The Airy equations with impulsive effect: multi-valued nonlinear perturbations
DOI:
https://doi.org/10.12775/TMNA.2021.033Keywords
Cauchy problem of a third-order (in space) PDE, multi-valued perturbation, impulsive effect, solution set, $R_{\delta}$-structure, $R_{\delta}$-mapAbstract
We study the topological regularity of solutions to the Cauchy problem of a (spatial) third-order partial differential equation with a multi-valued perturbation and an impulsive effect. In the framework of the functional space, the principal part of the differential operator corresponds to an Airy operator generating a noncompact $C_0$-group of unitary operators. Our attention is concerned with the $R_\delta$\text{-}structure of the solution set for the Cauchy problem. Geometric aspects of the corresponding solution map are also considered. In our main results, no any compactness condition on the impulsive functions is needed. Moreover, we give illustrating examples for the nonlinearity and impulsive functions.References
N. Abada, M. Benchohra and H. Hammouche, Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions, J. Differential Equations 246 (2009), 3834–3863.
Z. Agur, L. Cojocaru, G. Mazaur, R.M. Anderson and Y.L. Danon, Pulse mass measles vaccination across age cohorts, Proc. Natl. Acad. Sci. USA 90 (1993), 11698–11702.
J. Andres and L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems, Kluwer Academic Publishers, Dordrecht, 2003.
J. Andres and M. Pavlačková, Topological structure of solution sets to asymptotic boundary value problems, J. Differential Equations 248 (2010), 127–150.
N. Aronszajn, Le correspondant topologique de l’unicité dans la théorie des équations différentielles, Ann. of Math. 43 (1942), 730–738.
I. Benedetti and P. Rubbioni, Existence of solutions on compact and non-compact intervals for semilinear impulsive differential inclusions with delay, Topol. Methods Nonlinear Anal. 32 (2008), 227–245.
D. Bothe, Multi-valued perturbations of m-accretive differential inclusions, Israel J. Math. 108 (1998), 109–138.
K.C. Chang, Methods in Nonlinear Analysis, Springer Monographs in Mathematics, Springer–Verlag, Berlin, 2005.
D.H. Chen, R.N. Wang and Y. Zhou, Nonlinear evolution inclusions: Topological characterizations of solution sets and applications, J. Funct. Anal. 265 (2013), 2039–2073.
K. Deimling, Multivalued Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications, vol. 1, Walter de Gruyter & Co., Berlin, 1992.
J. Diestel, W.M. Ruess and W. Schachermayer, Weak compactness in L1 (µ; X), Proc. Amer. Math. Soc. 118 (1993), 447–453.
S. Djebali, L. Górniewicz and A. Ouahab, Filippov–Ważewski theorems and structure of solution sets for first order impulsive semilinear functional differential inclusions, Topol. Methods Nonlinear Anal. 32 (2008), 261–312.
S. Djebali, L. Górniewicz and A. Ouahab, First-order periodic impulsive semilinear differential inclusions: existence and structure of solution sets, Math. Comput. Modelling 52 (2010), 683–714.
S. Djebali, L. Górniewicz and A. Ouahab, Topological structure of solution sets for impulsive differential inclusions in Fréchet spaces, Nonlinear Anal. 74 (2011), 2141–2169.
S. Djebali, L. Górniewicz and A. Ouahab, Solution Sets for Differential Equations and Inclusions, Walter de Gruyter & Co., Berlin, 2013.
G. Gabor and A. Grudzka, Structure of the solution set to impulsive functional differential inclusions on the half-line, Nonlinear Differ. Equ. Appl. 19 (2012), 609–627.
L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, second edition, Topological Fixed Point Theory and Its Applications, vol. 4, Springer, Dordrecht, 2006.
J.R. Graef, J. Henderson and A. Ouahab, Impulsive Differential Inclusions, A Fixed Point Approach, De Gruyter Series in Nonlinear Analysis and Applications, vol. 20, De Gruyter, Berlin, 2013.
S.C. Hu and N.S. Papageorgiou, On the topological regularity of the solution set of differential inclusions with constraints, J. Differential Equations 107 (1994), 280–289.
D.M. Hyman, On decreasing sequence of compact absolute retracts, Fund. Math. 64 (1969), 91–97.
M. Kamenskiı̆, V. Obukhovskiı̆ and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Series in Nonlinear Analysis and Applications, vol. 7, Walter de Gruyter & Co., Berlin, 2001.
H. Kneser, Uber die Lösungen eine system gewöhnlicher differential Gleichungen das der lipschitzchen Bedingung nicht genügt, S.B. Preuss. Akad. Wiss. Phys. Math. Kl. 4 (1923), 171–174.
E. Kruger-Thiemr, Formal theory of drug dosage regiments I, J. Theoret. Biol. 13 (1966), 212–235.
E. Kruger-Thiemr, Formal theory of drug dosage regiments II, J. Theoret. Biol. 23 (1969), 169–190.
F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext, Springer, New York, 2009.
L. Malaguti and P. Rubbioni, Nonsmooth feedback controls of nonlocal dispersal models, Nonlinearity 29 (2016), 823–850.
V.D. Milman and A. Myshkis, On the stability of motion in the presence of impulses, Sib. Math. J. 1 (1960), 233–237.
A. Ülger, Weak compactness in L1 (µ; X), Proc. Amer. Math. Soc. 113 (1991), 143–149.
I.I. Vrabie, Compactness Methods for Nonlinear Evolutions, second edition, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 75, Longman and John Wiley & Sons, New York, 1995.
I.I. Vrabie, C0 -Semigroups and Applications, North-Holland Publishing Co., Amsterdam, 2003.
R.N. Wang, Q.H. Ma and Y. Zhou, Topological theory of non-autonomous parabolic evolution inclusions on a noncompact interval and applications, Math. Ann. 362 (2015), 173–203.
R.N. Wang and Z.X. Ma, The well-posedness for a class of nonlinear evolution equations with a Carathéodory type perturbation, submitted.
R.N. Wang, Z.X. Ma and A. Miranville, Topological structure of the solution sets for a nonlinear delay evolution, Int. Math. Res. Not. IMRN, (2021), DOI:10.1093/imrn/rnab176.
Q.J. Zhu, On the solution set of differential inclusions in Banach space, J. Differential Equations 93 (1991), 213–237.
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Zhong-Xin Ma, Rong-Nian Wang, Yang-Yang Yu
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0
