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SummaryThis paper is concerned with the recent developments in the solution of boundary value problems by integral equations of the first kind. Basic results for weakly singular and hypersingular boundary integral operators will be discussed. Emphases will be given to the mathematical foundation of the method' as well as to the physical interpretations of various side conditions derived for the unique solvability of the integral equations of the first kind.

This paper presents two simple models for nonlinear age-dependent population dynamics. In these models the basic equations of the theory reduce to systems of ordinary differential equations. We discuss certain qualitative aspects of these systems; in particular, we show that for many cases of interest periodic solutions are not possible. BASIC EQUATIONSRecently, Gurtin and MacCamy [l] introduced a nonlinear' theory of population dynamics with age dependence. This theory is based on the equations' p,(a,t)+p,(a,t)+CL(a,P(t))p(a,t)=O, B(t)=p(O,r)=lmp(a,P(r))p(a,r)da, (1.1) 0 where p(a,t) is the age distribution, that is, the number of individuals of age a at time t (a > 0, t > 0);is the total population; B(t)is the birth rate; ~(a, P)is the survival function; P(a, P)is the materniv function.'The linear theory is discussed in detail by Hoppensteadt[2]. 2Subscripts indicate partial differentiation.

The problem \[ u t ( x , t ) = ∫ 0 t a ( t − τ ) ∂ ∂ x σ ( u x ( x , τ ) ) d τ + f ( x , t ) , 0 > x > 1 , t > 0 , u ( 0 , t ) ≡ u ( 1 , t ) ≡ 0 u ( x , 0 ) = u 0 ( x ) {u_t}\left ( {x, t} \right ) = \int _0^t {} a\left ( {t - \tau } \right )\frac {\partial }{{\partial x}}\sigma \left ( {{u_x}\left ( {x,\tau } \right )} \right )d\tau + f\left ( {x, t} \right ), \qquad 0 > x > 1, \qquad t > 0, \\ u\left ( {0,t} \right ) \equiv u\left ( {1,t} \right ) \equiv 0 \qquad u\left ( {x, 0} \right ) = {u_0}\left ( x \right ) \] is considered. Asymptotic stability theorems for the solution are established under appropriate conditions on a a , σ \sigma and f f . The conditions on a a are of frequency domain type and are related to ones used previously in the study of Volterra integral equations, \[ u ˙ = − ∫ 0 t a ( t − τ ) g ( u ( τ ) ) d τ + f ( t ) \dot u = - \int _0^t a \left ( {t - \tau } \right )g\left ( {u\left ( \tau \right )} \right )d\tau + f\left ( t \right ) \] on a Hilbert space. An existence theorem for the problem is established under smallness assumptions on f f and u 0 {u_0} This theorem is related to one by Nishida for the damped non-linear wave equation, \[ u t t + α u t − ∂ ∂ x σ ( u x ) = 0 {u_{tt}} + \alpha {u_t} - \frac {\partial }{{\partial x}}\sigma \left ( {{u_x}} \right ) = 0 \] . It is shown that the problem is related to a theory of heat flow in materials with memory.