On common quasi-eigenvector problems

Nonlinear Analysis 69 (2008) 463–471 www.elsevier.com/locate/na

On common quasi-eigenvector problems Lai-Jiu Lin ∗ , Wei-Shih Du Department of Mathematics, National Changhua University of Education, Changhua, 50058, Taiwan Received 23 December 2006; accepted 29 May 2007

Abstract In this paper, we study four types of common quasi-eigenvector problems and two types of eigenvector problems. The existence theorems for the solutions of common quasi-eigenvector problems and eigenvector problems are also established. c 2008 Published by Elsevier Ltd

Keywords: Eigenvalue; Eigenvector; Common quasi-eigenvector problem; Maximal element theorem

1. Introduction In 2001, Li [3] first proved some eigenvector theorems by applying a Fan–KKM theorem [2, Lemma 1]. Subsequently, Li and Park [4] obtained new eigenvector theorems from their equilibrium theorems. In fact, in [3], Li study the following eigenvector problem (EIVP)Li in real normed linear spaces: (EIVP)Li Find x ∈ X with x 6= θ such that f (x) 6= θ and x ∈ R( f (x)) (that is, d(x, R( f (x)) = 0)), where R(x) := {r x : r ∈ R}. Taking motivation from the research due to Li [3] and Li and Park [4], we study the following four types of common quasi-eigenvector problems, [CQEIVP]-(I), [CQEIVP]-(II), [CQEIVP]-(III) and [CQEIVP]-(IV), and establish the existence theorems for the solutions of these problems in complex normed linear spaces. Definition 1.1. Let I be Q any index set. For Q each i ∈ I , let L i be a linear space with origin θi and X i be a nonempty subset of L i . Let L = i∈I L i and X = i∈I X i . Let K = R or C. For each i ∈ I , let f i : L → L i be a map. (a) The common quasi-eigenvector problem (I) (for short [CQEIVP]-(I)) is the following problem: • Find v = (vi )i∈I ∈ X and λ ∈ K such that f i (v) = λvi for all i ∈ I . In [CQEIVP]-(I), λ and v are called the common eigenvalue and the corresponding common weak quasieigenvector for { f i }i∈I , respectively. (b) The common quasi-eigenvector problem (II) (for short [CQEIVP]-(II)) is the following problem: • Find v = (vi )i∈I ∈ X with vi 6= θi and λ ∈ K such that f i (v) = λvi for all i ∈ I. ∗ Corresponding author.

E-mail addresses: [email protected] (L.-J. Lin), [email protected] (W.-S. Du). c 2008 Published by Elsevier Ltd 0362-546X/$ - see front matter doi:10.1016/j.na.2007.05.033

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In [CQEIVP]-(II), λ and v are called the common eigenvalue and the corresponding common (strong) quasieigenvector for { f i }i∈I , respectively. (c)The common quasi-eigenvector problem (III) (for short [CQEIVP]-(III)) is the following problem: • Find v = (vi )i∈I ∈ X and λ ∈ K such that λ f i (v) = vi for all i ∈ I. In [CQEIVP]-(III), λ and v are called the common quasi-eigenvalue and the corresponding common weak quasieigenvector for { f i }i∈I , respectively. (d)The common quasi-eigenvector problem (IV) (for short [CQEIVP]-(IV)) is the following problem: • Find v = (vi )i∈I ∈ X with vi 6= θi and λ ∈ K with λ 6= 0 such that f i (v) = λvi for all i ∈ I. In [CQEIVP]-(IV), λ and v are called the common strict eigenvalue and the corresponding common (strong) quasieigenvector for { f i }i∈I , respectively. [CQEIVP]-(IV) can take the following equivalent form: • Find v = (vi )i∈I ∈ X with vi 6= θi and λ ∈ K with λ 6= 0 such that λ f i (v) = vi for all i ∈ I. It is easy to see that every solution of [CQEIVP]-(II) and [CQEIVP]-(IV) is a solution of [CQEIVP]-(I), and every solution of [CQEIVP]-(IV) is a solution of [CQEIVP]-(II) and [CQEIVP]-(III). But the reverse implication does not hold. It is obvious that [CQEIVP]-(II) contains the following eigenvector problem (w-EIVP) as a special case: • Find x ∈ X with x 6= θ and λ ∈ K such that f (x) = λx. Also, [CQEIVP]-(IV) contains the following eigenvector problem (s-EIVP) as a special case: • Find x ∈ X with x 6= θ and λ ∈ K with λ 6= 0 such that f (x) = λx. In general, an eigenvalue for some map (or operator) may be a complex number or a zero. In this paper, we investigate the sufficient conditions of common quasi-eigenvector problems on complex normed linear spaces, instead of real normed linear spaces which Li [3] and Li and Park [4] studied. In Section 3, we establish the existence theorems for problems [CQEIVP]-(I), [CQEIVP]-(II) and [CQEIVP]-(IV) by using a maximal element theorem. Moreover, we give sufficient conditions for the existence of the solutions of eigenvector problems (w-EIVP) and (s-EIVP). Finally, in Section 4, we establish the existence theorems for the solutions of problems [CQEIVP]-(III), [CQEIVP]-(IV) and (s-EIVP). Notice that the existence of a common eigenvalue can be provided from our existence theorems for the solutions of these problems simultaneously. Since our definitions of eigenvalue and eigenvector are more general than and different from those of [3,4], some conditions that we discuss are different from those of [3,4] essentially. Consequently, our research concerning common quasi-eigenvector problems is new and our techniques and some results are original in the literature. 2. Preliminaries In what follows, K will denote the field of real numbers R or the field of complex numbers C. Let A and B B be nonempty S sets. A multivalued map T : A ( B is a function from A to the power set 2 of B. We define T (A) = {T (x) : x ∈ A} and let T − : B ( A be defined by the condition that x ∈ T − (y) if and only if y ∈ T (x). ¯ respectively. Recall that a point x ∈ A is called The convex hull of A and the closure of A are denoted by co A and A, a maximal element of a multivalued map T : A ( B if T (x) = ∅. Let (E, k·k) be a normed linear space with origin θ and X be a nonempty subset of E. Define d(u, X ) = inf{ku − xk : x ∈ X },

u ∈ E.

A real valued function ϕ : X → R is lower semicontinuous (for short l.s.c.) (resp. upper semicontinuous, for short u.s.c.) if {x ∈ X : ϕ(x) ≤ r } (resp. {x ∈ X : ϕ(x) ≥ r }) is closed for each r ∈ R. Let R : E ( E be a multivalued map defined by R(x) := {r x : r ∈ K},

x ∈ E.

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Clearly, for any x ∈ E, R(x) is closed and convex. Note that R(θ ) = {θ} and for any u ∈ E with u 6= θ, R(u) can be regarded as the line in E passing through θ and u. The vertical cone VC(X) of X is defined by [ R(x); VC(X ) = x∈X

see [3,4]. The kernel of a map f : X → E is the set ker f denoted by ker f = {x ∈ X : f (x) = θ}. Definition 2.1. Let X be a nonempty subset of a normed linear space E with origin θ and f : X → E a map. A λ ∈ K and a point x ∈ X with x 6= θ are called an eigenvalue and the corresponding eigenvector of f , respectively, if f (x) = λx. Remark 2.1. (a) In fact, a point x ∈ X with x 6= θ is an eigenvector of f in the sense of Definition 2.1, if f (x) ∈ R(x) or d( f (x), R(x)) = 0. (b) In [3], Li discuss the case K = R and called a point x ∈ X with x 6= θ, satisfying f (x) 6= θ, an eigenvector of f if x ∈ R( f (x)) (that is, d(x, R( f (x)) = 0)). Then it implies that there exists an eigenvalue λ ∈ R with λ 6= 0 such that f (x) = λx. As we see from [3], a point needs a stricter requirement to be able to become an eigenvector for some map in Li’s sense and [3] only study the nonzero eigenvalue problems under Li’s definition. It is quite obvious that Definition 2.1 and Li’s definition for the eigenvector are different. 3. The study of solutions of common quasi-eigenvector problems The following lemmas are crucial tools for our results and can be verified immediately from definitions. Lemma 3.1. Let L be a linear space. If f : L → R is convex and g : L → L is affine, then the composition f ◦ g is convex. Lemma 3.2. Let E be a normed linear space and A a nonempty subset of E. Let the function ϕ : E → R be defined by ϕ(x) = d(x, A). Then ϕ is continuous. Further, if A is convex, then ϕ is convex. The following maximal element theorem is a special case of [5, Theorem 4.4]; see also [1,6]. Theorem 3.1 ([5]). Let I be any index set. Let {X i }i∈I be a family of nonempty convex subsets, where each X i is Q contained in a Hausdorff t.v.s. E i . For each i ∈ I , let Si : X = i∈I X i ( X i be a multivalued map such that (i) for each x = (xi )i∈I ∈ X , xi 6∈ co Si (x); (ii) for each yi ∈ X i , Si− (yi ) is open in X ; (iii) there exist a nonempty compact subset K of X and a nonempty compact convex subset Mi of X i for all i ∈ I such that for each x ∈ X \ K , there exists j ∈ J such that M j ∩ S j (x) 6= ∅. ¯ = ∅ for all i ∈ I . Then there exists x¯ ∈ X such that Si (x) Now we establish an important result which plays a key role in the proof of the main results in this paper. Theorem 3.2. Let (E, k·k) be a normed linear space with origin θ, X a nonempty convex subset of E and f : E → E a continuous map with f (θ ) = θ. For any y ∈ X and λ ∈ R− := {r ∈ R : r ≤ 0}, let G : X × R− ( X be a multivalued map defined by G(y, λ) = {x ∈ X : d( f (y), R(x)) ≥ d( f (x), R(x)) + λ}. Then for each (y, λ) ∈ X × R− , G(y, λ) is a closed set in X . Proof. Clearly, θ ∈ G(y, λ) for all (y, λ) ∈ X × R− . Fix (y, λ) ∈ X × R− . Let xˆ ∈ G(y, λ). Then there exists a sequence {xn }n∈N in G(y, λ) such that xn → x. ˆ We consider two possible cases: Case (I). If xˆ = θ, then it is obvious that xˆ ∈ G(y, λ).

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Case (II). If xˆ 6= θ,

then

xˆ > 0. Since xn → xˆ as n → ∞, without loss of generality, we may assume that

xn 6= θ and xn − xˆ < xˆ for all n ∈ N. For these special subsets R(x) ˆ and R(xn ) of E, n ∈ N, there exist v ∈ R(x) ˆ and z n ∈ R(xn ) such that

f (x) ˆ − v = d( f (x), ˆ R(x)) ˆ (3.1) and

f (x) ˆ − z n = d( f (x), ˆ R(xn ))

(3.2)

for all n ∈ N. We claim that the inequality (3.3) holds, where



f (x)

ˆ + f (x) ˆ − v

xn − xˆ

d( f (x), ˆ R(x)) ˆ ≤ d( f (xn ), R(xn )) + f (x) ˆ − f (xn ) +

xˆ − xn − xˆ

(3.3)

for all n ∈ N. Since there exists wn ∈ R(xn ) such that k f (xn ) − wn k = d( f (xn ), R(xn )), we have from (3.2) and (3.4) that

f (x) ˆ R(xn )) ˆ − z n = d( f (x),

ˆ − wn ≤ f (x)

≤ f (x) ˆ − f (xn ) + d( f (xn ), R(xn ))



ˆ − z n , then, by (3.1) and (3.5), we obtain ˆ − v ≤ f (x) for all n ∈ N. Let n ∈ N be fixed. If f (x)

ˆ − v d( f (x), ˆ R(x)) ˆ = f (x)

≤ f (x) ˆ − zn

ˆ − f (xn ) + d( f (xn ), R(xn )). ≤ f (x)



Hence our claim (3.3) holds in this case. If f (x) ˆ − v > f (x) ˆ − z n , then z n 6= θ . Indeed, if z n = θ , then



f (x) ˆ − v = d( f (x), ˆ R(x)) ˆ ≤ f (x) ˆ − zn ,

(3.4)

(3.5)

(3.6)

which leads to a contradiction. Thus z n 6= θ. Since z n ∈ R(xn ), there exists tn ∈ K such that z n = tn xn . So kz n k |tn | = kx > 0. Let vn = tn xˆ ∈ R(x). ˆ Since nk



kz n k ≤ f (x) ˆ − zn ˆ + f (x)



< f (x) ˆ + f (x) ˆ − v , we have

kz n k

xˆ − xn kx k

n

f (x)

ˆ + f (x) ˆ − v

xˆ − xn < kxn k



f (x)

ˆ + f (x) ˆ − v

xˆ − xn .



≤

xˆ − xˆ − xn

kvn − z n k =

Combining (3.1), (3.2), (3.5) and (3.7), we get our claim (3.3) as follows:

d( f (x), ˆ R(x)) ˆ ≤ f (x) ˆ − vn

≤ f (x) ˆ − z n + kz n − vn k



f (x)

ˆ + f (x) ˆ − v

xˆ − xn



< d( f (x), ˆ R(xn )) +

xˆ − xˆ − xn





f (x) ˆ + f (x) ˆ − v

xˆ − xn .



≤ d( f (xn ), R(xn )) + f (x) ˆ − f (xn ) +

xˆ − xˆ − xn

(3.7)

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Next, we take a ∈ R(x) ˆ and u n ∈ R(xn ) such that k f (y) − ak = d( f (y), R(x)) ˆ

(3.8)

k f (y) − u n k = d( f (y), R(xn ))

(3.9)

and

for all n ∈ N. We also claim that the inequality (3.10) holds, where

kak d( f (y), R(xn )) ≤ k f (y) − ak + xn − xˆ

xˆ

(3.10)

for all n ∈ N. If a = θ, then we have from (3.9) that k f (y) − u n k = d( f (y), R(xn )) ≤ k f (y) − ak . > 0. For each n ∈ N, So our claim (3.10) holds in this case. If a 6= θ, then a = γ xˆ for some γ 6= 0. Thus |γ | = kkak xˆ k let cn = γ xn ∈ R(xn ). Consequently, it follows that d( f (y), R(xn )) ≤ k f (y) − cn k ≤ k f (y) − ak + ka − cn k

kak = k f (y) − ak + xˆ − xn ,

xˆ which prove our claim (3.10). Since xn ∈ G(y, λ), we have d( f (y), R(xn )) ≥ d( f (xn ), R(xn )) + λ.

(3.11)

From (3.3), (3.8), (3.10) and (3.11), we obtain d( f (y), R(x)) ˆ = k f (y) − ak

kak ≥ d( f (y), R(xn )) − xˆ − xn

xˆ

kak ≥ d( f (xn ), R(xn )) − xˆ − xn + λ

xˆ



f (x)

ˆ + f (x) ˆ − v

xˆ − xn



ˆ − f (xn ) − ≥ d( f (x), ˆ R(x)) ˆ − f (x)

xˆ − xˆ − xn

kak − xˆ − xn + λ

xˆ

(3.12)

for all n ∈ N. Since f is continuous and xn → xˆ as n → ∞, by (3.12), we have d( f (y), R(x)) ˆ ≥ d( f (x), ˆ R(x)) ˆ + λ, which means that xˆ ∈ G(y, λ). According to case (I) and case (II), we show that G(y, λ) is closed in X for each (y, λ) ∈ X × R− . The proof is completed.  Theorem 3.3. Let (E, k·k) be a normed linear space with origin θ, X a nonempty convex subset of E and f : E → E a map. Define two real valued functions g, h : X × X → R by g(x, y) = d( f (y), R(x)) − d( f (x), R(x)) and h(x, y) = d(y, R( f (x))) − d(x, R( f (x))). Then the following hold: (a) If f is affine and continuous, then for each x ∈ X , the function y → g(x, y) is convex and continuous. (b) For each x ∈ X , the function y → h(x, y) is convex and continuous.

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(c) Suppose that θ ∈ 6 X and f is continuous with f (θ ) = θ. Then for each y ∈ X , the function x → g(x, y) is u.s.c. (d) Suppose that f is continuous with θ 6∈ f (X ). Then for each y ∈ X , the function x → h(x, y) is u.s.c. Proof. By Lemmas 3.1 and 3.2, we prove conclusions (a) and (b). For any y ∈ X and λ ∈ R, let A : X × R ( X be a multivalued map defined by A(y, λ) = {x ∈ X : g(x, y) ≥ λ}. To see (c), it suffices to show that A(y, λ) is closed for each (y, λ) ∈ X × R. Fix (y, λ) ∈ X × R. If A(y, λ) = ∅, then we are done. Otherwise, let xˆ ∈ A(y, λ). Since θ 6∈ X , we have xˆ 6= θ. Following the same argument as in the proof of Case (II) in Theorem 3.2, we can verify that A(y, λ) is closed in X for each (y, λ) ∈ X × R. Hence for each y ∈ X , the function x → g(x, y) is u.s.c. Finally, following an argument very similar to that in [3, Theorem 1], one can show (d). The proof is completed.  Theorem 3.4. Let (E, k·k) be a normed linear space with origin θ, X be a nonempty convex subset of E and f : E → E be an affine and continuous map satisfying f (θ) = θ. Suppose that (H1 ) there exist a nonempty compact subset W of X and a nonempty compact convex subset M of X such that, for each y ∈ X \ W , there exists z ∈ M such that d( f (z), R(y)) < d( f (y), R(y)). Then there exists v ∈ X such that d( f (x), R(v)) ≥ d( f (v), R(v)) for all x ∈ X . Proof. Let g : X × X → R be defined by g(x, y) = d( f (y), R(x)) − d( f (x), R(x)) and C : X ( X be defined by C(x) = {y ∈ X : g(x, y) < 0}. Then, by Theorem 3.3(a), C(x) is convex from the convexity of g(x, ·) for each x ∈ X . Clearly, x 6∈ C(x) = co C(x) for each x ∈ X . By Theorem 3.2 with λ = 0, C − (y) = {x ∈ X : g(x, y) < 0} is open in X for each y ∈ X . By our coercivity condition (H1 ), there exist a nonempty compact subset W of X and a nonempty compact convex subset M of X such that for each y ∈ X \ W , M ∩ C(y) 6= ∅. Applying Theorem 3.1, there exists v ∈ X such that C(v) = ∅. That is, d( f (x), R(v)) ≥ d( f (v), R(v)) for all x ∈ X .  Remark 3.1. A linear map f is affine and satisfies f (θ ) = θ. Using Theorem 3.4, we have the following existence theorem which related to solve problem [CQEIVP]-(I). Theorem 3.5. Let I be any index set. For each Q i ∈ I , let (E i , k·ki ) be a normed linear space with origin θi and X i be a nonempty convex subset of Q E i . Let E = i∈I E i with origin θ = (θi )i∈I and the norm kxk = supi∈I kxi k, where x = (xi )i∈I ∈ E, and X = i∈I X i . For each i ∈ I , let f i : E → E i be an affine and continuous map satisfying f i (θ) = θi and ! \ X∩ ker f i 6= ∅. i∈I

Define f : E → E by f (x) = ( f i (x))i∈I ,

x ∈ E.

Suppose that the coercivity condition (H1 ) in Theorem 3.4 holds. Then there exists v = (vi )i∈I ∈ X such that f (v) ∈ R(v); that is, the problem [CQEIVP]-(I) has a solution. Proof. Obvious, the map f is affine and continuous satisfying f (θ) = θ for each T i ∈ I . By Theorem 3.4, there exists v ∈ X such that d( f (x), R(v)) ≥ d( f (v), R(v)) for all x ∈ X . Since X ∩ ( i∈I ker f i ) 6= ∅, there exists w ∈ X such that f i (w) = θi . Thus f (w) = θ and hence d( f (v), R(v)) ≤ d( f (w), R(v)) = 0.

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Therefore we have d( f (v), R(v)) = 0 or f (v) ∈ R(v). So there exists λ ∈ K such that f i (v) = λvi for all i ∈ I . This means that (λ, v) is a solution of [CQEIVP]-(I).  T Remark 3.2. (a) In Theorem 3.5, if X i = E i then the assumption “X ∩ ( i∈I ker f i ) 6= ∅” holds automatically. (b) Note that, in X may have θ 6∈ X . So the condition TTheorem 3.5, X i may have θi 6∈ X i for some i ∈ I and T “θ ∈ X ∩ ( i∈I ker f i )” cannot be guaranteed to hold. But if θ ∈ X ∩ ( i∈I ker f i ) holds, then for any λ ∈ R, (λ, θ ) is a solution of [CQEIVP]-(I). The following existence theorem related to solve problem [CQEIVP]-(II) is established immediately from Theorem 3.5. Theorem 3.6. In Theorem 3.5, if we assume further that θi 6∈ X i for each i ∈ I , then the problem [CQEIVP]-(II) has a solution (λ, v); that is, there exist a common eigenvalue λ ∈ K and the corresponding common quasi-eigenvector v = (vi )i∈I ∈ X with vi 6= θi such that f i (v) = λvi for each i ∈ I . The following result gives a sufficient condition for the existence of the solution of eigenvector problem (w- EIVP). Theorem 3.7 (Eigenvector Theorem (I)). Let (E, k·k) be a normed linear space with origin θ, X be a nonempty convex subset of E with θ 6∈ X and f : E → E be an affine and continuous map satisfying f (θ ) = θ and ker f ∩ X 6= ∅. Suppose that the coercivity condition (H1 ) in Theorem 3.4 holds. Then there exists v ∈ X with v 6= θ and λ ∈ K such that f (v) = λv. The following example tells us that the eigenvalue λ in Theorem 3.7 has the possibility of being nonzero. Example. Let E = R3 with the usual metric, a > 1 and K = R. Then X := {(0, y, z) ∈ R3 : 1 ≤ y ≤ a, −a ≤ z ≤ a} is a nonempty compact convex subset of E. Clearly, θ := (0, 0, 0) 6∈ X . Define f : E → E by f (x, y, z) = (x, y + z, y + z). Then f is linear and continuous. Let λ˜ := 2 and v˜ := (0, 1, 1) ∈ X . It is easy to see that f (v) ˜ = λ˜ v, ˜ so λ˜ is an eigenvalue and v˜ is the corresponding eigenvector of f . (In fact, for any 1 ≤ c ≤ a, f (vc ) = λ˜ vc , where vc := (0, c, c) ∈ X.) On the other hand, one can confirm ker f = span{(0, 1, −1)} 6= {θ} and ker f ∩ X 6= ∅ easily. By Theorem 3.7, we also show that there exist λ ∈ R and v ∈ X with v 6= θ such that f (v) = λv. We also have the following existence theorem for the solution of problem [CQEIVP]-(II) which is different from Theorem 3.6. Theorem 3.8. Let I be any index set. For each i ∈ I , let (E i , k·kQ i ) be a normed linear space with origin θi and X i be a nonempty convex subset of E i with θi 6∈ X i . Q Let E = i∈I E i with origin θ = (θi )i∈I and the norm kxk = supi∈I kxi k, where x = (xi )i∈I ∈ E, and X = i∈I X i . For each i ∈ I , let f i : E → E i be an affine and continuous map satisfying f i (θi ) = θi . Define f : E → E by f (x) = ( f i (x))i∈I ,

x ∈ E.

Suppose that X ⊂ V C( f (X )) :=

[

R( f (x))

x∈X

and the coercivity condition (H1 ) in Theorem 3.4 holds. Then there exists v = (vi )i∈I ∈ X with vi 6= θi for each i ∈ I such that f (v) ∈ R(v); that is the problem [CQEIVP]-(II) has a solution. Proof. By Theorem 3.4, there exists v ∈ X such that d( f (x), R(v)) ≥ d( f (v), R(v)) for all x ∈ X . Since X ⊂ V C( f (X )), there exists u ∈ X such that v ∈ R( f (u)). Thus there exists γ ∈ K such that v = γ f (u). Since θ 6∈ X , we have v 6= θ, γ 6= 0 and f (u) 6= θ. Hence f (u) = γ −1 v ∈ R(v). It follows that d( f (v), R(v)) ≤ d( f (u), R(v)) = 0 and we have d( f (v), R(v)) = 0 or f (v) ∈ R(v). Therefore there exists λ ∈ K such that f i (v) = λvi for all i ∈ I and [CQEIVP]-(II) has a solution (λ, v).  The following existence theorem for the solution of problem [CQEIVP]-(IV) is established immediately from Theorem 3.8.

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Theorem 3.9. In Theorem 3.8, if we assume further that θi 6∈ f i (X ) for each i ∈ I , then the problem [CQEIVP]-(IV) has a solution (λ, v); that is, there exist a common eigenvalue λ ∈ K with λ 6= 0 and the corresponding common quasi-eigenvector v = (vi )i∈I ∈ X with vi 6= θi such that f i (v) = λvi for each i ∈ I . The following existence theorem for the solution of the eigenvector problem (s-EIVP) gives nonzero eigenvalues. Theorem 3.10 (Eigenvector theorem (II)). Let (E, k·k) be a normed linear space with origin θ, X be a nonempty convex subset of E with θ 6∈ X and f : E → E be an affine and continuous map satisfying f (θ) = θ , θ 6∈ f (X ) and X ⊂ V C( f (X )). Suppose the coercivity condition (H1 ) in Theorem 3.4 holds. Then there exists v ∈ X with v 6= θ and λ ∈ K with λ 6= 0 such that f (v) = λv. 4. The existence theorems for solutions of problems [CQEIVP]-(III), [CQEIVP]-(IV) and (s-EIVP) In this section, we will establish the existence theorems for solutions of problems [CQEIVP]-(III), [CQEIVP]-(IV) and (s-EIVP). Theorem 4.1. Let (E, k·k) be a normed linear space with origin θ, X be a nonempty convex subset of E and f : E → E be a continuous map satisfying θ 6∈ f (X ). Suppose that (H2 ) there exist a nonempty compact subset W of X and a nonempty compact convex subset M of X such that, for each y ∈ X \ W , there exists z ∈ M such that d(z, R( f (y))) < d(y, R( f (y))). Then there exists v ∈ X such that d(x, R( f (v))) ≥ d(v, R( f (v))) for all x ∈ X . Proof. Let h : X × X → R be defined by h(x, y) = d(y, R( f (x))) − d(x, R( f (x))) and D : X ( X be defined by D(x) = {y ∈ X : h(x, y) < 0}. Then, by Theorem 3.3(b), D(x) is convex for each x ∈ X . It is obvious that x 6∈ co D(x) for each x ∈ X . By Theorem 3.3(d), D − (y) = {x ∈ X : h(x, y) < 0} is open in X for each y ∈ X . By our coercivity condition (H2 ), there exist a nonempty compact subset W of X and a nonempty compact convex subset M of X such that for each y ∈ X \ W , M ∩ D(y) 6= ∅. Applying Theorem 3.1, there exists v ∈ X such that D(v) = ∅. That is, d(x, R( f (v))) ≥ d(v, R( f (v))) for all x ∈ X .  Remark 4.1. Theorem 4.1 is comparable to [4, Theorem 5] in the following aspects: (a) In [4, Theorem 5], Li and Park only discuss the case K = R. (b) The coercivity condition (H2 ) in Theorem 4.1 and the coercivity condition (∗) in [4, Theorem 5] are different. The following existence theorem for solutions of problems [CQEIVP]-(III) can be established by using Theorem 4.1. Theorem 4.2. Let I be any index set. For each i ∈QI , let (E i , k·ki ) be a normed linear space with origin θi and X i be a nonempty open convex subsetQ of E i . Let E = i∈I E i with origin θ = (θi )i∈I and the norm kxk = supi∈I kxi k, where x = (xi )i∈I ∈ E, and X = i∈I X i . For each i ∈ I , let f i : E → E i be a continuous map. Define f : E → E by f (x) = ( f i (x))i∈I ,

x ∈ E.

Suppose that θ 6∈ f (X ) ⊂ V C(X ) :=

[

R(x)

x∈X

and the coercivity condition (H2 ) in Theorem 4.1 holds. Then there exists v = (vi )i∈I ∈ X such that v ∈ R( f (v)); that is, the problem [CQEIVP]-(III) has a solution.

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471

Proof. Clearly, the map f is continuous. By Theorem 4.1, there exists v ∈ X such that d(x, R( f (v))) ≥ d(v, R ( f (v))) for all x ∈ X . Since f (X ) ⊂ V C(X ), there exist u ∈ X and γ ∈ K such that f (v) = γ u. Since θ 6∈ f (X ), we have f (v) 6= θ, γ 6= 0 and u 6= θ. Hence u = γ −1 f (v) ∈ R( f (v)) and we deduce that d(v, R( f (v))) ≤ d(u, R( f (v))) = 0. So v ∈ R( f (v)) and hence there exists λ ∈ K such that vi = λ f i (v) for all i ∈ I . This means that (λ, v) is a solution of [CQEIVP]-(III).  The following existence theorem for the solution of problem [CQEIVP]-(IV) is established immediately from Theorem 4.2. Theorem 4.3. In Theorem 4.2, if we assume further that θi 6∈ X i for each i ∈ I , then the problem [CQEIVP]-(IV) has a solution (λ, v); that is, there exist a common eigenvalue λ ∈ K with λ 6= 0 and the corresponding common quasieigenvector v = (vi )i∈I ∈ X with vi 6= θi such that f i (v) = λvi for each i ∈ I . Proof. By Theorem 4.2, there exists v ∈ X and ρ ∈ K such that vi = ρ f i (v) for all i ∈ I . Since θi 6∈ X i , we have ρ 6= 0. Let λ := ρ −1 . Then f i (v) = λvi for each i ∈ I which means that (λ, v) is a solution of [CQEIVP]-(IV).  Remark 4.2. Note that some assumptions for Theorems 3.9 and 4.3 are different. Finally, from Theorem 4.3, we also establish an existence theorem for the solution of eigenvector problem (s-EIVP). Theorem 4.4 (Eigenvector Theorem (III)). Let (E, k·k) be a normed linear space with origin θ, X be a nonempty convex subset of E with θ 6∈ X and f : E → E be a continuous map satisfying θ 6∈ f (X ) ⊂ V C(X ). Suppose the coercivity condition (H2 ) in Theorem 4.1 holds. Then there exists v ∈ X with v 6= θ and λ ∈ K with λ 6= 0 such that f (v) = λv. Acknowledgment This research was supported by the National Science Council of the Republic of China. References [1] P. Deguire, K.K. Tan, G.X.Z. Yuan, The study of maximal elements, fixed point for Ls-majorized mappings and the quasi-variational inequalities in product spaces, Nonlinear Anal. 37 (1999) 933–951. [2] Ky Fan, A generalization of Tychonoff’s fixed point theorem, Math. Ann. 142 (1961) 305–310. [3] J. Li, Some eigenvector theorems proved by a Fan–KKM theorem, J. Math. Anal. Appl. 263 (2001) 738–747. [4] J. Li, S. Park, On solutions of generalized complementarity and eigenvector problems, Nonlinear Anal. 65 (2006) 12–24. [5] L.J. Lin, Q.H. Ansari, Collective fixed points and maximal elements with applications to abstract economies, J. Math. Anal. Appl. 296 (2004) 455–472. [6] L.J. Lin, W.S. Du, Systems of equilibrium problems with applications to new variants of Ekeland’s variational principle, fixed point theorems and parametric optimization problems, J. Global Optim. (in press).