Open Access

Eigenvalues of quasibounded maximal monotone operators

Journal of Inequalities and Applications20142014:21

https://doi.org/10.1186/1029-242X-2014-21

Received: 11 October 2013

Accepted: 16 December 2013

Published: 14 January 2014

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

0 T x + λ C x ,

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 L D ( 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 λ T + 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
T x + λ x = y , where  T x ( t ) = a b k ( t , s , x ( s ) ) d s .
Here, λ R 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
T x + λ C x = 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
0 T x + λ C x ,
(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 L D ( 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
0 T x + λ C x + ε J x

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 λ T + C , where λ 1 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 0 for every  x , y D ( T )  and every  u T x , v T y ,

where D ( T ) = { x X : T x } 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 0 for every  y D ( T )  and every  v T y

that x D ( T ) and u T x .

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 x D ( T ) with
x S and u , x S ,

where u T x , we have u K ( S ) .

We say that T : D ( T ) X 2 X satisfies the condition ( S q ) on a set M D ( 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 M D ( 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 ) x and J φ x = φ ( x ) for  x X ,

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, 0 D ( T ) and 0 T ( 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 0 D ( T ) D ( S ) and 0 T ( 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 0 D ( T ) and 0 T ( 0 ) . If { t n } is a sequence in ( 0 , ) and { x n } is a sequence in X such that
x n S and 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 n N
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 L D ( 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 y L { 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 : = C x h , 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 0 Ω 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 L D ( C ) such that

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

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

(c2) for every F F ( L ) and v L , the function c ( F , v ) : F R , defined by c ( F , v ) ( x ) = C x , v , is continuous on F, and

(c3) there exists a nondecreasing function ψ : [ 0 , ) [ 0 , ) such that
C x , x ψ ( x ) for all  x D ( 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
0 T x + λ C x + ε J x

has no solution in D ( T + C ) Ω .

Then there exists a ( λ ε , x ε ) ( 0 , Λ ] × ( D ( T + C ) Ω ) such that
0 T x ε + λ ε C x ε + ε J 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 , 0 T ( D ( T ) Ω ) , and the set C ( D ( C ) Ω ) is bounded, then the inclusion
    0 T x + λ C x
     

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:
0 T x + λ C x + ε J x for all  x D ( T + C ) Ω .
(2.1)
Considering a multi-valued map H given by
H ( s , x ) : = T x + s Λ C x + ε J x for  s [ 0 , 1 ] ,

the inclusion 0 H ( 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 + ε J with 0 ( T + ε J ) ( D ( T ) Ω ) .

Now we consider a single-valued map H 1 given by
H 1 ( t , s , x ) : = T t x + s Λ C x + ε J x for  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 + ε J ) 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 Λ C x n + ε J 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 0 Ω ; but x n Ω , which is a contradiction. Since we have the inequality
T t n x n , x n = s n Λ C x n , x n ε J x n , x n Λ ψ ( 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 = 0 for every  y L { 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 ) int D ( 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 + ε J is thus maximal monotone and T t n x n + ε J x n v + ε j , Lemma 1.2(a) says that
lim inf n T t n x n + ε J 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 Λ C x 0 + ε j = 0 or 0 T x 0 + s 0 Λ C x 0 + ε J 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 ) 0 for any  ( t , s ) ( 0 , t 0 ] × [ 0 , 1 ]  and all  x D ( 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 + s Λ C x + ε J x ,
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 F L { F j } and every v L { 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 0 and lim n A s n x n , y = 0
(2.7)
for every y L { 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 ε J x n , x n + T t x n , x n A s n x n , x n + s n Λ ψ ( S ) ,
which implies along with equation (2.7)
ε lim sup n x n 2 lim n s n Λ ψ ( S ) = 0 ,
where S is an upper bound for the sequence { x n } . Hence it follows that x n 0 , x 0 = 0 X = 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 n N . 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 = 0 for every  y L { F n } .
(2.9)
By the monotonicity of the operator T t + ε J , 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 ) and C 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 and J x n J x 0 = j
and hence
A s 0 x 0 = T t x 0 + s 0 Λ C x 0 + ε J 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 + ε J is maximal monotone and strongly quasibounded, 0 ( T + ε J ) ( 0 ) , and
H ( s , x ) = ( T + ε J ) x + s Λ C x 0 for all  s [ 0 , 1 ]  and all  x D ( 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 + s Λ C + ε J , 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 + s Λ C + ε J , Ω , 0 ) = d ( T t + s Λ C + ε J , 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
0 T x + s Λ C x + ε J x
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 Λ ψ ( 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 0 T x 0 , which contradicts the hypothesis that 0 T ( 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 = 0 for every  y L { 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
0 T 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 F F ( L ) and v L , the function c ( F , v ) : F R , defined by c ( F , v ) ( x ) = C x , v , is continuous on F, and

(c3) there exists a nondecreasing function ψ : [ 0 , ) [ 0 , ) such that
C x , x ψ ( x ) for all  x X .
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 0 T x ε + λ ε C x ε + ε J 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 0 T ( D ( T ) Ω ) , then the inclusion 0 T x + λ C x 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 y L { 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 X and C 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 0 T x 0 , which contradicts the hypothesis 0 T ( 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 M D ( A ) if A ( r x ) = r γ A x for all x M 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 p 1 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 L D ( C ) and C ( 0 ) = 0 such that

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

(t2) λ T x + C x + μ J φ x = 0 implies x = 0 for every μ 0 , where φ ( t ) = t γ ,

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

(c2) for every F F ( L ) and v L , the function c ( F , v ) : F R , defined by c ( F , v ) ( x ) = C x , v , is continuous on F, and

(c3) there exists a nondecreasing function ψ : [ 0 , ) [ 0 , ) such that
C x , x ψ ( x ) for all  x D ( C ) .

If the operators T and C are positively homogeneous of degree γ on L, then the operator λ T + 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 ) + ( 1 t ) 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
λ T 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 n N . Since the operators T, C, and J φ are positively homogeneous of degree γ, it follows from equation (3.1) that
λ T ( 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
λ T 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
0 λ T 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 = 0 for every  y L { F n } .
(3.3)
Since the operator λ T + q 0 ε J φ is maximal monotone, we have
lim inf n λ T 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 λ T u n + q n ε J φ u n , u n u 0 < 0 .
Hence it is clear that
lim sup n λ T u n + q n ε J φ u n , u n < λ v + q 0 ε j , u 0 .
(3.5)
For every u D ( T ) , we have, by the monotonicity of the operator λ T + 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 λ T + q 0 ε J φ , we have u 0 D ( T ) and ( λ T + 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 ) , and C 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
λ T u 0 + C u 0 + q 0 ε J φ u 0 = 0 and 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 : = λ T + 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 x F and C ˜ x , x ψ ˜ ( x ) for x D ( 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 x X .

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
λ T 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 : = λ T u n + C u n , we get
λ T u n + C u n + 1 n J φ u n = 1 x n γ p
and so w n 0 . Since λ T u n , u n = C u n , u n + w n , u n ψ ( 1 ) + w n for all n N , 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 0 and lim n C u n + λ v , y = 0
for every y L { F n } . Since the operator C satisfies the condition ( S + ) L , we obtain
u n u 0 , u 0 D ( C ) , and C u 0 + λ v = 0 .
By Lemma 1.2(b), we have u 0 D ( T ) and T u 0 = v and hence
λ T u 0 + C u 0 = 0 and 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 0 and lim n C x n + λ v 0 p , y = 0
for every y L { F n } . Since the operator C satisfies the condition ( S + ) L and T is maximal monotone, we conclude that
x 0 D ( λ T + C ) and λ T x 0 + C x 0 = p .

As p X was arbitrary, this says that the operator λ T + 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′) λ T x + C x , x 0 for all x L .

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

Proof Noting that
λ T x + C x + μ J φ x , x μ J φ x , x = μ x γ + 1 0

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

Declarations

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).

Authors’ Affiliations

(1)
Department of Mathematics, Sungkyunkwan University

References

  1. Krasnosel’skii MA: Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon, New York; 1964.Google Scholar
  2. Zeidler E: Nonlinear Functional Analysis and Its Applications III: Variational Methods and Optimization. Springer, New York; 1985.View ArticleMATHGoogle Scholar
  3. Minty G: Monotone operators in Hilbert spaces. Duke Math. J. 1962, 29: 341-346. 10.1215/S0012-7094-62-02933-2MathSciNetView ArticleMATHGoogle Scholar
  4. Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. In Nonlinear Functional Analysis. Am. Math. Soc., Providence; 1976:1-308. Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill 1968Google Scholar
  5. Zeidler E: Nonlinear Functional Analysis and Its Applications II/B: Nonlinear Monotone Operators. Springer, New York; 1990.View ArticleMATHGoogle Scholar
  6. Guan Z, Kartsatos AG: On the eigenvalue problem for perturbations of nonlinear accretive and monotone operators in Banach spaces. Nonlinear Anal. 1996, 27: 125-141. 10.1016/0362-546X(95)00016-OMathSciNetView ArticleMATHGoogle Scholar
  7. Kartsatos AG: New results in the perturbation theory of maximal monotone and m -accretive operators in Banach spaces. Trans. Am. Math. Soc. 1996, 348: 1663-1707. 10.1090/S0002-9947-96-01654-6MathSciNetView ArticleMATHGoogle Scholar
  8. Kartsatos AG, Skrypnik IV: Normalized eigenvectors for nonlinear abstract and elliptic operators. J. Differ. Equ. 1999, 155: 443-475. 10.1006/jdeq.1998.3592MathSciNetView ArticleMATHGoogle Scholar
  9. Kartsatos AG, Skrypnik IV: On the eigenvalue problem for perturbed nonlinear maximal monotone operators in reflexive Banach spaces. Trans. Am. Math. Soc. 2006, 358: 3851-3881. 10.1090/S0002-9947-05-03761-XMathSciNetView ArticleMATHGoogle Scholar
  10. Li H-X, Huang F-L: On the nonlinear eigenvalue problem for perturbations of monotone and accretive operators in Banach spaces. Sichuan Daxue Xuebao (J. Sichuan Univ.) 2000, 37: 303-309.MathSciNetMATHGoogle Scholar
  11. Browder FE: Fixed point theory and nonlinear problems. Bull. Am. Math. Soc. 1983, 9: 1-39. 10.1090/S0273-0979-1983-15153-4MathSciNetView ArticleMATHGoogle Scholar
  12. Kartsatos AG, Skrypnik IV:Topological degree theories for densely defined mappings involving operators of type ( S + ) . Adv. Differ. Equ. 1999, 4: 413-456.MathSciNetMATHGoogle Scholar
  13. Kartsatos AG, Skrypnik IV:A new topological degree theory for densely defined quasibounded ( S ˜ + ) -perturbations of multivalued maximal monotone operators in reflexive Banach spaces. Abstr. Appl. Anal. 2005, 2005: 121-158. 10.1155/AAA.2005.121MathSciNetView ArticleMATHGoogle Scholar
  14. Skrypnik IV: Nonlinear Higher Order Elliptic Equations. Naukova Dumka, Kiev; 1973. (Russian)MATHGoogle Scholar
  15. Skrypnik IV Transl., Ser. II. 139. In Methods for Analysis of Nonlinear Elliptic Boundary Value Problems. Am. Math. Soc., Providence; 1994.Google Scholar
  16. Browder FE, Hess P: Nonlinear mappings of monotone type in Banach spaces. J. Funct. Anal. 1972, 11: 251-294. 10.1016/0022-1236(72)90070-5MathSciNetView ArticleMATHGoogle Scholar
  17. Berkovits J:On the degree theory for densely defined mappings of class ( S + ) L .Abstr. Appl. Anal. 1999, 4: 141-152. 10.1155/S1085337599000111MathSciNetView ArticleMATHGoogle Scholar
  18. Kartsatos AG, Quarcoo J:A new topological degree theory for densely defined ( S + ) L -perturbations of multivalued maximal monotone operators in reflexive separable Banach spaces. Nonlinear Anal. 2008, 69: 2339-2354. 10.1016/j.na.2007.08.017MathSciNetView ArticleMATHGoogle Scholar
  19. Brézis H, Crandall MG, Pazy A: Perturbations of nonlinear maximal monotone sets in Banach space. Commun. Pure Appl. Math. 1970, 23: 123-144. 10.1002/cpa.3160230107View ArticleMathSciNetMATHGoogle Scholar
  20. Kim I-S, Bae J-H: Eigenvalue results for pseudomonotone perturbations of maximal monotone operators. Cent. Eur. J. Math. 2013, 11: 851-864. 10.2478/s11533-013-0211-2MathSciNetMATHGoogle Scholar
  21. Petryshyn WV: Approximation-Solvability of Nonlinear Functional and Differential Equations. Dekker, New York; 1993.MATHGoogle Scholar
  22. Adhikari DR, Kartsatos AG: Topological degree theories and nonlinear operator equations in Banach spaces. Nonlinear Anal. 2008, 69: 1235-1255. 10.1016/j.na.2007.06.026MathSciNetView ArticleMATHGoogle Scholar
  23. Browder FE: Degree of mapping for nonlinear mappings of monotone type. Proc. Natl. Acad. Sci. USA 1983, 80: 1771-1773. 10.1073/pnas.80.6.1771MathSciNetView ArticleMATHGoogle Scholar
  24. Schmitt K, Sim I: Bifurcation problems associated with generalized Laplacians. Adv. Differ. Equ. 2004, 9: 797-828.MathSciNetMATHGoogle Scholar

Copyright

© Kim and Bae; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.