Báo cáo hóa học: " FIXED POINT THEOREMS IN GENERATING SPACES OF QUASI-NORM FAMILY AND APPLICATIONS " pptx

FIXED POINT THEOREMS IN GENERATING SPACES OF QUASI-NORM FAMILY AND APPLICATIONS JIAN-ZHONG XIAO AND XING-HUA ZHU Received 11 April 2005; Accepted 21 November 2005 Some new concepts of generating spaces of quasi-norm family are introduced and their linear topological structures are studied. T hese spaces are not necessarily locally convex. By virtue of some properties in these spaces, several Schauder-type fixed point theorems are proved, which include the corresponding theorems in locally convex spaces as their special cases. As applications, some new fixed point theorems in Menger probabilistic normed spaces and fuzzy normed spaces are obtained. Copyright © 2006 J Z. Xiao and X H. Zhu. This is an open access article distributed un- der the Creative Commons Attribution License, which permits unrestricted use, distribu- tion, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The Schauder fixed point theorem and its generalizations which were obtained by Kras- noselskii et al. (see [3, 5], we call them the Schauder-type fixed point theorems), play important role in nonlinear analysis. In classical case, many interesting extensions and important applications of these theorems were presented by Fan [1] and others. In non- classical case, several extensions of these theorems in Menger probabilistic normed spaces were given under some conditions by Zhang-Guo [11]andLin[6]. Naturally, a subject is to consider their unified extensions both in classical case and in nonclassical case. In this paper, we introduce some new concepts of generating spaces of quasi-norm family, and establish some new unified versions of Schauder-type fixed point theorems in more general setting. As applications, we also study t he existence problems concerning the fixed points for operators on Menger probabilistic normed space and fuzzy normed space. Our results contain not only the former versions of the Schauder-type fixed point theorems but also the corresponding theorems in Menger probabilistic normed spaces and fuzzy normed spaces as their special cases. 2. Fixed point theorems in generating spaces of quasi-norm family Throughout this paper we denote the set of all positive integers by Z + and the field of real or complex numbers by E. Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 61623, Pages 1–10 DOI 10.1155/FPTA/2006/61623 2 Fixed point theorems in generating spaces Definit ion 2.1. Let X be a linear space over E and θ the origin of X.Let Q =  |·| α : α ∈ (0,1]  (2.1) be a family of mappings from X into [0,+ ∞). (X,Q) is called a generating space of quasi- norm family and Q a quasi-norm family if the following conditions are satisfied: (QN-1) |x| α = 0forallα ∈ (0,1] if and only if x = θ; (QN-2) |ex| α =|e||x| α for x ∈ X and e ∈ E; (QN-3) for any α ∈ (0, 1] there exists a β ∈ (0,α]suchthat |x + y| α ≤|x| β + |y| β for x, y ∈ X; (2.2) (QN-4) for any x ∈ X, |x| α is non-increasing and left-continuous for α ∈ (0,1]. (X,Q) is called a generating space of sub-strong quasi-norm family, strong quasi-norm family and semi-norm family respectively, if (QN-3) is st rengthened to (QN-3u), (QN- 3t) and (QN-3e), where (QN-3u) for any α ∈ (0, 1] there exists a β ∈ (0,α]suchthat      n  i=1 x i      α ≤ n  i=1   x i   β for any n ∈ Z + , x i ∈ X (i = 1, 2, ,n); (2.3) (QN-3t) for any α ∈ (0, 1] there exists a β ∈ (0,α]suchthat |x + y| α ≤|x| α + |y| β for x, y ∈ X; (2.4) (QN-3e) for any α ∈ (0, 1], it holds that |x + y| α ≤|x| α + |y| α for x, y ∈ X. Remark 2.2. Clearly, by Definition 2.1 we obtain the following assertions: (QN-3e) im- plies (QN-3t); (QN-3t) and (QN-4) imply (QN-3u); (QN-3u) implies (QN-3). Lemma 2.3. Let (X, Q) be a generating space of quasi-norm family, ε>0, α ∈ (0, 1], N(ε,α) ={x : |x| α <ε}. Then (i) e = 0 implies eN(1, α) = N(|e|,α); (ii) ε 1 ≤ ε 2 implies N(ε 1 ,α) ⊂ N(ε 2 ,α); (iii) α 1 ≤ α 2 implies N(ε,α 1 ) ⊂ N(ε, α 2 ). Proof. It follows immediately from (QN-2) and (QN-4).  Lemma 2.4. Let (X,Q) be a generating space of quasi-norm family. Then there exists a topology ᐀ Q on X such that (X,᐀ Q ) is a first-countable Hausdorff linear topological space (further, is metrizable) having {N(ε,α):ε>0, α ∈ (0, 1]} as a neighbourhood base of θ. Additionally, if (X,Q) is a generating space of semi-norm family, then (X,᐀ Q ) is a locally convex space. Sketch of proof. Applying Lemma 2.3, we have the following. (a) For N(ε 1 ,α 1 )andN(ε 2 ,α 2 ) there is a N(ε 0 ,α 0 )suchthat N  ε 0 ,α 0  ⊂ N  ε 1 ,α 1  ∩ N  ε 2 ,α 2  , (2.5) J Z. Xiao and X H. Zhu 3 where ε 0 = α 0 = min  ε 1 ,α 1 ,ε 2 ,α 2  . (2.6) (b) For N(ε,α), by (QN-3) and (QN-4), there is an N(ε/2,β)suchthat N(ε/2,β)+N(ε/2,β) ⊂ N(ε,α). (2.7) (c) For any e ∈ E, |e|≤1, it holds that eN(ε,α) ⊂ N(ε,α). (d) For any x ∈ X, there is a e = ε/(|x| α + ε) ∈ E such that ex ∈ N(ε,α). (e) For θ = x ∈ X, by (QN-1), there exist α 0 ∈ (0, 1] and ε 0 > 0suchthat |x| α 0 >ε 0 , that is, x/∈ N  ε 0 ,α 0  . (2.8) (f) {N(1/n,1/n):n ∈ Z + } is also a neighbourhood base of θ for ᐀ Q . (g) If (X,Q) satisfies (QN-3e), then N(ε,α)isconvex. Finally, by [4, pages 34-35, pages 45–49], the assertion is valid.  Remark 2.5. From Lemma 2.4 we see that the topology ᐀ Q can be described using se- quence instead of net or filter. Definit ion 2.6. Let (X,Q) be a generating space of quasi-norm family. (i) A sequence {x n } ∞ n=1 ⊂ X is said (a) to converge to x ∈ X denoted by lim n→∞ x n = x if lim n→∞ |x n − x| α = 0foreach α ∈ (0,1] (equivalently, for each α ∈ (0,1] there is a K ∈ Z + such that |x n − x| α <αfor all n ≥ K); (b) to be a Cauchy sequence if lim m,n→∞ |x m − x n | α = 0foreachα ∈ (0,1]. (ii) A subset B ⊂ X is said (a) to be complete if every Cauchy sequence in B converges in B; (b) to be bounded if for each α ∈ (0,1] there is a M = M(α) > 0suchthatB ⊂ N(M,α); (c) to be precompact (or totally bounded) if for each α ∈ (0, 1] there exist n α ∈ Z + and {x 1α ,x 2α , ,x n α α }⊂B such that B ⊂  n α i=1 x iα + N(α,α); (d) to be compact if every open cover of B has a finite subcover. (iii) An operator T from B ⊂ X into X is said to be continuous if for each x ∈ B, lim n→∞ x n = x implies lim n→∞ Tx n = Tx. Remark 2.7. From Definition 2.6 and Lemma 2.4 we get the following immediately: if B is compact, then it is precompact; If B is precompact, then it is bounded; If B is a subset of a precompact set, then it is also precompact. Lemma 2.8. Let (X,Q) be a generating space of quasi-norm family. (i) If Y ⊂ X is a finite dimensional subspace of X, then Y is topologically isomorphic to a finite dimensional Euclidean space and is therefore complete and closed in X. (ii) AsubsetofX is compact if and only if it is precompact and complete. Proof. It follows from Lemma 2.4 and [4, pages 59–61].  4 Fixed point theorems in generating spaces Lemma 2.9. Let (X,Q) be a generating space of strong quasi-norm family. Then for each α ∈ (0, 1], |x| α is a continuous function on X. Proof. By (QN-3t), for {x n }⊂X and x ∈ X,wehave |x| α ≤   x − x n   β +   x n   α ,   x n   α ≤   x n − x   β + |x| α , (2.9) that is, ||x n | α −|x| α |≤|x n − x| β , showing the assertion is true.  In the sequel, we denote the closure of a set B by B,theconvexhullofB by coB and the closure of the convex hull of B by coB.Now,wegiveourmaintheorems. Theorem 2.10. Let (X,Q) be a generating space of sub-strong quasi-norm family satisfying that each |·| α ∈ Q is continuous on X, C acompactconvexsubsetofX and T acontinuous operator from C into C. The n there exists an x 0 ∈ C such that Tx 0 = x 0 . Proof. For n ∈ Z + and α n ∈ (0, 1/n], by (QN-3u), there is β n ∈ (0, α n ]suchthat      k  i=1 x i      α n ≤ k  i=1   x i   β n ; ∀k ∈ Z + ,∀  x i  k i =1 ⊂ X. (2.10) Set α n+1 = min{β n ,1/(n +1)}. By (QN-4), we have that {α n } ∞ n=1 ⊂ (0,1] with α n+1 ≤ α n and α n ≤ 1/n such that      k  i=1 x i      α n ≤ k  i=1   x i   α n+1 ; ∀k ∈ Z + ,∀  x i  k i =1 ⊂ X. (2.11) Observe that a subset of a precompact set is also precompact. Since C is compact and TC ⊂ C, there exist p n ∈ Z + and {y in } p n i=1 ⊂ TC such that TC ⊂ p n  i=1 y in + N  α n+1 ,α n+1  . (2.12) Set g in (x) = max{0,α n+1 −|Tx− y in | α n+1 }, ∀x ∈ C, i = 1,2, , p n . Since the quasi-norms are continuous on X and T is continuous on C,wehavethatg in (x)iscontinuousonC.If x ∈ C,thenby(2.12), there exists i 0 (1 ≤ i 0 ≤ p n )suchthat|Tx− y i 0 n | α n+1 <α n+1 , that is, g i 0 n (x) > 0. Set g n (x) =  p n i=1 g in (x). Then for all x ∈ C, g n (x) > 0. Define T n x = p n  i=1 g in (x) g n (x) y in , x ∈ C. (2.13) Then T n is a continuous operator on C. Notice that g in (x) = 0ifandonlyif|Tx − y in | α n+1 <α n+1 .Foreachx ∈ C,by(2.11) and (QN-2) we have that   Tx− T n x   α n =      1 g n (x) p n  i=1 g in (x)  Tx− y in       α n ≤ 1 g n (x) p n  i=1 g in (x)   Tx− y in   α n+1 <α n+1 . (2.14) J Z. Xiao and X H. Zhu 5 Set C n = co{y in } p n i=1 , Y n = span{y in } p n i=1 .SinceC is compact and convex, by (2.13)wehave that T n C ⊂ C n and C n ⊂ C.Thus,T n C n ⊂ C n .SinceY n is a finite dimensional closed sub- space of X and the bounded convex closed set C n ⊂ Y n , by the Brouwer fixed point theo- rem and Lemma 2.8(i), there exists x n ∈ C n such that T n x n = x n .Since{x n } ∞ n=1 ⊂ C and C is compact, without loss of generality, we can suppose that lim n→∞ x n = x 0 ∈ C.Foreach α ∈ (0, 1], by (QN-3u) and (QN-4), there exists β ∈ (0,α/3] such that   x 0 − Tx 0   α ≤   x 0 − x n   β +   T n x n − Tx n   β +   Tx n − Tx 0   β . (2.15) Since T is continuous and lim n→∞ α n = 0, there exists a K ∈ Z + such that α n ≤ β, |x 0 − x n | β <βand |Tx n − Tx 0 | β <βfor all n ≥ K.By(2.14)wehavethat   T n x n − Tx n   β ≤   T n x n − Tx n   α n <α n+1 ≤ α n <β. (2.16) Hence (2.15) implies |x 0 − Tx 0 | α < 3β ≤ α, that is, Tx 0 = x 0 . This completes the proof.  As a direct consequence of Theorem 2.10, we can obtain the following by Lemma 2.9. Corollary 2.11. Let (X,Q) be a generating space of strong quasi-norm family, C acompact convex subset of X and T acontinuousoperatorfromC into C.Thenthereexistsanx 0 ∈ C such that Tx 0 = x 0 . Theorem 2.12. Let (X,Q) be a generating space of sub-strong quasi-norm family satisfying that each |·| α ∈ Q is continuous on X.LetC be a closed convex subset of X and T acontin- uous operator from C into C.IfX is complete and TC compact, then there exists an x 0 ∈ C such that Tx 0 = x 0 . Proof. Set B = TC.WewillprovethatcoB is compact. For each α ∈ (0,1], applying (QN- 3u) and (QN-4), there exists a β ∈ (0,α/3) such that |w 1 + w 2 + w 3 | α ≤|w 1 | β + |w 2 | β + |w 3 | β for all w 1 ,w 2 ,w 3 ∈ X.Thus, N(β,β)+N(β,β)+N(β,β) ⊂ N(α,α). (2.17) Applying (QN-3u) again, there exists a γ ∈ (0, β]suchthat      n  i=1 z i      β ≤ n  i=1   z i   γ ; ∀n ∈ Z + , ∀  z i  n i =1 ⊂ X. (2.18) Since TC is compact, we obtain that B is precompact. Thus, there exist an n γ ∈ Z + and {x iγ } n γ i=1 ⊂ B such that B ⊂ n γ  i=1 x iγ + N(γ,γ). (2.19) 6 Fixed point theorems in generating spaces Suppose that x ∈coB.Thenx =  k j =1 e j y j ,wherek ∈Z + , y j ∈ B and e j ≥0(j = 1,2, ,k),  k j =1 e j =1. By (2.19), there exists x j ∈{x iγ } n γ i=1 such that y j − x j ∈N(γ,γ), ( j = 1,2, ,k). By (2.18), we have that      x − k  j=1 e j x j      β ≤ k  j=1 e j   y j − x j   γ <γ≤ β. (2.20) Set C γ = co{x iγ } n γ i=1 , Y γ = span{x iγ } n γ i=1 .Since  k j =1 e j x j ∈ C γ ,from(2.20)wegetthat x ∈ C γ + N(β,β), that is, coB ⊂ C γ + N(β,β). (2.21) Since Y γ is a finite dimensional space and C γ a bounded set of Y γ ,wederivethatC γ is compact. Thus, there exist k γ ∈ Z + and {z iγ } k γ i=1 ⊂ C γ ⊂ coB such that C γ ⊂ C γ ⊂  k γ i=1 z iγ + N(β,β). Hence, by (2.17)and(2.21)wehavethat coB ⊂ coB + N(β,β) ⊂ C γ + N(β,β)+N(β,β) ⊂ k γ  i=1 z iγ + N(β,β)+N(β,β)+N(β,β) ⊂ k γ  i=1 z iγ + N(α,α), (2.22) showing coB is precompact. Since X is complete, we obtain that coB is complete. Ap- plying Lemma 2.8(ii), coB is compact. Since B ⊂ C and C isaclosedconvexset,wede- rive that coB ⊂ C.Clearly,T(coB) ⊂ TC = B ⊂ coB. Therefore, T is a continuous oper- ator from the convex compact set coB into itself. Applying Theorem 2.10, there exists x 0 ∈ coB ⊂ C such that Tx 0 = x 0 . This completes the proof.  As a direct consequence of Theorem 2.12, we can obtain the following by Lemma 2.9. Corollary 2.13. Let (X,Q) be a generating space of strong quasi-norm family. Let C be a closed convex subset of X and T acontinuousoperatorfromC into C.IfX is complete and TC compact, then there ex ists an x 0 ∈ C such that Tx 0 = x 0 . Theorem 2.14. Let (X, Q) be a generating space of strong quasi-norm family and C aclosed convex subset of X.LetT 1 and T 2 be continuous operators from C into C sat isfying the following conditions: (i) T 1 is contractive, that is, there exists a constant r ∈ [0,1) such that   T 1 x − T 1 y   α ≤ r|x − y| α ∀α ∈ (0,1], x, y ∈ C. (2.23) (ii) T 2 C is compact. (iii) For any x, y ∈ C it holds that T 1 x + T 2 y ∈ C. If X is complete, then there exists an x 0 ∈ C such that x 0 = (T 1 + T 2 )x 0 = T 1 x 0 + T 2 x 0 . Proof. Suppose that z ∈ T 2 C.Define T z : T z x = T 1 x + z, ∀x ∈ C. (2.24) J Z. Xiao and X H. Zhu 7 Then T z x − T z y = T 1 x − T 1 y.SinceC is closed, by (i) and (iii) we have that T z is a contractive operator from C into C.Sety n+1 = T z y n ,wheren ∈ Z + and y 0 ∈ C.Then {y n } ∞ n=0 ⊂ C. Since (QN-3t) and (QN-4) imply (QN-3u), for each α ∈ (0,1], by (i) and (QN-3u), there exists a β ∈ (0, α]suchthat   y n+p − y n   α =      p  i=1  y n+i − y n+i−1       α ≤ p  i=1   y n+i − y n+i−1   β ≤ p  i=1 r n+i−1   y 1 − y 0   β ≤ r n 1 − r   y 1 − y 0   β , (2.25) showing that {y n } is a Cauchy sequence in C.SinceX is complete and C closed, lim n→∞ y n = y ∈ C.ObservethatT is continuous. From y n+1 = T z y n we derive that T z y = y. By (i) we see that y is a unique fixed point of T z .DefineS : Sz = y.ThenS is an operator from T 2 C into C.By(2.24)wehavethat Sz = T 1 Sz + z. (2.26) By (i) and (QN-3t), for each α ∈ (0, 1] there is β ∈ (0,α]suchthat   Sz 1 − Sz 2   α =    T 1 Sz 1 − T 1 Sz 2  +  z 1 − z 2    α ≤    T 1 Sz 1 − T 1 Sz 2    α +   z 1 − z 2   β ≤ r   Sz 1 − Sz 2   α +   z 1 − z 2   β (2.27) for all z 1 ,z 2 ∈ T 2 C, that is, |Sz 1 − Sz 2 | α ≤ (1/(1 − r))|z 1 − z 2 | β , showing that S is contin- uous. Since ST 2 is a continuous operator from C into C, S(T 2 C) is compact by (ii), and ST 2 C ⊂ S(T 2 C), we derive that ST 2 C is compact. Observe t hat (QN-3t) and (QN-4) imply (QN-3u), and by Lemma 2.9, (QN-3t) implies the quasi-norms are continuous. Apply- ing Theorem 2.12, there exists an x 0 ∈ C such that ST 2 x 0 = x 0 .Settingz 0 = T 2 x 0 ,then Sz 0 = x 0 ,andby(2.26), Sz 0 = T 1 Sz 0 + z 0 . Therefore, x 0 = T 1 x 0 + T 2 x 0 . This completes the proof.  Remark 2.15. From Lemmas 3.5 and 2.9 we see that, if (X,Q) is a generating space of semi-norm family, then it is a locally convex Hausdorff linear topological space and its semi-norms are continuous. Noticing that (QN-3e) implies (QN-3u) and (QN-3t), The- orems 2.10, 2.12 (Corollaries 2.11, 2.13)andTheorem 2.14 areinsomesensethegen- eralizations of fixed point theorems (in locally convex space) of Schauder-Tychonoff, Schauder-Hukanare and Schauder-Krasnoselskii, respectively. 3. Applications: fixed point theorems in probabilistic case and fuzzy case Throughout this section, We denote by L, R, Δ the mappings from [0,1] × [0,1] into [0,1] which are symmetric and nondecreasing for both arguments and satisfy L(0,0) = 0, R(1,1) = 1, Δ(a,1) = a and Δ(a,Δ(b,c)) = Δ(Δ(a,b),c), respectively. we denote by Ᏺ the set of all fuzzy real numbers (see [9]). If η ∈ Ᏺ and η(t) = 0fort<0, then η is called a non-negative fuzzy real number and by Ᏺ + we mean the set of all them. For η ∈ Ᏺ + and α ∈ (0,1], α-level set [η] α ={t : η(t) ≥ α} is a closed interval and we write by [η] α = [η − α ,η + α ]. We also denote by Ᏸ the set of all left-continuous distribution functions (see 8 Fixed point theorems in generating spaces [7, 10]); by 0 the fuzzy number which satisfies 0(t) = 1fort = 0and0(t) = 0fort = 0; by H the distribution functions which satisfies H(t) = 1fort>0andH(t) = 0fort ≤ 0. Now we recall some basic concepts and facts about fuzzy normed space (briefly, FNS) and probabilistic normed space (briefly, PNS). Definit ion 3.1 (see [2, 9]). Let X be a real linear space and ·a mapping from X into Ᏺ + . Denote [x] α = [x − α ,x + α ]forx ∈ X and α ∈ (0, 1]. The quadruple (X,·,L,R) is called an FNS and ·a fuzzy norm if the following conditions are satisfied: (FN-1) x=0ifandonlyifx = θ; (FN-2) ex=|e|x for all x ∈ X and e ∈ (−∞,+∞); (FN-3) for all x, y ∈ X, (i) x + y(s + t) ≥ L(x(s),y(t)) whenever s ≤x − 1 , t ≤y − 1 and s + t ≤x + y − 1 ; (ii) x + y(s + t) ≤ R(x(s),y(t)) whenever s ≥x − 1 , t ≥y − 1 and s + t ≥x + y − 1 . Remark 3.2. By [9, Theorems 3.1–3.4] we know that the topology of (X, ·,L, R)isde- cided by {x + α : α∈(0,1]} or {N + (ε,α):ε>0, α ∈ (0,1]},whereN + (ε,α)={x : x + α <ε}. Lemma 3.3 (see [9]). Let (X, ·,L,R) be an FNS. Suppose that (R-1) R ≤ max; (R-2) for each α ∈ (0, 1] there exists a β ∈ (0,α] such that R(β,γ) <αfor γ ∈ (0,α); (R-3) lim a→0 + R(a,a) = 0. Then (X, {x + α : α ∈ (0,1]}) is (i) a generating space of quasi-norm family if (X, ·,L,R) satisfies (R-3); (ii) a generating space of strong quasi-norm family if (X, ·,L,R) satisfies (R-2); (iii) a generating space of semi-norm family if (X, ·,L,R) satisfies (R-1). Definit ion 3.4 (see [6, 7, 10]). Let X be a real linear space and F a mapping from X into Ᏸ. Denote F(x)(t) = f x (t)forx ∈ X and t ∈ (−∞,+∞). The triple (X,F,Δ)iscalleda Menger PNS if the following conditions are satisfied: (PN-1) f x (0) = 0; f x (t) = H(t)ifandonlyifx = θ; (PN-2) f ex (t) = f x (t/|e|)forallx ∈ X and 0 = e ∈ (−∞,+∞); (PN-3) f x+y (s + t) ≥ Δ( f x (s), f y (t)) for all x, y ∈ X and s,t ≥ 0. Lemma 3.5 (see [8, 10]). Let (X,F,Δ) be a Menger PNS. For any ε>0 and α ∈ (0,1] we define N ∗ (ε,α) ={x : f x (ε) > 1 − α} and x α = inf{t ≥ 0: f x (t) > 1 − α}. Then (i) N ∗ (ε,α) ={x : x α <ε}; (ii) f x (t) ≥ f y (t) for all t ≥ 0 if and only if x α ≤y α for all α ∈ (0,1]. Lemma 3.6 (see [8, 11]). Let (X,F,Δ) be a Menger PNS. Suppose that (Δ-1) Δ = min; (Δ-2) for each α ∈ (0, 1) there exists a β ∈ [α,1) such that Δ(β,γ) >αfor γ ∈ (α,1); (Δ-3) sup a<1 Δ(a,a) = 1. Then (X, {x α : α ∈ (0,1]}) is (i) a generating space of quasi-norm family if (X,F,Δ) satisfies (Δ-3); (ii) a generating space of strong quasi-norm family if (X, F,Δ) satisfies (Δ-2); (iii) a generating space of semi-norm family if (X,F, Δ) satisfies (Δ-1). J Z. Xiao and X H. Zhu 9 Remark 3.7. From Lemmas 3.5 and 3.3(i) we see that if (X,F, Δ) satisfies (Δ-3), then the (ε,α)-topology on (X,F,Δ)inducedby {N ∗ (ε,α):ε>0, α ∈ (0,1]} coincides with the topology on (X, {x α : α ∈ (0,1]}). Next we make use of Theorems 2.10, 2.12, 2.14 and Lemmas 3.3, 3.5, 3.6 to give some Schauder-type fixed point theorems in FNS and Menger PNS. The proofs are omitted here for the sake of brevity. Theorem 3.8. Let (X, ·,L,R) be an FNS with (R-2), C acompactconvexsubsetofX and T acontinuousoperatorfromC into C. Then there exists an x 0 ∈ C such that Tx 0 = x 0 . Theorem 3.9. Let (X, ·,L,R) be an FNS with (R-2), C aclosedconvexsubsetofX and T acontinuousoperatorfromC into C.IfX is complete and TC compact, then there exists an x 0 ∈ C such that Tx 0 = x 0 . Theorem 3.10. Let (X, ·,L,R) be an FNS with (R-2), and C a closed convex subset of X. Let T 1 and T 2 be continuous operators from C into C satisfying the following conditions: (i) T 1 is contractive, that is, there exists a constant r ∈ [0,1) such that   T 1 x − T 1 y   + α ≤ rx − y + α ∀α ∈ (0,1], x, y ∈ X. (3.1) (ii) T 2 C is compact. (iii) For any x, y ∈ C it holds that T 1 x + T 2 y ∈ C. If X is complete, then there exists an x 0 ∈ C such that x 0 = (T 1 + T 2 )x 0 = T 1 x 0 + T 2 x 0 . Theorem 3.11. Let (X,F,Δ) be a Menger PNS with (Δ-2), C acompactconvexsubsetofX and T acontinuousoperatorfromC into C. Then there exists an x 0 ∈ C such that Tx 0 = x 0 . Theorem 3.12. Let (X,F,Δ) be a Menger PNS with (Δ-2), C a closed convex subset of X and T acontinuousoperatorfromC into C.IfX is complete and TC is compact, then there exists an x 0 ∈ C such that Tx 0 = x 0 . Theorem 3.13. Let (X,F,Δ) be a Menger PNS with (Δ-2), and C a closed convex subset of X.LetT 1 and T 2 be continuous operators from C into C satis fying the following conditions: (i) T 1 is contractive, that is, there exists a constant r ∈ (0,1) such that centerline f T 1 x−T 1 y (t) ≥ f x−y (t/r) for all t ≥ 0; (ii) T 2 C is compact; (iii) for any x, y ∈ C,itholdsthatT 1 x + T 2 y ∈ C. If X is complete, then there exists an x 0 ∈ C such that x 0 = (T 1 + T 2 )x 0 = T 1 x 0 + T 2 x 0 . Remark 3.14. Since Δ(a,a) ≥ a for all a ∈ [0, 1] is equivalent to Δ = min, and T is non- expansive implies T is continuous, Theorem 3.12 presents an improved version of [11, Theorem 3.2], moreover, Theorems 3.11 and 3.13 present a complementary version of [6, Theorems 2 and 4]. 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Jian-Zhong Xiao: Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing 210044, China E-mail address: xiaojz@nuist.edu.cn Xing-Hua Zhu: Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing 210044, China E-mail address: zhuxh@nuist.edu.cn . FIXED POINT THEOREMS IN GENERATING SPACES OF QUASI-NORM FAMILY AND APPLICATIONS JIAN-ZHONG XIAO AND XING-HUA ZHU Received 11 April 2005; Accepted 21 November 2005 Some new concepts of generating. case and in nonclassical case. In this paper, we introduce some new concepts of generating spaces of quasi-norm family, and establish some new unified versions of Schauder-type fixed point theorems. cases. 2. Fixed point theorems in generating spaces of quasi-norm family Throughout this paper we denote the set of all positive integers by Z + and the field of real or complex numbers by E. Hindawi