0 Fixed Point Theory and Applications

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Sensitivity analysis for generalized quasi-variational relation problems in locally G-convex spaces

Nguyen Van Hung*

Correspondence: ngvhungdhdt@yahoo.com Department of Mathematics, Dong Thap University, 783 Pham Huu Lau Street, Ward 6, Cao Lanh City, Vietnam

Abstract

In this paper, we study generalized quasi-variational relation problems in locally G-convex spaces. Using the Kakutani-Fan-Glicksberg fixed-point theorem for upper semicontinuous set-valued mapping with nonempty closed acyclic values, we establish an existence theorem of a solution set for these problems. Moreover, the stability and closedness of the solution set for these problems are also obtained. The results presented in the paper improve and extend the main results in the literature. MSC: 47J20; 49J40

Keywords: generalized quasi-variational relation problems; G-convex spaces; Kakutani-Fan-Glicksberg fixed-point theorem; quasiconvexity; existence; closedness; upper semicontinuity; compactness

1 Introduction and preliminaries

The generalized quasi-variational relation problems include, as special cases, the generalized variational inclusion problems, the generalized vector equilibrium problems, the generalized vector variational inequality problems etc. In recent years, a lot of results for the existence and stability of solutions for variational relation problems, vector equilibrium problems and vector variational inequality problems have been established by many authors in different ways. For example, variational relation problems [1-4], vector equilibrium problems [5-18], vector variational inequality problems [19, 20] and the references therein.

For a set X, we shall denote by 2X and {X) the families of all subsets of X and the family of all nonempty finite subsets of X, respectively. For each A e {X), |A| denotes the cardinality of A. Let A« denote the standard «-dimensional simplex in Rn+1 with vertices {ei, e2,...,e«+i},thatis,

In+1 «+1 1

u e R«+1: u = ^ k(u)eit X(u) > 0, ^ X(u) = 1,

1=1 1=1 J

ft Spri

ringer

where et is the 1th unit vector in R«+1.

For any nonempty subset J of {0,1,2,...,«}, we denote Aj by the convex hull of the vertices {ej: j e J}.

© 2012 Hung; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A convex set A in a vector space is called a convex space if it is equipped with a topology which includes the Euclidean topology on convex hulls of any nonempty finite subsets of A.

The notion of a G-convex space was introduced by Park and Kim in [21]. Let X be a topological space, A c X be a nonempty subset and a function r : (A) ^ 2X \ {0} be such that the following conditions hold:

(a) for each M,N e (A), r(M) c r(N) if M C N,

(b) for each M e (A) with |M| = n + 1, there exists a continuous mapping

: An ^ r(M) such that, for each J e (M), (AJ) c r(J), where AJ denotes the face of An corresponding to J e (M).

Then (X, A, r) is called a generalized convex space (or a G-convex space). If A = X, we omit A simply write (X, r).

For a G-convex space (X, A, r), a subset B of X is said to be G-convex if, for eachM e (A), M c B implies r(M) c B.A space X is said to have a G-convex structure if and only if X is a G-convex space. A G-convex X is said to be a locally G-convex space if X is a uniform topological space with uniformity U, which has an open base B = {Vi: i e I} of symmetric entourages such that for each v e B, the set V(x) := {y e X: (y,x) e V} is a G-convex set for each x e X.

Now, we pass to our problem setting. Let X, Y, Z be real locally G-convex Hausdorff topological vector spaces, A c X, B c Y and D c Z be nonempty compact convex subsets. Let K1: A ^ 2A, K2: A ^ 2A, T : A ^ 2B be multifunctions and R(x, z,y) be a relation linking x e A, z e D and y e B. We adopt the following notations (see [2]). Letters w, m and s are used for weak, middle and strong kinds of considered problems respectively. For subsets U and V under consideration, we adopt the following notations:

(u, v)wU x V means Vu e U, 3v e V, (u, v)mU x V means 3v e V, Vu e U, (u, v)s U x V means Vu e U, Vv e V, p1(U, V) means U c V, P2 (U, V) means U n V = 0, p3(U, V) means U C V, p4(U, V) means U n V = 0.

Let a e {w, m, s} and p e {p1, p2, p3, p^}. We consider the following for a generalized quasi-variational relation problem (in short, (QVRa)):

(QVRa): Find x e A such that x e K1(x) and (y,z)aK2(x) x T(x) satisfying

R(x, z, y) holds.

Let (R) be the solution set of (QVRa).

Special cases of the problem (QVRa) are as follows:

(I) If we let A, D, B, X, Y, Z, Kl, K2, T be as in (QVRa) and F: A x D x B ^ 2Z be a multifunction, the relation R is defined as follows:

R(x, z, y) holds iff 0 e F(x, z, y).

Then (QVRa) becomes the generalized quasi-variational inclusion problem: Find xe A such that x e K (x) and (y, z)aK2 (x) x T(x) satisfying

0 e F (x, z, y).

(II) If we let A, D, B, X, Y, Z, Kl, K2, T, R be as in (QVRa) and F: A x D x B ^ 2Z and G: A x D ^ 2Z be multifunctions, the relation R is defined as follows:

R(x,z,y)holds iff p(F(x,z,y), G(x,z)).

Then (QVRa) becomes the generalized quasi-variational inclusion problem: Find x e A such that x e K (x) and (y, z)aK2 (x) x T(x) satisfying

p( F (x, z, y), G(x, z)).

(III) If we let A, D, B, X, Y, Z, Kl, K2, T be as in (QVRa) and F : A x D x B ^ 2Z, C: A ^ 2Z be multifunctions such that C(x) is a closed convex cone with int C(x) = 0, the relation R is defined as follows:

R(x,z,y)holds iff p(F(x,z,y), C(x)).

Then (QVRa) becomes the generalized vector quasi-equilibrium problem: Find x e A such that x e K (x) and (y, z)aK2 (x) x T(x) satisfying

p( F (x, z, y), C(x

(IV) If we let A, D, B, X, Y, Z, Kl, K2 be as in (QVRa), f: A x D x B ^ Z be a vector function, and C : A ^ 2Z be a multifunction such that C(x) is a closed convex cone with int C(x) = 0, the relation R is defined as follows:

R(x, z,y)holds iff f (x, z,y) e C(x).

Then (QVRa) becomes the vector quasi-equilibrium problem: Find x e A such that x e K (x) and (y, z)aK2 (x) x T(x) satisfying

f (x, z, y) e C(x).

(V) If we let D, A = B, X = Y, Z, Kl, K2, T be as in (QVRa), L(X, Z) be the space of all linear continuous operators from X to Z and H : L(X, Z) ^ L(X,Z), Q : A x A ^ X, F: A x A ^ Z be continuous single-valued mappings, C: A ^ 2Z be a multifunction such that C(x) is a closed convex cone with int C(x) = 0, the relation R is defined as follows:

R(x, z, y) holds iff H(z), Q(y, x)) + F(y, x) e C(x).

Then (QVRa) becomes the generalized mixed vector quasi-variational inequality problem: Find x e A such that x e K (x) and (y, z)aK2 (x) x T(x) satisfying

(H (z), Q(y, x ^ + F (y, x ) e C(x ).

Definition 1 ([22,23]) Let X, Y be two topological vector spaces, A be a nonempty subset of X and F: A ^ 2Z be a multifunction.

(i) F is said to be lower semicontinuous (lsc) at x0 e A if F(x0) n U = 0 for some open set U c Y implies the existence of a neighborhood N of x0 such that F(x) n U = 0, Vx e N. F is said to be lower semicontinuous in A if it is lower semicontinuous at all x0 e A.

(ii) F is said to be upper semicontinuous (usc) at x0 e A if for each open set U 2 F(x0), there is a neighborhood N of x0 such that U 2 F(x), Vx e N. F is said to be upper semicontinuous in A if it is upper semicontinuous at all x0 e A.

(iii) F is said to be continuous in A if it is both lsc and usc in A.

(iv) F is said to be closed if Graph(F) = {(x,y): x e A,y e F(x)} is a closed subset in A x Y.

Definition 2 ([22]) Let X, Y be two topological vector spaces, A be a nonempty subset of X, F: A ^ 2Z be a multifunction and C c Y be a nonempty closed convex cone.

(i) F is called upper C-continuous at x0 e A if for any neighborhood U of the origin in Y, there is a neighborhood V of x0 such that

F(x) c F(xo) + U + C, Vx e V.

(ii) F is called lower C-continuous at x0 e A if for any neighborhood U of the origin in Y, there is a neighborhood V of x0 such that

F(xo) c F(x) + U - C, Vx e V.

Definition 3 ([23]) Let X and Y be two topological vector spaces and A be a nonempty convex subset of X. A set-valued mapping F: A ^ 2Y is said to be properly C-quasiconvex if for any x, y e A and t e [0,1], we have

either F(x) c F(tx + (1 - t)y) + C or F(y) c F(tx +(1 - t)y) + C.

Lemma 4 ([23]) LetX, Y be two topological vector spaces, A be a nonempty convex subset ofX and F: A ^ 2Y be a multifunction.

(i) If F is upper semicontinuous at x0 e A with closed values, then F is closed at x0 e A.

(ii) If F is closed at x0 e A and Y is compact, then F is upper semicontinuous at x0 e A.

(iii) If F has compact values, then F is usc at x0 e A if and only if, for each net {xa }C A which converges to x0 e A and for each net {ya} with ya c F (xa), there are y0 e F (x0) and a subnet {y$} of {ya} such that ^ y0.

Definition 5 ([24]) Let X be a topological space. A subset A of X is called contractible at x0 e A, if there is a continuous F: A x [0,1] ^ A such that F(z,0) = z for all z e A and F(z,1) = x0 for all z e A.

A topological space X is said to be acyclic if all of its reduced Cech homology groups over the rationals vanish. In particular, each contractible space is acyclic, and thus any

nonempty convex or star-shaped set is acyclic. Moreover, by the definition of a contractible set, we see that each convex space is contractible.

We now have the following fixed-point theorem in locally G-convex spaces given by Yuan [25] which is a generalization of the Fan-Glickberg-type fixed-point theorem for an upper semicontinuous set-valued mapping with nonempty closed acyclic values.

Theorem 6 ([25], Theorem 2.1) Let X be a compact locally G-convex space and F: X ^ 2X be an upper semicontinuous set-valued mapping with nonempty closed acyclic values. Then F has a fixed-point; that is, there exists anx" e X such thatx" e F (x*).

2 Existence of solutions

In this section, we apply the Kakutani-Fan-Glicksberg fixed-point theorem for upper semi-continuous set-valued mapping with nonempty closed acyclic values to establish sufficient conditions for the existence of a solution set of generalized quasi-variational relation problems. Moreover, the closedness of the solution set for these problems is obtained.

Definition 7 Let X be a topological vector space, A be a nonempty convex subset of X and R(x) be a relation linking x e A. We say that R is quasiconvex at xo e A if Vx1,x2 e A, VX e [0,1] such that R(x1) holds and R(x2) holds, we have

R(Xx1 + (1 - X)x2) holds.

R is said to be quasiconvex in A if it is quasiconvex at all x0 e A.

Remark 8 In the Definition 7, if welet X = A = R, and let mapping F: R ^ R, then the relation R defined by R(x) holds iff F(x) с R_. We have Vxbx2 e A, VX e [0,1], if F(xO < 0, F(x2) < 0, then F((1 _ X)x1 + Xx2) < 0. This means that R is modified 0-level quasiconvex, since the classical quasiconvexity says that Vxi, x2 e A, VX e [0,1],

F((1 _ X)x1 + Xx2) < max{F(x1, F(x2)}.

Theorem 9 Assume for the problem (QVRa) that

(i) K1 is upper semicontinuous in A with nonempty closed contractible values, and K2 is lower semicontinuous A with nonempty closed values;

(ii) T is upper semicontinuous in A with nonempty closed acyclic values if a =w (or a = m) and lower semicontinuous in A with nonempty acyclic values if a = s;

(iii) for all (x,z) e A x D, R(x,z,K2(x)) holds;

(iv) for all (z,y) e D x B, R(-, z,y) is quasiconvex in A;

(v) the set {(x, z,y) e A x D x B: R(x, z,y) holds} is closed.

Then, the(QVRa) has a solution, i.e., there existx e Asuch thatx e K1(dc) and (y, z)aK2(x) x T(x) satisfying

R(x, z, y) holds.

Moreover, the solution set ofthe(QVRa) is closed.

Proof Since a = {w,m, s}, we have in fact three cases. However, the proof techniques are similar. We present only the proof for the case where a = m. Indeed, for all (x, z) e A x D, define a set-valued mapping: ^m : A x D ^ 2A by

^m(x,z) = {a e KL(x): R(a,z,y) holds, Vy e K2(x)}.

Since for any x e A, KL(x), K2(x) are nonempty. Thus, by assumption (iii), we have ^m (x, z) = 0.

(I) We show that ^m(x, z) is acyclic.

Since every contractible set is acyclic, it is enough to show that ^m(x, z) is contractible. Let a e Wm(x,z), thus a e KL(x) and R(a,z,y) holds,Vy e K2(x). Since KL(x) is contractible, there exists a continuous mapping f : KL(x) x [0,1] ^ KL(x) such that f (b,0) = b for all b e KL(x) and f (b,1) = a for all b e KL(x). Now, we set F(b, k) = ka + (1 - k)h(b, k) for all (b, k) e (x, z) x [0,1]. Then F is a continuous mapping, and we see that F(b,0) = b for all b e ^m(x,z) and F(b,1) = a for all b e ^m(x,z). Let (b, k) e ^m(x,z) x [0,1], we need to prove that F(b,k) e ^m(x,z). Since a,f (b,k) e Ki(x), and Ki(x) is contractible, thus, for a,f (b, k) e tym(x,z), it follows that

R(a, z, y) holds, Vy e K2 (x)

Rf (b, k), z, y) holds, Vy e K2(x). By (iv), R(-, z,y) is quasiconvex in A, we have R(ka + (1 - k)f (b, k), z,y) holds, Vy e K2(x),

i.e., F(b,k) e tym(x,z). Therefore, ^m(x,z) is contractible.

(II) We will prove is upper semicontinuous in A x D with nonempty closed values. Since A is a compact set and ^m(x, z) C A. Hence ^(x, z) is compact. We need to show

that is a closed mapping. Indeed, let a net {(xn, zn)} c A x D such that (xn, zn) ^ (x, z) e A x D, and let an e ^m(xn,zn) such that an ^ a0. Now, we need only prove that a0 e (x, z). Since an e K1(xn) andKL is upper semicontinuous at x e A with nonempty closed values, by Lemma 4(i), we have Kl is closed at x e A, thus a0 e KL(x). Suppose, to the contrary, a0 e ^m(x,z). Then 3y0 e K2(x) such that

R(a0, z,y0) does not hold. (1)

By the lower semicontinuity of K2, there is a net {yn} with yn e K2(xn) such that yn ^ y0. Since an e ^m(xn,zn), we have

R(an, zn, yn)holds. (2)

By the condition (v) and (2), we have

R(ao, z,yo) holds.

There is a contradiction between (3) and (1). Thus, ao e z). Hence, Фm is upper semicontinuous in A x D with nonempty closed values.

(III) Now, we shall show that the solution set Xm(R) = 0. Define the set-valued mapping Hm : A x D ^ 2AxD by

Hm(x, z) = ( Фт(х, z), T (x)), V(x, z) e A x D.

Then Hm is upper semicontinuous in A x D and V(x, z) e A x D, Hm(x, z) is a nonempty closed convex subset of A x D. By Theorem 6, there exists a point (x, z) e A x D such that (x,z) e Hm(x,z), that is,

x e ^m(x, z), z e T(x),

which implies that there exists xx e A and z e T(x) such that x e K1(x) and

R(x, z, y) holds,

i.e., xc e Xm(R).

(IV) Next, we prove that Xm(R) is closed.

Let a net {xa, a e I}e Sm(R) :xa ^ xo. We need to prove that xo e Sm(R). Indeed, by the lower semicontinuity of K2, for anyyo e K2(xo), there exists yn e K2(xa) suchthat yn ^ yo. As xa e Xm(R), there exists za e T(xa) such that

R(xa, za, y a ) holds.

Since K1 is upper semicontinuous with nonempty closed values, by Lemma 4(i), we have K1 is closed. Thus, xo e KL(xo). Since T is upper semicontinuous in A and T(xo) iscompact, there exists zo e T(xo) such that za ^ zo. By the condition (v), we have

R(xo, zo,yo) holds.

This means that xo e Sm(R). Thus Xm(R) is a closed set. □

Remark 10 If we let A = B, D, X = Y, Z, Kl = K2 = K, T, R, a = mas in (QVRa ), F : A x D x A ^ 2Z be a multifunction and C с Z be a nonempty closed convex cone, the relation R is defined as follows:

R(x, z, y)holds iff F (x, z, y) с C,

R(x, z, y)holds iff F (x, z, y) С C.

Then, (QVRa ) becomes the generalized strong vector quasi-equilibrium problem of type (I) and (II) (in short, (GSVQEP I) and (GSVQEPII)) studied in [17].

(GSVQEP I): Find x e A and z e T(x) such that x e K(x) and

F(x, z, y) C C, for all y e K(x);

(GSVQEP II): Find x e A and z e T(x) such that x e K(x) and

F (x, z, y) C C, for all y e K (x).

The following example shows that all the assumptions of Theorem 9 are satisfied, but Theorem 3.1 in [17] does not work. The reason is that F is not lower (-C)-continuous.

Example 11 Let X = Y = Z = R, A = B = D = [0,1], C = R+, Ki(x) = Kz(x) = [0,1] and

Ti(x) = T2(x) =

[0,1] if xo = 2, [0,2] otherwise,

F (x, z, y) =

[2.1] if x0 = z0 = y0 = 2,

[1.2] otherwise.

We let the relation R be defined by R(x, z, y) holds iff F(x, z, y) c R+. We can show that all the assumptions of Theorem 9 are satisfied. However, F is not lower (-C)-continuous at x0 = 2. Also, Theorem 3.1 in [17] does not work.

The following example shows that all the assumptions of Theorem 9 are satisfied, but Theorem 3.1 in [17] is not fulfilled. The reason is that F is not upper C-continuous.

Example 12 Let A, B, D, X, Y, Z, K, C be as in Example 11 and T(x) = {z} and

F (x, z, y) =

[1, 2] if x0 = z0 = y0 = 2, [i, |] otherwise.

We let the relation R be defined by R(x, z,y) holds iff F(x, z,y) c R+.It is easy to check that all the assumptions of Theorem 9 are satisfied. So, (QVRa) has a solution. However, F is not upper C-continuous at x0 = Also, Theorem 3.1 in [17] does not work.

The following example shows that all assumptions of Theorem 9 are satisfied, but Theorem 3.1 in [17] is not fulfilled. The reason is that F is not C-quasiconvex.

Example 13 Let A, B, D, X, Y, Z, K, C, T be as in Example 12 and

F (x, z, y) =

[1,2] if xo = zo = yo = [2,1] otherwise.

We let the relation R be defined by R(x, z,y) holds iff F(x, z,y) c R+.It is easy to check that all the assumptions of Theorem 9 are satisfied. However, F is not C-quasiconvex at x0 = 2. Thus, it gives also cases where Theorem 9 can be applied but Theorem 3.1 in [17] does not work.

If we let X, Y, Z be real locally convex Hausdorff topological vector spaces, then we have the following corollary.

Corollary 14 Assume for problem (QVRa) that

(i) K1 is upper semicontinuous in A with nonempty closed convex values, and K2 is lower semicontinuous A with nonempty closed values;

(ii) T is upper semicontinuous in A with nonempty closed convex values if a =w (or a = m) and lower semicontinuous in A with nonempty convex values if a = s;

(iii) for all (x,z) e A x D, R(x,z,K2(x)) holds;

(iv) for all (z,y) e D x B, R(-, z,y) is quasiconvex in A;

(v) the set {(x, z,y) e A x D x B: R(x, z,y) holds} is closed.

Then the(QVRa) has a solution, i.e., there existxe Asuch thatx e KL(x) and (y, z)aK2(x) x T(x) satisfying

R(x, z, y) holds.

Moreover, the solution set ofthe(QVRa) is closed.

Remark 15

(i) If we let X, Y, Z be real locally convex Hausdorff topological vector spaces, then (GSVQEP I) becomes the problem (GSVQEP) studied in [15].

(ii) If A = B, X = Y, Z, Kl = K2 = K, R as in (QVRa) and T (x) = {z}, F: A x A ^ 2Z is a multifunction, C c Z is a nonempty closed convex cone, the relation R is defined as follows:

R(x, z, y) holds iff F(x, y) c C.

Then (QVRa) becomes strong vector quasi-equilibrium problem (in short,

(SVQEP)) studied in [18].

Find x e A such that x e K(x) and

F(x, y) c C, for all y e K(x).

Remark 16

(i) Corollary 14 improves and extends Theorem 3.1 in [15] and Theorem 3.3 in [18].

(ii) Theorem 9 improves and extends Theorems 3.1 and 3.3 in [17].

3 Stability

In this section, we discuss the stability of the solutions for (QVRa). Throughout this section, let X, Y, Z be Banach spaces, N be a real locally G-convex Hausdorff topological vector space. Let A c X, B c Y and D c Z be nonempty compact convex subsets,

K2 = K2 = K: A ^ 2a, T: A ^ 2B be multifunctions, and R(x, z,y) be a relation linking x e A, z e D and y e B. Now, we let

3 := {(T, K)|T is upper semicontinuous in A with nonempty closed acyclic values, if a = w(or a = m), and lower semicontinuous in A with nonempty acyclic values, if a = s, and K is continuous with nonempty closed contractible values}.

Let El, E2 be compact sets in a normed space. Recall that the Hausdorff metric is defined by

H(Ei,E2) := max{H*(Ei,E2),H'(E2,Ei)},

whereH"(Ei,E2) := supeieEl d(ei,E2) and d(ei,E2) := infe2eE2 llei - e21|. For (T,K),(T',K') e 31, define

f ((T,K), (T',K')) := supHi(T(x), T'(x)) + supH2(K(x),K'(x)),

xeA xeA

where H2, H2 are the appropriate Hausdorff metrics. Obviously, (3, f )isa metric space.

Assume that R satisfies the conditions of Theorem 9. Then for each (T,K) e 3, (QVRa) has a solution x, i.e., there exists x e A such that x e K(x) and (y, z)aK(x) x T(x) satisfying

R(x, z, y) holds.

For (T,K) e 3, let

©a(T,K) := {x e A such that x e K(x)

and (y,z)aK(x) x T(x) satisfying R(x,z,y) holds}.

Then ©a(T,K) = 0, and so ©a(T,K) defines a set-valued mapping from 3 into A.

Lemma 17 ([26]) LetZ be a metric space and letM, Mn (n = 1,2,...) be compactsets in Z. Suppose that for any open set O D M, there exists n0 such that Mn c O, Vn > n0. Then any sequence {xn} satisfying xn e Mn has a convergent subsequence with limit in M.

Theorem 18 ©a : 3 ^ 2A is upper semicontinuous with compact values.

Proof Similar arguments can be applied to three cases. We present only the proof for the cases where a = m. Indeed, since A is compact, we need only show that &m is a closed mapping. Let a sequence {(Tn,Kn,xn)} c Graph(©m) be given such that (Tn,Kn,xn) ^ (T,K,x0). We now show that {(T,K,x0)} c Graph(©m).

Forany n,since xn e ©m(Tn, Kn), we have that xn e Kn(xn) and 3zn e Tn(xn), Vyn e Kn(xn) such that

R(xn, zn,yn) holds.

For any open set O D T(x0), since T(x0) is a compact set, there exists e > 0 such that

{z e Y: d(z, T(xo)) < e} c O, (5)

where d(z, T(xo)) = inf^r^) llz - ¡Z\\.

Since f ((Tn,Kn), (T,K)) ^ 0, xn ^ x0 and T is upper semicontinuous at x0, 3n0 such that

supHi(Tn(x), T(x)) < -, (6)

T(xn) c |z e Y: d(z, T(x0)) <|J, Vn > n0. (7)

From (5), (6) and (7), we have

T(xn) c jz e Y: d(z, T(x0)) <|J c {z e Y: d(z, T(x0)) < e} c O, Vn > n0. (8)

Since T(x0) c O and zn e Tn(xn), we can apply Lemma 17. There exists a subsequence {znk} of {zn} such that {znk} convergent to z0, it follows that z0 e T(x0). By using the same argument as above, we can show that x0 e K(x0).

Next, we need only show that R(x0, z0,y0) holds. Since xn ^ x0 and K is upper semicontinuous at x0, K(x0) is closed, there exists y0 e K(x0) such that yn ^ y0 (taking a subsequence if necessary). Since f ((Tn,Kn), (T,K)) ^ 0, we can chose a subsequence {Knk} of {Kn} such that

supH2(Knk(x),K(x)) <}. (9)

H2 (Knk (xnk), K(xnk )) < -1. This implies that there exist t' e Knk (xnk), k = 1,2,... such that

I . y 1

rnk tnk I < k '

Vnk -y0|| < Itnk - tnk II + \\tnk -y0 11 < 1 + Unk -y0\ ^ 0, and so we have t'nk ^ y0. Since xnk e Knk (xnk

), Znk e Tnk (xnk) and tnk e Knk (xnk), applying

(5), we have

R{xnk, Znk, t'nk) holds. Assumption (v) yields that

R(xo, zo,yo) holds.

Since x0 e K(x0) and z0 e T(x0) and (10) yields that (T,K,x0) e Graph(©m) and so Graph(©m) is closed. Therefore, ©m is closed. Since A is a compact set and ©m (T, K) c A. Hence ©m has a compact valued mapping. □

Remark 19 Theorem 18 improves and extends Theorems 3.1 and 3.3 in [17], Theorem 3.1 in [15].

Competing interests

The author declares that he has no competing interests.

Acknowledgements

The author thanks the two anonymous referees for their valuable remarks and suggestions, which helped him to improve

the article considerably.

Received: 22 July 2012 Accepted: 4 September 2012 Published: 19 September 2012

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doi:10.1186/1687-1812-2012-158

Cite this article as: Hung: Sensitivity analysis for generalized quasi-variational relation problems in locally G-convex spaces. Fixed Point Theory and Applications 2012 2012:158.

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