Mem. Dierential Equations Math. Phys. 6(1995), 113{115
M.DrakhlinandE.Litsyn
FUNCTIONAL DIFFERENTIAL EQUATIONS WITH PULSES
(Reported on September 22, 1995) The following equation is under consideration
x_(t) +Xk
j=1Aj(t)x(hj(t)) =f(t); t2[0;b]; (1) x() = 0; if <0
x(ti) =Bix(ti 0); i= 1;2;:::;m; (2) where
0 =t0< t1<< tm< tm+1=b; hi(t)t; t2[0;b]; detBi6= 0; i= 1;2;:::;m:
Under a solution of (1)-(2) we understand an absolutely continuous on every interval [ti 1;ti), i = 1;:::;m+ 1, function x : [0;b] !
R
n satisfying at the points ti the condition (2), and satisfying for almost allt2[0;b] the equation (1).Let us point out that equations of type (1)-(2) are intensively studied. A large number of works are devoted to such equations. Among them there are several monographs (see, for example [1], [2], [3]) which have appeared recently.
Dene by
D
(0
;t
1;:::;t
m;b
) the Banach space of functionsx: [0;b]!R
n absolutely continuous on every interval [ti;ti+1), i = 0;1;:::;m, and satisfying at the points ti, i = 1;2;:::;m, the condition (2). Denote byD
(0
;b
) the Banach space of absolutely continuous functionsy: [0;b]!R
n.Assume that thennmatriciesAj and the functions hj,j = 1;:::;k, are chosen such that the operatorL:
D
(0
;t
1;:::;t
m;b
)!L
p(0
;b
), 1p1, dened by(Lx)(t) = _x(t) +Xk
j=1Aj(t)x(hj(t)) =f(t); t2[0;b]; x() = 0; if <0
is continuous. The specialty of the equation
Lx=f (3)
in comparison with the types of the functional dierential equations studied before is that the domain of the operatorLconsists not of absolutely continuous functions but of piecewise absolutely continuous ones.
1991 Mathematics Subject Classication. 47H05.
Key words and phrases. Functional dierential equation with pulses, Cauchy Matrix.
114
Basing on this fact, we suggest the following scheme for investigating of the equation (3). Namely, between the two spaces
D
(0
;t
1;:::;t
m;b
) andD
(0
;b
) a linear isomor- phism can be established. Indeed, let J :D
(0
;b
) !D
(0
;t
1;:::;t
m;b
) be a linear isomorphism. Then the substitutionx=Jy (4)
transforms (3) to
e
Ly=f; (5)
whereLe=LJis a linear continuous operator acting from the space of absolutely conti- nuous functions
D
(0
;b
) into the Lebesgue spaceL
p(0
;b
), 1p1.Lemma 1.
The equalityx(t) =y(t) +Xm
i=1Qi(t)y(ti); (6)
where
Qi(t) =h[ti
;t
i+1
)(t) +m iX j=1[ti+j
;t
i+j+1 )(t)Yj
k=1Bi+j 1 ki(Bi E); i= 1;2;:::;m;
establishes linear isomorphism between the spaces
D
(0
;t
1;:::;t
m;b
) andD
(0
;b
).Here(;)is the characteristic function of the interval (;),Eis the identity matrix, B0=E. Substituting (6) into (1), we obtain
y_(t) +kX+m
j=1Aj(t)y(hj(t)) =f(t); t2[0;b]; (7) y() = 0; if <0;
where
Ak+i(t) =Xk
j=1Aj(t)Qi(hj(t)); hk+i(t) =ti; i= 1;2;:::;m:
Equation (7) is of the type of functional dierential equations with delayed argument.
Basics of general theory for such equations where introduced in [4]. The essential role in that investigations is assigned to the Cauchy matrix. Due to this, establishing connection between the Cauchy matrixC(t;s) of (1)-(2) and the Cauchy matrixCe(t;s) of (7) proves to be very useful.
Lemma 2.
The equalityC(t;s) =Ce(t;s) +Xm i=1[ti
;b)Qi(t)Ce(ti;s); 0stb;
determines a relation between the Cauchy matrices of the equations(1) (2) and (7).
115 The equations whichCe(t;s) satises as a function in both the rst and the second arguments are found in the works of V. P. Maksimov (see, for example, [5]). With the help of the last lemma, it is possible to to take advantage of those statements for constructing the corresponding equations forC(t;s).
References
1. V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of impulsive dif- ferential equations. World Scientic, Singapore, 1989.
2. S. G. Pandit and S. G. Deo, Dierential systems involving impulses. Lecture Notes in Math., Vol. 954, Springer-Verlag, New York, 1982.
3. A. M. Samolenko and N. A. Perestyuk, Dierential equations with impulse eect.
(Russian) Vishcha Shkola, Kiev, 1987.
4. N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatullina, Introduction to the theory of functional dierential equations. Nauka, Moscow, 1991.
5. V. P. Maksimov, On Cauchy formula for functional dierential equations. Diffe- rentsial'nye Uravnenija
13
(1977), No. 4, 601{606.Authors' addresses:
M. Drakhlin E. Litsyn
The Research Institute Department of Theoretical Mathematics The College of Judea and Samaria The Weizmann Institute of Science Kedumim-Ariel, D. N. Efraim, 44820 76100, Rehovot
Israel Israel
