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Nonlinear functional differential equations of monotone-type in Hilbert spaces. (English) Zbl 1191.34097
The paper is concerned with the initial value problem
\[ \begin{cases}\displaystyle\frac{dx}{dt}(t)=f(t,x_t),\quad t\in [0,T]\\ x(\theta)=\phi(\theta),\quad \theta\in[-r,0],\end{cases}\tag{1} \] where \(\phi\in C_H=C(-r,0;H)\), \(H\) is a real separable Hilbert space and for \(\theta\in [-r,0]\), \(x_t(\theta)=x(t+\theta)\), for every \(0\leq t\leq T\). The function \(f:[0,T]\times C_H\to H\) verifies a semimonotone condition, is bounded from above by a continuous and monotone increasing function, and is demicontinuous with respect to the functional variable. By using an approximation method and Schauder’s fixed point theorem, the author proves the existence and uniqueness of the generalized solution of \((1)\). Then it is investigated the semilinear functional evolution problem
\[ \begin{cases}\displaystyle\frac{dx}{dt}(t)=A(t)x(t)+f(t,x_t),\quad t\in [0,T]\\ x_0=\phi,\end{cases}\tag{2} \] where the family of linear operators \(A(t),\;t\in [0,T]\) verifies some assumptions. By a semigroup approach, the author studies the existence and uniqueness of the mild solution of \((2)\). If \(f:[0,T]\times \Omega\times C_H\to H\) is in addition jointly measurable (\((\Omega,{\mathcal F})\) is a measurable space), then he proves that the solutions of problems \((1)\) and \((2)\) are measurable. The existence and uniqueness of the global solutions for the above problems is also presented.

34K30 Functional-differential equations in abstract spaces
34K05 General theory of functional-differential equations
47N20 Applications of operator theory to differential and integral equations
Full Text: DOI
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