Scalar Asymptotic Contractivity and Fixed Points for Nonexpansive Mappings on Unbounded Sets George Isac Department of Mathematics Royal Military College of Canada, STN Forces Kingston, Ontario, Canada isac-g@rmc.ca Abstract Based on the notion of asymptotically contractive mapping due to Penot [16], we propose in this paper a new method for the study of existence of fixed points for nonexpansive mappings defined on unbounded sets. Key words: fixed-point theory, nonexpansive mappings, scalar asymptotically contractive mapping, scalar asymptotic derivability 1 Introduction The fixed-point theory is one of the most popular chapters considered in nonlinear functional analysis. Nonlinear functional analysis is an area of mathematics that has suddenly grown up over the past few decades, influenced by nonlinear problems posed in physics, mechanics, operations research, as well as in economics. In the fixed-point theory, an important chapter is the study of fixed points for nonexpansive mappings. Nonexpansive mappings are used in many practical problems. Many authors have studied the existence of fixed points for nonexpansive mappings in many papers as for example [1, 2, 4 6, 10 13, 16 21], among others. The nonexpansivity is related, in some sense, with the contractivity. For comparison of various definitions of contractive mapping, the reader is referred to the classic paper [17]. Generally, in many papers, the existence of fixed points for nonexpansive mappings have been considered with respect to bounded closed convex sets, or with respect to compact convex sets. In 1992, Luc [13] presented a fixed-point theorem for nonexpansive mappings with respect to unbounded sets using the notion of recessive compactness.
120 G. Isac Using the notion of asymptotically contractive mapping, Penot [16] generalized to unbounded sets some fixed-point theorems proved some time ago by Browder [1], Göhde [6], Kirk [10], and Luc [13]. Inspired by Penot s results, we present in this paper a new method for the study of existence of fixed points, for nonexpansive mappings, defined on unbounded sets. This method is based on the notion of scalar asymptotically contractive mapping. This method, which is somewhat related to the scalar asymptotic derivability [8, 9], seems to be an interesting method and it opens a new research direction in the study of existence of fixed points for nonexpansive mappings defined on a closed unbounded convex set. 2 Preliminaries We denote by (E, ) a Banach space and by (H,, ) a Hilbert space. Let C E be a nonempty unbounded closed convex set and h : C E be a mapping. We recall some known definitions. We say that h is nonexpansive if and only if, for any x, y C we have h(x) h(y) x y. The mapping h is said to be ρ-lipschitzian, if there exists a constant ρ>0 such that for any x, y C we have h(x) h(y) ρ x y. If0<ρ<1, then in this case we say that h is a contractive mapping. We recall that a Banach space (E, )isuniformly convex, if and only if for every ɛ [0, 2[ there is a real number δ(ɛ) ]0, 1] such that whenever x r, y r, x y ɛr, x, y E, r>0, then it follows that x + y 2 (1 δ(ɛ))r. Any Hilbert space is uniformly convex and any L p (Ω) space with 1 <p< and Ω a domain in R n is uniformly convex. For more details and results about uniformly convex Banach spaces, the reader is referred to [3, 21], and [22]. We say that a mapping h : C E is demi-closed on C if for any sequence {x n } n N C weakly convergent to an element x E and such that the sequence {h(x n )} n N is convergent in norm to an element y we have that x C and h(x ) = y. The demi-closedness is related to the notion of strongly continuous mapping [3, 22]. It is known that, if h is nonexpansive and E is uniformly convex, then I h is demi-closed. (We denoted by I the identity mapping.) For a proof of this result, see ([2], Theorem 8.4) and ([22], Proposition 10.9). It is remarked in [16] that the boundedness of C used in [2] and [22] is not necessary. We note that in some papers of Russian mathematicians, the demi-closed operator is called regular operator.
Scalar Asymptotic Contractivity and Fixed Points for NMU Sets 121 3 Scalar Asymptotically Contractive Mappings in Hilbert Spaces Let (H,, ) be a Hilbert space and C H be a nonempty unbounded closed convex set. Definition 1. We say that a mapping f : C H is scalar asymptotically contractive on C if and only if there exists an element x 0 C such that f(x) f(x 0 ),x x 0 x C, x x x 0 2 < 1. We have the following result. Theorem 1. Let (H,, ) be a Hilbert space, and let C H be an unbounded closed convex subset. Let f : C H be a mapping such that the following assumptions are satisfied: (i) f is nonexpansive, (ii) f(c) C, (iii) f is scalar asymptotically contractive on C. Then f has a fixed point in C. Proof. Let x 0 C be the element defined in assumption (iii) and let {α n } n N be a sequence in ]0, 1[ such that lim α n = 0. For any n N, we consider n the mapping f n : C H defined by f n (x) =(1 α n )f(x)+α n x 0. Because C is a convex set, we have that f n (x) C for any x C. (We used also assumption (ii)). For any n N, the mapping f n is a contraction with rate (1 α n ) (because f is nonexpansive). Applying the Banach contractive principle, we obtain an element x n C such that f n (x n )=x n. The sequence {x n } n N is bounded. Indeed, if this is not the case, considering a subsequence (if necessary), we may assume that {x n } n N is convergent to, asn. Let β ]0, 1[ and ρ>0 such that f(x) f(x 0 ),x x 0 β x x 0 2,for x C satisfying x >ρ.forn N, large enough, we have x n 2 x n x 0 x n,x n x 0 = (1 α n )f(x n )+α n x 0,x n x 0 = (1 α n )f(x n ) (1 α n )f(x 0 )+(1 α n )f(x 0 )+α n x 0,x n x 0 =(1 α n ) f(x n ) f(x 0 ),x n x 0 +(1 α n ) f(x 0 ),x n x 0 + α n x 0,x n x 0, which implies x n 2 x n x 0 (1 α n )β x n x 0 2 +(1 α n ) f(x 0 ) x n x 0 + α n x 0 x n x 0 (1 α n )β( x n 2 +2 x n x 0 + x 0 2 ) +(1 α n ) f(x 0 ) ( x n + x 0 )+α n x 0 ( x n + x 0 ).
122 G. Isac Dividing both sides by x n 2 and taking limits, we obtain 1 β, which is a contradiction. Thus {x n } n N is bounded and we can show that {f(x n )} n N is also bounded (using the fact that f is nonexpansive). Now, because for any n N we have we deduce that x n =(1 α n )f(x n )+α n x 0, x n f(x n ) = α n x 0 f(x n ) 0, as n. The space H being reflexive and {x n } n N a bounded sequence, we may assume (eventually considering a subsequence) that {x n } n N is weakly convergent to an element x C, (we used also Eberlein s Theorem). Because H is uniformly convex and f is nonexpansive, we have that I f is demi-closed. Therefore, because x n f(x n ) 0asn, we deduce that f(x )=x and the proof is complete. Corollary 1. Let (H,, ) be a Hilbert space, K H aclosedconvexcone, and f : K K a nonexpansive mapping. If f is scalar asymptotically contractive on K, then f has a fixed-point in K. Remark 1. Corollary 1 is an existence theorem for fixed points on a closed convex cone. The theory of fixed point on convex cones has many applications. Corollary 2. Let (H,, ) a Hilbert space, K H aclosedconvexcone,and h : K K a k 0 -Lipschitzian mapping (k 0 > 0) such that h(0) 0. If there exists an element x 0 K such that h(x) h(x 0 ),x x 0 x K, x x x 0 2 <k x K with k>k 0, then k is an eigenvalue of h associated with an eigenvector in K. Proof. We apply Theorem 1 taking f = 1 k h. Remark 2. J.P. Penot introduced in [16] the following notion. Let (E, ) be a Banach space, and let C E be an unbounded set. We say that f : C E is asymptotically contractive on C if there exists x 0 C such that f(x) f(x 0 ) < 1. x C, x x x 0 Several examples of asymptotically contractive mappings are given in [16]. We remark that in the case of Hilbert spaces, any asymptotically contractive mapping is scalar asymptotically contractive but the converse is not true. The method presented above, on Hilbert spaces, to obtain the existence of fixed points for nonexpansive mappings on unbounded sets, can be extended on Banach spaces. In the next section, we present this extension.
Scalar Asymptotic Contractivity and Fixed Points for NMU Sets 123 4 G-Scalar Asymptotically Contractive Mappings in Banach Spaces Let (E, ) be a reflexive Banach space and C E be an unbounded closed convex set. Let B : E E R be a bilinear mapping satisfying the following properties: (b1) there exists b>0 such that B(x, y) b x y for any x, y E (b2) there exists a>0 such that a x 2 B(x, x), for any x E. If we denote by G = 1 a B and M = b a, then we have that G(x, y) M x y, for any x, y E and x 2 G(x, x) for any x E. The function G used in this section will be a such function. Definition 2. We say that a mapping f : C E is G-scalar asymptotically contractive on C if there exists x 0 C such that G(f(x) f(x 0 ),x x 0 ) x C, x x x 0 2 < 1. Remark 3. 1. If E is a Hilbert space and G is the inner product,, defined on E, then in this case by Definition 2 we obtain the notion of scalar asymptotically contractive mapping introduced by Definition 1. 2. If the mapping G used in Definition 2 satisfies the property (b1 ) G(x, y) x y for any x, y E, then, in this case any asymptotically contractive mapping f (in Penot s sense) is G-scalar asymptotically contractive mapping. The main result of this section is the following: Theorem 2. Let (E, ) be a reflexive Banach space and C E be an unbounded closed convex set. Let f : C E be a mapping such that the following assumptions are satisfied: (i) f is nonexpansive, (ii) f(c) C, (iii) I f is demi-closed, (iv) f is G-scalar asymptotically contractive on C. Then f has a fixed point in C. Proof. The proof follows the same ideas used in the proof of Theorem 1, but we have some particular details. Let {α n } n N be a sequence of ]0, 1[ such that lim α n =0andletx 0 C n be the element used in assumption (iv). For every n N, we consider the mapping f n : C E defined by f n (x) =(1 α n )f(x)+α n x 0 for any x C.
124 G. Isac Obviously, the convexity of C with (ii) implies that f n (x) C, for any x C. Because f is nonexpansive, we have that for any n N, f n is a contraction. Applying, for any n N, the Banach contraction principle, we obtain an element x n C such that f n (x n )=x n. The sequence {x n } n N C is bounded. Indeed, if this is not the case, considering a subsequence (if necessary) we may assume that { x n } n N is convergent to +. Using assumption (iv), we find β ]0, 1[ and ρ>0 such that G(f(x) f(x 0 ),x x 0 ) β x x 0 2 for x C satisfying ρ< x. We have, x n 2 M x n x 0 G(x n,x n x 0 )=G((1 α n )f(x n )+α n x 0,x n x 0 ) = G((1 α n )f(x n )+α n x 0 (1 α n )f(x 0 )+(1 α n )f(x 0 ),x n x 0 ) =(1 α n )G(f(x n ) f(x 0 ),x n x 0 )+(1 α n )G(f(x 0 ),x n x 0 ) + α n G(x 0,x n x 0 ) (1 α n )β x n x 0 2 +(1 α n )M f(x 0 ) x n x 0 + α n M x 0 x n x 0 (1 α n )β[ x n 2 +2 x n x 0 + x 0 2 ] +(1 α n )M f(x 0 ) [ x n + x 0 ]+α n M x 0 [ x n + x 0 ]. Dividing both sides by x n 2 and taking limits, we obtain 1 β, which is a contradiction. Thus {x n } n N is bounded, and because f is nonexpansive, we can show that {f(x n )} n N is also bounded. Taking into consideration that we deduce that x n =(1 α n )f(x n )+α n x 0, for any n N, x n f(x n ) = α n x 0 f(x n ) 0asn. Because the space E is reflexive and the sequence {x n } n N is bounded, we may assume (eventually considering a subsequence) that {x n } n N is weakly convergent to an element x C. By the fact that I f is supposed to be demi-closed, we obtain that f(x )=x, and the proof is complete. Considering Remark 3 (2) of this section, we deduce from Theorem 2 the following corollary. Corollary 3 ([16]). Let (E, ) be a uniformly convex Banach space and C E be an unbounded closed convex subset. Let G : E E P be a bilinear mapping satisfying properties (b1) and (b2) with a = b =1.Letf : C E be a mapping such that the following assumptions are satisfied: (i) f is nonexpansive (ii) f(c) C, (iii) f is asymptotically contractive in Penot s sense. Then f has a fixed point in C.
Scalar Asymptotic Contractivity and Fixed Points for NMU Sets 125 5(G, A)-Scalar Asymptotically Contractive Mappings in Banach Spaces In this section, we put in evidence some relations between scalar asymptotically contractive, scalar asymptotically derivable, and asymptotic derivability. Let (H,, ) be a Hilbert space and C H be an unbounded closed convex set. Let x 0 C be an element and f : C H be a mapping. We introduced in [7] the following notion. If C is a closed convex cone and T : H H is a continuous linear mapping, we say that T is a scalar asymptotic derivative of f along C if f(x) T (x),x x C, x x 2 0. We recall that T is an asymptotic derivative of f along C if f(x) T (x) 0. x C, x x If T is an asymptotic derivative of f along C, then T is a scalar asymptotic derivative. M.A. Krasnoselskii introduced the concept of asymptotic derivative, which is much used in nonlinear analysis. We can generalize the concept of scalar asymptotic derivative, considering T a general mapping, not necessarily linear, eventually being an element of a particular class of nonlinear mappings. We consider the following notations: U = C x 0, u = x x 0, where x C and g(u) =f(u + x 0 ). Obviously, 0 U, g(0) = f(x 0 )andforanyu U we have f(u + x 0 )=f(x). If then we have g(u) g(0),u u U, u u 2 0, f(x) f(x 0 ),x x 0 g(u) g(0),u x C, x x x 0 2 = u U, u u 2 0. If we consider g(0) as a scalar asymptotic derivative of g along U, then we have that f is scalar asymptotically contractive on C. This fact implies the following generalization of the notion of G-scalar asymptotically contractivity. To do this, we need to recall some notions and to introduce some conditions. Let (E, ) be a Banach space and let C E be an unbounded closed convex set. We recall that a semi-inner-product in Lumer s sense [Trans. Amer. Math. Soc. 100, 29 43 (1961)], is a mapping satisfying the following properties: (s1) [x + y, z] =[x, z]+[y, z], for any x, y, z E,
126 G. Isac (s2) [λx, y] =λ[x, y], for any λ R, (s3) [x, x] > 0 for any x E, x 0, (s4) [x, y] 2 [x, x][y, y], for any x, y E. It is known that, for any Banach space we can define a semi-inner-product. Also, it is known that the mapping x [x, x] 1/2 is a norm on E. If this norm coincides with the norm given on E, we say that the semi-inner-product is compatible with the norm. We say that a mapping A : C E is φ-asymptotically bounded on C if there exist r, c > 0 such that: (α 1 ) A(x) cφ( x ) for all x C with x >r, φ(t) (α 2 ) lim t + t =0. Now, suppose that a mapping G : E E P satisfies the following properties: (β 1 ) G(x 1 + x 2,y)=G(x 1,y)+G(x 2,y), for any x 1,x 2,y E, (β 2 ) G(λx, y) =λg(x, y), for any λ>0 and any x, y E, (β 3 ) x 2 G(x, x), for any x E, (β 4 ) G(x, y) M x y, for some M>0and any x, y E. Obviously, any semi-inner-product compatible with the norm satisfies the properties (β 1 ), (β 2 ), (β 3 ), and (β 4 ). Definition 3. We say that a mapping f : C E is a (G, A)-scalar asymptotically contractive mapping on C, if there exists a mapping G : E E P satisfying the properties (β 1 ) (β 4 )andaρ-contractive mapping A : C E such that G(f(x) A(x),x) x C, x x 2 < 1. We have the following result. Theorem 3. Let (E, ) be a reflexive Banach space and C E be an unbounded closed convex set. Let f : C E be a nonexpansive mapping. If the following assumptions are satisfied: (1) f(c) C, (2) I f is demi-closed, (3) f is (G, A)-scalar asymptotically contractive on C, (4) A is φ-asymptotically bounded and A(C) C, then f has a fixed point in C. Proof. First, we observe that f and A are bounded mappings, i.e., f(d) and A(D) are bounded sets, whenever D C is bounded. Let {λ n } n N be a sequence in ]0, 1[ such that lim λ n = 0. For every n N, we consider the x mapping f n : C E defined by f n (x) =(1 λ n )f(x)+λ n A(x).
Scalar Asymptotic Contractivity and Fixed Points for NMU Sets 127 Because C is convex and considering the assumptions (1) and (4), we have that f n (C) C. Using the properties of f and A, we can show that, for any n N, f n is a contractive mapping with the rate k n =(1 λ n )+λ n ρ ]ρ, 1[. Applying the Banach contraction principle, we obtain an element x n C such that f n (x) =x n. The sequence {x n } n N is a bounded sequence. Indeed, if this is not the case considering (if necessary) a subsequence, we may assume that {x n } n N as n. Because f is (G, A)-scalar asymptotically contractive, there exist β ]0, 1[ and ρ 0 > 0 such that G(f(x) A(x),x) β x 2 for x C satisfying x >ρ 0. For n N, large enough we have x n 2 G(x n,x n )=G((1 λ n )f(x n )+λ n A(x n ),x n ) = G((1 λ n )f(x n ) (1 λ n )A(x n )+A(x n ),x n ) =(1 λ n )G(f(x n ) A(x n ),x n )+G(A(x n ),x n ) (1 λ n )β x n 2 + cmφ( x n ) x n. Dividing both sides by x n 2 and taking limits we obtain 1 β, which is a contradiction. Thus {x n } n N is bounded and consequently {f(x n )} n N and {A(x n )} n N are bounded sequences. Because for any n N we have and we obtain x n =(1 λ n )f(x n )+λ n A(x n ), x n f(x n ) = λ n A(x n ) f(x n ) 0asn. The space E being reflexive and {x n } n N being a bounded sequence, we may assume (eventually considering a subsequence and Eberlein s Theorem) that {x n } n N is a weakly convergent sequence to an element x C. Because I f is demi-closed, we have that f(x )=x and the proof is complete. Remark 4. 1. If the space (E, ) is a uniformly convex Banach space, then in this case in Theorem 3 it is not necessary to suppose that I f is demi-closed. 2. If in Theorem 3, C is a closed convex cone, we have a fixed-point theorem on closed convex cones. The fixed-points theorem on cones have many applications. Corollary 4. Let (E, ) be a uniformly convex Banach space and C E be a closed convex cone. Let h : C E be a k 0 -Lipschitzian mapping. If the following assumptions are satisfied:
128 G. Isac (1) h(c) C and h(0) 0, (2) the mapping f = 1 k 0 h is (G, A)-scalar asymptotically contractive on C, (3) A is φ-asymptotically bounded and A(C) C. Then k 0 is a positive eigenvalue of h associated with an eigenvector in C. Proof. We apply Theorem 3 to the mapping f = 1 k 0 h. Corollary 5. Let (E, ) be a uniformly convex Banach space and C E be a closed convex cone. Let h : C E be a k 0 -Lipschitzian mapping and ψ : C E be a ρ 0 -Lipschitzian mapping. If the following assumptions are satisfied: (1) h(c) C and h(0) 0and ψ(c) C, (2) ρ 0 <k 0, (3) ψ is φ-asymptotically bounded, G(h(x) ψ(x),x) (4) x k 2 0, x C, x then any k k 0 is an eigenvalue of h associated with an eigenvector in C. Proof. For any k we apply Theorem 3 taking f = 1 k h and A = 1 k ψ. Now, we can generalize the notion of scalar asymptotic derivative. Definition 4. We say that a mapping A : C E is a G-scalar asymptotic derivative of the mapping f : C E along C if G(f(x) A(x),x) x C, x x 2 0. Remark 5. If A is a ρ-contraction and a G-scalar asymptotic derivative for f along C, then f is a (G, A)-scalar asymptotically contractive mapping. From, Theorem 3, we deduce the following result. Corollary 6. Let (E, ) be a reflexive Banach space and C E be an unbounded closed convex set. Let f : C E be a nonexpansive mapping. If the following assumptions are satisfied: (1) f(c) C, (2) I f is demi-closed, (3) f has a G-scalar asymptotic derivative A : C E such that A is ρ-contractive, φ-asymptotically bounded and A(C) C, then f has a fixed point in C.
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