Monotone type operators in nonreflexive Banach spaces

Open Access
Research
  • 905 Downloads

Abstract

Let E be a real Banach space, E Open image in new window be the dual space of E, E Open image in new window be the dual space of E Open image in new window. Let T : D ( T ) E 2 E Open image in new window be a monotone type mapping. In this paper, first, we introduce the special case when T is the weak* sub-differential ϕ Open image in new window of a convex function ϕ and obtain a surjective result for the mapping ( ϕ + ϵ 2 ) Open image in new window, where ϵ > 0 Open image in new window. Second, we show the existence of solutions of the variational inequality problems for strictly quasi-monotone operators and semi-monotone operators. Finally, we construct a degree theory for mappings of the class ( S + ) Open image in new window and then construct a generalized degree for the weak* sub-differential of a convex function.

Keywords

Banach Space Convex Function Monotone Operator Maximal Monotone Variational Inequality Problem 

1 Introduction

Monotone operators in reflexive Banach spaces has many applications in nonlinear partial differential equations, nonlinear semi-group theory, variational inequality and so on (see [1, 2, 3, 4]). The theory for monotone operators in reflexive Banach spaces has been well developed. In recent years, many authors have generalized the monotone operator theory to nonreflexive Banach spaces. For example, maximal monotone operators in nonreflexive Banach spaces has been studied in [5, 6, 7, 8] and variational inequality problems related to monotone type mappings in nonreflexive Banach spaces have been studied in [9, 10, 11, 12, 13, 14]. For more references on variational inequality problems, see [15, 16, 17, 18, 19, 20, 21, 22, 23, 24] and [25]. Also, degree theory for monotone type mappings in nonreflexive separable Banach spaces has been studied in [26, 27]. Also, see [3, 28, 29, 30, 31, 32, 33, 34, 35, 36] for more references on degree theory of monotone type operators.

In this paper, we study variational inequality problems and degree theory for monotone type mappings in nonreflexive spaces. This paper is organized as follows:

Let E be a real Banach space, E Open image in new window be the dual space of E and E Open image in new window be the dual space of E Open image in new window. In Section 2, we introduce the weak* sub-differential ϕ Open image in new window of a convex function ϕ : E R { + } Open image in new window, which is a subset of the classical sub-differential, and we obtain ( ϕ + ϵ 2 ) = E Open image in new window for the sum of a lower semi-continuous convex function ϕ : E R { + } Open image in new window in the weak* topology and ϵ x 2 Open image in new window, where ϵ > 0 Open image in new window. In Section 3, we show the existence of solutions of variational inequality problems related to strictly quasi-monotone operators and semi-monotone operators. In Section 4, we construct a degree theory for mappings of class ( S + ) Open image in new window and then construct a generalized degree for the weak* sub-differential of a convex function and obtain some degree results.

Through this paper, we use ⇀ to represent the convergence in the weak* topology, ⇀ to represent the convergence in the weak topology and → represent the convergence in norm topology.

2 The weak* sub-differential of convex functions

In this section, let E be a real Banach space, E Open image in new window be the dual space of E and E Open image in new window be the dual space of E Open image in new window.

Now, we introduce the weak* sub-differential of a convex function and study the solvability problems related this mapping.

First, we recall that the classical sub-differential of a convex function ϕ : E R { + } Open image in new window at y is defined by
ϕ ( y ) = { f E : ϕ ( x ) ϕ ( y ) ( f , x y ) , x D ( ϕ ) } . Open image in new window

It is well known (Rockfellar [8]) that ∂ϕ is a maximal monotone mapping.

Definition 2.1 Let ϕ : E R { + } Open image in new window be a convex function. Then
ϕ ( y ) = { f E : ϕ ( x ) ϕ ( y ) ( f , x y ) , x D ( ϕ ) } Open image in new window

is called the weak* sub-differential of ϕ at y.

It is obvious that ϕ ( y ) ϕ ( y ) Open image in new window, but ϕ ( y ) = ϕ ( y ) Open image in new window when E is reflexive.

The following result is obvious.

Proposition 2.2 Let ϕ : E R { + } Open image in new window be a convex function. Then we have the following:
  1. (1)

    ϕ ( y ) Open image in new window is a weak closed convex subset of E Open image in new window;

     
  2. (2)

    0 ϕ ( y 0 ) Open image in new window if and only if ϕ ( y 0 ) = inf y D ( ϕ ) ϕ ( y ) Open image in new window;

     
  3. (3)

    ϕ : E E Open image in new window is monotone.

     

Definition 2.3 (see [37])

Let X be a topological space. A function f : X R Open image in new window is said to be sequentially lower semi-continuous from above at x 0 Open image in new window if, for any sequence { x n } Open image in new window with x n x 0 Open image in new window, f ( x n + 1 ) f ( x n ) Open image in new window implies that f ( x 0 ) lim n f ( x n ) Open image in new window.

Similarly, f is said to be sequentially upper semi-continuous from below at x 0 Open image in new window if, for any sequence { x n } Open image in new window with x n x 0 Open image in new window, f ( x n + 1 ) f ( x n ) Open image in new window implies that f ( x 0 ) lim n f ( x 0 ) Open image in new window.

Remark 1 It is well known that a lower semi-continuous function is a lower semi-continuous from above function, but the converse is not true and a lower semi-continuous from above and convex function with the coercive condition in a reflexive Banach space attains its minimum (see [37]). Also, it is well known that, for a convex function in a reflexive Banach space, lower semi-continuity in the strong topology is equivalent to lower semi-continuity in the weak topology, but this is not true for lower semi-continuity from above (see [38]). For more on lower semi-continuous from above functions with its generalizations and applications in nonconvex equilibrium problems, variational problems and fixed point problems, see [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50] and [51].

Proposition 2.4 Let ϕ : E R { + } Open image in new window be a convex function which is sequentially lower semi-continuous from above in the weak* topology and lim x + ϕ ( x ) = + Open image in new window, then there exists x 0 E Open image in new window such that ϕ ( x 0 ) = inf y D ( ϕ ) ϕ ( y ) Open image in new window.

Proof We take a sequence { x n } Open image in new window in E Open image in new window such that
ϕ ( x 1 ) ϕ ( x 2 ) ϕ ( x n ) , ϕ ( x n ) inf x D ( ϕ ) ϕ ( x ) . Open image in new window
Since lim x + ϕ ( x ) = + Open image in new window and { x n } Open image in new window is a bounded sequence in E Open image in new window, it follows that { x n } Open image in new window has a subsequence { x n k } Open image in new window of { x n } Open image in new window with x n k x 0 Open image in new window in E Open image in new window. By the assumption, since ϕ is sequentially lower semi-continuous from above, we have ϕ ( x 0 ) lim n ϕ ( x n ) Open image in new window and so it follows that
ϕ ( x 0 ) = inf y D ( ϕ ) ϕ ( y ) . Open image in new window

This completes the proof. □

Proposition 2.5 The function ϕ : E R Open image in new window defined by ϕ ( x ) = x 2 Open image in new window is sequentially lower semi-continuous in the weak* topology.

Proof Suppose x n x 0 Open image in new window. Then x 0 ( f ) = lim n x n ( f ) Open image in new window for all f E Open image in new window and so
| x 0 ( f ) | lim inf n x n f Open image in new window
for all f E Open image in new window. Thus we have
x 0 = sup f = 1 | x 0 ( f ) | lim inf n x n Open image in new window

and so x 0 2 lim inf n x n 2 Open image in new window. This completes the proof. □

Theorem 2.6 Let ϕ : E R { + } Open image in new window be a convex function which is sequentially lower semi-continuous in the weak* topology. Then we have
( ϕ + ϵ 2 ) ( E ) = E Open image in new window

for all ϵ > 0 Open image in new window.

Proof For any f E Open image in new window, we set ψ ( x ) = ϕ ( x ) + ϵ x 2 x ( f ) Open image in new window for all x D ( ϕ ) Open image in new window. It is obvious that ψ is sequentially lower semi-continuous in the weak* topology. Thus ψ is sequentially lower semi-continuous from above in the weak* topology and
lim x + ψ ( x ) = + . Open image in new window

By Proposition 2.4, there exists x 0 E Open image in new window such that ϕ ( x 0 ) = inf x D ( ψ ) ψ ( x ) Open image in new window. By (2) of Proposition 2.2, 0 ( ϕ + ϵ 2 ) x ( f ) ) ( x 0 ) Open image in new window, which is equivalent to f ( ϕ + ϵ 2 ) ( x 0 ) Open image in new window. This completes the proof. □

3 Existence of variational inequality problems

In this section, we study variational inequality problems related to monotone type operators in nonreflexive Banach spaces.

First, we recall the following.

Definition 3.1 ([11])

A mapping A ( u , v ) : E × E E Open image in new window is said to be semi-monotone if it satisfies the following conditions:
  1. (1)

    for each u E Open image in new window, A ( u , ) Open image in new window is monotone, i.e., ( A ( u , v ) A ( u , w ) , v w ) 0 Open image in new window for all v , w E Open image in new window;

     
  2. (2)

    for each fixed v E Open image in new window, A ( , v ) Open image in new window is completely continuous, i.e., if u j u 0 Open image in new window in weak* topology of E Open image in new window, then A ( u j , v ) Open image in new window has a subsequence A ( u j k , v ) Open image in new window with A ( u j k , v ) A ( u 0 , v ) Open image in new window in norm topology of E Open image in new window.

     

Definition 3.2 ([15])

Let E be a real Banach space and T : D E 2 E Open image in new window be a mapping. T is said to be strictly quasi-monotone if ( g , u v ) > 0 Open image in new window for all u , v D Open image in new window and for some g T v Open image in new window implies that ( f , u v ) > 0 Open image in new window for all f T u Open image in new window.

Remark 2 For quasi-monotone mappings, see [21].

Lemma 3.3 Let E be a real Banach space and C be a nonempty bounded closed convex subset of E Open image in new window. If A : C 2 E Open image in new window is a finite dimensional weak* upper semi-continuous (i.e. for each finite dimensional subspace F of E Open image in new window with F C Open image in new window, A : C F 2 E Open image in new window is upper semi-continuous in the weak topology) and strictly quasi-monotone mapping with bounded closed convex values, then ( f v , u 0 v ) 0 Open image in new window for all v C Open image in new window and for some f v T u 0 Open image in new window if and only if ( g , u 0 v ) 0 Open image in new window for all v C Open image in new window and g T v Open image in new window.

Proof The proof is similar to Lemma 2.3 in [15], we omit the details. □

Remark 3 For the results of Lemma 3.3 in monotone case, we refer to [10].

Theorem 3.4 Let E be a real Banach space and C be a nonempty weak* closed convex bounded subset of E Open image in new window. If A : C 2 E Open image in new window is a finite dimensional weakly upper semi-continuous and strictly quasi-monotone mapping with bounded closed convex values, then there exists u 0 C Open image in new window such that
( f v , u 0 v ) 0 Open image in new window

for all v C Open image in new window and for some f v T u 0 Open image in new window.

Proof For any finite dimensional subspace F of E with F C Open image in new window, let j F : F E Open image in new window be the natural inclusion and j F Open image in new window be the conjugate mapping of j F Open image in new window. Consider the following variational inequality problem:

Find u F C Open image in new window such that
( j F f v , u v ) 0 Open image in new window

for all v C F Open image in new window and for some f v T u Open image in new window.

Since T is finite dimensional weakly upper semi-continuous and j F T Open image in new window is upper semi-continuous on F C Open image in new window, there exists u F F C Open image in new window such that
( j F f v , u F v ) 0 Open image in new window
for all v C F Open image in new window and for some f v T u F Open image in new window, i.e., ( f v , u F v ) 0 Open image in new window for all v C F Open image in new window and for some f v T u F Open image in new window. By Lemma 3.3, we get
( g , u F v ) 0 Open image in new window
for all v C F Open image in new window and g T v Open image in new window. Now, we put
W F = { u C : ( g , u v ) 0 , v F C , g T v } . Open image in new window
It is obvious that W F Open image in new window is weak* closed convex. One can easily check that
W i = 1 n F i W F i , dim ( F i ) < + , F i C Open image in new window
for i = 1 , 2 , , n Open image in new window. Hence F F W F Open image in new window, where
F = { F E : F C , dim ( F ) < + } . Open image in new window
Take u 0 F F W F Open image in new window. We claim that u 0 Open image in new window satisfies the conclusion of Theorem 3.4. In fact, ( g , u 0 v ) 0 Open image in new window for all v C Open image in new window and g T v Open image in new window. By Lemma 3.3, it follows that
( f v , u 0 v ) 0 Open image in new window

for all v C Open image in new window and for some f v T u 0 Open image in new window. This completes the proof.

From Theorem 3.4, we have the following. □

Corollary 3.5 Let E be a real Banach space and C be a nonempty weak* closed convex unbounded subset of E Open image in new window. If A : C 2 E Open image in new window is a finite dimensional weakly upper semi-continuous and strictly quasi-monotone mapping with bounded closed convex values and there exist v 0 C Open image in new window and r > 0 Open image in new window such that
( f , u v 0 ) > 0 Open image in new window
for all f T u Open image in new window and u C Open image in new window with u > r Open image in new window, then there exists u 0 C Open image in new window such that
( f v , u 0 v ) 0 Open image in new window

for all v C Open image in new window and for some f v T u 0 Open image in new window.

Proof If C n = C B ( 0 , n ) Open image in new window, then, by Theorem 3.4, there exists u n C n Open image in new window such that
( f , u n v ) 0 Open image in new window
for all v C n Open image in new window and for some f v T u n Open image in new window. By Lemma 3.3, we know that
( g , u n v ) 0 Open image in new window
for all v C n Open image in new window and for some g T v Open image in new window. By the assumption, we know that u n r Open image in new window for each n = 1 , 2 , Open image in new window and thus we may assume that u n u 0 Open image in new window as n Open image in new window. Otherwise, we take a subsequence. Consequently, it follows that
( g , u 0 v ) 0 Open image in new window

for all v C Open image in new window and g T v Open image in new window. Again, if we use Lemma 3.3, we get the conclusion. This completes the proof. □

Corollary 3.6 Let E be a real Banach space, B ( 0 , R ) = { x < R : x X } E Open image in new window is the open ball centered at 0 with radius R. If A : B ( 0 , R ) ¯ E Open image in new window is a finite dimensional weakly continuous and strictly quasi-monotone mapping and
( A u , u ) > A u u Open image in new window

for all u B ( 0 , R ) Open image in new window, then there exists u 0 B ( 0 , r ) Open image in new window such that A u 0 = 0 Open image in new window.

Proof It is obvious that B ( 0 , R ) ¯ Open image in new window is weak* closed and convex. By Theorem 3.4, there exists u 0 B ( 0 , R ) ¯ Open image in new window such that
( A u 0 , u 0 v ) 0 Open image in new window
for all v B ( 0 , R ) Open image in new window. Now, we claim that A u 0 = 0 Open image in new window. First, we prove that u 0 < R Open image in new window. In fact, if u 0 = R Open image in new window, then, by the assumption, A u 0 0 Open image in new window and thus there exists v 0 B ( 0 , R ) Open image in new window such that ( A u 0 , v 0 ) = A u 0 v 0 Open image in new window. But we have
A u 0 u 0 < ( A u 0 , u 0 ) ( A u 0 , v 0 ) = A u 0 v 0 , Open image in new window
which is a contradiction. Therefore, we have u 0 < R Open image in new window. Since there exists r > 0 Open image in new window such that u 0 + v B ( 0 , R ) Open image in new window for all v E Open image in new window with v r Open image in new window, we have
( A u 0 , v ) 0 Open image in new window

for all v B ( 0 , r ) Open image in new window and so A u 0 = 0 Open image in new window. This completes the proof. □

Theorem 3.7 Let K E Open image in new window be a bounded weak* closed convex subset. Suppose that ϕ : E R { + } Open image in new window is a lower semi-continuous convex function in the weak* topology K D ( ϕ ) Open image in new window, A : K × K E Open image in new window is semi-monotone, and A ( u , ) Open image in new window is finite dimensional continuous for each u K Open image in new window. Then there exists w 0 K Open image in new window such that
( A ( w 0 , w 0 ) , u w 0 ) + ϕ ( u ) ϕ ( w 0 ) 0 Open image in new window

for all u K Open image in new window.

Proof For each finite dimensional subspace F of E Open image in new window with F K Open image in new window, set K F = K F Open image in new window and ϕ F ( x ) = ϕ ( x ) Open image in new window for x F D ( ϕ ) Open image in new window. By Theorem 2.5 in [11], there exists u F K F Open image in new window such that
( A ( u F , u F ) , u u F ) + ϕ F ( u ) ϕ F ( u F ) 0 Open image in new window
(3.1)
for all u K F Open image in new window. Let
F = { F E : F  is finite dimensional subspace with  F K } Open image in new window
and
W F = { w K : ( A ( w , u ) , u w ) + ϕ ( u ) ϕ ( w ) 0 } . Open image in new window
By (3.1) and the monotonicity of A ( u F , ) Open image in new window, W F Open image in new window is a nonempty bounded subset. Denote by W F ¯ Open image in new window the weak* closure of W F Open image in new window. For any F i F Open image in new window for each i = 1 , 2 , , n Open image in new window, it is easy to see that W i F i W F i Open image in new window for each i = 1 , 2 , , n Open image in new window. So, we have
F F W F ¯ . Open image in new window
Let w 0 F F W F ¯ Open image in new window. Now, we prove that
( A ( w 0 , w 0 ) , u w 0 ) + ϕ ( u ) ϕ ( w 0 ) 0 Open image in new window
for all u K Open image in new window. For each u K Open image in new window, take F F Open image in new window such that w 0 K F Open image in new window and u K F Open image in new window. There exists w j W F Open image in new window such that w j w 0 Open image in new window and
( A ( w j , u ) , u w j ) + ϕ ( u ) ϕ ( w j ) 0 Open image in new window
for each j = 1 , 2 , Open image in new window . By letting j Open image in new window, the complete continuity of A ( , u ) Open image in new window and weak* lower semi-continuity of ϕ imply that
( A ( w 0 , u ) , u w 0 ) + ϕ ( u ) ϕ ( w 0 ) 0 . Open image in new window
Set u = t w 0 + ( 1 t ) v Open image in new window for all t ( 0 , 1 ) Open image in new window and v K Open image in new window, by using the convexity of ϕ and letting t 1 Open image in new window, we get
( A ( w 0 , w 0 ) , v w 0 ) + ϕ ( v ) ϕ ( w 0 ) 0 . Open image in new window

This completes the proof. □

4 Degree theory for monotone type mapping

In this section, assume that E is always a real Banach space, E Open image in new window is the dual space of E and E Open image in new window is the dual space of E Open image in new window.

Definition 4.1 A set-valued operator T : D ( T ) E 2 E Open image in new window is said to be strong to weak upper semi-continuous at x 0 D ( T ) Open image in new window if, for each weak open neighborhood V of 0 in E Open image in new window (i.e., open in the weak topology of E Open image in new window), there exists an open neighborhood W of 0 in E Open image in new window such that T y ( T x 0 + V ) Open image in new window for all y x 0 + W Open image in new window.

Definition 4.2 A set-valued operator T : D ( T ) E 2 E Open image in new window is said to be a mapping of class ( S + ) Open image in new window if the following conditions are satisfied:
  1. (1)

    for each x D ( T ) Open image in new window, Tx is a bounded closed convex subset;

     
  2. (2)

    T is strong to weak upper semi-continuous;

     
  3. (3)
    if x n D ( T ) Open image in new window, f n T x n Open image in new window for each n 1 Open image in new window and x j x 0 Open image in new window such that
    lim ¯ n ( f n , x n x 0 ) 0 , Open image in new window
     

then x n x 0 D ( T ) Open image in new window and { f n } Open image in new window has a subsequence { f n k } Open image in new window with f n k f 0 T x 0 Open image in new window.

Definition 4.3 A family of set-valued operators T t : D E 2 E Open image in new window for all t [ 0 , 1 ] Open image in new window is said to be a homotopy of mappings of class ( S + ) Open image in new window if the following conditions are satisfied:
  1. (1)

    for each t [ 0 , 1 ] Open image in new window, x D Open image in new window, T t x Open image in new window is a bounded closed convex subset;

     
  2. (2)

    T t x : [ 0 , 1 ] × D E Open image in new window is strong to weak upper semi-continuous;

     
  3. (3)
    if x n D ( T ) Open image in new window, t n [ 0 , 1 ] Open image in new window, f n T t n x n Open image in new window for each n 1 Open image in new window, t n t 0 Open image in new window and x j x 0 Open image in new window such that
    lim ¯ n ( f n , x n x 0 ) 0 , Open image in new window
     

then x n x 0 D Open image in new window and { f n } Open image in new window has a subsequence { f n k } Open image in new window with f n k f 0 T t 0 x 0 Open image in new window.

Definition 4.4 Let T : D ( T ) E 2 E Open image in new window be a mapping satisfying the conditions (1) and (2) in Definition 4.1. Let { x j } D ( T ) Open image in new window with x j x 0 D ( T ) Open image in new window and f j T x j Open image in new window with f j f 0 Open image in new window. If lim sup j ( f j , x j x 0 ) 0 Open image in new window implies that
f 0 T x 0 , ( f 0 , x 0 ) = lim j ( f j , x j ) , Open image in new window

then T is called a generalized pseudo-monotone mapping.

Proposition 4.5 Let T : D ( T ) E 2 E Open image in new window be a mapping of class ( S + ) Open image in new window and S : E E Open image in new window be a mapping with closed convex values. Then the following conclusions hold:
  1. (1)

    if S is an upper semi-continuous and compact mapping, then T + S Open image in new window is a mapping of class ( S + ) Open image in new window;

     
  2. (2)

    if S is a generalized pseudo-monotone mapping and weak compact, i.e., S maps bounded subsets in E Open image in new window to weak compact subsets in E Open image in new window, then T + S Open image in new window is a mapping of class ( S + ) Open image in new window.

     

For any subspace F of E Open image in new window, let J F : F E Open image in new window be the natural inclusion and J F : E F Open image in new window be the conjugate mapping of j F Open image in new window. Note that, under the canonical injection mapping J : E E Open image in new window, i.e., J x ( f ) = f ( x ) Open image in new window for all f E Open image in new window and x E Open image in new window, E Open image in new window can be injected as a subspace of E Open image in new window and so, in the following, we always regard E Open image in new window as a subspace of E Open image in new window.

First, we need the following result from [36] (also, see [3]).

Lemma 4.6 Let F be a finite dimensional subspace, Ω F Open image in new window be an open bounded subset and let 0 Ω Open image in new window. Let T : Ω ¯ 2 F Open image in new window be an upper semi-continuous mapping with compact convex values, F 0 Open image in new window be a proper subspace of F, Ω F 0 = Ω F 0 Open image in new window and T F 0 = j F 0 T : Ω F 0 ¯ 2 F 0 Open image in new window be the Galerkin approximation of T, where j F 0 Open image in new window is the adjoint mapping of the natural inclusion j F 0 : F 0 F Open image in new window. If d ( T , Ω , 0 ) d ( T F 0 , Ω F 0 , 0 ) Open image in new window, then there exist x Ω Open image in new window and f T x Open image in new window such that ( f , x ) 0 Open image in new window and ( f , v ) = 0 Open image in new window for all v F 0 Open image in new window, where d ( , , ) Open image in new window is the topological degree for upper semi-continuous mappings with compact convex values in finite dimensional spaces (see Ma [52]).

Remark See [53, 54] for more references on degree theory of multivalued mappings.

Lemma 4.7 Let T : Ω ¯ 2 E Open image in new window be a bounded mapping of ( S + ) Open image in new window and let 0 T ( Ω ) Open image in new window. Then there exists a finite dimensional subspace F 0 Open image in new window of E Open image in new window such that
0 T F ( Ω F ) Open image in new window

for all finite dimensional subspace F of E Open image in new window with F 0 F Open image in new window, where T F = j F T Open image in new window.

Under the condition of Lemma 4.7, we know that deg ( T F , Ω F , 0 ) Open image in new window is well defined for the whole finite dimensional subspace F of E Open image in new window with F 0 F Open image in new window, where F 0 Open image in new window is the same as in Lemma 4.7.

Lemma 4.8 Under the condition of Lemma  4.7, there exists a finite dimensional subspace F 0 Open image in new window of E Open image in new window such that deg ( T F , Ω F , 0 ) Open image in new window does not depend on F.

Now, let Ω E Open image in new window be a nonempty open bounded subset and T : Ω ¯ 2 E Open image in new window be a mapping of class ( S + ) Open image in new window. Suppose that 0 T ( Ω ) Open image in new window. In view of Lemmas 4.6 and 4.8, we may define the topological degree as follows:
deg ( T , Ω D ( T ) , 0 ) = deg ( T F , Ω F , 0 ) , Open image in new window
(4.1)

where F is a finite dimensional subspace of E Open image in new window such that F 0 F Open image in new window and F 0 Open image in new window is the same as in Lemma 4.8.

Theorem 4.9 If deg ( T , Ω , 0 ) 0 Open image in new window, then 0 T x Open image in new window has a solution in Ω.

Proof The proof can be seen from the following proof of Theorem 4.10. □

Theorem 4.10 Let { T t } t [ 0 , 1 ] Open image in new window be a homotopy of mappings of class ( S + ) Open image in new window. If 0 T t ( Ω ) Open image in new window for all t [ 0 , 1 ] Open image in new window, then deg ( T t , Ω , 0 ) Open image in new window does not depends on t [ 0 , 1 ] Open image in new window.

Proof First, we claim that there exist finite dimensional subspaces F 0 Open image in new window of E Open image in new window such that 0 j F T t ( Ω F ) Open image in new window for all finite dimensional subspaces F with F 0 F Open image in new window. Suppose that this is not true. For any finite dimensional subspaces F, we define a set W F Open image in new window as follows:
W F = { ( t , x ) [ 0 , 1 ] × Ω :  there exists  f T t x { ( t , x ) such that  ( f , x ) 0  and  ( f , v ) = 0 , v F } . Open image in new window
Then W F Open image in new window is nonempty. Let W F ¯ Open image in new window be the closure of W F Open image in new window in [ 0 , 1 ] × E Open image in new window with E Open image in new window endowed with weak* topology. Consider the following family of sets:
F = { W F ¯ : F 0 F , dim ( F ) } . Open image in new window
It is easy to show that F F W F ¯ Open image in new window. Let ( t 0 , x 0 ) F F W F ¯ Open image in new window. If, for each v E Open image in new window, we take a finite dimensional subspace F such that v F Open image in new window and x 0 F Open image in new window, then there exist ( t j v , x j v ) W F Open image in new window and f j v T t j v x j v Open image in new window such that
t j v t 0 , x j v x 0 , ( f j v , x j v ) 0 , ( f j v , v ) = 0 Open image in new window
for each j 0 Open image in new window. Hence we have
lim sup j ( f j v , x j v x 0 ) 0 . Open image in new window

But, since { T t : t [ 0 , 1 ] } Open image in new window is a homotopy of mappings of class ( S + ) Open image in new window, it follows that x j v x 0 Ω Open image in new window and { f j v } Open image in new window has a subsequence { f j k v } Open image in new window that converges weakly to f 0 v T t 0 x 0 Open image in new window. Therefore, we have ( f 0 v , v ) = 0 Open image in new window. By Mazur’s separation theorem (see [55]), we get 0 T t 0 x 0 Open image in new window, which is a contradiction. The claim is completed. So, it follows that deg ( T t , F , Ω F , 0 ) Open image in new window is well defined for the whole finite dimensional subspace F with F 0 F Open image in new window.

Next, we prove that there exist a finite dimensional subspace F 1 Open image in new window and F 0 F 1 Open image in new window such that deg ( T t , F , Ω F , 0 ) Open image in new window does not depend on t [ 0 , 1 ] Open image in new window for all finite dimensional subspace F of E Open image in new window with F 1 F Open image in new window.

Suppose that this is not true. For any finite dimensional subspace F with F 0 F Open image in new window, we define
W F = { ( t , x ) [ 0 , 1 ] × Ω :  there exists  f T t x { ( t , x ) such that  ( f , x ) 0  and  ( f , v ) = 0 , v F } . Open image in new window
Then W F Open image in new window is nonempty by Lemma 4.6. Let W F ¯ Open image in new window be the closure of W F Open image in new window in [ 0 , 1 ] × E Open image in new window with E Open image in new window endowed with the weak∗∗ topology. Consider again the following family of sets:
F = { W F ¯ : F 0 F  with  dim ( F ) } . Open image in new window
It is easy to show that F F W F ¯ Open image in new window. Let ( t 0 , x 0 ) F F W F ¯ Open image in new window. Then, for each v E Open image in new window, we take a finite dimensional subspace F such that F 0 F Open image in new window, v F Open image in new window and x 0 F Open image in new window. Then there exist ( t j v , x j v ) W F Open image in new window and f j v T t j v x j v Open image in new window such that
t j v t 0 , x j v x 0 , ( f j v , x j v ) 0 , ( f j v , v ) = 0 Open image in new window
for j 0 Open image in new window. Hence we have
lim j ( f j v , x j v x 0 ) 0 . Open image in new window

But, since { T t : t [ 0 , 1 ] } Open image in new window is a homotopy of mappings of class ( S + ) Open image in new window, we have x j v x 0 Ω Open image in new window and f j v Open image in new window has a subsequence { f j k v } Open image in new window which converges weakly to f 0 v T t 0 x 0 Open image in new window. Therefore, we have ( f 0 v , v ) = 0 Open image in new window. Again, by Mazur’s separation theorem, 0 T t 0 x 0 Open image in new window, which is a contradiction. This completes the proof. □

Theorem 4.11 Let T : Ω ¯ 2 E Open image in new window be a mapping of class ( S + ) Open image in new window, where Ω E Open image in new window is an open bounded subset. If 0 Ω Open image in new window and ( f , x ) > 0 Open image in new window for all x Ω D ( T ) Open image in new window and f T x Open image in new window, then
deg ( T , Ω , 0 ) = 1 . Open image in new window
Proof Assume that F is a finite dimensional subspaces of E Open image in new window. It is straightforward to check that
( j F f , x ) > 0 Open image in new window
for all x Ω F Open image in new window and f T x Open image in new window. Therefore, we have deg ( T F , Ω F , 0 ) = 1 Open image in new window and so, by (4.1),
deg ( T , Ω , 0 ) = 1 . Open image in new window

 □

Theorem 4.12 Let T : E 2 E Open image in new window be a bounded mapping of class ( S + ) Open image in new window. If
lim x inf f T x ( f , x ) x = + , Open image in new window

then T E = E Open image in new window.

Proof For each p E Open image in new window, we set T 1 x = T x p Open image in new window for all x E Open image in new window. Then it is easy to see that T 1 Open image in new window is a mapping of class ( S + ) Open image in new window. One can easily see that ( f , x ) > 0 Open image in new window for all x B ( 0 , R ) Open image in new window, f T 1 x Open image in new window and sufficiently large R. Thus, by Theorem 4.11, deg ( T 1 , B ( 0 , R ) , 0 ) = 1 Open image in new window and so, by Theorem 4.9, 0 T 1 x Open image in new window has a solution in B ( 0 , R ) Open image in new window, i.e., p T x Open image in new window has a solution in B ( 0 , R ) Open image in new window. This completes the proof. □

In the following, we assume that E Open image in new window is separable and so we take any sequence { F n } Open image in new window of finite dimensional subspaces of E Open image in new window such that
F 1 F 2 F n , n = 1 F n ¯ = E . Open image in new window
(4.2)
Lemma 4.13 Let ϕ : D ( ϕ ) E R { + } Open image in new window be a lower semi-continuous convex function in the weak* topology, Ω E Open image in new window be open bounded and let x 1 D ( ϕ ) Open image in new window. Suppose that ϕ ( x 1 ) < ϕ ( x ) Open image in new window for all x Ω D ( ϕ ) Open image in new window. Then there exists a positive integer N such that
0 ϕ n ( Ω F n D ( ϕ n ) ) , Open image in new window

where ϕ n : F n R { + } Open image in new window is a mapping defined by ϕ n ( x ) = ϕ ( x ) Open image in new window for all x F n Open image in new window and F n = span ( F n { x 1 } ) Open image in new window for all n > N Open image in new window.

Proof Suppose that the conclusion is not true. There exists x n D ( ϕ ) Open image in new window such that 0 ϕ n ( x n ) Open image in new window and so we have ϕ ( x ) ϕ ( x n ) 0 Open image in new window for all x F n D ( ϕ ) Open image in new window, which contradicts ϕ ( x 1 ) < ϕ ( x ) Open image in new window for all x Ω D ( ϕ ) Open image in new window.

Under the assumption of Lemma 4.13, we know that there exists a positive integer N such that
0 ϕ n ( Ω F n D ( ϕ n ) ) Open image in new window
for all n > N Open image in new window and so, by [32], deg ( ϕ n , Ω F n , 0 ) Open image in new window is well defined. Now, we define a generalized degree as follows:
Deg ( ϕ , Ω D ( ϕ ) , 0 ) = { k :  there exists  F n , n 1 ,  satisfying  ( 4.2 ) such that  deg ( ϕ n j , Ω F n j , 0 ) k } . Open image in new window

 □

Remark For generalized degree theory, see [56].

Theorem 4.14 Let ϕ : D ( ϕ ) E R { + } Open image in new window be a lower semi-continuous convex function in the weak* topology. If lim x + ϕ ( x ) = + Open image in new window, then
Deg ( ϕ , B ( 0 , r ) D ( ϕ ) , 0 ) = { 1 } Open image in new window

for sufficiently large r.

Proof By the assumption lim x + ϕ ( x ) = + Open image in new window, it follows from Proposition 2.4 that there exists x 0 D ( ϕ ) Open image in new window such that ϕ ( x 0 ) = inf x D ( ϕ ) ϕ ( x ) Open image in new window if we take a large enough r such that ϕ ( x 0 ) < ϕ ( x ) Open image in new window for all x D ( ϕ ) B ( 0 , r ) Open image in new window.

For any F n Open image in new window ( n 1 Open image in new window) satisfying (4.2), we put F n = span ( F n { x 0 } ) Open image in new window. We may easily see that
ϕ n ( x 0 ) = inf x F n D ( ϕ ) ϕ ( x n ) Open image in new window
and so we have
( f , x ) 0 Open image in new window
for all x B ( 0 , r ) F n D ( ϕ n ) Open image in new window. Thus we have
deg ( ϕ n , B ( 0 , r ) F n D ( ϕ n ) , 0 ) = 1 Open image in new window
and, consequently, we have
Deg ( ϕ , B ( 0 , r ) D ( ϕ ) , 0 ) = { 1 } . Open image in new window

This completes the proof. □

Notes

Acknowledgements

The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2013053358).

References

  1. 1.
    Barbu V: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Leyden; 1976.CrossRefGoogle Scholar
  2. 2.
    Brezis H: Operateurs Maximaux Monotones. North-Holland, Amsterdam; 1973.Google Scholar
  3. 3.
    O’Regan D, Cho YJ, Chen YQ: Topological Degree Theory and Applications. Chapman and Hall/CRC Press, Boca Raton; 2006.Google Scholar
  4. 4.
    Pascali D, Sburlan S: Nonlinear Mappings of Monotone Type. Noordhoff, Leyden; 1978.CrossRefGoogle Scholar
  5. 5.
    Borwein JM: Maximality of sums of two maximal monotone operators in general Banach space. Proc. Am. Math. Soc. 2007, 135: 3917–3924. 10.1090/S0002-9939-07-08960-5CrossRefGoogle Scholar
  6. 6.
    Fitzpatrick SP, Phelps RR: Some properties of maximal monotone operators on nonreflexive Banach spaces. Set-Valued Anal. 1995, 3: 51–69. 10.1007/BF01033641CrossRefMathSciNetGoogle Scholar
  7. 7.
    Gossez JP: On the range of a coercive maximal monotone operator in a nonreflexive Banach space. Proc. Am. Math. Soc. 1972, 35: 88–92. 10.1090/S0002-9939-1972-0298492-7CrossRefMathSciNetGoogle Scholar
  8. 8.
    Rockafellar RT: On the maximal monotonicity of subdifferential mapping. Pac. J. Math. 1970, 33: 209–216. 10.2140/pjm.1970.33.209CrossRefMathSciNetGoogle Scholar
  9. 9.
    Beldiman M: Equilibrium problems with set-valued mappings in Banach spaces. Nonlinear Anal. 2008, 68: 3364–3371. 10.1016/j.na.2007.03.030CrossRefMathSciNetGoogle Scholar
  10. 10.
    Chang SS, Lee BS, Chen YQ: Variational inequalities for monotone operators in nonreflexive Banach spaces. Appl. Math. Lett. 1995, 8: 29–34.CrossRefMathSciNetGoogle Scholar
  11. 11.
    Chen YQ: On the semi-monotone operator theory and applications. J. Math. Anal. Appl. 1999, 231: 177–192. 10.1006/jmaa.1998.6245CrossRefMathSciNetGoogle Scholar
  12. 12.
    Domokos A, Kolumban J: Variational inequalities with operator solutions. J. Glob. Optim. 2002, 23: 99–110. 10.1023/A:1014096127736CrossRefMathSciNetGoogle Scholar
  13. 13.
    Verma RU: Variational inequalities involving strongly pseudomonotone hemicontinuous mappings in nonreflexive Banach spaces. Appl. Math. Lett. 1998, 11: 41–43.CrossRefGoogle Scholar
  14. 14.
    Watson PJ: Variational inequalities in nonreflexive Banach spaces. Appl. Math. Lett. 1997, 10: 45–48.CrossRefGoogle Scholar
  15. 15.
    Chen YQ, Cho YJ: On strictly quasi-monotone operators and variational inequalities. J. Nonlinear Convex Anal. 2007, 8: 391–396.MathSciNetGoogle Scholar
  16. 16.
    Fang YP, Huang NJ: Variational-like inequalities with generalized monotone mappings in Banach spaces. J. Optim. Theory Appl. 2003, 118: 327–338. 10.1023/A:1025499305742CrossRefMathSciNetGoogle Scholar
  17. 17.
    Fang Z: Vector variational inequalities with semi-monotone operators. J. Glob. Optim. 2005, 32: 633–642. 10.1007/s10898-004-2698-3CrossRefGoogle Scholar
  18. 18.
    Fang Z: A generalized vector variational inequality problem with a set-valued semi-monotone mapping. Nonlinear Anal. 2008, 69: 1824–1829. 10.1016/j.na.2007.07.025CrossRefMathSciNetGoogle Scholar
  19. 19.
    Guo JS, Yao JC: Variational inequalities with nonmonotone operators. J. Optim. Theory Appl. 1994, 80: 63–74. 10.1007/BF02196593CrossRefMathSciNetGoogle Scholar
  20. 20.
    Kassay G, Kolumban J, Pales Z: Factorization of Minty and Stampacchia variational inequality systems. Eur. J. Oper. Res. 2002, 143: 377–389. 10.1016/S0377-2217(02)00290-4CrossRefMathSciNetGoogle Scholar
  21. 21.
    Karamardian S, Schaible S: Seven kinds of monotone maps. J. Optim. Theory Appl. 1990, 66: 37–46. 10.1007/BF00940531CrossRefMathSciNetGoogle Scholar
  22. 22.
    Lin LJ, Yang MF, Ansari QH, Kassay G: Existence results for Stampacchia and Minty type implicit variational inequalities with multivalued maps. Nonlinear Anal. 2005, 61: 1–19. 10.1016/j.na.2004.07.038CrossRefMathSciNetGoogle Scholar
  23. 23.
    Minty GJ: On the generalization of a direct method of calculus of variations. Bull. Am. Math. Soc. 1967, 73: 315–321. 10.1090/S0002-9904-1967-11732-4CrossRefMathSciNetGoogle Scholar
  24. 24.
    Pascali D:On variational inequalities involving mappings of type ( S ) Open image in new window. In Nonlinear Analysis and Variational Problems. Springer, Berlin; 2010.Google Scholar
  25. 25.
    Plubtieng S, Sombut K: Existence results for system of variational inequality problems with semimonotone operators. J. Inequal. Appl. 2010., 2010: Article ID 251510Google Scholar
  26. 26.
    Wang FL, Chen YQ, O’Regan D:Degree theory for ( S + ) Open image in new window mappings in non-reflexive Banach spaces. Appl. Math. Comput. 2008, 202: 229–232. 10.1016/j.amc.2008.02.001CrossRefMathSciNetGoogle Scholar
  27. 27.
    Wang FL, Chen YQ, O’Regan D: Degree theory for monotone type mappings in non-reflexive Banach spaces. Appl. Math. Lett. 2009, 22: 276–279. 10.1016/j.aml.2008.03.022CrossRefMathSciNetGoogle Scholar
  28. 28.
    Adres J, Gorniewicz L: Note on topological degree for monotone type multivalued maps. Fixed Point Theory 2006, 7: 191–199.MathSciNetGoogle Scholar
  29. 29.
    Browder FE: Fixed point theory and nonlinear problems. Bull. Am. Math. Soc. 1983, 1: 1–39.CrossRefMathSciNetGoogle Scholar
  30. 30.
    Browder FE: Degree theory for nonlinear mappings. Proc. Symp. Pure Math. Soc. 1986, 45: 203–226.CrossRefMathSciNetGoogle Scholar
  31. 31.
    Chen YQ, Cho YJ:Topological degree theory for multi-valued mappings of class ( S + ) L Open image in new window. Arch. Math. 2005, 84: 325–333. 10.1007/s00013-004-1203-zCrossRefMathSciNetGoogle Scholar
  32. 32.
    Chen YQ, O’Donal D, Wang FL, Agarwal R: A note on the degree for maximal monotone mappings in finite dimensional spaces. Appl. Math. Lett. 2009, 22: 1766–1769. 10.1016/j.aml.2009.06.016CrossRefMathSciNetGoogle Scholar
  33. 33.
    Kartsatos AG, Skrypnik IV:Topological degree theories for densely defined mappings involving operators of type ( S + ) Open image in new window. Adv. Differ. Equ. 1999, 4: 413–456.MathSciNetGoogle Scholar
  34. 34.
    Kartsatos AG, Skrypnik IV:The index of a critical point for densely defined operators of type ( S + ) L Open image in new window in Banach spaces. Trans. Am. Math. Soc. 2001, 354: 1601–1630.CrossRefMathSciNetGoogle Scholar
  35. 35.
    Kartsatos AG, Skrypnik IV:A new topological degree theory foe densely defined quasibounded ( S ˜ + ) Open image in new window-perturbations of multivalued maxima monotone operators in reflexive Banach spaces. Abstr. Appl. Anal. 2005, 2005: 121–158. 10.1155/AAA.2005.121CrossRefMathSciNetGoogle Scholar
  36. 36.
    Zhang SS, Chen YQ:Degree theory for multivalued ( S ) Open image in new window type mappings and fixed point theorems. Appl. Math. Mech. 1990, 11: 441–454. 10.1007/BF02016374CrossRefGoogle Scholar
  37. 37.
    Chen YQ, Cho YJ, Yang L: Note on the results with lower semi-continuity. Bull. Korean Math. Soc. 2002, 39: 535–541.CrossRefMathSciNetGoogle Scholar
  38. 38.
    Aruffo A, Bottaro G: Generalizations of sequential lower semicontinuity. Boll. Uni. Mat. Ital. Serie 9 2008, 1: 293–318.MathSciNetGoogle Scholar
  39. 39.
    Aruffo AB, Bottaro G: Some variational results using generalizations of sequential lower semi-continuity. Fixed Point Theory Appl. 2010., 2010: Article ID 323487Google Scholar
  40. 40.
    Bugajewskia D, Kasprzak P: Fixed point theorems for weakly F -contractive and strongly F -expansive mappings. J. Math. Anal. Appl. 2009, 359: 126–134. 10.1016/j.jmaa.2009.05.024CrossRefMathSciNetGoogle Scholar
  41. 41.
    Castellania M, Pappalardob M, Passacantandob M: Existence results for nonconvex equilibrium problems. Optim. Methods Softw. 2010, 25: 49–58. 10.1080/10556780903151557CrossRefMathSciNetGoogle Scholar
  42. 42.
    Al-Homidan S, Ansari QH, Yao JC: Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory. Nonlinear Anal. 2008, 69: 126–139. 10.1016/j.na.2007.05.004CrossRefMathSciNetGoogle Scholar
  43. 43.
    Khanh PQ, Quy DN: A generalized distance and enhanced Ekeland’s variational principle for vector functions. Nonlinear Anal. 2010, 73: 2245–2259. 10.1016/j.na.2010.06.005CrossRefMathSciNetGoogle Scholar
  44. 44.
    Khanh PQ, Quy DN: On generalized Ekeland’s variational principle and equivalent formulations for set-valued mappings. J. Glob. Optim. 2011, 49: 381–396. 10.1007/s10898-010-9565-1CrossRefMathSciNetGoogle Scholar
  45. 45.
    Khanh PQ, Quy DN: On Ekeland’s variational principle for Pareto minima of set-valued mappings. J. Optim. Theory Appl. 2012, 153: 280–297. 10.1007/s10957-011-9957-5CrossRefMathSciNetGoogle Scholar
  46. 46.
    Lin LJ, Du WS: Ekeland’s variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces. J. Math. Anal. Appl. 2006, 323: 360–370. 10.1016/j.jmaa.2005.10.005CrossRefMathSciNetGoogle Scholar
  47. 47.
    Lin LJ, Du WS: On maximal element theorems, variants of Ekeland’s variational principle and their applications. Nonlinear Anal. 2008, 68: 1246–1262. 10.1016/j.na.2006.12.018CrossRefMathSciNetGoogle Scholar
  48. 48.
    Qiu JH: On Has version of set-valued Ekelands variational principle. Acta Math. Sin. 2012, 28: 717–726. 10.1007/s10114-011-0294-2CrossRefGoogle Scholar
  49. 49.
    Qiu JH, He F: P -distances, q -distances and a generalized Ekeland’s variational principle in uniform spaces. Acta Math. Sin. 2012, 28: 235–254. 10.1007/s10114-011-0629-zCrossRefMathSciNetGoogle Scholar
  50. 50.
    Qiu JH, He F: A general vectorial Ekeland’s variational principle with a p -distance. Acta Math. Sin. 2013, 29: 1655–1678. 10.1007/s10114-013-2284-zCrossRefMathSciNetGoogle Scholar
  51. 51.
    Chen YQ, Cho YJ, Kim JK, Lee BS: Note on KKM maps and applications. Fixed Point Theory Appl. 2006., 2006: Article ID 53286Google Scholar
  52. 52.
    Ma TW: Topological degree for set-valued compact vector fields in locally convex spaces. Diss. Math. 1972, 92: 1–43.Google Scholar
  53. 53.
    Gel’man BD, Obukhovskii VV: New results in the theory of multivalued mappings, II. Analysis and applications. J. Sov. Math. 1993, 25: 123–197.Google Scholar
  54. 54.
    Gorniewicz L Topological Fixed Point Theory and Its Applications 4. In Topological Fixed Point Theory of Multivalued Mappings. 2nd edition. Springer, Berlin; 2006.Google Scholar
  55. 55.
    Rudin W: Functional Analysis. MacGraw-Hill, New York; 1973.Google Scholar
  56. 56.
    Petryshyn WV: Generalized Topological Degree and Semilinear Equations. Cambridge University Press, Cambridge; 1995.CrossRefGoogle Scholar

Copyright information

© Chen and Cho; 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/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Authors and Affiliations

  1. 1.College of Applied MathematicsGuangdong University of TechnologyGuangzhouP.R. China
  2. 2.Department of Mathematics Education and the RINSGyongsang National UniversityJinjuKorea
  3. 3.Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia

Personalised recommendations