# Monotone type operators in nonreflexive Banach spaces

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## 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.

### 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, ${E}^{\ast}$ 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, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 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. □

## 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).

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