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On the neural functional differential inclusions. (English) Zbl 0977.34072

From the introduction: Consider the following initial value problem for a neutral functional-differential inclusion: \[ \begin{split} \dot y(t)\in F\biggl(t,y \bigl(\Delta_1(t)\bigr),\dots, y\bigl(\Delta_m(t)\bigr),\dot y\bigl( \tau_1(t)\bigr), \dots,\dot y\bigl(\tau_n(t)\bigr)\biggr),\;t>0,\\ y(t)=\psi (t);\;\dot y(t)= \dot\psi(t),\;t\leq 0.\end{split}\tag{1} \] Here, the unknown function \(y(\cdot)\) takes values in a Banach space \(B\), while the multimap \(F\) takes nonempty closed bounded values in \(B\). The deviations \(\Delta_i(t)\), \(\tau_j(t)\), \(i=1,\dots, m,j=1, \dots,n\), are of mixed type and unbounded in the general case. By a solution to (1) we mean an absolutely continuous function \(y(\cdot)\) satisfying (1) for a.e. \(t\). – Letting \(x(t)=\dot y(t)\) for \(t>0\) and \(z(t)=\dot \psi(t)\) for \(t\leq 0\) the authors reduce (1) to the problem (assuming \(\psi(0)=0)\) \[ \begin{split} x(t)\in F\biggl(t, \int^{\Delta_1(t)}_0x(s)ds, \dots,\int ^{\Delta_m(t)}_0x(s)ds, x\bigl(\tau_1(t)\bigr), \dots,x \bigl(\tau_n(t)\bigr)\biggr),\;t>0,\\ x(t)=z(t),\;t\leq 0\end{split}\tag{2} \] They are looking for \(x(\cdot)\in L^\infty_{\text{loc}}(\mathbb{R};B)\) the space of all locally essentially bounded strongly measurable functions, or \(x(\cdot)\in L^1_{\text{loc}}(\mathbb{R};B)\) consisting of all locally integrable functions. In order to define correctly the composition \(x(\tau_1(t))\) they replace \(x (\cdot)\) in the right-hand side of (2) by the Borel (strongly) measurable \(\tilde x(t)=x(t)\); \(t\in\mathbb{R} _{+N}\) for some null set \(N\). The space \(L^\infty_{\text{loc}}(\mathbb{R};B)\) [respectively \(L^1_{\text{loc}} (\mathbb{R}; B)]\) is a uniform Hausdorff space with a topology defined by a saturated family of pseudometrics. The authors consider the two cases separately and essentially use the techniques from V. G. Angelov [Czech. Math. J. 37(112), No. 1, 19-33 (1987; Zbl 0713.54045)], where a fixed-point approach is used to obtain the existence of solutions to the corresponding single-valued functional equation. Using the same approach they can show the continuous dependence of the solution set on the parameters and on the initial conditions.

MSC:

34K30 Functional-differential equations in abstract spaces
34K40 Neutral functional-differential equations

Citations:

Zbl 0713.54045
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References:

[1] Kisielewicz, M., Generalized functional differential equations of neutral type, Annls. Pol. Math., 41, 135-148 (1983) · Zbl 0543.34052
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