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


34K05 General theory of functional-differential equations
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