## Identification for a semilinear evolution equation in a Banach space.(English)Zbl 1205.34072

The following identification problem for a first-order semilinear differential equation in a general Banach space $$X$$ is considered
$u'(t)=Au(t)+f(t)+\varphi(u(t))z,\quad t\in(0,1),\;u(0)=\xi_0,\;\int^1_0 u(t)dt=\xi_1,$
where $$A$$ is the infinitesimal generator of a $$C_0$$-semigroup of contractions, $$f$$ is a $$C^1$$ function, $$\varphi$$ is a $$C^1$$ functional, $$\xi_0, \xi_1\in X$$, and $$z\in X$$ and $$u$$ are unknown. The authors prove the existence of a strict solution to the considered problem admitting an implicit representation. Moreover, under the additional assumption that the $$C_0$$-semigroup generated by the linear part has a sufficiently fast exponential decay, the uniqueness and the continuous dependence on the data of the solution are established. As an application, a semilinear parabolic problem is studied.

### MSC:

 34G20 Nonlinear differential equations in abstract spaces 47D06 One-parameter semigroups and linear evolution equations

### Keywords:

semilinear evolution; identification
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