# On the Solution Existence of Variational-Like Inequalities Problems for Weakly Relaxedη−α Monotone Mapping

Authors: Marwan Amin Kutbi; Wutiphol Sintunavarat
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

We introduce two new concepts of weakly relaxedη-α monotone mappings and weakly relaxed η-α semimonotone mappings. Using the KKM technique, the existence of solutions for variational-like problems with weakly relaxed η-α monotone mapping in reflexive Banach spaces is established. Also, we obtain the existence of solution for variational-like problems with weakly relaxed η-α semimonotone mappings in arbitrary Banach spaces by using the Kakutani-Fan-Glicksberg fixed-point theorem.


---

## Body

## 1. Introduction

The variational inequality theory provides us with a simple, natural, unified, and elegant framework to study a wide class of linear and nonlinear problems arising in many fields, such as mechanics, engineering sciences, elasticity, optimization, control, programming, economics, transportation, oceanography, and regional. Because of their wide applicability, various extensions and generalizations of the classical variational inequality problem have been proposed and studied in recent years. Variational-like inequalities problems is one of cornerstone in this field. Some special case of generalized variational inequalities and variational-like inequalities have been studied by several authors including Bai et al. [1], Chang et al. [2], dos Santos et al. [3], Xiao and Huang [4], Zhao and Xia [5, 6], and references therein.It is well known that the monotonicity and generalized monotonicity play an important role of the study in variational inequality theory. In recent years, a substantial number of papers on existence results for solving variational inequality problems and variational-like inequality problems based on different generalization of monotonicity such as pseudomonotonicity, quasimonotonicity, relaxed monotonicity, semimonotonicity, andp-monotonicity (see [7–15]) appeared. In [16], Fang and Huang introduced a new concept of relaxed η-α monotonicity and obtained the existence of solutions for variational-like inequalities with relaxed η-α monotone mappings in reflexive Banach spaces. Recently, Sintunavarat [17] established the existence of solution of mixed equilibrium problem with the weakly relaxed α-monotone bi-function in Banach spaces.Inspired and motivated by the work of Fang and Huang [16] and Sintunavarat [17], in this paper we introduce the two new concepts of weakly relaxed η-α monotone mappings and weakly relaxed η-α semimonotone mappings as well as two classes of variational-like inequalities with weakly relaxed η-α monotone mappings and weakly relaxed η-α semimonotone mappings. By using the KKM technique, we study some existence of solutions for variational-like inequalities with weakly relaxed η-α monotone mappings in reflexive Banach spaces. We also obtain the solvability of variational-like inequalities with weakly relaxed η-α semimonotone mappings in arbitrary Banach spaces by using the Kakutani-Fan-Glicksberg fixed-point theorem. Our results in this paper extend and improve the results of Fang and Huang [16] and many results in the literature.


## 2. Preliminaries

In this paper, unless otherwise specified,K is a nonempty closed convex subset of a real reflexive Banach space E with dual space E*. The following basic knowledge will be useful in our work.Definition 1 (see [18]).
LetT:K→E* and η:K×K→E be two mappings. A mapping T is said to be η-hemicontinuous if, for any fixed x,y∈K, the mapping f:[0,1]→ℝ defined by
(1)f(t)=〈T(ty+(1-t)x),η(y,x)〉
is continuous at 0+.Definition 2 (see [16]).
LetT:K→E* and η:K×K→E be two mappings. A mapping T is said to be η-coercive with respect to a proper function f:K→ℝ∪{∞} if there exists x0∈K such that
(2)[〈Tx-Tx0,η(x,x0)〉+f(x)-f(x0)]∥η(x,x0)∥⟶∞,
whenever ∥x∥ is large enough.Remark 3.
Iff=δK, where δK is the indicator function of K, then Definition 2 coincides with the definition of η-coercivity in the sense of Yang and Chen [15].Definition 4 (see [16]).
A mappingT:K→E* is said to be relaxed η-α monotone if there exist a function η:K×K→E and α:E→ℝ with α(tz)=tpα(z) for all t>0 and z∈E such that
(3)〈Tx-Ty,η(x,y)〉≥α(x-y),∀x,y∈K,
where p>1 is a constant.Remark 5.
(1)  Ifη(x,y)=x-y for all x,y∈K, then (3) reduces to
(4)〈Tx-Ty,x-y〉≥α(x-y),∀x,y∈K,
and T is said to be α-monotone.
(2) Ifη(x,y)=x-y for all x,y∈K and α(z)=k∥z∥p, where k>0 and p>1, then (3) becomes
(5)〈Tx-Ty,x-y〉≥k∥x-y∥p,∀x,y∈K,
and T is said to be p-monotone (see in [14, 19, 20]).
(3)  Every monotone mapping is relaxedη-α monotone with η(x,y)=x-y for all x,y∈K and α≡0.Definition 6.
LetF:K→2E be a set-valued mapping. Then, F is said to be KKM mapping if, for any finite subset {y1,y2,…,yn} of K, we have co{y1,y2,…,yn}⊂⋃i=1n‍F(yi), where co{y1,y2,…,yn} denotes the convex hull of {y1,y2,…,yn}.Remark 7.
LetF,G:K→2E. If F is KKM mapping and F(y)⊂G(y) for all y∈K, then G is also KKM mapping.Lemma 8 (see [21]).
LetM be a nonempty subset of a Hausdorff topological vector space X, and let F:M→2X be a KKM mapping. If F(y) is closed in X for all y∈M and compact for some y∈M, then ⋂y∈M‍F(y)≠∅.


## 3. Variational-Like Inequalities Problems with Weakly Relaxedη-α Monotone Mapping

In this section, we introduce the new class of mapping which generalizes several classes. Using KKM technique, we study and prove the existence of solutions for variational-like inequalities problems with mapping in such class in Banach spaces.Definition 9.
A mappingT:K→E* is said to be weakly relaxed η-α monotone if there exist a function η:K×K→E and α:E→ℝ with
(6)limt→0+α(tx)=0,(7)limt→0+ddtα(tx)=0,
for all t>0 and x∈E such that
(8)〈Tx-Ty,η(x,y)〉≥α(x-y),∀x,y∈K.Remark 10.
We obtain that the relaxedη-α monotonicity implies weakly relaxed η-α monotonicity. So, class of relaxed η-α monotone mapping is subclass of weakly relaxed η-α monotone mapping class. Also, we get that classes of relaxed α monotone mapping, p-monotone mapping, and monotone mapping are subclass of weakly relaxed η-α monotone mapping class.Theorem 11.
LetT:K→E* be an η-hemicontinuous and weakly relaxed η-α monotone and let f:K→ℝ∪{∞} be a proper convex function. Suppose that the following conditions hold:  (a)
η(x,x)=0 for all x∈K;(b)
for any fixedy,z∈K, the mapping x↦〈Tz,η(x,y)〉 is convex.
Then, the following problems (9) and (10) are equivalent:
(9)findx∈Ksuchthat〈Tx,η(y,x)〉+f(y)-f(x)≥0,55555555∀y∈K,(10)findx∈Ksuchthat〈Ty,η(y,x)〉+f(y)-f(x)≥α(y-x),5555555555∀y∈K.Proof.
Suppose that (9) has a solution. So, there exists x∈K such that
(11)〈Tx,η(y,x)〉+f(y)-f(x)≥0,∀y∈K.
Since T is weakly relaxed η-α monotone, we have
(12)〈Ty-Tx,η(y,x)〉≥α(y-x),∀y∈K,
and then
(13)〈Ty,η(y,x)〉+f(y)-f(x)≥〈Tx,η(y,x)〉+α(y-x)+f(y)-f(x)≥α(y-x),∀y∈K.
Therefore, x∈K is a solution of problem (10).
Conversely, suppose thatx∈K is a solution of problem (10) and y is any point in K with f(y)<∞. For t∈(0,1), we let yt:=(1-t)x+ty. From (10), we get that f(x)<∞. It follows from K being convex that yt∈K. From (10), we have
(14)〈Tyt,η(yt,x)〉+f(yt)-f(x)≥α(yt-x)=α(t(y-x)),
and thus
(15)f(yt)-f(x)≥α(t(y-x))-〈Tyt,η(yt,x)〉.
The convexity of f implies that
(16)f(yt)-f(x)=f((1-t)x+ty)-f(x)≤t(f(y)-f(x)).
From (15) and (16), we get
(17)t(f(y)-f(x))≥α(t(y-x))-〈Tyt,η(yt,x)〉.
By the assumption (b), we have
(18)〈Tyt,η(yt,x)〉=〈Tyt,η((1-t)x+ty,x)〉≤(1-t)〈Tyt,η(x,x)〉+t〈Tyt,η(x,y)〉=t〈Tyt,η(y,x)〉=t〈T((1-t)x+ty),η(y,x)〉.
It follows from (17) and (18) that
(19)t〈T((1-t)x+ty),η(y,x)〉+t(f(y)-f(x))≥α(t(y-x)),
that is
(20)〈T((1-t)x+ty),η(y,x)〉+f(y)-f(x)≥α(t(y-x))t,
for all y∈K. Taking t→0+ in the previous inequality and using η-hemicontinuity of T, we get
(21)〈Tx,η(y,x)〉+f(y)-f(x)≥limt→0+α(t(y-x))t.
From (6), we get limt→0+(α(t(y-x))/t) is indeterminate form. Using L’ Hôpital’s rule, we obtain that
(22)〈Tx,η(y,x)〉+f(y)-f(x)≥limt→0+(d/dt)α(t(y-x))1.
By property (7) of weakly relaxed η-α monotone of T, we have
(23)〈Tx,η(y,x)〉+f(y)-f(x)≥0,
for all y∈K with f(y)<∞. In case of f(y)=∞, the relation
(24)〈Tx,η(y,x)〉+f(y)-f(x)≥0
is trivial. Therefore, x∈K is also a solution of problem (9).Theorem 12.
LetK be a nonempty bounded closed convex subset of a real reflexive Banach space E, and let E* be the dual space of E. Suppose that T:K→E* is an η-hemicontinuous and weakly relaxed η-α monotone mapping and f:K→ℝ∪{∞} is a proper convex lower semicontinuous function. If the following conditions hold:  (a)
η(x,y)+η(y,x)=0 for all x,y∈K,(b)
for any fixedy,z∈K, the mapping x↦〈Tz,η(x,y)〉 is convex and lower semicontinuous function,(c)
α is weakly lower semicontinuous; that is, for any net {xβ}, we have that xβ converges to x in σ(E,E*) implies that α(x)≤liminfα(xβ),
then the problem (9) is solvable.Proof.
Consider the set valued mappingF:K→2E defined by
(25)F(y)={x∈K:〈Tx,η(y,x)〉+f(y)-f(x)≥0},
for all y∈K.
It is easy to see thatx∈K solves the problem (9); that is,
(26)〈Tx,η(y,x)〉+f(y)-f(x)≥0,∀y∈K,
if and only if x∈⋂y∈K‍F(y). Thus, it is sufficient to prove that ⋂y∈K‍F(y)≠∅.
Next, we show thatF is a KKM mapping. Assume the contrary, then there exists {y1,y2,…,ym}⊂K such that co{y1,y2,…,ym}⊄⋃i=1m‍F(yi). This implies that there exists y0∈co{y1,y2,…,ym} such that y0=∑i=1m‍tiyi, where ti≥0, i=1,2,…,m and ∑i=1m‍ti=1, but y0∉⋃i=1m‍F(yi).
From (25), we have
(27)〈Ty0,η(yi,y0)〉+f(yi)-f(y0)<0,fori=1,2,…,m.
By (b) and (25), we obtain that
(28)0=〈Ty0,η(y0,y0)〉=〈Ty0,η(∑i=1m‍tiyi,y0)〉≤∑i=1m‍ti〈Ty0,η(yi,y0)〉<∑i=1m‍ti[f(y0)-f(yi)]=f(y0)-∑i=1m‍tif(yi)≤f(y0)-f(y0)=0,
which is a contradiction. Therefore, F is a KKM mapping.
Now, we define another set-valued mappingG:K→2E by
(29)G(y)={x∈K:〈Tx,η(y,x)〉+f(y)-f(x)≥α(y-x)}
for all y∈K.
Next, we will claim thatF(y)⊂G(y) for all y∈K. For each y∈K, let x∈F(y); then,
(30)〈Tx,η(y,x)〉+f(y)-f(x)≥0.
From the weakly relaxed η-α monotonicity of T, we get
(31)〈Ty,η(y,x)〉+f(y)-f(x)≥α(y-x)+[〈Tx,η(y,x)〉+f(y)-f(x)]≥α(y-x).
This implies that x∈G(y) and hence F(y)⊂G(y) for all y∈K. So, G is also a KKM mapping.
By assumption,x↦〈Ty,η(x,y)〉 and f are two convex lower-semicontinuous functions. Then it is easy to see that they are both weakly lower semicontinuous. From the definition of G and the weakly lower semicontinuity of α, we get that G(y) is weakly closed for all y∈K. Since K is closed bounded and convex, it also is weakly compact, and then G(y) is weakly compact in K for each y∈K. From Lemma 8 and Theorem 11, we obtain that
(32)⋂y∈K‍F(y)=⋂y∈K‍G(y)≠∅.
So, there exists x∈K, such that
(33)〈Tx,η(y,x)〉+f(y)-f(x)≥0,∀y∈K;
that is, problem (9) has a solution. This completes the proof.We know that the relaxedη-α monotonicity implies the weakly relaxed η-α monotonicity. Therefore, Theorem 12 can be deduced to the following corollary.Corollary 13.
LetK be a nonempty bounded closed convex subset of a real reflexive Banach space E, and let E* be the dual space of E. Suppose that T:K→E* is an η-hemicontinuous and relaxed η-α monotone mapping and f:K→ℝ∪{∞} is a proper convex lower semicontinuous function. If the following conditions hold:  (a)
η(x,y)+η(y,x)=0 for all x,y∈K,(b)
for any fixedy,z∈K, the mapping x↦〈Tz,η(x,y)〉 is convex and lower semicontinuous function,(c)
α is weakly lower semicontinuous,
then the problem (9) is solvable.Remark 14.
Since the monotonicity,p-monotonicity, and relaxed α-monotonicity imply relaxed η-α monotonicity, we can be applying Corollary 13 to the other problems for the mapping satisfies these property.Next, we study and prove that result for the case ofK is unbounded set.Theorem 15.
LetK be a nonempty unbounded closed convex subset of a real reflexive Banach space E, and let E* be the dual space of E. Suppose that T:K→E* is an η-hemicontinuous and weakly relaxed η-α monotone mapping and f:K→ℝ∪{∞} is a proper convex lower semicontinuous function. If the following conditions hold:  (a)
η(x,y)+η(y,x)=0 for all x,y∈K,(b)
for any fixedy,z∈K, the mapping x↦〈Tz,η(x,y)〉 is convex and lower semicontinuous function,(c)
α is weakly lower semicontinuous,(d)
T is weakly η-coercive with respect to f; that is, there exists x0∈K such that
(34)〈Tx,η(x0,x)〉+f(x0)-f(x)<0,wheneverx∈K and ∥x∥ large enough,
then the problem (9) is solvable.Proof.
Forϵ>0, define Kϵ:={y∈K:∥y∥≤ϵ}. Consider the following problem:(35)findx∈Kϵsuchthat〈Txϵ,η(y,xϵ)〉+f(y)-f(xϵ)≥0,5555555∀y∈Kϵ.
SinceKϵ is bounded, by Theorem 12, we get that the problem (35) has at least one solution xϵ∈Kϵ.
Forx0 in the weakly η-coercivity condition (d), we fixed ϵ>∥x0∥. From (35), we can find that xϵ∈Kϵ such that
(36)〈Txϵ,η(x0,xϵ)〉+f(x0)-f(xϵ)≥0.
Since xϵ∈Kϵ, we have ∥xϵ∥≤ϵ. If ∥xϵ∥=ϵ for all ϵ, we may choose ϵ large enough so that the weakly η-coercivity condition (d) implies that
(37)〈Txϵ,η(x0,xϵ)〉+f(x0)-f(xϵ)<0,
which contradicts (36). Therefore, there exists ϵ such that ∥xϵ∥<ϵ.
For eachy∈K, we can choose 0<t<1 such that ty+(1-t)xϵ∈Kϵ. From (35) and the fact that ty+(1-t)xϵ∈Kϵ, we have
(38)〈Txϵ,η(ty+(1-t)xϵ,xϵ)〉+f(ty+(1-t)xϵ)-f(xϵ)≥0.
By the above inequality and the convexity of f and mapping in (b), we get
(39)t〈Txϵ,η(y,xϵ)〉+tf(y)-tf(xϵ)=t〈Txϵ,η(y,xϵ)〉+(1-t)〈Txϵ,η(xϵ,xϵ)〉+tf(y)+(1-t)f(xϵ)-f(xϵ)≥〈Txϵ,η(ty+(1-t)xϵ,xϵ)〉+f(ty+(1-t)xϵ)-f(xϵ)≥0,
for all y∈K. This implies that
(40)〈Txϵ,η(y,xϵ)〉+f(y)-f(xϵ)≥0,
for all y∈K. Therefore, xϵ∈K is a solution of the problem (9). This completes the proof.It is easy to see that the relaxedη-α monotonicity implies the weakly relaxed η-α monotonicity. So, Theorem 15 can be deduced to the following corollary.Corollary 16.
LetK be a nonempty unbounded closed convex subset of a real reflexive Banach space E, and let E* be the dual space of E. Suppose that T:K→E* is an η-hemicontinuous and relaxed η-α monotone mapping and f:K→ℝ∪{∞} is a proper convex lower semicontinuous function. If the following conditions hold:  (a)
η(x,y)+η(y,x)=0 for all x,y∈K,(b)
for any fixedy,z∈K, the mapping x↦〈Tz,η(x,y)〉 is convex and lower semicontinuous function,(c)
α is weakly lower semicontinuous,(d)
T is weakly η-coercive with respect to f,
then the problem (9) is solvable.Since theη-coercivity with respect to f implies that the weakly η-coercivity with respect to f, we can utilize Corollary 16 to the result of Fang and Huang [16].Corollary 17 (Theorem 2.3 [16]).
LetK be a nonempty unbounded closed convex subset of a real reflexive Banach space E, and let E* be the dual space of E. Suppose that T:K→E* is an η-hemicontinuous and relaxed η-α monotone mapping and f:K→ℝ∪{∞} is a proper convex lower semicontinuous function. If the following conditions hold:  (a)
η(x,y)+η(y,x)=0 for all x,y∈K,(b)
for any fixedy,z∈K, the mapping x↦〈Tz,η(x,y)〉 is convex and lower semicontinuous function,(c)
α is weakly lower semicontinuous,(d)
T is η-coercive with respect to f,
then the problem (9) is solvable.Remark 18.
Theorems11, 12, and 15 are improvement of the results of Fang and Huang [16] from the corresponding results of variational-like inequality problems for relaxed η-α monotone mapping to weakly relaxed η-α monotone mapping. Also, these results are extension of the known results of Hartman and Stampacchia [22] and corresponding results in [14, 19, 23].


## 4. Variational-Like Inequalities Problems with Weakly Relaxedη-α Semimonotone Mapping

Through this section, letE be an arbitrary Banach space with its dual space E*, let E** denote the dual space of E*, and let K be a nonempty closed convex subset of E**.Definition 19.
A mappingA:K×K→E** is said to be weakly relaxed η-α semimonotone if the following conditions hold:  (a)
for each fixedu∈K, A(u,·) is weakly relaxed η-α monotone; that is, there exist mappings η:K×K→E* and α:E**→ℝ such that(41)limt→0+α(tx)=0,limt→0+ddtα(tx)=0,for allt>0 and x∈E** and
(42)〈A(u,v)-A(u,w),η(v,w)〉≥α(v-w)5555155∀v,w∈K;(b)
for each fixedv∈K, A(·,v) is completely continuous; that is, for any net {uβ} with uβ converges to u0 in σ(E**,E*), we have that {A(uβ,v)} converges to A(u0,v) in the norm topology of E*.Remark 20.
We obtain that relaxedη-α semimonotonicity due to Fang and Huang [16] implies weakly relaxed η-α semimonotonicity. Therefore, the class of relaxed η-α semimonotone mappings is subclass of the class of weakly relaxed η-α semimonotone mappings.LetA:K×K→E** and η:K×K→E* be two mappings and f:K→ℝ∪{∞} is a proper convex lower-semicontinuous function. We consider the following problem:
(43)findu∈Ksuchthat〈A(u,u),η(v,u)〉+f(v)-f(u)≥0,55555555555555∀v∈K.Theorem 21.
LetE be a real Banach space and let K⊂E** be a nonempty bounded closed convex set. Suppose that A:K×K→E** is a weakly relaxed η-α semimonotone mapping and let f:K→ℝ∪{∞} be a proper convex lower semicontinuous function. If the following conditions hold:  (a)
η(x,y)+η(y,x)=0 for all x,y∈K,(b)
for any fixedy,z,w∈K, the mapping x↦〈A(z,w),η(x,y)〉 is convex and lower semicontinuous function,(c)
for eachx∈K, A(x,·):K→E** is finite-dimensional continuous; that is, for any finite-dimensional subspace F⊂E**, A(x,·):K∩F→E** is continuous,(d)
α is convex lower semicontinuous,
then the problem (43) is solvable.Proof.
LetF⊂E** be a finite-dimensional subspace with KF:=F∩K≠∅. For each w∈K, consider the following problem:(44)findu0∈KFsuchthat〈A(w,u0),η(v,u0)〉+f(v)-f(u0)≥0,55555∀v∈KF.
It follows from KF being bounded closed and convex and A(w,·) being continuous on KF and weakly relaxed η-α monotone for each fixed w∈K that from Theorem 12, we obtain that problem (44) has a solution u0∈KF.
Next, define a set-valued mappingT:KF→2KF as follows:
(45)Tw={u∈KF:〈A(w,u),η(v,u)〉+f(v)-f(u)≥0,∀v∈KF},∀w∈KF.
By Theorem 11, we get that, for each w∈KF,
(46)Tw={u∈KF:〈A(w,u),η(v,u)〉+f(v)-f(u)≥α(v-u),∀v∈KF}.
It is known that every convex lower-semicontinuous function in Banach spaces is weakly lower semicontinuous. Therefore, condition (b) and the proper convex lower semicontinuity off and α implies that T has nonempty bounded closed and convex values. By the complete continuity of A(·,u), we have that T is upper semicontinuous. Using the Kakutani-Fan-Glicksberg fixed-point theorem, we obtain that T has a fixed-point w0∈KF and thus
(47)〈A(w0,w0),η(v,w0)〉+f(v)-f(w0)≥0,∀v∈KF.
Now, define(48)ℱ:={F⊂E**:FisfinitedimensionalwithF∩K≠∅},WF:={u∈K:〈A(u,v),η(v,u)〉+f(v)-f(u)≥α(v-u),∀v∈KF},∀F∈ℱ.
From (47), using Theorem 11, we obtain that WF is nonempty and bounded. Here, we denote WF¯ by the σ(E**,E*)-closure of WF in E** and thus WF¯ is σ(E**,E*)-compact in E**.
It is known that, for anyFi∈ℱ, i∈ℕ, we have W∩iFi⊂∩WFi. Therefore, {WF¯:F∈ℱ} has the finite intersection property; that is,
(49)⋂F∈ℱ‍WF¯≠∅.
Next, we show that, foru∈⋂F∈ℱ‍WF¯,
(50)〈A(u,u),η(v,u)〉+f(v)-f(u)≥0,∀v∈K.
Indeed, for v∈K, let F∈ℱ be such that v∈KF and u∈KF. Then, there exists a net {uβ} in WF such that uβ converges to u in σ(E**,E*). From the definition of WF, we have
(51)〈A(uβ,v),η(v,uβ)〉+f(v)-f(uβ)≥α(v-uβ).
By the complete continuity of A(·,v) and the proper convex lower semicontinuity of f and α, we get
(52)〈A(u,v),η(v,u)〉+f(v)-f(u)≥α(v-u),∀v∈K.
Again, using Theorem 11, we conclude that
(53)〈A(u,v),η(v,u)〉+f(v)-f(u)≥0,∀v∈K.
This implies that u is a solution of the problem (43). This completes the proof.Theorem 22.
LetE be a real Banach space and let K⊂E** be a nonempty unbounded closed convex set. Suppose that A:K×K→E** is a weakly relaxed η-α semimonotone mapping and let f:K→ℝ∪{∞} be a proper convex lower semicontinuous function. If the following conditions hold:  (a)
η(x,y)+η(y,x)=0 for all x,y∈K,(b)
for any fixedy,z,w∈K, the mapping x↦〈A(z,w),η(x,y)〉 is convex and lower semicontinuous function,(c)
for eachx∈K, A(x,·):K→E* is finite-dimensional continuous,(d)
α is convex lower semicontinuous,(e)
there exists a pointu0∈K such that liminf∥u∥→∞[〈A(u,u),η(u,u0)〉+f(u)-f(u0)]>0,then the problem (43) is solvable.Proof.
Forϵ>0, we denote by Bϵ the closed ball with radius ϵ and center at 0 in E**. By Theorem 21, the problem
(54)〈A(u,u),η(v,u)〉+f(v)-f(u)≥0,∀v∈Bϵ
has a solution uϵ∈Bϵ.
Letϵ be large enough such that u0∈Bϵ. Therefore,
(55)〈A(uϵ,uϵ),η(u0,uϵ)〉+f(u0)-f(uϵ)≥0.
By condition (e), we get that {uϵ} is bounded. So, we may suppose that uϵ converges to u in  σ(E**,E*) as ϵ→∞. It follows from Theorem 11 that
(56)〈A(uϵ,v),η(v,uϵ)〉+f(v)-f(uϵ)≥α(v-uϵ),55555,5555∀v∈K.
Letting ϵ→∞, we have
(57)〈A(u,v),η(v,u)〉+f(v)-f(u)≥α(v-u),∀v∈K.
Again from Theorem 11, we get
(58)〈A(u,u),η(v,u)〉+f(v)-f(u)≥0,∀v∈K.
This show that u∈K is a solution of the problem (43). This completes the proof.Remark 23.
Theorems21 and 22 extend and improve Theorems 3.1 and 3.2 of Fang and Huang [16] and Theorems 2.1 to 2.6 of Chen [7].



---

*Source: 207845-2013-11-18.xml*