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A mixed neutral system. (English) Zbl 0553.34042
Consider a mixed type neutral system (1) $$x'(t)=f(t,x)+\sum^{m}_{j=1}F_ j(t,x)x'(t+P_ j)$$, (2) $$x(0)=x_ 0$$, where f is a continuous n-vector-valued functional, each $$F_ j$$ is a continuous $$n\times n$$ matrix-valued function defined on $$R\times C(R,R^ n)$$, each $$P_ j$$ is a constant real number and $$x_ 0\in R^ n$$. It is also assumed that $$| \cdot |$$ is a norm in $$R^ n$$, $$\| \cdot \|$$ the induced matrix norm and P, $$M_ f,M_ F,K_ f$$ and $$K_ F$$ are positive constants such that $$| f| \leq M_ f$$, each $$\| F_ j\| \leq M_ F$$ on $$R\times C(R,R^ n)$$, $$P=\max_{j} | P_ j|$$ and for all $$t\in R$$, with x, $$\tilde x\in C(R,R^ n)$$, $$| f(t,x)-f(t,\tilde x)| \leq K_ f\max_{t- p\leq s\leq t+p}| x(s)-\tilde x(s)|$$ and $$\| F(t,x)-F(t,\tilde x)\| \leq K_ F\max_{t-p\leq s\leq t+p}| x(s)-\tilde x(s)|.$$ The author proves that if $$P,M_ f,M_ F,K_ f$$ and $$K_ F$$ are sufficiently small and for any constant $$a>0$$ $e^{ap}[(1/a)(K_ f+(mK_ FM_ f)/(1-mM_ F))+mM_ F]<1$ then (1) and (2) have a unique solution such that $$\int^{t+1}_{t}| x'(s)| ds$$ is bounded for all t. An example is given, to illustrate the theory.
Reviewer: O.Akinyele

MSC:
 34K05 General theory of functional-differential equations
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References:
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