# Representation of the set of mild solutions to the relaxed semilinear differential inclusion

Irene Benedetti; Elena Panasenko

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2006)

- Volume: 26, Issue: 1, page 143-158
- ISSN: 1509-9407

## Access Full Article

top## Abstract

top## How to cite

topIrene Benedetti, and Elena Panasenko. "Representation of the set of mild solutions to the relaxed semilinear differential inclusion." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 26.1 (2006): 143-158. <http://eudml.org/doc/271136>.

@article{IreneBenedetti2006,

abstract = {We study the relation between the solutions set to a perturbed semilinear differential inclusion with nonconvex and non-Lipschitz right-hand side in a Banach space and the solutions set to the relaxed problem corresponding to the original one. We find the conditions under which the set of solutions for the relaxed problem coincides with the intersection of closures (in the space of continuous functions) of sets of δ-solutions to the original problem.},

author = {Irene Benedetti, Elena Panasenko},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {differential inclusion; mild solution; quasi-solution; convexified and perturbed problem; relaxation theorem; convexified problem; perturbation},

language = {eng},

number = {1},

pages = {143-158},

title = {Representation of the set of mild solutions to the relaxed semilinear differential inclusion},

url = {http://eudml.org/doc/271136},

volume = {26},

year = {2006},

}

TY - JOUR

AU - Irene Benedetti

AU - Elena Panasenko

TI - Representation of the set of mild solutions to the relaxed semilinear differential inclusion

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2006

VL - 26

IS - 1

SP - 143

EP - 158

AB - We study the relation between the solutions set to a perturbed semilinear differential inclusion with nonconvex and non-Lipschitz right-hand side in a Banach space and the solutions set to the relaxed problem corresponding to the original one. We find the conditions under which the set of solutions for the relaxed problem coincides with the intersection of closures (in the space of continuous functions) of sets of δ-solutions to the original problem.

LA - eng

KW - differential inclusion; mild solution; quasi-solution; convexified and perturbed problem; relaxation theorem; convexified problem; perturbation

UR - http://eudml.org/doc/271136

ER -

## References

top- [1] J.-P. Aubin and A. Cellina, Differential Inclusions, Grundlehren Math. Wiss. Vol. 264, Springer-Verlag, Berlin, Heidelberg 1984.
- [2] J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston-Basel-Berlin 1990.
- [3] Yu. Borisovich, B. Gelman, A. Myshkis and V. Obukhovskii, Introduction to the Theory of Multivalued Maps and Differential Inclusions, Editorial URSS, Moscow 2005 (in Russian).
- [4] A. Bulgakov, A. Efremov and E. Panasenko, Ordinary Differential Inclusions with Internal and External Perturbations, Differentsial'nye Uravneniya 36 (12) (2000), 1587-1598, translated in Differential Equations 36 (12) (2000), 1741-1753. Zbl0997.34009
- [5] K. Deimling, Multivalued Differential Equations, De Gruyter Sr. Nonlinear Anal. Appl. 1, Walter de Gruyter, Berlin-New York 1992.
- [6] A.F. Filippov, Differential Equations with Discontinuous Righthand Side, Dordrecht, Kluwer 1988.
- [7] H. Frankowska, A Priori Estimanes for Operational Differential Inclusions, J. Differential Equations 84 (1990), 100-128. Zbl0705.34016
- [8] A.D. Ioffe and V.M. Tihomirov, Theory of Extremal Problems, North Holland, Amsterdam 1979.
- [9] M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Sr. Nonlinear Anal. Appl. 7, Walter de Gruyter, Berlin-New York 2001. Zbl0988.34001
- [10] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New-York, Inc. 2000. Zbl0952.47036
- [11] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New-York, Inc. 1983. Zbl0516.47023
- [12] G. Pianigiani, On the Fundamental Theory of Multivalued Differential Equations, J. Differential Equations 25 (1) (1977), 30-38. Zbl0398.34017
- [13] A. Plis, On Trajectories of Orientor Fields, Bull. Acad. Polon. Sci, Ser. Math. 13 (8) (1965) 571-573. Zbl0138.34104
- [14] A.A. Tolstonogov, Differential Inclusions in Banach Space, Kluwer Acad. Publishers, Dordrecht 2000.