The method of upper and lower solutions for perturbed nth order differential inclusions

Bupurao C. Dhage; Adrian Petruşel

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

  • Volume: 26, Issue: 1, page 57-76
  • ISSN: 1509-9407

Abstract

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In this paper, an existence theorem for nth order perturbed differential inclusion is proved under the mixed Lipschitz and Carathéodory conditions. The existence of extremal solutions is also obtained under certain monotonicity conditions on the multi-functions involved in the inclusion. Our results extend the existence results of Dhage et al. [7,8] and Agarwal et al. [1].

How to cite

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Bupurao C. Dhage, and Adrian Petruşel. "The method of upper and lower solutions for perturbed nth order differential inclusions." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 26.1 (2006): 57-76. <http://eudml.org/doc/271177>.

@article{BupuraoC2006,
abstract = {In this paper, an existence theorem for nth order perturbed differential inclusion is proved under the mixed Lipschitz and Carathéodory conditions. The existence of extremal solutions is also obtained under certain monotonicity conditions on the multi-functions involved in the inclusion. Our results extend the existence results of Dhage et al. [7,8] and Agarwal et al. [1].},
author = {Bupurao C. Dhage, Adrian Petruşel},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {differential inclusion; method of upper and lower solutions; existence theorem; perturbed problem.},
language = {eng},
number = {1},
pages = {57-76},
title = {The method of upper and lower solutions for perturbed nth order differential inclusions},
url = {http://eudml.org/doc/271177},
volume = {26},
year = {2006},
}

TY - JOUR
AU - Bupurao C. Dhage
AU - Adrian Petruşel
TI - The method of upper and lower solutions for perturbed nth order differential inclusions
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2006
VL - 26
IS - 1
SP - 57
EP - 76
AB - In this paper, an existence theorem for nth order perturbed differential inclusion is proved under the mixed Lipschitz and Carathéodory conditions. The existence of extremal solutions is also obtained under certain monotonicity conditions on the multi-functions involved in the inclusion. Our results extend the existence results of Dhage et al. [7,8] and Agarwal et al. [1].
LA - eng
KW - differential inclusion; method of upper and lower solutions; existence theorem; perturbed problem.
UR - http://eudml.org/doc/271177
ER -

References

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  1. [1] R. Agarwal, B.C. Dhage and D. O'Regan, The upper and lower solution method for differential inclusions via a lattice fixed point theorem, Dynam. Systems Appl. 12 (2003), 1-7. Zbl1054.34020
  2. [2] J. Aubin and A. Cellina, Differential Inclusions, Springer Verlag 1984. Zbl0538.34007
  3. [3] M. Benchohra, Upper and lower solutions method for second order differential inclusions, Dynam. Systems Appl. 11 (2002), 13-20. Zbl1023.34008
  4. [4] B.C. Dhage, A fixed point theorem for multi-valued mappings with applications, preprint. 
  5. [5] B.C. Dhage, Multi-valued mappings and fixed points I. 
  6. [6] B.C. Dhage and S.M. Kang, Upper and lower solutions method for first order discontinuous differential inclusions, Math. Sci. Res. J. 6 (2002), 527-533. Zbl1028.34012
  7. [7] B.C. Dhage, T.L. Holambe and S.K. Ntouyas, Upper and lower solutions method for second order discontinuous differential inclusions, Math. Sci. Res. J. 7 (2003), 206-212. Zbl1055.34018
  8. [8] B.C. Dhage, T.L. Holambe and S.K. Ntouyas, The method of upper and lower solutions for Caratheodory nth order differential inclusions, Electronic J. Diff. Equ. 8 (2004), 1-9. Zbl1046.34023
  9. [9] N. Halidias and N. Papageorgiou, Second order multi-valued boundary value problems, Arch. Math. Brno 34 (1998), 267-284. Zbl0915.34021
  10. [10] S. Heikkila and V. Lakshmikantham, Monotone Iterative Technique for Nonlinear Discontinuous Differential Equations, Marcel Dekker Inc., New York 1994. Zbl0804.34001
  11. [11] S. Hu and N. Papageorgiu, Handbook of Multi-valued Analysis, Volume I, Kluwer Academic Publishers, Dordrecht 1997. 
  12. [12] A. Lasota and Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 781-786. Zbl0151.10703
  13. [13] M. Martelli, A Rothe's type theorem for non compact acyclic-valued maps, Boll. Un. Mat. Ital. 4 (Suppl. Fasc.) (1975), 70-76. Zbl0314.47035

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