EXISTENCE AND CONTINUOUS DEPENDENCE OF MILD SOLUTIONS TO SEMILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES Dedicated to Professor Junji Kato on his sixtieth birthday

This paper is concerned with a general existence and continuous dependence of mild solutions to semilinear functional differential equations with infinite delay in Banach spaces. In particular, our results are applicable to the equations whose Co-semigroups and nonlinear operators, defined on an open set, are noncompact. Introduction. Let £ be a Banach space over the real field R with norm | | and B a phase space satisfying the fundamental axioms given in [3], [4], [15]. If x : (—oo, σ + a) -> E, 0 < a < +oo, then for any t e (—oo, σ + a) define a mapping xt : (—oo, 0] -> E by χt(θ) = x(t + θ), -oo < θ < 0. Denote by C([a, b], E) the space of all continuous functions from [a, b] into E with the supremum norm. Let A be the infinitesimal generator of a Co-semigroup T(t) on E. In this paper we deal with the initial-value problem for the semilinear functional differential equation with infinite delay in E (for brevity, IP(σ, φ))\ d —u(t) = Au(t) + F(t, ut), σ < t < σ + a , at with uσ = φ G β, where (σ, φ) e R x B is given initial data and F is a (strongly) continuous function mapping an open subset D in R x B into E. \ϊu : (—oo, σ -j- £ i s a continuous function satisfying the integral equation I f' T(t σ)φ(0) + / T(t s)F(s, us)ds for t e [σ, σ + a] Jσ φ(t — σ) for t e (—oo, σ ] , then u is called a mild solution of IP(σ, φ). Roughly speaking, the study of the existence of mild solutions to IP(σ, φ) has been developed in two different directions. One direction is to find conditions to guarantee the existence and uniqueness of mild solutions for IP(σ, φ)\ for instance, refer to Iwamiya [8], Martin [11], Schumacher [16], Shin [21], [22] and Travis and Webb [24], etc. The other is to find conditions to ensure only the existence of mild solutions to IP(σ, φ), which is mainly 1991 Mathematics Subject Classification. Primary 34K30; Secondary 45K05, 34G20. 556 J. S. SHIN AND T. NAITO described in terms of the measure of noncompactness (a-measure for short) introduced by Kuratowskii; for instance, refer to [1], [7], [9], [10], [13], [14], [17], [19], [20]. In the present paper we will investigate the existence and the continuous dependence of mild solutions to IP(σ, φ) in the latter direction. First, we will establish a general existence theorem on mild solutions for IP(σ, φ). The fundamental results on the existence of mild solutions for the case of non-delay were established by Krasnoselskii, Krein and Sobolevskii [9] and Pazy [14] in which it is assumed that T(t) is a Co-compact semigroup on E or F is a compact operator. Recently, in the work of Henriquez [7] the above result was extended to IP(σ, φ). Thus, in the case that both T(t) and F are noncompact operators, we will develop an existence theorem of mild solutions to IP(σ, φ) in the present paper. In such a direction Bothe [1] showed a result on the existence of mild solutions to the multivalued semilinear differential equation on a closed set, which is a partial extension of the one due to Mόnch and Harten [13] for ordinary differential equations. However, even Bothe's result cannot directly extend to IP(σ, φ), because, contrary to the case of non-delay, it is difficult to obtain the compactness of a sequence {zn}neN C C([a, b], E) of approximate solutions for IP(σ, φ). To overcome this difficulty, we establish the following inequality (Theorem 1) on the a -measure: For a bounded subset U in C([α, b], E) a ( U Γ ( s)f(s)ds\[a, b ] \ f e l l X \ < γ τ sup a ( I f T(t s)f(s)ds \ f e u \ \ ,