International Journal of Analysis and Applications ISSN 2291-8639 Volume 9, Number 1 (2015), 54-67 http://www.etamaths.com EXISTENCE OF POSITIVE SOLUTIONS FOR A COUPLED SYSTEM OF (p,q)-LAPLACIAN FRACTIONAL HIGHER ORDER BOUNDARY VALUE PROBLEMS K. R. PRASAD1, B. M. B. KRUSHNA2,∗ AND L. T. WESEN1,3 Abstract. In this paper, we establish the existence of at least three positive solutions for a system of (p, q)-Laplacian fractional order two-point boundary value problems by applying five functionals fixed point theorem under suitable conditions on a cone in a Banach space. 1. Introduction In the universe, many real world problems can be formulated as mathematical models to analyze the situations and to predict future. Most of these models involve the rate of change of the dependent variable which leads to formation of the differential equations. One goal of differential equations is to understand the phenomena of nature by developing mathematical models. Fractional calculus is an extension of classical calculus and deals with the generalization of integration and differentiation to an arbitrary real order. A class of differential equations governed by nonlinear differential operators appears fre- quently and generated by great deal of interest in studying special types of problems. In this theory, the most applicable operator is the classical p-Laplacian operator. These types of problems arise in mathematical modeling of viscoelastic flows, turbulent filtration in porous media, biophysics, plasma physics and chemical reaction design. For a detailed description on applications of p-Laplacian operator, we refer [10]. The positivity of boundary value problems associated with ordinary differential equations were studied by many authors [14, 1, 2] and extended to p-Laplacian boundary value prob- lems [4, 22, 8]. Later these results are further extended to fractional order boundary value problems [6, 5, 9, 21, 12, 19] by utilizing various fixed point theorems on cones. Recently researchers are concentrating on the theory of fractional order boundary value problems associated with p-Laplacian operator. Yang and Yan [22] studied the existence of positive solutions for third order Sturm–Liouville boundary value problems with p-Laplacian operator by applying the fixed point index method. Chai [7] obtained the existence and multiplicity of positive solutions for a class of p-Laplacian fractional order boundary value problems by means of the fixed point theorem. Prasad and Krushna [18, 20] derived sufficient conditions for the existence of positive solutions to p-Laplacian fractional order boundary value problems. 2010 Mathematics Subject Classification. 34A08, 34B18, 35J05. Key words and phrases. fractional order derivative; (p, q)-Laplacian; boundary value problem; Green’s function, positive solution. c©2015 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 54 EXISTENCE OF POSITIVE SOLUTIONS FOR (p,q)-LAPLACIAN FBVPS 55 Motivated by the papers mentioned above, in this paper, we are concerned with estab- lishing the existence of positive solutions for a coupled system of (p,q)-Laplacian fractional order differential equations (1.1) D β1 0+ ( φp ( Dα1 0+ x(t) )) = f1 ( t,x(t),y(t) ) , t ∈ (0, 1), (1.2) D β2 0+ ( φq ( Dα2 0+ y(t) )) = f2 ( t,x(t),y(t) ) , t ∈ (0, 1), satisfying the boundary conditions (1.3) x(j)(0) = 0, j = 0, 1, · · ·,n− 2, x(n−2)(1) = 0, φp ( Dα1 0+ x(0) ) = 0, φp ( Dα1 0+ x(1) ) = 0,   (1.4) y(j)(0) = 0, j = 0, 1, · · ·,n− 2, y(n−2)(1) = 0, φq ( Dα2 0+ y(0) ) = 0, φq ( Dα2 0+ y(1) ) = 0,   where αi ∈ (n−1,n],n ≥ 3, βi ∈ (1, 2], φp(s) = |s|p−2s, φq(s) = |s|q−2s, p,q > 1, φ−1p = φq, φ−1q = φp, 1 p + 1 q = 1, fi : [0, 1] × R 2 → R+ are continuous and Dαi 0+ ,D βi 0+ , for i = 1, 2 are the standard Riemann–Liouville fractional order derivatives. By a positive solution of the coupled system of fractional order boundary value problem (1.1)-(1.4), we mean ( x(t),y(t) ) ∈ ( Cα1+β1 [0, 1] × Cα2+β2 [0, 1] ) satisfying the boundary value problem (1.1)-(1.4) with x(t) ≥ 0, y(t) ≥ 0, for all t ∈ [0, 1] and (x,y) 6= (0, 0). The rest of the paper is organized as follows. In Section 2, the solution of the boundary value problems (1.1), (1.3) and (1.2), (1.4) are expressed in terms of Green functions and the bounds for these Green functions are estimated. In Section 3, the existence of at least three positive solutions for a coupled system of (p,q)-Laplacian fractional order boundary value problem (1.1)-(1.4) are established, by using five functionals fixed point theorem. In Section 4, as an application, the results are demonstrated with an example. 2. Green Functions and Bounds In this section, the solution of the boundary value problems (1.1), (1.3) and (1.2), (1.4) are expressed in terms of the equivalent integral equations involving Green functions and the bounds for the Green functions are estimated, which are essential to establish the main results. Lemma 2.1. Let h1(t) ∈ C[0, 1]. Then the fractional order differential equation, (2.1) Dα1 0+ x(t) + h1(t) = 0, t ∈ (0, 1), satisfying (2.2) x(j)(0) = 0, j = 0, 1, · · ·,n− 2, x(n−2)(1) = 0, has a unique solution, x(t) = ∫ 1 0 G1(t,s)h1(s)ds, where G1(t,s) is the Green’s function for the problem (2.1)-(2.2) and is given by (2.3) G1(t,s) = { G11(t,s), 0 ≤ t ≤ s ≤ 1, G12(t,s), 0 ≤ s ≤ t ≤ 1, 56 PRASAD, KRUSHNA AND WESEN G11(t,s) = tα1−1(1 −s)α1−n+1 Γ(α1) , G12(t,s) = tα1−1(1 −s)α1−n+1 − (t−s)α1−1 Γ(α1) . For details refer to [19]. Lemma 2.2. Let h2(t) ∈ C[0, 1]. Then the fractional order differential equation, (2.4) D β1 0+ ( φp ( Dα1 0+ x(t) )) = h2(t), t ∈ (0, 1), satisfying (2.5) φp ( Dα1 0+ x(0) ) = 0, φp ( Dα1 0+ x(1) ) = 0, has a unique solution, (2.6) x(t) = ∫ 1 0 G1(t,s)φq (∫ 1 0 H1(s,τ)h2(τ)dτ ) ds, where (2.7) H1(t,s) =   [ t(1 −s) ]β1−1 Γ(β1) , 0 ≤ t ≤ s ≤ 1,[ t(1 −s) ]β1−1 − (t−s)β1−1 Γ(β1) , 0 ≤ s ≤ t ≤ 1. Proof. An equivalent integral equation for (2.4) is given by φp ( Dα1 0+ x(t) ) = 1 Γ(β1) ∫ t 0 (t− τ)β1−1h2(τ)dτ + k1tβ1−1 + k2tβ1−2. From (2.5), one gets that k2 = 0 and k1 = −1 Γ(β1) ∫ 1 0 (1 − τ)β1−1h2(τ)dτ. Then, φp ( Dα1 0+ x(t) ) = 1 Γ(β1) ∫ t 0 (t− τ)β1−1h2(τ)dτ − tβ1−1 Γ(β1) ∫ 1 0 (1 − τ)β1−1h2(τ)dτ = − ∫ 1 0 H1(t,τ)h2(τ)dτ. Therefore, Dα1 0+ x(t) + φq (∫ 1 0 H1(t,τ)h2(τ)dτ ) = 0. Hence x(t) in (2.6) is the solution to the fractional order boundary value problem (2.4), (1.3). � Lemma 2.3. The Green’s function G1(t,s) given in (2.3) is nonnegative, for all (t,s) ∈ [0, 1] × [0, 1]. Proof. Consider the Green’s function G1(t,s) given by (2.3). Let 0 ≤ t ≤ s ≤ 1. Then, we have G11(t,s) = 1 Γ(α1) [ tα1−1(1 −s)α1−n+1 ] ≥ 0. EXISTENCE OF POSITIVE SOLUTIONS FOR (p,q)-LAPLACIAN FBVPS 57 Let 0 ≤ s ≤ t ≤ 1. Then, we have G12(t,s) = 1 Γ(α1) [ tα1−1(1 −s)α1−n+1 − (t−s)α1−1 ] ≥ 1 Γ(α1) [ tα1−1(1 −s)α1−n+1 − (t− ts)α1−1 ] = tα1−1 Γ(α1) [( 1 + (n− 2)s + 1 2 (n2 − 3n + 2)s2 + · · · ) − 1 ] (1 −s)α1−1 ≥0. � Lemma 2.4. For t ∈ I = [ 1 4 , 3 4 ] , the Green’s function G1(t,s) given in (2.3) satisfies the following inequalities (P1) G1(t,s) ≤ G1(1,s), for all (t,s) ∈ [0, 1] × [0, 1], (P2) G1(t,s) ≥ (1 4 )α1−1 G1(1,s), for all (t,s) ∈ I × [0, 1]. Proof. Consider the Green’s function G1(t,s) given by (2.3). Let 0 ≤ t ≤ s ≤ 1. Then, we have ∂G11(t,s) ∂t = 1 Γ(α1) [ (α1 − 1)tα1−2(1 −s)α1−n+1 ] ≥ 0. Therefore, G11(t,s) is increasing in t, which implies G11(t,s) ≤ G11(1,s). Let 0 ≤ s ≤ t ≤ 1. Then, we have ∂G12(t,s) ∂t = 1 Γ(α1) [ (α1 − 1)tα1−2(1 −s)α1−n+1 − (α1 − 1)(t−s)α1−2 ] ≥ 1 Γ(α1) [ (α1 − 1)tα1−2(1 −s)α1−n+1 − (α1 − 1)(t− ts)α1−2 ] ≥ tα1−2 Γ(α1 − 1) [ 1 − ( 1 − (n− 3)s + 1 2 (n2 − 7n + 12)s2 + · · · )] (1 −s)α1−n+1 ≥ 0. Therefore, G12(t,s) is increasing in t, which implies G12(t,s) ≤ G12(1,s). Let 0 ≤ t ≤ s ≤ 1 and t ∈ I. Then G11(t,s) = 1 Γ(α1) [ tα1−1(1 −s)α1−n+1 ] ≥tα1−1 1 Γ(α1) [ (1 −s)α1−n+1 − (1 −s)α1−1 ] =tα1−1G11(1,s) ≥ (1 4 )α1−1 G11(1,s). 58 PRASAD, KRUSHNA AND WESEN Let 0 ≤ s ≤ t ≤ 1 and t ∈ I. Then G12(t,s) = 1 Γ(α1) [ tα1−1(1 −s)α1−n+1 − (t−s)α1−1 ] ≥ 1 Γ(α1) [ tα1−1(1 −s)α1−n+1 − (t− ts)α1−1 ] =tα1−1G12(1,s) ≥ (1 4 )α1−1 G12(1,s). � Lemma 2.5. For t,s ∈ [0, 1], the Green’s function H1(t,s) given in (2.7) satisfies the following inequalities (Q1) H1(t,s) ≥ 0, (Q2) H1(t,s) ≤ H1(s,s). For details refer to [20]. Lemma 2.6. Let ξ1 ∈ ( 14, 3 4 ). Then the Green’s function H1(t,s) holds the inequality, (2.8) min t∈I H1(t,s) ≥ ϑ∗1(s)H1(s,s), for 0 < s < 1, where ϑ∗1(s) =   [ 3 4 (1 −s)]β1−1 − ( 3 4 −s)β1−1 [s(1 −s)]β1−1 , s ∈ (0,ξ1], 1 (4s)β1−1 , s ∈ [ξ1, 1). For details refer to [20]. Lemma 2.7. Let g1(t) ∈ C[0, 1], then the fractional order differential equation, (2.9) Dα2 0+ y(t) + g1(t) = 0, t ∈ (0, 1), satisfying (2.10) y(j)(0) = 0, j = 0, 1, · · ·,n− 2, y(n−2)(1) = 0, has a unique solution, y(t) = ∫ 1 0 G2(t,s)g1(s)ds, where G2(t,s) is the Green’s function for the problem (2.9)-(2.10) and is given by (2.11) G2(t,s) = { G21(t,s), 0 ≤ t ≤ s ≤ 1, G22(t,s), 0 ≤ s ≤ t ≤ 1, G21(t,s) = tα2−1(1 −s)α2−n+1 Γ(α2) , G22(t,s) = tα2−1(1 −s)α2−n+1 − (t−s)α2−1 Γ(α2) . For details refer to [19]. EXISTENCE OF POSITIVE SOLUTIONS FOR (p,q)-LAPLACIAN FBVPS 59 Lemma 2.8. Let g2(t) ∈ C[0, 1]. Then the fractional order differential equation, (2.12) D β2 0+ ( φq ( Dα2 0+ y(t) )) = g2(t), t ∈ (0, 1), satisfying (2.13) φq ( Dα2 0+ y(0) ) = 0, φq ( Dα2 0+ y(1) ) = 0, has a unique solution, y(t) = ∫ 1 0 G2(t,s)φp (∫ 1 0 H2(s,τ)g2(τ)dτ ) ds, where (2.14) H2(t,s) =   [ t(1 −s) ]β2−1 Γ(β2) , 0 ≤ t ≤ s ≤ 1,[ t(1 −s) ]β2−1 − (t−s)β2−1 Γ(β2) , 0 ≤ s ≤ t ≤ 1. Proof. Proof is similar to Lemma 2.2. � Lemma 2.9. The Green’s function G2(t,s) given in (2.11) is nonnegative, for all (t,s) ∈ [0, 1] × [0, 1]. Proof. Proof is similar to Lemma 2.3. � Lemma 2.10. For I = [ 1 4 , 3 4 ] , the Green’s function G2(t,s) given in (2.11) satisfies the following inequalities (C1) G2(t,s) ≤ G2(1,s), for all (t,s) ∈ [0, 1] × [0, 1], (C2) G2(t,s) ≥ (1 4 )α2−1 G2(1,s), for all (t,s) ∈ I × [0, 1]. Proof. Proof is similar to Lemma 2.4. � Lemma 2.11. For t,s ∈ [0, 1], the Green’s function H2(t,s) given in (2.14) satisfies the following inequalities (D1) H2(t,s) ≥ 0, (D2) H2(t,s) ≤ H2(s,s). For details refer to [20]. Lemma 2.12. Let ξ2 ∈ ( 14, 3 4 ). Then the Green’s function H2(t,s) holds the inequality, (2.15) min t∈I H2(t,s) ≥ ϑ∗2(s)H2(s,s), for 0 < s < 1, where (2.16) ϑ∗2(s) =   [ 3 4 (1 −s)]β2−1 − ( 3 4 −s)β2−1 [s(1 −s)]β2−1 , s ∈ (0,ξ2], 1 (4s)β2−1 , s ∈ [ξ2, 1). For details refer to [20]. 60 PRASAD, KRUSHNA AND WESEN 3. Existence of Positive Solutions In this section, we establish sufficient conditions for the existence of at least three positive solutions for a system of (p,q)-Laplacian fractional order boundary value problem (1.1)-(1.4), by using five functionals fixed point theorem. Let γ,β,θ be nonnegative continuous convex functionals on P and α,ψ be nonnegative continuous concave functionals on P , then for nonnegative numbers h′,a′,b′,d′ and c′, convex sets are defined. P(γ,c′) = { y ∈ P : γ(y) < c′ } , P(γ,α,a′,c′) = { y ∈ P : a′ ≤ α(y); γ(y) ≤ c′ } , Q(γ,β,d′,c′) = { y ∈ P : β(y) ≤ d′; γ(y) ≤ c′ } , P(γ,θ,α,a′,b′,c′) = { y ∈ P : a′ ≤ α(y); θ(y) ≤ b′; γ(y) ≤ c′ } , Q(γ,β,ψ,h′,d′,c′) = { y ∈ P : h′ ≤ ψ(y); β(y) ≤ d′; γ(y) ≤ c′ } . In establishing the positive solutions for a coupled system of (p,q)-Laplacian fractional order boundary value problem (1.1)-(1.4), the following so called Five Functionals Fixed Point Theorem is fundamental. Theorem 3.1. [3] Let P be a cone in the real Banach space B. Suppose α and ψ are nonnegative continuous concave functionals on P and γ,β,θ are nonnegative continuous convex functionals on P, such that for some positive numbers c′ and e′, α(y) ≤ β(y) and ‖ y ‖≤ e′γ(y), for all y ∈ P(γ,c′). Suppose further that T : P(γ,c′) → P(γ,c′) is completely continuous and there exist constants h′,d′,a′ and b′ ≥ 0 with 0 < d′ < a′ such that each of the following is satisfied. (B1) { y ∈ P(γ,θ,α,a′,b′,c′) : α(y) > a′ } 6= ∅ and α(Ty) > a′ for y ∈ P(γ,θ,α,a′,b′,c′), (B2) { y ∈ Q(γ,β,ψ,h′,d′,c′) : β(y) > d′ } 6= ∅ and β(Ty) > d′ for y ∈ Q(γ,β,ψ,h′,d′,c′), (B3) α(Ty) > a′ provided y ∈ P(γ,α,a′,c′) with θ(Ty) > b′, (B4) β(Ty) < d′ provided y ∈ Q(γ,β,ψ,h′,d′,c′) with ψ(Ty) < h′. Then T has at least three fixed points y1,y2,y3 ∈ P(γ,c′) such that β(y1) < d′,a′ < α(y2) and d′ < β(y3) with α(y3) < a ′. Consider the Banach space B = E ×E, where E = { x : x ∈ C[0, 1] } equipped with the norm ‖(x,y)‖ = ‖x‖0 + ‖y‖0, for (x,y) ∈ B and the norm is defined as ‖x‖0 = max t∈[0,1] |x(t)|. Define a cone P ⊂ B by P = { (x,y) ∈ B : x(t) ≥ 0, y(t) ≥ 0, t ∈ [0, 1] and min t∈I [ x(t) + y(t) ] ≥ η‖(x,y)‖ } , where (3.1) η = min {(1 4 )α1−1 , (1 4 )α2−1} . EXISTENCE OF POSITIVE SOLUTIONS FOR (p,q)-LAPLACIAN FBVPS 61 Define the nonnegative continuous concave functionals α,ψ and the nonnegative contin- uous convex functionals β,θ,γ on P by α(x,y) = min t∈I { |x| + |y| } ,ψ(x,y) = min t∈I1 { |x| + |y| } , γ(x,y) = max t∈[0,1] { |x| + |y| } ,β(x,y) = max t∈I1 { |x| + |y| } ,θ(x,y) = max t∈I { |x| + |y| } , where I1 = [ 1 3 , 2 3 ] . For any (x,y) ∈ P , (3.2) α(x,y) = min t∈I { |x| + |y| } ≤ max t∈I1 { |x| + |y| } = β(x,y), (3.3) ‖(x,y)‖≤ 1 η min t∈I { |x| + |y| } ≤ 1 η max t∈[0,1] { |x| + |y| } = 1 η γ(x,y). Let (3.4) ϑ∗(s) = min { ϑ∗1(s),ϑ ∗ 2(s) } . Define L = min { 1∫ 1 0 G1(1,s)φq (∫ 1 0 H1(τ,τ)dτ ) ds , 1∫ 1 0 G2(1,s)φp (∫ 1 0 H2(τ,τ)dτ ) ds } , and M = max { 1∫ s∈I ηG1(1,s)φq (∫ τ∈I ϑ ∗(τ)H1(τ,τ)dτ ) ds , 1∫ s∈I ηG2(1,s)φp (∫ τ∈I ϑ ∗(τ)H2(τ,τ)dτ ) ds } . Theorem 3.2. Suppose there exist 0 < a′ < b′ < b′ η < c′ such that f1,f2 satisfies the following conditions: (A1) { f1 ( t,x(t),y(t) ) < φp (a′L 2 ) and f2 ( t,x(t),y(t) ) < φq (a′L 2 ) , t ∈ [0, 1] and x,y ∈ [ ηa′,a′ ] , (A2)   f1 ( t,x(t),y(t) ) > φp (b′M 2 ) and f2 ( t,x(t),y(t) ) > φq (b′M 2 ) , t ∈ I and x,y ∈ [ b′, b′ η ] , (A3)   f1 ( t,x(t),y(t) ) < φp (c′L 2 ) and f2 ( t,x(t),y(t) ) < φq (c′L 2 ) , t ∈ [0, 1] and x,y ∈ [ 0,c′ ] . Then the (p,q)-Laplacian fractional order boundary value problem (1.1)-(1.4) has at least three positive solutions, (x1,x2), (y1,y2) and (z1,z2) such that β(x1,x2) < a ′, b′ < α(y1,y2) and a ′ < β(z1,z2) with α(z1,z2) < b ′. 62 PRASAD, KRUSHNA AND WESEN Proof. Let T1,T2 : P → E and T : P → B be the operators defined by  T1(x,y)(t) = ∫ 1 0 G1(t,s)φq (∫ 1 0 H1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds, T2(x,y)(t) = ∫ 1 0 G2(t,s)φp (∫ 1 0 H2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds, and T(x,y)(t) = ( T1(x,y)(t),T2(x,y)(t) ) , for (x,y) ∈ B. It is obvious that a fixed point of T is the solution of the fractional order boundary value problem (1.1)-(1.4). Three fixed points of T are sought. First, it is shown that T : P → P . Let (x,y) ∈ P . Clearly, T1 ( x,y ) (t) ≥ 0 and T2 ( x,y ) (t) ≥ 0, for t ∈ [0, 1]. Also for (x,y) ∈ P,  ‖T1(x,y)‖0 ≤ ∫ 1 0 G1(1,s)φq (∫ 1 0 H1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds, ‖T2(x,y)‖0 ≤ ∫ 1 0 G2(1,s)φp (∫ 1 0 H2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds, and min t∈I T1(x,y)(t) = min t∈I ∫ 1 0 G1(t,s)φq (∫ 1 0 H1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds ≥ η ∫ 1 0 G1(1,s)φq (∫ 1 0 H1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds ≥ η‖T1(x,y)‖0. Similarly, min t∈I T2 ( x,y ) (t) ≥ η‖T2 ( x,y ) ‖0. Therefore, min t∈I { T1(x,y)(t) + T2(x,y)(t) } ≥ η‖T1(x,y)‖0 + η‖T2(x,y)‖0 = η ( ‖T1(x,y)‖0 + ‖T2(x,y)‖0 ) = η ∥∥∥(T1(x,y),T2(x,y))∥∥∥ = η‖T(x,y)‖. Hence T(x,y) ∈ P and so T : P → P. Moreover the operator T is completely continuous. From (3.2) and (3.3), for each (x,y) ∈ P, α(x,y) ≤ β(x,y) and ‖(x,y)‖ ≤ 1 η γ(x,y). It is shown that T : P(γ,c′) → P(γ,c′). Let (x,y) ∈ P(γ,c′). Then 0 ≤ |x| + |y| ≤ c′. The condition (A3) is used to obtain γ ( T(x,y)(t) ) = max t∈[0,1] [∫ 1 0 G1(t,s)φq (∫ 1 0 H1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds+∫ 1 0 G2(t,s)φp (∫ 1 0 H2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds ] ≤ ∫ 1 0 G1(t,s)φq (∫ 1 0 H1(s,τ)φp (c′L 2 ) dτ ) ds+∫ 1 0 G2(t,s)φp (∫ 1 0 H2(s,τ)φq (c′L 2 ) dτ ) ds EXISTENCE OF POSITIVE SOLUTIONS FOR (p,q)-LAPLACIAN FBVPS 63 < c′L 2 ∫ 1 0 G1(1,s)φq (∫ 1 0 H1(τ,τ)dτ ) ds+ c′L 2 ∫ 1 0 G2(1,s)φp (∫ 1 0 H2(τ,τ)dτ ) ds < c′ 2 + c′ 2 = c′. Therefore T : P(γ,c′) → P(γ,c′). Now the conditions (B1) and (B2) of Theorem 3.1 are to be verified. It is obvious that b′ ( η + 1 ) 2η ∈ { (x,y) ∈ P ( γ,θ,α,b′, b′ η ,c′ ) : α(x,y) > b′ } 6= ∅ and ηa′ + a′ 2 ∈ { (x,y) ∈ Q ( γ,β,ψ,ηa′,a′,c′ ) : β(x,y) < a′ } 6= ∅. Next, let (x,y) ∈ P ( γ,θ,α,b′, b′ η ,c′ ) or (x,y) ∈ Q ( γ,β,ψ,ηa′,a′,c′ ) . Then, b′ ≤ { |x(t)| + |y(t)| } ≤ b′ η and ηa′ ≤ { |x(t)| + |y(t)| } ≤ a′. Now the condition (A2) is applied to get α ( T(x,y)(t) ) = min t∈I [∫ 1 0 G1(t,s)φq (∫ 1 0 H1(s,τ)f1 ( τ,x(τ),y(τ))dτ ) ds+∫ 1 0 G2(t,s)φp (∫ 1 0 H2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds ] ≥ η [∫ 1 0 G1(1,s)φq (∫ 1 0 ϑ∗(τ)H1(τ,τ)φp (b′M 2 ) dτ ) ds+∫ 1 0 G2(1,s)φp (∫ 1 0 ϑ∗(τ)H2(τ,τ)φq (b′M 2 ) dτ ) ds ] > b′M 2 ∫ s∈I ηG1(1,s)φq (∫ τ∈I ϑ∗(τ)H1(τ,τ)dτ ) ds+ b′M 2 ∫ s∈I ηG2(1,s)φp (∫ τ∈I ϑ∗(τ)H2(τ,τ)dτ ) ds ≥ b′ 2 + b′ 2 = b′. Clearly the condition (A1) leads to β ( T(x,y)(t) ) = max t∈I1 [∫ 1 0 G1(t,s)φq (∫ 1 0 H1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds+∫ 1 0 G2(t,s)φp (∫ 1 0 H2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds ] ≤ ∫ 1 0 G1(1,s)φq (∫ 1 0 H1(s,τ)φp (a′L 2 ) dτ ) ds+∫ 1 0 G2(1,s)φp (∫ 1 0 H2(s,τ)φq (a′L 2 ) dτ ) ds 64 PRASAD, KRUSHNA AND WESEN < a′L 2 ∫ 1 0 G1(1,s)φq (∫ 1 0 H1(τ,τ)dτ ) ds+ a′L 2 ∫ 1 0 G2(1,s)φp (∫ 1 0 H2(τ,τ)dτ ) ds ≤ a′ 2 + a′ 2 = a′. To see that (B3) is satisfied, let (x,y) ∈ P ( γ,α,b′,c′ ) with θ ( T(x,y)(t) ) > b′ η . Then α ( T(x,y)(t) ) = min t∈I [∫ 1 0 G1(t,s)φq (∫ 1 0 H1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds+∫ 1 0 G2(t,s)φp (∫ 1 0 H2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds ] ≥ η [∫ 1 0 G1(1,s)φq (∫ 1 0 H1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds+∫ 1 0 G2(1,s)φp (∫ 1 0 H2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds ] ≥ η max t∈[0,1] [∫ 1 0 G1(t,s)φq (∫ 1 0 H1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds+∫ 1 0 G2(t,s)φp (∫ 1 0 H2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds ] ≥ η max t∈I [∫ 1 0 G1(t,s)φq (∫ 1 0 H1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds+∫ 1 0 G2(t,s)φq (∫ 1 0 H2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds ] = ηθ ( T(x,y)(t) ) > b′. Finally it is shown that (B4) holds. Let (x,y) ∈ Q ( γ,β,a′,c′ ) with ψ ( T(x,y) ) < ηa′. Then we have β ( T(x,y)(t) ) = max t∈I1 [∫ 1 0 G1(t,s)φq (∫ 1 0 H1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds+∫ 1 0 G2(t,s)φp (∫ 1 0 H2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds ] EXISTENCE OF POSITIVE SOLUTIONS FOR (p,q)-LAPLACIAN FBVPS 65 ≤ max t∈[0,1] [∫ 1 0 G1(t,s)φq (∫ 1 0 H1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds+∫ 1 0 G2(t,s)φp (∫ 1 0 H2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds ] ≤ 1 η min t∈I [∫ 1 0 G1(t,s)φq (∫ 1 0 H1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds+∫ 1 0 G2(t,s)φp (∫ 1 0 H2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds ] ≤ 1 η min t∈I1 [∫ 1 0 G1(t,s)φq (∫ 1 0 H1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds+∫ 1 0 G2(t,s)φp (∫ 1 0 H2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds ] = 1 η ψ ( T(x,y)(t) ) < a′. It is been proved that all the conditions of Theorem 3.1 are satisfied. Therefore the system of (p,q)-Laplacian fractional order boundary value problem (1.1)-(1.4) has at least three positive solutions (x1,x2), (y1,y2) and (z1,z2) such that β(x1,x2) < a ′,b′ < α(y1,y2) and a′ < β(z1,z2) with α(z1,z2) < b ′. This completes the proof. � 4. Example In this section, as an application, the results are demonstrated with an example. Consider the system of (p,q)-Laplacian fractional order differential equations (4.1) D1.80+ ( φp ( D3.80+ x(t) )) = f1(t,x,y), t ∈ (0, 1), (4.2) D1.70+ ( φq ( D3.90+ y(t) )) = f2(t,x,y), t ∈ (0, 1), satisfying the boundary conditions (4.3) x(0) = x′(0) = x′′(0) = 0 and x′′(1) = 0, φp ( D3.8 0+ x(0) ) = φp ( D3.8 0+ x(1) ) = 0, } (4.4) y(0) = y′(0) = y′′(0) = 0 and y′′(1) = 0, φq ( D3.9 0+ y(0) ) = φq ( D3.9 0+ y(1) ) = 0, } where f1(t,x,y) =   e2t 57 + cos(x + y) 9 + 11(x + y)3 9 , 0 ≤ x + y ≤ 4, cos(x + y) 9 + e2t 57 + 5632 9 , x + y > 4, f2(t,x,y) =   sin(x + y) 9 + 11(x + y)3 9 + e2t 56 , 0 ≤ x + y ≤ 4, e2t 56 + sin(x + y) 9 + 5632 9 , x + y > 4. 66 PRASAD, KRUSHNA AND WESEN Clearly fi, for i = 1, 2 are continuous and increasing on [0,∞). Let p = 2. By direct calculations, one can determine η = 0.0205, L = 27.9632 and M = 314.5214. Choosing a′ = 0.5,b′ = 4 and c′ = 200, then 0 < a′ < b′ < b′ η < c′ and f1,f2 satisfies (a) { f1 ( t,x,y ) < 0.5091 = φp (a′L 2 ) and f2 ( t,x,y ) < 0.5091 = φq (a′L 2 ) , t ∈ [0, 1] and x,y ∈ [ ηa′,a′ ] = [0.0103, 0.05], (b)   f1 ( t,x,y ) > 1356 = φp (b′M 2 ) and f2 ( t,x,y ) > 1356 = φq (b′M 2 ) , t ∈ I = [0.25, 0.75] and x,y ∈ [ b′, b′ η ] = [4, 195.12], (c)   f1 ( t,x,y ) < 2796.32 = φp (c′L 2 ) and f2 ( t,x,y ) < 2796.32 = φq (c′L 2 ) , t ∈ [0, 1] and x,y ∈ [ 0,c′ ] = [0, 200]. Then all the conditions of Theorem 3.2 are satisfied. Thus by Theorem 3.2, the (p,q)- Laplacian fractional order boundary value problem (4.1)-(4.4) has at least three positive solutions. References [1] R. P. Agarwal, D. O’Regan and P. J. Y. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999. [2] D. R. Anderson and J. M. Davis, Multiple positive solutions and eigenvalues for third order right focal boundary value problems, J. Math. Anal. Appl., 267(2002), 135–157. [3] R. I. Avery, A generalization of the Leggett-Williams fixed point theorem, Math. Sci. Res. Hot-line, 3(1999), 9–14. [4] R. I. Avery and J. Henderson, Existence of three positive pseudo-symmetric solutions for a one- dimensional p-Laplacian, J. Math. Anal. Appl., 277(2003), 395–404. [5] C. Bai, Existence of positive solutions for boundary value problems of fractional functional differential equations, Elec. J. Qual. Theory Diff. Equ., 30(2010), 1–14. [6] Z. Bai and H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311(2005), 495-505. [7] G. Chai, Positive solutions for boundary value problem of fractional differential equation with p- Laplacian operator, Bound. Value Probl., 2012(2012), 1–18. [8] T. Chen and W. Liu, An anti-periodic boundary value problem for the fractional differential equation with a p-Laplacian operator, Appl. Math. Lett., 25(2012), 1671–1675. [9] R. Dehghani and K. Ghanbari, Triple positive solutions for boundary value problem of a nonlinear fractional differential equation, Bulletin of the Iranian Mathematical Society, 33(2007), 1–14. [10] L. Diening, P. Lindqvist and B. Kawohl, Mini-Workshop: The p-Laplacian Operator and Applications, Oberwolfach Reports, 10(2013) 433–482. [11] L. H. Erbe and H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc., 120(1994), 743–748. [12] C. Goodrich, Existence of a positive solution to systems of differential equations of fractional order, Comput. Math. Appl., 62(2011), 1251–1268. [13] D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Acadamic Press, San Diego, 1988. [14] J. Henderson and S. K. Ntouyas, Positive solutions for systems of nonlinear boundary value problems, Nonlinear Stud., 15(2008), 51-60. [15] A. A. Kilbas, H. M. Srivasthava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science, Amsterdam, 2006. [16] L. Kong and J. Wang, Multiple positive solutions for the one-dimensional p-Laplacian, Nonlinear Anal., 42(2000), 1327–1333. [17] I. Podulbny, Fractional Diffrential Equations, Academic Press, San Diego, 1999. EXISTENCE OF POSITIVE SOLUTIONS FOR (p,q)-LAPLACIAN FBVPS 67 [18] K. R. Prasad and B. M. B. Krushna, Multiple positive solutions for a coupled system of p-Laplacian fractional order two-point boundary value problems, Int. J. Differ. Equ., 2014(2014), Article ID 485647, 1–10. [19] K. R. Prasad and B. M. B. Krushna, Multiple positive solutions for the system of (n, p)-type fractional order boundary value problems, Bull. Int. Math. Virtual Inst., 5(2015), 1–12. [20] K. R. Prasad and B. M. B. Krushna, Solvability of p-Laplacian fractional higher order two-point bound- ary value problems, Commun. Appl. Anal., 19(2015), 659–678. [21] X. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett., 22(2009), 64–69. [22] C. Yang, J. Yan, Positive solutions for third order Sturm–Liouville boundary value problems with p-Laplacian, Comput. Math. Appl., 59(2010), 2059–2066. 1Department of Applied Mathematics, Andhra University, Visakhapatnam, 530 003, India 2Department of Mathematics, MVGR College of Engineering, Vizianagaram, 535 005, India 3Department of Mathematics, Jimma University, Jimma, Oromia, 378, Ethiopia ∗Corresponding author