zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Impulsive differential inclusions with fractional order. (English) Zbl 1200.34006

The authors consider the Cauchy problem for a fractional impulsive differential inclusion:

D * α F(t,y(t))a.e.tJ{t 1 ,,t m },y(t k + )=I k (t k - ),k=1,,m,y ' (t k + )=I ¯ k (t k - ),k=1,,m,y(0)=a,y ' (0)=c,

the case of fractional differential equations and the following periodic one:

D * α F(t,y(t))a.e.tJ{t 1 ,,t m },y(t k + )=I k (t k - ),k=1,,m,y ' (t k + )=I ¯ k (t k - ),k=1,,m,y(0)=y(b);y ' (0)=y ' (b),

where J=[0,b],D * α denotes the Caputo fractional derivative (α(1,2]), and F is a set-valued map. The functions I k ,I ¯ k characterize the jump of the solutions at impulse point t k (k=1,,m). Under Lipschitz and Nagumo-type growth conditions on F, the authors prove existence of solutions via fixed point theory for multivalued mappings. Also, they study the topological structure of solution sets (compactness, R δ ; acyclicity; contractibility). The proofs use the general theory on topological structure of fixed point set for multi-valued operators. Note in case of first order and periodic differential inclusions, these results have been obtained in [S. Djebali, L. Górniewicz, A. Ouahab, Topol. Methods Nonlinear Anal. 32, No. 2, 261–312 (2008; Zbl 1182.34087)] for initial problems with delays and in [S. Djebali, L. Górniewicz, A. Ouahab, Math. Comput. Model. 52, 683–714 (2010)] for first-order periodic problems.

34A08Fractional differential equations
34A60Differential inclusions
26A33Fractional derivatives and integrals (real functions)
34A37Differential equations with impulses
47N20Applications of operator theory to differential and integral equations
[1]Milman, V. D.; Myshkis, A. A.: On the stability of motion in the presence of impulses, Siberian math. J. 1, 233-237 (1960)
[2]Milman, V. D.; Myshkis, A. A.: Random impulses in linear dynamical systems, , 64-81 (1963)
[3]Halanay, A.; Wexler, D.: Teoria calitativa a systeme cu impulduri, (1968) · Zbl 0176.05202
[4]Bainov, D. D.; Simeonov, P. S.: Systems with impulse effect, (1989) · Zbl 0683.34032
[5]Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations, (1989) · Zbl 0718.34011
[6]Pandit, S. G.; Deo, S. G.: Differential systems involving impulses, Lecture notes in mathematics 954 (1982) · Zbl 0539.34001
[7]Samoilenko, A. M.; Perestyuk, N. A.: Impulsive differential equations, (1995) · Zbl 0837.34003
[8]Agur, Z.; Cojocaru, L.; Mazaur, G.; Anderson, R. M.; Danon, Y. L.: Pulse mass measles vaccination across age cohorts, Proc. natl. Acad. sci. USA 90, 11698-11702 (1993)
[9]Kruger-Thiemr, E.: Formal theory of drug dosage regiments. I, J. theoret. Biol. 13, 212-235 (1966)
[10]Kruger-Thiemr, E.: Formal theory of drug dosage regiments. II, J. theoret. Biol. 23, 169-190 (1969)
[11]Aubin, J. P.: Impulse differential inclusions and hybrid systems: A viability approach, Lecture notes (2002)
[12]Benchohra, M.; Henderson, J.; Ntouyas, S. K.: Impulsive differential equations and inclusions, (2007)
[13]Henderson, J.; Ouahab, A.: Local and global existence and uniqueness results for second and higher order impulsive functional differential equations with infinite delay, Aust. J. Appl. math. 4, 1-26 (2007) · Zbl 1171.34053 · doi:http://ajmaa.org/cgi-bin/paper.pl?string=v4n2/V4I2P6.tex
[14]Graef, J. R.; Karsai, J.: On the oscillation of impulsively damped halflinear oscillators, Electron. J. Qual. theory differ. Equ., No. 14, 1-12 (2000) · Zbl 0971.34022
[15]Graef, J. R.; Karsai, J.: Oscillation and nonoscillation in nonlinear impulsive systems with increasing energy, Discrete contin. Dyn. syst., 161-173 (2000)
[16]Graef, J. R.; Shen, J. H.; Stavroulakis, I. P.: Oscillation of impulsive neutral delay differential equations, J. math. Anal. appl. 268, 310-333 (2002) · Zbl 1004.34054 · doi:10.1006/jmaa.2001.7836
[17]Graef, J. R.; Ouahab, A.: Some existence results and uniqueness solutions for functional impulsive differential equations with variable times in Fréchet spaces, Dyn. contin. Discrete impuls. Syst. ser. A math. Anal. 14, No. 1, 27-45 (2007) · Zbl 1120.34067
[18]Graef, J. R.; Ouahab, A.: Nonresonance impulsive functional dynamic equations on times scales, Int. J. Appl. math. Sci. 2, 65-80 (2005) · Zbl 1093.34043
[19]Graef, J. R.; Ouahab, A.: First order impulsive differential inclusions with periodic condition, Electron. J. Qual. theory differ. Equ. 31, 1-40 (2008) · Zbl 1183.34016 · doi:emis:journals/EJQTDE/2008/200831.html
[20]Graef, J. R.; Ouahab, A.: Extremal solutions for nonresonance impulsive functional dynamic equations on time scales, Appl. math. Comput. 196, No. 1, 333-339 (2008) · Zbl 1135.39009 · doi:10.1016/j.amc.2007.05.056
[21]Diethelm, K.; Freed, A. D.: On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, Scientifice computing in chemical engineering II–computational fluid dynamics, reaction engineering and molecular properties, 217-224 (1999)
[22]Gaul, L.; Klein, P.; Kempfle, S.: Damping description involving fractional operators, Mech. syst. Signal process. 5, 81-88 (1991)
[23]Glockle, W. G.; Nonnenmacher, T. F.: A fractional calculus approach of self-similar protein dynamics, Biophys. J. 68, 46-53 (1995)
[24]Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics, Fractals and fractional calculus in continuum mechanics, 291-348 (1997)
[25]Metzler, F.; Schick, W.; Kilian, H. G.; Nonnenmacher, T. F.: Relaxation in filled polymers: A fractional calculus approach, J. chem. Phys. 103, 7180-7186 (1995)
[26]Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, North-holland mathematics studies 204 (2006)
[27]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and differential equations, (1993)
[28]Podlubny, I.: Fractional differential equations, (1999)
[29]Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives, theory and applications, (1993) · Zbl 0818.26003
[30]Bai, Z.; Lu, H.: Positive solutions for boundary value problems of nonlinear fractional differential equations, J. math. Anal. appl. 311, 495-505 (2005) · Zbl 1079.34048 · doi:10.1016/j.jmaa.2005.02.052
[31]Diethelm, K.; Ford, N. J.: Analysis of fractional differential equations, J. math. Anal. appl. 265, 229-248 (2002) · Zbl 1014.34003 · doi:10.1006/jmaa.2001.7194
[32]Diethelm, K.; Walz, G.: Numerical solution of fractional order differential equations by extrapolation, Numer. algorithms 16, 231-253 (1997) · Zbl 0926.65070 · doi:10.1023/A:1019147432240
[33]El-Sayed, A. M. A.: Fractional order evolution equations, J. fract. Calc. 7, 89-100 (1995) · Zbl 0839.34069
[34]El-Sayed, A. M. A.: Fractional order diffusion-wave equations, Internat. J. Theoret. phys. 35, 311-322 (1996) · Zbl 0846.35001 · doi:10.1007/BF02083817
[35]El-Sayed, A. M. A.: Nonlinear functional differential equations of arbitrary orders, Nonlinear anal. 33, 181-186 (1998) · Zbl 0934.34055 · doi:10.1016/S0362-546X(97)00525-7
[36]El-Sayed, A. M. A.; Ibrahim, A. G.: Multivalued fractional differential equations, Appl. math. Comput. 68, 15-25 (1995) · Zbl 0830.34012 · doi:10.1016/0096-3003(94)00080-N
[37]Kilbas, A. A.; Trujillo, J. J.: Differential equations of fractional order: methods, results and problems II, Appl. anal. 81, 435-493 (2002) · Zbl 1033.34007 · doi:10.1080/0003681021000022032
[38]Momani, S. M.; Hadid, S. B.: Some comparison results for integro-fractional differential inequalities, J. fract. Calc. 24, 37-44 (2003) · Zbl 1057.45003
[39]Momani, S. M.; Hadid, S. B.; Alawenh, Z. M.: Some analytical properties of solutions of differential equations of noninteger order, Int. J. Math. math. Sci., 697-701 (2004) · Zbl 1069.34002 · doi:10.1155/S0161171204302231
[40]Nakhushev, A. M.: The Sturm–Liouville problems for a second order ordinary equations with fractional derivatives in the lower, Dokl. akad. Nauk SSSR 234, 308-311 (1977) · Zbl 0376.34015
[41]Podlubny, I.; Petraš, I.; Vinagre, B. M.; O’leary, P.; Dorčak, L.: Analogue realizations of fractional-order controllers: fractional order calculus and its applications, Nonlinear dynam. 29, 281-296 (2002) · Zbl 1041.93022 · doi:10.1023/A:1016556604320
[42]Yu, C.; Gao, G.: Existence of fractional differential equations, J. math. Anal. appl. 310, 26-29 (2005) · Zbl 1088.34501 · doi:10.1016/j.jmaa.2004.12.015
[43]Benchohra, M.; Henderson, J.; Ntouyas, S. K.; Ouahab, A.: Existence results for fractional order functional differential equations with infinite delay, J. math. Anal. appl. 338, 1340-1350 (2008) · Zbl 1209.34096 · doi:10.1016/j.jmaa.2007.06.021
[44]Lakshmikantham, V.: Theory of fractional functional differential equations, Nonlinear anal. 69, 3337-3343 (2008) · Zbl 1162.34344 · doi:10.1016/j.na.2007.09.025
[45]Lakshmikantham, V.; Vatsala, A. S.: Basic theory of fractional differential equations, Nonlinear anal. 69, 2677-2682 (2008) · Zbl 1161.34001 · doi:10.1016/j.na.2007.08.042
[46]Lakshmikantham, V.; Vatsala, A. S.: Theory of fractional differential inequalities and applications, Commun. appl. Anal. 11, No. 3–4, 395-402 (2007) · Zbl 1159.34006
[47]Lakshmikantham, V.; Vatsala, A. S.: General uniqueness and monotone iterative technique for fractional differential equations, Appl. math. Lett. 21, 828-834 (2008) · Zbl 1161.34031 · doi:10.1016/j.aml.2007.09.006
[48]Benchohra, M.; Graef, J. R.; Hamani, S.: Existence results for fractional differential inclusions with integral condition, Appl. anal. 87, 851-863 (2008)
[49]Benchohra, M.; Henderson, J.; Ntouyas, S. K.; Ouahab, A.: Existence results for fractional functional differential inclusions with infinite delay and application to control theory, Fract. calc. Appl. anal. 11, 35-56 (2008) · Zbl 1149.26010
[50]Henderson, J.; Ouahab, A.: Fractional functional differential inclusions with finite delay, Nonlinear anal. 70, 2091-2105 (2009) · Zbl 1159.34010 · doi:10.1016/j.na.2008.02.111
[51]Ouahab, A.: Some results for fractional boundary value problem of differential inclusions, Nonlinear anal. 69, 3877-3896 (2008) · Zbl 1169.34006 · doi:10.1016/j.na.2007.10.021
[52]Agarwal, R. P.; Benchohra, M.; Slimani, B. A.: Existence results for differential equations with fractional order and impulses, Mem. differential equations math. Phys. 44, 1-21 (2008) · Zbl 1178.26006
[53]Benchohra, M.; Slimani, B. A.: Existence and uniqueness of solutions to impulsive fractional differential equations, Electron. J. Differential equations 2009, No. 10, 1-11 (2009) · Zbl 1178.34004 · doi:emis:journals/EJDE/Volumes/2009/10/abstr.html
[54]J. Henderson, A. Ouahab, A Filippov’s theorem, some existence results and the compactness of solution sets of impulsive fractional order differential inclusions (submitted for publication)
[55]Górniewicz, L.: Topological fixed point theory of multivalued mappings, Mathematics and its applications 495 (1999) · Zbl 0937.55001
[56]Zhu, Q. J.: On the solution set of differential inclusions in Banach space, J. differential equations 93, 213-237 (1991) · Zbl 0735.34017 · doi:10.1016/0022-0396(91)90011-W
[57]Lasota, A.; Opial, Z.: An application of the Kakutani–Ky Fan theorem in the theory of ordinary differential equations, Bull. acad. Pol sci. Ser. sci. Math. astronom. Phys. 13, 781-786 (1965) · Zbl 0151.10703
[58]Kisielewicz, M.: Differential inclusions and optimal control, (1991)
[59]Aubin, J. P.; Cellina, A.: Differential inclusions, (1984)
[60]Aubin, J. P.; Frankowska, H.: Set-valued analysis, (1990)
[61]Deimling, K.: Multivalued differential equations, (1992) · Zbl 0760.34002
[62]Hu, Sh.; Papageorgiou, N.: Handbook of multivalued analysis, volume I: Theory, (1997)
[63]Kamenskii, M.; Obukhovskii, V.; Zecca, P.: Condensing multi-valued maps and semilinear differential inclusions in Banach spaces, (2001)
[64]Tolstonogov, A. A.: Differential inclusions in a Banach space, (2000)
[65]Caputo, M.: Elasticità e dissipazione, (1969)
[66]Caputo, M.: Linear models of dissipation whose Q is almost frequency independent, part II, Geophys. J. R. astron. Soc. 13, 529 (1967)
[67]Captuo, M.; Mainardi, F.: Linear models of dissipation in anelastic solids, Riv. nuovo cimento (Ser. II) 1, 161-198 (1971)
[68]Djebali, S.; Górniewicz, L.; Ouahab, A.: Filippov’s theorem and solution sets for first order impulsive semilinear functional differential inclusions, Topol. methods nonlinear anal. 32 (2008) · Zbl 1182.34087
[69]Hiai, F.; Umegaki, H.: Integral conditional expectations, and martingales of multivalued functions, J. multivariate anal. 7, 149-182 (1977) · Zbl 0368.60006 · doi:10.1016/0047-259X(77)90037-9
[70]Musielak, J.: Introduction to functional analysis, (1976)
[71]Castaing, C.; Valadier, M.: Convex analysis and measurable multifunctions, Lecture notes in mathematics 580 (1977) · Zbl 0346.46038
[72]Frankowska, H.: A priori estimates for operational differential inclusions, J. differential equations 84, 100-128 (1990) · Zbl 0705.34016 · doi:10.1016/0022-0396(90)90129-D
[73]A. Ouahab, Filippov’s theorem for impulsive differential inclusions with fractional order, Electron. J. Qual. Theory Differ. Equ. (in press) · Zbl 1214.34006 · doi:emis:journals/EJQTDE/sped1/123.pdf
[74]Granas, A.; Dugundji, J.: Fixed point theory, (2003)
[75]Brezis, H.: Analyse fonctionnelle thoire et applications, (1983) · Zbl 0511.46001
[76]Colombo, R. M.; Fryszkowski, A.; Rzežuchowski, T.; Staticu, V.: Continuous selection of solution sets of Lipschitzean differential inclusions, Funkcial. ekvac. 34, 321-330 (1991) · Zbl 0749.34008
[77]Bressan, A.; Colombo, G.: Extensions and selections of maps with decomposable values, Studia math. 90, 70-85 (1988) · Zbl 0677.54013
[78]Frigon, M.; Granas, A.: Théorèmes d’existence pour des inclusions différentielles sans convexité, CR acad. Sci. Paris ser. I 310, 819-822 (1990) · Zbl 0731.47048
[79]Ouahab, A.: Local and global existence and uniqueness results for impulsive differential equations with multiple delay, J. math. Anal. appl. 323, 456-472 (2006) · Zbl 1121.34084 · doi:10.1016/j.jmaa.2005.10.015
[80]A. Ouahab, Some contributions in impulsives differential equations and inclusions with fixed and variable times, Ph.D. Dissertation, University of Sidi-Bel-Abbès, Algeria, 2006
[81]Ouahab, A.: Existence and uniqueness results for impulsive functional differential equations with scalar multiple delay and infinite delay, Nonlinear anal. 67, 1027-1041 (2007) · Zbl 1126.34050 · doi:10.1016/j.na.2006.06.033
[82]Bielawski, R.; Górniewicz, L.; Plaskacz, S.: Topological approach to differential inclusions on closed sets of rn, Dyn. rep. 1, 225-250 (1992) · Zbl 0747.34013
[83]Górniewicz, L.: Homological methods in fixed point theory of multivalued maps, Dissertationes math. 129, 1-71 (1976) · Zbl 0324.55002
[84]Górniewicz, L.: On the solution sets of differential inclusions, J. math. Anal. appl. 113, 235-244 (1986) · Zbl 0609.34012 · doi:10.1016/0022-247X(86)90347-1
[85]Hyman, D. M.: On decreasing sequences of compact absolute retracts, Fund. math. 64, 91-97 (1969) · Zbl 0174.25804
[86]J.M. Lasry, R. Robert, Analyse Non Linéaire Multivoque, Publ. no. 7611, Centre de Recherche de Mathématique de la Décision, Université de Dauphine, Paris, pp. 1–190
[87]Andres, J.; Górniewicz, L.: Topological fixed point principles for boundary value problems, (2003)
[88]Browder, F. E.; Gupta, G. P.: Topological degree and nonlinear mappings of analytic type in Banach spaces, J. math. Anal. appl. 26, 390-402 (1969) · Zbl 0176.45401 · doi:10.1016/0022-247X(69)90162-0
[89]Haddad, G.; Lasry, J. M.: Periodic solutions of functional differential inclusions and fixed points of σ-selectionable correspondences, J. math. Anal. appl. 96, 295-312 (1983) · Zbl 0539.34031 · doi:10.1016/0022-247X(83)90042-2
[90]Henry, D.: Geometric theory of semilinear parabolic partial differential equations, (1989)
[91]Nieto, J. J.: Impulsive resonance periodic problems of first order, Appl. math. Lett. 15, 489-493 (2002) · Zbl 1022.34025 · doi:10.1016/S0893-9659(01)00163-X
[92]Nieto, J. J.: Periodic boundary value problems for first-order impulsive ordinary differential equations, Nonlinear anal. 51, 1223-1232 (2002) · Zbl 1015.34010 · doi:10.1016/S0362-546X(01)00889-6
[93]Nieto, J. J.: Basic theory for nonresonance impulsive periodic problems of first order, J. math. Anal. appl. 205, 423-433 (1997) · Zbl 0870.34009 · doi:10.1006/jmaa.1997.5207
[94]Dylawerski, G.; Górniewicz, L.: A remark on the kreasnosielskii’s translation operator, Serdica 9, 102-107 (1983)
[95]Górniewicz, L.: Topological degree of morphisme and its applications to differential inclusions, Racc. sem. Dip. mat. Univ. studi calabria 5, 1-48 (1985)
[96]Górniewicz, L.; Plaskacz, S.: Periodic solutions of differential inclusions in rn, Boll. unicone mat. Ital. 7, 409-420 (1993) · Zbl 0798.34018
[97]Krasnosel’skii, M. A.: Translation operator along the trajectories of differential equations, (1966)
[98]Krasnosel’skii, M. A.; Zabreiko, P.: Geometric mathods of nonlinear analysis, Grundlehren math. Wiss. 263 (1984)
[99]F. Zanolin, Continuation theorems for the periodic problem via the translation operator, University of Udine, Preprint, 1994
[100]Andres, J.: On the multivalued Poincaré operators, Topol. methods nonlinear anal. 10, 171-182 (1997) · Zbl 0909.47038
[101]Haddad, G.: Topological properties of the sets of solutions for functional differential inclusions, Nonlinear anal. 5, No. 12, 1349-1366 (1981) · Zbl 0496.34041 · doi:10.1016/0362-546X(81)90111-5
[102]Henderson, J.; Ouahab, A.: Impulsive version of Filippov’s theorem and the Filippov–wazewski theorem for second order impulsive semilinear functional differential inclusions, Int. J. Mod. math. 3, 111-133 (2008) · Zbl 1160.34073