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On the fractional powers of semidynamical systems. (English) Zbl 1152.45005

The authors investigate the following functional equation \[ \int_0^{\infty}f(s, \Phi(r,x))\eta_t^{\alpha}(dr)=f(s+t,x), \quad x,t\in\mathbb R^{+},\;x\in E, \] in which \(\Phi\) is a semidynamical system with state space \(E\) and \(\eta^{\alpha}=(\eta_t^{\alpha})_{(t>0)}\). They find that its solution \(f(s,t): (0,\infty)\times E\to [0,\infty)\), is the fractional power convolution semigroup of order \(\alpha\in(0,1)\). They also show that the considered equation relates to the so-called \(\alpha\)-Lyapunov functions defined by \(\Phi\) and \(\eta^{\alpha}\).

MSC:

45G05 Singular nonlinear integral equations
26A33 Fractional derivatives and integrals
37B25 Stability of topological dynamical systems
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