TWO-POINT BOUNDARY-VALUE PROBLEMS WITH IMPULSE EFFECTS * Lyudmil I. Karandzhulov

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1 МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 23 MATHEMATICS AND EDUCATION IN MATHEMATICS, 23 Proceedings of the Thirty Second Spring Conference of the Union of Bulgarian Mathematicians Sunny Beach, April 5 8, 23 TWO-POINT BOUNDARY-VALUE PROBLEMS WITH IMPULSE EFFECTS * Lyudmil I Karandzhulov A new scheme for investigation of boundary value problems of ordinary differential equations with impulse effects at finite number of points is suggested The problem is reduced to a two-point boundary value problem which dimension is greater than the dimension of the original problem Conditions for existence of unique solution and family of solutions are obtained Statement of the problem We consider a two-point boundary-value problem with generalized impulse conditions at finite number of points dy () dx = A(x)y + f(x), x a, b, x τ i, (2) By(a) + Cy(b) = d, (3) N i y(τ i + ) + M i y(τ i ) = v i, i =, p, a = τ < τ < τ 2 < < τ p < τ p+ = b, where the coefficients of the system () and the equalities (2), (3) are subordinate to the following conditions: (H) A(x) is (n n) matrix with continuous elements on a, b, f(x) is n-dimensional partially continuous vector function with break points of the first kind at τ i : f(x) = f (x), x a, τ, f(x) = f i (x), x (τ i, τ i+, i =, p, where f i (x) is continuous in τ i, τ i, i =, p + ; (H2) B and C are (k n) constant matrices, d R k and M i, N i i =, p, are (s n) constant matrices, v i R s Further, we give conditions for existence of the solution y( ) C (a, b \ {τ,, τ p }) of the impulsive boundary-value problem () (3) and construct this solution If instead of (3) we consider the following impulse conditions y(τ i + ) + (E n + S i )y(τ i ) = v i, i =, p, where (E n + S i ) are nonsingular matrices, then we obtain the problem investigated in 4 The system () with impulse and boundary conditions united in the form p+ l i y i ( ) * Math Sub Class: 34B5 Keywords: Boundary value problem, impulses, generalized inverse matrices 8 i=

2 where the functionals l i y i ( ) = is investigated in 2 τi τ i dσ(s) C i (s)y i (s), i =, p +, 2 Main results We denote x i = τ i, i =, p + and replace x x i (4) = t, h i = x i x i, i =, p + h i This means that x x i, x i is equivalent to t, Therefore after change of variables (4) the intervals x i, x i, i =, p + are represented in, Let y(x i + th i ) = z i (t), t, when x x i, x i Then dy dx = dz i(t) dt dt dx = dz i (t) Thus () takes the form h i dt dz i (t) = h i A(x i + th i )z i (t) + h i f(x i + th i ), i =, p + dt We introduce the next denotations A i (t) = h i A(x i + th i ), g i (t) = h i f(x i + th i ) Then the last system takes the form dz i (t) (5) = A i (t)z i (t) + g i (t), i =, p +, t, dt By means of the notations introduced above we find y(a) = z (), y(b) = z p+ () For this reason (2) takes the form (6) Bz () + Cz p+ () = d Since the solution y(x) of the problem () (3) is continuous on every interval a, x, (x i, x i, i = 2, p +, then as lim x τ i+ y(x) = y(τ i + ), we obtain y(τ i ) = z i (), y(τ i + ) = z i+ () On this way the impulse conditions (3) take the form (7) M i z i () + N i z i+ () = v i, i =, p Let z(t) and g(t) be (p + )n-dimensional vector functions z(t) = col(z (t),, z p+ (t)), g(t) = col(g (t),, g p+ (t)) Then we rewrite the differential systems (5) and conditions (6), (7) like a generalized two-point boundary-value problem (8) (9) where ż(t) = A(t)z(t) + g(t), t,, Bz() + Cz() = d, A(t) = diag (A (t),, A p+ (t))) is ((p + )n (p + )n) matrix, B = diag (B, N,, N p )) is ((k + ps) (p + )n)) constant matrix, 8

3 C = C M M 2 M p is ((k + ps) (p + )n) constant matrix, d = d, v,, v p T is (k + ps)- dimensional vector We are to solve the problem (8), (9) instead of the problem () (3) It is clear that y i (x) = z(t) ni when t = (x τ i )/h i Here index n i means successively n in number components of the solution z(t) of (8), (9) Let Φ(t), Φ() = E (p+)n be the normal fundamental matrix of the solutions of ż = A(t)z The generalized solution of the system (8) has the form () z(t) = Φ(t)c + η(t), c R (p+)n, where η(t) = t Φ(t)Φ (s)g(s)ds is a particular solution of (8) We substitute () in the boundary condition (9) and bearing in mind Φ() = E (p+)n, η() =, we obtain () Qc = d Cη(), where Q = B + CΦ() is ((k + ps) (p + )n) matrix We denote by Q + the ((p+)n (k +ps)) pseudoinverse matrix, 3 of the matrix Q, by P Q and P Q the orthoprojectors P Q : R (p+)n ker(q), P Q : R k+ps ker(q ), Q = Q T Theorem Let the conditions (H), (H2) be satisfied and rankq = n < min(k + ps, (p + )n) Then the boundary-value problem (8), (9) has one-parametric family of solutions (2) z(t, ξ) = Φ(t)P Q ξ + z(t), z(t) = Φ(t)Q + (d Cη()) + η(t) ( ) if and only if P Q d Cη() = Proof We have rankq = n < min(k + ps, (p + )n) Then the system () has a parametric solution (3) c = P Q ξ + Q + ( d Cη() ), ξ R (p+)n ( ) if and only if the condition of orthogonality P Q d Cη() = is fulfilled We substitute (3) in () and obtain (2) Corollary Let the conditions (H), (H2) be satisfied and P Q = Then the boundary-value problem (8), (9) has one-parametric solution of the form (2) In this case rankq = (p + )n and the system () is always solvable Corollary 2 Let the conditions (H), (H2) be satisfied and P Q = Then the boundary-value problem (8), (9) has an unique solution z(t) = z(t) if and only if 82

4 ( ) P Q d Cη() = In this case rankq = k + ps Corollary 3 Let the conditions (H), (H2) be satisfied and P Q =, P Q = Then the boundary-value problem (8), (9) has a unique solution z(t) = z(t) and Q + = Q In this case k + ps = (p + )n and detq Here we may supplement a case when k = s = n Keeping in mind the replacement (4) for the solution of the boundary-value problem (), (2) with impulse effects (3), we find ( ) ( ) x a x a y (x) = Φ P Q z (4) y(x, ξ) = y i (x) = Φ ξ + h n ( ) x τi P Q ξ + h i n i z, x a, τ, h n ( ) x τi, x (τ i, τ i, h i n i i = 2, p +, where n + n n p+ = (p + )n, n means the first n in number rows, n 2 - the second and etc, n p+ - the last n in number rows of the matrix Φ(t)P Q and the vector z(t) 3 Examples 3 We illustrate the theorem by the impulsive problem dy =, x, 2, x, y() + y(2) =, y( + ) + y( ) = dx In this case ż(t) = ż (t) ż 2 (t) C =, g(t) = Since Q has the representation Q =, Φ(t) =, d =, η(t) =, B =,, then rankq = Thus we find successively Q + =, P 4 Q =, P 2 Q = 2 Obviously, the condition of orthogonality P Q = is fulfilled This shows that the system c = is always solvable and according to (3) has one parametric solution c = ξ +, ξ R From (2) we find z(t, ξ) = 2 ξ + 2, ξ R 2 83

5 Finally from (4) we get y (x) = 2, ξ +, x,, 2 x(t, ξ) = y 2 (x) =, ξ + 2, x (, 2 2 y (x) = λ + 2, x,, x(t) = λ R y 2 (x) = λ + 2, x (, 2 32 We will find periodic solution of the system dy =, x, 3, x, y() = y(3), y( + ) = dx In this case h =, h 2 = 2 and the system (8) has the form ż (t) ż(t) = = ż 2 (t) 2 and we obtain Φ(t) = The boundary conditions (9) are z() + z() = The system () has the form Then In accordance with corollary 3 and from (4) we find 2 c = c = 2, T z(t) = z(t) = t + 2, 2t T y (x) = x + 2, x,, x(t, ξ) = y 2 (x) = x, x (, 3 We describe a scheme for investigation of boundary-value problems with impulse effects of the form (3) This scheme may be applied to multipoint boundary-value problems Then the matrices B and C are not so simple REFERENCES Generalized Inverses and Applications (Ed M Z Nached), New York, San Francisco, London Acad Press, 976, 54 2 L I Karandjulov Multipoint boundary-value problem with impulse effects Ukr Math J, 47 (995), No 7, R Penrose A generalized inverse for matrices Proc Cambridge Philos Soc, 5 (955),

6 4 A M Samoilenko, N A Perestyuk Impulsive differential equations Wored Scientific seris Nonlinear sciencee E, seriest Singapore; New Jerseu; London; Hong Kong: Word Sci, Vol 4, 994, 462 Людмил Иванов Каранджулов Факултет по приложна математика и информатика Технически университет София София, ПК likar@umeiacadbg ДВУТОЧКОВИ ГРАНИЧНИ ЗАДАЧИ С ИМПУЛСНО ВЪЗДЕЙСТИВИЕ Людмил Ив Каранджулов В работата е предложена нова схема за изследване на гранични задачи за обикновени диференциални уравнения с общи импулсни въздействия в краен брой точки чрез свеждане по двуточкова гранична задача с по-голяма размерност от първоначалната Получени са условията за съществуване както на единствено решение, така и на семейство от решения 85