The authors study the differential inclusion with multivalued perturbation and nonlocal initial condition
where is a nonlinear -dissipative operator which generates a contraction semigroup , is a real Banach space and is a set-valued function which is weakly upper semicontinuous in its second variable. is not assumed to be compact and is not assumed to be strongly upper semicontinuous.
In Theorem 3.1, existence of integral solutions is proven under the additional assumptions that is uniformly convex, is equicontinuous, is measurable in its first variable and satisfies a growth condition and a measure of noncompactness requirement and is continuous, compact and satisfies a growth condition. Note that is not assumed to be separable. The proof is an application of a fixed point theorem in [D. Bothe, “Multivalued perturbations of -accretive differential inclusions”, Isr. J. Math. 108, 109–138 (1998; Zbl 0922.47048)].
In Theorem 4.1, existence of integral solutions is proven under the additional assumptions that is separable, is uniformly convex, is measurable in its first variable, satisfies a Lipschitz condition and a growth condition and is Lipschitz continuous.
Note that is not assumed to be equicontinuous.
The proof is an application of a fixed point theorem found in [K. Deimling, Multivalued Differential Equations. De Gruyter Studies in Nonlinear Analysis and Applications. 1. Berlin: Walter de Gruyter (1992; Zbl 0760.34002)]. A result on asymptotic behavior as is proven in Theorem 5.1 in which a given solution is proven to be almost nonexpansive.
The implications of this property are given for each of several sets of assumptions on the Banach space .
Finally, a partial differential equations example is given in which Theorem 3.1 is applied.