ArticleOriginal scientific text
Title
A viability result for nonconvex semilinear functional differential inclusions
Authors 1, 2
Affiliations
- "Constantin Brâncusi" University, Department of Mathematics, Târgu-Jiu, 210152, Romania
- "Al. I. Cuza" University of Iasi, Faculty of Mathematics, Iasi, 700506, Romania
Abstract
We establish some sufficient conditions in order that a given locally closed subset of a separable Banach space be a viable domain for a semilinear functional differential inclusion, using a tangency condition involving a semigroup generated by a linear operator.
Keywords
viability, invariance, tangency condition, semilinear differential inclusions
Bibliography
- J.P. Aubin, Viability Theory. Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA 1991.
- R.J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965), 1-12.
- H. Brézis and F.E. Browder, A general principle on ordered sets in nonlinear functional analysis, Advances in Math. 21 (3) (1976), 355-364.
- C. Castaing and M.D.P. Monteiro Marques, Topological properties of solution sets for sweeping processes with delay, Portugal. Math. 54 (4) (1997), 485-507.
- O. Cârja and M.D.P. Monteiro Marques, Viability for nonautonomous semilinear differential equations, J. Differential Equations 166 (2) (2000), 328-346.
- O. Cârja and C. Ursescu, The characteristics method for a first order partial differential equation, An. Stiint. Univ. Al. I. Cuza Iasi Sect. I a Mat. 39 (4) (1993), 367-396.
- O. Cârja and I.I. Vrabie, Some new viability results for semilinear differential inclusions, NoDEA Nonlinear Differential Equations Appl. 4 (3) (1997), 401-424.
- K. Deimling, Multivalued differential equations, De Gruyter Series in Nonlinear Analysis and Applications, 1. Walter de Gruyter & Co., Berlin 1992.
- W.E. Fitzgibbon, Semilinear functional differential equations in Banach space, J. Differential Equations 29 (1) (1978), 1-14.
- A. Fryszkowski, Existence of solutions of functional-differential inclusion in nonconvex case, Ann. Polon. Math. 45 (2) (1985), 121-124.
- A. Gavioli and L. Malaguti, Viable solutions of differential inclusions with memory in Banach spaces, Portugal. Math. 57 (2) (2000), 203-217.
- L. Górniewicz, Topological fixed point theory of multivalued mappings, Mathematics and its Applications, 495. Kluwer Academic Publishers, Dordrecht 1999.
- G. Haddad, Monotone trajectories of differential inclusions and functional-differential inclusions with memory, Israel J. Math. 39 (1-2) (1981), 83-100.
- G. Haddad, Monotone viable trajectories for functional-differential inclusions, J. Differential Equations 42 (1) (1981), 1-24.
- J.K. Hale and S.M. Verduyn Lunel, Introduction to functional-differential equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York 1993.
- F. Iacob and N.H. Pavel, Invariant sets for a class of perturbed differential equations of retarded type, Israel J. Math. 28 (3) (1977), 254-264.
- M. Kisielewicz, Differential inclusions and optimal control, Mathematics and its Applications (East European Series), 44, Kluwer Academic Publishers Group, Dordrecht; PWN-Polish Scientific Publishers, Warsaw 1991.
- V. Lakshmikantham and S. Leela, Nonlinear differential equations in abstract spaces, International Series in Nonlinear Mathematics: Theory, Methods and Applications, 2, Pergamon Press, Oxford-New York 1981.
- V. Lakshmikantham, S. Leela and V. Moauro, Existence and uniqueness of solutions of delay differential equations on a closed subset of a Banach space, Nonlinear Anal. 2 (3) (1978), 311-327.
- S. Leela and V. Moauro, Existence of solutions in a closed set for delay differential equations in Banach spaces, Nonlinear Anal. 2 (1) (1978), 47-58.
- V. Lupulescu and M. Necula, Viability and local invariance for non-convex semilinear differential inclusions, Nonlinear Funct. Anal. Appl. 9 (3) (2004), 495-512.
- E. Mitidieri and I.I. Vrabie, A class of strongly nonlinear functional differential equations, Universita degli Studi di Trieste, Instituto di Matematica, Quaderni Matematici II Serie 122 (1986), 1-19.
- E. Mitidieri and I.I. Vrabie, Existence for nonlinear functional differential equations, Hiroshima Math. J. 17 (3) (1987), 627-649.
- N.H. Pavel, Differential equations, flow invariance and applications, Research Notes in Mathematics, 113, Pitman (Advanced Publishing Program), Boston, MA 1984.
- A. Syam, Contribution Aux Inclusions Différentielles, Doctoral thesis, Université Montpellier II, 1993.
- A.A. Tolstonogov and I.A. Finogenko, On functional differential inclusions in Banach space with a nonconvex right-hand side, Soviet. Math. Dokl. 22 (1980), 320-324.
- I.I. Vrabie, C₀-semigroups and applications, North-Holland Mathematics Studies, 191, North-Holland Publishing Co., Amsterdam 2003.
- G.F. Webb, Autonomous nonlinear functional differential equations and nonlinear semigroups, J. Math. Anal. Appl. 46 (1974), 1-12.
- Q.J. Zhu, On the solution set of differential inclusions in Banach space, J. Differential Equations 93 (2) (1991), 213-237.