The antipodal mapping theorem and difference equations in Banach spaces
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1 Journal of Difference Equations and Applications Vol., No.,, 1 13 The antipodal mapping theorem and difference equations in Banach spaces Daniel Franco, Donal O Regan and Juan Peran * Departamento de Matemática Aplicada, Universidad Nacional de Educación a Distancia, Apartado 60149, Madrid 8080, Spain. Department of Mathematics. National University of Ireland. Galway. Ireland. () We employ the Borsuk-Krasnoselskii antipodal theorem to prove a new fixed point theorem in ordered Banach spaces. Then, the applicability of the result is shown by presenting sufficient conditions for the existence of solutions to initial value problems for first-order difference equations in Banach spaces. To prove that result we shall employ set valued analysis techniques. Keywords: difference equations; antipodal theorem; set valued analysis AMS Subject Classification (000): 39A05; 47H10 1 Introduction Let E be a Banach space and f a compact map from B = {x E : x 1} to E. Theorem 1.1 Borsuk-Krasnoselskii antipodal theorem. Suppose that f has no fixed points on B and that the antipodal condition f(x) x f(x) x f( x) + x f( x) + x, x B is satisfied. Then f has a fixed point in B. The above result has been taken from section 16.3 in [] and it was first This research has been supported in part by Ministerio de Ciencia y Tecnología (Spain), project MTM C03-03 * Corresponding author. jperan@ind.uned.es
2 Franco, O Regan and Peran proved by Borsuk in 1933 in the finite dimensional case [7] and this generalization to the infinite dimensional setting is due to Krasnoselskii [16]. There is a vast literature on applications of the Borsuk-Krasnoselskii antipodal theorem and its variants and generalizations to different fields of mathematics (we recommend the interested reader to the nice survey paper [1]). In particular, it has been employed extensively to prove existence and multiple results for nonlinear integral and differential equations (see [19, 0] and the references therein). In the first part of this paper, using Borsuk-Krasnoselskii s result we shall present a new fixed point theorem in Banach spaces with an order structure given by an normal order cone. More precisely, we shall introduce new abstract definitions of lower and upper solutions for the equation u = f(u) in the Banach space E. Then, we shall show the existence of maximal and minimal fixed points in a sector defined by the lower and the upper solution without assuming monotone conditions on the nonlinearity f. The second part of the paper deals with the application of the previous result to prove new sufficient conditions for the existence of solutions of initial and boundary value problems for first and higher order difference equations in Banach spaces. Evidently, if the dimension of the Banach space is finite we just have a difference equation or a system of difference equations (see [1, 15] for a general theory). Existence of solutions for difference equations, difference equations in Banach spaces or systems of difference equations have been considered by several authors using different techniques [, 8, 9, 11, 13, 17]. We note that in [9] the authors also introduce a cone in order to prove existence of single and multiple fixed sign solutions. In [9] the cone is needed so that a fixed point theorem of cone expansion and compression of norm type can be used. The approach in [9] is completely different from our approach. In [8,10,13,14] the lower and upper solution technique was employed. In Section 3 we prove an existence result for first order initial value problems in Banach spaces, assuming that there exist suitably defined upper an lower solutions. Then, we study higher order initial and boundary problems by reducing these problems to initial value ones. Finally, we note that our technique was inspired by methods employed to solve the first order scalar case in [10]. Existence principle Let E be an ordered Banach space whose order cone K E is normal and has non-empty interior. That is, K is a closed, convex, proper ({ x, x} K x = 0}), normal (there is a number r 0 > 0 such that 0 u v u r 0 v ) cone and int(k).
3 Antipodal theorem and difference equations 3 As usual, we will write u v for v u int(k) and [u, v] = (u + K) (v K) = {w E : u w v} (u, v) = int([u, v]) = (u + int(k)) (v int(k)) = {w E : u w v}. Notice that [u, v] is a bounded set, because for each w [u, v] one has w w u + u r 0 v u + u. We consider on E m, with m N, the norm x = x x m and the partial order relation correspondent to the cone K m (which is closed, convex, proper, normal and with non-empty interior), that is: x y if and only if x k y k for all k {1,..., m}. A map is said to be compact if it is continuous and maps bounded sets into relatively compact sets. Remark 1 A compact subset S E always has maximal and minimal elements. To see that, suppose J is a linearly ordered subset of S and consider the filter base on S formed by the sets F v = {u S : u v} with v J. Since S is a compact set, this filter base has a cluster point u 0 S, that is u 0 v J F v. Therefore, u 0 is an upper bound of J. By Zorn s Lemma, S has a maximal element. A slight change in the proof actually shows the minimal element existence. Theorem.1 If a b in E and f : [a, b] E is a compact map such that a f(a + v); b f(b v) for all v K with v b a, then f has a fixed point in (a, b) but none in [a, b]. Furthermore, the set of fixed points {u [a, b] : f(u) = u} is a non-empty compact subset of (a, b), having maximal and minimal elements. Proof First suppose that f has a fixed point u [a, b] such that u [a, b] = ((a + K) (b K)) ((a + K) (b K)). Then one has u a K with u a b a or b u K with b u b a. If u a K with u a b a, then a f(a + (u a)) = f(u) = u, so u a int(k), which is impossible. Similarly, if we assume b u K with b u b a, we have b u int(k), once more a contradiction. Now consider the Minkowski functional defined on E by p(w) = inf{r > 0 : w r [ a b, b a ]}.
4 4 Franco, O Regan and Peran The functional p is a norm on E, since [ a b, b a ] is a symmetric, closed, bounded, convex set with 0 ] a b, b a [. Furthermore, p and are equivalent norms. To see this notice 1 r 0 b a w p(w) 1 r 1 w, where r 0 comes from the normal cone condition and r 1 is the radius of a ball with centre 0 and contained in ( a b, b a ). Since ( a b, b a ) = {w E : p(w) < 1}, the result follows from the antipodal theorem if the antipodal condition holds for all u [ a b Recall that, b a f (u) u f (u) u f ( u) + u f ( u) + u ]; here f (u) = f ( u + b+a ) b+a. [ a b, b a ] = ( ( a b + K) ( b a K) ) ( ( a b + K) ( b a K) ). Suppose the antipodal condition does not hold. Then there is a number r > 0 and v K with v b a such that that is This gives f ( a b + v ) ( a b + v ) = r ( f ( b a v ) ( b a v )), f (a + v) (a + v) = r (f (b v) (b v)). v = a (a + v) f (a + v) (a + v) = r (f (b v) (b v)) r (b (b v)) = rv, and then (r + 1)v int(k), which is impossible because v K. Therefore S = {u [a, b] : f(u) = u} is a non-empty compact set. To see the compactness notice that f continuous implies S is closed, and also since S is bounded we have that S = f(s) is a relatively compact set. Finally, the minimal and maximal existence follows from Remark 1. In view of the last Theorem we make the following definition.
5 Antipodal theorem and difference equations 5 Definition. We say that a b in E are a pair of lower and upper solution for equation u = f(u) iff a f(a + v); b f(b v) for all v K with v b a. 3 Applications to difference equations Before presenting our results in this section we need to recall some definitions and results of the theory of set valued analysis [4]. Let X be a non-empty set and let P(X) denote the set of all subsets of X. We define a multifunction F from X to an arbitrary set Y, written F : X Y, to be a map F : P(X) P(Y ) such that F (A) = a A F ({a}). Of course, given F ({x}) for each x X, the multifunction F is determined. We denote F ({x}) by F (x). The map F : P(Y ) P(X) defined by F (B) = {x X : F (x) B } is a multifunction, called the inverse multifunction of F. It is easily proved that the composition of two multifunctions F : X Y, G: Y Z is a multifunction G F : X Z such that G F (x) = y F (x) G(y). As usual, we identify each map f : X Y with the multifunction A X f(a). Obviously, the map usually denoted as f 1 : P(Y ) P(X) coincides with the multifunction f. Let X and Y be Hausdorff spaces. A multifunction F : X Y is called upper semicontinuous (u.s.c.) if F (C) is a closed subset of X for each closed subset C of Y. An u.s.c. multifunction with non-empty compact values is called an usco map. Lemma 3.1 Let D E be a closed bounded set, f : D E a compact map and h: D E defined by h(u) = u f(u). Then h is a closed map, that is, h 1 : E D is an u.s.c. multifunction. Proof Let C D be a closed set, (y n ) y 0 a convergent sequence with y n h(c) and x n C such that y n = h(x n ). Since cl(f(c)) is a compact set, there exists a convergent subsequence (f(x ni )) of (f(x n )) with limit w 0 cl(f(c)). Now, so w 0 + y 0 C and lim x ni = lim f(x ni ) + lim h(x ni ) = w 0 + y 0 f(w 0 + y 0 ) = f(lim x ni ) = lim f(x ni ) = w 0.
6 6 Franco, O Regan and Peran Finally, h(w 0 + y 0 ) = w 0 + y 0 f(w 0 + y 0 ) = w 0 + y 0 w 0 = y 0, so y 0 h(c) and h(c) = ( h 1) (C) is a closed subset of E. 3.1 First order difference equations. Consider the following difference equation: y k = f k (y k+1 ), k {1,..., m 1} (1) where y = (y 1,..., y m ) E m, y k = y k+1 y k and f k : E E is a compact map for each k {1,..., m 1}. Define h k : E E by h k (u) = u f k (u) for k {1,..., m 1}. Then rewrite (1) as y k = h k (y k+1 ), k {1,..., m 1}. () We define a continuous map H = (H 1,..., H m ): E E m as follows: H m (u) = u (3) H k (u) = h k (H k+1 (u)), k {1,..., m 1}. (4) It is easy to see that y = H(u) is a solution of (1) for each u E, and conversely, for each solution y of (1) there is a u E with y = H(u). Definition 3. We say that α β in E m are a pair of lower and upper solutions for equation (1) if for each k = 1,..., m 1 and v K such that v β k+1 α k+1 one has α k f k (α k+1 + v) and β k f k (β k+1 v). Remark 1 If α β are a pair of lower and upper solutions, we observe by considering v = 0 above that α is a subsolution and β is a supersolution for (1), in accordance with standard definitions (see for example [10]). But an arbitrary subsolution with an arbitrary supersolution, do not necessarily form a pair of lower and upper solutions in the sense of Definition 3.. Finally, notice that y E m is a solution of (1) if and only if it forms with itself a pair of lower and upper solutions. Remark Observe that α β are a pair of lower and upper solutions if and only if, for all k = 1,..., m 1 and v, v K such that v, v β k+1 α k+1, h k (α k+1 + v) v α k β k h k (β k+1 v ) + v (5)
7 Antipodal theorem and difference equations 7 Lemma 3.3 Let α β be a pair of lower and upper solutions for (1) and w k (α k, β k ) for each k {1,..., m 1}. Then, the set S k (w k ) = h 1 k (w k) [α k+1, β k+1 ] is a non-empty compact set contained in (α k+1, β k+1 ). Furthermore, the multifunction S k : (α k, β k ) (α k+1, β k+1 ) is an usco map. Proof For each v K with v β k+1 α k+1 the compact map f from [α k+1, β k+1 ] to E defined by f(u) = f k (u) + w k satisfies α k f(α k+1 + v) w k and β k f(β k+1 v) w k. This gives and α k+1 α k+1 + (w k α k ) f(α k+1 + v) β k+1 β k+1 + (w k β k ) f(β k+1 v). By Theorem.1, the set {u [α k+1, β k+1 ] : f(u) = u} is a non-empty compact subset of (α k+1, β k+1 ). But this is the desired conclusion, since from h k (u) = u f k (u) = u f(u) + w k we have S k (w k ) = {u [α k+1, β k+1 ] : f(u) = u}. It remains to be proved that S k is an u.s.c. multifunction. The set D = [α k+1, β k+1 ] is closed and bounded, the map f k : D E is compact and, by Lemma 3.1, h k : D E is a closed map. Suppose B int(d) = (α k+1, β k+1 ) is closed for the subspace topology induced on int(d), that is, there is a closed (in E) B D such that B = B int(d). Then S k (B) = S k (B ) = h k (B ) (α k, β k ) is a closed subset of (α k, β k ) for the induced topology on this set. Theorem 3.4 Let α β be pair of lower and upper solutions for (1). For each w (α 1, β 1 ), denote by S(w) the subset of [α, β] formed by the solutions y E m of (1) satisfying the initial value condition y 1 = w. Then S(w) is a non-empty compact subset of (α, β) having maximal and minimal elements. Furthermore, the multifunction S : (α 1, β 1 ) (α, β) is an usco map satisfying S = H S m 1 S m S 1. Proof The proof falls in four parts:
8 8 Franco, O Regan and Peran a) We claim S k S k 1 S 1 (w) is a non-empty subset of (α k+1, β k+1 ) for k {1,..., m 1}. For k = 1 it follows directly from Lemma 3.3. Assuming the assertion holds for k 1, apply Lemma 3.3 for each w k S k 1 S 1 (w). b) We claim S(w) H S m 1 S m S 1 (w). If y S(w), then y 1 = w and y k = h k (y k+1 ) for k {1,..., m 1}. This gives y = H(y m ) and y m S m 1 (y m 1 ) S m 1 S m (y m )... S m 1 S m S 1 (w). Thus y H S m 1 S m S 1 (w) and so S(w) H S m 1 S m S 1 (w). c) We claim H S m 1 S m S 1 (w) S(w) (α, β). If y H S m 1 S m S 1 (w), then y = H(y m ), with y m S m 1 S m S 1 (w), which means that y is a solution of (1). Since y m S m 1 S m S 1 (w), there exist w, w 3..., w m 1 such that w S 1 (w) (α, β ),..., w k+1 S k (w k ) (α k+1, β k+1 ),... and y m S m 1 (w m 1 ) (α m, β m ). But, by the definition of S k, one has w m 1 = h m 1 (y m ) = y m 1, w m = h m (y m 1 ) = y m, w = h (y 3 ) = y, w = h 1 (y ) = y 1. Therefore, y S(w) (α, β). d) A continuous map, considered as a single-valued multifunction, is an usco map and finite compositions of usco maps are usco maps (see 6. of [6]). By Lemma 3.3 and the result above, S is an usco map. The maximal and minimal existence is a consequence of Remark 1. Remark 3 As we have mentioned in section 1, some authors have proposed various definitions of upper and lower solutions notions in the scalar case and have developed related existence results. Notice Theorem 3.4 is stated for the multidimensional case. The crucial, but simple, idea is that one can reduce the order of a problem by increasing its dimension (see Hartman s paper [1] for the linear case). Therefore, Definition 3. and Theorem 3.4 have a wide range of applications, as the following sections show. Observe that order reduction
9 Antipodal theorem and difference equations 9 and Definition 3. gives new and useful notions of upper and lower solutions for higher order problems. 3. Higher order difference equations. Let p > n be natural numbers and consider the following n-th order initial value problem on an ordered Banach space F whose order cone is normal and has non-empty interior: n z k = ϕ k (z k+1, z k+1, z k+1,..., n 1 z k+1 ), k {1,..., p n} (6) z 1 = w 1,..., z n = w n, (7) here n z k = ( n 1 z k ). As we have seen, this problem can be written as z k = ψ k (z k+1, z k+, z k+3,..., z k+n ), k {1,..., p n} (8) z 1 = w 1,..., z n = w n. (9) Assume p be a multiple of n, p = mn (if p wasn t a multiple of n, we would define ψ k (u 1,..., u n ) = u n for k {p n + 1,..., mn}, where mn is the smallest multiple of n greater than p). Consider the first order initial value problem in E = F n : y k = h k (y k+1 ), k {1,..., m 1} (10) y 1 = (w 1,..., w n ), (11) where the maps h k : E E are defined as follows: the n-th component of h k is defined by (h k ) n (u) = ψ nk (u). Assuming that (h k ) n, (h k ) n 1,..., (h k ) j+1 have been defined, let (h k ) j be (h k ) j (u) = ψ n(k 1)+j ( (h k ) j+1 (u), (h k ) j+ (u),..., (h k ) n (u), u 1, u,..., u j ). It is easily checked that, for each solution z 1, z,..., z p of (8)-(9), we obtain a solution of (10)-(11) by y k = (z n(k 1)+1, z n(k 1)+,..., z nk ) with k {1,,..., m} and vice versa. Now Section 3.1 can be applied to problem (10)-(11).
10 10 Franco, O Regan and Peran Example 3.5 The discrete analogous of the autonomous boundary value problem z + F (z) = 0; z(0) = z (0) = 0 can be formulated as z k = ψ(z k+1, z k+ ), k {1,..., m } (1) z 1 = z = 0 (13) where m N, ψ(u 1, u ) = g(u 1 ) u and g(x) = x F (x) 4m. Setting as above y k = (z k 1, z k ) for k {1,,..., m}, the equivalent bidimensional first order equation is y k = f(y k+1 ), k {1,..., m 1} (14) where f(u) = L(u) + γ(u), with γ(u 1, u ) = 1 4m (F (u 1) + F (g(u 1 ) u ), F (u 1 )) ( ) ( ) u1 and L(u 1, u ) =. u Consider the order relation given by K = {(x, x ) R : x x 1 0} and suppose F is such that there exist d m 0 with dm m w m γ(u) dm m w m, where w m = ( m 1/m, 3m 1/m 1 ). Remark gives us the condition for the upper and lower solutions to be inductively constructed from greater to smaller indexes. In such a way, we obtain α k = α k+1 L(α k+1 ) + d m m w m β k = α k + d mc k m w m, where c k = m k/m + m 1/m 1. Observe that c k+1 c k = m1/m 1 m c 1/m k+1 and that m1/m 1 m w 1/m m L(w m ). It is obvious that β k α k int(k). Furthermore,
11 Antipodal theorem and difference equations 11 for all v K, 0 v (β k+1 ) (α k+1 ), since L(v) (0, 0), one has α k = L(α k+1 ) d m m w m L(α k+1 + v) + γ(α k+1 + v) = f(α k+1 + v) β k = α k + d m(c k+1 c k ) m w m = L(α k+1 ) d m m w m + d m(c k+1 c k ) m w m = = L(α k+1 ) + d m m w m + d m(c k+1 c k ) m w m = = L(α k+1 ) + d m m w m + d mc k+1 (m 1/m 1) m m 1/m w m L(α k+1 ) L(v) + d m m w m + d mc k+1 m L(w m ) L(β k+1 v) + γ(β k+1 v) = f(β k+1 v). If we begin with α m = dm m ( m 1/m m + m + m 1/m 1, m 1/m m + m + 3m 1/m 1 ), we obtain α 1 = (I L) m 1 (α m ) + d m m ( m ) (I L) k (w m ) = k=0 = d ( ) ( m m 1 m + m 1/m m + m + m 1/m 1 m m m + 3 m 1/m m + m + 3m 1/m 1 + d ( ) ( ) m m 1 m m 1/m m m 3 m 3m 1/m 1 ) + (0, 0) α 1 + d mc 1 m w m = β 1 We conclude by Theorem 3.4 that there exists a solution satisfying y 1 = (0, 0), thus problem 1-13 has a solution located by α and β. 3.3 Other boundary conditions. Some boundary value problems for discrete equations can be changed into initial value problems with a greater order. For instance, consider the following first order problem with a periodic boundary condition: c k = φ k (w k+1 ), k {1,..., m 1} (15) w 1 = w m, (16)
12 1 Franco, O Regan and Peran with φ k : E E. Let the maps ψ k : E E, k {1,..., m 1}, be defined as follows: ψ 1 (u) = 0, ψ (u) = φ 1 (u ) u 1, ψ j 1 (u) = u, j {,..., m 1} ψ j (u) = φ j (u ), j {,..., m 1} ψ m 1 (u) = u 1. It can be easily shown that if z 1,..., z m, z m+1 is a solution for the initial value problem z k = ψ k (z k+1, z k+ ), k {1,..., m 1} (17) z 1 = z = 0, (18) then a solution for the problem (15)-(16) is obtained by setting w 1 = z m, c k = z k for k {,..., m}. Observe that (17)-(18) can be handle like (8)-(9). Notice that in order for ψ k to be a compact map, E must be a finite dimensional space. References [1] R.P. Agarwal, Difference equations and inequalities. Theory, methods, and applications, Marcel Dekker Inc., New York, 000. [] R.P. Agarwal and D. O Regan, Difference equations in abstract spaces, J. Austral. Math. Soc. (Series A) 64 (1998) [3] D.R. Anderson, A fourth-order nonlinear difference equation, J. Difference Equ. Appl. 9 (003), no. 1, [4] J.P. Aubin and H. Frankowska, Set-valued analysis, Birkhäuser, Boston, [5] R. I. Avery, Chuan Jen Chyan, J. Henderson, Twin solutions of boundary value problems for ordinary differential equations and finite difference equations, Comput. Math. Appl. 4 (001), no. 3-5, [6] G. Beer, Topologies on closed and closed convex sets, Kluwer Academic Publishers, Dordrecht, [7] K. Borsuk, Über stetige Abbildungen der euklidischen Räume, Fund. Math. 1 (1933) [8] A. Cabada and V. Otero-Espinar, Optimal existence results for nth order periodic boundary value difference equations, J. Math. Anal. Appl. 47 (000), no. 1, [9] C.J. Chyan, J. Henderson and H.C. Lo, Existence of triple solutions of discrete (n, p) boundary value problems, Appl. Math. Lett. 14 (001), no. 3,
13 Antipodal theorem and difference equations 13 [10] D. Franco, D. O Regan and J. Perán, Upper and lower solution theory for first and second order difference equations, Dynamic Systems and Applications, (to appear). [11] C. González and A. Jiménez-Melado, Asymptotic behavior of solutions of difference equations in Banach spaces, Proc. Amer. Math. Soc. 18 (000), no. 6, [1] P. Hartman, Difference equations: disconjugacy, principal solutions, Green s functions, complete monotonicity, Trans. Amer. Math. Soc. 46 (1978), [13] J. Henderson and H.B. Thompson, Existence of multiple solutions for secondorder discrete boundary value problems, Comput. Math. Appl. 43 (00), no , [14] J. Henderson and P.J.Y. Wong, On multiple solutions of a system of m discrete boundary value problems, Z. Angew. Math. Mech. 81 (001), no. 4, [15] W.G. Kelley and A.C. Peterson, Difference equations. An introduction with applications, Harcourt/Academic Press, San Diego CA, 001. [16] M.A. Krasnoselskii, On a fixed point principle for completely continous operators in functional spaces, Dokl. Akad. Nauk SSSR 73 (1950) (Russian) [17] C.V. Pao, Monotone methods for a finite difference system of reaction diffusion equation with time delay, Comput. Math. Appl. 36 (1998) [18] A. Peterson, Existence and uniqueness theorems for nonlinear difference equations, J. Math. Anal. Appl. 15 (1987), no. 1, [19] W.B. Song, On the solvability of a nonselfadjoint quasilinear elliptic boundary value problem, Taiwanese J. Math. 6 (00), no. 4, [0] S. Staněk, On some boundary value problems for second order functionaldifferential equations, Nonlinear Anal. 8 (1997), no. 3, [1] H. Steinlein, Borsuk s antipodal theorem and its generalizations and applications: a survey, Topological Methods in Nonlinear Analysis, , Sém. Math. Sup., 95, Presses Univ. Montréal, Montreal, [] E. Zeidler, Nonlinear functional analysis and its applications. Vol I. Fixed-point theorems, Springer-Verlag, New York, 1986.
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