Nonlinear Analysis: Hybrid Systems 3 (2009) 734–737 Contents lists available at ScienceDirect Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs Fractional differential equations with a Krasnoselskii–Krein type condition T. Gnana Bhaskar a,∗, V. Lakshmikantham a, S. Leela b a Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA b SUNY, Geneseo, USA a r t i c l e i n f o Article history: Received 29 June 2009 Accepted 29 June 2009 Keywords: Fractional differential equations Caputo derivative Krasnoselskii–Krein type conditions a b s t r a c t We consider an initial value problem for a fractional differential equation of Caputo type. The convergence of the Picard successive approximations is established by first showing that the Caputo derivatives of these approximations converge. The method utilized, originally introduced in [O. Kooi, The method of successive approximations and a uniqueness theorem of Krasnoselskii–Krein in the theory of differential equations, Nederi. Akad. Wetensch, Proc. Ser. A61; Indag. Math. 20 (1958) 322–327], is interesting in itself. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction Recently, the Krasnoselskii–Krein type of uniqueness result has been extended to fractional differential equations [5], also obtaining the convergence of successive approximations to the unique solution. Themethod thatwas followed in [5] is based on employing the theory of differential and integral inequalities and showing directly that the successive approximations converge. There is another method that is used for ordinary differential equations in which the sequence of derivatives of successive approximations converges uniformly [2] and the nice idea used in that paper is not yet popular. In this article, we plan to utilize the ideas presented in [2] to extend Krasnoselskii–Krein type results to fractional differential equations of Caputo type. This extension demands appropriate nontrivial modifications of the method and, therefore, is interesting and useful. For various uniqueness results for ODE, see [1] and [3]. 2. Main result Consider the IVP with a Caputo fractional derivative cDqφ(x) = f (x, φ(x)), φ(x0) = y0, x0 ≥ 0 (2.1) where 0 < q < 1, f ∈ C(Ω,R), |f (x, y)| ≤ M onΩ , Ω = {(x, y) : 0 ≤ x− x0 ≤ a, |y− y0| ≤ b}, and aM ≤ b. Suppose that f satisfies the following conditions onΩ: |f (x, y1)− f (x, y2)| ≤ KηΓ (q)|y1 − y2| (x− x0)q , x 6= x0, with Kη < q, K > 1; (A) |f (x, y1)− f (x, y2)| ≤ β|y1 − y2|α, 0 < α < 1, K(1− α) < 1 and β > 0 is a constant. (B) ∗ Corresponding author. E-mail addresses:
[email protected],
[email protected] (T.G. Bhaskar),
[email protected] (V. Lakshmikantham). 1751-570X/$ – see front matter© 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2009.06.010 T.G. Bhaskar et al. / Nonlinear Analysis: Hybrid Systems 3 (2009) 734–737 735 By a solution of the IVP (2.1), we mean a function φ ∈ Cq([x0, x0 + a],R) for which the Caputo derivative cDqφ(x) exists and satisfies (2.1). Any solution φ(x) = φ(x, x0, y0) of (2.1) also satisfies the equivalent Volterra integral equation φ(x) = y0 + 1 Γ (q) ∫ x x0 (x− s)q−1f (s, φ(s))ds, 0 < q < 1 on [x0, x0 + a] and vice versa [4]. We shall now state our main result. Theorem 1. Assume that the function f in (2.1) satisfies the Krasnoselskii–Krein type conditions (A) and (B). Then, there exists a unique solutionφ(x) = φ(x, x0, y0) of (2.1) on [x0, x0+r]where r = min { a, ( bΓ (1+q) M ) 1 q } and the sequence of approximations {φn(x)}, φn(x) ∈ Ω for n = 0, 1, 2, . . . , defined by φn+1(x) = y0 + 1 Γ (q) ∫ x x0 (x− s)q−1f (s, φn(s))ds, (2.2) converge uniformly to the unique solution φ(x) of (2.1) on J = [x0, x0 + r]. Proof. Choose a function φ0(x), φ0 ∈ C1([x0, x0 + a],R) with φ0(x0) = y0 and φ0(x) ∈ Ω . The successive approximations {φn(x)}, n = 1, 2, . . ., given by (2.2) are well defined and continuous on [x0, x0 + r] where r = min { a, ( bΓ (1+q) M ) 1 q } . Indeed, for n = 1, 2, . . . , by induction, we get |φn(x)− y0| ≤ 1 Γ (q) ∫ x x0 (x− s)q−1|f (s, φn−1(s))|ds ≤ M qΓ (q) (x− x0)q ≤ Ma q Γ (1+ q) ≤ b. Note that (2.2) is equivalent to cDqφn+1(x) = f (x, φn(x)), φn(x0) = y0, (2.3) for all n = 1, 2, . . . . We wish to show that lim n→∞ | cDqφn(x)− f (x, φn(x))| = 0, on J. (2.4) From condition (B), we have K(1 − α) < 1, i.e. K < 1 + α∑∞i=0 αi, and therefore, there exists an integer N > 1 such that K < 1+ α∑N−1i=0 αi. We have from (2.3), and the fact that |f (x, y)| ≤ M onΩ , |cDqφ1(x)− f (x, φ1(x))| = |f (x, φ0(x))− f (x, φ1(x))| ≤ 2M (2.5) and for n = 1, 2, . . ., |cDqφn(x)− f (x, φn(x))| = |cDqφn+1(x)− cDqφn(x)|. In view of (2.3) and condition (B), we get, for i = 1, 2, . . . , |cDqφi+1(x)− f (x, φi+1(x))| = |f (x, φi(x))− f (x, φi+1(x))| ≤ β(|φi(x)− φi+1(x)|)α = β [ 1 Γ (q) ∫ x x0 (x− s)q−1|(cDqφi(s)− f (s, φi(s)))|ds ]α . From this and (2.5), and by induction, we obtain the following estimate on J: |cDqφN+1(x)− f (x, φN+1(x))| ≤ R(x− x0)λ (2.6) where λ = qr, r = α∑N−1i=0 αi, N > 1 such that K < 1+ r and R = ( β1+α+α 2+···+αN−1 ) (2M)α N ( 1 qΓ (q) )α+α2+···+αN . (2.7) 736 T.G. Bhaskar et al. / Nonlinear Analysis: Hybrid Systems 3 (2009) 734–737 Now, utilizing the conditions (A) and (2.3) we arrive at, for j = 0, 1, 2, . . . and 0 < x− x0 ≤ a, |cDqφN+j+1(x)− f (x, φN+j+1(x))| ≤ KηΓ (q) (x− x0)q |φN+j+1(x)− φN+j(x)| ≤ Kη (x− x0)q [∫ x x0 (x− s)q−1| (cDqφN+j(s)− f (s, φN+j(s))) |ds] and therefore, using the estimate (2.6) and the induction argument we get, for 0 ≤ x− x0 ≤ a and j = 0, 1, . . . , |cDqφN+j+1(x)− f (x, φN+j+1(x))| ≤ ( Kη q )j R(x− x0)λ. Since Kη < q, the claim (2.4) is proved. Thus, we have the sequence {φn(x)}with φn(x0) = y0, |φn(x)− y0| ≤ b, cDqφn(x) is continuous on |x− x0| ≤ a, satisfying the estimate |cDqφN+j+1(x)− f (x, φN+j+1)| ≤ Rρ j(x− x0)λ (2.8) where ρ = Kηq < 1 and the constants R, λ and N are determined by (2.7). We shall now show that the sequence {φn(x)} converges uniformly on 0 ≤ x−x0 ≤ a. By (2.8), there exist on |x−x0| ≤ a continuous functions α˜N+j+1(x), j = 0, 1, . . . , such that cDqφN+j+1(x) = f (x, φN+j+1(x))+ α˜N+j+1(x), (2.9) where |α˜N+j+1(x)| ≤ Rρ j(x− x0)λ ≤ Rρ jaλ (2.10) and φN+j+1(x) = y0 + 1 Γ (q) ∫ x x0 (x− s)q−1 [f (s, φN+j(s))+ α˜N+j+1(s)] ds. (2.11) In view of (2.3) and (2.9)–(2.11) and condition (B), we have ∆ ≤ β (|φN+j+1(x)− φN+i+1(x)|)α + R(ρ i + ρ j)(x− x0)λ, (2.12) where for convenience, we define∆ = |cDqφN+j+1(x)− cDqφN+i+1(x)|. Since |φN+j+1(x)− φN+i+1(x)| = 1 Γ (q) {∫ x x0 [ (x− s)q−1 (f (s, φN+j(s))− f (s, φN+i(s)))+ |α˜N+j+1(s)− α˜N+i+1(s)|] ds} , we get from the above, (2.12) and (ρ i + ρ j) ≤ 2 ∆ ≤ β ( 2M + 2Raλ qΓ (q) )α (x− x0)qα + 2R(x− x0)λ. Furthermore, since λ = qr and qα < qr we obtain the estimate ∆ ≤ β [( 2M + 2Raλ qΓ (q) )α + 2R ] aqr ≡ R1. (2.13) Again using (2.13) and proceeding as before, we have a new estimate given by ∆ ≤ β [( R1 + 2Raλ qΓ (q) )α + 2R ] (x− x0)λ ≡ R2(x− x0)λ, where R2 = β [( R1+2Raλ qΓ (q) )α + 2R]. Repeating this process (N − 1) times and letting RN−1 be an aptly chosen constant Rˆ, we get for i, j = 0, 1, 2, . . . and 0 ≤ x− x0 ≤ a, ∆ ≤ Rˆ(x− x0)λ. (2.14) Now, condition (A), (2.9), (2.10) and (2.14) yield ∆ ≤ |f (x, φN+j+1(x))− f (x, φN+i+1(x))| + R(ρ i + ρ j)(x− x0)λ ≤ KηΓ (q) (x− x0)q |φN+j+1(x)− φN+i+1(x)| + R(ρ i + ρ j)(x− x0)λ. T.G. Bhaskar et al. / Nonlinear Analysis: Hybrid Systems 3 (2009) 734–737 737 which leads to ∆ ≤ [( Kη q ) (Rˆ+ 2Raλ)+ R(ρ i + ρ j) ] (x− x0)λ. Identifying Rˆ+ 2Raλ as R∗ we can write the above in the form ∆ ≤ [ρR∗ + R(ρ i + ρ j)] (x− x0)λ. By induction we arrive at ∆ ≤ [ρm−1R∗ + (ρ i + ρ j)R(ρm−1 + ρm−2 + · · · + 1)] (x− x0)λ ≤ [ ρm−1R∗ + (ρ i + ρ j) R 1− ρ ] (x− x0)λ. This last estimate shows that ∆ → 0 as i, j,m → ∞ since ρ = Kηq < 1. Hence the sequence {f (x, φn(x))} satisfies the Cauchy criterion, the sequence {f (x, φn(x))} is uniformly convergent on |x − x0| ≤ a and, as a consequence, the sequence {φn(x)} is also uniformly convergent on |x− x0| ≤ a. Let φ(x) be the limit function of {φn(x)}; it is easy to see that φ(x) is the solution of (2.1). If there are two solutions φ(x), ψ(x), consider the sequence φ(x), ψ(x), φ(x), ψ(x), . . . . It is easy to see that this sequence is uniformly convergent and hence φ(x) ≡ ψ(x). This implies that the solution is unique. � We can also prove uniqueness of solutions directly as in [5]. See also Theorem 1.7.1 in [1]. References [1] R.P. Agarwal, V. Lakshmikantham, Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations, World Scientific, Singapore, 1993. [2] O. Kooi, The method of successive approximations and a uniqueness theorem of Krasnoselskii–Krein in the theory of differential equations, Nederi. Akad. Wetensch, Proc. Ser. A61; Indag. Math. 20 (1958) 322–327. [3] M.A. Krasnoselskii, S.G. Krein, On a class of uniqueness theorems for the equation y′ = f (x, y), Usphe. Mat. Nauk (N.S) 11 (1) (1956) 209–213. (Russian: Math. Reviews 18, p. 38). [4] V. Lakshmikantham, S. Leela, J. Vasundhara Devi, Theory of Fractional Dynamical Systems, Cambridge Academic Publishers, Cambridge, 2009. [5] V. Lakshmikantham, S. Leela, Krasnoselskii–Krein type uniqueness result for fractional differential equations, Nonlinear Anal. TMA 71 (7–8) (2009) 3421–3424. Fractional differential equations with a Krasnoselskii--Krein type condition Introduction Main result References