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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions

Xinan Hao / Huaqing Wang
Published Online: 2018-06-07 | DOI: https://doi.org/10.1515/math-2018-0055

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 2010: 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, 2, 3, 4, 5, 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, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] and the references therein for fractional differential equations, and [21, 22, 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, 35, 36, 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, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 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 D0+αβ(D0+βu(t))+λf1(t,u(t),D0+βu(t),v(t))=0,0<t<1,D0+γδ(D0+δv(t))+λf2(t,u(t),v(t),D0+δv(t))=0,0<t<1,D0+βu(0)=D0+β+1u(0)=0,D0+β+1u(1)=01D0+βu(s)dA(s),D0+δv(0)=D0+δ+1v(0)=0,D0+δ+1v(1)=01D0+δv(s)dB(s),(1)

where D0+αβ,D0+β,D0+γδ and D0+δ are the standard Riemann-Liouville fractional derivatives, λ > 0 is a parameter, 2 < α, γ ≤ 3, 0 < β, δ < 1, αβ > 2, γδ > 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, 01D0+βu(s)dA(s) and 01D0+δv(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 D0+αu(t)+f(t,u(t),v(t))=0,0<t<1,1<α2,D0+βv(t)+g(t,u(t),v(t))=0,0<t<1,1<β2,u(0)=0,u(1)=01u(s)dA(s),v(0)=0,v(1)=01v(s)dB(s),(2)

where D0+α and D0+β are the standard Riemann-Liouville fractional derivatives, A, B are nondecreasing functions of bunded variations, 01u(s)dA(s) and 01v(s)dB(s) are Riemann-Stieltjes integrals, f(t, x, y), g(t, x, y) : (0, 1) × (0, ∞)2 → [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 and Schauder fixed point theorem. For the special boundary conditions u(1) = 01ϕ(s)u(s)ds, v(1) = 01φ(s)v(s)ds, where ϕ, φL(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)(t, u(t)), g = b(t)(t, v(t)), u(1) = 01ϕ(s)u(s)ds, v(1) = 01φ(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 D0+αu(t)+λf(t,u(t),v(t))=0,0<t<1,n1<αn,D0+βv(t)+μg(t,u(t),v(t))=0,0<t<1,m1<βm,u(0)=u(0)==u(n2)(0)=0,u(1)=01u(s)dA(s),v(0)=v(0)==v(m2)(0)=0,v(1)=01v(s)dB(s)(3)

where D0+α and D0+β are the standard Riemann-Liouville fractional derivatives, f, g : [0, 1]×[0, ∞)2 → [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 λ and μ via the Guo-Krasnosel’skii fixed point theorem. For the multi-point boundary conditions u(1) = i=1paiu(ξi),v(1)=i=1qbiv(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 = (t, v), 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 D0+αu(t)+λf(t,u(t),D0+βu(t),v(t))=0,0<t<1,2<α3,D0+γv(t)+λg(t,u(t))=0,0<t<1,2<γ3,D0+βu(0)=D0+β+1u(0)=0,D0+βu(1)=01D0+βu(s)dA(s),v(0)=v(0)=0,v(1)=01v(s)dB(s),(4)

where λ > 0 is a parameter, 0 < β < 1, α-β > 2, D0+α,D0+β and D0+γ are the standard Riemann-Liouville fractional derivatives, A, B are nondecreasing functions of bunded variations, 01D0+βu(s)dA(s) and 01v(s)dB(s) are Riemann-Stieltjes integrals, f : (0, 1) × [0, +∞)3 → (−∞, +∞) 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: D0+αu(t)+p(t)f(t,u(t),v(t))q1(t)=0,0<t<1,2<α3,D0+βv(t)+q(t)g(t,u(t),v(t))q2(t)=0,0<t<1,2<β3,u(0)=u(0)=0,u(1)=01u(s)dA(s),v(0)=v(0)=0,v(1)=01v(s)dB(s),(5)

where D0+α and D0+β are the standard Riemann-Liouville fractional derivatives, f, g : [0, 1]×[0, ∞)2 → [0, ∞) are continuous, q1, q2 : (0, 1)→ [0, +∞) are Lebesgue integrable, A, B are suitable functions of bounded variation, 01u(t)dA(t) and 01v(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, 23, 24, 25, 26, 27, 28, 29, 30, 31, 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, multi-point and some nonlocal conditions as special cases.

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 I0+αu(t)=1Γ(α)0t(ts)α1u(s)ds,

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 D0+αu(t)=1Γ(nα)ddtn0t(ts)nα1u(s)ds,

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 uL(0, 1), ρ > σ > 0 and n is a natural number, then I0+ρI0+σu(t)=I0+ρ+σu(t),D0+σI0+ρu(t)=I0+ρσu(t),D0+σI0+σu(t)=u(t),ddtn(D0+σu(t))=D0+n+σu(t).

Lemma 2.4

([1, 2]). Assume that uC(0, 1) ∩ L(0, 1) with a fractional derivative of order α > 0 that belongs to C(0, 1) ∩ L(0, 1). Then I0+αD0+αu(t)=u(t)+c1tα1+c2tα2++cNtαN,

for some c1, c2, ⋯, cN ∈ (−∞, +∞), where N is the smallest integer greater than or equal to α.

Lemma 2.5

([33]). Given hC(0, 1) ∩ L(0, 1), then the BVP D0+αβx(t)+h(t)=0,0<t<1,2<αβ3,x(0)=x(0)=0,x(1)=0,

has a unique solution x(t)=01G1(t,s)h(s)ds,

where G1(t,s)=1Γ(αβ)tαβ1(1s)αβ2,0ts1,tαβ1(1s)αβ2(ts)αβ1,0st1,

and the Green function G1 (t, s) has the following properties:

  1. G1(t, s) > 0, t, s ∈(0, 1).

  2. k1(t) G1(1, s) ≤ G1(t, s) ≤ G1(1, s), t, s ∈ [0, 1].

  3. G1(t,s)k1(t)Γ(αβ),t,s[0,1], where k1(t) = tαβ−1.

By Lemma 2.4, the unique solution of the BVP D0+αβx(t)=0,0<t<1,x(0)=x(0)=0,x(1)=1,

is γ1(t)=tαβ1αβ1. By [33], the Green function for the BVP D0+αβx(t)+h(t)=0,0<t<1,x(0)=x(0)=0,x(1)=01x(s)dA(s),

is given by G1(t,s)=G1(t,s)+γ1(t)1ȷ11(s),(6)

where ȷ1=01γ1(t)dA(t)1,1(s)=01G1(t,s)dA(t),s[0,1].

Similarly, the Green function for the BVP D0+γδy(t)+h(t)=0,0<t<1,y(0)=y(0)=0,y(1)=01y(s)dB(s),

is given by G2(t,s)=G2(t,s)+γ2(t)1ȷ22(s),(7)

where G2(t,s)=1Γ(γδ)tγδ1(1s)γδ2,0ts1,tγδ1(1s)γδ2(ts)γδ1,0st1,ȷ2=01γ2(t)dB(t)1,2(s)=01G2(t,s)dB(t),γ2(t)=tγδ1γδ1.

Lemma 2.6

([33]). The Green function G2(t, s) has the following properties:

  1. G2(t, s) > 0, t, s ∈(0, 1).

  2. k2(t) G2(1, s) ≤ G2(t, s) ≤ G2(1, s), t, s ∈ [0, 1].

  3. G2(t,s)k2(t)Γ(γδ),t,s[0,1], where k2(t) = tγδ−1.

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. Gi(t, s) ≥ Gi(t, s) > 0, t, s ∈ (0, 1), i = 1, 2.

  2. ki(t) Gi(1, s) ≤ Gi(t, s) ≤ ρi Gi(1, s), t, s ∈[0, 1], i = 1, 2.

  3. G1(t,s)ρ1Γ(αβ)k1(t),G2(t,s)ρ2Γ(γδ)k2(t),t,s[0,1], where ρ1=1+01dA(t)1ȷ1,ρ2=1+01dB(t)1ȷ2.

Now let us consider the following modified problem of systems (1) D0+αβω(t)+λf1(t,I0+βω(t),ω(t),I0+δz(t))=0,0<t<1,D0+γδz(t)+λf2(t,I0+βω(t),I0+δz(t),z(t))=0,0<t<1,ω(0)=ω(0)=0,ω(1)=01ω(s)dA(s),z(0)=z(0)=0,z(1)=01z(s)dB(s).(8)

Lemma 2.8

If (ω, z) ∈ C[0, 1] × C[0, 1] is a positive solution of systems (8), then (I0+βω,I0+δz) 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)=I0+βω(t),v(t)=I0+δz(t), then D0+αβ(D0+βu(t))=D0+αβ(D0+βI0+βω(t))=D0+αβω(t)=λf1(t,I0+βω(t),ω(t),I0+δz(t))=λf1(t,u(t),D0+βu(t),v(t)),D0+γδ(D0+δv(t))=D0+γδ(D0+δI0+δz(t))=D0+γδz(t)=λf2(t,I0+βω(t),I0+δz(t),z(t))=λf2(t,u(t),v(t),D0+δv(t)), D0+βu(0)=ω(0)=0,D0+β+1u(0)=ddt(D0+βu(t))|t=0=ω(0)=0,D0+β+1u(1)=ddt(D0+βu(t))|t=1=ω(1)=01ω(s)dA(s)=01D0+βu(s)dA(s),D0+δv(0)=z(0)=0,D0+δ+1v(0)=ddt(D0+δv(t))|t=0=z(0)=0,D0+δ+1v(1)=ddt(D0+δv(t))|t=1=z(1)=01z(s)dB(s)=01D0+δv(s)dB(s).

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) = (I0+βω,I0+δz) 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 giC([0, 1] × [0, +∞)3, [0, +∞)), qi, piC((0, 1), [0, +∞)) and 0<01pi(s)ds<+,0<01qi(s)ds<+,i=1,2.

  • (H3)

    There exists a constant r>max2ρ12Γ(αβ)01q1(s)ds,2ρ22Γ(γδ)01q2(s)ds

    such that f1(t,u1,u2,u3)0,(t,u1,u2,u3)(0,1)×0,r2Γ(β+1)×0,r2×[0,+),f2(t,v1,v2,v3)0,(t,v1,v2,v3)(0,1)×[0,+)×0,r2Γ(δ+1)×0,r2.

    Define a modified function [z(t)]* for any zC[0, 1] by [z(t)]=z(t),z(t)0,0,z(t)<0.

    Next we consider the following systems: D0+αβx(t)+λ[f1(t,I0+β[x(t)a(t)],[x(t)a(t)],I0+δ[y(t)b(t)])+q1(t)]=0,0<t<1,D0+γδy(t)+λ[f2(t,I0+β[x(t)a(t)],I0+δ[y(t)b(t)],[y(t)b(t)])+q2(t)]=0,0<t<1,x(0)=x(0)=0,x(1)=01x(s)dA(s),y(0)=y(0)=0,y(1)=01y(s)dB(s),(9)

    where a(t) = λ 01G1(t, s)q1(s)ds and b(t) = λ 01G2(t, s)q2(s)ds are the solutions of the following BVPs (10) and (11), respectively, D0+αβa(t)+λq1(t)=0,0<t<1,a(0)=a(0)=0,a(1)=01a(s)dA(s),(10) D0+γδb(t)+λq2(t)=0,0<t<1,b(0)=b(0)=0,b(1)=01b(s)dB(s).(11)

Lemma 2.9

([33]). Assume that condition (H1) holds. Then the positive solutions a(t) and b(t) of BVPs (10) and (11) satisfy a(t)λρ1Γ(αβ)k1(t)01q1(t)dt,b(t)λρ2Γ(γδ)k2(t)01q2(t)dt,t[0,1].

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)=I0+βω(t),v(t)=I0+δz(t)) is a positive solution of systems (1).

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 D0+αβω(t)=D0+αβ(x(t)a(t))=D0+αβx(t)D0+αβa(t)=λ[f1(t,I0+β[x(t)a(t)],[x(t)a(t)],I0+δ[y(t)b(t)])+q1(t)][λq1(t)]=λf1(t,I0+β[x(t)a(t)],[x(t)a(t)],I0+δ[y(t)b(t)])=λf1(t,I0+βω(t),ω(t),I0+δz(t)), ω(0)=x(0)a(0)=0,ω(0)=x(0)a(0)=0,ω(1)=x(1)a(1)=01x(s)dA(s)01a(s)dA(s)=01ω(s)dA(s),

and D0+γδz(t)=D0+γδ(y(t)b(t))=D0+γδy(t)D0+γδb(t)=λ[f2(t,I0+β[x(t)a(t)],I0+δ[y(t)b(t)],[y(t)b(t)])+q2(t)][λq2(t)]=λf2(t,I0+β[x(t)a(t)],I0+δ[y(t)b(t)],[y(t)b(t)])=λf2(t,I0+βω(t),I0+δz(t),z(t)), z(0)=y(0)b(0)=0,z(0)=y(0)b(0)=0,z(1)=y(1)b(1)=01y(s)dB(s)01b(s)dB(s)=01z(s)dB(s).

So (ω, z) is a positive solution of systems (8). It follows from Lemma 2.8 that (u(t)=I0+βω(t),v(t)=I0+βz(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,u=max0t1|u(t)|,v=max0t1|v(t)|,(u,v)X.

Let P=(u,v)X:u(t)ρ11k1(t)u,v(t)ρ21k2(t)v,t[0,1],

then P is a cone of X. Define an operator A : PX by A(x,y)=(A1(x,y),A2(x,y)),

where A1, A2 : PC[0, 1] are defined by A1(x,y)(t)=λ01G1(t,s)(f1(s,I0+β[x(s)a(s)],[x(s)a(s)],I0+δ[y(s)b(s)])+q1(s))ds,A2(x,y)(t)=λ01G2(t,s)(f2(s,I0+β[x(s)a(s)],I0+δ[y(s)b(s)],[y(s)b(s)])+q2(s))ds.

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 : PP is a completely continuous operator.

Proof

For any (x, y) ∈ P, there exists a constant L > 0 such that ∥(x, y)∥1L, then [x(s)a(s)]x(s)x(x,y)1L,s[0,1],[y(s)b(s)]y(s)y(x,y)1L,s[0,1], I0+β[x(s)a(s)]=0s(st)β1[x(t)a(t)]Γ(β)dtLΓ(β+1),I0+δ[y(s)b(s)]=0s(st)δ1[y(t)b(t)]Γ(δ)dtLΓ(δ+1).

It follows from Lemma 2.7 that A1(x,y)(t)λ01ρ1G1(1,s)(p1(s)g1(s,I0+β[x(s)a(s)],[x(s)a(s)],I0+δ[y(s)b(s)])+q1(s))dsλρ1(M+1)01G1(1,s)(p1(s)+q1(s))ds<+,A2(x,y)(t)λ01ρ2G2(1,s)(p2(s)g2(s,I0+β[x(s)a(s)],I0+δ[y(s)b(s)],[y(s)b(s)])+q2(s))dsλρ2(M+1)01G2(1,s)(p2(s)+q2(s))ds<+,

where M=maxmaxt[0,1],0u1LΓ(β+1),0u2L,0u3LΓ(δ+1)g1(t,u1,u2,u3),maxt[0,1],0v1LΓ(β+1),0v2LΓ(δ+1),0v3Lg2(t,v1,v2,v3).

Thus, A : PX is well defined.

Next, we prove A(P) ⊂ P. Denote F1(s)=f1(s,I0+β[x(s)a(s)],[x(s)a(s)],I0+δ[y(s)b(s)])+q1(s),s[0,1],F2(s)=f2(s,I0+β[x(s)a(s)],I0+δ[y(s)b(s)],[y(s)b(s)])+q2(s),s[0,1].

For any (x, y) ∈ P, we have A1(x,y)=max0t1|A1(x,y)(t)|λρ101G1(1,s)F1(s)ds.

So A1(x,y)(t)λ01k1(t)G1(1,s)F1(s)dsρ11k1(t)A1(x,y),t[0,1].

Similarly, A2(x,y)(t)ρ21k2(t)A2(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 : PP 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, θΩ1, Ω1Ω2, A : P ∩ (Ω2Ω1) → P is a completely continuous operator. If the following conditions are satisfied: Axx,xPΩ1,Axx,xPΩ2,orAxx,xPΩ1,Axx,xPΩ2,

then A has at least one fixed point in P ∩(Ω2Ω1).

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 limu2+mint[a,b]u1,u30f1(t,u1,u2,u3)u2=+,limv3+mint[a,b]v1,v20f2(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¯=min1,r2ρ1(g1+1)01G1(1,s)[p1(s)+q1(s)]ds,r2ρ2(g2+1)01G2(1,s)[p2(s)+q2(s)]ds,

where g1=maxg1(s,u1,u2,u3):0s1,0u1rΓ(β+1),0u2r,0u3rΓ(δ+1),g2=maxg2(s,v1,v2,v3):0s1,0v1rΓ(β+1),0v2rΓ(δ+1),0v3r.

Suppose 0 < λ < λ, then for any (x, y) ∈ P∂Ω1, s ∈[0, 1], we have 0[x(s)a(s)]x(s)xr,0[y(s)b(s)]y(s)yr,I0+β[x(s)a(s)]=0s(sτ)β1[x(τ)a(τ)]Γ(β)dτrΓ(β+1),I0+δ[y(s)b(s)]=0s(sτ)δ1[y(τ)b(τ)]Γ(δ)dτrΓ(δ+1).

Thus, A1(x,y)λ01ρ1G1(1,s)[f1(s,I0+β[x(s)a(s)],[x(s)a(s)],I0+δ[y(s)b(s)])+q1(s)]dsλρ101G1(1,s)[p1(s)g1(s,I0+β[x(s)a(s)],[x(s)a(s)],I0+δ[y(s)b(s)])+q1(s)]ds<λ¯ρ1(g1+1)01G1(1,s)[p1(s)+q1(s)]dsr2,

and A2(x,y)λ01ρ2G2(1,s)[f2(s,I0+β[x(s)a(s)],I0+δ[y(s)b(s)],[y(s)b(s)])+q1(s)]dsλρ201G2(1,s)[p2(s)g2(s,I0+β[x(s)a(s)],I0+δ[y(s)b(s)],[y(s)b(s)])+q2(s)]ds<λ¯ρ2(g2+1)01G2(1,s)[p2(s)+q2(s)]dsr2.

So A(x,y)1=A1(x,y)+A2(x,y)r=(x,y)1,(x,y)PΩ1.(12)

On the other hand, let ki~=mint[a,b]ki(t) and L~>max4ρ1λk1~2abG1(1,s)ds,4ρ2λk2~2abG2(1,s)ds. By (H4), there exists N > 0 such that f1(t,u1,u2,u3)L~u2,t[a,b],u2N,u1,u30,f2(t,v1,v2,v3)L~v3,t[a,b],v3N,v1,v20.

Let R>max2r,4ρ1Nk1~,4ρ2Nk2~,Ω2={(x,y)P:(x,y)1<R}.

For any (x, y) ∈ P∂Ω2, ∥(x, y)∥1 = R, we have xR2 or yR2. If xR2, we deduce x(t)a(t)ρ11k1(t)xρ1Γ(αβ)k1(t)01q1(s)ds=k1(t)xρ1ρ1Γ(αβ)01q1(s)dsk1~R2ρ1R4ρ1=k1~4ρ1RN,t[a,b],

then f1(s,I0+β[x(s)a(s)],[x(s)a(s)],I0+δ[y(s)b(s)])=f1(s,I0+β(x(s)a(s)),(x(s)a(s)),I0+δ[y(s)b(s)])L~[x(s)a(s)]k1~L~R4ρ1.

Thus A1(x,y)=maxt[0,1]λ01G1(t,s)[f1(s,I0+β[x(s)a(s)],[x(s)a(s)],I0+δ[y(s)b(s)])+q1(s)]dsmaxt[0,1]λabk1(t)G1(1,s)L~(x(s)a(s))dsλk1~L~abG1(1,s)k1~4ρ1Rds=λk1~2L~4ρ1RabG1(1,s)dsR=(x,y)1.

If yR2, in a similar manner, we have y(t)b(t)k2~4ρ2RN,t[a,b], and A2(x,y)λk2~2L~4ρ2RabG2(1,s)dsR=(x,y)1.

Thus A(x,y)1=A1(x,y)+A2(x,y)(x,y)1,(x,y)PΩ2.(13)

By (12), (13) and Lemma 2.12, A has a fixed point (, ) with r ≤ ∥(, )∥1R. Now, we will prove (t) > a(t), (t) ≥ b(t)(or (t) ≥ a(t), (t) > b(t)), t ∈ (0, 1). We shall divide the proof into three cases: (i) x~r2,y~r2;(ii)x~>r2,y~<r2;(iii)x~<r2,y~>r2.

  • Case (i)

    If ∥∥ ≥ r2, from (H3) and Lemma 2.9, we get x~(t)ρ11k1(t)x~ρ11k1(t)r2>ρ11k1(t)ρ12Γ(αβ)01q1(s)ds=ρ1Γ(αβ)k1(t)01q1(s)dsa(t),t(0,1).

    Similarly, if ∥∥ ≥ r2 we obtain (t) > b(t), t ∈ (0, 1).

  • Case (ii)

    For ∥∥ > r2, similar to (i), we have (t) > a(t), t ∈ (0, 1). For ∥∥ < r2, we have 0[y~(s)b(s)]y~(s)y~r2,0I0+β[x~(s)a(s)]=I0+β(x~(s)a(s))RΓ(β+1),0I0+δ[y~(s)b(s)]r2Γ(δ+1).

    It follows from (H2) that f2(t,v1,v2,v3)0,(t,v1,v2,v3)(0,1)×0,RΓ(β+1)×0,r2Γ(δ+1)×0,r2,

    then y~(t)=λ01G2(t,s)[f2(s,I0+β[x~(s)a(s)],I0+δ[y~(s)b(s)],[y~(s)b(s)])+q2(s)]dsλ01G2(t,s)q2(s)ds=b(t),t(0,1).

  • Case (iii)

    If ∥∥ < r2 and ∥∥ > r2, similar to (ii), we have (t) ≥ a(t), (t) > b(t), t ∈ (0, 1).

So by Lemma 2.10 we know that (u¯(t),v¯(t))=(I0+β(x~(t)a(t)),I0+δ(y~(t)b(t))) is a positive solution of systems (1). □

4 An Example

Example 4.1

Consider the following problem: D0+52(D0+14u(t))+λπt(1t)(u(t)90)2+(D0+14u(t)3)4+v2(t)685t4148=0,t(0,1),D0+94(D0+18v(t))+λΓ(94)1t4u2(t)+(v(t)30)2+(D0+18v(t)3)230t112=0,t(0,1),D0+14u(0)=D0+54u(0)=0,D0+54u(1)=9697D0+14u116,D0+18v(0)=D0+98v(0)=0,D0+98v(1)=4041D0+18v116.(14)

Problem (14) can be regarded as a problem of the form (1) with α=114,β=14,γ=198,δ=18, A(t)=0,t0,116,9697,t116,1,B(t)=0,t0,116,4041,t116,1, f1(t,u1,u2,u3)=πt341t(u190)2+(u23)4+u32685π48t(1t),f2(t,v1,v2,v3)=Γ(94)t1t4v12+(v230)2+(v33)230Γ(94)121t4.

Evidently, f1, f2 : (0, 1) × [0, +∞)3 → (−∞, +∞) are continuous and singular at t = 0, 1,

By direct calculations, we get k1(t)=t32,k2(t)=t54,γ1(t)=23t32,γ2(t)=45t54,ρ1=2,ρ2=2,ȷ1=01γ1(t)dA(t)=9697×13×11632=197,1(s)=9697G1116,s0,ȷ2=01γ2(t)dB(t)=4041×45×11654=141,2(s)=4041G2116,s0,

so condition (H1) is satisfied. Let q1(t)=π48t(1t),p1(t)=πt341t,g1(t,u2,u2,u3)=(u190)2+(u23)4+u32685,q2(t)=Γ(94)121t4,p2(t)=Γ(94)t1t4,g2(t,v2,v2,v3)=v12+(v230)2+(v33)230,

then 01q1(t)dt=π48B12,12=ππ48,01q2(t)dt=Γ(94)9,01p1(t)dt=πB14,12,01p2(t)dt=Γ94B34,12,

so condition (H2) is satisfied. On the other hand, max2ρ12Γ(αβ)01q1(s)ds,2ρ22Γ(γδ)01q2(s)ds=89.

Select r = 2, then f1(t,u1,u2,u3)πt(1t)(u190)2+(u23)4+u32685148>0,(t,u1,u2,u3)(0,1)×0,1Γ(54)×[0,1]×[0,+), f2(t,v1,v2,v3)Γ(94)1t4v12+(v230)2+(v33)230112>0,(t,v1,v2,v3)(0,1)×[0,+)×0,1Γ(98)×[0,1].

Hence, condition (H3) is satisfied. In addition, for any [a, b] ⊂ (0, 1), limu2+mint[a,b]u1,u30f1(t,u1,u2,u3)u2=+,limv3+mint[a,b]v1,v20f2(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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About the article

Received: 2017-11-19

Accepted: 2018-04-19

Published Online: 2018-06-07


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 581–596, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0055.

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© 2018 Hao and Wang, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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