1. bookVolumen 79 (2021): Heft 2 (December 2021)
Zeitschriftendaten
License
Format
Zeitschrift
eISSN
1338-9750
Erstveröffentlichung
12 Nov 2012
Erscheinungsweise
3 Hefte pro Jahr
Sprachen
Englisch
access type Uneingeschränkter Zugang

Controllability of Nonlocal Impulsive Functional Differential Equations with Measure of Noncompactness in Banach Spaces

Online veröffentlicht: 01 Jan 2022
Volumen & Heft: Volumen 79 (2021) - Heft 2 (December 2021)
Seitenbereich: 59 - 80
Eingereicht: 16 Oct 2019
Zeitschriftendaten
License
Format
Zeitschrift
eISSN
1338-9750
Erstveröffentlichung
12 Nov 2012
Erscheinungsweise
3 Hefte pro Jahr
Sprachen
Englisch
Abstract

This paper is concerned with the controllability of impulsive differential equations with nonlocal conditions. First, we establish a property of measure of noncompactness in the space of piecewise continuous functions. Then, by using this property and Darbo-Sadovskii’s Fixed Point Theorem, we get the controllability of nonlocal impulsive differential equations under compactness conditions, Lipschitz conditions and mixed-type conditions, respectively.

[1] BYSZEWSKI, L.: Theorems about the existence and uniqueness of solutions of a semi-linear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162 (1991), 494–505.10.1016/0022-247X(91)90164-U Search in Google Scholar

[2] FITZGIBBON, W.: Semilinear interodifferential equations in Banach space, Nonlinear Anal., Theory, Methods Appl. 4 (1980), 745–760. Search in Google Scholar

[3] LAKSHMIKANTHAM, V.—BAINOV, D. D.—SIMEONOV, P. S. D.: Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.10.1142/0906 Search in Google Scholar

[4] BENCHOHRA, M.—HENDERSON, J.—NTOUYAS, S. K.: Impulsive Differential Equations and Inclusions, Hindawi Publishing, New York, 2006.10.1155/9789775945501 Search in Google Scholar

[5] BAINOV, D. D.—SIMEONOV, P. S.: Impulsive differential equations: Periodic solutions and applications, In: Pitman Monographs and Surveys in Pure and Applied Mathematics Vol. 66. Harlow: Longman Scientific & Technical; New York, NY: John Wiley & Sons, Inc. 1993. Search in Google Scholar

[6] BANAS, J.—GOEBEL, K.: Measure of Noncompactness in Banach Spaces, In:Lect. Notes Pure Appl. Math. Vol. 60, Marcel Dekker, New York, 1980, Search in Google Scholar

[7] GUO, M.—XUE, X.—LI, R.: Controllability of impulsive evolution inclusions with nonlocal conditions, J. Optim. Theory Appl. 120 (2004), 355–374.10.1023/B:JOTA.0000015688.53162.eb Search in Google Scholar

[8] HERNÁNDEZ, E.— RABELO, M.—HENRIQUEZ, H. R.: Existence of solutions for impulsive partial neutral functional differential equations, J. Math. Anal. Appl. 331 (2007), 1135–1158,10.1016/j.jmaa.2006.09.043 Search in Google Scholar

[9] AHMAD, B.—MALAR, K.—KARTHIKEYAN, K.: A study of nonlocal problems of impulsive integro differential equations with measure of noncompactness, Adv. Difference Equ. 2013 (2013), paper no. 205, 11 p. Search in Google Scholar

[10] JI, S.—LI, G.: A unified approach to nonlocal impulsice differential equations with measure of noncompactness, Adv. Difference Equ. 2012 (2012), paper no. 182. Search in Google Scholar

[11] CHANG, Y. K.: Controllability of impulsive functional differential systems with infinite delay in Banach spaces, Chaos, Solitons Fractals, 33 (2007), no. 5, 1601–1609. Search in Google Scholar

[12] LIU, B.: Controllability of impulsive neutral functional differential inclusions with infinite delay, Nonlinear Anal. 60 (2005), 1533–1552.10.1016/j.na.2004.11.022 Search in Google Scholar

[13] LI, S.—LI, G.—WANG, M.: , Appl. Math. Comput. 217 (2011), 6981–6989. Search in Google Scholar

[14] CHALISHAJAR, D.N.: Controllability of impulsive partial neutral functional differential equations with infinite delay, Int. J. Math. Anal. 5 (2011), no. 8, 369–380. Search in Google Scholar

[15] CHALISHAJAR, D. N.—ACHARYA, F. S.: Controllability of neutral impulsive differential inclusions with non-local conditions, Appl. Math. 2 (2011), 1486–1496.10.4236/am.2011.212211 Search in Google Scholar

[16] SELVI, S.—MALLIKA ARJUNAN, M.: Controllability results for impulsive differential systems with infinite delay, Appl. Math. 42 (2012), 6187–6192. Search in Google Scholar

[17] SELVI, S.—MALLIKA ARJUNAN, M.: Controllability results for impulsive differential systems with finite delay, J. Nonlinear Sci. Appl. 5 (2012), no. 3, 206–219. Search in Google Scholar

[18] CHEN, L.—LI, G.: Approximate Controllability of impulsive differential equations with nonlocal conditions, Int. J. Nonlinear Sci. 10 (2010), no. 4, 438–446. Search in Google Scholar

[19] JI, S.—WEN, S.: Nonlocal Cauchy problem for impulsive differential equations in Banach spaces, Int. J. Nonlinear Sci. 10 (2010), no. 1, 88–95. Search in Google Scholar

[20] JI, S.—LI, G.: Existence results for impulsive differential inclusions with nonlocal conditions, Comput. Math. Appl. 62 (2011), 1908–1915.10.1016/j.camwa.2011.06.034 Search in Google Scholar

[21] LI, J.—NIETO, J. J.— SHEN, J.: Impulsive periodic boundary value problems of firstorder differential equations, J. Math. Anal. Appl. 325 (2007), 226–236.10.1016/j.jmaa.2005.04.005 Search in Google Scholar

[22] BENCHOHRA, M.—HENDERSON, J.—NTOUYAS, S. K.: An existence result for firstorder impulsive functional differential equations in Banach spaces, Comput. Math. Appl. 42 (2001), 1303–1310.10.1016/S0898-1221(01)00241-3 Search in Google Scholar

[23] ABADA, N.—BENCHOHRA, M.—HAMMOUCHE, H.: Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions, J. Differ. Equ. 246 (2009), 3834–3863.10.1016/j.jde.2009.03.004 Search in Google Scholar

[24] XUE, X.: Nonlocal nonlinear differential equations with a measure of noncompactness in Banach spaces, Nonlinear Anal. 70 (2009), 2593–2601.10.1016/j.na.2008.03.046 Search in Google Scholar

[25] DONG, Q.—LI, G.: Existence of solutions for semilinear differential equations with nonlocal conditions in Banach spaces, Electron. J. Qual. Theory Differ. Equ. 2009 (2009), paper no. 47, 13 p. Search in Google Scholar

[26] AGARWAL, R. P.—BENCHOHRA, M.—SEBA, D.: On the application of measure of noncompactness to the existence of solutions for fractional differential equations, Results Math. 55 (2009), 221–230.10.1007/s00025-009-0434-5 Search in Google Scholar

[27] LIANG, J.—LIU, J. H.—XIAO, T. J.: Nonlocal impulsive problems for nonlinear differential equations in Banach spaces, Math. Comput. Model. 49 (2009), 798–804.10.1016/j.mcm.2008.05.046 Search in Google Scholar

[28] ZHU, L.—DONG, Q.—LI, G.: Impulsive differential equations with nonlocal conditions in general Banach spaces, Adv. Difference Equ. 2012 (2012), paper no. 10, 11 p. Search in Google Scholar

[29] SUN, J.—ZHANG, X.: The fixed point theorem of convex-power condensing operator and applications to abstract semilinear evolution equations, Acta Math. Sinica (Chin. Ser.) 48 (2005), 439–446. Search in Google Scholar

[30] OBUKHOVSKI, V.—ZECCA, P.: Controllability for systems governed by semilinear differential inclusions in a Banach space with a noncompact semigroup, Nonlinear Anal. 70 (2009), 3424–3436.10.1016/j.na.2008.05.009 Search in Google Scholar

[31] LIU, L.S.—GUO, F.—WU, C.X.—WU, Y. H.: Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. Math. Anal. Appl. 309 (2005), 638–649.10.1016/j.jmaa.2004.10.069 Search in Google Scholar

[32] QUINN, M. D.—CARMICHAEL, N.: An approach to non-linear control problems using fixed point methods, degree theory and pseudo inverses, Numer. Func. Anal. Optim. 7 (1984–85), 197–219.10.1080/01630568508816189 Search in Google Scholar

[33] DUTKIEWICZ, A.: On the Kneser-Hukuhara property for an integro-differential equation in Banach spaces, Tatra Mt. Math. Publ. 48 (2011), 51–59, doi: 10.2478/v10127--011–0005–5. Search in Google Scholar

[34] SCHMEIDEL, E.: An application of measures of noncompactness in investigation of boundedness of solutions of second order neutral difference equations, Adv. Difference Equ. 91 (2013), 11 p, doi:10.1186/1687–1847–2013–91.10.1186/1687-1847-2013-91 Search in Google Scholar

Empfohlene Artikel von Trend MD

Planen Sie Ihre Fernkonferenz mit Scienceendo