In this paper we investigate the formulation of a class of boundary value problems of fractional order with the Riemann-Liouville fractional derivative and integral-type boundary conditions. The existence of solutions is established by applying a fixed point theorem of Krasnosel'skii and Zabreiko for asymptotically linear mappings

Kosmatov, Nickolai

Electronic Journal of Qualitative Theory of Differential Equations

Kosmatov Nickolai

University of Szeged

Electronic Journal of Qualitative Theory of Differential Equations2009, No. 20, 1-10; http://www.math.u-szeged.hu/ejqtde/Solutions to a class of nonlinear differential equationsof fractional orderNickolai KosmatovDepartment of Mathematics and StatisticsUniversity of Arkansas at Little RockLittle Rock, AR 72204-1099, USAnxkosmatov@ualr.eduAbstractIn this paper we investigate the formulation of a class of boundary valueproblems of fractional order with the Riemann-Liouville fractional derivative andintegral-type boundary conditions. The existence of solutions is established byapplying a fixed point theorem of Krasnosel’skĭı and Zabreiko for asymptoticallylinear mappings.2000 Mathematics Subject Classifications: 34A12, 34A34, 45D05Key words: differential equation of fractional order, fixed point theorem, Riemann-Liouville derivativeEJQTDE, 2009 No. 20, p. 11 PreliminariesIn this section we introduce basic facts of fractional calculus involving the Riemann-Liouville fractional differential operator and list several recent and classical resultsdealing with initial and boundary value problems of fractional order. In the presentsection we also formulate the boundary value problems and establish the equivalence ofthose to certain integral equations. The existence of solutions of the integral equationsis shown in Section 2 by applying a fixed point theorem of Krasnosel’skĭı and Zabreiko[10].The study of primarily initial value problems for differential equations of fractionalorder with various types of integro-differential operators such as Riemann-Liouvilleis extensive. It includes several well-known monographs [8, 13, 14, 15] and papers[2, 3, 4, 5, 6, 9, 11, 12]. Boundary value problems for fractional order have receivedless attention than initial value problems [1]. In this work we obtain several existenceresults for a Riemann-Liouville differential equation fractional order 1 < α < 2. Wepoint out that if the solutions are sought in the class of continuous functions, one of theconditions loses its meaning as a boundary condition and should instead be interpretedas a well-posedness condition. In this regard the paper clears certain misconceptionsabout formulation of boundary value problems for differential equations of fractionalorder.The Riemann-Liouville fractional integral of order α > 0 of a function u ∈ Lp(0, 1),EJQTDE, 2009 No. 20, p. 21 ≤ p < ∞, is the integralIα0+u(t) =1Γ(α)∫ t0(t − s)α−1u(s) ds.The Riemann-Liouville fractional derivative of order α > 0, n = [α] + 1, is defined byDα0+u(t) =1Γ(n − α)(ddt)n ∫ t0(t − s)n−α−1u(s) ds.It is well-known [9, 13] that if the fractional derivative Dα0+u, n − 1 < α < n, of afunction u is integrable, thenIα0+Dα0+u(t) = u(t) −n∑k=1Dα−k0+ u(0)Γ(α − k + 1)tα−k. (1)For an integrable function g, it holds thatDα0+Iα0+g(t) = g(t). (2)For β < 0, it is convenient to introduce the notation Iβ0+ = D−β0+ . If α > 0 andβ + α > 0, then a composition rule (also called a semigroup property)Iβ0+Iα0+g(t) = Iα+β0+ g(t)holds (see, e.g., [9, 13]).If u ∈ C[0, 1], then (1) becomesu(t) = Dα−10+ u(0)tα−1Γ(α)+ Iα0+Dα0+u(t) (3)since Dα−20+ u(0) = I2−α0+ u(0) = 0.We study the Riemann-Liouville integro-differential equationDα0+u(t) = f(t, u(t)), t ∈ (0, 1), (4)EJQTDE, 2009 No. 20, p. 3of fractional order 1 < α < 2. We assume throughout the note that(H1) f ∈ C([0, 1] ×R,R).We seek solutions of (4) in the class of continuous functions satisfying the boundaryconditionIβ0+u(1) = 0, (5)where β > −α. In addition, we assume that γ > 1 − α and impose the conditionIγ0+u(0) = 0, (6)to which we refer as the well-posedness condition. The condition (6) plays the role ofthe second boundary condition.By a solution of the boundary value problem (4)-(6) we understand a functionu ∈ C[0, 1] with Dα0+u ∈ C[0, 1] satisfying the equation (4) and the conditions (5) and(6).Since α + γ > 1, setting g = Dα0+u ∈ C[0, 1], we have by the semigroup property(3) thatIγ0+u(t) = Dα−10+ u(0)tα+γ−1Γ(α + γ)+ Iα+γ0+ g(t).In particular, the right side of the above equation vanishes at t = 0 thus justifying theuse of (6) and the applicability of the term “well-posedness” condition to the scope ofthis work.Similarly,Iβ0+u(t) = Dα−10+ u(0)tα+β−1Γ(α + β)+ Iα+β0+ g(t).EJQTDE, 2009 No. 20, p. 4It follows from (5) thatDα−10+ u(0) = −Γ(α + β)Iα+β0+ g(1)andu(t) = −Γ(α + β)Γ(α)Iα+β0+ g(1)tα−1 + Iα0+g(t).Replacing g with the inhomogeneous term of (4), we obtain that if u ∈ C[0, 1] is asolution of the problem (4)-(6), then u ∈ C[0, 1] is the solution of the integral equationu(t) = −Γ(α + β)Γ(α)Iα+β0+ f(·, u(·))(1)tα−1 + Iα0+f(·, u(·))(t). (7)The converse is also true in view of (2).Since the solvability of (4)-(6) is equivalent the existence of a solution of the integralequation (7), we will seek a fixed point of the integral mapping, for t ∈ [0, 1],Tu(t) = −Γ(α + β)Γ(α)tα−1∫ 10(1− s)α+β−1f(s, u(s)) ds +1Γ(α)∫ t0(t− s)α−1f(s, u(s)) ds.(8)Note that the parameter γ is absent from the integral equation (7) simply because theshape of the mapping above is dictated by the class of admissible functions rather thanthe condition (6) at t = 0 (see, e. g., [1] with u(0) = 0 which is the case γ = 0 in thisnote).Let X = C[0, 1] be endowed with the sup-norm denoted by ‖ · ‖. It is clear thatT : X → X is a completely continuous mapping. The existence of a fixed point of Twill follow from the Krasnosel’skĭı-Zabreiko fixed point theorem [10]:EJQTDE, 2009 No. 20, p. 5Theorem 1.1 Let X be a Banach space. Assume that T : X → X is completelycontinuous mapping and L : X → X is a bounded linear mapping such that 1 is not aneigenvalue of L andlim‖u‖→∞‖Tu − Lu‖‖u‖= 0. (9)Then T has a fixed point in X.For applications of Theorem 1.1 we refer the reader to [7] and the references therein.Although the inhomogeneous term in the above mentioned work was considered in theform φ(t)g(u(t)), we observe that there is no need in separating the variables so thata slightly more general scenario f(t, u(t)) is considered in the present note.2 The existence resultThe first existence result is obtained for the problem (4)-(6).In addition to the hypothesis (H1) we impose the following conditions:(H2) lim|u|→∞f(t, u)u= φ(t) uniformly in [0, 1];(H3) f(t, 0) does not vanish identically in [0, 1].Theorem 2.1 Let the assumptions (H1)-(H3) be satisfied. Assume thatΓ(α + β)Γ(α)∫ 10(1 − s)α+β−1|φ(s)| ds +1Γ(α)∫ 10(1 − s)α−1|φ(s)| ds < 1. (10)Then the boundary value problem (4)-(6) has a nontrivial solution.EJQTDE, 2009 No. 20, p. 6Proof. Define a bounded linear mapping, L : X → X, byLu(t) = −Γ(α + β)Γ(α)tα−1∫ 10(1 − s)α+β−1φ(s)u(s) ds +1Γ(α)∫ t0(t − s)α−1φ(s)u(s) ds,for t ∈ [0, 1]. Suppose that λ = 1 is an eigenvalue of L. Since (10) holds,‖Lu‖ ≤(Γ(α + β)Γ(α)∫ 10(1 − s)α+β−1|φ(s)| ds +1Γ(α)∫ 10(1 − s)α−1|φ(s)| ds)‖u‖< ‖u‖,together with Lu = u with u 6= 0 lead to a contradiction. Thus λ = 1 is not aneigenvalue of L.Let ε > 0. There exists an A > 0 such that∣∣∣∣f(t, z)z− φ(s)∣∣∣∣< ε (11)for all t ∈ [0, 1] provided |z| > A. SetB = sup{|f(t, z)| : t ∈ [0, 1], |z| ∈ [0, A|}.If |u(t)| ≤ A, then |f(t, u(t)) − φ(t)u(t)| ≤ B + ‖φ‖A. Choose M > A so thatB + ‖φ‖A ≤ Mε. Then, for u ∈ X with ‖u‖ > M , either |u(s)| ≤ A, in which case|f(t, u(t)) − φ(t)u(t)| < ε‖u‖, or |u(s)| > A, in which case, by (11), we also have|f(t, u(t)) − φ(t)u(t)| < ε‖u‖.Now, if ‖u‖ > M , then‖Tu − Lu‖ = supt∈[0,1]|Tu(t) − Lu(t)|= supt∈[0,1]∣∣∣∣∣−Γ(α + β)Γ(α)tα−1∫ 10(1 − s)α+β−1(f(s, u(s)) − φ(s)u(s)) dsEJQTDE, 2009 No. 20, p. 7+1Γ(α)∫ t0(t − s)α−1(f(s, u(s))− φ(s)u(s)) ds∣∣∣∣∣≤(Γ(α + β)Γ(α)∫ 10(1 − s)α+β−1 ds +1Γ(α)∫ 10(1 − s)α−1 ds)ε‖u‖≤(Γ(α + β)(α + β)Γ(α)+1Γ(α + 1))ε‖u‖.We see that the condition (9) is verified. Thus, by Theorem 1.1, T has a fixed pointin X which is a solution of the boundary value problem (4)-(6). It follows from (H3)that the boundary value problem does not possess the trivial solution.References[1] Z. Bai and H. Lü, Positive solutions for boundary value problem of nonlinearfractional differential equation, J. Math. Anal. Appl., 311 (2005), 495-505.[2] M. Benchohra, J. Henderson, S. K. Ntoyuas and A. Ouahab, Existence results forfractional order functional differential equations with infinite delay, J. Math. Anal.Appl., 338 (2008), 1340-1350.[3] D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractionaldifferential equation, J. Math. Anal. Appl., 204 (1996), 609-625.[4] K. Diethelm and A. D. Freed, On the solution of nonlinear fractional order differ-ential equations used in the modelling of viscoplasticity, in “Scientific Computing inChemical Engineering II - Computational Fluid Dynamics, Reaction Engineering,EJQTDE, 2009 No. 20, p. 8and Molecular Properties,” (eds. F. Keil, W. Mackens, H. Voß and J. Werther),Springer-Verlag, Heidelberg, 1999, 217-224.[5] W. G. El-Sayed and A. M. A. El-Sayed, On the functional integral equationsof mixed type and integro-differential equations of fractional orders, Appl. Math.Comput., 154 (2004), 461-467.[6] O. K. Jaradat, A. Al-Omari and S. Momani, Existence of the mild solution forfractional semilinear initial value problems, Nonlinear Anal., 69 (2007), 3153-3159.[7] B. Karna and B. Lawrence, An existence result for a multipoint boundary valueproblem on a time scale, Adv. Difference Equ., (2006), Art. ID 63208, 1-8.[8] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, “Theory and Applications ofFractional Differential Equations,” North Holland Mathematics Studies, 204, El-sevier, 2006.[9] A. A. Kilbas and J. J. Trujillo, Differential equations of fractional order: methods,results and problems, Appl. Anal., 78 (2001), 153-192.[10] M. A. Krasnosel’skĭı and P. P. Zabreiko, “Geometrical Methods of Nonlinear Anal-ysis,” Springer-Verlag, New York, 1984.[11] V. Lakshmikantham, Theory of fractional functional differential equations, Non-linear Anal., 69 (2008), 3337-3343.EJQTDE, 2009 No. 20, p. 9[12] V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equa-tions, Nonlinear Anal., 69 (2008), 2677-2682.[13] I. Podlubny, “Fractional Differential Equations, Mathematics in Sciences and Ap-plications,” Academic Press, New York, 1999.[14] J. Sabatier, O. P. Agrawal and J. A. Tenreiro-Machado, “Advances in FractionalCalculus: Theoretical Developments and Applications in Physics and Engineer-ing,” Springer, The Netherlands, 2007.[15] S. G. Samko, A. A. Kilbas and O. I. Mirichev, “Fractional Integral and Derivatives(Theory and Applications),” Gordon and Breach, Switzerland, 1993.(Received February 25, 2009)EJQTDE, 2009 No. 20, p. 10

Solutions to a class of nonlinear differential equations of fractional order

Nickolai Kosmatov

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Electronic journal of qualitative theory of differential equations

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Electronic Journal of Qualitative Theory of Differential Equations2009, No. 20, 1-10; http://www.math.u-szeged.hu/ejqtde/Solutions to a class of nonlinear differential equationsof fractional orderNickolai KosmatovDepartment of Mathematics and StatisticsUniversity of Arkansas at Little RockLittle Rock, AR 72204-1099, USAnxkosmatov@ualr.eduAbstractIn this paper we investigate the formulation of a class of boundary valueproblems of fractional order with the Riemann-Liouville fractional derivative andintegral-type boundary conditions. The existence of solutions is established byapplying a fixed point theorem of Krasnosel’ski˘ı and Zabreiko for asymptoticallylinear mappings.2000 Mathematics Subject Classifications: 34A12, 34A34, 45D05Key words: differential equation of fractional order, fixed point theorem, Riemann-Liouville derivativeEJQTDE, 2009 No. 20, p. 11 PreliminariesIn this section we introduce basic facts of fractional calculus involving the Riemann-Liouville fractional differential operator and list several recent and classical resultsdealing with initial and boundary value problems of fractional order. In the presentsection we also formulate the boundary value problems and establish the equivalence ofthose to certain integral equations. The existence of solutions of the integral equationsis shown in Section 2 by applying a fixed point theorem of Krasnosel’ski˘ı and Zabreiko[10].The study of primarily initial value problems for differential equations of fractionalorder with various types of integro-differential operators such as Riemann-Liouvilleis extensive. It includes several well-known monographs [8, 13, 14, 15] and papers[2, 3, 4, 5, 6, 9, 11, 12]. Boundary value problems for fractional order have receivedless attention than initial value problems [1]. In this work we obtain several existenceresults for a Riemann-Liouville differential equation fractional order 1 < α < 2. Wepoint out that if the solutions are sought in the class of continuous functions, one of theconditions loses its meaning as a boundary condition and should instead be interpretedas a well-posedness condition. In this regard the paper clears certain misconceptionsabout formulation of boundary value problems for differential equations of fractionalorder.The Riemann-Liouville fractional integral of order α > 0 of a function u ∈ Lp(0, 1),EJQTDE, 2009 No. 20, p. 21 ≤ p <∞, is the integralIα0+u(t) =1Γ(α)∫ t0(t− s)α−1u(s) ds.The Riemann-Liouville fractional derivative of order α > 0, n = [α] + 1, is defined byDα0+u(t) =1Γ(n− α)(ddt)n ∫ t0(t− s)n−α−1u(s) ds.It is well-known [9, 13] that if the fractional derivative Dα0+u, n − 1 < α < n, of afunction u is integrable, thenIα0+Dα0+u(t) = u(t)−n∑k=1Dα−k0+ u(0)Γ(α− k + 1)tα−k. (1)For an integrable function g, it holds thatDα0+Iα0+g(t) = g(t). (2)For β < 0, it is convenient to introduce the notation Iβ0+ = D−β0+ . If α > 0 andβ + α > 0, then a composition rule (also called a semigroup property)Iβ0+Iα0+g(t) = Iα+β0+ g(t)holds (see, e.g., [9, 13]).If u ∈ C[0, 1], then (1) becomesu(t) = Dα−10+ u(0)tα−1Γ(α)+ Iα0+Dα0+u(t) (3)since Dα−20+ u(0) = I2−α0+ u(0) = 0.We study the Riemann-Liouville integro-differential equationDα0+u(t) = f(t, u(t)), t ∈ (0, 1), (4)EJQTDE, 2009 No. 20, p. 3of fractional order 1 < α < 2. We assume throughout the note that(H1) f ∈ C([0, 1]×R,R).We seek solutions of (4) in the class of continuous functions satisfying the boundaryconditionIβ0+u(1) = 0, (5)where β > −α. In addition, we assume that γ > 1− α and impose the conditionIγ0+u(0) = 0, (6)to which we refer as the well-posedness condition. The condition (6) plays the role ofthe second boundary condition.By a solution of the boundary value problem (4)-(6) we understand a functionu ∈ C[0, 1] with Dα0+u ∈ C[0, 1] satisfying the equation (4) and the conditions (5) and(6).Since α + γ > 1, setting g = Dα0+u ∈ C[0, 1], we have by the semigroup property(3) thatIγ0+u(t) = Dα−10+ u(0)tα+γ−1Γ(α+ γ)+ Iα+γ0+ g(t).In particular, the right side of the above equation vanishes at t = 0 thus justifying theuse of (6) and the applicability of the term “well-posedness” condition to the scope ofthis work.Similarly,Iβ0+u(t) = Dα−10+ u(0)tα+β−1Γ(α+ β)+ Iα+β0+ g(t).EJQTDE, 2009 No. 20, p. 4It follows from (5) thatDα−10+ u(0) = −Γ(α + β)Iα+β0+ g(1)andu(t) = −Γ(α + β)Γ(α)Iα+β0+ g(1)tα−1 + Iα0+g(t).Replacing g with the inhomogeneous term of (4), we obtain that if u ∈ C[0, 1] is asolution of the problem (4)-(6), then u ∈ C[0, 1] is the solution of the integral equationu(t) = −Γ(α + β)Γ(α)Iα+β0+ f(·, u(·))(1)tα−1 + Iα0+f(·, u(·))(t). (7)The converse is also true in view of (2).Since the solvability of (4)-(6) is equivalent the existence of a solution of the integralequation (7), we will seek a fixed point of the integral mapping, for t ∈ [0, 1],Tu(t) = −Γ(α + β)Γ(α)tα−1∫ 10(1− s)α+β−1f(s, u(s)) ds+1Γ(α)∫ t0(t− s)α−1f(s, u(s)) ds.(8)Note that the parameter γ is absent from the integral equation (7) simply because theshape of the mapping above is dictated by the class of admissible functions rather thanthe condition (6) at t = 0 (see, e. g., [1] with u(0) = 0 which is the case γ = 0 in thisnote).Let X = C[0, 1] be endowed with the sup-norm denoted by ‖ · ‖. It is clear thatT : X → X is a completely continuous mapping. The existence of a fixed point of Twill follow from the Krasnosel’ski˘ı-Zabreiko fixed point theorem [10]:EJQTDE, 2009 No. 20, p. 5Theorem 1.1 Let X be a Banach space. Assume that T : X → X is completelycontinuous mapping and L : X → X is a bounded linear mapping such that 1 is not aneigenvalue of L andlim‖u‖→∞‖Tu− Lu‖‖u‖= 0. (9)Then T has a fixed point in X.For applications of Theorem 1.1 we refer the reader to [7] and the references therein.Although the inhomogeneous term in the above mentioned work was considered in theform φ(t)g(u(t)), we observe that there is no need in separating the variables so thata slightly more general scenario f(t, u(t)) is considered in the present note.2 The existence resultThe first existence result is obtained for the problem (4)-(6).In addition to the hypothesis (H1) we impose the following conditions:(H2) lim|u|→∞f(t, u)u= φ(t) uniformly in [0, 1];(H3) f(t, 0) does not vanish identically in [0, 1].Theorem 2.1 Let the assumptions (H1)-(H3) be satisfied. Assume thatΓ(α+ β)Γ(α)∫ 10(1− s)α+β−1|φ(s)| ds+1Γ(α)∫ 10(1− s)α−1|φ(s)| ds < 1. (10)Then the boundary value problem (4)-(6) has a nontrivial solution.EJQTDE, 2009 No. 20, p. 6Proof. Define a bounded linear mapping, L : X → X, byLu(t) = −Γ(α + β)Γ(α)tα−1∫ 10(1− s)α+β−1φ(s)u(s) ds+1Γ(α)∫ t0(t− s)α−1φ(s)u(s) ds,for t ∈ [0, 1]. Suppose that λ = 1 is an eigenvalue of L. Since (10) holds,‖Lu‖ ≤(Γ(α + β)Γ(α)∫ 10(1− s)α+β−1|φ(s)| ds+1Γ(α)∫ 10(1− s)α−1|φ(s)| ds)‖u‖< ‖u‖,together with Lu = u with u 6= 0 lead to a contradiction. Thus λ = 1 is not aneigenvalue of L.Let  > 0. There exists an A > 0 such that∣∣∣∣f(t, z)z − φ(s)∣∣∣∣ <  (11)for all t ∈ [0, 1] provided |z| > A. SetB = sup{|f(t, z)| : t ∈ [0, 1], |z| ∈ [0, A|}.If |u(t)| ≤ A, then |f(t, u(t)) − φ(t)u(t)| ≤ B + ‖φ‖A. Choose M > A so thatB + ‖φ‖A ≤ M. Then, for u ∈ X with ‖u‖ > M , either |u(s)| ≤ A, in which case|f(t, u(t)) − φ(t)u(t)| < ‖u‖, or |u(s)| > A, in which case, by (11), we also have|f(t, u(t))− φ(t)u(t)| < ‖u‖.Now, if ‖u‖ > M , then‖Tu− Lu‖ = supt∈[0,1]|Tu(t)− Lu(t)|= supt∈[0,1]∣∣∣∣∣−Γ(α + β)Γ(α)tα−1∫ 10(1− s)α+β−1(f(s, u(s))− φ(s)u(s)) dsEJQTDE, 2009 No. 20, p. 7+1Γ(α)∫ t0(t− s)α−1(f(s, u(s))− φ(s)u(s)) ds∣∣∣∣∣≤(Γ(α + β)Γ(α)∫ 10(1− s)α+β−1 ds+1Γ(α)∫ 10(1− s)α−1 ds)‖u‖≤(Γ(α + β)(α + β)Γ(α)+1Γ(α + 1))‖u‖.We see that the condition (9) is verified. Thus, by Theorem 1.1, T has a fixed pointin X which is a solution of the boundary value problem (4)-(6). It follows from (H3)that the boundary value problem does not possess the trivial solution.References[1] Z. Bai and H. Lu¨, Positive solutions for boundary value problem of nonlinearfractional differential equation, J. Math. Anal. Appl., 311 (2005), 495-505.[2] M. Benchohra, J. Henderson, S. K. Ntoyuas and A. Ouahab, Existence results forfractional order functional differential equations with infinite delay, J. Math. Anal.Appl., 338 (2008), 1340-1350.[3] D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractionaldifferential equation, J. Math. Anal. Appl., 204 (1996), 609-625.[4] K. Diethelm and A. D. Freed, On the solution of nonlinear fractional order differ-ential equations used in the modelling of viscoplasticity, in “Scientific Computing inChemical Engineering II - Computational Fluid Dynamics, Reaction Engineering,EJQTDE, 2009 No. 20, p. 8and Molecular Properties,” (eds. F. Keil, W. Mackens, H. Voß and J. Werther),Springer-Verlag, Heidelberg, 1999, 217-224.[5] W. G. El-Sayed and A. M. A. El-Sayed, On the functional integral equationsof mixed type and integro-differential equations of fractional orders, Appl. Math.Comput., 154 (2004), 461-467.[6] O. K. Jaradat, A. Al-Omari and S. Momani, Existence of the mild solution forfractional semilinear initial value problems, Nonlinear Anal., 69 (2007), 3153-3159.[7] B. Karna and B. Lawrence, An existence result for a multipoint boundary valueproblem on a time scale, Adv. Difference Equ., (2006), Art. ID 63208, 1-8.[8] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, “Theory and Applications ofFractional Differential Equations,” North Holland Mathematics Studies, 204, El-sevier, 2006.[9] A. A. Kilbas and J. J. Trujillo, Differential equations of fractional order: methods,results and problems, Appl. Anal., 78 (2001), 153-192.[10] M. A. Krasnosel’ski˘ı and P. P. Zabreiko, “Geometrical Methods of Nonlinear Anal-ysis,” Springer-Verlag, New York, 1984.[11] V. Lakshmikantham, Theory of fractional functional differential equations, Non-linear Anal., 69 (2008), 3337-3343.EJQTDE, 2009 No. 20, p. 9[12] V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equa-tions, Nonlinear Anal., 69 (2008), 2677-2682.[13] I. Podlubny, “Fractional Differential Equations, Mathematics in Sciences and Ap-plications,” Academic Press, New York, 1999.[14] J. Sabatier, O. P. Agrawal and J. A. Tenreiro-Machado, “Advances in FractionalCalculus: Theoretical Developments and Applications in Physics and Engineer-ing,” Springer, The Netherlands, 2007.[15] S. G. Samko, A. A. Kilbas and O. I. Mirichev, “Fractional Integral and Derivatives(Theory and Applications),” Gordon and Breach, Switzerland, 1993.(Received February 25, 2009)EJQTDE, 2009 No. 20, p. 10

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Solutions to a class of nonlinear differential equations of fractional order

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