Existence of Solutions to First-Order Dynamic Boundary Value Problems

Transcription

1 Interntionl Journl of Difference Equtions. ISSN Volume Number (2006), pp. 7 c Reserch Indi Publictions Existence of Solutions to First-Order Dynmic Boundry Vlue Problems Qiuyi Di nd Christopher C. Tisdell Deprtment of Mthemtics, Hunn Norml University, Chngsh Hunn 4008, P R Chin E-mil: diqiuyi@yhoo.com.cn School of Mthemtics nd Sttistics, The University of New South Wles, Sydney NSW 2052, Austrli E-mil: cct@mths.unsw.edu.u Abstrct This rticle investigtes the existence of solutions to boundry vlue problems (BVPs) involving systems of first-order dynmic equtions on time scles subject to two-point boundry conditions. The methods involve novel dynmic inequlities nd fixed-point theory to yield new theorems gurnteeing the existence of t lest one solution. AMS subject clssifiction: 39A2, 34B5. Keywords: Existence of solutions, boundry vlue problems, dynmic equtions, fixed point methods, time scles.. Introduction This pper considers the existence of solutions to the first-order dynmic eqution of the type x + b(t)x = h(t, x), t, c] T :=, c] T, (.) subject to the boundry conditions G(x(), x(σ(c))) = 0,, c T, (.2) This reserch ws funded by The Austrlin Reserch Council s Discovery Projects (DP ). Received April 3, 2006; Accepted April 23, 2006

2 2 Qiuyi Di nd Christopher C. Tisdell where h :, c] T R n R n is continuous, nonliner function; t is from soclled time scle T (which is nonempty closed subset of R); x is the generlized derivtive of x; the function b :, c] T R; < c re given constnts in T; nd G is some known function describing liner set of boundry conditions. Eqution (.) subject to (.2) is known s dynmic boundry vlue problem (BVP) on time scles. If T = R, then x = x nd (.), (.2) become the following BVP for ordinry differentil equtions x + b(t)x = h(t, x), t, c], (.3) G(x(), x(c)) = 0, (.4) If T = Z, then x = x nd (.), (.2) become the following BVP for difference equtions x + b(t)x = h(t, x), t {, +,..., c}, (.5) G(x(), x(c + )) = 0,, c Z. (.6) There re mny more time scles thn just T = R nd T = Z nd hence mny more dynmic equtions. The field of dynmic equtions on time scles provides nturl frmework for:. estblishing new insight into the theories of non-clssicl difference equtions; 2. forming novel knowledge bout differentil-difference equtions; 3. dvncing, in their own right, ech of the theories of differentil equtions nd (clssicl) difference equtions. In cses nd 2, interested reserchers tend to nlyze known results for differentil equtions nd/or (clssicl) difference equtions nd then extend these ides to the more generl time scle setting. Above, non-clssicl difference equtions include, for exmple, the rpidly developing q-difference equtions ], used in physics. Differentil-difference equtions feture both differentil equtions nd difference equtions. These type of equtions pper in models where time flows continuously nd discretely t different periods, see Exmple 5.. In sitution 3, reserchers desire to formulte new results in the generl time scle setting, with prticulr significnce being found when specil cses of the new results re novel, even for the differentil or difference eqution cse. Motivted by the bove, nd lso by 7, 8], this rticle investigtes the existence of solutions to systems of dynmic equtions in the generl time scle setting. Some sufficient conditions, in terms of dynmic inequlities on h, re presented tht ensure the existence of t lest one solution to the dynmic BVP under considertion. The min tools involve fixed-point methods nd the Nonliner Alterntive. This rticle dvnces ll three situtions rised bove. In Sections 2 nd 3, the interest is in the first two situtions. In Section 4, the interest lies in the third sitution.

3 Dynmic BVPs 3 To understnd the nottion used bove, some preliminry definitions re needed. Definition.. A time scle T is nonempty closed subset of the rel numbers R. 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.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 R. Definition.3. 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.4. 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 =,..., n. Cll x (t) the delt derivtive of x(t) nd sy tht x is delt differentible. Definition.5. If K (t) = k(t), then define the delt integrl by If T = R, then k(s) s = k(s) s = K(t) K(). k(s)ds, while if T = Z, then t k(s) s = k(s). Once gin, there re mny more time scles thn just R nd Z nd hence there re mny more delt integrls. For more generl definition of the delt integrl see 2]. Theorem.6. 9] Assume tht k : T R n nd let t T κ. (i) If k is delt differentible t t, then k is continuous t t.

4 4 Qiuyi Di nd Christopher C. Tisdell (ii) If k is continuous t t nd t is right-scttered, then k is delt differentible t t with k k(σ(t)) k(t) (t) =. σ(t) t (iii) If k is delt differentible nd t is right-dense, then k k(t) k(s) (t) = lim. s t t s (iv) If k is delt differentible t t, then k(σ(t)) = k(t) + µ(t)k (t). The reltively young theory of time scles dtes bck to Hilger 9]. The monogrphs 2] nd 2] lso provide n excellent introduction. For more recent developments in dynmic equtions on time scles, the reder is referred to, 3, 5 8, 0, 4, 5, 9, 20]. A solution to (.), (.2) is continuous function x :, σ(c)] T R n (denoted by x C(, σ(c)] T ; R n )) tht stisfies (.) nd (.2). In wht follows, if y, z R n, then y, z denotes the usul inner product nd z denotes the Eucliden norm of z on R n. 2. Existence for the Non-Periodic Cse This section considers the existence of solutions to the first-order dynmic eqution subject to the boundry conditions x = f(t, x), t, c] T, (2.) Mx() + Rx(σ(c)) = 0,, c T, (2.2) where f :, c] T R n R n is continuous, nonliner function; < c re given constnts in T; nd M, R re given constnts in R. Throughout this section, ssume M + R 0. (2.3) Lemm 2.. Suppose (2.3) holds. The BVP (2.), (2.2) is equivlent to the integrl eqution x(t) = f(s, x(s)) s (M + R) R f(s, x(s)) s, t, c] T. (2.4) Proof. Let x :, σ(c)] T R n stisfy (2.) nd (2.2). It is esy to see tht x(t) = x() + f(s, x(s)) s, t, σ(c)] T, (2.5)

5 Dynmic BVPs 5 nd x(σ(c)) = x() + So (2.2) gives ( 0 = Mx() + R x() + nd rerrnging (2.6) yields f(s, x(s)) s. f(s, x(s)) s ) (2.6) x() = (M + R) R So substituting (2.7) into (2.5) gives, for t, σ(c)] T, x(t) = (M + R) R f(s, x(s)) s + f(s, x(s)) s. (2.7) f(s, x(s)) s. (2.8) If x is solution to (2.4), then is it esy to show tht (2.) nd (2.2) hold by direct clcultion. The following is the min result of this section. Theorem 2.2. Suppose (2.3) holds nd f :, c] T R n R n is continuous. If there exist non-negtive constnts α nd K such tht f(t, q) 2α q, f(t, q) + K, (t, q), c] T R n, (2.9) nd M/R, (2.0) then the BVP (2.), (2.2) hs t lest one solution. Proof. By Lemm 2., we wnt to show tht there exists t lest one solution to (2.4), which is equivlent to showing tht (2.), (2.2) hs t lest one solution. To do this, we use the Nonliner Alterntive. Consider the mp T : C(, σ(c)] T ; R n ) C(, σ(c)] T ; R n ) defined by T x(t) = (M + R) R f(s, x(s)) s + f(s, x(s)) s, t, σ(c)] T. Thus, our problem is reduced to proving the existence of t lest one x such tht x = T x. (2.) With this in mind, consider the fmily of equtions ssocited with (2.) given by x = λt x, λ 0, ]. (2.2)

6 6 Qiuyi Di nd Christopher C. Tisdell We show tht x λt x, x B P, λ 0, ], (2.3) for some suitble bll B P C(, σ(c)] T ; R n ) with rdius P > 0. Let { B P = x C(, σ(c)] T ; R n ) mx x(t) < P t,σ(c)] T P = + (M + R) R ] K(σ(c) ) +. Let x be solution to (2.2) nd see tht, by Lemm 2., x must lso be solution to }, x = λf(t, x), t, c] T, λ 0, ], (2.4) Mx() + Rx(σ(c)) = 0. (2.5) Consider r(t) := x(t) 2 for ll t, σ(c)] T. By the product rule 2, Theorem.20 (iii)] nd Theorem.6 (iv) we hve From (2.6) nd (2.9) obtin Also, (2.0) implies since (2.2) gives r (t) = 2 x(t), x (t) + µ(t) x (t) 2, t, c] T, = 2 x(t), λf(t, x(t)) + µ(t) λf(t, x(t)) 2 2 x(t), λf(t, x(t)). (2.6) λf(t, q) 2α q, λf(t, q) + λk αr (t) + K, (2.7) r(σ(c)) r() (2.8) x(σ(c)) M/R x() x(). Let H := + (M + R) R. We show tht λt x < P for ll x P nd thus (2.3) will hold. With this in mind, consider λt x(t) = (M + R) R λf(s, x(s)) s + λf(s, x(s)) s H H λf(s, x(s)) s αr (s) + K ] s, from (2.7) = H α(r(σ(c)) r()) + K(σ(c) )] H K(σ(c) )], from (2.8) < P

7 Dynmic BVPs 7 nd thus T : B P C(, σ(c); R n ) stisfies (2.3). The opertor T : B P C(, σ(c); R n ) is compct by the Arzel Ascoli theorem (becuse it is completely continuous mp restricted to closed bll). The Nonliner Alterntive ensures the existence of t lest one solution in B P to (2.4) nd hence to (2.), (2.2). If M = = N, then the boundry conditions (2.2) become the so-clled ntiperiodic boundry conditions x() = x(σ(c)) (2.9) nd the following corollry to Theorem 2.2 esily follows. Corollry 2.3. Let f :, c] T R n R n be continuous. If there exist non-negtive constnts α nd K such tht (2.9) holds, then the nti-periodic BVP (2.), (2.9) hs t lest one solution. Proof. It is esy to see tht for M = = R, ll of the conditions of Theorem 2.2 hold. Thus the result follows from Theorem 2.2. Now consider (2.), (2.2) with n =. For this cse the following corollry to Theorem 2.2 is obtined. Corollry 2.4. Suppose (2.3) holds nd let f :, c] T R R be continuous. If there exist non-negtive constnts α nd K such tht f(t, q) 2αqf(t, q) + K, (t, q), c] T R, (2.20) nd M/R, (2.2) then, for n =, the BVP (2.), (2.2) hs t lest one solution. Proof. It is esy to see tht for n = : (2.9) becomes (2.20); nd the result follows from Theorem Existence for the Periodic Cse This section considers the existence of solutions to the first-order system of periodic BVPs x + b(t)x = g(t, x), t, c] T, (3.) x() = x(σ(c)), (3.2) where g :, c] T R n R n nd b :, c] T R re both continuous functions, with b hving no zeros on, c] T. Also considered herein, is the existence of solutions to the first-order system x = f(t, x), t, c] T, (3.3)

8 8 Qiuyi Di nd Christopher C. Tisdell subject to (3.2), where f :, c] T R n R n is continuous function. We need few more definitions to ssist with our investigtion. The following gives generlized ide of continuity on time scles. Definition 3.. Assume k : T R. Define nd denote k C rd (T; R) s right-dense continuous (rd-continuous) if: k is continuous t every right-dense point t T; nd lim k(s) exists nd is finite t every left-dense point t T. s t Now define the so-clled set of regressive functions, R, by R = {p C rd (T; R) nd + p(t)µ(t) 0 on T}. The methods in this section rely on generlized exponentil function e p (, t 0 ) on time scle T. For p R, we define (see 2, Theorem 2.35]) the exponentil function e p (, t 0 ) on the time scle T s the unique solution to the IVP x = p(t)x, x(t 0 ) = x 0. If p R, then e p (, t 0 ) hs no zeros. More explicitly, the exponentil function e p (, t 0 ) is given by e p (t, t 0 ) = ( ) exp p(s) s, for t T, µ = 0, t 0 ( exp t 0 where Log is the principl logrithm function. Throughout this section ssume ) Log( + µ(s)p(s)) s, for t T, µ > 0, µ(t) b R, nd e b (σ(c), ). (3.4) Lemm 3.2. Let (3.4) hold. The BVP (3.), (3.2) is equivlent to the integrl eqution x(t) = e b (t, ) e b (σ(c), ) for t, σ(c)] T. ] g(s, x(s)) t e b (σ(s), ) s + g(s, x(s)) e b (σ(s), ) s (3.5) Proof. Let x be solution to (3.), (3.2). By the quotient rule 2, Theorem.20 (v)], consider ] x(t) = x (t) + b(t)x(t) e b (t, ) e b (σ(t), ) = g(t, x(t)) e b (σ(t), ), t, c] T,

9 Dynmic BVPs 9 nd hence integrting the bove on, t] T obtin ] g(s, x(s)) x(t) = e b (t, ) x() + e b (σ(s), ) s, t, σ(c)] T. (3.6) Using (3.2) nd (3.6), obtin x() = x(σ(c)) = e b (σ(c), ) nd rerrngement leds to x() = x() + ] g(s, x(s)) e b (σ(s), ) s σ(c) g(s, x(s)) s. (3.7) e b (σ(c), ) e b (σ(s), ) So by substituting (3.7) into (3.6), the result follows. If x is solution to (3.5), then x stisfies (3.), (3.2), which my be verified by direct computtion. The following is the min result of this section. Theorem 3.3. Let (3.4) hold nd g :, c] T R n R n be continuous. If there exist non-negtive constnts α nd K such tht λg(t, q) e b (σ(t), ) α 2 q, λg(t, q) b(t)q ] + K, then the BVP (3.), (3.2) hs t lest one solution. (t, q, λ), c] T R n 0, ], (3.8) Proof. From Lemm 3.2 we see tht the BVP (3.), (3.2) is equivlent to the integrl eqution (3.5). Define the mp T : C(, σ(c)] T ; R n ) C(, σ(c)] T ; R n ) by T x(t) = e b (t, ) e b (σ(c), ) ] g(s, x(s)) t e b (σ(s), ) s + g(s, x(s)) e b (σ(s), ) s for t, σ(c)] T. Thus, our problem is reduced to proving the existence of t lest one x such tht With this in mind, it is sufficient to show tht x = T x. (3.9) λt x x, x B P, λ 0, ], (3.0)

10 0 Qiuyi Di nd Christopher C. Tisdell for some suitble bll B P C(, σ(c)] T ; R n ) with rdius P > 0. Let { } B P = x C(, σ(c)] T ; R n ) mx x(t) < P, t,σ(c)] T ( )] P = e b (t, ) e b (σ(c), ) + K +. sup t,σ(c)] T The rest of the proof follows similr lines to the proof of Theorem 2.2 nd so is only briefly sketched. Let x be solution to λt x = x. Consider r(t) := x(t) 2 for ll t, σ(c)] T. By the product rule 2, Theorem.20 (iii)] we hve Let r (t) = 2 x(t), x (t) + µ(t) x (t) 2, t, c] T, = 2 x(t), λg(t, x(t)) b(t)x(t) + µ(t) λg(t, x(t)) b(t)x(t) 2. 2 x(t), λg(t, x(t)) b(t)x(t). (3.) ( )] H (t) := e b (t, ) e b (σ(c), ) +, t, σ(c)] T. Now, for ech t, σ(c)] T, consider λt x(t) e b (t, ) e b (σ(c), ) λg(s, x(s)) H (t) e b (σ(s), ) s H (t) H (t) 2α x(s), λg(s, x(s)) b(t)x(s) + K] s αr (s) + K ] s = H (t) α( x(σ(c)) 2 x() 2 ) + K ] = H (t)k < P ] λg(s, x(s)) t e b (σ(s), ) s + λg(s, x(s)) e b (σ(s), ) s nd so (3.0) holds. The result now follows from the Nonliner Alterntive. The following two corollries give less technicl conditions implying (3.8) which re esy to verify in prctice. Corollry 3.4. Let (3.4) hold with b < 0 nd let g :, c] T R n R n be continuous. If there exist non-negtive constnts α nd K such tht g(t, q) e b (σ(t), ) 2α q, g(t, q) + K, (t, q), c] T R n, (3.2)

11 Dynmic BVPs then the BVP (3.), (3.2) hs t lest one solution. Proof. It is not difficult to show tht if (3.2) holds, then, since b < 0, (3.8) must lso hold nd the result follows from Theorem 3.3. Corollry 3.5. Let (3.4) hold nd let g :, c] T R n R n be continuous. If g is bounded on, c] T R n, then the BVP (3.), (3.2) hs t lest one solution. Proof. Since g is bounded on, c] T R n, there exists non-negtive constnt L such tht sup (t,q),c] T R n g(t, q) L. Thus, (3.8) will hold for the choices α = 0 nd K = sup t,c] T e b (σ(t), ) L. The result then follows from Theorem 3.3. Now consider (3.), (3.2) with n =. For this cse, the following corollry to Theorem 3.3 is obtined. Corollry 3.6. Let g be continuous, sclr-vlued function nd let (3.4) hold. If there exist non-negtive constnts α nd K such tht λg(t, q) e b (σ(t), ) 2αqλg(t, q) b(t)q2 ] + K, (t, q, λ), c] T R 0, ], (3.3) then, for n =, the BVP (3.), (3.2) hs t lest one solution. Proof. It is esy to see tht for n = : (3.8) becomes (3.3); nd the result follows from Theorem 3.3. Attention is now turned to (3.3), (3.2). As it stnds, the BVP (3.3), (3.2) is not gurnteed to be invertible. Tht is, we my be unble to reformulte it s n equivlent delt integrl eqution. However, the BVP x x = f(t, x) x, t, c] T, (3.4) x() = x(σ(c)), (3.5) is invertible since Lemm 3.2 holds for the specil cses b = nd g(t, p) = f(t, p) p. We will use this to now formulte some existence theorems for (3.3), (3.2). Theorem 3.7. Let f :, c] T R n R n be continuous nd let (3.4) hold for b =. If there exist non-negtive constnts α nd K such tht f(t, q) q e (σ(t), ) 2α q, f(t, q) + K, (t, q), c] T R n, (3.6)

12 2 Qiuyi Di nd Christopher C. Tisdell then the BVP (3.3), (3.2) hs t lest one solution. Proof. Consider the BVP (3.3), (3.2) rewritten s x x = f(t, x) x, t, c] T, (3.7) x() = x(σ(c)). (3.8) The BVP (3.7), (3.8) is in the form (3.), (3.2) with b = nd g(t, p) = f(t, p) p. It is not difficult to show tht (3.8) reduces to (3.6) for these specil cses. Hence the result follows from Theorem 3.3. Corollry 3.8. Let f :, c] T R n R n be continuous nd let (3.4) hold for b =. If there exist non-negtive constnts α nd K such tht f(t, q) q e (σ(t), ) 2α q, f(t, q) + K, (t, q), c] T R n, (3.9) then the BVP (3.3), (3.2) hs t lest one solution. Proof. The result follows from Corollry 3.8 with b =, g(t, q) = f(t, q) q. Corollry 3.9. Let f :, c] T R n R n be continuous. If f(t, q) q is bounded for (t, q), c] T R n, then the BVP (3.3), (3.2) hs t lest one solution. Proof. This is specil cse of Corollry 3.9 with b =, g(t, q) = f(t, q) q. 4. More on Existence for the Periodic Cse This section considers the existence of solutions to the first-order system of periodic BVPs x + b(t)x σ = g(t, x σ ), t, c] T, (4.) x() = x(σ(c)), (4.2) where g :, c] T R n R n nd b :, c] T R re both continuous functions, with b hving no zeros on, c] T nd x σ (t) := x(σ(t)). The reder my wonder why x σ ppers in (4.). There re two resons: firstly, the theory developed in this section is, in generl, inpplicble to (3.); nd secondly, by studying (4.), dvncement in the theory of is still ttinble (by tking T = R). Throughout this section ssume x + b(t)x = g(t, x), t, c], (4.3) x() = x(c). (4.4) b R, nd e b (σ(c), ). (4.5)

13 Dynmic BVPs 3 Lemm 4.. Let (4.5) hold. The BVP (4.), (4.2) is equivlent to the integrl eqution σ(c) x(t) = g(s, x σ (s))e b (s, ) s e b (t, ) e b (σ(c), ) ] + g(s, x σ (s))e b (s, ) s (4.6) for t, σ(c)] T. Proof. Let x be solution to (3.), (3.2). By the product rule, consider x(t)e b (t, )] = x (t) + b(t)x σ (t)]e b (t, ) = g(t, x(t))e b (t, ), t, c] T. Integrting the bove on, t] T nd then using the boundry conditions, the result follows in similr wy to the proof of Lemm 3.2. If x is solution to (4.6), then x stisfies (4.), (4.2), which my be verified by direct computtion. The following is the min result of this section. Theorem 4.2. Let (4.5) hold nd g :, c] T R n R n be continuous. If there exist non-negtive constnts α nd K such tht λg(t, q)e b (t, ) 2α q, λg(t, q) b(t)q + K, (t, q, λ), c] T R n 0, ], (4.7) then the BVP (4.), (4.2) hs t lest one solution. Proof. From Lemm 4. we see tht the BVP (4.), (4.2) is equivlent to the integrl eqution (4.6). Define the mp T 2 : C(, σ(c)] T ; R n ) C(, σ(c)] T ; R n ) by T 2 x(t) = e b (t, ) e b (σ(c), ) + g(s, x σ (s))e b (s, ) s ] g(s, x σ (s))e b (s, ) s for t, σ(c)] T. Thus, our problem is reduced to proving the existence of t lest one x such tht With this in mind, it is sufficient to show tht x = T 2 x. (4.8) λt 2 x x, x B P2, λ 0, ], (4.9)

14 4 Qiuyi Di nd Christopher C. Tisdell for some suitble bll B P2 C(, σ(c)] T ; R n ) with rdius P 2 > 0. Let { } B P2 = x C(, σ(c)] T ; R n ) mx x(t) < P 2, t,σ(c)] T ( )] P 2 = e b (t, ) e b (σ(c), ) + K +. sup t,σ(c)] T The rest of the proof follows similr lines to the proof of Theorem 3.3 nd so is only briefly sketched. Let x be solution to λt 2 x = x. Consider r(t) := x(t) 2 for ll t, σ(c)] T. By the product rule 2, Theorem.20 (iii)] nd Theorem.6 (vi) we hve, for α 0, Let αr (t) = α 2 x σ (t), x (t) µ(t) x (t) 2], t, c] T, 2α x(t), λg(t, x σ (t)) b(t)x σ (t). (4.0) ( )] H 2 (t) := e b (t, ) e b (σ(c), ) +, t, σ(c)] T. Now, for ech t, σ(c)] T, consider λt x(t) e b (t, ) e b (σ(c), ) + H 2 (t) H 2 (t) λg(s, x(s)) e b (s, ) s] λg(s, x(s)) e b (s, ) s 2α x(s), λg(s, x(s)) b(t)x(s) + K] s αr (s) + K ] s = H 2 (t) α( x(σ(c)) 2 x() 2 ) + K ] = H 2 (t)k < P 2 nd so (4.9) holds. The result now follows from the Nonliner Alterntive. As specil cse of Theorem 4.2, the following result ppers to be new. Corollry 4.3. Let e R c b(t)dt nd g :, c] R n R n be continuous. If there exist non-negtive constnts α nd K such tht λg(t, q)e R t b(s)ds 2α q, λg(t, q) b(t)q + K, (t, q, λ), c] R n 0, ], (4.)

15 Dynmic BVPs 5 then the BVP (4.3), (4.4) hs t lest one solution. Proof. Tke T = R in Theorem An Exmple An exmple is now provided to highlight some of the theory from previous sections. Exmple 5.. Consider the sclr dynmic BVP (n = ) given by x = tx 3 +, t, c] T, (5.) x() = x(σ(c)),, c T. (5.2) The clim is tht (5.), (5.2) hs t lest one solution for rbitrry T. Proof. Let f(t, q) = tq 3 +. It needs to be shown tht (2.20) holds for non-negtive constnts α nd K. First note tht f(t, q) c q 3 + for (t, q), c] T R. Also note tht for (t, q), c] T R c(q 4 + q + 0) c( q 3 + ) c q 3 + f(t, q). Hence for α nd K to be chosen below 2αqf(t, q) + K = 2αq(tq 3 + ) + K 2α(q 4 + q) + K = c(q 4 + q) + 0c, for the choice α = c/2, K = 0c g(t, q), for ll (t, q), c] T R nd thus (2.20) holds for the choices α = c/2 nd K = 0c. Thus, ll of the conditions of Corollry 2.4 hold nd the BVP (5.), (5.2) hs t lest one solution. To give the reder flvour of the bove result, consider (5.), (5.2) on the time scle T := {2 /i} i= 2, ) = {, 32, 53 },... 2, ). It is not difficult, using Definition.2, to show tht for the bove time scle, 4 t σ(t) = 3 t, for t, 2) T, t, for t 2, ) T.

16 6 Qiuyi Di nd Christopher C. Tisdell For this time scle, let = nd c = 3. So our intervls of interest for (5.), (5.2) become, c] T =, 3] T =, σ(c)] T. Note tht by using Theorem.6 we obtin x(σ(t)) x(t), for t, 2) T, x σ(t) t (t) = x (t), for t 2, ) T. For this specil time scle, (5.) nd (5.2) hve t lest one solution. References ] 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(2): , ] M. Bohner nd A. Peterson, Dynmic equtions on time scles, An introduction with pplictions. Birkhäuser Boston, Inc., Boston, MA, ] M. Bohner nd C.C. Tisdell, Oscilltion nd nonoscilltion of forced second order dynmic equtions. Pcific J. Mth. (in press) 4] F.B. Christinsen nd T.M. Fenchel, Theories of Popultions in Biologicl Communities, Lecture Notes in Ecologicl Studies, 20, Springer-Verlg, Berlin, ] L. Erbe, A. Peterson, nd C.C. Tisdell, Existence of solutions to second-order BVPs on time scles, Appl. Anl., 84(0): , ] Johnny Henderson, Alln Peterson, nd C.C. Tisdell, On the existence nd uniqueness of solutions to boundry vlue problems on time scles, Adv. Difference Equ., 2:93 09, ] J. Henderson nd C.C. Tisdell, Topologicl trnsverslity nd boundry vlue problems on time scles, J. Mth. Anl. Appl., 289():0 25, ] J. Henderson, C.C. Tisdell, nd W.K.C. Yin, Uniqueness implies existence for three-point boundry vlue problems for dynmic equtions, Appl. Mth. Lett., 7(2):39 395, ] S. Hilger, Anlysis on Mesure Chins A Unified Approch to Continuous nd Discrete Clculus, Res. Mth., 8:8 56, ] J. Hoffcker nd C.C. Tisdell, Stbility nd instbility for dynmic equtions on time scles, Comput. Mth. Appl., 49(9-0), , ] V. Kc, nd P. Cheung, Quntum clculus. Universitext. Springer-Verlg, New York, ] B. Kymkçln, V. Lkshmiknthm, nd S. Sivsundrm, Dynmicl Systems on Mesure Chins, Kluwer Acdemic Publishers, Boston, ] D. Otero, D. Giulini, nd M. Sssno, Temporl dimension nd trnsition out of chos, Physic A: Sttisticl nd Theoreticl Physics, 78: , 99.

17 Dynmic BVPs 7 4] A.C. Peterson, Y.N. Rffoul, nd C.C. Tisdell, Three point boundry vlue problems on time scles, J. Difference Equ. Appl., 0(9): , ] Alln C. Peterson nd C.C. Tisdell, Boundedness nd uniqueness of solutions to dynmic equtions on time scles, J. Difference Equ. Appl., 0(3-5): , ] V. Spedding, Tming Nture s Numbers, New Scientist, July 2003, ] C.C. Tisdell, Existence of solutions to first-order periodic boundry vlue problems, J. Mth. Anl. Appl., 323(2): , ] C.C. Tisdell, Existence nd uniqueness to nonliner dynmic boundry vlue problems. Cubo (in press) 9] C.C. Tisdell, Pvel Drbek, nd Johnny Henderson, Multiple solutions to dynmic equtions on time scles, Comm. Appl. Nonliner Anl., (4):25 42, ] C.C. Tisdell nd H.B. Thompson, On the existence of solutions to boundry vlue problems on time scles, Dyn. Contin. Discrete Impuls. Syst. Ser. A Mth. Anl., 2(5): , 2005.