Some Applications of Fixed Point Theorem in Economics and Nonlinear Functional Analysis

Transcription

1 International Mathematical Forum, 5, 2010, no. 49, Some Applications of Fixed Point Theorem in Economics and Nonlinear Functional Analysis S. A. R. Hosseiniun Facualty of Mathematical Sciences Shahid Beheshti University, Tehran, Iran Mona Nabiei Facualty of Mathematical Sciences Shahid Beheshti University, Tehran, Iran mona Abstract In this paper we obtain some applications of fixed point theorem in hyperconvex metric spaces in economics and use it to solve some functional equations by reducing the set of convexhull to the set of all convex combinations. Mathematics Subject Classification: 47A15, 46A32, 47D20 Keywords: Hyperconvex space, Ky Fan theorem, KKM principle, Fixed point, A-compact set, Multifunction, Convex combination 1 Introduction The concept of convexity is based on some algebric operations such as summation and scalar multiplication. In hyperconvex metric spaces, instead of convexity, we define convex combinations with absolute retraction [8]. So if we have a metric space without this operations we can t discuss its convexity. In [5], Ky Fan generalized the classical Knaster-Kuratowski-Mazurkiewicz theorem [4] (simply KKM theorem), and obtained a matching theorem [5], for open covering of convex sets. Later in [10], Park obtained a Ky-Fan type matching theorem for a hyperconvex space. In this paper, we obtain some applications of Ky Fan type matching theorem in economics and use it to solve some functional equations.

2 2408 S. A. R. Hosseiniun and M. Nabiei 2 Preliminary Notes A metric space (M,d) is called a hyperconvex space, if it satisfies the following conditions [8]. 1. If a collection of balls intersect pairwise, the intersection of all balls in the collection is not empty. 2. For any x, y M and α [0, 1], there exists z M such that d(y, z) = (1 α)d(x, y) and d(x, z) =αd(x, y). The spaces l and L are examples of hyperconvex spaces[8]. A mapping T : M M is called nonexpansive if d(tx,ty) d(x, y) for all x, y M. Let E M be a nonempty bounded subset of M. For each x M define r x (E) = sup{d(x, y) : y E} cov(e) = {B : B is a closed ball and E B} A(M) ={E M : E = cov(e)}. cov(e) is called the cover of E, and if E A(M) we say that E is an admissible subset of M [8]. Next theorem is stated in [8]. Proposition 2.1 Let (M,d) be a metric space. 1. There exists an index set I and a natural isometric embedding from M into l (I). 2. If for any metric space N, which containes isometrically M, there exists a nonexpansive retract r : N M (i.e. r is nonexpansive and r(x) =x for any x M) then M is a hyperconvex space. 3. M is hyperconvex iff for any subspace Y of a metric space X and any nonexpansive function f : Y M, there exists a nonexpansive extension f : X M. 4. Assume that M is a hyperconvex space, and E a nonempty bounded subset of M, then (a) cov(e) = {B(x, r x (E)) : x M}; (b) E A(M) iff E is an intersection of balls.

3 Fixed point theorem 2409 The proof of the first part of above proposition is used to define convex combinations in a metric space. We have to show that the natural isometric embedding of M into l (I) exists. First set I = M and define i : M l (I) by i(x) ={d(x, y) d(x 0,y)} y I, where x 0 is a fixed element of M. It is easy to see that i(x) i(y) = sup{ d(x, z) d(x, y) : z M} = d(x, y) for every x, y M, which implies(1). Now, if M is a hyperconvex space, by (2) of proposition 2.1. there exists a nonexpansive retract r : l (I) M. For x 1,..., x n M, and α 1,..., α n, nonnegative real numbers such that α j = 1, we define convex combination of x 1,..., x n as follows α j x j = r( α j i(x j )). This definition is a relation between algebric operator and topological structure of this space, so it is similar to convex combination in a topological vector space of elements x 1,..., x n. For example, we have d( α j x j, α j y j ) α j d(x j,y j ) for any x 1,..., x n and y 1,..., y n in M [8]. Every convex combination of elements in a bounded subset E of M belongs to cov(e), so cov(e) behaves just like convexhull of a subset in a topological vector space. If X and Y are two sets and X Y their Cartesian product, let E X Y. We define E x = {y : (x, y) E}, x X, E y = {x : (x, y) E}, y Y. We call E x and E y, the x-section and y-section of E respectively. Note that E x Y and E y X. IfX is a metric space, for each A X and ε>0, B(A, ε) = B(x, ε). x A Let D and X be two sets, a multifunction F : D 2 X is a function from D into the power set of X. Let

4 2410 S. A. R. Hosseiniun and M. Nabiei F (D) = {Fx : x D} F y = {x D : y Fx} y X Graph(F )={(x, y) D X : y Fx} Let M be a hyperconvex space and D M, a multifunction F : D 2 M is called KKM,if n cov{x 1,..., x n } Fx j for each finite subset {x 1,..., x n } D [10]. Let e j be the jth unit in R n+1, denote the standard n-simplex by Δ n, that is Δ n = {u R n+1 : u = n+1 j=1 j=1 λ j (u)e j, λ j (u) 0, n+1 j=1 λ j (u) =1} We say that L is an A compact subset of M, if for each finite subset S M, there exists a compact admissible set K LS M, such that L S K LS. For example every line segment in R is an A-compact subset of R. Let C(X, Y ) be the class of all continuous maps from a topological space X into a topological space Y. The following theorem is a version of Ky Fan type matching theorem for open covers [10]. Theorem 2.2 Let M be a compact hyperconvex space, X M, Y a topological space and F : X 2 Y be an open valued multifunction satisfying F (X) =Y. Then, for each f C(M,Y ), there exist a nonempty finite subset A X and x 0 cov(a), such that f(x 0 ) Fa, for each a A. Remark 2.3 In this paper the symbols M and Y refer to hyperconvex metric spaces and topological spaces respectively. You can see the proof of next theorems in [6]. Theorem 2.4 Let X be a nonempty subset of M, and F : X 2 Y be an open valued multifunction. If f C(M,Y ), and for each finite subset A X, f(cov(a)) Fa then the collection {F x : a A x X} has nonempty intersection. Theorem 2.5 Let G : M 2 Y is a closed valued multifunction, and G(A) = Y for a finite subset A = {x 1,..., x n } of M. If F : M 2 Y is an admissible inverse image multifunction and Gx Fx for each x M, then for any f C(M,Y ), there exists x 0 M with f(x 0 ) Fx 0.

5 Fixed point theorem Fixed point theorem The following theorem is another version of theorem 2.4. and we use the dual of it to prove theorem 3.1. Theorem 3.1 Let α β and f,g : M Y R { } are two functions such that g(x, y) f(x, y), f 1 [β,+ ) y is admissible and g 1 [α, + ) x is a closed set for each x M and y Y. If there exists a finite subset A of M such that Y = a A g 1 [α, + ) a, then for any h C(M,Y ) there exists x 0 M with f(x 0,h(x 0 )) β. Proof. For each x M, define Fx = f 1 [β,+ ) x and Gx = g 1 [α, + ) x, then Y = G(A). For any y Y, F y is admissible, Gx is closed subset of Y and for each x M, Gx Fx. So using theorem 2.4., for any h C(M,Y ) there exists x 0 M such that h(x 0 ) Fx 0, that is f(x 0,h(x 0 )) β and the proof is completed. The dual form of theorem 3.1. in the case M = Y and f = g could be stated as follows. Corollary 3.2 Let β α, g : M M R { } is a function such that g 1 (,β] x is admissible and g 1 (,α] x is a closed set, for each x M. If there exists a finite subset A of M such that M = a A g 1 (,α] a, then for any h C(M,M) there exists x 0 M with g(x 0,h(x 0 )) β. Theorem 3.3 Every totally bounded space M has the fixed point property. Proof. In corollary 3.2, put g = d : M M R + {0}, then for any n N, M is coverd by a finite closed 1/n balls. But for any y M, d 1 (, 1/n] y is admissible and closed in M, so by corollary 3.2, for any f C(M,M), there exists x n M such that d(x n,f(x n )) 1/n. Since M is hyperconvex, there exists y M such that d(y, f(y)) = 0, and the proof is completed. 4 Application in economics and nonlinear functional equations In [9], Z.D.Mitrovic has used some results by S.Park [11] and drived a sufficient condition for existance of an equilibrium point in the economic model of supply and demand for finite dimensional topologicalvector space. In this section we use the results in section 4 and extend the main results of [9], to any hyperconvex space. Suppose H is a hyperconvex space, let l 1 (H) be the set of

6 2412 S. A. R. Hosseiniun and M. Nabiei function ϕ : H R such that x H ϕ(x) < and D 0 = {ϕ l 1 (H) : x H ϕ(x) =0}, D 1 = {ϕ l 1 (H) : rang(ϕ) R + {0}, ϕ(x) =1}. x H Since, R is complete, x H ϕ(x) < implies that x H ϕ(x) <, and so the above definitions are well-defined. Let D, S : l 1 + (H) l1 (H), where l 1 +(H) ={ϕ l 1 (H) : range(ϕ) R + {0}}. The function of supply and demand ξ : l 1 + (H) l1 (H) is defined by ξ(ϕ) = D(ϕ) S(ϕ) for each ϕ l 1 + (H). The economic interpretation of equilibrium point is to find the point ϕ 0 for which ξ(ϕ 0 ) = 0, i.e. the amounts of D and S at ϕ 0 are equal. For each ξ C(D 1,D 0 ) and x H, define ξ x : D 1 R by ξ x (ϕ) =ξ(ϕ)(x). Theorem 4.1 Let ξ C(D 1,D 0 ) and f C(H, D 1 ),suppose that for each finite subset A of H, and for each x cov(a), there exists a A such that ξ x (f(a)) 0. Then there exists ϕ 0 D 1 such that ξ(ϕ 0 )=0. Proof. For each x H, define the set F (x) D 1,by F (x) :={ϕ D 1 : ξ x (ϕ) 0} We shall prove that all conditions of theorem 2.4. are satisfied. Since ξ x is continuous and ξ(ϕ) D 0 for all ϕ D 1, F (x) is closed and nonempty subset of D 1, for each x H. And for every nonempty finite subset {x 1,..., x n } of H, we have n f(cov{x 1,..., x n } Fx j. So by theorem 4.4, x H F (x), i.e, there exists ϕ 0 D 1, such that ξ x (ϕ 0 ) 0 for each x H. But ξ(ϕ 0 ) D 0 so x H ξ x (ϕ 0 ) = 0 Therefore ξ x (ϕ 0 ) = 0 for each x H, and it shows that ξ(ϕ 0 )=0. In the rest of this section, again we use some results of section 4 to solve functional equations, which arise in modeling of numerous phenomena in sciences and engineering. We saw in section 4 that, for multifunction F from a subset X of a hyperconvex space to the power set of a topological space Y, and a continuous function f from X to Y, with some conditions, there exists x 0 in X, such that f(x 0 ) Fx 0. We know that for each compact set K, C(K) is a hyperconvex metric space[8]. Let for each g C(K), Fg be the set of all solutions of a functional j=1

7 Fixed point theorem 2413 equation depend to g (or differentional eq. or integral eq,), then depending to each continuous function f from C(K) to C(K), there exists a solution of this equation. Example 6.2. One important applications is in the initial value problem for a system of ordinary differential equations. This system may be writhen as f (t) =ψ[t, f(t)] t [0,a] for some a > 0 (1) f(0) = α, with ψ :[0, ) C n C n is a function and f taking values in a compact subset K of C n, and the solution with continuous first derivative is required. By integration (1) becomes f(t) =α + t 0 ψ[s, f(s)]ds which is a nonlinear Volterra integral equation. Set ϕ, h : C[0,a] C[0,a] with ϕ(u)(t) =α + t 0 u(s)ds and h(u)(t) =ψ(t, u(t)) for u C([0,a]) and t [0,a]. Now, if we define F (g) ={u C([0,a]) : ϕ(u) =g} for each g C([0,a]), it is enough to check the conditions of theorem 4.3 for existence of f C([0,a]), with h(f) F (f), (i.e ϕ(h(f)) = f). Hence f is a solution of equation(1). Example 6.3.The equation f(x) = Ω k(x, y, f(y))dy + g(x) where k, g are given real functions and Ω is a compact subset of R n is known as Urysohn integral equation. Special case of this equation which has been extensively studied is the Hammerstein integral equation f(x) = k(x, y, f(y))ψ[y, f(y)]dy (2) Ω The importance of these equations is due to the fact that they include the integral equations derived like the previous example as a particular cases. Set ϕ : C(Ω Ω) C(Ω), h : C(Ω) C(Ω Ω) with ϕ(u)(x) := u(x, y)dy + g(x) and h(f)(x, y) = k(x, y, f(y)), Ω for u C(Ω Ω), f C(Ω) and x, y Ω. If we define F (v) ={u C(Ω Ω) : ϕ(u) =v} for each v C(Ω), it is enough to check the conditions of theorem 2.4. for existence of f C(Ω), with h(f) F (f). In this case f is a solution of equation (2).

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