Fractional calculus and fractional differential equations in nonreflexive Banach spaces. (English) Zbl 1311.34010
Summary: We establish an existence result for the fractional differential equation
\[
\begin{cases} D_p^\alpha y(t) = f(t, y(t)), \\ y(0) = y_0, \end{cases}
\]
where \(D_p^\alpha y(\cdot)\) is a fractional pseudo-derivative of a weakly absolutely continuous and pseudo-differentiable function \(y(\cdot):T\to E\), the function \(f(t,\cdot):T\times E\to E\) is weakly-weakly sequentially continuous for every \(t\in T\) and \(f(\cdot, y(\cdot))\) is Pettis integrable for every weakly absolutely continuous function \(y(\cdot):T\to E, T\) is a bounded interval of real numbers and \(E\) is a nonreflexive Banach space.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
| 34G20 | Nonlinear differential equations in abstract spaces |
References:
| [1] | Ambrosetti, A., Un teorema di esistenza per le equazioni differenziali negli spazi di Banach, Rend Sem Mat Univ Padova, 39, 349-369 (1967) · Zbl 0174.46001 |
| [2] | Arendt, W.; Batty, C.; Hieber, M.; Neubrander, F., Vector-valued laplace transforms and cauchy problems, (Monographs in mathematics, vol. 96 (2001), Birkhauser: Birkhauser Basel) · Zbl 0978.34001 |
| [3] | Arino, O.; Gautier, S.; Penot, J. P., A fixed point theorem for sequentially continuous mappings with application to ordinary differential equations, Funkcialaj Ekvacioj, 27, 273-279 (1984) · Zbl 0599.34008 |
| [4] | Benchohra, M.; Mostefai, F., Weak solutions for nonlinear fractional differential equations with integral boundary conditions in Banach spaces, Opuscula Math, 32, 1, 31-40 (2012) · Zbl 1252.34002 |
| [5] | Benchohra, M.; Graef, J. R.; Mostefai, F., Weak solutions for nonlinear fractional differential equations on reflexive Banach spaces, Electron J Qual Theory Differ Equ, 54, 1-10 (2010) · Zbl 1206.26006 |
| [6] | Cichoń, M., Weak solutions of ordinary differential equations in Banach spaces, Discuss Math Differ Inc Control Optimal, 15, 5-14 (1995) · Zbl 0829.34051 |
| [7] | Cichoń, M., On solutions of differential equations in Banach spaces, Nonlinear Anal, 60, 651-667 (2005) · Zbl 1061.34043 |
| [8] | Cichoń, M.; Kubiaczyk, I., Kneser’s theorem for strong, weak and pseudo-solutions of ordinary differential equations in Banach spaces, Ann Polon Math, 52, 13-21 (1995) · Zbl 0836.34062 |
| [9] | Cramer, E.; Lakshmikantham, V.; Mitchell, A. R., On the existence of weak solutions of differential equations in nonreflexive Banach spaces, Nonlinear Anal, 2, 259-262 (1978) · Zbl 0379.34041 |
| [10] | Chow, S.; Schuur, J. D., An existence theorem for ordinary differential equations in Banach spaces, Bull Am Math Soc, 77, 1018-1020 (1971) · Zbl 0264.34072 |
| [11] | De Blasi, F., On a property of the unit sphere in a Banach space, Bull Math Soc Sci Math RS Roumanie, 21, 259-262 (1977) · Zbl 0365.46015 |
| [12] | Deimling, K., Ordinary differential equations in Banach spaces, (LNM, vol. 596 (1977), Springer-Verlag Publisher: Springer-Verlag Publisher Berlin, Heidelberg, New York) · Zbl 0364.34030 |
| [13] | Diestel, J.; Uhl, J. J., Vector measures, Mathematical surveys, vol. 15 (1977), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0369.46039 |
| [14] | Dilworth, S. J.; Girardi, M., Nowhere weak differentiability of the Pettis integral, Quaestiones Math, 18, 4, 365-380 (1995) · Zbl 0856.28006 |
| [15] | Dieudonné, J., Deux exemples singuliers d’équations différentielles, Acta Sci Math, 12, 38-40 (1950) · Zbl 0037.06002 |
| [16] | Dinuleanu, N., Vector measures (1967), Pergamon Press: Pergamon Press New York |
| [17] | Dutkiewicz, A.; Szufla, S., Kneser’s theorem for weak solution of and integral equation with weakly singular kernal, Publ Inst Math, 77, 91, 87-92 (2005) · Zbl 1142.45308 |
| [18] | Edwards, R. E., Functional analysis (1965), Holt Rinehart and Wiston: Holt Rinehart and Wiston New York · Zbl 0182.16101 |
| [20] | Geitz, R. F., Geometry and the Pettis integral, Trans Am Math Soc, 169, 2, 535-548 (1982) · Zbl 0498.28005 |
| [21] | Gripenberg, G.; Londen, S.-O.; Staffans, O., Volterra integral and functional equations (1900), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0695.45002 |
| [22] | Gripenberg, C., Unique solutions of some Volterra integral equations, Math Scand, 48, 59-67 (1981) · Zbl 0463.45002 |
| [23] | Gomaa, A. M., Weak and strong solutions for differential equations in Banach spaces, Chaos Solutions Fract, 18, 687-692 (2003) · Zbl 1058.34077 |
| [24] | Hashem, H. H.G., Weak solutions of differential equations in Banach spaces, J Fract Calc Appl, 3, 1, 1-9 (2012) |
| [25] | Kadets, V. M., Non-differentiable indefinite Pettis integrals, Quaestiones Math., 17, 137-139 (1994) · Zbl 0816.46035 |
| [26] | Kato, S., On existence of solutions of ordinary differential equations in Banach spaces, Funkcial Ekvac, 19, 239-245 (1976) · Zbl 0358.34064 |
| [27] | Knight, W. J., Absolute continuity of some vector functions and measures, Can J Math, 24, 5, 737-746 (1972) · Zbl 0235.46034 |
| [28] | Knight, W. J., Solutions of differential equations in Banach spaces, Duke Math J, 41, 437-442 (1974) · Zbl 0288.34063 |
| [29] | Kuratowski, K., Sur les espaces complets, Fundam Math, 15, 301-309 (1930) · JFM 56.1124.04 |
| [30] | Lakshmikantham, V.; Leela, S.; Vasundhara Devi, J., Theory of fractional dynamic systems (2009), Cambridge Academic Publishers: Cambridge Academic Publishers Cambridge · Zbl 1188.37002 |
| [31] | Munroe, M. E., A note on weak differentiability of Pettis integrals, Bull Am Math Soc, 52, 167-174 (1946) · Zbl 0061.25102 |
| [33] | O’Regan, D., Fixed point theory for weakly sequentially continuous mapping, Math Comput Model., 27, 5, 1-14 (1998) · Zbl 1185.34026 |
| [34] | O’Regan, D., Weak solutions of ordinary differential equations in Banach spaces, Appl Math Lett, 12, 101-105 (1999) · Zbl 0933.34068 |
| [35] | Papageargiou, N. S., Weak solutions of differential equations in Banach spaces, Bull Aust Math Soc, 33, 407-418 (1986) · Zbl 0578.34039 |
| [36] | Perri, E., On a characterization of reflexive Banach spaces, Rend Sem Mat Univ Padova, 69, 211-219 (1983) · Zbl 0529.46009 |
| [37] | Pettis, J. P., On integration in vector spaces, Trans Am Math Soc, 44, 277-304 (1938) · JFM 64.0371.02 |
| [38] | Philips, R. S., Integration in a convex linear topological space, Trans Am Math Soc, 47, 114-145 (1940) · Zbl 0022.31902 |
| [39] | Pianigiani, G., Existence of solutions of ordinary differential equations in Banach spaces, Bull Acad Pol Math, 23, 853-857 (1975) · Zbl 0317.34050 |
| [40] | Salem, H. A.H., On the fractional calculus in abstract spaces and their applications to the Dirichlet-type problem of fractional orders, Comput Math Appl, 59, 1278-1293 (2010) · Zbl 1241.34011 |
| [41] | Salem, H. A.H.; El-Sayed, A. M.A., Weak solution for fractional order integral equations in reflexive Banach space, Math Slov, 55, 2, 169-181 (2005) · Zbl 1111.26011 |
| [42] | Salem, H. A.H.; El-Sayed, S. M.A., A note of the fractional Calculus in Banach spaces, Stud Sci Math Hungarian, 42, 2, 115-130 (2005) · Zbl 1086.45004 |
| [43] | Samko, S.; Kilbas, A.; Marichev, O., Fraction integrals and derivatives (1993), Gordon and Breach Science Publisher · Zbl 0818.26003 |
| [44] | Schwabik, S.; Guoju, Y., Topics in Banach space integration (2005), World Scientific: World Scientific Singapore · Zbl 1088.28008 |
| [45] | Singer, I., Linear functionals on the space of continuous mappings of a compact space into a Banach space, Rev Roum Math Pures Appl, 2, 301-315 (1957) |
| [46] | Solomon, B. W., On differentiability of vector-valued functions of a real variables, Stud Math, 29, 1-4 (1967) · Zbl 0179.19102 |
| [47] | Solomon, B. W., Denjoy integration in abstract spaces, Memories of the American mathematical society (1969), American Mathematical Society: American Mathematical Society Providence (RI) · Zbl 0175.34301 |
| [48] | Stefansson, G. F., Pettis integrability, Trans Am Math Soc, 330, 401-417 (1993) · Zbl 0823.28004 |
| [49] | Szep, A., Existence theorem for weak solutions of differential equations in Banach spaces, Stud Sci Math Hungarian, 6, 197-203 (1971) · Zbl 0238.34100 |
| [50] | Talagrand, M., Pettis integral and measure theory, Memories of the American mathematical society, vol. 307 (1984), Am. Math. Soc.: Am. Math. Soc. Providence (RI) · Zbl 0582.46049 |
| [51] | Teixeira, E. V., Strong solutions for differential equations in abstract spaces, J Differ Equ, 214, 65-91 (2005) · Zbl 1094.34043 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
