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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.


MSC:
34A08Fractional differential equations
34A60Differential inclusions
26A33Fractional derivatives and integrals (real functions)
34A37Differential equations with impulses
47N20Applications of operator theory to differential and integral equations
References:
[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