doi:10.5899/2012/jnaa-00135 Research Article
The Existence and the stability of solutions for
equilibrium problems with lower and upper
bounds
Congjun Zhang1
, Jinlu Li2
and Zhenmin Feng 3∗
(1)College of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210046, P. R. China.
(2)Department of Mathematics,Shawnee State University 940 Second Street portsmouth,Ohio, 45662, USA.
(3)College of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210046, P. R. China.
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Copyright 2012 c⃝Congjun Zhang, Jinlu Li and Zhenmin Feng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and re-production in any medium, provided the original work is properly cited.
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Abstract
In this paper, we study a class of equilibrium problems with lower and upper bounds. We obtain some existence results of solutions for equilibrium problems with lower and upper bounds by employing some classical fixed-point theorems. We investigate the stability of the solution sets for the problems, and establish sufficient conditions for the upper semicon-tinuity, lower semicontinuity and continuity of the solution set mappingS: Λ1×Λ2 →2X
in a Hausdorff topological vector space, in the case where a set K and a mapping f are perturbed respectively by parametersλand µ.
Keywords: Equilibrium problem, Upper semicontinuity, Lower semicontinuity, Solution set map-ping.
1
Introduction
Let X be a Hausdorff topological vector space, K a nonempty subset of X, and let f :
K×K →Rbe a function. The equilibrium problem is to find x∈K, such that
f(x, y)≥0, ∀y ∈K (1.1)
Equilibrium problems include variational inequality problems as well as fixed point problems, complementarity problems, optimization, saddle point problems and Nash equi-librium problems as special cases. Equiequi-librium problems provide us with a systematic framework to study a wide class of problems arising in finance economics, optimization and operation research et al., which motivate the extensive concern. In recent years, equi-librium problems have been deeply and thoroughly researched. See, for example, [1]-[9].
In 1999, Isac, Sehgal and Singh [10] raised the following open problem, which is the equilibrium problem with lower and upper bounds.
LetX be a locally convex topological vector space,K a nonempty closed subset ofX,
f :K×K →R a functional, and letα, β be real numbers withα≤β. The problem is to
find x∈K, such that
α≤f(x, y)≤β, ∀y∈K (1.2)
As to the equilibrium problem with lower and upper bounds, the existence, in particu-lar, the stability of solution sets, are of considerate interest at present. For the existence of solutions of such problems, Li [11] gave some answers to the open problem (1.2) by intro-ducing and using the concept of extremal subsets. Chadli et al. [12] derived some results by employing a fixed point theorem and the FKKM theorem. Zhang [13] obtained some existence theorems by using the concept of (α, β)−convexity. Ding [14] also drew some conclusions in this aspect. As to the stability of solution sets for equilibrium problems, Bianchi and Pini [15] considered equilibrium problems in vector metric spaces, where the functional f and the setK are perturbed by the parameters εand η respectively. Li et al. [16] studied the stability of solutions of generalized vector quasi-variational inequality problems. Recently, Anh and Khanh [17] as well as Huang et al. [18] established suffi-cient conditions for the solution set of parametric multi-valued vector quasi-equilibrium problems with fixed cone to be semicontinuous.
However, the stability of solution sets for equilibrium problems with lower and upper bounds have been rarely studied up to now.
In this paper, we obtain some existence results of equilibrium problems with lower and upper bounds by employing some classical fixed point theorems. We investigate the stability of the solution sets for the problems in a Hausdorff topological vector space, in the case where a set K and a mapping f are perturbed respectively by parametersλand
µ. Finally, we study the stability of the solution sets in a vector metric space, in the particular case where K is fixed, andf is perturbed by a parameterε.
2
Preliminaries
Let X, Y denote topological vector spaces. For a nonempty subset K of X, let co(K) denote the convex hull of K, and ⟨K⟩ denote the family of all nonempty finite subsets of
K. 2X denotes the family of all nonempty subsets of X, and R denotes the set of real
numbers.
Definition 2.1. Let F :X → 2Y be a set-valued mapping, the subset {(x, y) ∈ X×Y :
y∈F(x)} of X×Y is called the graph ofF, denoted by graph(F).
Definition 2.2. Let F : X → 2Y be a set-valued mapping, F−1
: Y → X is defined as follows: x∈F−1
(y) if and only if y ∈F(x), F−1
Definition 2.3. Let F :X→2Y be a set-valued mapping,F is called closed, if the graph
of F is a closed subset of X×Y.
Definition 2.4. Let F :X→2Y be a set-valued mapping,F is called upper
semicontinu-ous at x0, if for every open setW containing F(x0), there exists a neighborhoodU of x0, such that for every x∈U, we haveF(x)⊂W. F is called upper semicontinuous in X, if
F is upper semicontinuous at every point of X.
Definition 2.5. Let F :X →2Y be a set-valued mapping,F is called lower
semicontin-uous at x0, if for every open set V with V ∩F(x0)̸= Ø, there exists a neighborhood U of
x0, such that for every x∈ U, we have F(x)∩V ̸= Ø, F is called lower semicontinuous in X, if F is lower semicontinuous at every point of X.
Definition 2.6. [13] Let K be a nonempty convex subset of X, f : K × K → R a
functional, and let α, β ∈ R with α ≤ β, f(x, y) is called (α, β)−convex related to y, if
for every finite set {y1, . . . , yn} ⊂ K, and for every y0 ∈ co{y1, . . . , yn}, there exists an
element i∈ {1,2, . . . , n}, such that α≤f(y0, yi)≤β.
Definition 2.7. [16] Let f :K×K →R be a functional,f is called pseudomonotone, if f(x, y)≥0 implies f(y, x)≤0.
Definition 2.8. Let f :X→Rbe a functional,f is called quasiconcave(qcv), if for every x, y∈X, 0≤t≤1, we have f((1−t)x+ty)≥min{f(x), f(y)}. f is called quasiconvex
(qcv), if −f is quasiconcave.
The following lemma and the concept of upper semicontinuity, see [19].
Lemma 2.1. Let F :X →2Y be a set-valued mapping, if Y is a compact space, and F
is closed, then F is upper semicontinuous in X.
Lemma 2.2. Let F :X→2Y be a set-valued mapping, andF is compact-valued, then F
is upper semicontinuous at x0 ∈X if and only if for every {xn} ⊂X, xn →x0, and for every yn∈F(xn), there exists y0∈F(x0), {yni} ⊂ {yn}, such thatyni →y0.
Lemma 2.3. Let F :X →2Y be a set-valued mapping, then F is lower semicontinuous
at x0 ∈ X if and only if for every {xn} ⊂ X, xn → x0, and for every y0 ∈F(x0), there exists yn∈F(xn), such thatyn→y0.
Lemma 2.4. Letf :X→Rbe a functional, f is lower semicontinuous if and only if for
every c∈R,{x∈X:f(x)≤c} is a closed subset ofX. f is upper semicontinuous if and
only if−f is lower semicontinuous.
Lemma 2.5. Let f : K×K → R be a functional, if for every y ∈ Y, f(·, y) is upper
semicontinuous, then u(x) = inf
y∈Y f(x, y) is upper semicontinuous in X. If for every y ∈
Y, f(·, y) is lower semicontinuous, then u(x) = sup
y∈Y
f(x, y) is lower semicontinuous in X.
Lemma 2.6. LetF, G:X→2Y be set-valued mappings. If the following conditions hold:
(i) For everyx∈X,F(x)∩G(x)̸= Ø;
(ii) F is upper semicontinuous at x0;
(iii) F(x0) is compact;
(iv) The graph of G is closed,
Lemma 2.7. Let X be a Hausdorff topological vector space, K a nonempty subset of X,
D a nonempty compact convex subset of K, and let F : K → 2D be a nonempty
set-valued mapping, if for every y ∈ K, F−1
(y) is open, then there exists xˆ ∈K, such that
ˆ
x∈co(F(ˆx)).
Lemma 2.8. Let X be a Hausdorff topological vector space, K a nonempty compact convex subset of X, and let F : K → 2K be a set-valued mapping. If the following
conditions hold:
(i) For everyx∈K, F(x) is convex;
(ii) For every y∈K,F−1
(y) contains an open subset Oy of K (Oy may be Ø);
(iii) K=∪y∈KOy,
Then there exists xˆ∈K, such thatxˆ∈F(ˆx).
Lemma 2.9. [22]Let X be a Hausdorff topological vector space, C, K nonempty convex subsets of X, and let F : C → 2X be a set-valued mapping. If the following conditions
hold:
(i) C⊂K ⊂F(C);
(ii) F(C) is a compact subset of X;
(iii) For everyx∈C, F(x) is open,
Then there exists xˆ∈C, such that xˆ∈co(F−1
(ˆx)).
Lemma 2.10. [14] LetX be a Hausdorff topological vector space,K a nonempty compact convex subset of X, α, β ∈ R, α ≤ β, and let f : K ×K → R be a functional. If the following conditions hold:
(i) For everyy ∈K, f(x, y) is continuous related to x;
(ii) For every x∈K, f(x, y) is(α, β)−convex related to y, Then there exists x∈K, such thatα≤f(x, y)≤β for every y∈K.
3
The existence of solutions
In this section, the solution existence of Problem (1.2) will be studied by employing some fixed point theorems.
Throughout this section, all topological spaces are assumed to be Hausdorff.
Theorem 3.1. Let K be a nonempty subset of X, α, β ∈ R, α ≤ β, and let f, g1, g2 :
K×K →Rbe functionals. Assume that the following conditions hold:
(i) For everyx∈K, g1(x, x)≥α and g2(x, x)≤β;
(ii) For eachx∈K, co({y∈K :f(x, y)< α or f(x, y)> β})
⊂ {y∈K :g1(x, y)< α or g2(x, y)> β};
(iii) There exists a compact convex subsetD of K, such that for every x∈K,
{y∈K:f(x, y)< αor f(x, y)> β} ⊂D;
(iv) For every y∈K, {x∈K :α≤f(x, y)≤β} is a closed subset of K, Then there exists xˆ∈K, such thatα≤f(ˆx, y)≤β for every y∈K.
Proposition 3.1. Let the set-valued mappings F, G:K→2K be defined by
F(x) =K\ {y∈K:α≤f(x, y)≤β}, ∀x∈K
G(x) ={y∈K :g1(x, y)< α} ∪ {y∈K :g2(x, y)> β}, ∀x∈K
By Condition (iii), there exists a compact convex setD⊂K, such thatF(x)⊂Dfor each
x ∈ K. Condition (iv) indicates that F−1
(y) ={x ∈K :f(x, y) < α or f(x, y) > β} is open in K for every x ∈ K. Then from Lemma 2.10, it follows that there exists xˆ∈ K, such that xˆ ∈ co(F(ˆx)). Since co(F(x)) ⊂ G(x) for every x ∈ K, so xˆ ∈ G(ˆx) which implies that g1(ˆx,xˆ) < α or g2(ˆx,xˆ) > β. This contradicts Condition (i). Thus, there exists xˆ∈K, such that α≤f(ˆx, y)≤β, for every y∈K.
Corollary 3.1. Let K be a nonempty compact convex subset ofX, α, β∈R, α≤β, and
let f :K×K →R be a functional. Assume that the following conditions hold:
(i) For everyx∈K, α≤f(x, x)≤β;
(ii) For eachx∈K, {y∈K:f(x, y)< αor f(x, y)> β} is convex;
(iii) For everyy ∈K, f(x, y) is continuous related to x,
Then there exists xˆ∈K, such thatα≤f(ˆx, y)≤β for every y∈K.
Proposition 3.2. Letg1 =g2 =f, then Condition(i)and(ii)are equivalent to Condition
(i) and (ii) of Theorem 3.1. Clearly, Condition (iii) of Theorem 3.1 is satisfied, since K
is a nonempty compact convex subset of X. Since for every y ∈K, f(x, y) is continuous related to x, Condition (iv) of Theorem 3.1 is also satisfied. Hence, by Theorem 3.1, the proof is completed.
Theorem 3.2. Let K be a nonempty compact convex subset of X, α, β∈R, α≤β, and
let f, g1, g2 :K×K →R be functionals. Assume that the following conditions hold:
(i) For eachx∈K, g1(x, x)≥α and g2(x, x)≤β;
(ii) For every x∈K, {y ∈K :f(x, y)< α or f(x, y)> β} ⊂ {y ∈K :g1(x, y)< α or g2(x, y)> β};
(iii) For everyx∈K, {y∈K:f(x, y)< αor f(x, y)> β} is convex;
(iv) For each y∈K, there exists an open subset Oy of K, such that
Oy ⊂ {x∈K :f(x, y)< α or f(x, y)> β};
(v) K=∪y∈KOy,
Then there exists xˆ∈K, such thatα≤f(ˆx, y)≤β for every y∈K.
Proposition 3.3. Let the set-valued mappings F, G:K→2K be defined by
F(x) =K\ {y∈K:α≤f(x, y)≤β}, ∀x∈K
G(x) ={y∈K :g1(x, y)< α} ∪ {y∈K :g2(x, y)> β}, ∀x∈K
Suppose to the contrary that the result of the theorem is not true. Equivalently, F(x) is nonempty for everyx∈K. Condition (ii)implies that F(x)⊂G(x) for every x∈K. By Condition (iii), F(x) is convex for each x ∈ K. From Condition (iv), for every y ∈ K, there exists an open set Oy ⊂ K, such that Oy ⊂F−1(y). Since K = ∪y∈KOy, Lemma
2.8 indicates that there exists xˆ ∈ K, such that xˆ ∈ F(ˆx), and since F(x) ⊂ G(x) for all x ∈K, so xˆ ∈G(ˆx) which shows that g1(ˆx,xˆ) < α or g2(ˆx,xˆ) > β. This contradicts Condition (i). Thus, there existsxˆ∈K, such thatα≤f(ˆx, y)≤β for every y∈K.
Remark 3.1. In Theorem 3.2, if g1 = g2 = f, then Condition (i) implies that α ≤
f(x, x)≤β for all x∈K and Condition (ii) can be omitted.
Theorem 3.3. Let K be a nonempty convex subset of X, α, β ∈ R, α ≤ β, and let f, g1, g2 :K×K →R be functionals. Assume that the following conditions hold:
(i) For everyx∈K, g1(x, x)≥α and g2(x, x)≤β;
⊂ {x∈K :g1(x, y)< α or g2(x, y)> β};
(iii) For everyy ∈K, {x∈K:f(x, y)< αor f(x, y)> β} is nonempty;
(iv) ∪x∈K{y∈K:f(x, y)< αor f(x, y)> β} is compact in X;
(v) For each x∈K, {y ∈K :f(x, y)< α or f(x, y)> β} is open in X, Then there exists xˆ∈K, such thatα≤f(ˆx, y)≤β for every y∈K.
Proposition 3.4. Let the set-valued mappings F, G:K→2K be defined by
F(x) =K\ {y∈K:α≤f(x, y)≤β}, ∀x∈K
G(x) ={y∈K :g1(x, y)< α} ∪ {y∈K :g2(x, y)> β}, ∀x∈K
Suppose to the contrary that the result of the theorem is not true. Equivalently, F(x) is nonempty for every x ∈ K. By Condition (ii), co(F−1
(y)) ⊂ G−1
(y) for every y ∈ K. Condition (iii) indicates that F−1
(y) is nonempty for each y ∈K. From Condition (iv),
F(K) is compact in X. Condition (v) implies that F(x) is an open subset of X for all
x ∈ K. Then by Lemma 2.9, there exists xˆ ∈ K, such that xˆ ∈ co(F−1
(ˆx)), and since
co(F−1
(y)) ⊂ G−1
(y) for all y ∈ K, so xˆ ∈ G−1
(ˆx) which shows that g1(ˆx,xˆ) < α or
g2(ˆx,xˆ) > β. This contradicts Condition (i). Therefore, there exists xˆ ∈ K, such that
α≤f(ˆx, y)≤β for everyy∈K.
Remark 3.2. In Theorem 3.3, if g1 = g2 = f, then Condition (i) implies that α ≤
f(x, x) ≤ β for all x ∈ K, and Condition (ii) indicates that {x ∈ K : f(x, y) < α or
f(x, y)> β} is convex.
4
The Stability of solution sets
LetX,Λ1,Λ2 be topological vector spaces,K: Λ1→2X a nonempty set-valued mapping,
α, β∈R,α≤β, and let f :X×X×Λ2 →Rbe a functional. Given (λ, µ)∈Λ1×Λ2, we
consider:
Find x∈K(λ), such that α≤f(x, y, µ)≤β, ∀y∈K(λ).
For every given (λ, µ)∈Λ1×Λ2, we denote byS(λ, µ) the solution set of this problem.
In this section, We will discuss the stability of the solution sets for Problem (1.2) in a Hausdorff topological vector space, in the case where a set K and a mapping f are perturbed respectively by parametersλandµ, that is the semicontinuity and the continuity of the solution mapping S : Λ1×Λ2 → 2X. Then, we study the upper semicontinuity of
the solution set mapping in a vector metric space, in the particular case whereK is fixed, and f is perturbed by a parameter ε.
In the following, all topological spaces are assumed to be Hausdorff.
First, we discuss the upper semicontinuity of the solution set mapping Problem (1.2).
Theorem 4.1. Let K : Λ1 →2X be a nonempty set-valued mapping,α, β ∈R,α≤β and let f :X×X×Λ2 →R be a functional. If the following assumptions are satisfied:
(i)K(·)is continuous inΛ1, andK(x)is nonempty compactly convex for everyx∈Λ1;
(ii) f(·,·,·) is continuous in X×X×Λ2;
(iii) For each (λ, µ) ∈ Λ1 ×Λ2, and for each x ∈ K(λ), f(x, y, µ) is (α, β)−convex related to y,
Then (a) For every (λ, µ)∈Λ1×Λ2, S(λ, µ)̸= Ø;
Proposition 4.1. (a)For every(λ, µ)∈Λ1×Λ2,K(λ)andf(·,·, µ)satisfy the conditions of Lemma 2.10, then by Lemma 2.10, it follows that S(λ, µ)̸= Ø.
(b) For every (λ, µ)∈Λ1×Λ2, S(λ, µ) ={x∈K(λ) :α≤f(x, y, µ)≤β,∀y∈K(λ)}. By Condition(ii),S(λ, µ)is a closed subset ofK(λ), and sinceK(λ)is compact, soS(λ, µ)
is compact.
Next, we prove S is upper semicontinuous. Let {(λn, µn)} ⊂ Λ1×Λ2, (λn, µn) −→
(λ, µ), xn∈S(λn, µn). By Lemma 2.2, we only need to prove that there existsx∈S(λ, µ),
and{xni} ⊂ {xn}, such thatxni −→x. Sincexn∈K(λn)andK is upper semicontinuous,
then Lemma 2.2 implies that there existsx∈K(λ), and{xni} ⊂ {xn}, such thatxni −→x.
Next, we prove x∈S(λ, µ). Suppose x /∈S(λ, µ), then there exists y∈K(λ), such that
f(x, y, µ)< α or f(x, y, µ)> β (4.3)
By the lower semicontinuity of K and Lemma 2.3, for the above y, we know that there exists {yni}, such that yni ∈ K(λni), and yni −→ y. Since xni ∈ S(λni, µni), then α ≤ f(xni, yni, µni) ≤ β, and from the continuity of f, we have α ≤ f(x, y, µ) ≤ β. This
contradicts (4.3).The proof is completed.
When f is fixed, K is perturbed by a parameter ε, we have:
Corollary 4.1. Let Λ be a Hausdorff topological vector space, K : Λ →2X a nonempty
set-valued mapping, α, β ∈ R, α ≤ β, and let f : X ×X → R be a functional. If the
following assumptions are satisfied:
(i) K(·) is continuous in Λ, and K(x) is nonempty compactly convex for everyx∈Λ;
(ii) f(·,·) is continuous in X×X;
(iii) For everyε∈Λ, and for every x∈K(ε), f(x, y) is(α, β)−convex related to y, Then (a) For every ε∈Λ, S(ε)̸= Ø;
(b) The solution set mapping S: Λ→2X is upper semicontinuous inΛ.
When K is fixed, andf is perturbed by a parameter ε, we have:
Corollary 4.2. Let Λ be a Hausdorff topological vector space, K a nonempty compact convex subset of X, α, β ∈R, α ≤β, and let f :X×X×Λ→ R be a functional. If the
following assumptions are satisfied:
(i) f(·,·,·) is continuous in X×X×Λ;
(ii) For every ε∈Λ, and for every x∈K, f(x, y, ε) is(α, β)−convex related to y; Then (a) For every ε∈Λ, S(ε)̸= Ø;
(b) The solution set mapping S: Λ→2X is continuous in Λ.
Theorem 4.2. Let K : Λ1 →2X be a nonempty set-valued mapping,α, β ∈R,α≤β and let f :X×X×Λ2 →R be a functional. If the following assumptions are satisfied:
(i)K(·)is continuous inΛ1, andK(x)is nonempty compactly convex for everyx∈Λ1;
(ii) For each(λ, µ)∈Λ1×Λ2, and for eachx∈K(λ), α≤f(x, x, µ)≤β;
(iii) For every(λ, µ)∈Λ1×Λ2, and for every x∈K(λ),
{y∈K(Λ) :f(x, y, µ)< α or f(x, y, µ)> β} is convex;
(iv) f(·,·,·) is continuous in X×X×Λ2, Then (a) For every (λ, µ)∈Λ1×Λ2, S(λ, µ)̸= Ø;
Proposition 4.2. For every (λ, µ) ∈ Λ1 ×Λ2, K(λ) and f(·,·, µ) satisfy the conditions of Corollary 3.1, then by Corollary 3.1, we have S(λ, µ)̸= Ø.
(b) The proof is similar to the proof of Theorem 4.1.
Similar to Theorem 4.1, we have the following two corollaries:
Corollary 4.3. Let Λ be a Hausdorff topological vector space, K : Λ →2X a nonempty
set-valued mapping, α, β ∈ R, α ≤ β, and let f : X ×X → R be a functional. If the
following assumptions are satisfied:
(i) K(·) is continuous in Λ, and K(x) is nonempty compactly convex for everyx∈Λ;
(ii) For every ε∈Λ, and for every x∈K(ε), α≤f(x, x)≤β;
(iii) For eachε∈Λ, and for each x∈K(ε),
{y∈K(ε) :f(x, y)< α or f(x, y)> β} is convex;
(iv) f(·,·) is continuous in X×X, Then (a) For every ε∈Λ, S(ε)̸= Ø;
(b) The solution set mapping S: Λ→2X is upper semicontinuous inΛ.
Corollary 4.4. Let Λ be a Hausdorff topological vector space, K a nonempty compact convex subset of X, α, β ∈R, α ≤β, and let f :X×X×Λ→ R be a functional. If the following assumptions are satisfied:
(i) For everyε∈Λ, and for every x∈K, α≤f(x, x, ε)≤β;
(ii) For eachε∈Λ, and for each x∈K,
{y∈K:f(x, y, ε)< αor f(x, y, ε)> β} is convex;
(iii) f(·,·,·) is continuous in X×X×Λ, Then (a) For every ε∈Λ, S(ε)̸= Ø;
(b) The solution set mapping S: Λ→2X is upper semicontinuous inΛ.
Next, the lower semicontinuity and continuity of the solution set mapping S(·,·) at (λ0, µ0)∈Λ1×Λ2 will be studied. Suppose that S(·,·) is nonempty-valued in a
neighbor-hood of (λ0, µ0), that isS(λ, µ)̸= Ø for every λ∈U(λ0) and for every µ∈V(µ0).
Theorem 4.3. Let K : Λ1 →2X be a nonempty set-valued mapping,α, β ∈R,α≤β and let f :X×X×Λ2 →R be a functional. If the following assumptions are satisfied:
(i) K(·) is lower semicontinuous at λ0;
(ii) For every x0 ∈S(λ0, µ0), and for every neighborhood W(x0) of x0,
K(λ)∩W(x0)̸= Ø implies S(λ, µ)∩W(x0)̸= Øfor every µ∈V(µ0), Then S(·,·) is lower semicontinuous at (λ0, µ0).
Proposition 4.3. For a given (λ, µ)∈Λ1×Λ2, we have
S(λ, µ) ={x∈K(λ) :α≤f(x, y, µ)≤β,∀y∈K(λ)}.
Let{(λn, µn)} ⊂Λ1×Λ2 satisfying(λn, µn)→(λ0, µ0), for anyx0 ∈S(λ0, µ0), by Lemma 2.3, we only need to prove: there exists xn∈S(λn, µn), such thatxn→x0.
In fact, since x0 ∈ K(λ0), and K(·) is lower semicontinuous at λ0, then, by defi-nition 2.5, for every neighborhood W(x0) of x0 with W(x0)∩K(λ0) ̸= Ø, there exists a neighborhood I(λ0) of λ0 with I(λ0) ⊂ U(λ0), such that K(λ)∩W(x0) ̸= Ø for ev-ery λ ∈ I(λ0). Since λn → λ0 and µn → µ0, so there exists a neighborhood J(µ0) of µ0 with J(µ0) ⊂ V(µ0), and there exists N0, such that when n ≥ N0, we have
λn ∈ I(λ0) and µn ∈ J(µ0). Hence, K(λn)∩W(x0) ̸= Ø. Then, from Condition (ii),
S(λn, µn)∩W(x0) ̸= Ø which implies that there exists xn ∈ S(λn, µn)∩W(x0). Since
Theorem 4.4. Let K : Λ1 →2X be a nonempty set-valued mapping,α, β ∈R,α≤β and let f :X×X×Λ2 →R be a functional. If the following assumptions are satisfied:
(i) K(·) is continuous atλ0, and K(λ0) is compact;
(ii) f(·,·,·) is continuous in X×X× {µ0};
(iii) For everyx0∈S(λ0, µ0), and for every neighborhood W(x0) of x0,
K(λ)∩W(x0)̸= Ø implies S(λ, µ)∩W(x0)̸= Øfor every µ∈V(µ0), Then S(·,·) is continuous at(λ0, µ0).
Proposition 4.4. First, similar to the proof of Theorem 4.1, Condition (i) and (ii) indi-cate that S(·,·) is upper semicontinuous at (λ0, µ0).
Second, similar to Theorem 4.3, the lower semicontinuity of S(·,·) at (λ0, µ0) can be proved.
This completes the proof.
Theorem 4.5. Let K : Λ1 →2X be a nonempty set-valued mapping,α, β ∈R,α≤β and let f :X×X×Λ2 →R be a functional. If the following assumptions are satisfied:
(i) K(·) is continuous atλ0, and K(λ0) is compact;
(ii) f(·,·,·) is continuous in X×X× {µ0};
(iii) For everyx∈S(λ0, µ0), and for every y ∈K(λ0),α < f(x, y, µ0)< β, Then S(·,·) is continuous at(λ0, µ0).
Proposition 4.5. For a given (λ, µ)∈Λ1×Λ2, we have
S(λ, µ) ={x∈K(λ) :α≤f(x, y, µ)≤β,∀y∈K(λ)}.
First, we claim that S(·,·) is upper semicontinuous at (λ0, µ0). This can be proved by Condition (i) and(ii), similar to the proof of Theorem 4.1.
Next, we prove the lower semicontinuity of S(·,·) at (λ0, µ0). Suppose that S(·,·) is not lower semicontinuous at (λ0, µ0) which shows that there exists x0 ∈ S(λ0, µ0), and there exists {(λn, µn)} ⊂ Λ1 ×Λ2 with (λn, µn) → (λ0, µ0), for each xn ∈ S(λn, µn),
we have xn doesn’t converge to x0. Since K(·) is lower continuous at λ0, and then for
{λn} ⊂ Λ1 with λn → λ0, and for x0 ∈ S(λ0, µ0) ⊂ K(λ0), there exists xˆn ∈ K(λn), such that xˆn → x0. By the hypothesis, we know that there exists {xˆnj} ⊂ {xˆn}, such
that xˆnj ∈/ S(λnj, µnj) for all j ∈ N, which implies that there exists ynj ∈ K(λnj, µnj),
we have f(ˆxnj, ynj, µnj) < α or f(ˆxnj, ynj, µnj) > β. From the upper semicontinuity
of K(·) at λ0 and the compactness of K(λ0), it follows that there exists y0 ∈ K(λ0), such that ynj → y0. By Condition (iii), α < f(x0, y0, µ0) < β, and by Condition (ii), f( ˆxnj, ynj, µnj)→f(x0, y0, µ0). Contradiction! The proof is completed.
At Last, we will study the upper semicontinuity of the solution set mapping in a vector metric space, in the particular case where K is fixed, and f is perturbed by a parameter
ε.
Let X, Y be metric spaces, K a nonempty compact convex subset of X,U(ε0) ⊂Y,
α, β ∈R,α ≤β, and let W :K×K×U(ε0) →R be a functional. For every ε∈U(ε0),
consider the problem: Find x∈K, such that
α≤W(x, y, ε)≤β, ∀y∈K (4.4)
For every ε ∈ U(ε0), denote by Tαβ(ε) ⊂ K the solution set of Problem (4.4), Tαβ :
U(ε0)→2K the solution set mapping of Problem (4.4). And we suppose thatW satisfies:
g(x, y, ε)≤W(x, y, ε)≤f(x, y, ε), ∀x, y∈K,∀ε∈U(ε0)
Now, we consider the equilibrium problems related to f and g respectively: Find x∈K, such that
f(x, y, ε)≤β, ∀y∈K (4.5)
For everyε∈U(ε0), denote by Tβ(ε)⊂K the solution set of Problem (4.5),Tβ :U(ε0)→
2K the solution set mapping of Problem (4.5).
Tβ(ε) =∩y∈K{x∈K :f(x, y, ε)≤β}={x∈K : sup y∈K
f(x, y, ε)≤β}
graph(Tβ) ={(ε, x)∈U(x0)×K:x∈Tβ(ε)}
Find x∈K, such that
g(x, y, ε)≥α, ∀y∈K (4.6)
For everyε∈U(ε0), denote byTα(ε)⊂K the solution set of Problem (4.6),Tα:U(ε0)→
2K the solution set mapping of Problem (4.6).
Tα(ε) =∩y∈K{x∈K :g(x, y, ε)≥α}={x∈K : inf
y∈Kg(x, y, ε)≥α}
graph(Tα) ={(ε, x)∈U(ε0)×K :x∈Tα(ε)}
Clearly, for everyε∈U(ε0),Tα(ε)∩Tβ(ε)⊂Tαβ(ε).
Define T :U(ε0)→2K as follows:
T(ε) =Tα(ε)∩Tβ(ε), ∀ε∈U(ε0).
Theorem 4.6. Let X, Y be metric spaces, K a nonempty compact convex subset of X,
U(ε0)⊂Y, α, β ∈R, α≤β, and let f, g:K×K×U(ε0)→R be functionals, Tα, Tβ, T :
U(ε0) → 2K defined as above, satisfying Tα(ε)∩Tβ(ε) ̸= Ø for every ε∈ U(ε0). If the following conditions hold:
(i) For eachy∈K,g(·, y,·) is upper semicontinuous;
(ii) For every y ∈K, and for every ε∈U(ε0), g(·, y, ε) :K →R is upper semicontin-uous;
(iii) For everyy ∈K, f(·, y,·) is lower semicontinuous,
Then T =Tα∩Tβ :ε→Tα(ε)∩Tβ(ε) is upper semicontinuous for every ε∈U(ε0).
Proposition 4.6. By Condition (i) and Lemma 2.5, inf
y∈Kg(x, y, ε) is upper
semicontin-uous for each x ∈ K and for each ε ∈ U(ε0), then graph(Tα) = {(ε, x) ∈ U(ε0)×K :
inf
y∈Kg(x, y, ε)≥α}is closed. SinceK is compact, then from Lemma 2.1,Tα is upper
semi-continuous. Condition (ii) and Lemma 2.4 imply that for all ε∈U(ε0), Tα(ε) is closed,
and sinceK is compact, so Tα(ε) is compact for every ε∈U(ε0). As above, by Condition
(iii), sup
y∈K
f(x, y, ε) is lower semicontinuous for each x∈ K and for each ε∈U(ε0), then
graph(Tβ) ={(ε, x) ∈U(ε0)×K : sup
y∈K
f(x, y, ε)≤β} is closed. Therefore, from Lemma
2.6, we haveT =Tα∩Tβ :ε→Tα(ε)∩Tβ(ε) is upper semicontinuous for everyε∈U(ε0).
Theorem 4.7. Let X, Y be metric spaces, K a nonempty compact convex subset of X,
U(ε0)⊂Y, α, β ∈R, α≤β, and let f, g:K×K×U(ε0)→R be functionals, Tα, Tβ, T :
(1) For every ε∈U(ε0), g(·,·, ε)−α is pseudomonotone;
(2) For each x∈K, g(x,·,·) is lower semicontinuous;
(3) For every y∈K, and for every ε∈U(ε0), g(·, y, ε) is upper semicontinuous;
(4) For each x∈K, and for each ε∈U(ε0),g(x,·, ε) is convex;
(5) For all t∈K, g(t, t, ε) =α;
(6) For every y∈K, f(·, y,·) is lower semicontinuous,
Then T =Tα∩Tβ :ε→Tα(ε)∩Tβ(ε) is upper semicontinuous for every ε∈U(ε0).
Proposition 4.7. First, we prove Tα : U(ε0) → 2K is upper semicontinuous. Since K is compact, then from Lemma 2.1, we only need to prove Tα is closed. Let {(εn, xn)}
satisfy xn∈Tα(εn) and (εn, xn) →(ε, x). Next, we prove x ∈T(ε). xn∈Tα(εn) implies
g(xn, y, εn) ≥ α for each y ∈ K. By Condition (1), for every y ∈ K, g(y, xn, εn) ≤ α,
and by Condition (2), g(y, x, ε) ≤lim inf
n→∞ g(y, xn, εn) ≤ α for every y ∈ K. Define yt = ty+ (1−t)x, then Condition(4) and (5) indicate that
α=g(yt, yt, ε) =tg(yt, y, ε) + (1−t)g(yt, x, ε)≤max{g(yt, y, ε), g(yt, x, ε)}
Suppose g(yt, y, ε) < g(yt, x, ε) ≤ α, then g(yt, y, ε) < α. g(yt, x, ε) = α implies
g(yt, yt, ε)< α, which clearly contradicts Condition (5). Thus,g(yt, y, ε)≥g(yt, x, ε), and
then g(yt, y, ε) ≥ α. By Condition (3), we know that α ≤ lim sup t→0+
g(yt, y, ε) ≤ g(x, y, ε),
then x ∈Tα(ε). Hence, Tα is closed. Condition (3) and Lemma 2.4 show that for every
ε∈U(ε0),Tα(ε)is closed, and sinceKis compact, soTα(ε)is compact for eachε∈U(ε0). From Condition (6), it follows that
graph(Tβ) ={(ε, x)∈U(ε0)×K: sup
y∈K
f(x, y, ε)≤β}
is closed. Thus, by Lemma 2.6, we can prove T =Tα∩Tβ : ε→ Tα(ε)∩Tβ(ε) is upper
semicontinuous for everyε∈U(ε0).
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