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Lipschitz perturbation to evolution inclusion driven by time-dependent maximal monotone operators. (English) Zbl 1494.34135

The authors consider existence of solutions for the problem \[ -\dot{u}(t)\in A(t)u(t)+F(t,u(t))\text{ for almost all }t\in [0,1] \] \[ u(t)\in D\left( A(t)\right) \text{ for almost all }t\in [0,1] \] \[ u(0)=u_{0}\in D(A(0)), \]
in a separable Hilbert space, where \(A\) is a maximal monotone operator that is absolutely continuous with respect to pseudo-distance and \(F\) is a (possibly unbounded) perturbation that is closed set valued, measurable in its first variable and Lipschitz in its second variable. Techniques from [D. Azzam-Laouir et al., J. Fixed Point Theory Appl. 21, No. 2, Paper No. 40, 32 p. (2019; Zbl 1418.34122); A. A. Tolstonogov, J. Math. Anal. Appl. 447, No. 1, 269–288 (2017; Zbl 1351.49017)] are applied. The relaxation problem in which \(F\) is replaced by \(\overline{co}F\) is also investigated. Finally, applications to fractional order boundary problems and control/viscosity are given.

MSC:

34G25 Evolution inclusions
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
49J45 Methods involving semicontinuity and convergence; relaxation
49J52 Nonsmooth analysis
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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