Hausdorff product measures and $C^1$-solution sets of abstract semilinear functional differential inclusions

Jian-Zhong Xiao, Zhi-Yong Wang, Juan Liu

DOI: http://dx.doi.org/10.12775/TMNA.2016.076

Abstract


A second order semilinear neutral functional differential inclusion with nonlocal conditions and multivalued impulse characteristics in a separable Banach space is considered. By developing appropriate computing techniques for the Hausdorff product measures of noncompactness, the topological structure of $C^1$-solution sets is established; and some interesting discussion is offered when the multivalued nonlinearity of the inclusion is a weakly upper semicontinuous map satisfying a condition expressed in terms of the Hausdorff measure.

Keywords


Impulsive functional inclusion; cosine family of operators; fixed point for multivalued map; measure of noncompactness

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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.

C.D. Aliprantis and K.C. Border, Infinite Dimensional Analysis, Springer–Verlag, Berlin, Heidelberg, 2006.

J.P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.

G. Babesku, Regularity and uniform continuity properties of cosine and sine class of operators, Lucr. Semin. Math. Fis. Inst. Politehn. Timisoara, November (1983), 47–50.

M. Benchohra, L. Górniewicz, S.K. Ntouyas and A. Ouahab, Controllability results for impulsive functional differential inclusions, Rep. Math. Phys. 54 (2004), 211–228.

M. Benchohra, J. Henderson and S.K. Ntouyas, Existence results for impulsive multivalued semilinear neutral functional differential inclusions in Banach spaces, J. Math. Anal. Appl. 263 (2001), 763–780.

D. Bothe, Multivalued perturbations of m-accretive differential inclusions, Israel J. Math. 108 (1998), 109–138.

J. Diestel, W.M. Ruess and W. Schachemayer, Weak compactness in L1 (µ, X), Proc. Amer. Math. Soc. 118 (1993), 447–453.

H.O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, NorthHolland, Amsterdam, 1985.

E. Hernández M, H.R. Henrı́quez and M.A. McKibben, Existence results for abstract impulsive second-order neutral functional differential equations, Nonlinear Anal. 70 (2009), 2736–2751.

E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society Colloquium Publications, Vol. 31, Fourth Printing of Revised Edition, 1981.

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, Berlin, New York, 2001.

J. Kisyński, On cosine operator functions and one parameter group of operators, Studia Math. 49 (1972), 93–105.

J. Liang and T.J. Xiao, A characterization of norm continuity of propagators for second order abstract differential equations, Computers Math. Applic. 36 (2) (1998), 87–94.

B. Liu, Controllability of impulsive neutral functional differential inclusions with infinite delay, Nonlinear Anal. 60 (2005), 1533–1552.

S.K. Ntouyas and D. O’Regan, Existence results for semilinear neutral functional differential inclusions via analytic semigroups, Acta Appl. Math. 98 (2007), 223–253.

V. Obukhovskiı̆ and P. Zecca, Controllability for systems governed by semilinear differential inclusions in a Banach space with a noncompact semigroup, Nonlinear Anal. 70 (2009), 3424–3436.

N.S. Papageorgiou and S.Th. Kyritsi-Yiallourou, Handbook of Applied Analysis, Advances in Mechanics and Mathematics, Vol. 19, Springer Science Business Media, LLC, 2009.

J.Y. Park, Y.C. Kwun and H.J. Lee, Controllability of second-order neutral functional differential inclusions in Banach spaces, J. Math. Anal. Appl. 285 (2003), 37–49.

W.V. Petryshyn and P.M. Fitzpatrick, A degree theory, fixed point theorems, and mapping theorems for multivalued noncompact mappings, Trans. Amer. Math. Soc. 194 (1974), l–25.

C.C. Travis and G.F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hungaricae 32 (1978), 75-96.

A. Ülger, Weak compactness in L1 (µ, X), Proc. Amer. Math. Soc. 113 (1991), 143–149.

V.V. Vasil’ev, S.G. Krein and S.I. Piskarev, Semigroups of operators, cosine operator functions, and linear differential equations, J. Soviet Math. 54 (4) (1991), 1042–1129.


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