Introduction

This monograph consists of four chapters. The first two chapters are concerned with integro-differential equation of the form

8z(t, s) 16(1) ~ =c(t,s)z(t,s)+ G k(t,s,u)z(t,u)du+/(t,s). Here c: Jx[a,b] -+ lR, k : Jx[a,b]x[a,b] -+ lR, and I: Jx[a,b] -+ lR are given functions, where J is a bounded or unbounded interval; the function z is unknown. The equation (1) has been studied first by Barbashin and his pupils (see BARBASHIN [1957, 1958J, BARBASHIN-BISJARINA [1963J, BARBASH IN-LIBERMAN [1958], BISJARINA [1963, 1964, 1964a, 1967J, BvKOV [1953, 1953a, 1961, 1962, 1962a, 1962b, 1962c, 1983], BVKOVKULTAEV [1984J, BVKOV-TANKIEV [1982], BVKOVA [1967, 1968, 1971], BVKOVA-BvKOVA [1969J, KAMVTOV [1976], KHOTEEV [1976, 1984], KRIVOSHEJN-BVKOVA [1974], LIBERMAN [1958], MAMVTOV [1985], SALGAPAROV [1962], ATAMANOV [1977], BOJKOV [1985], IMANDIEVDZHURAEV-PANKOV [1974], KALIEV [1974], and SAMEDOVA [1958]. For this reason this equation is nowadays called integro-differential equation 01 Barbashin type or simply Barbashin equation. Identifying a real function z = z(t, s) of two variables (t, s) with the abstract function z = z(t) of one variable t E J which takes its values in some Banach space X of real functions on [a, b) and is defined by z(t)( s) = z(t, s), one may write equation (1) as an ordinary differential equation

(2) dzdt = A(t)z + I(t) in X. This-identification which is a common device in the theory of partial differential equations when passing from a parabolic equation to an abstract evolution equation turns out to be useful also here. Observe that the operator function A(t) in (2) has a very special form: it is the sum

(3) A(t) = G(t) +K(t)

of a multipliC4tion operator

(4) C(t)z(s) = c(t, s)z(s) generated by the multiplier c, and an integral operator

(5) K(t)z(s) = t. k(t,s,u)z(u)du generated by the kernel kj we sha.11 ca.11 operators of this form Barbashin operators in what follows. This pleasant fact a.llows us to combine methods of the theory of differential equations in Banach spaces with methods of the theory of linear integral operators. As a consequence, we obtain a theory which is far more complete and richer than the theory of differential equations (2) containing an arbitrary operator function A(t). In particular, in many cases it is useful to study first the "reduced" equation

(6) dzdt = C(t)z + f(t)

with C(t) being defined by (4), and then to consider the "full" equation (2) as an integral perturbation of (6). This philosophy will be used over and over again in our study of the Barbashin equation (1). Let us illustrate this by means of a simple example. It is a. well-known fact that, under some natural hypotheses, the differential equation (2) defines a continuous operator function U = U(t, T) of two variables, ca.11ed the evolution operator (or Cauchy operator) of (2). This operator may be defined as the unique solution of the integral equation

U(t,T) = 1+ l' A(T)U(t, T) dT (t,T E J). The integral operator with kemel U may then be considered as the right inverse of the differential operator i - A(t)j in particular, the unique solution x of (2) satisfying the initial condition (7) zeal = Zll

Now, the evolution operator Uo = Uo(t, r) of the reduced equation (6) may be calculated explicitly: as one expects, it is just the multiplication operator

(8) Uo(t,r)z(s) =e(t,r,s)z(s)

(9)

(11)

generated by the multiplier

e(t, r, s) = exp {i' c(e, s) de} . On the other hand, since the difference K(t) = A(t) - C(t) is an integral operator, it is natural to expect that also the difference H(t, r) = U(t, r) - Uo(t, r) of the corresponding evolution operators is an integral operator

H(t,r)z(s) = 1"h(t,r,s,u)z(u)du containing some kernel h which may be evaluated in terms of the kernel k. This is in fact true in many Banach spaces X which occur in applications. A particularly simple, though important, special case is that of the stationary integra-differential equation of Barbashin type

8z(t,s) 1"(10) at =c(s)z(t,s)+ II k(s,u)z(t,u)du+f(t,s), i.e. the multiplier c and the kernel k do not depend on t. In this case the differential equation (2) reduces to

dz dt = Az + f(t),

where A = C +K is a stationary Barbashin operator with

(12) ez(s) =c(s)z(s)

and

(13)

Xx(s) = 1·k(a, O")x(O") dO". Of course, much more can be said in this case; for example, the evolution operator of the full equation (14) is then simply the exponential operator U(t, T) =eA(C-.,.). We point out that the operator X in (5) is not a usual integral operator, since the kernel k depends on a parameter t, but the unknown solution x does not. Such operators are called partial integral operators, inasmuch as the integration is carried out only with respect to some arguments, while the other arguments of the integrand are "frozen". This is of course analogous to partial differential operators.