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Averaging of perturbed one sided Lipschitz differential inclusions. (English) Zbl 1067.34063

The authors study an averaging method for upper semicontinuous differential inclusions in Banach spaces of the form \[ \dot x(t)\in F(t,x)+G\left(\frac{t}{\varepsilon},x\right), \,\, x(0)=x_0, \quad t\in [0,a], \,\, x\in E, \] where \(F(\cdot,\cdot), G(\cdot,\cdot)\) have nonempty convex and compact values, \(F(t,\cdot)\) is one-sided Lipschitz and maps bounded sets into bounded sets and \(E\) a Banach space with uniformly convex dual. The averaging of functional-differential inclusions is studied, too.

MSC:

34G25 Evolution inclusions
34C29 Averaging method for ordinary differential equations
34E13 Multiple scale methods for ordinary differential equations
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[1] Aubin, J.: Viability Theory. Boston: Birghäuser 1991. · Zbl 0755.93003
[2] Couchouron, J. and M. Kamenski: An abstract topological point of view and a general averaging principle in the theory of differential inclusions. Nonlinear Analysis 42 (2000), 625 - 651. · Zbl 0972.34049 · doi:10.1016/S0362-546X(99)00181-9
[3] Deimling, K.: Multivalued Differential Equations. Berlin: De Gruyter 1992. · Zbl 0760.34002
[4] Donchev, T.: Semicontinuous differential inclusions. Rend. Sem. Mat. Univ. di Padova 101 (1999), 147 - 160. · Zbl 0936.34010
[5] Donchev, T.: Functional differential inclusions involving dissipative and compact multifunctions. Glasnik Matemati\check cki 33 (1998)(53), 51 - 60. · Zbl 0913.34015
[6] Donchev, T.: Averaging of differential inclusions in Banach spaces. Ann. UACG XL (1998/99)II, 53 - 63. · Zbl 1226.34054
[7] Hu, S. and N. Papageorgiou: Handbook of Multivalued Analysis. Vol. I: Theory. Dordrecht: Kluwer 1997. Vol. II: Applications. Dordrecht: Kluwer 2000.
[8] Janiak, T. and E. Luczak-Kumorek: The theorem of midding for functional differential equations of neutral type. Disc. Math. Diff. Incl. 11 (1991), 63 -73. · Zbl 0754.34040
[9] Kamenski, M., Obukhovskii, V. and P. Zecca: Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. Berlin: De Gruyter 2001. · Zbl 0988.34001
[10] Kisielewicz, M.: Method of averaging for differential equations with compact convex valued solutions. Rend. Math. 9 (1976), 397 - 408. · Zbl 0364.34017
[11] Lakshmikantham, V. and S. Leela: Nonlinear Differential Equations in Ab- stract Spaces. Oxford: Pergamon Press 1981. · Zbl 0456.34002
[12] Plotnikov, V.: Averaging Method in Control Problems. Kiev-Odessa: Libid 1992 (in Russian). · Zbl 0595.49022
[13] Plotnikov, V., Plotnikov, A. and A. Vityuk: Differential Equations with Multivalued Right-Hand Side. Asymptotic Methods. Odessa : Astro Print 1999 (in Russian). · Zbl 0939.34015
[14] Plotnikov, V. and V. Savchenko: About averaging of differential inclusions. Ukr. Math. Journal 48 (1996), 1572 - 1575 (in Russian).
[15] Zverkova, T.: Ground of the averaging method when the Lipschitz condition is violated. Ann. Higher Schools - Sofia 23 (1987), 41 - 48 (in Russian). · Zbl 0678.34046
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