International Journal of Analysis and Applications ISSN 2291-8639 Volume 5, Number 2 (2014), 136-146 http://www.etamaths.com EIGENVALUES FOR ITERATIVE SYSTEMS OF (n,p)-TYPE FRACTIONAL ORDER BOUNDARY VALUE PROBLEMS K. R. PRASAD1, B. M. B. KRUSHNA2,∗ AND N. SREEDHAR3 Abstract. In this paper, we determine the eigenvalue intervals of λ1,λ2, · · ·,λn for which the iterative system of (n,p)-type fractional order two-point boundary value problem has a positive solution by an application of Guo- Krasnosel’skii fixed point theorem on a cone. 1. Introduction The study of fractional order differential equations has emerged as an impor- tant area of mathematics. It has wide range of applications in various fields of science and engineering such as physics, mechanics, control systems, flow in porous media, electromagnetics and viscoelasticity. Recently, much interest has been cre- ated in establishing positive solutions and multiple positive solutions for two-point, multi-point boundary value problems (BVPs) associated with ordinary and frac- tional order differential equations. To mention the related papers along these lines, we refer to Erbe and Wang [4], Davis, Henderson, Prasad and Yin [3] for ordi- nary differential equations, Henderson and Ntouyas [6, 7], Henderson, Ntouyas and Purnaras [8, 9] for systems of ordinary differential equations, Bai and Lu [1], Zhang [17], Kauffman and Mboumi [10], Benchohra, Henderson, Ntoyuas and Ouahab [2], Su and Zhang [16], Khan, Rehman and Henderson[11], Prasad and Krushna [15] for fractional order differential equations. This paper concerned with determining the eigenvalues λi, 1 ≤ i ≤ n, for which there exist positive solutions for the iterative system of (n,p)-type fractional order boundary value problems (1.1) Dα0+yi(t) + λiai(t)fi(yi+1(t)) = 0, 1 ≤ i ≤ n, 0 < t < 1, yn+1(t) = y1(t), 0 < t < 1, } (1.2) y (j) i (0) = 0, 0 ≤ j ≤ n− 2, y (p) i (1) = 0, where Dα 0+ is the standard Riemann-Liouville fractional order derivative, n− 1 < α ≤ n and n ≥ 3, 1 ≤ p ≤ α− 1 is a fixed integer. By a positive solution of the fractional order BVP (1.1)-(1.2), we mean (y1(t),y2(t), ·· ·,yn(t)) ∈ ( C[α]+1[0, 1] )n satisfying (1.1)-(1.2) with yi(t) ≥ 0, i = 1, 2, 3, · · ·n, for 2010 Mathematics Subject Classification. 26A33, 34B15, 34B18. Key words and phrases. Fractional derivative, Boundary value problem, Iterative system, Two- point, Green’s function, Eigenvalues, Positive solution. c©2014 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 136 EIGENVALUES FOR ITERATIVE SYSTEMS 137 all t ∈ [0, 1] and (y1(t),y2(t), · · ·,yn(t)) 6= (0, 0, · · ·, 0). We assume the following conditions hold throughout the paper: (A1) fi : R + → R+ is continuous, for 1 ≤ i ≤ n, (A2) ai : [0, 1] → R + is continuous and ai does not vanish identically on any closed subinterval of [0, 1], for 1 ≤ i ≤ n, (A3) each of fi0 = lim x→0+ fi(x) x and fi∞ = lim x→∞ fi(x) x , for 1 ≤ i ≤ n, exists as positive real numbers. The rest of the paper is organized as follows. In Section 2, we construct the Green’s function for the homogeneous BVP and estimate the bounds for the Green’s function. In Section 3, we establish criteria to determine the eigenvalues for which the fractional order BVP (1.1)-(1.2) has at least one positive solution in a cone by using Guo-Krasnosel’skii fixed point theorem. In Section 4, as an application, we demonstrate our results with an example. 2. Green’s function and Bounds In this section, we construct the Green’s function for the homogeneous BVP and estimate the bounds for the Green’s function which are needed in establishing the main results. Lemma 2.1. If h(t) ∈ C[0, 1], then the fractional order BVP, (2.1) Dα0+y1(t) + h(t) = 0, t ∈ (0, 1), (2.2) y (j) 1 (0) = 0, 0 ≤ j ≤ n− 2, y (p) 1 (1) = 0 has a unique solution, y1(t) = ∫ 1 0 G(t,s)h(s)ds, where (2.3) G(t,s) = { tα−1(1−s)α−1−p Γ(α) , 0 ≤ t ≤ s ≤ 1, tα−1(1−s)α−1−p−(t−s)α−1 Γ(α) , 0 ≤ s ≤ t ≤ 1. Proof. Assume that y1(t) ∈ C[α]+1[0, 1] is a solution of fractional order BVP (2.1)- (2.2) and is uniquely expressed as Iα0+D α 0+y1(t) = −I α 0+h(t) y1(t) = −1 Γ(α) ∫ t 0 (t−s)α−1h(s)ds + c1tα−1 + c2tα−2 + c3tα−3 + · · · + cntα−n. From y (j) 1 (0) = 0, 0 ≤ j ≤ n− 2, we have cn = cn−1 = cn−2 = · · · = c2 = 0. Then y1(t) = −1 Γ(α) ∫ t 0 (t−s)α−1h(s)ds + c1tα−1, y (p) 1 (t) = c1 p∏ i=1 (α− i)tα−1−p − p∏ i=1 (α− i) 1 Γ(α) ∫ 1 0 (t−s)α−1−ph(s)ds. 138 PRASAD, KRUSHNA AND SREEDHAR From y (p) 1 (1) = 0, we have c1 p∏ i=1 (α− i) − p∏ i=1 (α− i) 1 Γ(α) ∫ 1 0 (1 −s)α−1−ph(s)ds = 0. Therefore, c1 = 1 Γ(α) ∫ 1 0 (1 −s)α−1−ph(s)ds. Thus, the unique solution of (2.1)-(2.2) is y1(t) = −1 Γ(α) ∫ t 0 (t−s)α−1h(s)ds + tα−1 Γ(α) ∫ 1 0 (1 −s)α−1−ph(s)ds = ∫ 1 0 G(t,s)h(s)ds, where G(t,s) is given in (2.3). � Lemma 2.2. The Green’s function G(t,s) satisfies the following inequalities, (i) G(t,s) ≥ 0, for all (t,s) ∈ [0, 1] × [0, 1], (ii) G(t,s) ≤ G(1,s), for all (t,s) ∈ [0, 1] × [0, 1], (iii) G(t,s) ≥ 1 4α−1 G(1,s), for all (t,s) ∈ I × [0, 1], where I = [ 1 4 , 3 4 ] . Proof. The Green’s function G(t,s) is given in (2.3). For 0 ≤ t ≤ s ≤ 1. G(t,s) = 1 Γ(α) [tα−1(1 −s)α−1−p] ≥ 0. For 0 ≤ s ≤ t ≤ 1, G(t,s) = 1 Γ(α) [tα−1(1 −s)α−1−p − (t−s)α−1] ≥ 1 Γ(α) [tα−1(1 −s)α−1−p − tα−1(1 −s)α−1] = 1 Γ(α) [tα−1(1 −s)α−1−p][1 − (1 −s)p] ≥ 0. Hence the inequality (i) is proved. We prove the inequality (ii). For 0 ≤ t ≤ s ≤ 1, ∂ ∂t G(t,s) = 1 Γ(α) [(α− 1)tα−2(1 −s)α−1−p] ≥ 0. For 0 ≤ s ≤ t ≤ 1, ∂ ∂t G(t,s) = 1 Γ(α) [(α− 1)tα−2(1 −s)α−1−p − (α− 1)(t−s)α−2] = (α− 1) Γ(α) [ tα−2(1 −s)α−2(1 −s)1−p − (t−s)α−2 ] ≥ (α− 1) Γ(α) [ tα−2(1 −s)α−2(1 −s)1−p − (t− ts)α−2 ] = (α− 1) Γ(α) [ (1 −s)1−p − 1 ] (t− ts)α−2 ≥ 0. EIGENVALUES FOR ITERATIVE SYSTEMS 139 Therefore G(t,s) is increasing with respect to t ∈ [0, 1]. Hence the inequality (ii) is proved. Now, we establish the inequality (iii). For 0 ≤ t ≤ s ≤ 1 and t ∈ I, G(t,s) G(1,s) = tα−1(1 −s)α−1−p (1 −s)α−1−p = tα−1 ≥ 1 4α−1 . For 0 ≤ s ≤ t ≤ 1 and t ∈ I, G(t,s) G(1,s) = tα−1(1 −s)α−1−p − (t−s)α−1 (1 −s)α−1−p − (1 −s)α−1 ≥ tα−1(1 −s)α−1−p − (t− ts)α−1 (1 −s)α−1−p − (1 −s)α−1 =tα−1 ≥ 1 4α−1 . Hence the inequality (iii) is proved. � An n-tuple (y1(t),y2(t), ···,yn(t)) is a solution of the BVP (1.1)-(1.2) if and only if yi(t) ∈ C[α]+1[0, 1] satisfies the following equations y1(t) =λ1 ∫ 1 0 G(t,s1)a1(s1)f1 ( λ2 ∫ 1 0 G(s1,s2)a2(s2) · · · fn−1 ( λn ∫ 1 0 G(sn−1,sn)an(sn)fn(y1(sn))dsn ) · · ·ds2 ) ds1 and yi(t) = λi ∫ 1 0 G(t,s)ai(s)fi(yi+1(s))ds, 0 ≤ t ≤ 1, 2 ≤ i ≤ n, where yn+1(t) = y1(t), 0 ≤ t ≤ 1. In establishing our main result, we will employ the following fixed point theorem due to Guo-Krasnosel’skii [5, 13]. Theorem 2.3. [5, 13] Let X be a Banach Space, P ⊆ X be a cone and suppose that Ω1, Ω2 are open subsets of X with 0 ∈ Ω1 and Ω1 ⊂ Ω2. Suppose further that T : P ∩ (Ω2\Ω1) → P is completely continuous operator such that either (i) ‖ Tu ‖≤‖ u ‖, u ∈ P ∩∂Ω1 and ‖ Tu ‖≥‖ u ‖, u ∈ P ∩∂Ω2, or (ii) ‖ Tu ‖≥‖ u ‖, u ∈ P ∩∂Ω1 and ‖ Tu ‖≤‖ u ‖, u ∈ P ∩∂Ω2 holds. Then T has a fixed point in P ∩ (Ω2\Ω1). 3. Positive Solutions in a Cone In this section, we establish criteria to determine the eigenvalues for which the fractional order BVP (1.1)-(1.2) has at least one positive solution in a cone. Let X = {x : x ∈ C[0, 1]} be the Banach space equipped with the norm ‖x‖ = max 0≤t≤1 |x(t)|. Define a cone P ⊂ X by P = { x ∈ X | x(t) ≥ 0 on [0, 1] and min t∈I x(t) ≥ 1 4α−1 ‖x‖ } . 140 PRASAD, KRUSHNA AND SREEDHAR Now, we define an integral operator T : P → X, for y1 ∈ P , by (3.1) Ty1(t) = λ1 ∫ 1 0 G(t,s1)a1(s1)f1 ( λ2 ∫ 1 0 G(s1,s2)a2(s2) · ·· fn−1 ( λn ∫ 1 0 G(sn−1,sn)an(sn)fn(y1(sn))dsn ) · · ·ds2 ) ds1. Notice from (A1), (A2) and Lemma 2.2 that, for y1 ∈ P , Ty1(t) ≥ 0 on [0, 1]. And also, we have Ty1(t) ≤ λ1 ∫ 1 0 G(1,s1)a1(s1)f1 ( λ2 ∫ 1 0 G(s1,s2)a2(s2) · ·· fn−1 ( λn ∫ 1 0 G(sn−1,sn)an(sn)fn(y1(sn))dsn ) · · ·ds2 ) ds1 so that (3.2) ‖Ty1‖≤ λ1 ∫ 1 0 G(1,s1)a1(s1)f1 ( λ2 ∫ 1 0 G(s1,s2)a2(s2) · ·· fn−1 ( λn ∫ 1 0 G(sn−1,sn)an(sn)fn(y1(sn))dsn ) · · ·ds2 ) ds1. Next, if y1 ∈ P , we have from Lemma 2.2 and (3.2) that min t∈I Ty1(t) = min t∈I λ1 ∫ 1 0 G(t,s1)a1(s1)f1 ( λ2 ∫ 1 0 G(s1,s2)a2(s2) · ·· fn−1 ( λn ∫ 1 0 G(sn−1,sn)an(sn)fn(y1(sn))dsn ) · · ·ds2 ) ds1 ≥ λ1 1 4α−1 ∫ 1 0 G(1,s1)a1(s1)f1 ( λ2 ∫ 1 0 G(s1,s2)a2(s2) · · · fn−1 ( λn ∫ 1 0 G(sn−1,sn)an(sn)fn(y1(sn))dsn ) · · ·ds2 ) ds1 ≥ 1 4α−1 ‖Ty1‖. Therefore, min t∈I Ty1(t) ≥ 1 4α−1 ‖Ty1‖. Hence, Ty1 ∈ P and so T : P → P . Further, the operator T is a completely con- tinuous operator by an application of the Arzela-Ascoli Theorem. Now, we seek suitable fixed point of T belonging to the cone P. For our first result, we define positive numbers N1 and N2, by N1 = max 1≤i≤n {[ 1 4α−1 ∫ s∈I G(1,s)ai(s)dsfi∞ ]−1} and N2 = min 1≤i≤n {[∫ 1 0 G(1,s)ai(s)dsfi0 ]−1} . EIGENVALUES FOR ITERATIVE SYSTEMS 141 Theorem 3.1. Assume that the conditions (A1)-(A3) are satisfied. Then, for each λ1,λ2, · · ·,λn satisfying (3.3) N1 < λi < N2, 1 ≤ i ≤ n, there exists an n-tuple (y1,y2, · · ·,yn) satisfying (1.1)-(1.2) such that yi(t) > 0, 1 ≤ i ≤ n on (0, 1). Proof. Let λi, 1 ≤ i ≤ n be given as in (3.3). Now, let � > 0 be chosen such that max 1≤i≤n {[ 1 4α−1 ∫ s∈I G(1,s)ai(s)ds(fi∞ − �) ]−1} ≤ min 1≤i≤n λi and max 1≤i≤n λi ≤ min 1≤i≤n {[∫ 1 0 G(1,s)ai(s)ds(fi0 + �) ]−1} . We seek fixed point of the completely continuous operator T : P → P defined by (3.1). Now, from the definitions of fi0, 1 ≤ i ≤ n, there exists an H1 > 0 such that, for each 1 ≤ i ≤ n, fi(x) ≤ (fi0 + �)x, 0 < x ≤ H1. Let y1 ∈ P with ‖y1‖ = H1. We first have from Lemma 2.2 and the choice of �, for 0 ≤ sn−1 ≤ 1, λn ∫ 1 0 G(sn−1,sn)an(sn)fn(y1(sn))dsn ≤ λn ∫ 1 0 G(1,sn)an(sn)(fn0 + �)y1(sn)dsn ≤ λn ∫ 1 0 G(1,sn)an(sn)dsn(fn0 + �)‖y1‖ ≤‖y1‖ = H1. It follows in a similar manner from Lemma 2.2 and the choice of � that, for 0 ≤ sn−2 ≤ 1, λn−1 ∫ 1 0 G(sn−2,sn−1)an−1(sn−1) fn−1 ( λn ∫ 1 0 G(sn−1,sn)an(sn)fn(y1(sn))dsn ) dsn−1 ≤ λn−1 ∫ 1 0 G(sn−1,sn−1)an−1(sn−1)dsn−1(fn−1,0 + �)‖y1‖ ≤‖y1‖ = H1. Continuing with this bootstrapping argument, we have, for 0 ≤ t ≤ 1, λ1 ∫ 1 0 G(t,s1)a1(s1)f1 ( λ2 ∫ 1 0 G(s1,s2)a2(s2) · · · fn(y1(sn))dsn ) · · ·ds2 ) ds1 ≤ H1, so that, for 0 ≤ t ≤ 1, Ty1(t) ≤ H1. 142 PRASAD, KRUSHNA AND SREEDHAR Hence, ‖Ty1‖≤ H1 = ‖y1‖. If we set Ω1 = {x ∈ X | ‖x‖ < H1}, then (3.4) ‖Ty1‖≤‖y1‖, for y1 ∈ P ∩∂Ω1. Next, from the definitions of fi∞, 1 ≤ i ≤ n, there exists H2 > 0 such that, for each 1 ≤ i ≤ n, fi(x) ≥ (fi∞ − �)x, x ≥ H2. Choose H2 = max{2H1, 4α−1H2}. Let y1 ∈ P and ‖y1‖ = H2. Then, min t∈I y1(t) ≥ 1 4α−1 ‖y1‖≥ H2. Then, from Lemma 2.2 and choice of �, for 0 ≤ sn−1 ≤ 1, we have that λn ∫ 1 0 G(sn−1,sn)an(sn)fn(y1(sn))dsn ≥ λn ∫ s∈I G(1,sn)an(sn)fn(y1(sn))dsn ≥ 1 4α−1 λn ∫ s∈I G(1,sn)an(sn)(fn∞ − �)y1(sn)dsn ≥ 1 4α−1 λn ∫ s∈I G(1,sn)an(sn)dsn(fn∞ − �)‖y1‖ ≥‖y1‖ = H2. It follows in a similar manner from Lemma 2.2 and choice of �, for 0 ≤ sn−2 ≤ 1, λn−1 ∫ 1 0 G(sn−2,sn−1)an−1(sn−1) fn−1 ( λn ∫ 1 0 G(sn−1,sn)an(sn)fn(y1(sn))dsn ) dsn−1 ≥ 1 4α−1 λn−1 ∫ s∈I G(1,sn−1)an−1(sn−1)dsn−1(fn−1,∞ − �)‖y1‖ ≥‖y1‖ = H2. Again, using a bootstrapping argument, we have λ1 ∫ 1 0 G(t,s1)a1(s1)f1 ( λ2 ∫ 1 0 G(s1,s2)a2(s2) · ·· fn(y1(sn))dsn ) · · ·ds2 ) ds1 ≥ H2, so that Ty1(t) ≥ H2 = ‖y1‖. Hence, ‖Ty1‖≥‖y1‖. So if we set Ω2 = {x ∈ X | ‖x‖ < H2}, then (3.5) ‖Ty1‖≥‖y1‖, for y1 ∈ P ∩∂Ω2. Applying Theorem 2.3 to (3.4) and (3.5), we obtain that T has a fixed point y1 ∈ P ∩ (Ω2\Ω1). Setting y1 = yn+1, we obtain a positive solution (y1,y2, · · ·,yn) of (1.1)-(1.2) given iteratively by yi(t) = λi ∫ 1 0 G(t,s)ai(s)fi(yi+1(s))ds, i = n,n− 1, · · ·, 1. The proof is completed. � EIGENVALUES FOR ITERATIVE SYSTEMS 143 Prior to our next result, we define the positive numbers N3 and N4 by N3 = max 1≤i≤n {[ 1 4α−1 ∫ s∈I G(1,s)ai(s)dsfi0 ]−1} and N4 = min 1≤i≤n {[∫ 1 0 G(1,s)ai(s)dsfi∞ ]−1} . Theorem 3.2. Assume that the conditions (A1)-(A3) are satisfied. Then, for each λ1,λ2, · · ·,λn satisfying (3.6) N3 < λi < N4, 1 ≤ i ≤ n, there exists an n-tuple (y1,y2, · · ·,yn) satisfying (1.1)-(1.2) such that yi(t) > 0, 1 ≤ i ≤ n on (0, 1). Proof. Let λi, 1 ≤ i ≤ n be given as in (3.6). Now, let � > 0 be chosen such that max 1≤i≤n {[ 1 4α−1 ∫ s∈I G(1,s)ai(s)ds(fi0 − �) ]−1} ≤ min 1≤i≤n λi and max 1≤i≤n λi ≤ min 1≤i≤n {[∫ 1 0 G(1,s)ai(s)ds(fi∞ + �) ]−1} . Let T be the cone preserving, completely continuous operator that was defined by (3.1). From the definition of fi0, 1 ≤ i ≤ n there exists H3 > 0 such that, for each 1 ≤ i ≤ n, fi(x) ≥ (fi0 − �)x, 0 < x ≤ H3. Also, from the definitions of fi0, it follows that fi0(0) = 0, 1 ≤ i ≤ n, and so there exist 0 < Kn < Kn−1 < · · · < K2 < H3 such that λifi(t) ≤ Ki−1∫ 1 0 G(1,s)ai(s)ds , t ∈ [0,Ki], 3 ≤ i ≤ n, and λ2f2(t) ≤ H3∫ 1 0 G(1,s)a2(s)ds , t ∈ [0,K2]. Choose y1 ∈ P with ‖y1‖ = Kn. Then, we have λn ∫ 1 0 G(sn−1,sn)an(sn)fn(y1(sn))dsn ≤ λn ∫ 1 0 G(1,sn)an(sn)fn(y1(sn))dsn ≤ ∫ 1 0 G(1,sn)an(sn)Kn−1dsn∫ 1 0 G(1,sn)an(sn)dsn ≤ Kn−1. 144 PRASAD, KRUSHNA AND SREEDHAR Continuing with this bootstrapping argument, it follows that λ2 ∫ 1 0 G(1,s2)a2(s2)f2 ( λ3 ∫ 1 0 G(s2,s3)a3(s3) · ·· fn(y1(sn))dsn ) · · ·ds3 ) ds2 ≤ H3. Then, Ty1(t) = λ1 ∫ 1 0 G(t,s1)a1(s1)f1 ( λ2 ∫ 1 0 G(s1,s2)a2(s2) · ·· fn(y1(sn))dsn ) · · ·ds2 ) ds1 ≥ 1 4α−1 λ1 ∫ s∈I G(1,s1)a1(s1)(f10 − �)‖y1‖ds1 ≥‖y1‖. So, ‖Ty1‖≥‖y1‖. If we set Ω1 = {x ∈ X | ‖x‖ < Kn}, then (3.7) ‖Ty1‖≥‖y1‖, for y1 ∈ P ∩∂Ω1. Since each fi∞ is assumed to be a positive real number, it follows that fi, 1 ≤ i ≤ n, is unbounded at ∞. For each 1 ≤ i ≤ n, set f∗i (x) = sup 0≤s≤x fi(s). Then, it is straightforward that, for each 1 ≤ i ≤ n, f∗i is a nondecreasing real- valued function, fi ≤ f∗i and lim x→∞ f∗i (x) x = fi∞. Next, by definition of fi∞, 1 ≤ i ≤ n, there exists H4 such that, for each 1 ≤ i ≤ n, f∗i (x) ≤ (fi∞ + �)x, x ≥ H4. It follows that there exists H4 = max{2H3,H4} such that, for each 1 ≤ i ≤ n, f∗i (x) ≤ f ∗ i (H4), 0 < x ≤ H4. Choose y1 ∈ P with ‖y1‖ = H4. Then, using the usual bootstrapping argument, we have Ty1(t) = λ1 ∫ 1 0 G(t,s1)a1(s1)f1(λ2 · ··)ds1 ≤ λ1 ∫ 1 0 G(t,s1)a1(s1)f ∗ 1 (λ2 · ··)ds1 ≤ λ1 ∫ 1 0 G(1,s1)a1(s1)f ∗ 1 (H4)ds1 ≤ λ1 ∫ 1 0 G(1,s1)a1(s1)ds1(f1∞ + �)H4 ≤ H4 = ‖y1‖, and so ‖Ty1‖≤‖y1‖. So, if we let Ω2 = {x ∈ X | ‖x‖ < H4}, then (3.8) ‖Ty1‖≤‖y1‖, for y1 ∈ P ∩∂Ω2. Applying Theorem 2.3 to (3.7)-(3.8), we obtain that T has a fixed point y1 ∈ P ∩ (Ω2\Ω1), which in turn with y1 = yn+1, yields an n-tuple (y1,y2, · · ·,yn) EIGENVALUES FOR ITERATIVE SYSTEMS 145 satisfying the BVP (1.1)-(1.2) for the chosen values of λi, 1 ≤ i ≤ n. The proof is thus completed. � 4. Example In this section, as an application, we demonstrate our results with an example. Consider the fractional order boundary value problem (4.1) D2.50+ y1(t) + λ1 1 + t y2(46 − 27.5e−2y2 )(500 − 487e−3y2 ) = 0, t ∈ (0, 1), D2.50+ y2(t) + λ2 1 + t y3(37 − 25.5e−5y3 )(400 − 368e−y3 ) = 0, t ∈ (0, 1), D2.50+ y3(t) + λ3 1 + t y1(79 − 75e−y1 )(800 − 749.5e−2y1 ) = 0, t ∈ (0, 1),   (4.2) yi(0) = 0, y ′ i(0) = 0 and y ′ i(1) = 0, i = 1, 2, 3. The Green’s function G(t,s) of corresponding homogeneous BVP is given by G(t,s) = { t1.5(1−s)0.5 Γ(2.5) , 0 ≤ t ≤ s ≤ 1, t1.5(1−s)0.5−(t−s)1.5 Γ(2.5) , 0 ≤ s ≤ t ≤ 1. By direct calculations, we found that f10 = 299,f20 = 368,f30 = 202, f1∞ = 23000,f2∞ = 14800,f3∞ = 63200, N1 = max {[ (0.25)1.5 ∫ 0.75 0.25 G(1,s)a1(s)ds(23000) ]−1 , [ (0.25)1.5 ∫ 0.75 0.25 G(1,s)a2(s)ds(14800) ]−1 , [ (0.25)1.5 ∫ 0.75 0.25 G(1,s)a3(s)ds(63200) ]−1 } , = max{0.0009634, 0.0014972, 0.0003506} = 0.0014972. Similarly, N2 = min{0.0307737, 0.0250037, 0.0455512} = 0.0250037. Applying The- orem 3.1, we get an optimal eigenvalue interval 0.0014972355 < λi < 0.0250037, for i = 1, 2, 3 in which the fractional order BVP (4.1)-(4.2) has at least one positive solution. References [1] Z. Bai and H. Lu, Positive solutions for boundary value problems of nonlinear fractional differential equations, J. Math. Anal. Appl., 311(2005), 495-505. [2] M. Benchohra, J. Henderson, S. K. Ntoyuas and A. Ouahab, Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl., 338(2008), 1340-1350. [3] J. M. Davis, J. Henderson, K. R. Prasad and W. Yin, Eigenvalue intervals for non-linear right focal problems, Appl. Anal., 74(2000), 215-231. [4] L. H. Erbe and H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc., 120(1994), 743-748. [5] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Orlando, 1988. 146 PRASAD, KRUSHNA AND SREEDHAR [6] J. Henderson and S. K. Ntouyas, Positive solutions for systems of nth order three-point nonlocal boundary value problems, Electronic Journal of Qualitative Theory of Differential Equations, 18(2007), 1-12. [7] J. Henderson and S. K. Ntouyas, Positive solutions for systems of nonlinear boundary value problems, Nonlinear Studies, 15(2008), 51-60. [8] J. Henderson, S. K. Ntouyas and I. K. Purnaras, Positive solutions for systems of general- ized three-point nonlinear boundary value problems, Comment. Math. Univ. Carolin., 49, 1(2008), 79-91. [9] J. Henderson, S. K. Ntouyas and I. K. Purnaras, Positive solutions for systems of second order four-point nonlinear boundary value problems, Commu. Appl. Anal., 12(2008), No.1, 29-40. [10] E. R. Kauffman and E. Mboumi, Positive solutions of a boundary value problem for a nonlin- ear fractional differential equation, Electronic Journal of Qualitative Theory of Differential Equations, 3(2008), 1-11. [11] R. A. Khan, M. Rehman and J. Henderson, Existence and uniqueness of solutions for nonlin- ear fractional differential equations with integral boundary conditions, Fractional Differential Calculus, 1(2011), 29-43. [12] A. A. Kilbas, H. M. Srivasthava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier, Amsterdam, 2006. [13] M. A. Krasnosel’skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. [14] I. Podulbny, Fractional Differential Equations, Academic Press, San Diego, 1999. [15] K. R. Prasad and B. M. B. Krushna, Multiple positive solutions for a coupled system of Riemann-Liouville fractional order two-point boundary value problems, Nonlinear Studies, Vol. 20, No.4(2013), 501-511. [16] X. Su and S. Zhang, Solutions to boundary value problems for nonlinear differential equations of fractional order, Electronic Journal of Differential Equations, 26(2009), 1-15. [17] S. Zhang, Existence of solutions for a boundary value problem of fractional order, Acta Math. Sci., 26B(2006), 220-228. 1Department of Applied Mathematics, Andhra University, Visakhapatnam, 530 003, India 2Department of Mathematics, MVGR College of Engineering, Vizianagaram, 535 005, India 3Department of Mathematics, GITAM University, Visakhapatnam, 530 045, India ∗Corresponding author