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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

${u}^{\text{'}}\left(t\right)\in Au\left(t\right)+F\left(t,u\left(t\right)\right),\phantom{\rule{4pt}{0ex}}u\left(0\right)=g\left(u\right)$

where $A:D\left(A\right)\subseteq X\to X$ is a nonlinear $m$-dissipative operator which generates a contraction semigroup $S\left(t\right)$, $X$ is a real Banach space and $F$ is a set-valued function which is weakly upper semicontinuous in its second variable. $S\left(t\right)$ 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, $S\left(t\right)$ 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 $S\left(t\right)$ 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 $t\to \infty$ 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:
 34G25 Evolution inclusions 47H06 Accretive operators, dissipative operators, etc. (nonlinear) 47H08 Measures of noncompactness and condensing mappings, $K$-set contractions, etc. 34D05 Asymptotic stability of ODE 47N20 Applications of operator theory to differential and integral equations