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Integrated solutions of non-densely defined semilinear integro-differential inclusions: existence, topology and applications
R. Pietkun Toruń, Poland
Abstract:
Given a linear closed but not necessarily densely defined operator $A$ on a Banach space $E$ with nonempty resolvent set and a multivalued map $F\colon I\times E\multimap E$ with weakly sequentially closed graph, we consider the integro-differential inclusion
$$
\dot{u}\in Au+F\biggl(t,\int u\biggr)\quad\text{on } I,\qquad u(0)=x_0.
$$
We focus on the case when $A$ generates an integrated semigroup and obtain existence of integrated solutions if $E$ is weakly compactly generated and $F$ satisfies
$$
\beta(F(t,\Omega))\leqslant \eta(t)\beta(\Omega) \quad\text{for all bounded } \Omega\subset E,
$$
where $\eta\in L^1(I)$ and $\beta$ denotes the De Blasi measure of noncompactness. When $E$ is separable, we are able to show that the set of all integrated solutions is a compact $R_\delta$-subset of the space $C(I,E)$ endowed with the weak topology. We use this result to investigate a nonlocal Cauchy problem described by means of a nonconvex-valued boundary condition operator. We also include some applications to partial differential equations with multivalued terms are.
Bibliography: 26 titles.
Keywords:
convergence theorem, De Blasi measure of noncompactness, integrated semigroup, integrated solution, $R_\delta$-set, semilinear integro-differential inclusion.
Received: 26.09.2019 and 20.03.2021
Citation:
R. Pietkun, “Integrated solutions of non-densely defined semilinear integro-differential inclusions: existence, topology and applications”, Mat. Sb., 212:7 (2021), 122–162; Sb. Math., 212:7 (2021), 1001–1039
Linking options:
https://www.mathnet.ru/eng/sm9331https://doi.org/10.1070/SM9331 https://www.mathnet.ru/eng/sm/v212/i7/p122
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Abstract page: | 192 | Russian version PDF: | 20 | English version PDF: | 8 | Russian version HTML: | 42 | References: | 43 | First page: | 4 |
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