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Existence and asymptotic properties of solutions of nonlinear multivalued differential inclusions with nonlocal conditions. (English) Zbl 1246.34059

The authors study the differential inclusion with multivalued perturbation and nonlocal initial condition

u ' (t)Au(t)+F(t,u(t)),u(0)=g(u)

where A:D(A)XX is a nonlinear m-dissipative operator which generates a contraction semigroup S(t), X is a real Banach space and F is a set-valued function which is weakly upper semicontinuous in its second variable. S(t) is not assumed to be compact and F is not assumed to be strongly upper semicontinuous.

In Theorem 3.1, existence of integral solutions is proven under the additional assumptions that X * is uniformly convex, S(t) is equicontinuous, F is measurable in its first variable and satisfies a growth condition and a measure of noncompactness requirement and g is continuous, compact and satisfies a growth condition. Note that X is not assumed to be separable. The proof is an application of a fixed point theorem in [D. Bothe, “Multivalued perturbations of m-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 X is separable, X * is uniformly convex, F is measurable in its first variable, satisfies a Lipschitz condition and a growth condition and g is Lipschitz continuous.

Note that S(t) 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 t 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 X.

Finally, a partial differential equations example is given in which Theorem 3.1 is applied.

MSC:
34G25Evolution inclusions
47H06Accretive operators, dissipative operators, etc. (nonlinear)
47H08Measures of noncompactness and condensing mappings, K-set contractions, etc.
34D05Asymptotic stability of ODE
47N20Applications of operator theory to differential and integral equations
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