A General Class of Impulsive Evolution Equations

JinRong Wang, Michal Feckan

DOI: http://dx.doi.org/10.12775/TMNA.2015.072

Abstract


One of the novelty of this paper is the study of a
general class of impulsive differential equations, which is more
reasonable to show dynamics of evolution processes in
Pharmacotherapy. This fact reduces many difficulties in applying
analysis methods and techniques in Bielecki's normed Banach spaces
and thus makes the study of existence and uniqueness theorems
interesting. The other novelties of this paper are new concepts of
Ulam's type stability and Ulam-Hyers-Rassias stability results on
compact and unbounded intervals.

Keywords


Impulsive evolution equations; mild solutions; existence; stability

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References


S. Afonso, E. M. Bonotto, M. Federson and L. Gimenes, Boundedness of solutions of retarded functional differential equations with variable impulses via generalized ordinary differential equations, Math. Nachr. 285 (2012), 545-561.

S. Afonso, E. M. Bonotto, M. Federson and L. Gimenes, Stability of functional differential equations with variable impulsive perturbations via generalized ordinary differential equations, Bull. Sci. Math. 137 (2013), 189-214.

S .M. Afonso, E. M. Bonotto, M. Federson and S. Schwabik, Discontinuous local semi ows for Kurzweil equations leading to LaSalle's invariance principle for differential systems with impulses at variable times, J. Differential Equations 250 (2011), 2969-3001.

N. U. Ahmed, Existence of optimal controls for a general class of impulsive systems on Banach space, SIAM J. Control Optimal 42 (2003), 669-685.

N.U. Ahmed, K. L. Teo and S.H. Hou, Nonlinear impulsive systems on infinite dimensional spaces, Nonlinear Anal. 54 (2003), 907-925.

Sz. Andras and J. J. Kolumban, On the Ulam-Hyers stability of first order differential systems with nonlocal initial conditions, Nonlinear Anal. 82 (2013), 1-11.

Sz. Andras and A. R. Meszaros, Ulam-Hyers stability of dynamic equations on time scales via Picard operators, Appl. Math. Comput. 219 (2013), 4853-4864.

D.D. Bainov, V. Lakshmikantham and P.S. Simeonov, Theory of impulsive differential equations, Series in Modern Applied Mathematics, vol. 6, World Scientific, Singapore, 1989.

D.D. Bainov and P. S. Simeonov, Integral inequalities and applications, Kluwer Academic Publishers, Dordrecht, 1992.

M. Benchohra, J. Henderson and S. Ntouyas, Impulsive differential equations and inclusions, Contemporary Mathematics and Its Applications, vol. 2, Hindawi, New York, USA, 2006.

L. Cadariu, Stabilitatea Ulam-Hyers-Bourgin pentru ecuatii functionale, Ed. Univ. Vest Timisoara, Timisara, 2007.

Y. K. Chang, and W. T. Li, Existence results for second order impulsive functional differential inclusions, J. Math. Anal. Appl. 301 (2005), 477-490.

D. S. Cimpean and D. Popa, Hyers-Ulam stability of Euler's equation, Appl. Math. Lett. 24 (2011), 1539-1543.

Z. Fan, Impulsive problems for semilinear differential equations with nonlocal conditions, Nonlinear Anal. 72 (2010), 1104-1109.

Z. Fan and G. Li, Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. Funct. Anal. 258 (2010), 1709-1727.

M. Feckan, Y. Zhou and J. Wang, On the concept and existence of solutions for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simulat. 17 (2012), 3050-3060.

M. Frigon and D. O'Regan, Existence results for first-order impulsive differential equations, J. Math. Anal. Appl. 193 (1995), 96-113.

M. Frigon and D. O'Regan, Impulsive differential equations with variable times, Nonlinear Anal. 26 (1996), 1913-1922.

M. Frigon and D. O'Regan, First order impulsive initial and periodic problems with variable moments, J. Math. Anal. Appl. 233 (1999), 730-739.

B. Hegyi and S. M. Jung, On the stability of Laplace's equation, Appl. Math. Lett. 26 (2013), 549-552.

E. Hernandez and D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc. 141 (2013), 1641-1649.

D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. 27 (1941), 222-224.

D. H. Hyers, G. Isac and Th.M. Rassias, Stability of functional equations in several variables, Birkhauser, 1998.

S. M. Jung, Hyers-Ulam-Rassias stability of functional equations in mathematical analysis, Hadronic Press, Palm Harbor, 2001.

S. M. Jung, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett. 17 (2004), 1135-1140.

J. Liu, Nonlinear impulsive evolution equations, Dyn. Contin. Discrete Impuls. Syst. 6 (1999), 77-85.

N. Lungu and D. Popa, Hyers-Ulam stability of a first order partial differential equation, J. Math. Anal. Appl. 385 (2012), 86-91.

A. Pazy, Semigroup of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983.

M. Pierri, D. O'Regan and V. Rolnik, Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses, Appl. Math. Comput. 219 (2013), 6743-6749.

Th. M. Rassias, On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

I. A. Rus, Ulam stability of ordinary differential equations, Studia Univ. Babes Bolyai Mathematica 54 (2009), 125-133.

I. A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math. 26 (2010), 103-107.

A. M. Samoilenko and N.A. Perestyuk, Impulsive differential equations, World Scienti fic Series on Nonlinear Science. Series A: Monographs and Treatises, vol. 14, World Scientific, Singapore, 1995.

P. Sattayatham, Strongly nonlinear impulsive evolution equations and optimal control, Nonlinear Anal. 57 (2004), 1005-1020.

S. M. Ulam, A collection of mathematical problems, Interscience Publishers, New York, 1968.

J. Wang, M. Feckan and Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolution equations, Dyn. Partial Differ. Equ. 8 (2011), 345-361.

J. Wang, M. Feckan and Y. Zhou, Ulam's type stability of impulsive ordinary differential equations, J. Math. Anal. Appl. 395 (2012), 258-264.


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