×

Implicit fractional differential equations via the Liouville-Caputo derivative. (English) Zbl 1322.34012

Summary: We study an initial value problem for an implicit fractional differential equation with the Liouville-Caputo fractional derivative. By using fixed point theory and an approximation method, we obtain some existence and uniqueness results.

MSC:

34A08 Fractional ordinary differential equations
34A09 Implicit ordinary differential equations, differential-algebraic equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Gaul, Damping description involving fractional operators, Mech. Syst. Signal Process. 5 pp 81– (1991)
[2] Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanis, Fractals and Fractional Calculus in Continuum Mechanics pp 291– (1997)
[3] Metzler, Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys. 103 pp 7180– (1995)
[4] Abbas, Topics in Fractional Differential Equations (2012)
[5] Kilbas, Theory and Applications of Fractional Differential Equations 204 (2006) · Zbl 1138.26300
[6] Miller, An Introduction to the Fractional Calculus and Differential Equations (1993) · Zbl 0789.26002
[7] Podlubny, Fractional Differential Equations (1999)
[8] Samko, Fractional Integrals and Derivatives, Theory and Applications (1993)
[9] Diethelm, The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type 2004 (2010) · Zbl 1215.34001
[10] Aghajani, Existence of solutions for a class of nonlinear Volterra singular integral equations, Comput. Math. Appl. 62 pp 1215– (2011) · Zbl 1228.45002
[11] Ahmad, Anti-periodic fractional boundary value problems with nonlinear term depending on lower order derivative, Fract. Calc. Appl. Anal. 15 pp 451– (2012) · Zbl 1281.34005
[12] Benchohra, Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl. 338 pp 1340– (2008) · Zbl 1209.34096
[13] Chalishajar, Existence and uniqueness results for boundary value problems of higher order fractional integro-differential equations involving Gronwall’s inequality in Banach spaces, Acta Math. Sci. 33 pp 758– (2013) · Zbl 1299.34059
[14] Chalishajar, Boundary value problems for impulsive fractional evolution integro-differential equations with Gronwall’s inequality in Banach spaces, Discontin. Nonlinearity Complex. 3 pp 33– (2014) · Zbl 1300.34151
[15] Henderson, Fractional functional differential inclusions with finite delay, Nonlinear Anal. 70 pp 2091– (2009) · Zbl 1159.34010
[16] Henderson, Impulsive differential inclusions with fractional order, Comput. Math. Appl. 59 pp 1191– (2010) · Zbl 1200.34006
[17] Ouahab, Some results for fractional boundary value problem of differential inclusions, Nonlinear Anal. 69 pp 3877– (2008) · Zbl 1169.34006
[18] Ouahab, Semilinear fractional differential inclsions, Comput. Math. Appl. 64 pp 3235– (2012) · Zbl 1268.34024
[19] Benavides, An existence theorem for implicit differential equations in a Banach space, Ann. Mat. Pura Appl. 4 pp 119– (1978) · Zbl 0418.34058
[20] Emmanuele, Convergence of successive approximations for implicit ordinary differential equations in Banach spaces, Funkc. Ekvac. 24 pp 325– (1981) · Zbl 0486.34051
[21] Emmanuele, On the existence of solutions of ordinary differential equations in implicit form in Banach spaces, Ann. Mat. Pura Appl. 129 pp 367– (1981) · Zbl 0499.34039
[22] Hokkanen, Continuous dependence for an implicit nonlinear equation, J. Differ. Equ. 110 pp 67– (1994) · Zbl 0802.34074
[23] Hokkanen, Existence of a periodic solution for implicit nonlinear equations, Differ. Integral Equ. 9 pp 745– (1996)
[24] Li, Peano’s theorem for implicit differential equations, J. Math. Anal. Appl. 258 pp 591– (2001) · Zbl 0982.34003
[25] Ricceri, Lipschitz solutions of the implicit Cauchy problem g(x’) = f(t, x), x(0) = 0, with f discontinuous in x, Rend. Circ. Mat. Palermo 34 pp 127– (1985) · Zbl 0578.34002
[26] Seikkala, Uniqueness, comparison and existence results for discontinuous implicit differential equations, Nonlinear Anal. 30 pp 1771– (1997) · Zbl 0893.34001
[27] Benchohra, Nonlinear fractional implicit differential equations, Commun. Appl. Anal. 17 pp 471– (2013)
[28] Benchohra, Existence and uniqueness results for nonlinear implicit fractional differential equations with boundary conditions, Rom. J. Math. Comput. Sci. 4 pp 60– (2014)
[29] Vityuk, The Darboux problem for an implicit differential equation of fractional order, Ukr. Mat. Visn. 7 pp 439– (2010)
[30] Caputo, Elasticità e Dissipazione (1969)
[31] Caputo, Linear models of dissipation whose Q is almost frequency independent, Part II, Geophys. J. R. Astr. Soc. 13 pp 529– (1967)
[32] Caputo, Linear models of dissipation in anelastic solid, Riv. Nuovo Cimento (Ser. II) 1 pp 161– (1971)
[33] Henry, Geometric Theory of Semilinear Parabolic Partial differential Equations (1989)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.