This paper establishes the existence of at least three positive solutions for a coupled system of p-Laplacian fractional order two-point boundary value problems, , t ∈ (0,1), , t ∈ (0,1), , , , , , , by applying five functionals fixed point theorem.

1. Introduction

The theory of differential equations offers a broad mathematical basis to understand the problems of modern society which are complex and interdisciplinary by nature. Fractional order differential equations have gained importance due to their applications to almost all areas of science, engineering, and technology. Among all the theories, the most applicable operator is the classical p-Laplacian, given by ϕ_p(s) = |s|^p−2s, p > 1. These types of problems have a wide range of applications in physics and related sciences such as biophysics, plasma physics, and chemical reaction design.

The positive solutions of boundary value problems associated with ordinary differential equations were studied by many authors [1–3] and extended to p-Laplacian boundary value problems [4–6]. Later, these results are further extended to fractional order boundary value problems [7–15] by applying various fixed point theorems on cones. Recently, researchers are concentrating on the theory of fractional order boundary value problems associated with p-Laplacian operator.

In 2012, Chai [16] investigated the existence and multiplicity of positive solutions for a class of boundary value problem of fractional differential equation with p-Laplacian operator,

(1)

by means of the fixed point theorem on cones.

This paper is concerned with the existence of positive solutions for a coupled system of p-Laplacian fractional order boundary value problems:

(2)

(3)

(4)

(5)

where ϕ_p(s) = |s|^p−2s, p > 1,

, 1/p + 1/q = 1, γ, δ are positive real numbers, 2 < α_i ≤ 3, 1 < β_i, q_i ≤ 2, f_i : [0,1] × ℝ² → ℝ⁺ are continuous functions, and

, for i = 1,2 are the standard Riemann-Liouville fractional order derivatives.

The rest of the paper is organized as follows. In Section 2, the Green functions for the homogeneous BVPs corresponding to (2), (4) are constructed and the bounds for the Green functions are estimated. In Section 3, sufficient conditions for the existence of at least three positive solutions for a coupled system of p-Laplacian fractional order BVP (2)–(5) 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 Green functions for the homogeneous BVPs are constructed and the bounds for the Green functions are estimated, which are essential to establish the main results.

Let G₁(t, s) be Green’s function for the homogeneous BVP:

(6)

(7)

Lemma 1. Let d = δΓ(α₁) + γΓ(α₁ − q₂) ≠ 0. If h ∈ C[0,1], then the fractional order BVP

(8)

with (7) has a unique solution

(9)

where

(10)

(11)

Proof. Let be the solution of fractional order BVP (8), (7). Then

(12)

and hence

(13)

Using the boundary conditions (7), c₁, c₂, and c₃ are determined as

(14)

Hence, the unique solution of (8), (7) is

(15)

Lemma 2. Let y(t) ∈ C[0,1] and 2 < α₁ ≤ 3, 1 < β₁ ≤ 2. Then the fractional order BVP

(16)

with (4) has a unique solution

(17)

where

(18)

Proof. An equivalent integral equation for (16) is given by

(19)

Using the conditions

, c₁, and c₂ are determined as

and c₂ = 0. Then,

(20)

Therefore,

(21)

Consequently,

(22)

Hence,

is the solution of fractional order BVP (16) and (4).

Lemma 3. Assume that δ(q₂ − 1) > γΓ(α₁ − q₂)/Γ(α₁). Then Green’s function G₁(t, s) satisfies the following inequalities:

(i)
G₁(t, s) ≥ 0, for all (t, s)∈[0,1]×[0,1],
(ii)
G₁(t, s) ≤ G₁(1, s), for all (t, s)∈[0,1]×[0,1],
(iii)
, for all (t, s) ∈ I × [0,1],

where I = [1/4,3/4].

Proof. Green’s function G₁(t, s) is given in (10). For 0 ≤ t ≤ s ≤ 1,

(23)

For 0 ≤ s ≤ t ≤ 1,

(24)

Hence, the inequality (i) is proved. For 0 ≤ t ≤ s ≤ 1,

(25)

Therefore, G₁₁(t, s) is increasing with respect to t, which implies that G₁₁(t, s) ≤ G₁₁(1, s). Now, for 0 ≤ s ≤ t ≤ 1,

(26)

Therefore, G₁₂(t, s) is increasing with respect to t, which implies that G₁₂(t, s) ≤ G₁₂(1, s). Hence, the inequality (ii) is proved. Now, the inequality (iii) can be established.

Let 0 ≤ t ≤ s ≤ 1 and t ∈ I. Then

(27)

Let 0 ≤ s ≤ t ≤ 1 and t ∈ I. Then

(28)

Hence the inequality (iii) is proved.

Lemma 4. For t, s ∈ [0,1], Green’s function H₁(t, s) satisfies the following inequalities:

(i)
H₁(t, s) ≥ 0,
(ii)
H₁(t, s) ≤ H₁(s, s).

Proof. Green’s function H₁(t, s) is given in (18). Clearly, it is observed that, for 0 ≤ t ≤ s ≤ 1, H₁(t, s) ≥ 0.

For 0 ≤ s ≤ t ≤ 1,

(29)

Hence, the inequality (i) is proved. Now we establish the inequality (ii), for 0 ≤ t ≤ s ≤ 1,

(30)

Therefore, H₁(t, s) is increasing with respect to t, for s ∈ [0,1), which implies that H₁(t, s) ≤ H₁(s, s). Similarly, it can be proved that H₁(t, s) ≤ H₁(s, s) for 0 ≤ s ≤ t ≤ 1. Hence the inequality (ii) is proved.

Lemma 5. Green’s function H₁(t, s) satisfies the following inequality: (A) there exists a positive function such that

(31)

Proof. Since H₁(t, s) is monotonic function, for all t, s ∈ [0,1], we have

(32)

From (i) of Lemma 4, H₁(t, s) ≥ 0, for t, s ∈ [0,1]. For s ∈ (0,1/4), H₁(t, s) is increasing with respect to t for

and decreasing with respect to t for

. For s ∈ (1/4,1), H₁(t, s) is decreasing with respect to t for s ≤ t and increasing with respect to t for s ≥ t. If we define

(33)

Then,

(34)

where

(35)

and ξ ∈ (1/4,3/4) satisfy the equation

. In particular, ξ = 0.5 if β₁ = 2; ξ → 0.5 as β₁ → 2; and ξ → 0.76 as β₁ → 1. Hence the inequality in (31) holds.

In a similar manner, the results of the Green functions G₂(t, s) and H₂(t, s) for the homogeneous BVPs corresponding to the fractional order BVP (3) and (5) are obtained.

Remark 6. Consider the following.

G₁(t, s) ≥ ηG₁(1, s) and G₂(t, s) ≥ ηG₂(1, s), for all (t, s) ∈ I × [0,1], where .

Remark 7. Consider the following.

H₁(t, s) ≥ γ^*(s)H₁(s, s) and H₂(t, s) ≥ γ^*(s)H₂(s, s), for all (t, s) ∈ I × [0,1], where .

3. Existence of Multiple Positive Solutions

In this section, the existence of at least three positive solutions for a coupled system of p-Laplacian fractional order BVP (2)–(5) is established by using five functionals fixed point theorem.

Let γ, β, θ be nonnegative continuous convex functionals on P and let α, ψ be nonnegative continuous concave functionals on P; then for nonnegative numbers h^′, a^′, b^′, d^′, and c^′, convex sets are defined:

(36)

In obtaining multiple positive solutions of the p-Laplacian fractional order BVP (2)-(5), the following so-called five functionals fixed point theorem is fundamental.

Theorem 8 (see [17].)Let P be a cone in the real Banach space B. Suppose that 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 . Suppose further that 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^′} ≠ ∅ 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^′} ≠ ∅ and β(Ty) > d^′ for y ∈ Q(γ, β, ψ, h^′, d^′, c^′),
(B3)
α(Ty) > a^′ provided that y ∈ P(γ, α, a^′, c^′) with θ(Ty) > b^′,
(B4)
β(Ty) < d^′ provided that y ∈ Q(γ, β, ψ, h^′, d^′, c^′) with ψ(Ty) < h^′.

Then, T has at least three fixed points

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 = {u : u ∈ C[0,1]} equipped with the norm ∥(u, v)∥ = ∥u∥₀ + ∥v∥₀, for (u, v) ∈ B and the norm, is defined as

(37)

Define a cone P ⊂ B by

(38)

Define the nonnegative continuous concave functionals α, ψ and the nonnegative continuous convex functionals β, θ, γ on P by

(39)

where I₁ = [1/3,2/3]. For any (u, v) ∈ P,

(40)

Let

(41)

Theorem 9. Suppose that there exist 0 < a^′ < b^′ < b^′/η < c^′ such that f_i, for i = 1,2 satisfies the following conditions:

(A1)
f_i(t, u(t), v(t)) < ϕ_p(a^′L/2), t ∈ [0,1] and u, v ∈ [ηa^′, a^′],
(A2)
f_i(t, u(t), v(t)) > ϕ_p(b^′M/2), t ∈ I and u, v ∈ [b^′, b^′/η],
(A3)
f_i(t, u(t), v(t)) < ϕ_p(c^′L/2), t ∈ [0,1] and u, v ∈ [0, c^′].

Then, the fractional order BVP (2)–(5) 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^′.

Proof. Let T₁, T₂ : P → E and T : P → B be the operators defined by

(42)

It is obvious that a fixed point of T is the solution of the fractional order BVP (2)–(5). Three fixed points of T are sought. First, it is shown that T : P → P. Let (u, v) ∈ P. Clearly, T₁(u, v)(t) ≥ 0 and T₂(u, v)(t) ≥ 0, for t ∈ [0,1]. Also, for (u, v) ∈ P,

(43)

Similarly,

. Therefore,

(44)

Hence, T(u, v) ∈ P and so T : P → P. Moreover, T is completely continuous operator. From (40), for each (u, v) ∈ P, α(u, v) ≤ β(u, v), and ∥(u, v)∥ ≤ (1/η)γ(u, v). It is shown that

. Let

. Then 0 ≤ |u| + |v| ≤ c^′. Condition (A3) is used to obtain

(45)

Therefore

. Now conditions (B1) and (B2) of Theorem 8 are to be verified. It is obvious that

(46)

Next, let (u, v) ∈ P(γ, θ, α, b^′, b^′/η, c^′) or (u, v) ∈ Q(γ, β, ψ, ηa^′, a^′, c^′). Then, b^′ ≤ |u(t)| + |v(t)| ≤ b^′/η and ηa^′ ≤ |u(t)| + |v(t)| ≤ a^′. Now, condition (A2) is applied to get

(47)

Clearly, condition (A1) leads to

(48)

To see that (B3) is satisfied, let (u, v) ∈ P(γ, α, b^′, c^′) with θ(T(u, v)(t)) > b^′/η. Then

(49)

Finally, it is shown that (B4) holds. Let (u, v) ∈ Q(γ, β, a^′, c^′) with ψ(T(u, v)) < ηa^′. Then, we have

(50)

It has been proved that all the conditions of Theorem 8 are satisfied. Therefore, the fractional order BVP (2)–(5) 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 of the theorem.

4. Example

In this section, as an application, the result is demonstrated with an example.

Consider a coupled system of p-Laplacian fractional order BVP:

(51)

where

(52)

Then the Green functions G_i(t, s) and H_i(t, s), for i = 1,2, are given by

(53)

Clearly, the Green functions G_i(t, s) and H_i(t, s), for i = 1,2, are positive and f₁, f₂ are continuous and increasing on [0, ∞). By direct calculations, η = 0.08, p = 2, L = 33.16, and M = 1677.73. Choosing a^′ = 1, b^′ = 10 and c^′ = 900 and then 0 < a^′ < b^′ < b^′/η ≤ c^′ and f_i, for i = 1,2 satisfies

(i)
f_i(t, u, v) < 16.5845 = ϕ_p(a^′L/2), t ∈ [0,1] and u, v ∈ [0.08,1],
(ii)
f_i(t, u, v) > 8388.65 = ϕ_p(b^′M/2), t ∈ [1/4,3/4] and u, v ∈ [10,125],
(iii)
f_i(t, u, v) < 14926.09 = ϕ_p(c^′L/2), t ∈ [0,1] and u, v ∈ [0,900].

Then, all the conditions of Theorem 9 are satisfied. Therefore, it follows from Theorem 8 that the p-Laplacian fractional order BVP (51) has at least three positive solutions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors thank the referees for their valuable comments and suggestions.

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Multiple Positive Solutions for a Coupled System of p-Laplacian Fractional Order Two-Point Boundary Value Problems

Abstract

1. Introduction

2. Green Functions and Bounds

3. Existence of Multiple Positive Solutions

4. Example

Conflict of Interests

Acknowledgment

References

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References

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Multiple Positive Solutions for a Coupled System of p-Laplacian Fractional Order Two-Point Boundary Value Problems

Abstract

1. Introduction

2. Green Functions and Bounds

3. Existence of Multiple Positive Solutions

4. Example

Conflict of Interests

Acknowledgment

References

Citing Literature

References

Related

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