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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.
[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.
[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
[8] Diethelm, K; Ford, NJ, Analysis of fractional differential equations, Journal of Mathematical Analysis and Applications, 265, 229-248, (2002) · Zbl 1014.34003
[9] El-Sayed, AMA, Fractional order evolution equations, Journal of Fractional Calculus, 7, 89-100, (1995) · Zbl 0839.34069
[10] El-Sayed, AMA, Fractional-order diffusion-wave equation, International Journal of Theoretical Physics, 35, 311-322, (1996) · Zbl 0846.35001
[11] El-Sayed, AMA, Nonlinear functional-differential equations of arbitrary orders, Nonlinear Analysis: Theory, Methods & Applications, 33, 181-186, (1998) · Zbl 0934.34055
[12] Gaul, L; Klein, P; Kempfle, S, Damping description involving fractional operators, Mechanical Systems and Signal Processing, 5, 81-88, (1991)
[13] Glockle, WG; Nonnenmacher, TF, A fractional calculus approach to self-similar protein dynamics, Biophysical Journal, 68, 46-53, (1995)
[14] Lakshmikantham, V; Devi, JV, Theory of fractional differential equations in a Banach space, European Journal of Pure and Applied Mathematics, 1, 38-45, (2008) · 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
[16] Metzler, F; Schick, W; Kilian, HG; Nonnenmacher, TF, Relaxation in filled polymers: a fractional calculus approach, Journal of Chemical Physics, 103, 7180-7186, (1995)
[17] Momani, SM; Hadid, SB, Some comparison results for integro-fractional differential inequalities, Journal of Fractional Calculus, 24, 37-44, (2003) · 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, 697-701, (2004) · Zbl 1069.34002
[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, 29, 281-296, (2002) · Zbl 1041.93022
[20] Yu, C; Gao, G, Existence of fractional differential equations, Journal of Mathematical Analysis and Applications, 310, 26-29, (2005) · Zbl 1088.34501
[21] El-Borai, MM, On some fractional evolution equations with nonlocal conditions, International Journal of Pure and Applied Mathematics, 24, 405-413, (2005) · 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, 197-211, (2004) · Zbl 1081.34053
[23] Jaradat, OK; Al-Omari, A; Momani, S, Existence of the mild solution for fractional semilinear initial value problems, Nonlinear Analysis: Theory, Methods & Applications, 69, 3153-3159, (2008) · Zbl 1160.34300
[24] Goldstein JA: Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs. Clarendon Press/Oxford University Press, New York, NY, USA; 1985:x+245.
[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.
[26] Travis, CC; Webb, GF, Second order differential equations in Banach spaces, 331-361, (1978), New York, NY, USA
[27] Travis, CC; Webb, GF, Cosine families and abstract nonlinear second order differential equations, Acta Mathematica Academiae Scientiarum Hungaricae, 32, 75-96, (1978) · Zbl 0388.34039
[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.
[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
[30] Kisielewicz M: Differential Inclusions and Optimal Control, Mathematics and Its Applications. Volume 44. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1991:xx+240.
[31] Deimling K: Multivalued Differential Equations, de Gruyter Series in Nonlinear Analysis and Applications. Volume 1. Walter de Gruyter, Berlin, Germany; 1992:xii+260.
[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.
[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.
[34] Henry D: Geometric Theory of Semilinear Parabolic Partial Differential Equations. Springer, Berlin, Germany; 1989.
[35] Covitz, H; Nadler, SB, Multi-valued contraction mappings in generalized metric spaces, Israel Journal of Mathematics, 8, 5-11, (1970) · Zbl 0192.59802
[36] Granas A, Dugundji J: Fixed Point Theory, Springer Monographs in Mathematics. Springer, New York, NY, USA; 2003:xvi+690. · Zbl 1025.47002
[37] Martelli, M, A Rothe’s type theorem for non-compact acyclic-valued maps, Bollettino della Unione Matematica Italiana. Serie 4, 11, 70-76, (1975) · Zbl 0314.47035
[38] Burton, TA; Kirk, C, A fixed point theorem of Krasnoselskii-Schaefer type, Mathematische Nachrichten, 189, 23-31, (1998) · Zbl 0896.47042
[39] Dhage, BC, Multi-valued mappings and fixed points. I, Nonlinear Functional Analysis and Applications, 10, 359-378, (2005) · Zbl 1100.47040
[40] Dhage, BC, Multi-valued mappings and fixed points. II, Tamkang Journal of Mathematics, 37, 27-46, (2006) · Zbl 1108.47046
[41] Hale, JK; Kato, J, Phase space for retarded equations with infinite delay, Funkcialaj Ekvacioj, 21, 11-41, (1978) · 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.
[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
[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
[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
[47] Belarbi, A; Benchohra, M; Hamani, S; Ntouyas, SK, Perturbed functional differential equations with fractional order, Communications in Applied Analysis, 11, 429-440, (2007) · 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, 85, 1459-1470, (2006) · Zbl 1175.34080
[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, 11, 35-56, (2008) · 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, 338, 1340-1350, (2008) · Zbl 1209.34096
[51] Belmekki, M; Benchohra, M, Existence results for fractional order semilinear functional differential equations, Proceedings of A. Razmadze Mathematical Institute, 146, 9-20, (2008) · Zbl 1175.26006
[52] Belmekki, M; Benchohra, M; Górniewicz, L, Functional differential equations with fractional order and infinite delay, Fixed Point Theory, 9, 423-439, (2008) · Zbl 1162.26302
[53] Heymans, N; Podlubny, I, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheologica Acta, 45, 765-772, (2006)
[54] Podlubny, I, Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus & Applied Analysis, 5, 367-386, (2002) · Zbl 1042.26003
[55] Prüss J: Evolutionary Integral Equations and Applications, Monographs in Mathematics. Volume 87. Birkhäuser, Basel, Switzerland; 1993:xxvi+366.
[56] Hilfe R (Ed): Applications of Fractional Calculus in Physics. World Scientific, River Edge, NJ, USA; 2000:viii+463.
[57] Hernández, E; Henríquez, HR, Existence results for partial neutral functional differential equations with unbounded delay, Journal of Mathematical Analysis and Applications, 221, 452-475, (1998) · Zbl 0915.35110
[58] Hernández, E; Henríquez, HR, Existence of periodic solutions of partial neutral functional differential equations with unbounded delay, Journal of Mathematical Analysis and Applications, 221, 499-522, (1998) · Zbl 0926.35151
[59] El-Sayed, AMA; Ibrahim, A-G, Multivalued fractional differential equations, Applied Mathematics and Computation, 68, 15-25, (1995) · Zbl 0830.34012
[60] Ouahab, A, Some results for fractional boundary value problem of differential inclusions, Nonlinear Analysis: Theory, Methods & Applications, 69, 3877-3896, (2008) · Zbl 1169.34006
[61] Agarwal, RP; Benchohra, M; Hamani, S, Boundary value problems for differential inclusions with fractional order, Advanced Studies in Contemporary Mathematics, 16, 181-196, (2008) · Zbl 1152.26005
[62] Chang, Y-K; Nieto, JJ, Some new existence results for fractional differential inclusions with boundary conditions, Mathematical and Computer Modelling, 49, 605-609, (2009) · Zbl 1165.34313
[63] Yosida K: Functional Analysis, Grundlehren der Mathematischen Wissenschaften. Volume 123. 6th edition. Springer, Berlin, Germany; 1980:xii+501.
[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
[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.
[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.
[67] Dhage, BC; Henderson, J, Existence theory for nonlinear functional boundary value problems, Electronic Journal of Qualitative Theory of Differential Equations, 2004, 1-15, (2004) · Zbl 1082.34054
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