Abstract
The paper gives a necessary and sufficient condition for the existence of monotone trajectories to differential inclusionsdx/dt ∈S[x(t)] defined on a locally compact subsetX ofR p, the monotonicity being related to a given preorder onX. This result is then extended to functional differential inclusions with memory which are the multivalued case to retarded functional differential equations. We give a similar necessary and sufficient condition for the existence of trajectories which reach a given closed set at timet=0 and stay in it with the monotonicity property fort≧0.
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References
J. P. Aubin and F. Clarke,Monotone invariant solutions to differential inclusions, J. London Math. Soc. (2)16 (1977), 357–366.
J. P. Aubin, A. Cellina and J. Nohel,Monotone trajectories of multivalued dynamical systems, Ann. Mat. Pura Appl.115 (1977), 99–117.
J. M. Bony,Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier19 (1969), 277–304.
H. Brézis,On a characterization of flow invariant sets, Comm. Pure Appl. Math.23 (1970), 261–263.
F. Clarke,Generalized gradients and applications, Trans. Amer. Math. Soc.205 (1975), 247–262.
M. Crandall,A generalization of Peano’s existence theorem and flow invariance, Proc. Amer. Math. Soc.36 (1972), 151–155.
S. Gautier,Equations différentielles multivoques sur un fermé, Publication interne Université de Pau, 1976.
G. Haddad,Monotone invariant trajectories for hereditary functional differential inclusions, Cahier MD No 7901.
J. Hale,Theory of Functional Differential Equations, Springer, 1977.
S. Leela and V. Moauro,Existence of solutions in a closed set for delay differential equations in Banach spaces, J. Nonlinear Analysis Theory Math. Appl.2 (1978), 391–423.
R. H. Martin,Differential equations on closed subsets of a Banach space, Trans. Amer. Math. Soc.179 (1973), 399–414.
M. Nagumo,Uber die Laga der integralkurven gewöhnlicher differential Gleichungen, Proc. Phys. Math. Soc. Japan24 (1942), 551–559.
R.M. Redheffer,The theorems of Bony and Brézis on flow invariant sets, Amer. Math. Monthly79 (1972), 790–797.
G. Seifert,Positively invariant closed sets for systems of delay differential equations, J. Differential Equations22 (1976), 292–304.
J. A. Yorke,Invariance for ordinary differential equations, Math. Systems Theory1 (1967), 353–372.
J. A. Yorke,Differential inequalities and non-Lipschitz scalar functions, Math. Systems Theory4 (1970), 140–153.
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Haddad, G. Monotone trajectories of differential inclusions and functional differential inclusions with memory. Israel J. Math. 39, 83–100 (1981). https://doi.org/10.1007/BF02762855
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DOI: https://doi.org/10.1007/BF02762855