(2.5)

From equations (2.3), (2.5), and the equality

C x n + u , x n = C x n + 1 s n Λ ( T t n x n + ε J x n ) , x n 1 s n Λ ( T t n x n + ε J x n ) , x n x 0 1 s n Λ ( T t n x n + ε J x n ) , x 0 + u , x n

it follows that

lim sup n C x n + u , x n lim inf n 1 s n Λ T t n x n + ε J x n , x n x 0 0 .
(2.6)

Since the operator C satisfies the condition ( S + ) D ( C ) , we find from equations (2.4) and (2.6) that x n x 0 D(C) and C x 0 + u =0. Since lim n T t n x n , x n x 0 =0, Lemma 1.2(b) tells us that x 0 D(T) and v T x 0 . From J x n J x 0 = j , we get

v + s 0 ΛC x 0 +ε j =0or0T x 0 + s 0 ΛC x 0 +εJ x 0 ,

which contradicts our boundary condition equation (2.1). Consequently, we have proven our first assertion: that there exists a number t 0 >0 such that

H 1 (t,s,x)0for any (t,s)(0, t 0 ]×[0,1] and all xD(C)Ω.

In the next step, we want to show that for each fixed t(0, t 0 ], the degree d( H 1 (t,s,),Ω,0) is independent of s[0,1], where d denotes the Kartsatos-Skrypnik degree from [12]. Fix t(0, t 0 ]. For s[0,1], let A s :D( A s )X X be defined by

A s x:= H 1 (t,s,x)= T t x+sΛCx+εJx,

where D( A s )=X for s=0 and D( A s )=D(C) for s(0,1]. First of all, for every finite-dimensional space FL{ F j } and every vL{ F j }, the function a ˜ (F,v):F×[0,1]R, defined by a ˜ (F,v)(x,s)= A s x,v, is continuous on F×[0,1] because the operators T t and J are continuous and C satisfies the condition (c2). To show that the family { A s } satisfies the condition ( S + ) 0 , L ( s ) , we assume that { s n } is a sequence in [0,1] and { x n } is a sequence in L{ F n } such that s n s 0 , x n x 0 , and

lim sup n A s n x n , x n 0and lim n A s n x n ,y=0
(2.7)

for every yL{ F n }, where s 0 [0,1] and x 0 X. By Lemma 1.1(a), the sequence { T t x n } is bounded in X . So we may suppose without loss of generality that T t x n v and J x n j for some v , j X . There are two cases to consider. If s 0 =0, then we have

ε x n 2 εJ x n , x n + T t x n , x n A s n x n , x n + s n Λψ(S),

which implies along with equation (2.7)

ε lim sup n x n 2 lim n s n Λψ(S)=0,

where S is an upper bound for the sequence { x n }. Hence it follows that x n 0, x 0 =0X=D( A s 0 ), and A s 0 x 0 =0. Now let s 0 (0,1]. We may suppose that s n >0 for all nN. Set s ˜ n :=1/( s n Λ) and s ˜ :=1/( s 0 Λ). The relation (2.7) can be expressed in the form

lim sup n C x n + s ˜ n ( T t + ε J ) x n , x n 0 , lim n C x n + s ˜ n ( T t + ε J ) x n , y = 0 for every  y L { F n } .
(2.8)

From the second part of equation (2.8), it is obvious that

lim n C x n + s ˜ ( v + ε j ) , y =0for every yL{ F n }.
(2.9)

By the monotonicity of the operator T t +εJ, we have

lim inf n ( T t + ε J ) x n , x n x 0 lim inf n ( T t + ε J ) x 0 , x n x 0 =0.
(2.10)

Hence it follows from the first part of equation (2.8) and from equation (2.10) that

lim sup n C x n + s ˜ ( v + ε j ) , x n lim inf n s ˜ n ( T t + ε J ) x n , x n x 0 0 .
(2.11)

Since the operator C satisfies the condition ( S + ) L , we find from equations (2.9) and (2.11) that

x n x 0 , x 0 D(C)=D( A s 0 )andC x 0 + s ˜ ( v + ε j ) =0.

By the demicontinuity of the operators T t and J, we have

T t x n T t x 0 = v andJ x n J x 0 = j

and hence

A s 0 x 0 = T t x 0 + s 0 ΛC x 0 +εJ x 0 =0.

Consequently, the family { A s } satisfies the condition ( S + ) 0 , L ( s ) , as required.

Since A s (x)0 for all (s,x)[0,1]×(D( A s )Ω), we see, in view of Theorem A of [18], that the degree d( A s ,Ω,0) is independent of the choice of s[0,1]. Until now, we have shown that for each fixed t(0, t 0 ], the degree d( H 1 (t,s,),Ω,0) is constant for all s[0,1]. Notice that T+εJ is maximal monotone and strongly quasibounded, 0(T+εJ)(0), and

H(s,x)=(T+εJ)x+sΛCx0for all s[0,1] and all xD(T+C)Ω.

Combining this with our first assertion above, Theorem 2 of [18] says that for each fixed s[0,1], the degree d( T t +sΛC+εJ,Ω,0) is constant for all t(0, t 0 ]. If deg denotes the degree introduced in [18], then for every s[0,1], we have

deg(T+sΛC+εJ,Ω,0)=d( T t +sΛC+εJ,Ω,0)for t(0, t 0 ]

and hence

deg ( T + s Λ C + ε J , Ω , 0 ) = d ( A s , Ω , 0 ) = d ( A 0 , Ω , 0 ) = d ( T t + ε J , Ω , 0 ) = 1 ,

where the last equality follows from Theorem 3 in [23]. Thus, for all s(0,1], the inclusion

0Tx+sΛCx+εJx

has a solution in D(T+C)Ω, which contradicts property (P). We conclude that statement (a) is true.

  1. (b)

    Let { ε n } be a sequence in (0, ε 0 ] such that ε n 0. According to statement (a), there exists a sequence {( λ ε n , x ε n )} in (0,Λ]×(D(T+C)Ω) such that

    u ε n + λ ε n C x ε n + ε n J x ε n =0,

where u ε n T x ε n . If we set λ n := λ ε n , x n := x ε n , and u n := u ε n , it can be rewritten in the form

u n + λ n C x n + ε n J x n =0.
(2.12)

Notice that the sequence { u n } is bounded in X . This follows from the strong quasiboundedness of the operator T together with the inequality

u n , x n = λ n C x n , x n ε n J x n , x n Λψ(S),

where S is an upper bound for the sequence { x n }. From equation (2.12), { λ n C x n } is bounded in X . Without loss of generality, we may suppose that

λ n λ 0 , x n x 0 ,and u n u 0 ,
(2.13)

where λ 0 [0,Λ], x 0 X, and u 0 X . Note that the limit λ 0 belongs to (0,Λ]. In fact, if λ 0 =0, then the boundedness of the set C(D(C)Ω) implies that λ n C x n 0 and so by equation (2.12) u n 0. Since the maximal monotone operator T satisfies the condition ( S q ) on D(T)Ω, we find from equation (2.13) and Lemma 1.2(b) that x n x 0 Ω, x 0 D(T), and 0T x 0 , which contradicts the hypothesis that 0T(D(T)Ω). As C x n (1/ λ 0 ) u 0 , we have

lim n C x n + 1 λ 0 u 0 , y =0for every yL{ F n }.
(2.14)

From equation (2.12) it follows that

lim sup n C x n + 1 λ 0 u 0 , x n 1 λ 0 lim inf n u n + ε n J x n , x n x 0 0 ,
(2.15)

where the last inequality follows from Lemma 1.2(a). Since the operator C satisfies the condition ( S + ) D ( C ) , we obtain from equations (2.14) and (2.15) x n x 0 D(C) and λ 0 C x 0 + u 0 =0. By the maximal monotonicity of the operator T, we have x 0 D(T) and u 0 T x 0 . We conclude that

0T x 0 + λ 0 C x 0 and x 0 D(T+C)Ω.

This completes the proof. □

Remark 2.2 (a) In Theorem 2.1, it is inevitable that the set C(D(C)Ω) is assumed to be bounded because it does not hold in general that if λ n 0 then λ n C x n 0.

  1. (b)

    When C is quasibounded and satisfies the condition ( S ˜ + ), it was studied in [[9], Theorem 4] by using Kartsatos-Skrypnik degree theory for ( S ˜ + )-perturbations of maximal monotone operators developed in [13]. For the case where C is generalized pseudomonotone in place of the condition ( S ˜ + ), we refer to [[20], Theorem 2.1].

From Theorem 2.1, we get the following eigenvalue result in the case when the operator C satisfies the condition ( S + ).

Corollary 2.3 Let T, Ω, L, Λ, ε 0 be as in Theorem  2.1. Suppose that C:X X is a strongly quasibounded demicontinuous operator such that

(c1′) C satisfies the condition ( S + ) on X,

(c2) for every FF(L) and vL, the function c(F,v):FR, defined by c(F,v)(x)=Cx,v, is continuous on F, and

(c3) there exists a nondecreasing function ψ:[0,)[0,) such that

Cx,xψ ( x ) for all xX.

Then the following statements hold:

  1. (a)

    If property (P) is fulfilled for a given ε>0, then there exists a ( λ ε , x ε )(0,Λ]×(D(T)Ω) such that 0T x ε + λ ε C x ε +εJ x ε .

  2. (b)

    If property (P) is fulfilled for every ε(0, ε 0 ], T satisfies the condition ( S q ) on D(T)Ω and 0T(D(T)Ω), then the inclusion 0Tx+λCx has a solution in (0,Λ]×(D(T)Ω).

Proof Statement (a) follows immediately from Theorem 2.1 if we only show that the operator C satisfies the condition ( S + ) D ( C ) with D(C)=X. To do this, let h X be given and suppose that { x n } is any sequence in X such that

x n x 0 , lim sup n C x n h, x n 0,and lim n C x n h,y=0
(2.16)

for every yL{ F n }. Then {C x n , x n } is obviously bounded from above. By the strong quasiboundedness of the operator C, the sequence {C x n } is bounded in X . Since L{ F n } is dense in the reflexive Banach space X, it follows from the third one of equation (2.16) that C x n h. Hence we obtain from the first and second one of equation (2.16)

lim sup n C x n , x n x 0 lim sup n C x n h , x n lim n C x n h , x 0 + lim n h , x n x 0 0 .

Since C satisfies the condition ( S + ) on X and is demicontinuous, we have

x n x 0 XandC x 0 h=0.

Thus, the operator C satisfies the condition ( S + ) D ( C ) with D(C)=X.

  1. (b)

    Let { ε n } be a sequence in (0, ε 0 ] such that ε n 0. In view of (a), there exists a sequence {( λ n , x n )} in (0,Λ]×(D(T)Ω) such that

    u n + λ n C x n + ε n J x n =0,
    (2.17)

where u n T x n . Notice that the sequence {C x n } is bounded in X and so is { u n }. This follows from the strong quasiboundedness of the operator C and the inequality

C x n , x n = 1 λ n u n , x n ε n λ n J x n , x n 0.

We may suppose that λ n λ 0 , x n x 0 , and u n u 0 , where λ 0 [0,Λ], x 0 X, and u 0 X . Note that λ 0 belongs to (0,Λ]. Indeed, if λ 0 =0, then we have by the boundedness of {C x n } and equation (2.17) u n 0 and hence by the condition ( S q ) x n x 0 D(T) and 0T x 0 , which contradicts the hypothesis 0T(D(T)Ω). The rest of the proof proceeds analogously as in the proof of Theorem 2.1. □

Remark 2.4 (a) The boundedness assumption on the set C(D(C)Ω) is unnecessary in Corollary 2.3, provided that the operator C is strongly quasibounded.

  1. (b)

    An analogous result to Corollary 2.3 can be found in [[9], Corollary 1], where the operator C is supposed to be bounded.

We close this section by exhibiting a simple example of operators A satisfying the condition ( S + ) D ( A ) .

Let G be a bounded open set in R N . Let 1<p< and X= W 0 1 , p (G). Define the two operators A 1 , A 2 :X X by

A 1 u , v = i = 1 N G | u x i | p 2 u x i v x i d x , A 2 u , v = G | u | p 2 u v d x .

Then the operator A 1 is clearly bounded and continuous, and it satisfies the condition ( S + ) on X. The operator A 2 is compact; see [[24], Theorem 2.2] and [[5], Proposition 26.10]. In particular, the sum A:= A 1 + A 2 satisfies the condition ( S + ) D ( A ) with D(A)=X.

3 Fredholm alternative

In this section, we present a variant of the Fredholm alternative for strongly quasibounded maximal monotone operators, by applying Kartsatos-Quarcoo degree theory as in Section 2.

Given γ>0, an operator A:D(A)X X is said to be positively homogeneous of degree γ on a set MD(A) if A(rx)= r γ Ax for all xM and all r>0. For example, the duality operator J φ :X X is positively homogeneous of degree γ on X if φ(t)= t γ for t[0,). In addition, the operators A 1 and A 2 given at the end of Section 2 are positively homogeneous of degree p1 on X= W 0 1 , p (G).

Theorem 3.1 Let L be a dense subspace of X and let λ,γ[1,) be given. Suppose that T:D(T)=L X is an operator and C:D(C)X X is an operator with LD(C) and C(0)=0 such that

(t1) T is maximal monotone and strongly quasibounded with T(0)=0,

(t2) λTx+Cx+μ J φ x=0 implies x=0 for every μ0, where φ(t)= t γ ,

(c1) C satisfies the condition ( S + ) L ,

(c2) for every FF(L) and vL, the function c(F,v):FR, defined by c(F,v)(x)=Cx,v, is continuous on F, and

(c3) there exists a nondecreasing function ψ:[0,)[0,) such that

Cx,xψ ( x ) for all xD(C).

If the operators T and C are positively homogeneous of degree γ on L, then the operator λT+C is surjective.

Proof Let p be an arbitrary but fixed element of X . For each fixed ε>0, consider a family of operators A t :D( A t )X X , t[0,1] given by

A t (x):=H(t,x):=t ( λ T x + C x + ε J φ x p ) +(1t)ε J φ x,

where D( A t )=X for t=0 and D( A t )=L for t(0,1]. The first aim is to prove that the set of all solutions of the equation H(t,x)=0 is bounded, independent of t[0,1]. If t=0, then H(0,x)=ε J φ x=0 implies x=0. It suffices to show that {(t,x)(0,1]×L:H(t,x)=0} is bounded. Assume the contrary; then there exist sequences { t n } in (0,1] and { x n } in L such that t n t 0 [0,1], x n , and

t n ( λ T x n + C x n + ε J φ x n p ) +(1 t n )ε J φ x n =0,

which can be written as

λT x n +C x n p + ε t n J φ x n =0.
(3.1)

We may suppose that x n 1 for all nN. Since the operators T, C, and J φ are positively homogeneous of degree γ, it follows from equation (3.1) that

λT ( x n x n ) +C ( x n x n ) 1 x n γ p + ε t n J φ ( x n x n ) =0.

Setting u n := x n / x n and q n :=1/ t n , we have u n =1, q n >0, and

λT u n +C u n 1 x n γ p + q n ε J φ u n =0.
(3.2)

Then we obtain from equation (3.2) and (c3)

λ T u n , u n = C u n , u n + 1 x n γ p , u n q n ε J φ u n , u n ψ ( 1 ) + p .

Hence the strong quasiboundedness of T implies that the sequence {T u n } is bounded in  X . There are two cases to consider. If t 0 =0, then q n , J φ u n , u n =1, and the monotonicity of T with T(0)=0 implies

0λT u n , u n ψ(1)+ p q n ε,

which is a contradiction. Now let t 0 >0 and set q 0 :=1/ t 0 . Without loss of generality, we may suppose that

u n u 0 ,T u n v ,and J φ u n j ,

where u 0 X, v X , and j X . By equation (3.2), we have C u n λ v q 0 ε j and hence

lim n C u n + λ v + q 0 ε j , y =0for every yL{ F n }.
(3.3)

Since the operator λT+ q 0 ε J φ is maximal monotone, we have

lim inf n λT u n + q n ε J φ u n , u n u 0 0.
(3.4)

In fact, if equation (3.4) is false, then there is a subsequence of { u n }, denoted again by { u n }, such that

lim n λT u n + q n ε J φ u n , u n u 0 <0.

Hence it is clear that

lim sup n λT u n + q n ε J φ u n , u n < λ v + q 0 ε j , u 0 .
(3.5)

For every uD(T), we have, by the monotonicity of the operator λT+ q n ε J φ ,

lim inf n λ T u n + q n ε J φ u n , u n lim inf n [ λ T u n + q n ε J φ u n , u + λ T u + q n ε J φ u , u n u ] = λ v + q 0 ε j , u + λ T u + q 0 ε J φ u , u 0 u ,

which implies along with equation (3.5)

λ v + q 0 ε j ( λ T u + q 0 ε J φ u ) , u 0 u >0.
(3.6)

By the maximal monotonicity of λT+ q 0 ε J φ , we have u 0 D(T) and (λT+ q 0 ε J φ ) u 0 =λ v + q 0 ε j . Letting u= u 0 D(T) in equation (3.6), we get a contradiction. Thus, equation (3.4) is true.

Furthermore, equation (3.4) implies, because of (1/ x n γ ) p 0, that

lim inf n λ T u n 1 x n γ p + q n ε J φ u n , u n u 0 0.
(3.7)

From equations (3.2), (3.7), and the equality

C u n + λ v + q 0 ε j , u n = C u n + λ T u n 1 x n γ p + q n ε J φ u n , u n λ T u n 1 x n γ p + q n ε J φ u n , u n u 0 λ T u n 1 x n γ p + q n ε J φ u n , u 0 + λ v + q 0 ε j , u n

it follows that

lim sup n C u n + λ v + q 0 ε j , u n lim inf n λ T u n 1 x n γ p + q n ε J φ u n , u n u 0 0 .
(3.8)

Since the operator C satisfies the condition ( S + ) L , we obtain from equations (3.3) and (3.8)

u n u 0 , u 0 D(C),andC u 0 +λ v + q 0 ε j =0.

Since T is maximal monotone and J φ is continuous, Lemma 1.2(b) implies that

u 0 D(T),T u 0 = v ,and J φ u 0 = j .

Therefore, we obtain

λT u 0 +C u 0 + q 0 ε J φ u 0 =0and u 0 =1,

which contradicts hypothesis (t2) with μ= q 0 ε. Thus, we have shown that {(t,x)[0,1]×L:H(t,x)=0} is bounded.

So we can choose an open ball B r (0) in X of radius r>0 centered at the origin 0 so that

{ x L : H ( t , x ) = 0  for some  t [ 0 , 1 ] } B r (0).

This means that H(t,x)= A t (x)0 for all (t,x)[0,1]×(D( A t ) B r (0)). Note that the operator T ˜ ε :=λT+ε J φ is maximal monotone, strongly quasibounded, T ˜ ε (0)=0, and the operator C ˜ :=C p satisfies the condition ( S + ) L and other conditions with c ˜ (F,v)(x):= C ˜ x,v for xF and C ˜ x,x ψ ˜ (x) for xD( C ˜ ), where ψ ˜ (t):=(1+ p )max{ψ(t),t}. Moreover, we know from Section 1 that the operator ε J φ is continuous, bounded and strictly monotone, and that it satisfies the condition ( S + ), and ε J φ x,x=ε x γ + 1 for xX.

Using the homotopy invariance property of the degree stated in [[18], Theorem 3], we have

deg ( λ T + C + ε J φ p , B r ( 0 ) , 0 ) =deg ( ε J φ , B r ( 0 ) , 0 ) =1.
(3.9)

Applying equation (3.9) with ε=1/n, there exists a sequence { x n } in L such that

λT x n +C x n + 1 n J φ x n = p .
(3.10)

Next, we show that the sequence { x n } is bounded in X. Indeed, assume on the contrary that there is a subsequence of { x n }, denoted by { x n }, such that x n . Dividing both sides of equation (3.10) by x n γ and setting u n := x n / x n and w n :=λT u n +C u n , we get

λT u n +C u n + 1 n J φ u n = 1 x n γ p

and so w n 0. Since λT u n , u n =C u n , u n + w n , u n ψ(1)+ w n for all nN, it follows from (t1) that the sequence {T u n } is bounded in X . We may suppose that u n u 0 and T u n v for some u 0 X and some v X . As in the proof of equations (3.3) and (3.8) above, we can show that

lim sup n C u n + λ v , u n 0and lim n C u n + λ v , y =0

for every yL{ F n }. Since the operator C satisfies the condition ( S + ) L , we obtain

u n u 0 , u 0 D(C),andC u 0 +λ v =0.

By Lemma 1.2(b), we have u 0 D(T) and T u 0 = v and hence

λT u 0 +C u 0 =0and u 0 =1,

which contradicts hypothesis (t2) with μ=0. Therefore, the sequence { x n } is bounded in X.

Combining this with equation (3.10), we know from (c3) and (t1) that the sequence {T x n } is also bounded in X . Thus we may suppose that x n x 0 and T x n v 0 for some x 0 X and some v 0 X . From C x n λ v 0 + p and the maximal monotonicity of the operator T, we get as before

lim sup n C x n + λ v 0 p , x n 0and lim n C x n + λ v 0 p , y =0

for every yL{ F n }. Since the operator C satisfies the condition ( S + ) L and T is maximal monotone, we conclude that

x 0 D(λT+C)andλT x 0 +C x 0 = p .

As p X was arbitrary, this says that the operator λT+C is surjective. This completes the proof. □

Remark 3.2 An analogous result to Theorem 3.1 was investigated in [[20], Theorem 4.1], where the method was to use Kartsatos-Skrypnik degree theory for quasibounded densely defined ( S ˜ + )-perturbations of maximal monotone operators, developed in [13]; see also [[9], Theorem 5].

As a particular case of Theorem 3.1, we have another surjectivity result.

Corollary 3.3 Let L, T, and C be the same as in Theorem  3.1, except that hypothesis (t2) is replaced by

(t2′) λTx+Cx,x0 for all xL.

If λ is not an eigenvalue for the pair (T,C), that is, λTx+Cx=0 implies x=0, then the operator λT+C is surjective.

Proof Noting that

λTx+Cx+μ J φ x,xμ J φ x,x=μ x γ + 1 0

for every xL and μ>0, it is clear that hypothesis (t2) in Theorem 3.1 is satisfied. Apply Theorem 3.1. □

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Acknowledgements

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2012-0008345).

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KI conceived of the study and drafted the manuscript. BI participated in coordination. All authors approved the final manuscript.

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Kim, I., Bae, J. Eigenvalues of quasibounded maximal monotone operators. J Inequal Appl 2014, 21 (2014). https://doi.org/10.1186/1029-242X-2014-21

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Keywords

  • Maximal Monotone
  • Maximal Monotone Operator
  • Degree Theory
  • Duality Operator
  • Dense Subspace