Positive solutions of semipositone singular fractional ...

Open Math. 2018; 16: 581–596

Open Mathematics Research Article Xinan Hao* and Huaqing Wang

Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions https://doi.org/10.1515/math-2018-0055 Received November 19, 2017; accepted April 19, 2018.

Abstract: In this paper, the existence of positive solutions for systems of semipositone singular fractional differential equations with a parameter and integral boundary conditions is investigated. By using fixed point theorem in cone, sufficient conditions which guarantee the existence of positive solutions are obtained. An example is given to illustrate the results. Keywords: Positive solutions, Semipositone, Singular fractional differential systems, Integral boundary conditions MSC: 26A33, 34A08, 34B18

1 Introduction The subject of fractional calculus has gained considerable popularity and importance during the past decades, mainly due to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. In recent years, fractional differential equations have been widely used in optics and thermal systems, electromagnetics, control engineering and robotic, and many other fields, see [1-6] and the references therein. The research on fractional differential equations is very important in both theory and applications. By using nonlinear analysis tools, some scholars established the existence, uniqueness, multiplicity and qualitative properties of solutions, we refer the readers to [7-20] and the references therein for fractional differential equations, and [21-33] for fractional differential systems. Boundary value problems (BVPs for short) with integral boundary conditions for ordinary differential equations represent a very interesting and important class of problems, and arise in the study of various biological, physical and chemical processes [34-37], such as heat conduction, thermo-elasticity, chemical engineering, underground water flow, and plasma physics. The existence of solutions or positive solutions for such class of problems has attracted much attention (see [38-52] and the references therein). In this paper, we study the systems of semipositone singualr fractional differential equations with a parameter and integral boundary conditions

*Corresponding Author: Xinan Hao: School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China, E-mail: [email protected] Huaqing Wang: Department of Mathematics, Jining University, Qufu, 273155, Shandong, China Open Access. © 2018 Hao and Wang, published by De Gruyter. NonCommercial-NoDerivs 4.0 License.

This work is licensed under the Creative Commons Attribution-

582 | X. Hao −β β β Dα 0+ (D 0+ u(t)) + λ f 1 (t, u(t), D 0+ u(t), v(t)) = 0,

Dγ0+−δ (Dδ0+ v(t)) + λf2 (t, u(t), v(t), Dδ0+ v(t)) = 0,

0 < t < 1, 0 < t < 1,

1

Dβ0+ u(0) = Dβ0++1 u(0) = 0,

Dβ0++1 u(1) = ∫ Dβ0+ u(s)dA(s),

(1)

0 1

Dδ0+ v(0) = Dδ0++1 v(0) = 0,

D0δ++1 v(1) = ∫ Dδ0+ v(s)dB(s), 0

−β Dα 0+ ,

Dβ0+ ,

Dγ0+−δ

Dδ0+

where and are the standard Riemann-Liouville fractional derivatives, λ > 0 is a parameter, 2 < α, γ ≤ 3, 0 < β , δ < 1, α − β > 2, γ − δ > 2. f1 , f2 ∶ (0, 1) × [0, +∞)3 → (−∞, +∞) are continuous and may be singular at t = 0, 1. A and B are nondecreasing functions of bunded variations, 1 β 1 δ ∫0 D0+ u(s)dA(s) and ∫0 D0+ v(s)dB(s) are Riemann-Stieltjes integrals. The study of nonlinear fractional differential systems is important as this kind of systems occur in various problems of applied mathematics. Recently, Wu et al. [26] considered the fractional differential systems involving nonlocal boundary conditions Dα 0+ u(t) + f (t, u(t), v(t)) = 0,

0 < t < 1, 1 < α ≤ 2,

Dβ0+ v(t) +

0 < t < 1, 1 < β ≤ 2,

g(t, u(t), v(t)) = 0, 1

u(0) = 0,

u(1) = ∫ u(s)dA(s),

(2)

0 1

v(0) = 0,

v(1) = ∫ v(s)dB(s), 0

β where Dα 0+ and D 0+ are the standard Riemann-Liouville fractional derivatives, A, B are nondecreas1 1 ing functions of bunded variations, ∫0 u(s)dA(s) and ∫0 v(s)dB(s) are Riemann-Stieltjes integrals, 2 f (t, x, y), g(t, x, y) ∶ (0, 1) × (0, ∞) → [0, ∞) are two continuous functions and may be singular at t = 0, 1 and x = y = 0. The existence of positive solutions is established by the upper and lower solutions technique 1 and Schauder fixed point theorem. For the special boundary conditions u(1) = ∫0 φ(s)u(s)ds, v(1) = 1 ∫0 ϕ(s)v(s)ds, where φ, ϕ ∈ L(0, 1) are nonnegative, Liu et al. [27] investigated the existence of a pair of positive solutions for nonlocal fractional differential systems (2) by constructing two cones and computing the fixed point index in product cone. For the case f = a(t)̃ f (t, u(t)), g = b(t)̃ g (t, v(t)), u(1) = 1 1 ∫0 φ(s)u(s)ds, v(1) = ∫0 ϕ(s)v(s)ds, Yang [28] established sufficient conditions for the existence and nonexistence of positive solutions to fractional differential systems (2) by the Banach fixed point theorem, nonlinear differentiation of Leray-Schauder type and the fixed point theorems of cone expansion and compression of norm type. In [29], Henderson, Luca and Tudorache discussed the systems of nonlinear fractional differential equations with integral boundary conditions

Dα 0+ u(t) + λ f (t, u(t), v(t)) = 0,

0 < t < 1, n − 1 < α ≤ n,

Dβ0+ v(t) + µg(t, u(t), v(t))

0 < t < 1, m − 1 < β ≤ m,

= 0,

1

u(0) = u′ (0) = ⋯ = u(n−2) (0) = 0,

u(1) = ∫ u(s)dA(s),

(3)

0 1

v(0) = v′ (0) = ⋯ = v(m−2) (0) = 0,

v(1) = ∫ v(s)dB(s) 0

β 2 where Dα 0+ and D 0+ are the standard Riemann-Liouville fractional derivatives, f , g ∶ [0, 1] × [0, ∞) → [0, ∞) are continuous. Under different combinations of superlinearity and sublinearity of the functions f and g, various existence and nonexistence results for positive solutions are derived in terms of different value of

Positive solutions of semipositone nonlocal fractional differential systems

| 583

λ and µ via the Guo-Krasnosel’skii fixed point theorem. For the multi-point boundary conditions u(1) = p q ∑i=1 a i u(ξi ), v(1) = ∑i=1 b i v(ηi ), Henderson and Luca [30] proved the existence theorems for the positive

solutions with respect to a cone for nonlinear fractional differential systems (3) by the Guo-Krasnosel’skii fixed point theorem. Under sufficient conditions on functions f and g, the authors [31] studied the existence and multiplicity of positive solutions of nonlinear fractional differential systems (3) when λ = µ = 1 and f =̃ f (t, v), g = ̃ g (t, u) by using the Guo-Krasnosel’skii fixed point theorem and some theorems from the fixed point index theory. In [32], Wang et al. investigated the fractional differential systems involving integral boundary conditions arising from the study of HIV infection models β Dα 0+ u(t) + λ f (t, u(t), D 0+ u(t), v(t)) = 0,

Dγ0+ v(t) + λg(t, u(t))

0 < t < 1, 2 < α ≤ 3,

0 < t < 1, 2 < γ ≤ 3,

= 0,

1

Dβ0+ u(0) = Dβ0++1 u(0) = 0,

Dβ0+ u(1) = ∫ Dβ0+ u(s)dA(s),

(4)

0 1

v(0) = v′ (0) = 0,

v(1) = ∫ v(s)dB(s), 0

β γ where λ > 0 is a parameter, 0 < β < 1, α − β > 2, Dα 0+ , D 0+ and D 0+ are the standard Riemann1 Liouville fractional derivatives, A, B are nondecreasing functions of bunded variations, ∫0 Dβ0+ u(s)dA(s) 1 3 and ∫0 v(s)dB(s) are Riemann-Stieltjes integrals, f ∶ (0, 1) × [0, +∞) → (−∞, +∞) and g ∶ (0, 1) × [0, +∞) → (−∞, +∞) are two continuous functions and may be singular at t = 0, 1. By using the fixed point theorem in cone, existence results of positive solutions for systems (4) are established. In [33], Jiang, Liu and Wu considered the following semipositone singular fractional differential systems:

Dα 0+ u(t) + p(t)f (t, u(t), v(t)) − q 1 (t) = 0,

0 < t < 1, 2 < α ≤ 3,

Dβ0+ v(t) +

0 < t < 1, 2 < β ≤ 3,

q(t)g(t, u(t), v(t)) − q2 (t) = 0, 1

u(0) = u′ (0) = 0,

u′ (1) = ∫ u(s)dA(s),

(5)

0 1

v(0) = v′ (0) = 0,

v′ (1) = ∫ v(s)dB(s), 0

β 2 where Dα 0+ and D 0+ are the standard Riemann-Liouville fractional derivatives, f , g ∶ [0, 1] × [0, ∞) → [0, ∞) are continuous, q1 , q2 ∶ (0, 1) → [0, +∞) are Lebesgue integrable, A, B are suitable functions of bounded 1 1 variation, ∫0 u(t)dA(t) and ∫0 v(t)dB(t) involving Stieltjes integrals with signed measures. The existence and multiplicity of positive solutions to systems (5) are obtained by using a well known fixed point theorem. It should be noted that the nonlinearity in most of the previous works needs to be nonnegative to get the positive solutions [22-32]. When the nonlinearity is allowed to take on both positive and negative values, such problems are called semipositone problems in the literature. Motivated by the works mentioned above, we consider the semipositone singular fractional differential systems (1). The existence of positive solutions is established by applying the fixed point theorem in cone. In comparison with previous works, this paper has several new features. Firstly, nonlinearities are allowed to change sign and tend to negative infinity. Secondly, systems (1) involves a parameter and f1 , f2 involve fractional derivatives of unknown functions. Finally, the nonlocal conditions are given by Riemann-Stieltjes integrals, which include two-point, three-point, multipoint and some nonlocal conditions as special cases.

584 | X. Hao

2 Preliminaries and lemmas For convenience of the reader, we present here some necessary definitions and properties about fractional calculus theory. Definition 2.1. The Riemann-Liouville fractional integral of order α of a function u ∶ (0, +∞) → (−∞, +∞) is given by t

I0α+ u(t) =

1 α−1 u(s)ds, ∫ (t − s) Γ (α) 0

provided the right-hand side is pointwise defined on (0, +∞). Definition 2.2 ([1,2]). The Riemann-Liouville fractional derivative of order α > 0 of a continuous function u ∶ (0, +∞) → (−∞, +∞) is given by t

Dα 0+ u(t) =

1 d n ( ) ∫ (t − s)n−α−1 u(s)ds, Γ (n − α) dt 0

where n = [α] + 1, [α] denotes the integer part of the number α, provided the right-hand side is pointwise defined on (0, +∞). Lemma 2.3 ([1,2]). If u ∈ L(0, 1), ρ > σ > 0 and n is a natural number, then I0ρ+ I0σ+ u(t) = I0ρ++σ u(t), Dσ0+ I0σ+ u(t) = u(t),

(

Dσ0+ I0ρ+ u(t) = I0ρ+−σ u(t),

d n σ σ + u(t). ) (D0+ u(t)) = D0n+ dt

Lemma 2.4 ([1,2]). Assume that u ∈ C(0, 1) ∩ L(0, 1) with a fractional derivative of order α > 0 that belongs to C(0, 1) ∩ L(0, 1). Then α−1 I0α+ Dα + c2 tα−2 + ⋯ + c N tα−N , 0+ u(t) = u(t) + c 1 t

for some c1 , c2 , ⋯, c N ∈ (−∞, +∞), where N is the smallest integer greater than or equal to α. Lemma 2.5 ([33]). Given h ∈ C(0, 1) ∩ L(0, 1), then the BVP −β Dα 0+ x(t) + h(t) = 0, ′

0 < t < 1, 2 < α − β ≤ 3, ′

x(0) = x (0) = 0,

x (1) = 0,

has a unique solution 1

x(t) = ∫ G∗1 (t, s)h(s)ds, 0

where G∗1 (t, s)

⎧ ⎪ tα−β −1 (1 − s)α−β −2 , 0 ≤ t ≤ s ≤ 1, 1 ⎪ = ⎨ α−β −1 Γ (α − β ) ⎪ ⎪ (1 − s)α−β −2 − (t − s)α−β −1 , 0 ≤ s ≤ t ≤ 1, ⎩t

and the Green function G∗1 (t, s) has the following properties: (1) G∗1 (t, s) > 0, t, s ∈ (0, 1). (2) k1 (t)G∗1 (1, s) ≤ G∗1 (t, s) ≤ G∗1 (1, s), t, s ∈ [0, 1]. , t, s ∈ [0, 1], where k1 (t) = tα−β −1 . (3) G∗1 (t, s) ≤ Γ k(1α(t) −β )


| 585

By Lemma 2.4, the unique solution of the BVP −β Dα 0+ x(t) = 0,

0 < t < 1,

′

x′ (1) = 1,

x(0) = x (0) = 0, is γ1 (t) =

tα−β −1 α−β −1 .

By [33], the Green function for the BVP −β Dα 0+ x(t) + h(t) = 0,

0 < t < 1, 1

′

x(0) = x (0) = 0,

′

x (1) = ∫ x(s)dA(s), 0

is given by G1 (t, s) = G∗1 (t, s) + where

1

γ1 (t) `1 (s), 1 − ȷ1

(6)

1

ȷ1 = ∫ γ1 (t)dA(t) ≠ 1,

∗

`1 (s) = ∫ G1 (t, s)dA(t),

0

s ∈ [0, 1].

0

Similarly, the Green function for the BVP Dγ0+−δ y(t) + h(t) = 0,

0 < t < 1, 1

y(0) = y′ (0) = 0,

y′ (1) = ∫ y(s)dB(s), 0

is given by G2 (t, s) = G∗2 (t, s) + where G∗2 (t, s) =

γ2 (t) `2 (s), 1 − ȷ2

(7)

γ −δ −1 ⎧ ⎪ (1 − s)γ −δ−2 , 0 ≤ t ≤ s ≤ 1, 1 ⎪t ⎨ γ −δ−1 Γ (γ − δ ) ⎪ ⎪ (1 − s)γ −δ−2 − (t − s)γ −δ−1 , 0 ≤ s ≤ t ≤ 1, ⎩t

1

1

ȷ2 = ∫ γ2 (t)dB(t) ≠ 1,

∗

`2 (s) = ∫ G2 (t, s)dB(t),

0

γ2 (t) =

0

tγ −δ−1 . γ−δ−1

Lemma 2.6 ([33]). The Green function G∗2 (t, s) has the following properties: (1) G∗2 (t, s) > 0, t, s ∈ (0, 1). (2) k2 (t)G∗2 (1, s) ≤ G∗2 (t, s) ≤ G∗2 (1, s), t, s ∈ [0, 1]. (3) G∗2 (t, s) ≤ Γk(2γ(t) , t, s ∈ [0, 1], where k2 (t) = tγ −δ−1 . −δ ) Lemma 2.7 ([33]). Let ȷ1 , ȷ2 ∈ [0, 1) and `1 (s), `2 (s) ≥ 0 for s ∈ [0, 1], the functions G1 (t, s) and G2 (t, s) given by (6) and (7) satisfy: (1) G i (t, s) ≥ G∗i (t, s) > 0, t, s ∈ (0, 1), i = 1, 2. (2) k i (t)G∗i (1, s) ≤ G i (t, s) ≤ ρi G∗i (1, s), t, s ∈ [0, 1], i = 1, 2. (3) G1 (t, s) ≤ Γ (αρ1−β ) k1 (t), G2 (t, s) ≤ Γ (ργ2−δ) k2 (t), t, s ∈ [0, 1], where 1

ρ1 = 1 +

∫0 dA(t) , 1 − ȷ1

1

ρ2 = 1 +

∫0 dB(t) . 1 − ȷ2

586 | X. Hao Now let us consider the following modified problem of systems (1) −β β δ Dα 0+ ω (t) + λ f 1 (t, I 0+ ω (t), ω (t), I 0+ z(t)) = 0,

Dγ0+−δ z(t) + λf2 (t, I0β+ ω (t), I0δ+ z(t), z(t)) 1 ′ ′

ω (0) = ω (0) = 0,

= 0,

0 < t < 1, 0 < t < 1, (8)

ω (1) = ∫ ω (s)dA(s), 0 1

′

z(0) = z (0) = 0,

′

z (1) = ∫ z(s)dB(s). 0

Lemma 2.8. If (ω , z) ∈ C[0, 1] × C[0, 1] is a positive solution of systems (8), then (I0β+ ω , I0δ+ z) is a positive solution of systems (1). Proof. Suppose (ω , z) ∈ C[0, 1] × C[0, 1] is a positive solution of systems (8), denote u(t) = I0β+ ω (t), v(t) = I0δ+ z(t), then −β β α−β β β α−β Dα 0+ (D 0+ u(t)) = D 0+ (D 0+ I 0+ ω (t)) = D 0+ ω (t) = − λf1 (t, I0β+ ω (t), ω (t), I0δ+ z(t)) = − λf1 (t, u(t), Dβ0+ u(t), v(t)), Dγ0+−δ (Dδ0+ v(t)) = Dγ0+−δ (Dδ0+ I0δ+ z(t)) = Dγ0+−δ z(t) = − λf2 (t, I0β+ ω (t), I0δ+ z(t), z(t)) = − λf2 (t, u(t), v(t), Dδ0+ v(t)), Dβ0+ u(0) = ω (0) = 0,

d (Dβ+ u(t))∣t=0 = ω ′ (0) = 0, dt 0

Dβ0++1 u(0) =

1

Dβ0++1 u(1) =

1

d (Dβ+ u(t))∣t=1 = ω ′ (1) = ∫ ω (s)dA(s) = ∫ Dβ0+ u(s)dA(s), dt 0 0

Dδ0+ v(0) = z(0) = 0,

0

d (Dδ0+ v(t))∣t=0 = z′ (0) = 0, dt

D0δ++1 v(0) =

1

Dδ0++1 v(1) =

1

d (Dδ0+ v(t))∣t=1 = z′ (1) = ∫ z(s)dB(s) = ∫ Dδ0+ v(s)dB(s). dt 0

0

On the other hand, if ω (t) > 0, z(t) > 0, by Definition 2.1, we have u(t) > 0, v(t) > 0, t ∈ (0, 1), then (u, v) = (I0β+ ω , I0δ+ z) is a positive solution of systems (1). We impose the following assumptions: (H1 ) A, B are increasing functions of bounded variations such that `1 (s) ≥ 0, `2 (s) ≥ 0 for s ∈ [0, 1] and 0 ≤ ȷ1 , ȷ2 < 1. (H2 ) f1 , f2 ∶ (0, 1) × [0, +∞)3 → (−∞, +∞) are continuous and satisfy −q1 (t) ≤ f1 (t, u1 , u2 , u3 ) ≤ p1 (t)g1 (t, u1 , u2 , u3 ), −q2 (t) ≤ f2 (t, v1 , v2 , v3 ) ≤ p2 (t)g2 (t, v1 , v2 , v3 ), where g i ∈ C([0, 1] × [0, +∞)3 , [0, +∞)), q i , p i ∈ C((0, 1), [0, +∞)) and 1

0 < ∫ p i (s)ds < +∞, 0

1

0 < ∫ q i (s)ds < +∞, 0

i = 1, 2.


| 587

(H3 ) There exists a constant ⎧ ⎫ 1 1 ⎪ ⎪ ⎪ ⎪ 2ρ22 ⎪ 2ρ21 ⎪ r > max ⎨ q (s)ds, q (s)ds ⎬ 1 2 ∫ ∫ ⎪ ⎪ Γ ( α − β ) Γ ( γ − δ ) ⎪ ⎪ ⎪ ⎪ 0 0 ⎩ ⎭ such that r r ] × [0, ] × [0, +∞), 2Γ (β + 1) 2

f1 (t, u1 , u2 , u3 ) ≥ 0,

(t, u1 , u2 , u3 ) ∈ (0, 1) × [0,

f2 (t, v1 , v2 , v3 ) ≥ 0,

(t, v1 , v2 , v3 ) ∈ (0, 1) × [0, +∞) × [0,

r r ] × [0, ] . 2Γ (δ + 1) 2

Define a modified function [z(t)]∗ for any z ∈ C[0, 1] by ⎧ ⎪ ⎪ z(t), z(t) ≥ 0, [z(t)]∗ = ⎨ ⎪ z(t) < 0. ⎪ ⎩ 0, Next we consider the following systems: D0α+−β x(t) + λ[f1 (t, I0β+ [x(t) − a(t)]∗ , [x(t) − a(t)]∗ , I0δ+ [y(t) − b(t)]∗ ) + q1 (t)] = 0, 0 < t < 1, Dγ0+−δ y(t) + λ[f2 (t, I0β+ [x(t) − a(t)]∗ , I0δ+ [y(t) − b(t)]∗ , [y(t) − b(t)]∗ ) + q2 (t)] = 0, 0 < t < 1, 1

x(0) = x′ (0) = 0,

x′ (1) = ∫ x(s)dA(s),

(9)

0 1

y(0) = y′ (0) = 0,

y′ (1) = ∫ y(s)dB(s), 0

1

1

where a(t) = λ ∫0 G1 (t, s)q1 (s)ds and b(t) = λ ∫0 G2 (t, s)q2 (s)ds are the solutions of the following BVPs (10) and (11), respectively, −β Dα 0+ a(t) + λ q 1 (t) = 0,

0 < t < 1, 1

a(0) = a′ (0) = 0,

(10)

a′ (1) = ∫ a(s)dA(s), 0

Dγ0+−δ b(t) + λq2 (t) = 0,

0 < t < 1, 1

b(0) = b′ (0) = 0,

(11)

b′ (1) = ∫ b(s)dB(s). 0

Lemma 2.9 ([33]). Assume that condition (H1 ) holds. Then the positive solutions a(t) and b(t) of BVPs (10) and (11) satisfy 1

a(t) ≤

λρ1 k1 (t) ∫ q1 (t)dt, Γ (α − β ) 0

1

b(t) ≤

λρ2 k2 (t) ∫ q2 (t)dt, Γ (γ − δ )

t ∈ [0, 1].

0

Lemma 2.10. If (x, y) ∈ C[0, 1] × C[0, 1] with x(t) > a(t), y(t) > b(t) for any t ∈ (0, 1) is a positive solution of systems (9), then (ω (t) = x(t) − a(t), z(t) = y(t) − b(t)) is a positive solution of systems (8), and (u(t) = I0β+ ω (t), v(t) = I0δ+ z(t)) is a positive solution of systems (1).

588 | X. Hao Proof. In fact, if (x, y) ∈ C[0, 1] × C[0, 1] is a solution of systems (9) with x(t) > a(t), y(t) > b(t), then from systems (9) and the definition of []∗ , we get −β α−β α−β α−β Dα 0+ ω (t) = D 0+ (x(t) − a(t)) = D 0+ x(t) − D 0+ a(t)

= − λ[f1 (t, I0β+ [x(t) − a(t)]∗ , [x(t) − a(t)]∗ , I0δ+ [y(t) − b(t)]∗ ) + q1 (t)] − [−λq1 (t)] = − λf1 (t, I0β+ [x(t) − a(t)]∗ , [x(t) − a(t)]∗ , I0δ+ [y(t) − b(t)]∗ ) = − λf1 (t, I0β+ ω (t), ω (t), I0δ+ z(t)), ω (0) = x(0) − a(0) = 0,

′

′

1 ′

′

′

ω (0) = x (0) − a (0) = 0, 1

1

′

ω (1) = x (1) − a (1) = ∫ x(s)dA(s) − ∫ a(s)dA(s) = ∫ ω (s)dA(s), 0

and

0

0

Dγ0+−δ z(t) = Dγ0+−δ (y(t) − b(t)) = Dγ0+−δ y(t) − D0γ+−δ b(t) = − λ[f2 (t, I0β+ [x(t) − a(t)]∗ , I0δ+ [y(t) − b(t)]∗ , [y(t) − b(t)]∗ ) + q2 (t)] − [−λq2 (t)] = − λf2 (t, I0β+ [x(t) − a(t)]∗ , I0δ+ [y(t) − b(t)]∗ , [y(t) − b(t)]∗ ) = − λf2 (t, I0β+ ω (t), I0δ+ z(t), z(t)), z(0) = y(0) − b(0) = 0,

z′ (0) = y′ (0) − b′ (0) = 0,

1

1

1

z′ (1) = y′ (1) − b′ (1) = ∫ y(s)dB(s) − ∫ b(s)dB(s) = ∫ z(s)dB(s). 0

0

0

So (ω , z) is a positive solution of systems (8). It follows from Lemma 2.8 that (u(t) = I0β+ ω (t), v(t) = I0β+ z(t)) is a positive solution of systems (1). Let X = C[0, 1] × C[0, 1] , then X is a Banach space with the norm ∥(u, v)∥1 = ∥u∥ + ∥v∥, Let

∥u∥ = max ∣u(t)∣, 0≤t≤1

∥v∥ = max ∣v(t)∣, 0≤t≤1

(u, v) ∈ X.

−1 P = {(u, v) ∈ X ∶ u(t) ≥ ρ−1 1 k 1 (t)∥u∥, v(t) ≥ ρ2 k 2 (t)∥v∥, t ∈ [0, 1]} ,

then P is a cone of X. Define an operator A ∶ P → X by A(x, y) = (A1 (x, y), A2 (x, y)), where A1 , A2 ∶ P → C[0, 1] are defined by ⎧ 1 ⎪ ⎪ ⎪ ⎪ A1 (x, y)(t) = λ ∫ G1 (t, s)(f1 (s, I0β+ [x(s) − a(s)]∗ , [x(s) − a(s)]∗ , I0δ+ [y(s) − b(s)]∗ ) + q1 (s))ds, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎨ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ A (x, y)(t) = λ G2 (t, s)(f2 (s, I0β+ [x(s) − a(s)]∗ , I0δ+ [y(s) − b(s)]∗ , [y(s) − b(s)]∗ ) + q2 (s))ds. 2 ∫ ⎪ ⎪ ⎪ ⎪ 0 ⎩ Clearly, if (x, y) ∈ P is a fixed point of A, then (x, y) is a solution of systems (9). Lemma 2.11. Assume that conditions (H1 ) − (H3 ) hold, then A ∶ P → P is a completely continuous operator. Proof. For any (x, y) ∈ P, there exists a constant L > 0 such that ∥(x, y)∥1 ≤ L, then [x(s) − a(s)]∗ ≤ x(s) ≤ ∥x∥ ≤ ∥(x, y)∥1 ≤ L,

s ∈ [0, 1],

[y(s) − b(s)]∗ ≤ y(s) ≤ ∥y∥ ≤ ∥(x, y)∥1 ≤ L,

s ∈ [0, 1],

Positive solutions of semipositone nonlocal fractional differential systems s

(s − t)β −1 [x(t) − a(t)]∗ L dt ≤ , Γ (β ) Γ (β + 1)

s

(s − t)δ−1 [y(t) − b(t)]∗ L dt ≤ . Γ (δ ) Γ (δ + 1)

I0β+ [x(s) − a(s)]∗ = ∫ 0

I0δ+ [y(s) −

∗

b(s)] = ∫ 0

| 589

It follows from Lemma 2.7 that A1 (x, y)(t) 1

≤λ ∫ ρ1 G∗1 (1, s)(p1 (s)g1 (s, I0β+ [x(s) − a(s)]∗ , [x(s) − a(s)]∗ , I0δ+ [y(s) − b(s)]∗ ) + q1 (s))ds 0 1

≤λρ1 (M + 1) ∫ G∗1 (1, s)(p1 (s) + q1 (s))ds < +∞, 0

A2 (x, y)(t) 1

≤λ ∫ ρ2 G∗2 (1, s)(p2 (s)g2 (s, I0β+ [x(s) − a(s)]∗ , I0δ+ [y(s) − b(s)]∗ , [y(s) − b(s)]∗ ) + q2 (s))ds 0 1

≤λρ2 (M + 1) ∫ G∗2 (1, s)(p2 (s) + q2 (s))ds < +∞, 0

where

⎧ ⎪ ⎪ g1 (t, u1 , u2 , u3 ), M = max ⎨ max L L ⎪ ⎪ ⎩t∈[0,1], 0≤u1 ≤ Γ (β+1) , 0≤u2 ≤L, 0≤u3 ≤ Γ (δ+1) ⎫ ⎪ ⎪ max g2 (t, v1 , v2 , v3 )⎬ . L L ⎪ t∈[0,1], 0≤v1 ≤ Γ (β +1) , 0≤v2 ≤ Γ (δ+1) , 0≤v3 ≤L ⎪ ⎭

Thus, A ∶ P → X is well defined. Next, we prove A(P) ⊂ P. Denote F1 (s) = f1 (s, I0β+ [x(s) − a(s)]∗ , [x(s) − a(s)]∗ , I0δ+ [y(s) − b(s)]∗ ) + q1 (s),

s ∈ [0, 1],

F2 (s) = f2 (s, I0β+ [x(s) − a(s)]∗ , I0δ+ [y(s) − b(s)]∗ , [y(s) − b(s)]∗ ) + q2 (s),

s ∈ [0, 1].

For any (x, y) ∈ P, we have 1

∥A1 (x, y)∥ = max ∣A1 (x, y)(t)∣ ≤ λρ1 ∫ G∗1 (1, s)F1 (s)ds. 0≤t≤1

So

0

1

A1 (x, y)(t) ≥ λ ∫ k1 (t)G∗1 (1, s)F1 (s)ds ≥ ρ−1 1 k 1 (t)∥A 1 (x, y)∥,

t ∈ [0, 1].

0

Similarly, A2 (x, y)(t) ≥ ρ−1 2 k 2 (t)∥A 2 (x, y)∥, t ∈ [0, 1]. Thus, A(P) ⊂ P. According to the Ascoli-Arzela theorem and the Lebesgue dominated convergence theorem, we can easily get that A ∶ P → P is completely continuous. Lemma 2.12 ([53]). Let P be a cone in Banach space E, Ω1 and Ω2 are bounded open sets in E, θ ∈ Ω1 , Ω 1 ⊂ Ω2 , A ∶ P ∩ (Ω 2 /Ω1 ) → P is a completely continuous operator. If the following conditions are satisfied: ∥Ax∥ ≤ ∥x∥, ∀x ∈ P ∩ ∂Ω1 , ∥Ax∥ ≥ ∥x∥, ∀x ∈ P ∩ ∂Ω2 , or ∥Ax∥ ≥ ∥x∥, ∀x ∈ P ∩ ∂Ω1 , ∥Ax∥ ≤ ∥x∥, ∀x ∈ P ∩ ∂Ω2 , then A has at least one fixed point in P ∩ (Ω 2 /Ω1 ).

590 | X. Hao

3 Main results Theorem 3.1. Assume that conditions (H1 ) − (H3 ) are satisfied. Further assume that the following condition holds: (H4 ) There exists [a, b] ⊂ (0, 1) such that lim

u2 →+∞

min

t∈[a,b] u1 ,u3 ≥0

f1 (t, u1 , u2 , u3 ) = +∞, u2

lim

v3 →+∞

min

t∈[a,b] v1 ,v2 ≥0

f2 (t, v1 , v2 , v3 ) = +∞. v3

Then there exists λ > 0 such that for any 0 < λ < λ, systems (1) have at least one positive solution (u, v). Proof. Let Ω1 = {(x, y) ∈ X ∶ ∥(x, y)∥1 < r},

⎧ ⎪ ⎪ ⎪ ⎪ ⎩

λ = min ⎨1,

⎫ ⎪ ⎪ ⎬, 1 ∗ 1 ∗ ∗ ∗ ⎪ 2ρ1 (g1 + 1) ∫0 G1 (1, s)[p1 (s) + q1 (s)]ds 2ρ2 (g2 + 1) ∫0 G2 (1, s)[p2 (s) + q2 (s)]ds ⎪ ⎭ r

r

,

where g1∗ = max {g1 (s, u1 , u2 , u3 ) ∶ 0 ≤ s ≤ 1, 0 ≤ u1 ≤

g2∗ = max {g2 (s, v1 , v2 , v3 ) ∶ 0 ≤ s ≤ 1, 0 ≤ v1 ≤

r Γ (β + 1)

r Γ (β + 1)

, 0 ≤ u2 ≤ r, 0 ≤ u3 ≤

, 0 ≤ v2 ≤

r Γ (δ + 1)

r Γ (δ + 1)

},

, 0 ≤ v3 ≤ r} .

Suppose 0 < λ < λ, then for any (x, y) ∈ P ∩ ∂Ω1 , s ∈ [0, 1], we have 0 ≤ [x(s) − a(s)]∗ ≤ x(s) ≤ ∥x∥ ≤ r, I0β+ [x(s) −

s

(s − τ )β −1 [x(τ ) − a(τ )]∗ r dτ ≤ , Γ (β ) Γ (β + 1)

s

(s − τ )δ−1 [y(τ ) − b(τ )]∗ r dτ ≤ . Γ (δ ) Γ (δ + 1)

∗

a(s)] = ∫ 0

I0δ+ [y(s) − b(s)]∗ = ∫ 0

0 ≤ [y(s) − b(s)]∗ ≤ y(s) ≤ ∥y∥ ≤ r,

Thus, ∥A1 (x, y)∥ 1

≤λ ∫ ρ1 G∗1 (1, s)[f1 (s, I0β+ [x(s) − a(s)]∗ , [x(s) − a(s)]∗ , I0δ+ [y(s) − b(s)]∗ ) + q1 (s)]ds 0 1

≤λρ1 ∫ G∗1 (1, s)[p1 (s)g1 (s, I0β+ [x(s) − a(s)]∗ , [x(s) − a(s)]∗ , I0δ+ [y(s) − b(s)]∗ ) + q1 (s)]ds 0 1

0 such that

∀(x, y) ∈ P ∩ ∂Ω1 .

4ρ2 4ρ1 , }. 2 2 λk̃1 ∫ab G∗1 (1,s)ds λk̃2 ∫ab G∗2 (1,s)ds

f1 (t, u1 , u2 , u3 ) ≥ ̃ Lu2 ,

t ∈ [a, b], u2 ≥ N, u1 , u3 ≥ 0,

f2 (t, v1 , v2 , v3 ) ≥ ̃ Lv3 ,

t ∈ [a, b], v3 ≥ N, v1 , v2 ≥ 0.

Let R > max {2r,

4ρ1 N 4ρ2 N , }, k̃1 k̃2

Ω2 = {(x, y) ∈ P ∶ ∥(x, y)∥1 < R}.

For any (x, y) ∈ P ∩ ∂Ω2 , ∥(x, y)∥1 = R, we have ∥x∥ ≥

R 2

or ∥y∥ ≥ R2 . If ∥x∥ ≥ R2 , we deduce 1

x(t) − a(t)

≥ρ−1 1 k 1 (t)∥x∥ −

ρ1 k1 (t) ∫ q1 (s)ds Γ (α − β ) 0

1 ⎡ ⎤ ⎢ ∥x∥ ⎥ ρ1 ⎢ =k1 (t) ⎢ − q1 (s)ds⎥ ∫ ⎥ ρ Γ ( α − β ) ⎢ 1 ⎥ ⎣ ⎦ 0 R R ≥k̃1 ( − ) 2ρ 1 4ρ 1 k̃1 R ≥ N, t ∈ [a, b], = 4ρ 1

then

f1 (s, I0β+ [x(s) − a(s)]∗ , [x(s) − a(s)]∗ , I0δ+ [y(s) − b(s)]∗ ) =f1 (s, I0β+ (x(s) − a(s)), (x(s) − a(s)), I0δ+ [y(s) − b(s)]∗ ) k̃1 ̃ LR ≥̃ L[x(s) − a(s)] ≥ . 4ρ 1

(12) By (H4 ), there

592 | X. Hao Thus ∥A1 (x, y)∥ RRR 1 RRR R R = max RRRRRλ ∫ G1 (t, s)[f1 (s, I0β+ [x(s) − a(s)]∗ , [x(s) − a(s)]∗ , I0δ+ [y(s) − b(s)]∗ ) + q1 (s)]dsRRRRR t∈[0,1] RR RRR RR 0 R b

L(x(s) − a(s))ds ≥ max λ ∫ k1 (t)G∗1 (1, s)̃ t∈[0,1] a

b

≥λk̃1 ̃ L ∫ G∗1 (1, s) a

k̃1 Rds 4ρ 1

b

2

λk̃1 ̃ L = R ∫ G∗1 (1, s)ds ≥ R = ∥(x, y)∥1 . 4ρ 1 a

If ∥y∥ ≥ R2 , in a similar manner, we have y(t) − b(t) ≥

≥ N, t ∈ [a, b], and

b

2

∥A2 (x, y)∥ ≥

k̃2 R 4ρ2

λk̃2 ̃ L R ∫ G∗2 (1, s)ds ≥ R = ∥(x, y)∥1 . 4ρ 2 a

Thus ∥A(x, y)∥1 = ∥A1 (x, y)∥ + ∥A2 (x, y)∥ ≥ ∥(x, y)∥1 ,

∀(x, y) ∈ P ∩ ∂Ω2 .

(13)

By (12), (13) and Lemma 2.12, A has a fixed point (̃ x, ̃ y) with r ≤ ∥(̃ x, ̃ y)∥1 ≤ R. Now, we will prove ̃ x(t) > a(t), ̃ y(t) ≥ b(t)(or ̃ x(t) ≥ a(t), ̃ y(t) > b(t)), t ∈ (0, 1). We shall divide the proof into three cases: (i) ∥̃ x∥ ≥ 2r , ∥̃ y∥ ≥ 2r ; (ii) ∥̃ x∥ > 2r , ∥̃ y∥ < 2r ; (iii) ∥̃ x∥ < 2r , ∥̃ y∥ > 2r . Case (i). If ∥̃ x∥ ≥ 2r , from (H3 ) and Lemma 2.9, we get 1

̃ x(t) ≥ρ−1 x∥ ≥ ρ−1 1 k 1 (t)∥̃ 1 k 1 (t)

ρ21 r > ρ−1 1 k 1 (t) ∫ q1 (s)ds 2 Γ (α − β ) 0

1

=

ρ1 k1 (t) ∫ q1 (s)ds ≥ a(t), Γ (α − β )

t ∈ (0, 1).

0

y(t) > b(t), t ∈ (0, 1). Similarly, if ∥̃ y∥ ≥ 2r we obtain ̃ Case (ii). For ∥̃ x∥ > 2r , similar to (i), we have ̃ x(t) > a(t), t ∈ (0, 1). For ∥̃ y∥ < 2r , we have r , 2

0 ≤ [̃ y(s) − b(s)]∗ ≤ ̃ y(s) ≤ ∥̃ y∥ ≤ 0 ≤ I0β+ [̃ x(s) − a(s)]∗ = I0β+ (̃ x(s) − a(s)) ≤ 0 ≤ I0δ+ [̃ y(s) − b(s)]∗ ≤

R , Γ (β + 1)

r . 2Γ (δ + 1)

It follows from (H2 ) that f2 (t, v1 , v2 , v3 ) ≥ 0,

(t, v1 , v2 , v3 ) ∈ (0, 1) × [0,

R Γ (β + 1)

] × [0,

r r ] × [0, ] , 2Γ (δ + 1) 2

then 1

̃ y(t) =λ ∫ G2 (t, s)[f2 (s, I0β+ [̃ x(s) − a(s)]∗ , I0δ+ [̃ y(s) − b(s)]∗ , [̃ y(s) − b(s)]∗ ) + q2 (s)]ds 0 1

≥λ ∫ G2 (t, s)q2 (s)ds = b(t), 0

t ∈ (0, 1).

Positive solutions of semipositone nonlocal fractional differential systems |

593

Case (iii). If ∥̃ x∥ < 2r and ∥̃ y∥ > 2r , similar to (ii), we have ̃ x(t) ≥ a(t), ̃ y(t) > b(t), t ∈ (0, 1). So by Lemma 2.10 we know that (u(t), v(t)) = (I0β+ (̃ x(t) − a(t)), I0δ+ (̃ y(t) − b(t))) is a positive solution of systems (1).

4 An Example Example 4.1. Consider the following problem: 1 ⎡ ⎤ ⎢ (u(t) − 90)2 + (D 4+ u(t) − 3)4 + v2 (t) 1 ⎥ ⎢ ⎥ 0 ⎢ ⎥ = 0, t ∈ (0, 1), D (D u(t)) + √ − √ 4 ⎢ ⎥ 48 685 t t(1 − t) ⎢ ⎥ ⎣ ⎦ 1 ⎡ ⎤ 9 2 2 2 8 9 1 λΓ ( 4 ) ⎢ u (t) + (v(t) − 30) + (D0+ v(t) − 3) 1 ⎥ ⎢ ⎥ = 0, t ∈ (0, 1), − D04+ (D08+ v(t)) + √ √ ⎢ 4 12 ⎥ ⎥ 30 t 1−t ⎢ ⎣ ⎦ 5 5 1 1 96 1 D 4+ u ( ) , D04+ u(0) = D04+ u(0) = 0, D04+ u(1) = 97 0 16 9 9 1 1 40 1 D 8+ v ( ) . D08+ v(0) = D08+ v(0) = 0, D08+ v(1) = 41 0 16 5 2 0+

√

λ π

1 4 0+

Problem (14) can be regarded as a problem of the form (1) with α =

11 , 4

β = 14 , γ =

19 , 8

δ = 18 ,

⎧ ⎧ 1 1 ⎪ ⎪ ⎪ ⎪ ), 0, t ∈ [0, ), 0, t ∈ [0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 16 16 A(t) = ⎨ B(t) = ⎨ 1 40 ⎪ ⎪ 96 1 ⎪ ⎪ ⎪ ⎪ , t ∈ [ , 1] , , t ∈ [ , 1] , ⎪ ⎪ ⎪ ⎪ 16 16 ⎩ 97 ⎩ 41 √ √ (u1 − 90)2 + (u2 − 3)4 + u23 π π f1 (t, u1 , u2 , u3 ) = √ , − √ √ 4 3 685 t 1−t 48 t(1 − t) Γ ( 94 ) v21 + (v2 − 30)2 + (v3 − 3)2 Γ ( 94 ) f2 (t, v1 , v2 , v3 ) = √ √ − . √ 30 t 4 1−t 12 4 1 − t

Evidently, f1 , f2 ∶ (0, 1) × [0, +∞)3 → (−∞, +∞) are continuous and singular at t = 0, 1, By direct calculations, we get 5

3

k1 (t) = t 2 , k2 (t) = t 4 , γ1 (t) = 1

2 23 4 5 t , γ2 (t) = t 4 , ρ1 = 2, ρ2 = 2, 3 5 3

ȷ1 = ∫ γ1 (t)dA(t) = 0

96 1 1 2 1 × ×( ) = , 97 3 16 97

1

`1 (s) =

96 ∗ 1 G1 ( , s) ≥ 0, 97 16

`2 (s) =

40 ∗ 1 G2 ( , s) ≥ 0, 41 16

5

40 4 1 4 1 ȷ2 = ∫ γ2 (t)dB(t) = × ×( ) = , 41 5 16 41 0

so condition (H1 ) is satisfied. Let q1 (t) =

√ √ (u1 − 90)2 + (u2 − 3)4 + u23 π π , g1 (t, u2 , u2 , u3 ) = , , p1 (t) = √ √ √ 4 685 t3 1 − t 48 t(1 − t)

q2 (t) =

Γ ( 94 )

Γ ( 94 ) v21 + (v2 − 30)2 + (v3 − 3)2 , p (t) = , g (t, v , v , v ) = , √ √ √ 2 2 2 2 3 30 12 4 1 − t t 4 1−t

then

1

√ ππ 1 1 ∫ q1 (t)dt = 48 B ( 2 , 2 ) = 48 , 0

√

π

1

∫ q2 (t)dt = 0

Γ ( 49 )

9

,

(14)

594 | X. Hao 1

∫ p1 (t)dt =

√

1 1 ), 4 2

πB ( ,

0

1

9 3 1 ∫ p2 (t)dt = Γ ( 4 ) B ( 4 , 2 ) , 0

so condition (H2 ) is satisfied. On the other hand, ⎫ ⎧ 1 1 ⎪ ⎪ ⎪ ⎪ 2ρ22 ⎪ 8 ⎪ 2ρ21 q1 (s)ds, q2 (s)ds⎬ = . max ⎨ ∫ ∫ ⎪ ⎪ Γ ( α − β ) Γ ( γ − δ ) 9 ⎪ ⎪ ⎪ ⎪ 0 0 ⎭ ⎩ Select r = 2, then √

(u1 − 90)2 + (u2 − 3)4 + u23 1 − ] > 0, 685 48 t(1 − t) ⎡ 1 ⎤ ⎢ ⎥ ⎥ × [0, 1] × [0, +∞), (t, u1 , u2 , u3 ) ∈ (0, 1) × ⎢0, ⎢ Γ(5)⎥ ⎣ 4 ⎦

f1 (t, u1 , u2 , u3 ) ≥ √

π

Γ ( 94 )

[

v21 + (v2 − 30)2 + (v3 − 3)2 1 − ] > 0, 30 12 1−t 1 (t, v1 , v2 , v3 ) ∈ (0, 1) × [0, +∞) × [0, ] × [0, 1]. Γ ( 89 )

f2 (t, v1 , v2 , v3 ) ≥ √ 4

[

Hence, condition (H3 ) is satisfied. In addition, for any [a, b] ⊂ (0, 1), lim

min

u2 →+∞ t∈[a,b] u1 ,u3 ≥0

f1 (t, u1 , u2 , u3 ) = +∞, u2

lim

min

v3 →+∞ t∈[a,b] v1 ,v2 ≥0

f2 (t, v1 , v2 , v3 ) = +∞, v3

condition (H4 ) is satisfied. Hence, problem (14) has at least one positive solution by Theorem 3.1. Acknowledgement: The authors would like to thank the referees for their pertinent comments and valuable suggestions. This work is supported financially by the National Natural Science Foundation of China (11501318) and the China Postdoctoral Science Foundation (2017M612230).

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