An existence and uniqueness of mild solutions of fractional evolution problems. (English) Zbl 07931040
Summary: In this present paper, we concern to investigate the existence and uniqueness of mild solutions for a new class of fractional evolution problems involving sectorial operators and nonlocal conditions by means of fixed point theory. Here we use Caputo fractional derivative of order \(\boldsymbol{\alpha}\) with \(\mathbf{2} < \boldsymbol{\alpha} < \mathbf{3}\). In this sense, in order to elucidate the results obtained, we end the article with examples.
MSC:
| 34G20 | Nonlinear differential equations in abstract spaces |
| 26A33 | Fractional derivatives and integrals |
References:
| [1] | Burton, TA; Kirk, C., A fixed point theorem of Krasnoselskii-Schaefer type, Math Nachr, 189, 1, 23-31, 1998 · Zbl 0896.47042 · doi:10.1002/mana.19981890103 |
| [2] | Chefnaj, N.; Taqbibt, A.; Hilal, K.; Melliani, S.; kajouni, A., Boundary value problems for differential equations involving the generalized Caputo-Fabrizio fractional derivative in \(b\)-metric spaces, Turk J Sci, 8, 1, 24-36, 2023 |
| [3] | Cui, Z.; Zhou, Z., Existence of solutions for Caputo fractional delay differential equations with nonlocal and integral boundary conditions, Fixed Point Theory Algorithms Sci Eng, 2023, 1, 1, 2023 · Zbl 1541.34100 · doi:10.1186/s13663-022-00738-3 |
| [4] | Derbazi, C.; Hammouche, H., Boundary value problems for Caputo fractional differential equations with nonlocal and fractional integral boundary conditions, Arab J Math, 9, 3, 531-544, 2020 · Zbl 1456.34004 · doi:10.1007/s40065-020-00288-9 |
| [5] | Diethelm K, Freed AD (1999) On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity. In: Scientific computing in chemical engineering II. Springer, Berlin, pp 217-224. doi:10.1007/978-3-642-60185-9-24 |
| [6] | Ding, X-L; Ahmad, B., Analytical solutions to fractional evolution equations with almost sectorial operators, Adv Differ Equ, 2016, 1, 1-25, 2016 · Zbl 1419.35204 · doi:10.1186/s13662-016-0927-y |
| [7] | Glöckle, WG; Nonnenmacher, TF, A fractional calculus approach to self-similar protein dynamics, Biophys J, 68, 1, 46-53, 1995 · doi:10.1016/S0006-3495(95)80157-8 |
| [8] | Kexue, L.; Jigen, P., Laplace transform and fractional differential equations, Appl Math Lett, 24, 12, 2019-2023, 2011 · Zbl 1238.34013 · doi:10.1016/j.aml.2011.05.035 |
| [9] | Kilbas, AA; Srivastava, HM; Trujillo, JJ, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, 2006, Amsterdam: Elsevier, Amsterdam · Zbl 1092.45003 |
| [10] | Lakshmikantham, V.; Leela, S.; Vasundhara Devi, J., Theory of fractional dynamic systems, 2009, Cambridge: Cambridge Scientific Publishers, Cambridge · Zbl 1188.37002 |
| [11] | Metzler, R.; Schick, W.; Kilian, HG; Nonnenmacher, TF, Relaxation in filled polymers: a fractional calculus approach, J Chem Phys, 103, 16, 7180-7186, 1995 · doi:10.1063/1.470346 |
| [12] | Miller, KS; Ross, B., An introduction to the fractional calculus and differential equations, 1993, New York: Wiley, New York · Zbl 0789.26002 |
| [13] | Oldham, K.; Spanier, J., The fractional calculus; theory and applications of differentiation and integration to arbitrary order. Math. Sci. Engine, 11, 1974, New York: Academic Press, New York · Zbl 0292.26011 |
| [14] | Periago, F.; Straub, B., A functional calculus for almost sectorial operators and applications to abstract evolution equations, J Evol Equ, 2, 1, 41-68, 2002 · Zbl 1005.47015 · doi:10.1007/s00028-002-8079-9 |
| [15] | Sousa, JVC; Santos Oliveira, D.; Capelas de Oliveira, E., A note on the mild solutions of Hilfer impulsive fractional differential equations, Chaos Solit Fract, 147, 2021 · Zbl 1486.34116 · doi:10.1016/j.chaos.2021.110944 |
| [16] | Sousa, JVC; Jarad, F.; Abdeljawad, T., Existence of mild solutions to Hilfer fractional evolution equations in Banach space, Ann Funct Anal, 12, 1-16, 2021 · Zbl 1458.34032 · doi:10.1007/s43034-020-00095-5 |
| [17] | Sousa, JVC; Kishor, D.; Kucche, D.; Capelas de Oliveira, E., Stability of mild solutions of the fractional nonlinear abstract Cauchy problem, Electr Res Arch, 30, 1, 272-288, 2022 · Zbl 1500.34054 · doi:10.3934/era.2022015 |
| [18] | Sousa, JVC; Gala, S.; Capelas De Oliveira, E., On the uniqueness of mild solutions to the time-fractional Navier-Stokes equations in \(L^N(\mathbb{R}^N)^N \), Comput Appl Math, 42, 1, 41, 2023 · Zbl 1538.35263 · doi:10.1007/s40314-023-02185-1 |
| [19] | Sousa, JVC; Aurora, M.; Pulido, P.; Govindaraj, V.; Capelas de Oliveira, E., On the \(\varepsilon \)-regular mild solution for fractional abstract integro-differential equations, Soft Comput, 27.21, 15533-15548, 2023 · doi:10.1007/s00500-023-09172-y |
| [20] | Su, X., Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl Math Lett, 22, 1, 64-69, 2009 · Zbl 1163.34321 · doi:10.1016/j.aml.2008.03.001 |
| [21] | Taqbibt, A.; Elomari, M.; Melliani, S., Nonlocal semilinear \(\phi \)-Caputo fractional evolution equation with a measure of noncompactness in Banach space, Filomat, 37, 20, 6877-6890, 2023 · doi:10.2298/FIL2320877T |
| [22] | Wang, RN; Chen, DH; Xiao, TJ, Abstract fractional Cauchy problems with almost sectorial operators, J Differ Equ, 252, 1, 202-235, 2012 · Zbl 1238.34015 · doi:10.1016/j.jde.2011.08.048 |
| [23] | Zhou, Y.; He, JW, New results on controllability of fractional evolution systems with order \(\alpha \in (1, 2)\), Evol Equ Control Theory, 10, 3, 491-509, 2021 · Zbl 1481.34081 · doi:10.3934/eect.2020077 |
| [24] | Zhou, Y.; Jiao, F., Existence of mild solutions for fractional neutral evolution equations, Comput Math Appl, 59, 3, 1063-1077, 2010 · Zbl 1189.34154 · doi:10.1016/j.camwa.2009.06.026 |
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