PROPERTIES OF L P (K)-SOLUTIONS OF LINEAR NONHOMOGENEOUS IMPULSIVE DIFFERENTIAL EQUATIONS WITH UNBOUNDED LINEAR OPERATOR

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1 PROPERTIES OF L P (K)-SOLUTIONS OF LINEAR NONHOMOGENEOUS IMPULSIVE DIFFERENTIAL EQUATIONS WITH UNBOUNDED LINEAR OPERATOR Atanaska Georgieva Abstract. Sufficient conditions for the existence of L (k)-solutions of linear nonhomogeneous imulsive differential euations with unbounded linear oerator are found. An examle of the theory of the linear nonhomogeneous artial imulsive differential euations of arabolic tye is given. Keywords: Linear nonhomogeneous imulsive differential euations, L (k)-solutions, artial imulsive differential euations of arabolic tye 2 Mathematics Subject Classification: 34A37, 47H. Introction We study the existence of solutions in the saces L (k) ( ) of linear nonhomogeneous imulsive differential euations with unbounded linear oerator. In Theorem we rove the existence of L (k)-solutions. Further we give an examle. We consider a artial imulsive differential euation with ellitic linear art and rece it to an ordinary imulsive differential euation. This ordinary euation satisfy the conditions of Theorem and therefore there exist L (k)-solutions of the considered ordinary euation. By this way, we establish L (k)-solutions also of the given artial imulsive euation. 2. Statement of the roblem Let X be a Banach sace with norm. and identity I. By D(T ) X we will denote the domain of the oerator T : D(T ) X. We consider the following linear nonhomogeneous imulsive differential euation () dt = A(t)u + f(t) for t t n (2) u(t + n ) = Q n (u(t n )) + h n for n =, 2,..., where A(t): D(A(t)) X (t R + ) and Q n : D(Q n ) D(A(t n )) (n =, 2,...) are linear unbounded oerators. The sets D(A(t)) and D(Q n ) (t, n =, 2,...) are dense in X. The function f(.) : R + X is continuous and h = { } h n n= is a seuence of elements of X. The oints of jum t n satisfy the following conditions = t o < t <... < t n <..., lim t n =. We set n Q = I, h =.

2 3 Anniversary International Conference REMIA2 Furthermore, we assume that all considered functions are left continuous and there exist the Cauchy oerator U(t, s) ( s t) of the linear ordinary euation (3) dt = A(t)u. Remark.. Sufficient conditions for the existence of U(t, s) can be found in ([2], [3], [4]). It is easy to rove that the functions u(t) = V (t, s)ξ for ξ D(A(s)) with (4) V (t, s) = U(t, t n )Q n U(t n, t n )Q n... Q k U(t k, s) ( s t k t n < t) satisfy the linear imulsive Cauchy roblem (5) dt = A(t)u for t t n (6) u(t + n ) = Q n (u(t n )) for n =, 2,... (7) u(s) = ξ. Let us note that the oerator V (t, s) is bounded if one of the following conditions holds (B) Q n U(t n, t n ) are bounded oerators (n =, 2,...). (B2) U(t n+, t n )Q n are bounded oerators (n =, 2,...). Let the following condition be fulfilled. (H) There exists a continuous function k(.,. ) : R + R + R + such that V (t, s)ξ k(t, s) ξ, where s < t and ξ D(A(s)). We introce the following saces with norms L (k) = l (k) = g L (k) = su { } g(.) : R + X : su k(t, s) g(s) ds < { g = { g n } n= X : su k(t, s) g(s) ds < t n <t We introce the following conditions. (H) There exists constant M > such that (H2) There exists constant M 2 > such that, g l (k) = su } k(t, t + n ) g n < ( t k(t, s)ds M. k(t, t + n ) M 2. < t n<t k(t, t + n ) g n ).

3 -2 December 2, Plovdiv, Bulgaria 3 3. Main results Lemma.. Let the following conditions be fulfilled:. Condition (B) or (B2) holds. 2. Conditions (H), (H) and (H2) hold. Then for any function f L (k) and for any seuence h = {h n } n= l (k) the linear nonhomogeneous imulsive euation (), (2) has a bounded solution u(t) (t R + ) such that (8) u(t) = V (t, )u() + V (t, s)f(s)ds + < t n <t V (t, t + n )h n. Proof: It is immediately verified that the function u(t) is a solution of the linear nonhomogeneous imulsive euation (), (2). We shall estimate the norm of the integral and the sum in (8). Let =. We use Holder s ineuality. For the norm of the integral in (8) we obtain the estimate V (t, s)f(s)ds <t n <t t k(t, s) f(s) ds ) k(t, s)ds k (t, s)k (t, s) f(s) ds k(t, s) f(s) ds M f L (k). For the norm of the sum in (8) we obtain the estimate V (t, t + n )h n <t n <t ( <t n <t k(t, t + n ) h n ) ( k(t, t + n ) <t n <t <t n <t k (t, t + n )k (t, t + n )h n Lemma 2.. Let the following conditions be fulfilled:. Condition (B) or (B2) holds. 2. Conditions (H) and (H) hold. Then the oerator G, defined by the formula (9) G f(t) = V (t, s)f(s)ds mas L (k) into L (k) and the following estimate is valid () G f L (k) M f L (k), ) k(t, t + n ) h n M 2 h l (k).

4 32 Anniversary International Conference REMIA2 where + =. Proof: Let f L (k). From Lemma we have the estimate () G f(t) M f L. (k) We shall rove G f L (k). From () and condition (H) we obtain G f L (k) = su k(t, s) G f(s) ds su = M f L su (k) k(t, s)ds M f. L(k) Hence ineuality () holds. Lemma 3.. Let the following conditions be fulfilled:. Condition (B) or (B2) holds. 2. Conditions (H) and (H2) hold. Then the oerator G 2, defined by the formula (2) G 2 h(t) = V (t, t + n )h n <t n <t mas l (k) into L (k) and the following estimate is valid (3) G 2 h L(k) M where + =. M 2 h l (k), ) k(t, s)m f L ds (k) = Proof: Let h = {h n } n= be an arbitrary secuence of l (k). From Lemma we have the estimate (4) G 2 h(t) M2 h l. (k) We shall rove G 2 h L (k). From (4) and condition (H2) we obtain G 2 h L (k) = su k(t, s) G 2 h(s) ds su ) k(t, s)m2 h l (k) ds = = M 2 h l su (k) k(t, s)ds M 2 M h l. (k) Hence ineuality (3) holds.

5 -2 December 2, Plovdiv, Bulgaria 33 Theorem.. Let the following conditions be fulfilled:. Condition (B) or (B2) holds. 2. Conditions (H), (H) and (H2) hold. 3. The function V (t, )ξ L (k) (t R +, ξ D(A())). Then for any function f L (k) and for any seuence h = {h n } n= l (k) the linear nonhomogeneous imulsive euation (), (2) has in L (k) a uniue solution and this solution is bounded. Proof: Let the function f L (k) and the seuence h = {h n } n= l (k). Then we write down euality (8) in the form (5) u(t) = V (t, )u() + G f(t) + G 2 h(t), where the oerators G and G 2 are defined by (9), (2). From (5), Lemma 2, Lemma 3 and condition 3 of Theorem it follows that the solution of the linear nonhomogeneous imulsive euation (), (2) is in the sace L (k). We shall illustrate Theorem by an examle from the ualitative theory of the linear nonhomogeneous artial imulse differential euations. Examle. In this examle we consider a artial imulse differential euation and rece it to an ordinary imulse differential euation. For this ordinary imulsive differential euation, the conditions of Theorem are fulfilled. Several notations and results for ordinary differential euations, used in the examle, are given in caite 5 7 of [4]. Note the introtion to the theory of artial imulse differential euations is considered in []. Let Ω be a bounded domain with smooth boundary Ω in R n, Q = (, ) Ω and Γ = (, ) Ω. We denote P n = {(t n, x) : x Ω}, P = P n, Λ n = {(t n, x) : x Ω}, Λ = n= Λ n. Consider the linear nonhomogeneous imulse arabolic euation with initial and smooth conditions n= (6) u t = Ã(t, x, D)u + f(t, x), (t, x) Q \ P (7) D α u(t, x) =, α < m, (t, x) Γ \ Λ (8) u(, x) = v(x), x Ω (9) u(t + n, x) = Q n (u(t n, x)) + h n (x)), x Ω, n =, 2,...,

6 34 Anniversary International Conference REMIA2 where Ã(t, x, D) = α 2m a α (t, x)d α, Q n : D( Q n ) D(Ã(t n, x, D)) (n =, 2,...) are linear oerators, f(.,. ) : R + R n R and h n (.) : R n R are continuous functions. Let X = L (Ω, R) ( < < ), where L (Ω, R) = ( with norm v = v(x) dx Ω { v : Ω R; ). Ω } v(x) dx < With the family Ã(t, x, D), (t R +) of strongly ellitic oerators we associate a family of linear oerators A(t), (t R + ) acting in X by A(t)u = Ã(t, x, D)u, for u D. This is done as follows D = D(A(t)) = W 2m, (Ω) W m, (Ω), (t R + ). Let v X. We set f(t)(x) = f(t, x), Q n (u(t n ))(x) = Q n (u(t n, x)), h n (x) = h n (x), (t R +, x Ω), where Q n : D(Q n ) D are linear oerators, the sets D(Q n ) X lie dense in X, the function f(.) : R + X is continuous and { } h n n= X. Let U(t, s) is the Cauchy oerator of the linear euation dt = A(t)u. Sufficient conditions for the validity of the estimate U(t, s) Ce k(t s) ( s t; C, k > constants) are given in [4]. We shall consider the concrete case when t n = n (n =, 2,...), f(t, x) = e γt ψ(x), Qn ξ = kn C( + n 2 )e C+k ξ, hn (x) = e αn ϕ(x), where the functions ψ, ϕ X and α, γ are ositive constants. Then for ξ X f(t) = e γt kn ξ, Qn ξ = C( + n 2 )e C+k ξ, h n = e αn ξ. Let V (t, s) ( s t) is the Cauchy oerator of the linear imulse euation dt = A(t)u for t t n u(t + n ) = Q n (u(t n )) for n =, 2,...

7 -2 December 2, Plovdiv, Bulgaria 35 Then for < s k < n < t, ξ D the following estimate is valid V (t, s)ξ kte k(t s) ξ. We set k(t, s) = kte k(t s). In this case the conditions of Theorem hold. Hence the ordinary euation (), (2) has L (k)-solution, which inced L (k)-solution of the artial euation (6) (9) for any x Ω. References [] L. Erbe, H. Freedman, X. Liu, J. Wu: Comarison rincile for imulsive arabolic euations with alications to models of single secies growth, J. Austral. Math. Soc. Ser. B 32, (99), [2] J. A. Goldstein: Semigrous of Linear Oerators and Alications, Oxford University Press, New York, (985), 235. [3] M. Krasnoselski, P. Zabreiko, E. Pustilnic, P. Sobolevski: Integrable Oerators in Saces of Integrable Functions, Moskva, (966), 499 (In Russian). [4] A. Pazy: Semigrous of Linear Oerators, Sringer-Verlag, (983), 279. Atanaska Georgieva Faculty of Mathematics and Informatics Plovdiv University 236 Bulgaria Blvd. 43 Plovdiv, Bulgaria, afi2@abv.bg

8 36 Anniversary International Conference REMIA2