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Monotonicity and existence of periodic orbits for projected dynamical systems on Hilbert spaces. (English) Zbl 1085.37014

Let \(X\) be a Hilbert space of arbitrary dimension and let \(K\) be a nonempty, closed and convex subset of \(X\). For any \(z\in X\) denote by \(P_K(z)\) a unique element in \(K\) such that \(\|P_K(z)-z\|=\inf\{\|x-z\|: x\in K\}\). The mapping \(P_K:X\to K\) given by \(z\to P_K(z)\) is called the projector operator. For any \(x\in K\) and for any \(v\in X\), there exists the limit \(\pi_K(x,v)=\lim_{\delta\to 0+}[P_K(x+\delta v)-x]/\delta\). A mapping \(\Phi:\mathbb{R}_+\times K\to K\) being a solution of the initial value problem \(\Phi(t,x)=\Pi_K(\Phi(t,x),-F(\Phi(t, x)))\), \(\Phi(0,x)=x\in K\) is called a projected dynamical system. The paper presents some results on the existence of periodic orbits for projected dynamical systems under monotonicity conditions of Minty-Browder type. The main tool used in the considerations is the approach through fixed-point theory.

MSC:

37C27 Periodic orbits of vector fields and flows
34C25 Periodic solutions to ordinary differential equations
49J40 Variational inequalities
37N40 Dynamical systems in optimization and economics
34A60 Ordinary differential inclusions
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