Journl of Computtions & Modelling, vol.3, no.4, 2013, 1-10 ISSN: 1792-7625 (print), 1792-8850 (online) Scienpress Ltd, 2013 On the Continuous Dependence of Solutions of Boundry Vlue Problems for Dely Differentil Equtions Evngeli S. Athnssidou 1 Abstrct We prove generl theorem on the continuous dependence of solutions of boundry vlue of boundry vlue problems for dely differentil equtions with nonliner problems boundry condition. The proof is bsed on the continuity of the Brouwer topologicl degree. Approprite remrks on the convergence of sequences of functions improve some known results. Keywords: Dely differentil equtions, boundry conditions, continuous convergence. 1 Introduction In this work we consider the most generl boundry vlue problem for dely differentil equtions. In prticulr we study boundry vlue problems of the form 1 Deprtment of Mthemtics, University of Athens, GR-15784, Pnepistimiopolis, Athens, Greece. E-mil: ethn@mth.uo.gr Article Info: Received : July 1, 2013. Revised : August 30, 2013. Published online : December 1, 2013.
2 On the Continuous Dependence of Solutions of Dely Differentil Equtions x (t) = f(t, x t ), T(x) =, where f is vector function, T is continuous opertor nd is constnt vector. We prove generl theorem bout continuous dependence of solutions of the bove boundry vlue problem. Existence, uniqueness nd continuous dependence of solutions of boundry vlue problems of this type hve been proved in [1], [2] nd [3]. Corresponding results for boundry vlue problems for ordinry differentil equtions re included in [4], [5] nd [6].More detils for problems of this type cn be find in the books [7], [8], [9] for ordinry differentil equtions nd [7],[10], [11] nd [12] for dely differentil equtions. The proof here is quite different from the method, in the ppers [1] nd [2], where the Schuder s theorem is employed. In our work the continuity of the Brouwer topologicl degree is pplied [13], [14]. In [1] the results re proved in the spce of continuous functions but in [2] this spce hve been replced by the spce of Lipschitzin functions. In [1] nd [2] the theorems require the hypothesis of unrestricted uniqueness tht is condition which fulfills the function f such tht the corresponding boundry vlue problem hs exctly one solution. This condition is needed only for the limit problem. In this pper we will use the continuous dependence of the Brouwer topologicl degree [13], [14]. Also we will pply some properties of the convergences of sequences of functions in order to generlize known results. 2 Preliminries nd nottions Let τ be positive number. The spce of ll continuous functions φ: [ τ, 0] R Ν, will be denoted by C 0 = C 0 ([ τ, 0], R Ν ) endowed with the supremum norm φ C 0 = sup{ φ(t) : t [ τ, 0]}.
E. Athnssidou 3 For function x: [ τ, b] R Ν, b > 0nd t [0, b], we define the function x t : [ τ, 0] R Ν by x t (s) = x(t + s), s [ τ, 0]. Especilly the condition x 0 = φ is equivlent to x(s) = φ(s), s [ τ, 0]. Let A = A([0, b], R Ν )be the spce of bsolutely continuous functions from [0, b] into R Ν endowed with the supremum norm. We denote by C(A, R N ) the spce of continuous opertors from A to R Ν. We sy tht function f: [0, b] C 0 R Ν stisfies the Crthéodory conditions if the following re vlid: (i) for every fixed φ, f is mesurble with respect to t, (ii) for every fixed t, f is continuous with respect to φ nd (iii) for every bounded set D C 0 there exists n integrble function m such tht f(s, φ) m(s), fors [0, b], φ D. A fmily Φ of the functions f: [0, b] C 0 R Ν we sy tht stisfies the Crthéodory conditions uniformly if every function f fulfils the conditions (i), (ii) nd lso (iv) for every bounded set D C 0 there exists n integrble function Msuch tht f(s, φ) M(s) for every s [0, b], φ D nd f Φ.With C we will denote the spce of functions f: [0, b] C 0 R Ν which stisfy the Crthéodory conditions. Now we present some new results regrding convergence or continuous convergence tht we will need. Let (X, d), (Y, ρ) be rbitrry metric spces. In prticulr X = [0, b] C 0 Y = R N in our cse. Also for the remining of this section, let f n, f: X Y, n = 1,2,. We recll the following definitions (See lso [15], [16]).
4 On the Continuous Dependence of Solutions of Dely Differentil Equtions () We sy tht (f n )converges to f f n f iff, for ech x X nd for ech sequence (x n )in X, with x n x it holds tht f n (x n ) f(x). (b) We sy tht the sequence (f n ) is exhustive, iff x X ε > 0 δ = δ(x, ε) > 0 n 0 = n 0 (x, ε): d(x, t) < δ ρ f n (x), f n (t) < ε, for n n 0. (c) We sy tht (f n ) is wekly-exhustiveiff, x X ε > 0 δ = δ(x, ε) > 0 d(x, t) < δ n t N: ρ f n (x), f n (t) < ε, for n n t. Obviously if (f n ) is exhustive then (f n ) is wekly-exhustive. It is not hrd to see tht the inverse impliction fils ([15]). Now, we formulte some new results on convergence. The proofs of Propositions 2.1 nd 2.2 cn be found in [15] nd the proof of Propositions 2.3 nd 2.4 in [16]. Proposition 2.1 The following re equivlent. () f n f, (b) (f n )convergespointwise tofnd(f n ) is exhustive. Proposition 2.2 Suppose tht(f n )converges pointwisetof. Then the following re equivlent. () fis continuous, (b) (f n )is wekly exhustive. We note tht the functions f n, n = 1,2, need not to be continuous in the bove theorem. Also s corollry from proposition 2.1, 2.2 we get tht the limit of ny sequence of functions is necessrily continuous. With the next theorems we see how convergence nd uniform convergence re relted. Proposition 2.3 Suppose thtf n f. Then(fn )converges uniformly to f on compct subsets of X.
E. Athnssidou 5 For detils nd concrete exmples regrding the difference of convergence nd uniform locl convergence see [16].In the inverse direction, the continuity of the functionsf n, n = 1,2, is necessry. Proposition 2.4 Suppose tht {f n } C(X, Y) {f: X Y f is continuous}. If for ech x X there is neighborhood A of x such tht (f n ) converges uniformly to fon A, then, f n f. In cse tht X is loclly compct nd {f n } C(X, Y) s corollry from propositions 2.3 nd 2.4 we get tht f n f (fn ) converges uniformly on compct to f. In view of the bove propositions, some comments re in order: () Suppose {f n : n = 1,2, } C nd tht f n f. Then by propositions 2.1 nd 2.2 it follows tht f is continuous, hence f C. (b) In theorems on continuous dependence of solutions e.g. theorem 5.1 of Hle [11], we require tht f n (t, ) f(t, ) for ech t [0, b]. From propositions 2.1 nd 2.2, it follows gin tht f(t, ) is continuous for ech t [0, b]. Also, since f n (t, φ) f(t, φ), n we get tht f(, φ)is mesurble for ech φ C 0. Hence condition (i) nd (ii) re utomticlly stisfied by f. 3 Continuous dependence In this section we prove generl theorem on continuous dependence of solutions of boundry vlue problems for differentil equtions with dely. We consider the following boundry vlue problem x (t) = f(t, x t ), (1)
6 On the Continuous Dependence of Solutions of Dely Differentil Equtions T(x) =, (2) wheref C nd T L. The continuous function x: [ τ, b] R N nd R N is clled the solution of (1) in Crthéodory sense if x stisfies (1) lmost everywhere on [0, b], x is constnt on [ τ, 0]nd x is bsolutely continuous on [0, b]. The sme function is clled the solution of boundry vlue problem (1), (2) if this is solution of eqution (1) in Crthéodory sense nd stisfies boundry condition (2). From Theorem 3.1, Theorem 5.1 nd 7 of Hle [11] (see lso Theorem 3.2 of [8]) with some modifictions we get the following Theorem 3.1 nd Theorem 3.2. Theorem 3.1 If f C, then for ech (t, φ) [0, b] C 0 with φ constnt, there is solution of (1) pssing through (t, φ). Theorem 3.2Let f, f n C, φ, φ n Α, n = 1,2, with φ n p.w φ,{f n : n N} stisfies uniformly Crthéodory conditions ndf n f. If x n is ny solution of the problem x (t) = f n (t, x t ), x 0 = φ n, nd the problem x (t) = f(t, x t ), x 0 = φ hs unique solution, let x, then x n converges uniformly to x. Now we prove generl theorem on the continuous dependence of boundry vlue problems for dely differentil equtions of type (1), (2). Theorem 3.3Let f 0, f n C nd T 0, T n C(A, R N ), n = 1,2,. Also we suppose tht {f n : n N} stisfies uniformly Crthéodory conditions. (i) f n f0 nd T n T0. (ii) The boundry vlue problem x (t) = f n (t, x t ), x 0 = u (constnt function), where u R N, hs t most one solution for every u R N, n = 0,1,2 (iii) The boundry vlue problem
E. Athnssidou 7 x (t) = f 0 (t, x t ), T 0 (x) = r, hs t most one solution for echr R N. Let v, v n R N with lim n v n = v. If x 0 x 0 (t, f, v, T) is the solution of x (t) = f(t, x t ), T(x) = v, then for ech ε > 0 there exists n 0 = n 0 (ε) such tht for n n 0 the boundry vlue problem x (t) = f n (t, x t ), T n (x) = v n hs solution x n x n (t, f n, v n, T n ) stisfying x n x 0 < ε. Proof. Let ε > 0. We consider the following problem P n (u) : x (t) = f n (t, x t ), x 0 = u, where u R N, n = 0,1,2,. Assertion: There exist neighborhood V of x 0 (0) in R N nd m N such tht for every u V, n m the problem P n (u) hs solution u n stisfying u n x 0 < ε. Proof of Assertion: Suppose not, then in view of Theorem 3.1, it follows tht for ech m, there is u m B x 0 (0), 1 nd n m m m such tht the solution u n m of problem P nm (u m ) stisfies u n m x 0 ε. But, since u m x 0 (0), s m, it follows from Theorem 3.2 tht u n m converges uniformly to x 0. Hence, we rrive to contrdiction. We observe tht if f n = f 0 for n = 1,2, then the conclusion of the ssertion holds for the solutions of P 0 (u) on some neighbourhood of x 0 (0). Now let (V, m) be pir stisfying the Assertion where V stisfies lso the u bove observtion. For ech u V we set σ n to be the solution of problem P n (u) such tht
8 On the Continuous Dependence of Solutions of Dely Differentil Equtions σ u n x 0 ε for n m, or n = 0. (3) We fix bll B 1 with centrex 0 (0) in R N such tht B 1 V nd define F n : B 1 R N, F n (u) = T n (σ u n )(n = 0,1, ). We observe tht, if u n u 0, u n B 1, n = 1,2, then by Theorem 3.2 it follows u tht σ n u n converges uniformly to 0 σ0. Hence by hypothesis (i) we get tht F n F0, nd Proposition 2.3 implies the uniform convergence of F n to F 0, u F n F0. (4) Also by the uniqueness of solution (hypothesis (ii)), it follows gin from Theorem 3.2 tht ech F n (n = 0,1, ) u is continuous (since the mpping u σ n is continuous). In view of (iii), F is injective nd we hve v F 0 ( B 1 ). Hence dist(f 0 ( B 1 ), v) = d > 0, sincef 0 ( B 1 ) is closed. Tking into ccount tht v n v nd (4), we get the existence of n 0 N, such tht dist(f n ( B 1 ), v n ) d 3, for n n 0. Consequently, the Brouwer topologicl degree deg(f n, B 1, v n ) is well defined for n n 0 nd in view of the continuous dependence of topologicl degree there is n 1 N, n 1 n 0 such tht deg(f n, B 1, v n ) = deg(f 0, B 1, v) for n n 1. Since F 0 is injective, we hve (see [13], [14]) deg(f 0, B 1, v) = ±1. Hence for n n 1 we hve deg(f n, B 1, v n ) = ±1 forn n 1 nd the eqution F n (u) = v n hs t lest one solutionu n in B 1. From property (3) of the solution of the problem P n (u n ) it follows tht the solution x n σ n u n fulfils the condition x n x < ε, tht mens the Theorem is proven by tking n 0 = n 1.
E. Athnssidou 9 Now from Theorem 3.2 we hve the following. Corollry 3.3 Under the ssumptions of Theorem 3.2 we ssume dditionl tht the boundry vlue problem x (t) = f n (t, x t ), T n (x) = u n hs unique solution x n. Then we hve lim n x n = x uniformly on [0, b]. Proof. It follows directly from Theorem 3.3. Corollry 3.4 If the problems corresponding to eqution x (t) = f(t, x t ) hve locl uniqueness nd if the boundry vlue problem x (t) = f(t, x t ), T(x) = v (5) hve t most one solution for every v R N, then the set V of ll v R N for which (5) hs t most one solution is n open subset of R N. Proof. If v 0 V it is not n interior point of V, then there exists sequence (v n ), n = 1,2, in R N such tht lim n v n = v nd x (t) = f(t, x t ), T(x) = v n hs no solution. This contrdicts the conclusion of Theorem 3.3 by tking f n = f 0, T n = T 0. References [1] K. Twrdowsk, On existence, uniqueness nd continuous dependence of solutions of boundry vlue problems for differentil equtions with delyed prmeter, Zeszyty Nukowe UJ, Prce Mt.,18, (1977), 143-153. [2] M. Jblonski nd K. Twrdowsk, On boundry Vlue problems for differentil equtions with retrded rgument, 1987. [3] J.W. Myjk, A boundry vlue problem for non-liner differentil equtions with retrded rgument, Ann. Polon. Mth., 27, (1973), 133-142.
10 On the Continuous Dependence of Solutions of Dely Differentil Equtions [4] S.N. Chow nd A. Lsot, On boundry vlue problems for ordinry differentil equtions, J. Diff. Eqs., 14, (1973), 326-337. [5] G. Vidossich, On the continuous dependence of solutions of boundry vlue problems for ordinry differentil equtions, J. Diff. Eqs., 82, (1989), 1-14. [6] G.A. Klsen, Continuous dependence for N-point boundry vlue problems, SIAM J. Appl. Mth, 29, (1975), 99-102. [7] J. Mwhin,: Nonliner boundry vlue problems for ordinry differentil equtions: From Schuder theorem to stble homotopy, in Nonliner Anlysis, Acdemic Press, New York, 1978. [8] P. Hrtmn, Ordinry Differentil Equtions, Wiley, New York, 1964. [9] L.C. Piccinini, G. Stmpcchi nd G. Vidossich, Ordinry Differentil Equtions in R N (problems nd methods), Springer-Verlg, New York, 1984. [10] R.D. Driver, Ordinry nd Dely Differentil Equtions, Springer-Verlg New York, 1977. [11] J. K. Hle, Functionl differentil equtions, Applied Mthemticl Sciences, 3, Springer-Verlg, New York, 1971. [12] O. Diekmnn, S. A. Vn Gils, S. M. VerduynLunel nd H. O. Wlther, Dely Equtions: Functionl-Complex nd Nonliner Anlysis, New York, 1995. [13] N. G. Lloyd, Degree Theory, Cmbridge Univ. Press, Cmbridge, 1978. [14] Qin Min nd Li Zhengyun, Rotting Vector-Field nd Degree Theory, Publishing House of Beijing University, 1982. [15] V. Gregorides nd N. Ppnstssiou, A notion of exhustiveness nd Ascoli-type theorems, Topology nd its Applictions, 155, (2008), 1111-1128. [16] E. Athnssidou, C. Ppchristodoulos nd N. Ppnstssiou, nd Hyper convergence in Function Spces, (submitted).