Existence and Uniqueness to Nonlinear Dynamic Boundary Value Problems

Transcription

1 A Mthemticl Journl Vol. 8, N o 3, (11-23). December Existence nd Uniqueness to Nonliner Dynmic Boundry Vlue Problems Christopher C. Tisdell 1 School of Mthemtics, The University of New South Wles, Sydney 2052, Austrli cct@mths.unsw.edu.u ABSTRACT This rticle investigtes the existence nd uniqueness of solutions to the nonliner dynmic system x = f(t, x), (1) where: t comes from so-clled time scle T; x is generlized derivtive of x; nd (1) is subjected to some two-point boundry conditions. The system (1) my ccurtely describe dynmic processes where time my flow continuously nd discretely t different stges in the one model. Some sufficient conditions re presented, involving dynmic inequlities on f, tht gurntee the existence nd uniqueness of solutions to (1). Some exmples re given to highlight the new results, including continuous-discrete hybrid system. RESUMEN Este rtículo indg sobre l existenci y unicidd de soluciones del sistem dinámico no linel x = f(t, x), en donde t tom vlores en un llmd escl de tiempo T; x es l derivd generlizd de x; y (1) está sujet un condición de borde de dos-puntos. El sistem (1) puede describir con exctitud procesos dinámicos en donde el tiempo fluye de mner contínu y discret en distintos estdos en un mismo modelo. Se presentn lguns condiciones suficientes sobre f, que incluyen desigulddes dinámics, que grntizn existenci y unicidd de soluciones de (1). 1 The Author grtefully cknowledges the reserch funding from The Austrlin Reserch Council s Discovery Projects (DP ).

2 12 Christopher C. Tisdell 8, 3(2006) Se dn ejemplos que ilustrn los nuevos resultdos, incluyendo un sistem híbrido continuo-discreto. Key words nd phrses: AMS 2000 Clssifiction: existence of solutions, uniqueness of solutions, time scle, dynmic equtions, boundry vlue problems 39A12 1 Introduction The theory of dynmic equtions on time scles generlizes the theory of differentil nd difference equtions, reveling the links nd nomlies between them in the process. The concept is prticulrly useful in modelling stop-strt processes where continuous nd discrete time my be present t different stges. For exmple: insect popultion models in biology 17, p.7ff; the periodic dischrge of cpcitor in circuit theory 2, p.15; nd hybrid systems 15 ll feture continuous nd discrete time in the modelling process. This pper considers the existence nd uniqueness of solutions to the first-order dynmic eqution x = f(t, x), t, c T, (2) subject to the boundry conditions Mx() + Rx(σ(c)) = α,, c T, α IR n, (3) where f :, c T IR n IR n is continuous, nonliner function; t is from soclled time scle T (which is nonempty closed subset of IR); x is the generlized derivtive of x; nd c re given constnts; M nd R re given constnts in IR; nd α is given constnt in IR n. Eqution (2) subject to (3) is known s n boundry vlue problem (BVP) on time scles. If T = IR then x = x nd (2), (3) become the following BVP for ordinry differentil equtions x = f(t, x), t, c, (4) Mx() + Rx(c) = α. (5) If T = Z then x = x nd (2), (3) become the following BVP for difference equtions x(t) = f(t, x(t)), t =, + 1,..., c 1, c, (6) Mx() + Rx(c + 1) = α. (7) There re mny more time scles thn just T = IR nd T = Z (see Remrk 2.10) nd hence there re mny more dynmic equtions.

3 8, 3(2006) Existence nd Uniqueness to Nonliner Dynmic Boundry Motivted by the bove nd by M 14 (see lso references therein), this rticle investigtes the existence nd uniqueness of solutions to systems of dynmic equtions in the more generl time scle setting. Approprite dynmic inequlities on f re formulted to ensure the existence nd uniqueness of solutions to (2), (3) for the time scle environment. Hence, the min contribution tht this work mkes is to provide quite generl results tht hve mny potentil pplictions in the discrete-continuous time relm. To understnd the nottion used bove nd the ide of time scles, some preliminry definitions re needed. Definition 1.1 A time scle T is nonempty closed subset of the rel numbers IR. We ssume throughout tht, c T. Denote the time scle intervl by, c T =, c T. Since time scle my or my not be connected, the concept of the jump opertor is useful, nd we will use it to define the generlized derivtive x of the function x. Definition 1.2 The forwrd (bckwrd) jump opertor σ(t) t t for t < sup T (respectively ρ(t) t t for t > inf T) is given by σ(t) = inf{τ > t : τ T}, (ρ(t) = sup{τ < t : τ T}, ) for ll t T. Define the grininess function µ : T 0, ) s µ(t) = σ(t) t. Throughout this work the ssumption is mde tht T hs the topology tht it inherits from the stndrd topology on the rel numbers IR. The jump opertors σ nd ρ llow the clssifiction of points in time scle in the following wy: If σ(t) > t, then the point t is clled right-scttered; while if ρ(t) < t, then t is termed left-scttered. If t < sup T nd σ(t) = t, then the point t is clled right-dense; while if t > inf T nd ρ(t) = t, then we sy t is left-dense. If T hs left-scttered mximum vlue m, then we define T κ = T {m}. Otherwise T κ = T. Definition 1.3 Fix t T κ nd let x : T R n. Define x (t) to be the vector (if it exists) with the property tht given ɛ > 0 there is neighbourhood U of t with x i (σ(t)) x i (s) x i (t)σ(t) s ɛ σ(t) s, for ll s U nd ech i = 1,..., n. Cll x (t) the (delt) derivtive of x(t) nd sy tht x is (delt) differentible. Definition 1.4 If G (t) = g(t) then define the delt integrl by g(s) s = G(t) G(). If T = IR then g(s) s = g(s)ds, nd if T = Z then g(s) s = t 1 g(s).

4 14 Christopher C. Tisdell 8, 3(2006) Definition 1.5 A function g :, σ(c) T IR is clled right dense continuous, denoted by g C rd (, σ(c) T ), if: g is continuous t ll right dense points in, σ(c) T ; lim t v g(t) exists nd is finite t ll left dense points v, σ(c) T. Theorem 1.6 If g C rd (, σ(c) T ) then exists for ll t, σ(c) T. g(s) s For more discussion on the delt integrl see 2, Chp.1, 3, Chp.1. The following theorem is from 6. Theorem 1.7 Assume tht g : T R n nd let t T κ. (i) If g is delt differentible t t then g is continuous t t. (ii) If g is continuous t t nd t is right-scttered then g is delt differentible t t with g g(σ(t)) g(t) (t) =. σ(t) t (iii) If g is delt differentible nd t is right-dense then g g(t) g(s) (t) = lim. s t t s (iv) If g is delt differentible t t then g(σ(t)) = g(t) + µ(t)g (t). A solution to (2), (3) is function x C(, σ(c) T ; R n ) tht stisfies (2) for ech t, c T nd lso stisfies (3). The reltively young theory of time scles dtes bck to Hilger 6. The monogrphs 2, 3, 10, lso provide n introduction. For more on boundry vlue problems on time scles, including existence nd uniqueness results, see 1, 5, 8, 7, 16, 18, 19 2 Existence nd Uniqueness In wht follows, if y, z IR n then y, z denotes the usul inner product nd z denotes Eucliden norm of z on IR n. Throughout this work we ssume tht M + R 0. (8) Lemm 2.1 Suppose (8) holds. If x C(, c T ; IR n ) stisfies (2) nd (3) then x(t) = f(s, x(s)) s + A, t, σ(c) T,

5 8, 3(2006) Existence nd Uniqueness to Nonliner Dynmic Boundry where A = (M + R) 1 α R f(s, x(s)) s. Proof It is esy to see tht x(t) = x() + nd x(σ(c)) = x() + So (3) gives ( f(s, x(s)) s, t, σ(c) T, (9) α = Mx() + R x() + nd rerrnging (11) we obtin x() = (M + R) 1 α R f(s, x(s)) s. (10) f(s, x(s)) s So substituting (12) into (9) we obtin, for t, σ(c) T, x(t) = (M + R) α 1 R nd the proof is finished. f(s, x(s)) s f(s, x(s)) s + ) (11). (12) f(s, x(s)) s, (13) Theorem 2.2 Suppose (8) holds nd f C(, c T IR n ; IR n ). constnt Q > 0 such tht If there exists 2 q, f(t, q) + λµ(t) f(t, q) 2 > 0, for ll t, c T, q Q, λ (0, 1, (14) then the BVP (2), (3) hs t lest one solution in C(, c T ; IR n ). Proof We wnt to show tht there exists t lest one solution to (13), which is equivlent to showing tht (2), (3) hs t lest one solution. Consider the mp T : C(, σ(c) T ; IR n ) C(, σ(c) T ; IR n ) defined by T (x(t)) = (M + R) 1 α R f(s, x(s)) s + f(s, x(s)) s, (15) for ll t, σ(c) T. Thus, our problem is reduced to proving the existence of t lest one x such tht x = T x. (16)

6 16 Christopher C. Tisdell 8, 3(2006) Now consider the following fmily of problems ssocited with (16), nmely nd define the open bll with centre 0 by x = λt x, λ 0, 1, (17) B P = {x C(, σ(c) T ; IR n ) : x(t) < P, t, σ(c) T } P = mx{ Q, R 1 ( α + M Q)}. We show tht ll possible solutions to (17) stisfy x(t) < P for ll t, σ(c) T. For λ = 0 see tht (17) gives only the zero solution, so obviously x(t) < P for ll t, σ(c) T in this cse. Let λ (0, 1. In view of (13) nd (15), we my equivlently rewrite (17) s the fmily of BVPs x = λf(t, x), t, c T, λ (0, 1, (18) Mx() + Rx(σ(c)) = λα, (19) Consider r(t) = x(t) 2 for ll t, σ(c) T nd let t 0, σ(c) T be such tht r(t 0 ) = mx t,σ(c)t r(t). First we show tht r(t) < Q 2 for t, σ(c)) T. Aruge by contrdiction by ssuming t 0, σ(c)) T nd r(t 0 ) Q 2. Then 0 r (t 0 ) = x(t 0 ) + x(σ(t 0 )), x (t 0 ), by the product rule, 2, p.8 = 2 x(t 0 ), x (t 0 ) + µ(t 0 ) x (t 0 ) 2, by Theorem 1.7, (iv) = 2 x(t 0 ), λf(t 0, x(t 0 )) + µ(t 0 )λ 2 f(t 0, x(t 0 )) 2, from (18) = λ 2 x(t 0 ), f(t 0, x(t 0 )) + µ(t 0 )λ f(t 0, x(t 0 )) 2 > 0 by (14), contrdiction. Thus we must hve r(t) < Q 2 for t, σ(c)) T. Secondly, consider the cse t 0 = σ(c). Since x() < Q from the cse t 0, σ(c)) T, the boundry conditions (19) give x(σ(c)) < R 1 α + M B. Thus ll possible solutions to (17) stisfy x(t) < P for ll t, σ(c) T. Since f is continuous, T : B P C(, σ(c) T ; IR n ) is compct mp by the Arzel-Ascoli theorem. Thus, the following Lery-Schuder degrees re defined nd homotopy principle is pplicble 13, Chp. 4 d LS (I T, B P, 0) = d LS (I λt, B P, 0) I = identity mp = d LS (I, B P, 0) = 1, since 0 B P, nd thus the non-zero property of the Lery-Schuder degree 13, Chp. 4 ensures the existence of t lest one solution in B P to (16) nd hence to (2), (3).

7 8, 3(2006) Existence nd Uniqueness to Nonliner Dynmic Boundry Corollry 2.3 Suppose (8) holds nd f C(, c T IR n ; IR n ). constnt Q > 0 such tht If there exists q, f(t, q) > 0, for ll t, c T, q Q, (20) then the BVP (2), (3) hs t lest one solution in C(, c T ; IR n ). Proof See tht if (20) holds then (14) holds (s µ 0) nd thus the result follows from Theorem 2.2. Exmple 2.4 An exmple of two-dimensionl function f = (f 1, f 2 ) stisfying (20) is (f 1 (t, x 1, x 2 ), f 2 (t, x 1, x 2 )) = (x x 2 t + 1, x 3 2 x 1 t + 1), Q = 2, s for x x x, f(t, x) = x x 1 + x x 2 > 0 Theorem 2.5 Let f C(, c T IR n ; IR n ). If 2 u 1 u 2, f(t, u 1 ) f(t, u 2 ) + µ(t) f(t, u 1 ) f(t, u 2 ) 2 > 0, t, c T, u 1 u 2, (21) then the BVP (2), (3) hs, t most, one solution in C(, c T ; IR n ). Proof Suppose tht there re two solutions u 1 u 2 to (2), (3). Let z = u 1 u 2 nd consider the BVP z = f(t, u 1 ) f(t, u 2 ), t, c T, Mz() + Rz(σ(c)) = 0. Arguing similrly s in the proof of Theorem 2.2 we obtin tht r(t) = z(t) 2 cnnot hve positive mximum on, σ(c)) T becuse this would contrdict (21). If r(σ(c)) > 0 then contrdiction is reched vi (22). Hence z = 0 on, σ(c) T nd the solutions re unique. Theorem 2.6 Suppose (8) holds nd f C(, c T IR n ; IR n ). If there exist functions p,r C rd (, c T ; 0, )) such tht f(t, q) p(t) q + r(t), for ll t, c T, q IR n, (22) 1 + (M + R) 1 R p(s) s < 1, (23) then the BVP (2), (3) hs t lest one solution in C(, c T ; IR n ). Proof Consider (15), (16) nd the ssocited fmily (17). Let H := 1 + (M + R) 1 R.

8 18 Christopher C. Tisdell 8, 3(2006) From (17) see tht for ll t, σ(c) T, x(t) λ T x(t) T x(t) = (M + R) 1 α R H H H f(s, x(s)) s + f(s, x(s)) s + (M + R) 1 α (p(s) x(s) + r(s)) s + (M + R) 1 α mx x(t) t,σ(c) T p(s) s + so rerrnging nd tking the mximum we obtin mx x(t) H t,σ(c) T Define the open bll with centre 0 by r(s) s r(s) s + (M + R) 1 α 1 H p(s) s f(s, x(s)) s + (M + R) 1 α, := L. (24) B L+1 = {x C(, σ(c) T ; IR n ) : x(t) < L + 1, t, σ(c) T }. We cn see from (24) tht ll possible solutions to (17) stisfy x(t) < L + 1 for ll t, σ(c) T. Since f is continuous, T : B L+1 C(, σ(c) T ; IR n ) is compct mp by the Arzel-Ascoli theorem. Thus, the following Lery-Schuder degrees re defined nd homotopy principle is pplicble 13, Chp. 4 d LS (I T, B L+1, 0) = d LS (I λt, B L+1, 0) I = identity mp = d LS (I, B L+1, 0) = 1, since 0 B L+1, nd thus the non-zero property of the Lery-Schuder degree 13, Chp. 4 ensures the existence of t lest one solution in B L+1 to (16) nd hence to (2), (3). Remrk 2.7 Condition (22) my be interpreted s liner growth condition on f(t, x) in x. Exmples of two-dimensionl functions f = (f 1, f 2 ) stisfying (22) re (f 1 (t, x 1, x 2 ), f 2 (t, x 1, x 2 )) = (e t (x x 2 2) 1/2, e t (x x 2 2) 1/2 ), p(t) = 2e t (f 1 (t, x 1, x 2 ), f 2 (t, x 1, x 2 )) = (t 2 x 1 cos x 2, t 2 x 2 sin x 1 ), p(t) = 2t 2, with r(t) = 0 in both cses. Corollry 2.8 Suppose (8) holds nd f C(, c T IR n ; IR n ). If f(t, q) is bounded on, c T IR n then the BVP (2), (3) hs t lest one solution in C(, c T ; IR n ).

9 8, 3(2006) Existence nd Uniqueness to Nonliner Dynmic Boundry Proof By ssumption, there exists constnt K 1 0 such tht f(t, q) K 1, for ll (t, q), c T IR n, nd therefore (22) holds with p = 0 nd r = K 1. Obviously then (23) holds nd the result follows from Theorem 2.6. Theorem 2.9 Suppose (8) holds nd f C(, c T IR n ; IR n ). function p C rd (, c T ; 0, )) such tht If there exists f(t, u) f(t, v) p(t) u v, for ll t, c T, u, v IR n, (25) nd (23) holds then the BVP (2), (3) hs unique solution in C(, c T ; IR n ). Proof See tht (25) implies tht f(t, q) p(t) q + f(t, 0), for ll t, c T, q IR n nd therefore by Theorem 2.6, the BVP (2), (3) hs t lest one solution. Now suppose tht there re two solutions u 1 nd u 2 to (2), (3). Let z = u 1 u 2 nd consider the BVP z = f(t, u 1 ) f(t, u 2 ), t, c T, Mz() + Rz(σ(c)) = 0. Arguing s in the proofs of Lemm 2.1 nd Theorem 2.6, for t, σ(c) T we obtin z(t) = (M + R) 1 R + nd rerrnging we obtin f(s, u 1 (s)) f(s, u 2 (s)) s (1 + (M + R) 1 R ) f(s, u 1 (s)) f(s, u 2 (s)) s mx z(t) t,σ(c) T mx z(t) ( 1 (1 + (M + R) 1 R ) t,σ(c) T p(s) s p(s) s) 0 so we hve z(t) = 0 for ll t, c T by (23) nd hence the solution is unique. Remrk 2.10 In view of the rther generl delt integrl inequlity (23) in Theorems 2.6 nd 2.9, it seems nturl to explicity clrify this condition for the benefit of the reder when considering exmples of common time scles.

10 20 Christopher C. Tisdell 8, 3(2006) 1. If T = IR then (23) becomes 2. If T = Z then (23) becomes 1 + (M + R) 1 R c p(s) ds < (M + R) 1 R c p(s) ds < 1. s= 3. If T = h Z = {0, ±h, ±2h,...}, h > 0, then (23) becomes 1 + (M + R) 1 R c h 1 s= h h p(sh) < If T, σ(c) consists only of isolted points then (23) becomes 1 + (M + R) 1 R s,σ(c)) T µ(s)p(s) < If T = q IN {0} = {q 1, q 2, q 3,...} {0}, q > 1, then µ(t) = (q 1)t nd (23) becomes 1 + (M + R) 1 R s,qc) T (q 1)s p(s) < 1. This type of time scle ppers in the so-clled quntum clculus nd quntum difference equtions 9, 11. Exmple 2.11 Consider the following discrete BVP in T = Z. x = ( x 1, x 2 ) = (t 2 x 1 cos x 2, t 2 x 2 sin x 1 ), t 10, 19 Z, (26) = (f 1 (t, x 1, x 2 ), f 2 (t, x 1, x 2 )) = f(t, x), x(10) + x(20) = (1, 1). (27) The BVP (26), (27) hs t lest one solution. Proof We will use Theorem 2.6 with n = 2, M = 1 = R nd α = (1, 1). We hve x 2 1 f(t, x) = cos2 x 2 + x 2 2 sin2 x 1 t 4 with p(t) = t 2, r(t) = 0 nd thus (22) holds. t 2 x x2 2 = p(t) x,

11 8, 3(2006) Existence nd Uniqueness to Nonliner Dynmic Boundry Condition (23) becomes 1 + (M + R) 1 R p(s) s = 1 + 1/2 = s 2 s= < 1. Thus ll of the conditions of Theorem 2.6 hold nd we conclude tht the BVP hs t lest one solution. 3 A Hybrid Exmple A hybrid continuous-discrete exmple is now presented. defined by T := 2, 3 {4} 5, 6. Here Consider the time scle σ(t) = { t, for t 2, 3) 5, 6, t + 1, for t = 3, 4, nd so µ(t) = { 0, for t 2, 3) 5, 6, 1, for t = 3, 4, nd thus x (t) = { x (t), for t 2, 3) 5, 6, x(t), for t = 3, 4. For the bove time scle, consider the two-dimensionl system of dynmic BVPs ( x x = (x 1, x 2 2 ) = 1 + x 2 2 cos x ) 1 x 2 t 6, 1 + x 2 2 t 6, t 2, 6 T, (28) = (f 1 (t, x 1, x 2 ), f 2 (t, x 1, x 2 )) = f(t, x), x(2) + x(6) = (1, 1). (29) We clim tht (28), (29) dmits t lest one solution. We wnt to use Theorem 2.6. See tht f(t, x) = (x x 2 2 ) cos2 x 1 + (x x2 2 ) t 12 t 6 (x x 2 2) 1/2 = p(t) x, with p(t) = t 6, r(t) = 0. Hence (22) holds.

12 22 Christopher C. Tisdell 8, 3(2006) We need to show tht (23) holds. As in Exmple 2.11, 1 + (M + R) 1 R = 3/2. Also, p(s) s = = p(s) s + s 6 ds p(s) s + 4 s s= p(s) s s 6 ds < 2/3, so tht (23) holds. All of the conditions of Theorem 2.6 hold nd we conclude tht the BVP hs t lest one solution. 4 Open Problems It seems nturl to extend the workings of this pper in two possible directions: 1. by ttempting to remove condition (8) from the theorems of Section 2. (For T = IR, 12 imposed the condition M + R = 0, which is n extension of (8).) 2. by considering wider clsses of BVPs with three- or multipoint boundry conditions. Currently very few rticles pper on systems of first-order multipoint BVPs, even for the cse T = IR, see 14, p.212, 12, p Received: My Revised: Sept References 1 P. Amster, C. Rogers nd C. C. Tisdell, Existence of Solutions to Boundry Vlue Problems for Dynmic Systems on Time Scles, J. Mth. Anl. Appl., 308 (2005), no. 2, M. Bohner nd A. Peterson, Dynmic Equtions on Time Scles, Birkhuser, Boston, M. Bohner nd A. Peterson, (Eds), Advnces in Dynmic Equtions on Time Scles, Birkhuser, Boston, F. B. Christinsen nd T. M. Fenchel, Theories of Popultions in Biologicl Communities, Lecture Notes in Ecologicl Studies, 20. Springer- Verlg, Berlin, 1977.

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