Nonlinear evolution inclusions: Topological characterizations of solution sets and applications

Journal of Functional Analysis

Volumn 265, Issue 9, 2013, Pages 2039-2073

  Chen, De Han ^a   Wang, Rong Nian ^a   Zhou, Yong ^b  

^a NANCHANG UNIVERSITY   (China)

^b XIANGTAN UNIVERSITY   (China)

Author keywords

Fr chet space; Inverse limit; Nonlinear delay evolution inclusion; Nonlocal condition; R structure

Indexed keywords

EID: 84881375584     PISSN: 00221236     EISSN: 10960783     Source Type: Journal    
DOI: 10.1016/j.jfa.2013.05.033     Document Type: Article

References (54)

1. Abada N., Benchohra M., Hammouche H. Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions. J. Differential Equations 2009, 246:3834-3863.
2. Afanasiev V.N., Kolmanovskii V.B., Nosov V.R. Mathematical Theory of Control Systems Design. Math. Appl. 1995, vol. 341. Kluwer Academic Publishers, Dordrecht.
3. Aizicovici S., McKibben M. Existence results for a class of abstract nonlocal Cauchy problems. Nonlinear Anal. 2000, 39:649-668.
4. Aizicovici S., Staicu V. Multivalued evolution equations with nonlocal initial conditions in Banach spaces. NoDEA Nonlinear Differential Equations Appl. 2007, 14:361-376.
5. Andres J., Gabor G., Górniewicz L. Boundary value problems on infinite intervals. Trans. Amer. Math. Soc. 1999, 351:4861-4903.
6. Andres J., Gabor G., Górniewicz L. Topological structure of solution sets to multi-valued asymptotic problems. Z. Anal. Anwend. 2000, 19(1):35-60.
7. Andres J., Gabor G., Górniewicz L. Acyclicity of solution sets to functional inclusions. Nonlinear Anal. 2002, 49:671-688.
8. Andres J., Pavlačková M. Topological structure of solution sets to asymptotic boundary value problems. J. Differential Equations 2010, 248:127-150.
9. Aronszajn N. Le correspondant topologique de l'unicité dans la théorie des équations différentielles. Ann. of Math. 1942, 43:730-738.
10. Bader R., Kryszewski W. On the solution sets of differential inclusions and the periodic problem in Banach spaces. Nonlinear Anal. 2003, 54:707-754.
11. Bakowska A., Gabor G. Topological structure of solution sets to differential problems in Fréchet spaces. Ann. Polon. Math. 2009, 95(1):17-36.
12. Baras P. Compacité de l'opérateur definissant la solution d'une équation d'évolution non linéaire (du/dt)+Au∋f. C. R. Acad. Sci. Sér. I Math. 1978, 286:1113-1116.
13. Barbu V. Nonlinear Semigroups and Differential Equation in Banach Spaces 1976, Editura Academiei/Noordhoff, Bucuresti.
14. Borsuk K. Theory of Retracts. Monogr. Mat. 1967, vol. 44. PWN, Warsaw.
15. Brezis H. Problèmes unilatéraux. J. Math. Pures Appl. 1972, 51:1-168.
16. Cârjǎ O., Necula M., Vrabie I.I. Viability, Invariance and Applications. Math. Stud. 2007, vol. 207. Elsevier/North-Holland.
17. Ceron S.S., Lopes O. α-Contractions and attractors for dissipative semilinear hyperbolic equations and systems. Ann. Mat. Pura Appl. 1991, 160:193-206.
18. Conti G., Obukhovskii V., Zecca P. On the topological structure of the solution set for a semilinear functional-differential inclusion in a Banach space. Banach Center Publ. 1996, vol. 35:159-169. Inst. of Math. Polish Acad. of Sci., Warsaw.
19. De Blasi F.S., Myjak J. On the solutions sets for differential inclusions. Bull. Pol. Acad. Sci. Math. 1985, 12:17-23.
20. De Blasi F.S., Myjak J. On the structure of the set of solutions of the Darboux problem for hyperbolic equations. Proc. Edinb. Math. Soc. 1986, 29:7-14.
21. Deimling K. Multivalued Differential Equations 1992, de Gruyter, Berlin.
22. Deng K. Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions. J. Math. Anal. Appl. 1993, 179:630-637.
23. 1(μ;X). Proc. Amer. Math. Soc. 1993, 118:447-453.
24. Diestel J., Uhl J.J. Vector Measures. Math. Surveys 1977, vol. 15. American Mathematical Society, Providence, RI.
25. Dugundji J. An extension of Tietze's theorem. Pacific J. Math. 1951, 1:353-367.
26. Gabor G. Acyclicity of solution sets of inclusions in metric spaces. Topol. Methods Nonlinear Anal. 1999, 14:327-343.
27. Gabor G. Some results on existence and structure of solution sets to differential inclusions on the halfline. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 2002, 5:431-446.
28. Gabor G., Grudzka A. Structure of the solution set to impulsive functional differential inclusions on the half-line. NoDEA Nonlinear Differential Equations Appl. 2012, 19:609-627.
29. García-Falset J. Existence results and asymptotic behaviour for nonlocal abstract Cauchy problems. J. Math. Anal. Appl. 2008, 338:639-652.
30. Górniewicz L. Topological Fixed Point Theory of Multivalued Mappings 2006, Springer, Berlin. second ed.
31. δ-maps. Topology Appl. 1994, 55(3):239-250.
32. Górniewicz L., Pruszko T. On the set of solutions of the Darboux problem for some hyperbolic equations. Bull. Acad. Polon. Math. 1980, 28(5-6):279-286.
33. Hu S.C., Papageorgiou N.S. On the topological regularity of the solution set of differential inclusions with constraints. J. Differential Equations 1994, 107(2):280-289.
34. Hukuhara M. Sur les systémes des équations differentielles ordinaires. Jpn. J. Math. 1928, 5:345-350.
35. Hyman D.M. On decreasing sequence of compact absolute retracts. Fund. Math. 1969, 64:91-97.
36. Kamenskii M., Obukhovskii V., Zecca P. Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. de Gruyter Ser. Nonlinear Anal. Appl. 2001, vol. 7. Walter de Gruyter, Berlin, New York.
37. Ke T.D., Obukhovskii V., Wong N.C., Yao J.C. Approximate controllability for system governed by nonlinear Volterra type equations. Differ. Equ. Dyn. Syst. 2012, 20(1):35-52.
38. Kneser H. Uber die Lösungen eine system gewöhnlicher differential Gleichungen das der lipschitzchen Bedingung nicht genügt. S. B. Preuss. Akad. Wiss. Phys. Math. Kl. 1923, 4:171-174.
39. Lakshmikantham V., Leela S. Nonlinear Differential Equations in Abstract Spaces. Int. Ser. Nonlinear Math. Theory Methods Appl. 1981, vol. 2. Pergamon Press.
40. O'Regan D. Topological structure of solution sets in Fréchet spaces: The projective limit approach. J. Math. Anal. Appl. 2006, 324:1370-1380.
41. Obukhovskii V., Rubbioni P. On a controllability problem for systems governed by semilinear functional differential inclusions in Banach spaces. Topol. Methods Nonlinear Anal. 2000, 15:141-151.
42. Paicu A., Vrabie I.I. A class of nonlinear evolution equations subjected to nonlocal initial conditions. Nonlinear Anal. 2010, 72:4091-4100.
43. Papageorgiou N.S. Optimal control of nonlinear evolution inclusions. J. Optim. Theory Appl. 1990, 67:321-354.
44. Rudin W. Functional Analysis 1991, McGraw-Hill Science. second ed.
45. Šeda V., Belohorec Š. A remark on second order functional differential systems. Arch. Math. (Brno) 1993, 36(3-4):169-176.
46. Staicu V. On the solution sets to nonconvex differential inclusions of evolution type. Discrete Contin. Dyn. Syst. 1998, 2:244-252.
47. Vrabie I.I. Compactness Methods for Nonlinear Evolutions. Pitman Monogr. Surv. Pure Appl. Math. 1995, vol. 75. Longman, John Wiley & Sons. second ed.
48. Vrabie I.I. Existence for nonlinear evolution inclusions with nonlocal retarded initial conditions. Nonlinear Anal. 2011, 74:7047-7060.
49. Vrabie I.I. Existence in the large for nonlinear delay evolution inclusions with nonlocal initial conditions. J. Funct. Anal. 2012, 262:1363-1391.
50. Wang R.N., Chen D.H., Xiao T.J. Abstract fractional Cauchy problems with almost sectorial operators. J. Differential Equations 2012, 252:202-235.
51. Wang R.N., Xiao T.J., Liang J. A note on the fractional Cauchy problems with nonlocal initial conditions. Appl. Math. Lett. 2011, 24:1435-1442.
52. Wang R.N., Zhu P.X. Non-autonomous evolution inclusions with nonlocal history conditions: Global integral solutions. Nonlinear Anal. 2013, 85:180-191.
53. Yorke J. Spaces of Solutions. Lect. Notes Op. Res. Math. Econ. 1969, vol. 12:383-403. Springer-Verlag.
54. Zhu Q.J. On the solution set of differential inclusions in Banach space. J. Differential Equations 1991, 93:213-237.