Теоремы типа Бохера для динамических уравнений на временных шкалах

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1 Теоремы типа Бохера для динамических уравнений на временных шкалах c Владимир Шепселевич Бурд Ярославский государственный университет им. П.Г. Демидова Аннотация. Найдены условия, при которых все решения систем динамических уравнений на временных шкалах стремятся к конечным пределам при t. rxiv: v1 [mth.ca] 16 Sep 2016 Theorems of Bôcher s type for dynmic equtions on time scles V.Sh. Burd Demidov Yroslvl Stte University Abstrct The conditions re found tht ll solutions of systems dynmic equtions on time scles tends to finite limits s t. A. Wintner [1,2] hs nmed by theorems of type Bôher ssertions which is gurnteed tht the ech nontrivil solution of the liner system of ordinry differentil equtions dx dt = A(t)x, t 0 t <, where A(t) is n n mtrix, tends to nontrivil finite limit s t. It hs plce, if the elements of mtrix A(t) belong to L 1 [t 0, ]. This result is connected with nme of Bôcher. [3] (see lso [4]). А. Wintner hs receive series of less restrictive conditions. P, Hrtmn [5] hs estblish nlogue of Bˆоher s result for nonliner system. In this pper nlogicl problem is studied for dynmicl equtions on time scles. We consider the liner eqution x = A(t)x, t T, (1) where T R is time scle nd supt =, A(t) be n n n-vlued function on T. It is ssumed tht A(t) is rd-continuous nd regressive (see [6]). The system(1) will be sid to be of clss (S) if (i) every solution x(t) of (1) hs limit x s t, nd (ii) for every constnt vector x there is solution x(t) of (1) such tht x(t) x s t. Evidently tht (1) is of clss (S) if nd only if for every fundmentl mtrix X(t) of (1), X = limx(t) exists s t nd is nonsingulr. 1

2 Let s note still tht (1) is of clss (S), if it hs fundmentl mtrix of form s t. Let T, > 0. Let integrl well known. (S). Theorem 1. If integrl X 0 (t) = I +o(1) A(s)ds is convergent bsolutely. Following theorem A(s) s is convergent bsolutely. Then system (1) is of clss Proof. The solution of system (1) re represented in the form We hve the inequlity x(t) = x()+ x(t) x() + The Gronwll s inequlity for times scles gives x(t) x(0) e A (t,), t T, where e A (t,) is exponentil function of eqution x = A(t) x. A(s)x(s) s. (2) A(s) x(s) s. (3) Consequently ll solutions of system (1) is bounded. Let t 2 > t 1. Let s evlute norm x(t 2 ) x(t 1 ) : x(t 2 ) x(t 1 ) 2 t 1 A(s) x(s) s. (4) From inequlity (4) follows tht x(t 2 ) x(t 1 ) cn be mde s much s smll if to tke t 1 big enough, Therefore x(t) converges to the finite limits s t. From Gronwll s inequlity follows x(t) x(s) (e A (t,s)) 1, t,s T. (5) The inequlity (5) implies tht x 0 unless x(n) 0. This proves theorem 1. Let now integrl mke the chnge of vribles A(s) s is convergent (possibly just conditionlly). In system (1) x(t) = y(t)+y(t)y(t), 2

3 where N N mtrix Y(t) we shll choose lter. We obtin [I +Y(σ(t))]y +Y (t)y(t) = A(t)y(t)+A(t)Y(t)y(t). (6) Let Then Y (t) = A(t). Y(t) = A(s) s. t Evidently Y(t) 0 s t. Therefore mtrix I + Y(σ(t)) hs bounded converse for sufficiently lrge t. Therefore we obtin for sufficiently lrge t y (t) = (I +Y(t)) 1 A(t)Y(t)y(t). (7) Theorem 2. Let integrl A(s) s is convergent nd integrl A(t)( A(s) s) t (8) is convergent bsolutely. Then system (1) is of clss (S). Proof. From conditions theorem 2 follows tht right prt of system (7) stisfy conditions of theorem 1. Remrk. If X(t) is fundmentl mtrix of system (1), then X 1 (t) is fundmentl mtrix of the system x = x ( A)(t), where (see [6]) ( A)(t) = A(t)[I +µ(t)a(t)] 1. Therefore the theorem 2 remins fir, if the requirement of bsolute convergence integrl (8) to replce with the requirement of bsolute convergence of integrl ( Further we receive the theorem. Theorem 3. Let integrls A(s) s nd integrl A(t)( ( A)(s) s)a(t) t. A(s)( A(t)( A(τ) τ) s) t A(s) s) t re convergent nd τ=s 3

4 is convergent bsolutely. Then system (1) is of clss (S). Proof. In system (1) mke now the chnge of vribles x(t) = y(t)+y 1 (t)y(t)+y 2 (t)y(t). where mtrices Y 1 (t), Y 2 (t) re we shll choose lter. We obtin (I +Y 1 (σ(t))+y 2 (σ(t))y(t) +(Y1 (t))y(t)+ +(Y2 (t))y(t) = (A(t)+A(t)Y 1 (t)+a(t)y 2 (t))y(t). (9) Let Then Y 1 (t) = Y 1 (t) = A(t), Y 2 (t) = A(t)Y 1 (t). A(s) s. Y 2 (t) = A(s)( A(τ) τ) s (10) t From the formuls (10) follows tht Y 1 (σ(t)), Y 2 (σ(t)) re converges to 0 s t. Therefore mtrices I + Y 1 (σ(t)), I + Y 2 (σ(t)) hve bounded converse for sufficiently lrge t. Hence system (9) cn be written in following form t y (t) = (I +Y 1 (σ(t))+y 2 (σ(t))) 1 A(t)Y 2 (t)y(t). (11) From conditions theorem 3 follows tht right prt of system (6) stisfy conditions of theorem 1. Generlly, if integrls τ=s A(t) t, A(t)( A(s) s) t,..., A(t 1 )( re convergent nd integrl A(t 1 ) t 2 =t 1 A(t 2 ) t 2 t 2 =t 1 A(t 2 ) t 2 t k =t k 1 A(t k ) t k ) t 1 t k+1 =t k A(t k+1 ) t k+1 ) t 1 is convergent bsolutely, then system (1) is of clss (S). Consider now the nonliner system of difference equtions x (t) = f(t,x(t)), t T, x R N. (12) 4

5 Theorem 4. Let f(t, x) be defined for t T x < δ( ) nd stisfy inequlity f(t,x) K(t) x, where K(t) t <. If x 0 is sufficiently smll, let us sy Then for solution x(t) of (12) stisfying x() = x 0 exists nd x 0 unless x(t) 0. x 0 e K (t,t 0 ) < δ, (13) x = lim t x(t) Proof. The solutions of system (12) re represented in the form x(t) = x()+ f(s,x(s)) s. (14) If x() = x 0 stisfies (14), then from (13) nd conditions of theorem 4 follows tht The Gronwll s inequlity gives x(t) [ x() + K(t) x(t). x(t) x()e K (t,t 0 ). Consequently ll solutions of system (11) is bounded. Let t 2 > t 1. Let s evlute norm x(t 2 ) x(t 1 ) : x(t 2 ) x(t 1 ) 2 t 1 K(s) x(s) s. (15) From inequlity (4) follows tht x(t 2 ) x(t 1 ) cn be mde s much s smll if to tke t 1 big enough, Therefore x(t) converges to the finite limits s t. From Gronwll s inequlity follows x(t) x(s) (e K (t,s)) 1, t,s T. (16) The inequlity (16) implies tht x 0 unless x(n) 0. This proves theorem 4. Let T = R. Then the theorem 2 together with the remrk 1 return to the results Wintner [1]. 5

6 Let T = Z. We consider discrete dibtic oscilltor x(n+2) (2cosα)x(n+1)+(1+g(n))x(n) = 0, (17) where 0 < α < π, g(n) 0 s n. For g(n) 0 the eqution (16) hs the form x(n) = C 1 cosnα+c 2 sinnα. We convert (16) into system of equtions by introducing new vribles C 1 (n), C 2 (n) We obtin the system x(n) = C 1 (n)cosnα+c 2 (n)sinnα, x(n+1) = C 1 (n)cos(n+1)α+c 2 (n)sin(n+1)α. u(n) = B(n)g(n)u(n), where u 1 (n) = C 1 (n+1) C 1 (n), u 2 (n) = C 2 (n+1) C 2 (n). The mtrix B(n) hs the form B(n) = 1 sinα (A 0 +A 1 (n)), where A 0 = ( sinα cosα cosα sinα ) ( sin(2n+1)α cos(2n+1)α, A 1 (n) = cos(2n+1)α sin(2n+1)α ). References 1. Wintner A. On theorem of Bôcher in the theory of ordinry liner differentil equtions, Amer. J. Mth., v. 76, 1954, pp Wintner A. Addend to the pper on Bôcher s theorem, Amer. J. Mth., v. 78, 1956, pp Bôcher M. On regulr singulr points of liner differentil equtions of the second order whose coefficients re not necessrily nlytic, Trns. Amer. Mth. Soc., v.1, 1900, pp Dunkel O. Regulr singulr points of system of homogeneous liner differentil equtions of the first order, Proc. Amer. Acd. Arts sci., v. 38, , pp Hrtmn P. Ordinry differentil equtions,new York, Wiley. 1964, 612 p. 6. Bohner M., Peterson A. Dynmic equtions on time scles, Boston, Birkhuser, 2000, 358 p. 6