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A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative. (English) Zbl 1182.34103

This review paper focuses on semilinear functional differential equations containing Riemann-Liouville fractional derivatives. The paper’s main contribution is that it brings together and establishes a range of results on basic theory for different classes of problem. For semilinear functional differential equations of both classical and neutral type, consideration is given first to equations with finite delay and then to equations with infinite delay. There is a review of existence results and an example in each case. Finally the authors consider perturbed differential equations and inclusions, and provide some existence results on ordered Banach spaces.

MSC:

34K37 Functional-differential equations with fractional derivatives
34K40 Neutral functional-differential equations
34K30 Functional-differential equations in abstract spaces
34K09 Functional-differential inclusions
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[1] Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Volume 204. Elsevier Science, Amsterdam, The Netherlands; 2006:xvi+523. · Zbl 1092.45003
[2] Kiryakova V: Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics Series. Volume 301. Longman Scientific & Technical, Harlow, UK; John Wiley & Sons, New York, NY, USA; 1994:x+388. · Zbl 0882.26003
[3] Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication. John Wiley & Sons, New York, NY, USA; 1993:xvi+366. · Zbl 0789.26002
[4] Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340. · Zbl 0924.34008
[5] Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon, Switzerland; 1993:xxxvi+976. · Zbl 0818.26003
[6] Agarwal RP, Benchohra M, Hamani S: Boundary value problems for fractional differential equations. to appear in Georgian Mathematical Journal · Zbl 1179.26011
[7] Diethelm, K.; Freed, AD; Keil, F. (ed.); Mackens, W. (ed.); Voß, H. (ed.); Werther, J. (ed.), On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, 217-224 (1999), Heidelberg, Germany · doi:10.1007/978-3-642-60185-9_24
[8] Diethelm K, Ford NJ: Analysis of fractional differential equations.Journal of Mathematical Analysis and Applications 2002,265(2):229-248. 10.1006/jmaa.2000.7194 · Zbl 1014.34003 · doi:10.1006/jmaa.2000.7194
[9] El-Sayed AMA: Fractional order evolution equations.Journal of Fractional Calculus 1995, 7: 89-100. · Zbl 0839.34069
[10] El-Sayed AMA: Fractional-order diffusion-wave equation.International Journal of Theoretical Physics 1996,35(2):311-322. 10.1007/BF02083817 · Zbl 0846.35001 · doi:10.1007/BF02083817
[11] El-Sayed AMA: Nonlinear functional-differential equations of arbitrary orders.Nonlinear Analysis: Theory, Methods & Applications 1998,33(2):181-186. 10.1016/S0362-546X(97)00525-7 · Zbl 0934.34055 · doi:10.1016/S0362-546X(97)00525-7
[12] Gaul L, Klein P, Kempfle S: Damping description involving fractional operators.Mechanical Systems and Signal Processing 1991,5(2):81-88. 10.1016/0888-3270(91)90016-X · doi:10.1016/0888-3270(91)90016-X
[13] Glockle WG, Nonnenmacher TF: A fractional calculus approach to self-similar protein dynamics.Biophysical Journal 1995,68(1):46-53. 10.1016/S0006-3495(95)80157-8 · doi:10.1016/S0006-3495(95)80157-8
[14] Lakshmikantham V, Devi JV: Theory of fractional differential equations in a Banach space.European Journal of Pure and Applied Mathematics 2008,1(1):38-45. · Zbl 1146.34042
[15] Mainardi, F.; Carpinteri, A. (ed.); Mainard, F. (ed.), Fractional calculus: some basic problems in continuum and statistical mechanis, 291-348 (1997), Vienna, Austria · Zbl 0917.73004 · doi:10.1007/978-3-7091-2664-6_7
[16] Metzler F, Schick W, Kilian HG, Nonnenmacher TF: Relaxation in filled polymers: a fractional calculus approach.Journal of Chemical Physics 1995,103(16):7180-7186. 10.1063/1.470346 · doi:10.1063/1.470346
[17] Momani SM, Hadid SB: Some comparison results for integro-fractional differential inequalities.Journal of Fractional Calculus 2003, 24: 37-44. · Zbl 1057.45003
[18] Momani SM, Hadid SB, Alawenh ZM: Some analytical properties of solutions of differential equations of noninteger order.International Journal of Mathematics and Mathematical Sciences 2004,2004(13-16):697-701. · Zbl 1069.34002 · doi:10.1155/S0161171204302231
[19] Podlubny I, Petráš I, Vinagre BM, O’Leary P, Dorčák L’: Analogue realizations of fractional-order controllers. Fractional order calculus and its applications.Nonlinear Dynamics 2002,29(1-4):281-296. · Zbl 1041.93022 · doi:10.1023/A:1016556604320
[20] Yu C, Gao G: Existence of fractional differential equations.Journal of Mathematical Analysis and Applications 2005,310(1):26-29. 10.1016/j.jmaa.2004.12.015 · Zbl 1088.34501 · doi:10.1016/j.jmaa.2004.12.015
[21] El-Borai MM: On some fractional evolution equations with nonlocal conditions.International Journal of Pure and Applied Mathematics 2005,24(3):405-413. · Zbl 1090.35006
[22] El-Borai MM: The fundamental solutions for fractional evolution equations of parabolic type.Journal of Applied Mathematics and Stochastic Analysis 2004,2004(3):197-211. 10.1155/S1048953304311020 · Zbl 1081.34053 · doi:10.1155/S1048953304311020
[23] Jaradat OK, Al-Omari A, Momani S: Existence of the mild solution for fractional semilinear initial value problems.Nonlinear Analysis: Theory, Methods & Applications 2008,69(9):3153-3159. 10.1016/j.na.2007.09.008 · Zbl 1160.34300 · doi:10.1016/j.na.2007.09.008
[24] Goldstein JA: Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs. Clarendon Press/Oxford University Press, New York, NY, USA; 1985:x+245. · Zbl 0592.47034
[25] Fattorini HO: Second Order Linear Differential Equations in Banach Spaces, North-Holland Mathematics Studies. Volume 108. North-Holland, Amsterdam, The Netherlands; 1985:xiii+314. · Zbl 0564.34063
[26] Travis, CC; Webb, GF, Second order differential equations in Banach spaces, 331-361 (1978), New York, NY, USA · Zbl 0455.34044
[27] Travis CC, Webb GF: Cosine families and abstract nonlinear second order differential equations.Acta Mathematica Academiae Scientiarum Hungaricae 1978,32(1-2):75-96. 10.1007/BF01902205 · Zbl 0388.34039 · doi:10.1007/BF01902205
[28] Ahmed NU: Semigroup Theory with Applications to Systems and Control, Pitman Research Notes in Mathematics Series. Volume 246. Longman Scientific & Technical, Harlow, UK; John Wiley & Sons, New York, NY, USA; 1991:x+282. · Zbl 0727.47026
[29] Pazy A: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences. Volume 44. Springer, New York, NY, USA; 1983:viii+279. · Zbl 0516.47023 · doi:10.1007/978-1-4612-5561-1
[30] Kisielewicz M: Differential Inclusions and Optimal Control, Mathematics and Its Applications. Volume 44. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1991:xx+240. · Zbl 0731.49001
[31] Deimling K: Multivalued Differential Equations, de Gruyter Series in Nonlinear Analysis and Applications. Volume 1. Walter de Gruyter, Berlin, Germany; 1992:xii+260. · Zbl 0760.34002
[32] Górniewicz L: Topological Fixed Point Theory of Multivalued Mappings, Mathematics and Its Applications. Volume 495. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1999:x+399. · Zbl 0937.55001 · doi:10.1007/978-94-015-9195-9
[33] Hu S, Papageorgiou NS: Handbook of Multivalued Analysis. Volume I: Theory, Mathematics and Its Applications. Volume 419. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997:xvi+964. · Zbl 0887.47001 · doi:10.1007/978-1-4615-6359-4
[34] Henry D: Geometric Theory of Semilinear Parabolic Partial Differential Equations. Springer, Berlin, Germany; 1989.
[35] Covitz H, Nadler SB Jr.: Multi-valued contraction mappings in generalized metric spaces.Israel Journal of Mathematics 1970,8(1):5-11. 10.1007/BF02771543 · Zbl 0192.59802 · doi:10.1007/BF02771543
[36] Granas A, Dugundji J: Fixed Point Theory, Springer Monographs in Mathematics. Springer, New York, NY, USA; 2003:xvi+690. · Zbl 1025.47002 · doi:10.1007/978-0-387-21593-8
[37] Martelli M: A Rothe’s type theorem for non-compact acyclic-valued maps.Bollettino della Unione Matematica Italiana. Serie 4 1975,11(3, supplement):70-76. · Zbl 0314.47035
[38] Burton TA, Kirk C: A fixed point theorem of Krasnoselskii-Schaefer type.Mathematische Nachrichten 1998, 189: 23-31. 10.1002/mana.19981890103 · Zbl 0896.47042 · doi:10.1002/mana.19981890103
[39] Dhage BC: Multi-valued mappings and fixed points. I.Nonlinear Functional Analysis and Applications 2005,10(3):359-378. · Zbl 1100.47040
[40] Dhage BC: Multi-valued mappings and fixed points. II.Tamkang Journal of Mathematics 2006,37(1):27-46. · Zbl 1108.47046
[41] Hale JK, Kato J: Phase space for retarded equations with infinite delay.Funkcialaj Ekvacioj 1978,21(1):11-41. · Zbl 0383.34055
[42] Hino Y, Murakami S, Naito T: Functional-Differential Equations with Infinite Delay, Lecture Notes in Mathematics. Volume 1473. Springer, Berlin, Germany; 1991:x+317. · Zbl 0732.34051
[43] Hale JK: Theory of Functional Differential Equations, Applied Mathematical Sciences. Volume 3. 2nd edition. Springer, New York, NY, USA; 1977:x+365. · Zbl 0352.34001 · doi:10.1007/978-1-4612-9892-2
[44] Hale JK, Verduyn Lunel SM: Introduction to Functional-Differential Equations, Applied Mathematical Sciences. Volume 99. Springer, New York, NY, USA; 1993:x+447. · Zbl 0787.34002 · doi:10.1007/978-1-4612-4342-7
[45] Kolmanovskii V, Myshkis A: Introduction to the Theory and Applications of Functional-Differential Equations, Mathematics and Its Applications. Volume 463. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1999:xvi+648. · Zbl 0917.34001 · doi:10.1007/978-94-017-1965-0
[46] Wu J: Theory and Applications of Partial Functional-Differential Equations, Applied Mathematical Sciences. Volume 119. Springer, New York, NY, USA; 1996. · Zbl 0870.35116 · doi:10.1007/978-1-4612-4050-1
[47] Belarbi A, Benchohra M, Hamani S, Ntouyas SK: Perturbed functional differential equations with fractional order.Communications in Applied Analysis 2007,11(3-4):429-440. · Zbl 1148.34042
[48] Belarbi A, Benchohra M, Ouahab A: Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces.Applicable Analysis 2006,85(12):1459-1470. 10.1080/00036810601066350 · Zbl 1175.34080 · doi:10.1080/00036810601066350
[49] Benchohra M, Henderson J, Ntouyas SK, Ouahab A: Existence results for fractional functional differential inclusions with infinite delay and applications to control theory.Fractional Calculus & Applied Analysis 2008,11(1):35-56. · Zbl 1149.26010
[50] Benchohra M, Henderson J, Ntouyas SK, Ouahab A: Existence results for fractional order functional differential equations with infinite delay.Journal of Mathematical Analysis and Applications 2008,338(2):1340-1350. 10.1016/j.jmaa.2007.06.021 · Zbl 1209.34096 · doi:10.1016/j.jmaa.2007.06.021
[51] Belmekki M, Benchohra M: Existence results for fractional order semilinear functional differential equations.Proceedings of A. Razmadze Mathematical Institute 2008, 146: 9-20. · Zbl 1175.26006
[52] Belmekki M, Benchohra M, Górniewicz L: Functional differential equations with fractional order and infinite delay.Fixed Point Theory 2008,9(2):423-439. · Zbl 1162.26302
[53] Heymans N, Podlubny I: Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives.Rheologica Acta 2006,45(5):765-772. 10.1007/s00397-005-0043-5 · doi:10.1007/s00397-005-0043-5
[54] Podlubny I: Geometric and physical interpretation of fractional integration and fractional differentiation.Fractional Calculus & Applied Analysis 2002,5(4):367-386. · Zbl 1042.26003
[55] Prüss J: Evolutionary Integral Equations and Applications, Monographs in Mathematics. Volume 87. Birkhäuser, Basel, Switzerland; 1993:xxvi+366. · Zbl 0784.45006 · doi:10.1007/978-3-0348-8570-6
[56] Hilfe R (Ed): Applications of Fractional Calculus in Physics. World Scientific, River Edge, NJ, USA; 2000:viii+463. · Zbl 0998.26002
[57] Hernández E, Henríquez HR: Existence results for partial neutral functional differential equations with unbounded delay.Journal of Mathematical Analysis and Applications 1998,221(2):452-475. 10.1006/jmaa.1997.5875 · Zbl 0915.35110 · doi:10.1006/jmaa.1997.5875
[58] Hernández E, Henríquez HR: Existence of periodic solutions of partial neutral functional differential equations with unbounded delay.Journal of Mathematical Analysis and Applications 1998,221(2):499-522. 10.1006/jmaa.1997.5899 · Zbl 0926.35151 · doi:10.1006/jmaa.1997.5899
[59] El-Sayed AMA, Ibrahim A-G: Multivalued fractional differential equations.Applied Mathematics and Computation 1995,68(1):15-25. 10.1016/0096-3003(94)00080-N · Zbl 0830.34012 · doi:10.1016/0096-3003(94)00080-N
[60] Ouahab A: Some results for fractional boundary value problem of differential inclusions.Nonlinear Analysis: Theory, Methods & Applications 2008,69(11):3877-3896. 10.1016/j.na.2007.10.021 · Zbl 1169.34006 · doi:10.1016/j.na.2007.10.021
[61] Agarwal RP, Benchohra M, Hamani S: Boundary value problems for differential inclusions with fractional order.Advanced Studies in Contemporary Mathematics 2008,16(2):181-196. · Zbl 1152.26005
[62] Chang Y-K, Nieto JJ: Some new existence results for fractional differential inclusions with boundary conditions.Mathematical and Computer Modelling 2009,49(3-4):605-609. 10.1016/j.mcm.2008.03.014 · Zbl 1165.34313 · doi:10.1016/j.mcm.2008.03.014
[63] Yosida K: Functional Analysis, Grundlehren der Mathematischen Wissenschaften. Volume 123. 6th edition. Springer, Berlin, Germany; 1980:xii+501. · Zbl 0435.46002
[64] Castaing C, Valadier M: Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics. Volume 580. Springer, Berlin, Germany; 1977:vii+278. · Zbl 0346.46038 · doi:10.1007/BFb0087685
[65] Heikkilä S, Lakshmikantham V: Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics. Volume 181. Marcel Dekker, New York, NY, USA; 1994:xii+514. · Zbl 0804.34001
[66] Joshi MC, Bose RK: Some Topics in Nonlinear Functional Analysis, A Halsted Press Book. John Wiley & Sons, New York, NY, USA; 1985:viii+311. · Zbl 0596.47038
[67] Dhage BC, Henderson J: Existence theory for nonlinear functional boundary value problems.Electronic Journal of Qualitative Theory of Differential Equations 2004,2004(1):1-15. · Zbl 1082.34054
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