This
paper establishes the existence, uniqueness and continuous
dependence on initial data of the strong solution of
an abstract quasi-linear implicit integrodifferential
equation in real reflexive Banach space X, whose dual
is uniformly convex, using the Rothe's method.Murphy
showed that the approximate solutions converge to a
`limit solution', which becomes a unique solution
to this homogeneous case of Equation 1. Kartsatos (1978)
established a theorem concerning the existence of a
unique strong solution to Equation 2 under the assumption
that A(t,u)v is Lipschitzian in
t, u and m-accretive in v. In
the present paper, the conditions on A(t,u)v
are motivated by Kartsatos.
Kato (1985b) has proved the
existence, uniqueness and continuous dependence on the
initial data of the solution of abstract quasi-linear,
and showed that these results are applicable to different
kinds of the quasi-linear equations such as symmetric
hyperbolic systems of the first order, wave equations, Korteweg-de Vries equation, Navier-Stokes and Euler
equation, magnetohydrodynamics equation, coupled Maxwell
and Dirac equations, etc. Kato (1985a) has proved the
existence of a strong solution to Equation 3 under the
conditions on A(t,u) and f(t,u)
for (t,u) Î J ´
W similar to that of Crandal and Sougandis (1986).
Amann (1986) has treated various cases of Equation 3
in interpolation spaces using the theory of analytic
semigroups.
in
a Hilbert space H with the assumption that A
: V ® V* is a coercive maximal monotone
operator, where V is a reflexive Banach space
with the dual V* such that V Ç
H is dense in V and H, and the
function f and the Volterra operator F(u)(t)
satisfy certain Lipschitz-like conditions. The existence,
uniqueness and continuous dependence on initial data
of the solutions to the abstract non-linear implicit
and explicit integrodifferential equations have been
studied by Bahuguna and Raghavendra (1989 and 1994)
using Rothe's method under certain appropriate conditions. |
