# Monotone type operators in nonreflexive Banach spaces

- Yuqing Chen
^{1}and - Yeol Je Cho
^{2, 3}Email author

**2014**:119

https://doi.org/10.1186/1687-1812-2014-119

© Chen and Cho; licensee Springer. 2014

**Received: **27 February 2014

**Accepted: **30 April 2014

**Published: **15 May 2014

## Abstract

Let *E* be a real Banach space, ${E}^{\ast}$ be the dual space of *E*, ${E}^{\ast \ast}$ be the dual space of ${E}^{\ast}$. Let $T:D(T)\subseteq {E}^{\ast \ast}\to {2}^{{E}^{\ast}}$ be a monotone type mapping. In this paper, first, we introduce the special case when *T* is the weak* sub-differential ${\partial}^{\ast}\varphi $ of a convex function *ϕ* and obtain a surjective result for the mapping ${\partial}^{\ast}(\varphi +\u03f5{\parallel \cdot \parallel}^{2})$, where $\u03f5>0$. 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 $({S}_{+})$ and then construct a generalized degree for the weak* sub-differential of a convex function.

## 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 [1–4]). 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 [5–8] and variational inequality problems related to monotone type mappings in nonreflexive Banach spaces have been studied in [9–14]. For more references on variational inequality problems, see [15–24] and [25]. Also, degree theory for monotone type mappings in nonreflexive separable Banach spaces has been studied in [26, 27]. Also, see [3, 28–36] 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, ${E}^{\ast}$ be the dual space of *E* and ${E}^{\ast \ast}$ be the dual space of ${E}^{\ast}$. In Section 2, we introduce the weak* sub-differential ${\partial}^{\ast}\varphi $ of a convex function $\varphi :{E}^{\ast \ast}\to R\cup \{+\mathrm{\infty}\}$, which is a subset of the classical sub-differential, and we obtain ${\partial}^{\ast}(\varphi +\u03f5{\parallel \cdot \parallel}^{2})={E}^{\ast}$ for the sum of a lower semi-continuous convex function $\varphi :{E}^{\ast \ast}\to R\cup \{+\mathrm{\infty}\}$ in the weak* topology and $\u03f5{\parallel x\parallel}^{2}$, where $\u03f5>0$. 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 $({S}_{+})$ 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, ${E}^{\ast}$ be the dual space of *E* and ${E}^{\ast \ast}$ be the dual space of ${E}^{\ast}$.

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

*y*is defined by

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

**Definition 2.1**Let $\varphi :{E}^{\ast \ast}\to R\cup \{+\mathrm{\infty}\}$ be a convex function. Then

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

It is obvious that ${\partial}^{\ast}\varphi (y)\subseteq \partial \varphi (y)$, but ${\partial}^{\ast}\varphi (y)=\partial \varphi (y)$ when *E* is reflexive.

The following result is obvious.

**Proposition 2.2**

*Let*$\varphi :{E}^{\ast \ast}\to R\cup \{+\mathrm{\infty}\}$

*be a convex function*.

*Then we have the following*:

- (1)
${\partial}^{\ast}\varphi (y)$

*is a weak closed convex subset of*${E}^{\ast}$; - (2)
$0\in {\partial}^{\ast}\varphi ({y}_{0})$

*if and only if*$\varphi ({y}_{0})={inf}_{y\in D(\varphi )}\varphi (y)$; - (3)
${\partial}^{\ast}\varphi :{E}^{\ast \ast}\to {E}^{\ast}$

*is monotone*.

**Definition 2.3** (see [37])

Let *X* be a topological space. A function $f:X\to R$ is said to be *sequentially lower semi-continuous from above* at ${x}_{0}$ if, for any sequence $\{{x}_{n}\}$ with ${x}_{n}\to {x}_{0}$, $f({x}_{n+1})\le f({x}_{n})$ implies that $f({x}_{0})\le {lim}_{n\to \mathrm{\infty}}f({x}_{n})$.

Similarly, *f* is said to be *sequentially upper semi-continuous from below* at ${x}_{0}$ if, for any sequence $\{{x}_{n}\}$ with ${x}_{n}\to {x}_{0}$, $f({x}_{n+1})\ge f({x}_{n})$ implies that $f({x}_{0})\le {lim}_{n\to \mathrm{\infty}}f({x}_{0})$.

**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 [38–50] and [51].

**Proposition 2.4** *Let* $\varphi :{E}^{\ast \ast}\to R\cup \{+\mathrm{\infty}\}$ *be a convex function which is sequentially lower semi*-*continuous from above in the weak** *topology and* ${lim}_{\parallel x\parallel \to +\mathrm{\infty}}\varphi (x)=+\mathrm{\infty}$, *then there exists* ${x}_{0}\in {E}^{\ast \ast}$ *such that* $\varphi ({x}_{0})={inf}_{y\in D(\varphi )}\varphi (y)$.

*Proof*We take a sequence $\{{x}_{n}\}$ in ${E}^{\ast \ast}$ such that

*ϕ*is sequentially lower semi-continuous from above, we have $\varphi ({x}_{0})\le {lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n})$ and so it follows that

This completes the proof. □

**Proposition 2.5** *The function* $\varphi :{E}^{\ast \ast}\to R$ *defined by* $\varphi (x)={\parallel x\parallel}^{2}$ *is sequentially lower semi*-*continuous in the weak** *topology*.

*Proof*Suppose ${x}_{n}{\rightharpoonup}^{\ast}{x}_{0}$. Then ${x}_{0}(f)={lim}_{n\to \mathrm{\infty}}{x}_{n}(f)$ for all $f\in {E}^{\ast}$ and so

and so ${\parallel {x}_{0}\parallel}^{2}\le {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\parallel {x}_{n}\parallel}^{2}$. This completes the proof. □

**Theorem 2.6**

*Let*$\varphi :{E}^{\ast \ast}\to R\cup \{+\mathrm{\infty}\}$

*be a convex function which is sequentially lower semi*-

*continuous in the weak**

*topology*.

*Then we have*

*for all* $\u03f5>0$.

*Proof*For any $f\in {E}^{\ast}$, we set $\psi (x)=\varphi (x)+\u03f5{\parallel x\parallel}^{2}-x(f)$ for all $x\in D(\varphi )$. 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

By Proposition 2.4, there exists ${x}_{0}\in {E}^{\ast \ast}$ such that $\varphi ({x}_{0})={inf}_{x\in D(\psi )}\psi (x)$. By (2) of Proposition 2.2, $0\in {\partial}^{\ast}(\varphi +\u03f5{\parallel \cdot \parallel}^{2})-x(f))({x}_{0})$, which is equivalent to $f\in {\partial}^{\ast}(\varphi +\u03f5{\parallel \cdot \parallel}^{2})({x}_{0})$. 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])

*semi-monotone*if it satisfies the following conditions:

- (1)
for each $u\in {E}^{\ast \ast}$, $A(u,\cdot )$ is monotone,

*i.e.*, $(A(u,v)-A(u,w),v-w)\ge 0$ for all $v,w\in {E}^{\ast \ast}$; - (2)
for each fixed $v\in {E}^{\ast \ast}$, $A(\cdot ,v)$ is completely continuous,

*i.e.*, if ${u}_{j}\rightharpoonup {u}_{0}$ in weak* topology of ${E}^{\ast \ast}$, then $A({u}_{j},v)$ has a subsequence $A({u}_{{j}_{k}},v)$ with $A({u}_{{j}_{k}},v)\to A({u}_{0},v)$ in norm topology of ${E}^{\ast}$.

**Definition 3.2** ([15])

Let *E* be a real Banach space and $T:D\subseteq {E}^{\ast \ast}\to {2}^{{E}^{\ast}}$ be a mapping. *T* is said to be *strictly quasi-monotone* if $(g,u-v)>0$ for all $u,v\in D$ and for some $g\in Tv$ implies that $(f,u-v)>0$ for all $f\in Tu$.

**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* ${E}^{\ast \ast}$. *If* $A:C\to {2}^{{E}^{\ast}}$ *is a finite dimensional weak** *upper semi*-*continuous* (*i*.*e*. *for each finite dimensional subspace* *F* *of* ${E}^{\ast \ast}$ *with* $F\cap C\ne \mathrm{\varnothing}$, $A:C\cap F\to {2}^{{E}^{\ast}}$ *is upper semi*-*continuous in the weak topology*) *and strictly quasi*-*monotone mapping with bounded closed convex values*, *then* $({f}_{v},{u}_{0}-v)\le 0$ *for all* $v\in C$ *and for some* ${f}_{v}\in T{u}_{0}$ *if and only if* $(g,{u}_{0}-v)\le 0$ *for all* $v\in C$ *and* $g\in Tv$.

*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*${E}^{\ast \ast}$.

*If*$A:C\to {2}^{{E}^{\ast}}$

*is a finite dimensional weakly upper semi*-

*continuous and strictly quasi*-

*monotone mapping with bounded closed convex values*,

*then there exists*${u}_{0}\in C$

*such that*

*for all* $v\in C$ *and for some* ${f}_{v}\in T{u}_{0}$.

*Proof* For any finite dimensional subspace *F* of *E* with $F\cap C\ne \mathrm{\varnothing}$, let ${j}_{F}:F\to E$ be the natural inclusion and ${j}_{F}^{\ast}$ be the conjugate mapping of ${j}_{F}$. Consider the following variational inequality problem:

for all $v\in C\cap F$ and for some ${f}_{v}\in Tu$.

*T*is finite dimensional weakly upper semi-continuous and ${j}_{F}^{\ast}T$ is upper semi-continuous on $F\cap C$, there exists ${u}_{F}\in F\cap C$ such that

*i.e.*, $({f}_{v},{u}_{F}-v)\le 0$ for all $v\in C\cap F$ and for some ${f}_{v}\in T{u}_{F}$. By Lemma 3.3, we get

for all $v\in C$ and for some ${f}_{v}\in T{u}_{0}$. 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*${E}^{\ast \ast}$.

*If*$A:C\to {2}^{{E}^{\ast}}$

*is a finite dimensional weakly upper semi*-

*continuous and strictly quasi*-

*monotone mapping with bounded closed convex values and there exist*${v}_{0}\in C$

*and*$r>0$

*such that*

*for all*$f\in Tu$

*and*$u\in C$

*with*$\parallel u\parallel >r$,

*then there exists*${u}_{0}\in C$

*such that*

*for all* $v\in C$ *and for some* ${f}_{v}\in T{u}_{0}$.

*Proof*If ${C}_{n}=C\cap B(0,n)$, then, by Theorem 3.4, there exists ${u}_{n}\in {C}_{n}$ such that

for all $v\in C$ and $g\in Tv$. 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*, $B(0,R)=\{\parallel x\parallel <R:x\in {X}^{\ast \ast}\}\subset {E}^{\ast \ast}$

*is the open ball centered at*0

*with radius*

*R*.

*If*$A:\overline{B(0,R)}\to {E}^{\ast}$

*is a finite dimensional weakly continuous and strictly quasi*-

*monotone mapping and*

*for all* $u\in \partial B(0,R)$, *then there exists* ${u}_{0}\in B(0,r)$ *such that* $A{u}_{0}=0$.

*Proof*It is obvious that $\overline{B(0,R)}$ is weak* closed and convex. By Theorem 3.4, there exists ${u}_{0}\in \overline{B(0,R)}$ such that

for all $v\in B(0,r)$ and so $A{u}_{0}=0$. This completes the proof. □

**Theorem 3.7**

*Let*$K\subset {E}^{\ast \ast}$

*be a bounded weak**

*closed convex subset*.

*Suppose that*$\varphi :{E}^{\ast \ast}\to R\cup \{+\mathrm{\infty}\}$

*is a lower semi*-

*continuous convex function in the weak**

*topology*$K\subseteq D(\varphi )$, $A:K\times K\to {E}^{\ast}$

*is semi*-

*monotone*,

*and*$A(u,\cdot )$

*is finite dimensional continuous for each*$u\in K$.

*Then there exists*${w}_{0}\in K$

*such that*

*for all* $u\in K$.

*Proof*For each finite dimensional subspace

*F*of ${E}^{\ast \ast}$ with $F\cap K\ne \mathrm{\varnothing}$, set ${K}_{F}=K\cap F$ and ${\varphi}_{F}(x)=\varphi (x)$ for $x\in F\cap D(\varphi )$. By Theorem 2.5 in [11], there exists ${u}_{F}\in {K}_{F}$ such that

*ϕ*imply that

*ϕ*and letting $t\to 1$, we get

This completes the proof. □

## 4 Degree theory for monotone type mapping

In this section, assume that *E* is always a real Banach space, ${E}^{\ast}$ is the dual space of *E* and ${E}^{\ast \ast}$ is the dual space of ${E}^{\ast}$.

**Definition 4.1** A set-valued operator $T:D(T)\subseteq {E}^{\ast \ast}\to {2}^{{E}^{\ast}}$ is said to be *strong to weak upper semi-continuous* at ${x}_{0}\in D(T)$ if, for each weak open neighborhood *V* of 0 in ${E}^{\ast}$ (*i.e.*, open in the weak topology of ${E}^{\ast}$), there exists an open neighborhood *W* of 0 in ${E}^{\ast \ast}$ such that $Ty\cap (T{x}_{0}+V)\ne \mathrm{\varnothing}$ for all $y\in {x}_{0}+W$.

**Definition 4.2**A set-valued operator $T:D(T)\subseteq {E}^{\ast \ast}\to {2}^{{E}^{\ast}}$ is said to be a

*mapping of class*$({S}_{+})$ if the following conditions are satisfied:

- (1)
for each $x\in D(T)$,

*Tx*is a bounded closed convex subset; - (2)
*T*is strong to weak upper semi-continuous; - (3)if ${x}_{n}\in D(T)$, ${f}_{n}\in T{x}_{n}$ for each $n\ge 1$ and ${x}_{j}{\rightharpoonup}^{\ast}{x}_{0}$ such that$\underset{n\to \mathrm{\infty}}{\overline{lim}}({f}_{n},{x}_{n}-{x}_{0})\le 0,$

then ${x}_{n}\to {x}_{0}\in D(T)$ and $\{{f}_{n}\}$ has a subsequence $\{{f}_{{n}_{k}}\}$ with ${f}_{{n}_{k}}\rightharpoonup {f}_{0}\in T{x}_{0}$.

**Definition 4.3**A family of set-valued operators ${T}_{t}:D\subseteq {E}^{\ast \ast}\to {2}^{{E}^{\ast}}$ for all $t\in [0,1]$ is said to be a

*homotopy of mappings of class*$({S}_{+})$ if the following conditions are satisfied:

- (1)
for each $t\in [0,1]$, $x\in D$, ${T}_{t}x$ is a bounded closed convex subset;

- (2)
${T}_{t}x:[0,1]\times D\to {E}^{\ast}$ is strong to weak upper semi-continuous;

- (3)if ${x}_{n}\in D(T)$, ${t}_{n}\in [0,1]$, ${f}_{n}\in {T}_{{t}_{n}}{x}_{n}$ for each $n\ge 1$, ${t}_{n}\to {t}_{0}$ and ${x}_{j}{\rightharpoonup}^{\ast}{x}_{0}$ such that$\underset{n\to \mathrm{\infty}}{\overline{lim}}({f}_{n},{x}_{n}-{x}_{0})\le 0,$

then ${x}_{n}\to {x}_{0}\in D$ and $\{{f}_{n}\}$ has a subsequence $\{{f}_{{n}_{k}}\}$ with ${f}_{{n}_{k}}\rightharpoonup {f}_{0}\in {T}_{{t}_{0}}{x}_{0}$.

**Definition 4.4**Let $T:D(T)\subseteq {E}^{\ast \ast}\to {2}^{{E}^{\ast}}$ be a mapping satisfying the conditions (1) and (2) in Definition 4.1. Let $\{{x}_{j}\}\subset D(T)$ with ${x}_{j}{\rightharpoonup}^{\ast}{x}_{0}\in D(T)$ and ${f}_{j}\in T{x}_{j}$ with ${f}_{j}\rightharpoonup {f}_{0}$. If ${lim\hspace{0.17em}sup}_{j\to \mathrm{\infty}}({f}_{j},{x}_{j}-{x}_{0})\le 0$ implies that

then *T* is called a *generalized pseudo-monotone mapping*.

**Proposition 4.5**

*Let*$T:D(T)\subseteq {E}^{\ast \ast}\to {2}^{{E}^{\ast}}$

*be a mapping of class*$({S}_{+})$

*and*$S:{E}^{\ast \ast}\to {E}^{\ast}$

*be a mapping with closed convex values*.

*Then the following conclusions hold*:

- (1)
*if**S**is an upper semi*-*continuous and compact mapping*,*then*$T+S$*is a mapping of class*$({S}_{+})$; - (2)
*if**S**is a generalized pseudo*-*monotone mapping and weak compact*,*i*.*e*.,*S**maps bounded subsets in*${E}^{\ast \ast}$*to weak compact subsets in*${E}^{\ast}$,*then*$T+S$*is a mapping of class*$({S}_{+})$.

For any subspace *F* of ${E}^{\ast \ast}$, let ${J}_{F}:F\to {E}^{\ast \ast}$ be the natural inclusion and ${J}_{F}^{\ast}:{E}^{\ast \ast \ast}\to {F}^{\ast}$ be the conjugate mapping of ${j}_{F}$. Note that, under the canonical injection mapping $J:{E}^{\ast}\to {E}^{\ast \ast \ast}$, *i.e.*, $Jx(f)=f(x)$ for all $f\in {E}^{\ast \ast}$ and $x\in {E}^{\ast}$, ${E}^{\ast}$ can be injected as a subspace of ${E}^{\ast \ast \ast}$ and so, in the following, we always regard ${E}^{\ast}$ as a subspace of ${E}^{\ast \ast \ast}$.

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

**Lemma 4.6** *Let* *F* *be a finite dimensional subspace*, $\mathrm{\Omega}\subset F$ *be an open bounded subset and let* $0\in \mathrm{\Omega}$. *Let* $T:\overline{\mathrm{\Omega}}\to {2}^{{F}^{\ast}}$ *be an upper semi*-*continuous mapping with compact convex values*, ${F}_{0}$ *be a proper subspace of* *F*, ${\mathrm{\Omega}}_{{F}_{0}}=\mathrm{\Omega}\cap {F}_{0}\ne \mathrm{\varnothing}$ *and* ${T}_{{F}_{0}}={j}_{{F}_{0}}^{\ast}T:\overline{{\mathrm{\Omega}}_{{F}_{0}}}\to {2}^{{F}_{0}^{\ast}}$ *be the Galerkin approximation of* *T*, *where* ${j}_{{F}_{0}}^{\ast}$ *is the adjoint mapping of the natural inclusion* ${j}_{{F}_{0}}:{F}_{0}\to F$. *If* $d(T,\mathrm{\Omega},0)\ne d({T}_{{F}_{0}},{\mathrm{\Omega}}_{{F}_{0}},0)$, *then there exist* $x\in \partial \mathrm{\Omega}$ *and* $f\in Tx$ *such that* $(f,x)\le 0$ *and* $(f,v)=0$ *for all* $v\in {F}_{0}$, *where* $d(\cdot ,\cdot ,\cdot )$ *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*$T:\overline{\mathrm{\Omega}}\to {2}^{{E}^{\ast}}$

*be a bounded mapping of*$({S}_{+})$

*and let*$0\notin T(\partial \mathrm{\Omega})$.

*Then there exists a finite dimensional subspace*${F}_{0}$

*of*${E}^{\ast \ast}$

*such that*

*for all finite dimensional subspace* *F* *of* ${E}^{\ast \ast}$ *with* ${F}_{0}\subseteq F$, *where* ${T}_{F}={j}_{F}^{\ast}T$.

Under the condition of Lemma 4.7, we know that $deg({T}_{F},\mathrm{\Omega}\cap F,0)$ is well defined for the whole finite dimensional subspace *F* of ${E}^{\ast \ast}$ with ${F}_{0}\subseteq F$, where ${F}_{0}$ is the same as in Lemma 4.7.

**Lemma 4.8** *Under the condition of Lemma * 4.7, *there exists a finite dimensional subspace* ${F}_{0}$ *of* ${E}^{\ast \ast}$ *such that* $deg({T}_{F},\mathrm{\Omega}\cap F,0)$ *does not depend on* *F*.

where *F* is a finite dimensional subspace of ${E}^{\ast \ast}$ such that ${F}_{0}\subset F$ and ${F}_{0}$ is the same as in Lemma 4.8.

**Theorem 4.9** *If* $deg(T,\mathrm{\Omega},0)\ne 0$, *then* $0\in Tx$ *has a solution in* Ω.

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

**Theorem 4.10** *Let* ${\{{T}_{t}\}}_{t\in [0,1]}$ *be a homotopy of mappings of class* $({S}_{+})$. *If* $0\notin {T}_{t}(\partial \mathrm{\Omega})$ *for all* $t\in [0,1]$, *then* $deg({T}_{t},\mathrm{\Omega},0)$ *does not depends on* $t\in [0,1]$.

*Proof*First, we claim that there exist finite dimensional subspaces ${F}_{0}$ of ${E}^{\ast \ast}$ such that $0\notin {j}_{F}^{\ast}{T}_{t}(\partial \mathrm{\Omega}\cap F)$ for all finite dimensional subspaces

*F*with ${F}_{0}\subset F$. Suppose that this is not true. For any finite dimensional subspaces

*F*, we define a set ${W}_{F}$ as follows:

*F*such that $v\in F$ and ${x}_{0}\in F$, then there exist $({t}_{j}^{v},{x}_{j}^{v})\in {W}_{F}$ and ${f}_{j}^{v}\in {T}_{{t}_{j}^{v}}{x}_{j}^{v}$ such that

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

Next, we prove that there exist a finite dimensional subspace ${F}_{1}$ and ${F}_{0}\subset {F}_{1}$ such that $deg({T}_{t,F},{\mathrm{\Omega}}_{F},0)$ does not depend on $t\in [0,1]$ for all finite dimensional subspace *F* of ${E}^{\ast \ast}$ with ${F}_{1}\subset F$.

*F*with ${F}_{0}\subset F$, we define

^{∗∗}topology. Consider again the following family of sets:

*F*such that ${F}_{0}\subset F$, $v\in F$ and ${x}_{0}\in F$. Then there exist $({t}_{j}^{v},{x}_{j}^{v})\in {W}_{F}$ and ${f}_{j}^{v}\in {T}_{{t}_{j}^{v}}{x}_{j}^{v}$ such that

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

**Theorem 4.11**

*Let*$T:\overline{\mathrm{\Omega}}\to {2}^{{E}^{\ast}}$

*be a mapping of class*$({S}_{+})$,

*where*$\mathrm{\Omega}\subset {E}^{\ast \ast}$

*is an open bounded subset*.

*If*$0\in \mathrm{\Omega}$

*and*$(f,x)>0$

*for all*$x\in \partial \mathrm{\Omega}\cap D(T)$

*and*$f\in Tx$,

*then*

*Proof*Assume that

*F*is a finite dimensional subspaces of ${E}^{\ast \ast}$. It is straightforward to check that

□

**Theorem 4.12**

*Let*$T:{E}^{\ast \ast}\to {2}^{{E}^{\ast}}$

*be a bounded mapping of class*$({S}_{+})$.

*If*

*then* $T{E}^{\ast \ast}={E}^{\ast}$.

*Proof* For each $p\in {E}^{\ast}$, we set ${T}_{1}x=Tx-p$ for all $x\in {E}^{\ast \ast}$. Then it is easy to see that ${T}_{1}$ is a mapping of class $({S}_{+})$. One can easily see that $(f,x)>0$ for all $x\in \partial B(0,R)$, $f\in {T}_{1}x$ and sufficiently large *R*. Thus, by Theorem 4.11, $deg({T}_{1},B(0,R),0)=1$ and so, by Theorem 4.9, $0\in {T}_{1}x$ has a solution in $B(0,R)$, *i.e.*, $p\in Tx$ has a solution in $B(0,R)$. This completes the proof. □

**Lemma 4.13**

*Let*$\varphi :D(\varphi )\subseteq {E}^{\ast \ast}\to R\cup \{+\mathrm{\infty}\}$

*be a lower semi*-

*continuous convex function in the weak**

*topology*, $\mathrm{\Omega}\subset {E}^{\ast \ast}$

*be open bounded and let*${x}_{1}\in D(\varphi )$.

*Suppose that*$\varphi ({x}_{1})<\varphi (x)$

*for all*$x\in \partial \mathrm{\Omega}\cap D(\varphi )$.

*Then there exists a positive integer*

*N*

*such that*

*where* ${\varphi}_{n}:{F}_{n}^{\prime}\to R\cup \{+\mathrm{\infty}\}$ *is a mapping defined by* ${\varphi}_{n}(x)=\varphi (x)$ *for all* $x\in {F}_{n}^{\prime}$ *and* ${F}_{n}^{\prime}=span({F}_{n}\cup \{{x}_{1}\})$ *for all* $n>N$.

*Proof* Suppose that the conclusion is not true. There exists ${x}_{n}\in D(\varphi )$ such that $0\in \partial {\varphi}_{n}({x}_{n})$ and so we have $\varphi (x)-\varphi ({x}_{n})\ge 0$ for all $x\in {F}_{n}^{\prime}\cap D(\varphi )$, which contradicts $\varphi ({x}_{1})<\varphi (x)$ for all $x\in \partial \mathrm{\Omega}\cap D(\varphi )$.

*N*such that

□

**Remark** For generalized degree theory, see [56].

**Theorem 4.14**

*Let*$\varphi :D(\varphi )\subseteq {E}^{\ast \ast}\to R\cup \{+\mathrm{\infty}\}$

*be a lower semi*-

*continuous convex function in the weak**

*topology*.

*If*${lim}_{\parallel x\parallel \to +\mathrm{\infty}}\varphi (x)=+\mathrm{\infty}$,

*then*

*for sufficiently large* *r*.

*Proof* By the assumption ${lim}_{\parallel x\parallel \to +\mathrm{\infty}}\varphi (x)=+\mathrm{\infty}$, it follows from Proposition 2.4 that there exists ${x}_{0}\in D(\varphi )$ such that $\varphi ({x}_{0})={inf}_{x\in D(\varphi )}\varphi (x)$ if we take a large enough *r* such that $\varphi ({x}_{0})<\varphi (x)$ for all $x\in D(\varphi )\cap \partial B(0,r)$.

This completes the proof. □

## Declarations

### 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).

## Authors’ Affiliations

## References

- Barbu V:
*Nonlinear Semigroups and Differential Equations in Banach Spaces*. Noordhoff, Leyden; 1976.View ArticleGoogle Scholar - Brezis H:
*Operateurs Maximaux Monotones*. North-Holland, Amsterdam; 1973.Google Scholar - O’Regan D, Cho YJ, Chen YQ:
*Topological Degree Theory and Applications*. Chapman and Hall/CRC Press, Boca Raton; 2006.Google Scholar - Pascali D, Sburlan S:
*Nonlinear Mappings of Monotone Type*. Noordhoff, Leyden; 1978.View ArticleGoogle Scholar - Borwein JM: Maximality of sums of two maximal monotone operators in general Banach space.
*Proc. Am. Math. Soc.*2007, 135: 3917–3924. 10.1090/S0002-9939-07-08960-5View ArticleGoogle Scholar - Fitzpatrick SP, Phelps RR: Some properties of maximal monotone operators on nonreflexive Banach spaces.
*Set-Valued Anal.*1995, 3: 51–69. 10.1007/BF01033641View ArticleMathSciNetGoogle Scholar - 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/S0002-9939-1972-0298492-7View ArticleMathSciNetGoogle Scholar - Rockafellar RT: On the maximal monotonicity of subdifferential mapping.
*Pac. J. Math.*1970, 33: 209–216. 10.2140/pjm.1970.33.209View ArticleMathSciNetGoogle Scholar - Beldiman M: Equilibrium problems with set-valued mappings in Banach spaces.
*Nonlinear Anal.*2008, 68: 3364–3371. 10.1016/j.na.2007.03.030View ArticleMathSciNetGoogle Scholar - Chang SS, Lee BS, Chen YQ: Variational inequalities for monotone operators in nonreflexive Banach spaces.
*Appl. Math. Lett.*1995, 8: 29–34.View ArticleMathSciNetGoogle Scholar - Chen YQ: On the semi-monotone operator theory and applications.
*J. Math. Anal. Appl.*1999, 231: 177–192. 10.1006/jmaa.1998.6245View ArticleMathSciNetGoogle Scholar - Domokos A, Kolumban J: Variational inequalities with operator solutions.
*J. Glob. Optim.*2002, 23: 99–110. 10.1023/A:1014096127736View ArticleMathSciNetGoogle Scholar - Verma RU: Variational inequalities involving strongly pseudomonotone hemicontinuous mappings in nonreflexive Banach spaces.
*Appl. Math. Lett.*1998, 11: 41–43.View ArticleGoogle Scholar - Watson PJ: Variational inequalities in nonreflexive Banach spaces.
*Appl. Math. Lett.*1997, 10: 45–48.View ArticleGoogle Scholar - Chen YQ, Cho YJ: On strictly quasi-monotone operators and variational inequalities.
*J. Nonlinear Convex Anal.*2007, 8: 391–396.MathSciNetGoogle Scholar - 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:1025499305742View ArticleMathSciNetGoogle Scholar - Fang Z: Vector variational inequalities with semi-monotone operators.
*J. Glob. Optim.*2005, 32: 633–642. 10.1007/s10898-004-2698-3View ArticleGoogle Scholar - 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.2007.07.025View ArticleMathSciNetGoogle Scholar - Guo JS, Yao JC: Variational inequalities with nonmonotone operators.
*J. Optim. Theory Appl.*1994, 80: 63–74. 10.1007/BF02196593View ArticleMathSciNetGoogle Scholar - 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)00290-4View ArticleMathSciNetGoogle Scholar - Karamardian S, Schaible S: Seven kinds of monotone maps.
*J. Optim. Theory Appl.*1990, 66: 37–46. 10.1007/BF00940531View ArticleMathSciNetGoogle Scholar - 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.2004.07.038View ArticleMathSciNetGoogle Scholar - Minty GJ: On the generalization of a direct method of calculus of variations.
*Bull. Am. Math. Soc.*1967, 73: 315–321. 10.1090/S0002-9904-1967-11732-4View ArticleMathSciNetGoogle Scholar - Pascali D:On variational inequalities involving mappings of type $(S)$. In
*Nonlinear Analysis and Variational Problems*. Springer, Berlin; 2010.Google Scholar - Plubtieng S, Sombut K: Existence results for system of variational inequality problems with semimonotone operators.
*J. Inequal. Appl.*2010., 2010: Article ID 251510Google Scholar - Wang FL, Chen YQ, O’Regan D:Degree theory for $(S+)$ mappings in non-reflexive Banach spaces.
*Appl. Math. Comput.*2008, 202: 229–232. 10.1016/j.amc.2008.02.001View ArticleMathSciNetGoogle Scholar - 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.2008.03.022View ArticleMathSciNetGoogle Scholar - Adres J, Gorniewicz L: Note on topological degree for monotone type multivalued maps.
*Fixed Point Theory*2006, 7: 191–199.MathSciNetGoogle Scholar - Browder FE: Fixed point theory and nonlinear problems.
*Bull. Am. Math. Soc.*1983, 1: 1–39.View ArticleMathSciNetGoogle Scholar - Browder FE: Degree theory for nonlinear mappings.
*Proc. Symp. Pure Math. Soc.*1986, 45: 203–226.View ArticleMathSciNetGoogle Scholar - Chen YQ, Cho YJ:Topological degree theory for multi-valued mappings of class ${({S}_{+})}_{L}$.
*Arch. Math.*2005, 84: 325–333. 10.1007/s00013-004-1203-zView ArticleMathSciNetGoogle Scholar - 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.2009.06.016View ArticleMathSciNetGoogle Scholar - Kartsatos AG, Skrypnik IV:Topological degree theories for densely defined mappings involving operators of type $({S}_{+})$.
*Adv. Differ. Equ.*1999, 4: 413–456.MathSciNetGoogle Scholar - Kartsatos AG, Skrypnik IV:The index of a critical point for densely defined operators of type ${({S}_{+})}_{L}$ in Banach spaces.
*Trans. Am. Math. Soc.*2001, 354: 1601–1630.View ArticleMathSciNetGoogle Scholar - Kartsatos AG, Skrypnik IV:A new topological degree theory foe densely defined quasibounded $({\tilde{S}}_{+})$-perturbations of multivalued maxima monotone operators in reflexive Banach spaces.
*Abstr. Appl. Anal.*2005, 2005: 121–158. 10.1155/AAA.2005.121View ArticleMathSciNetGoogle Scholar - Zhang SS, Chen YQ:Degree theory for multivalued $(S)$ type mappings and fixed point theorems.
*Appl. Math. Mech.*1990, 11: 441–454. 10.1007/BF02016374View ArticleGoogle Scholar - Chen YQ, Cho YJ, Yang L: Note on the results with lower semi-continuity.
*Bull. Korean Math. Soc.*2002, 39: 535–541.View ArticleMathSciNetGoogle Scholar - Aruffo A, Bottaro G: Generalizations of sequential lower semicontinuity.
*Boll. Uni. Mat. Ital. Serie 9*2008, 1: 293–318.MathSciNetGoogle Scholar - Aruffo AB, Bottaro G: Some variational results using generalizations of sequential lower semi-continuity.
*Fixed Point Theory Appl.*2010., 2010: Article ID 323487Google Scholar - 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.2009.05.024View ArticleMathSciNetGoogle Scholar - Castellania M, Pappalardob M, Passacantandob M: Existence results for nonconvex equilibrium problems.
*Optim. Methods Softw.*2010, 25: 49–58. 10.1080/10556780903151557View ArticleMathSciNetGoogle Scholar - 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.2007.05.004View ArticleMathSciNetGoogle Scholar - 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.2010.06.005View ArticleMathSciNetGoogle Scholar - 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/s10898-010-9565-1View ArticleMathSciNetGoogle Scholar - 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/s10957-011-9957-5View ArticleMathSciNetGoogle Scholar - 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.2005.10.005View ArticleMathSciNetGoogle Scholar - 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.2006.12.018View ArticleMathSciNetGoogle Scholar - Qiu JH: On Has version of set-valued Ekelands variational principle.
*Acta Math. Sin.*2012, 28: 717–726. 10.1007/s10114-011-0294-2View ArticleGoogle Scholar - 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-zView ArticleMathSciNetGoogle Scholar - 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-zView ArticleMathSciNetGoogle Scholar - Chen YQ, Cho YJ, Kim JK, Lee BS: Note on KKM maps and applications.
*Fixed Point Theory Appl.*2006., 2006: Article ID 53286Google Scholar - Ma TW: Topological degree for set-valued compact vector fields in locally convex spaces.
*Diss. Math.*1972, 92: 1–43.Google Scholar - 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 - 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 - Rudin W:
*Functional Analysis*. MacGraw-Hill, New York; 1973.Google Scholar - Petryshyn WV:
*Generalized Topological Degree and Semilinear Equations*. Cambridge University Press, Cambridge; 1995.View ArticleGoogle Scholar

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