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### formerly Central European Journal of Mathematics

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

Xinan Hao
/ Huaqing Wang
Online erschienen: 07.06.2018 | 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.

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+α−β(D0+βu(t))+λf1(t,u(t),D0+βu(t),v(t))=0,0(1)

where $\begin{array}{}{D}_{{0}^{+}}^{\alpha -\beta },{D}_{{0}^{+}}^{\beta },{D}_{{0}^{+}}^{\gamma -\delta }\text{\hspace{0.17em}and\hspace{0.17em}}{D}_{{0}^{+}}^{\delta }\end{array}$ 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, $\begin{array}{}{\int }_{0}^{1}{D}_{{0}^{+}}^{\beta }u\left(s\right)dA\left(s\right)\text{\hspace{0.17em}and\hspace{0.17em}}{\int }_{0}^{1}{D}_{{0}^{+}}^{\delta }v\left(s\right)dB\left(s\right)\end{array}$ 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(2)

where $\begin{array}{}{D}_{{0}^{+}}^{\alpha }\text{\hspace{0.17em}and\hspace{0.17em}}{D}_{{0}^{+}}^{\beta }\end{array}$ are the standard Riemann-Liouville fractional derivatives, A, B are nondecreasing functions of bunded variations, $\begin{array}{}{\int }_{0}^{1}\end{array}$u(s)dA(s) and $\begin{array}{}{\int }_{0}^{1}\end{array}$v(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) = $\begin{array}{}{\int }_{0}^{1}\end{array}$ϕ(s)u(s)ds, v(1) = $\begin{array}{}{\int }_{0}^{1}\end{array}$φ(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) = $\begin{array}{}{\int }_{0}^{1}\end{array}$ϕ(s)u(s)ds, v(1) = $\begin{array}{}{\int }_{0}^{1}\end{array}$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 $D0+αu(t)+λf(t,u(t),v(t))=0,0(3)

where $\begin{array}{}{D}_{{0}^{+}}^{\alpha }\text{\hspace{0.17em}and\hspace{0.17em}}{D}_{{0}^{+}}^{\beta }\end{array}$ 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) = $\begin{array}{}\sum _{i=1}^{p}{a}_{i}u\left({\xi }_{i}\right),\phantom{\rule{thinmathspace}{0ex}}v\left(1\right)=\sum _{i=1}^{q}{b}_{i}v\left({\eta }_{i}\right),\end{array}$ 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(4)

where λ > 0 is a parameter, 0 < β < 1, α-β > 2, $\begin{array}{}{D}_{{0}^{+}}^{\alpha },{D}_{{0}^{+}}^{\beta }\text{\hspace{0.17em}and\hspace{0.17em}}{D}_{{0}^{+}}^{\gamma }\end{array}$ are the standard Riemann-Liouville fractional derivatives, A, B are nondecreasing functions of bunded variations, $\begin{array}{}{\int }_{0}^{1}{D}_{{0}^{+}}^{\beta }u\left(s\right)dA\left(s\right)\end{array}$ and $\begin{array}{}{\int }_{0}^{1}\end{array}$v(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(5)

where $\begin{array}{}{D}_{{0}^{+}}^{\alpha }\text{\hspace{0.17em}and\hspace{0.17em}}{D}_{{0}^{+}}^{\beta }\end{array}$ 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, $\begin{array}{}{\int }_{0}^{1}\end{array}$u(t)dA(t) and $\begin{array}{}{\int }_{0}^{1}\end{array}$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, 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 $I0+αu(t)=1Γ(α)∫0t(t−s)α−1u(s)ds,$

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 $D0+αu(t)=1Γ(n−α)ddtn∫0t(t−s)n−α−1u(s)ds,$

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 $I0+ρI0+σu(t)=I0+ρ+σu(t),D0+σI0+ρu(t)=I0+ρ−σu(t),D0+σI0+σu(t)=u(t),ddtn(D0+σu(t))=D0+n+σu(t).$

#### 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 $I0+αD0+αu(t)=u(t)+c1tα−1+c2tα−2+⋯+cNtα−N,$

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+α−βx(t)+h(t)=0,0

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

where $G1∗(t,s)=1Γ(α−β)tα−β−1(1−s)α−β−2,0≤t≤s≤1,tα−β−1(1−s)α−β−2−(t−s)α−β−1,0≤s≤t≤1,$

and the Green function $\begin{array}{}{G}_{1}^{\ast }\end{array}$ (t, s) has the following properties:

1. $\begin{array}{}{G}_{1}^{\ast }\end{array}$(t, s) > 0, t, s ∈(0, 1).

2. k1(t) $\begin{array}{}{G}_{1}^{\ast }\end{array}$(1, s) ≤ $\begin{array}{}{G}_{1}^{\ast }\end{array}$(t, s) ≤ $\begin{array}{}{G}_{1}^{\ast }\end{array}$(1, s), t, s ∈ [0, 1].

3. $\begin{array}{}{G}_{1}^{\ast }\left(t,s\right)\le \frac{{k}_{1}\left(t\right)}{\mathit{\Gamma }\left(\alpha -\beta \right)},\phantom{\rule{thinmathspace}{0ex}}t,s\in \left[0,1\right],\end{array}$ where k1(t) = tαβ−1.

By Lemma 2.4, the unique solution of the BVP $D0+α−βx(t)=0,0

is $\begin{array}{}{\gamma }_{1}\left(t\right)=\frac{{t}^{\alpha -\beta -1}}{\alpha -\beta -1}.\end{array}$ By [33], the Green function for the BVP $D0+α−βx(t)+h(t)=0,0

is given by $G1(t,s)=G1∗(t,s)+γ1(t)1−ȷ1ℓ1(s),$(6)

where $ȷ1=∫01γ1(t)dA(t)≠1,ℓ1(s)=∫01G1∗(t,s)dA(t),s∈[0,1].$

Similarly, the Green function for the BVP $D0+γ−δy(t)+h(t)=0,0

is given by $G2(t,s)=G2∗(t,s)+γ2(t)1−ȷ2ℓ2(s),$(7)

where $G2∗(t,s)=1Γ(γ−δ)tγ−δ−1(1−s)γ−δ−2,0≤t≤s≤1,tγ−δ−1(1−s)γ−δ−2−(t−s)γ−δ−1,0≤s≤t≤1,ȷ2=∫01γ2(t)dB(t)≠1,ℓ2(s)=∫01G2∗(t,s)dB(t),γ2(t)=tγ−δ−1γ−δ−1.$

#### Lemma 2.6

([33]). The Green function $\begin{array}{}{G}_{2}^{\ast }\end{array}$(t, s) has the following properties:

1. $\begin{array}{}{G}_{2}^{\ast }\end{array}$(t, s) > 0, t, s ∈(0, 1).

2. k2(t) $\begin{array}{}{G}_{2}^{\ast }\end{array}$(1, s) ≤ $\begin{array}{}{G}_{2}^{\ast }\end{array}$(t, s) ≤ $\begin{array}{}{G}_{2}^{\ast }\end{array}$(1, s), t, s ∈ [0, 1].

3. $\begin{array}{}{G}_{2}^{\ast }\left(t,s\right)\le \frac{{k}_{2}\left(t\right)}{\mathit{\Gamma }\left(\gamma -\delta \right)},\phantom{\rule{thinmathspace}{0ex}}t,s\in \left[0,1\right],\end{array}$ 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) ≥ $\begin{array}{}{G}_{i}^{\ast }\end{array}$(t, s) > 0, t, s ∈ (0, 1), i = 1, 2.

2. ki(t) $\begin{array}{}{G}_{i}^{\ast }\end{array}$(1, s) ≤ Gi(t, s) ≤ ρi $\begin{array}{}{G}_{i}^{\ast }\end{array}$(1, s), t, s ∈[0, 1], i = 1, 2.

3. $\begin{array}{}{G}_{1}\left(t,s\right)\le \frac{{\rho }_{1}}{\mathit{\Gamma }\left(\alpha -\beta \right)}{k}_{1}\left(t\right),{G}_{2}\left(t,s\right)\le \frac{{\rho }_{2}}{\mathit{\Gamma }\left(\gamma -\delta \right)}{k}_{2}\left(t\right),\phantom{\rule{thinmathspace}{0ex}}t,s\in \left[0,1\right],\end{array}$ where $ρ1=1+∫01dA(t)1−ȷ1,ρ2=1+∫01dB(t)1−ȷ2.$

Now let us consider the following modified problem of systems (1) $D0+α−βω(t)+λf1(t,I0+βω(t),ω(t),I0+δz(t))=0,0(8)

#### Lemma 2.8

If (ω, z) ∈ C[0, 1] × C[0, 1] is a positive solution of systems (8), then $\begin{array}{}\left({I}_{{0}^{+}}^{\beta }\omega ,\phantom{\rule{thinmathspace}{0ex}}{I}_{{0}^{+}}^{\delta }z\right)\end{array}$ 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 $\begin{array}{}u\left(t\right)={I}_{{0}^{+}}^{\beta }\omega \left(t\right),\phantom{\rule{thinmathspace}{0ex}}v\left(t\right)={I}_{{0}^{+}}^{\delta }z\left(t\right),\end{array}$ then $D0+α−β(D0+βu(t))=D0+α−β(D0+βI0+βω(t))=D0+α−βω(t)=−λf1(t,I0+βω(t),ω(t),I0+δz(t))=−λf1(t,u(t),D0+βu(t),v(t)),D0+γ−δ(D0+δv(t))=D0+γ−δ(D0+δI0+δz(t))=D0+γ−δz(t)=−λf2(t,I0+βω(t),I0+δz(t),z(t))=−λf2(t,u(t),v(t),D0+δv(t)),$ $D0+βu(0)=ω(0)=0,D0+β+1u(0)=ddt(D0+βu(t))|t=0=ω′(0)=0,D0+β+1u(1)=ddt(D0+βu(t))|t=1=ω′(1)=∫01ω(s)dA(s)=∫01D0+βu(s)dA(s),D0+δv(0)=z(0)=0,D0+δ+1v(0)=ddt(D0+δv(t))|t=0=z′(0)=0,D0+δ+1v(1)=ddt(D0+δv(t))|t=1=z′(1)=∫01z(s)dB(s)=∫01D0+δv(s)dB(s).$

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) = $\begin{array}{}\left({I}_{{0}^{+}}^{\beta }\omega ,{I}_{{0}^{+}}^{\delta }z\right)\end{array}$ 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 $r>max2ρ12Γ(α−β)∫01q1(s)ds,2ρ22Γ(γ−δ)∫01q2(s)ds$

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+α−βx(t)+λ[f1(t,I0+β[x(t)−a(t)]∗,[x(t)−a(t)]∗,I0+δ[y(t)−b(t)]∗)+q1(t)]=0,0(9)

where a(t) = λ $\begin{array}{}{\int }_{0}^{1}\end{array}$G1(t, s)q1(s)ds and b(t) = λ $\begin{array}{}{\int }_{0}^{1}\end{array}$G2(t, s)q2(s)ds are the solutions of the following BVPs (10) and (11), respectively, $D0+α−βa(t)+λq1(t)=0,0(10) $D0+γ−δb(t)+λq2(t)=0,0(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 $a(t)≤λρ1Γ(α−β)k1(t)∫01q1(t)dt,b(t)≤λρ2Γ(γ−δ)k2(t)∫01q2(t)dt,t∈[0,1].$

#### 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 $\begin{array}{}\left(u\left(t\right)={I}_{{0}^{+}}^{\beta }\omega \left(t\right),\phantom{\rule{thinmathspace}{0ex}}v\left(t\right)={I}_{{0}^{+}}^{\delta }z\left(t\right)\right)\end{array}$ 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 $D0+α−βω(t)=D0+α−β(x(t)−a(t))=D0+α−βx(t)−D0+α−βa(t)=−λ[f1(t,I0+β[x(t)−a(t)]∗,[x(t)−a(t)]∗,I0+δ[y(t)−b(t)]∗)+q1(t)]−[−λq1(t)]=−λf1(t,I0+β[x(t)−a(t)]∗,[x(t)−a(t)]∗,I0+δ[y(t)−b(t)]∗)=−λf1(t,I0+βω(t),ω(t),I0+δz(t)),$ $ω(0)=x(0)−a(0)=0,ω′(0)=x′(0)−a′(0)=0,ω′(1)=x′(1)−a′(1)=∫01x(s)dA(s)−∫01a(s)dA(s)=∫01ω(s)dA(s),$

and $D0+γ−δz(t)=D0+γ−δ(y(t)−b(t))=D0+γ−δy(t)−D0+γ−δb(t)=−λ[f2(t,I0+β[x(t)−a(t)]∗,I0+δ[y(t)−b(t)]∗,[y(t)−b(t)]∗)+q2(t)]−[−λq2(t)]=−λf2(t,I0+β[x(t)−a(t)]∗,I0+δ[y(t)−b(t)]∗,[y(t)−b(t)]∗)=−λf2(t,I0+βω(t),I0+δz(t),z(t)),$ $z(0)=y(0)−b(0)=0,z′(0)=y′(0)−b′(0)=0,z′(1)=y′(1)−b′(1)=∫01y(s)dB(s)−∫01b(s)dB(s)=∫01z(s)dB(s).$

So (ω, z) is a positive solution of systems (8). It follows from Lemma 2.8 that $\begin{array}{}\left(u\left(t\right)={I}_{{0}^{+}}^{\beta }\omega \left(t\right),\phantom{\rule{thinmathspace}{0ex}}v\left(t\right)={I}_{{0}^{+}}^{\beta }z\left(t\right)\right)\end{array}$ is a positive solution of systems (1).

Let X = C[0, 1] × C[0, 1], then X is a Banach space with the norm $∥(u,v)∥1=∥u∥+∥v∥,∥u∥=max0≤t≤1|u(t)|,∥v∥=max0≤t≤1|v(t)|,(u,v)∈X.$

Let $P=(u,v)∈X:u(t)≥ρ1−1k1(t)∥u∥,v(t)≥ρ2−1k2(t)∥v∥,t∈[0,1],$

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 $A1(x,y)(t)=λ∫01G1(t,s)(f1(s,I0+β[x(s)−a(s)]∗,[x(s)−a(s)]∗,I0+δ[y(s)−b(s)]∗)+q1(s))ds,A2(x,y)(t)=λ∫01G2(t,s)(f2(s,I0+β[x(s)−a(s)]∗,I0+δ[y(s)−b(s)]∗,[y(s)−b(s)]∗)+q2(s))ds.$

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 $[x(s)−a(s)]∗≤x(s)≤∥x∥≤∥(x,y)∥1≤L,s∈[0,1],[y(s)−b(s)]∗≤y(s)≤∥y∥≤∥(x,y)∥1≤L,s∈[0,1],$ $I0+β[x(s)−a(s)]∗=∫0s(s−t)β−1[x(t)−a(t)]∗Γ(β)dt≤LΓ(β+1),I0+δ[y(s)−b(s)]∗=∫0s(s−t)δ−1[y(t)−b(t)]∗Γ(δ)dt≤LΓ(δ+1).$

It follows from Lemma 2.7 that $A1(x,y)(t)≤λ∫01ρ1G1∗(1,s)(p1(s)g1(s,I0+β[x(s)−a(s)]∗,[x(s)−a(s)]∗,I0+δ[y(s)−b(s)]∗)+q1(s))ds≤λρ1(M+1)∫01G1∗(1,s)(p1(s)+q1(s))ds<+∞,A2(x,y)(t)≤λ∫01ρ2G2∗(1,s)(p2(s)g2(s,I0+β[x(s)−a(s)]∗,I0+δ[y(s)−b(s)]∗,[y(s)−b(s)]∗)+q2(s))ds≤λρ2(M+1)∫01G2∗(1,s)(p2(s)+q2(s))ds<+∞,$

where $M=maxmaxt∈[0,1],0≤u1≤LΓ(β+1),0≤u2≤L,0≤u3≤LΓ(δ+1)g1(t,u1,u2,u3),maxt∈[0,1],0≤v1≤LΓ(β+1),0≤v2≤LΓ(δ+1),0≤v3≤Lg2(t,v1,v2,v3).$

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 $∥A1(x,y)∥=max0≤t≤1|A1(x,y)(t)|≤λρ1∫01G1∗(1,s)F1(s)ds.$

So $A1(x,y)(t)≥λ∫01k1(t)G1∗(1,s)F1(s)ds≥ρ1−1k1(t)∥A1(x,y)∥,t∈[0,1].$

Similarly, $\begin{array}{}{A}_{2}\left(x,y\right)\left(t\right)\ge {\rho }_{2}^{-1}{k}_{2}\left(t\right)\parallel {A}_{2}\left(x,y\right)\parallel ,\phantom{\rule{thinmathspace}{0ex}}t\in \left[0,1\right].\end{array}$ 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: $∥Ax∥≤∥x∥,∀x∈P∩∂Ω1,∥Ax∥≥∥x∥,∀x∈P∩∂Ω2,or∥Ax∥≥∥x∥,∀x∈P∩∂Ω1,∥Ax∥≤∥x∥,∀x∈P∩∂Ω2,$

then A has at least one fixed point in P ∩(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 $limu2→+∞mint∈[a,b]u1,u3≥0f1(t,u1,u2,u3)u2=+∞,limv3→+∞mint∈[a,b]v1,v2≥0f2(t,v1,v2,v3)v3=+∞.$

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

where $g1∗=maxg1(s,u1,u2,u3):0≤s≤1,0≤u1≤rΓ(β+1),0≤u2≤r,0≤u3≤rΓ(δ+1),g2∗=maxg2(s,v1,v2,v3):0≤s≤1,0≤v1≤rΓ(β+1),0≤v2≤rΓ(δ+1),0≤v3≤r.$

Suppose 0 < λ < λ, then for any (x, y) ∈ P∂Ω1, s ∈[0, 1], we have $0≤[x(s)−a(s)]∗≤x(s)≤∥x∥≤r,0≤[y(s)−b(s)]∗≤y(s)≤∥y∥≤r,I0+β[x(s)−a(s)]∗=∫0s(s−τ)β−1[x(τ)−a(τ)]∗Γ(β)dτ≤rΓ(β+1),I0+δ[y(s)−b(s)]∗=∫0s(s−τ)δ−1[y(τ)−b(τ)]∗Γ(δ)dτ≤rΓ(δ+1).$

Thus, $∥A1(x,y)∥≤λ∫01ρ1G1∗(1,s)[f1(s,I0+β[x(s)−a(s)]∗,[x(s)−a(s)]∗,I0+δ[y(s)−b(s)]∗)+q1(s)]ds≤λρ1∫01G1∗(1,s)[p1(s)g1(s,I0+β[x(s)−a(s)]∗,[x(s)−a(s)]∗,I0+δ[y(s)−b(s)]∗)+q1(s)]ds<λ¯ρ1(g1∗+1)∫01G1∗(1,s)[p1(s)+q1(s)]ds≤r2,$

and $∥A2(x,y)∥≤λ∫01ρ2G2∗(1,s)[f2(s,I0+β[x(s)−a(s)]∗,I0+δ[y(s)−b(s)]∗,[y(s)−b(s)]∗)+q1(s)]ds≤λρ2∫01G2∗(1,s)[p2(s)g2(s,I0+β[x(s)−a(s)]∗,I0+δ[y(s)−b(s)]∗,[y(s)−b(s)]∗)+q2(s)]ds<λ¯ρ2(g2∗+1)∫01G2∗(1,s)[p2(s)+q2(s)]ds≤r2.$

So $∥A(x,y)∥1=∥A1(x,y)∥+∥A2(x,y)∥≤r=∥(x,y)∥1,∀(x,y)∈P∩∂Ω1.$(12)

On the other hand, let $\begin{array}{}\stackrel{~}{{k}_{i}}=\underset{t\in \left[a,b\right]}{min}{k}_{i}\left(t\right)\text{\hspace{0.17em}and\hspace{0.17em}}\stackrel{~}{L}>max\left\{\frac{4{\rho }_{1}}{\lambda {\stackrel{~}{{k}_{1}}}^{2}{\int }_{a}^{b}{G}_{1}^{\ast }\left(1,s\right)ds},\frac{4{\rho }_{2}}{\lambda {\stackrel{~}{{k}_{2}}}^{2}{\int }_{a}^{b}{G}_{2}^{\ast }\left(1,s\right)ds}\right\}.\end{array}$ By (H4), there exists N > 0 such that $f1(t,u1,u2,u3)≥L~u2,t∈[a,b],u2≥N,u1,u3≥0,f2(t,v1,v2,v3)≥L~v3,t∈[a,b],v3≥N,v1,v2≥0.$

Let $R>max2r,4ρ1Nk1~,4ρ2Nk2~,Ω2={(x,y)∈P:∥(x,y)∥1

For any (x, y) ∈ P∂Ω2, ∥(x, y)∥1 = R, we have $\begin{array}{}\parallel x\parallel \ge \frac{R}{2}\text{\hspace{0.17em}or\hspace{0.17em}}\parallel y\parallel \ge \frac{R}{2}.\text{\hspace{0.17em}If\hspace{0.17em}}\parallel x\parallel \ge \frac{R}{2},\end{array}$ we deduce $x(t)−a(t)≥ρ1−1k1(t)∥x∥−ρ1Γ(α−β)k1(t)∫01q1(s)ds=k1(t)∥x∥ρ1−ρ1Γ(α−β)∫01q1(s)ds≥k1~R2ρ1−R4ρ1=k1~4ρ1R≥N,t∈[a,b],$

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 $∥A1(x,y)∥=maxt∈[0,1]λ∫01G1(t,s)[f1(s,I0+β[x(s)−a(s)]∗,[x(s)−a(s)]∗,I0+δ[y(s)−b(s)]∗)+q1(s)]ds≥maxt∈[0,1]λ∫abk1(t)G1∗(1,s)L~(x(s)−a(s))ds≥λk1~L~∫abG1∗(1,s)k1~4ρ1Rds=λk1~2L~4ρ1R∫abG1∗(1,s)ds≥R=∥(x,y)∥1.$

If $\begin{array}{}\parallel y\parallel \ge \frac{R}{2},\end{array}$ in a similar manner, we have $\begin{array}{}y\left(t\right)-b\left(t\right)\ge \frac{\stackrel{~}{{k}_{2}}}{4{\rho }_{2}}R\ge N,\phantom{\rule{thinmathspace}{0ex}}t\in \left[a,b\right],\end{array}$ and $∥A2(x,y)∥≥λk2~2L~4ρ2R∫abG2∗(1,s)ds≥R=∥(x,y)∥1.$

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 (, ) 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) $\begin{array}{}\parallel \stackrel{~}{x\phantom{\rule{thinmathspace}{0ex}}}\parallel \ge \frac{r}{2},\phantom{\rule{thinmathspace}{0ex}}\parallel \stackrel{~}{y\phantom{\rule{thinmathspace}{0ex}}}\parallel \ge \frac{r}{2};\left(\text{ii}\right)\parallel \stackrel{~}{x\phantom{\rule{thinmathspace}{0ex}}}\parallel >\frac{r}{2},\phantom{\rule{thinmathspace}{0ex}}\parallel \stackrel{~}{y\phantom{\rule{thinmathspace}{0ex}}}\parallel <\frac{r}{2};\left(\text{iii}\right)\parallel \stackrel{~}{x\phantom{\rule{thinmathspace}{0ex}}}\parallel <\frac{r}{2},\phantom{\rule{thinmathspace}{0ex}}\parallel \stackrel{~}{y\phantom{\rule{thinmathspace}{0ex}}}\parallel >\frac{r}{2}.\end{array}$

• Case (i)

If ∥∥ ≥ $\begin{array}{}\frac{r}{2}\end{array}$, from (H3) and Lemma 2.9, we get $x~(t)≥ρ1−1k1(t)∥x~∥≥ρ1−1k1(t)r2>ρ1−1k1(t)ρ12Γ(α−β)∫01q1(s)ds=ρ1Γ(α−β)k1(t)∫01q1(s)ds≥a(t),t∈(0,1).$

Similarly, if ∥∥ ≥ $\begin{array}{}\frac{r}{2}\end{array}$ we obtain (t) > b(t), t ∈ (0, 1).

• Case (ii)

For ∥∥ > $\begin{array}{}\frac{r}{2}\end{array}$, similar to (i), we have (t) > a(t), t ∈ (0, 1). For ∥∥ < $\begin{array}{}\frac{r}{2}\end{array}$, we have $0≤[y~(s)−b(s)]∗≤y~(s)≤∥y~∥≤r2,0≤I0+β[x~(s)−a(s)]∗=I0+β(x~(s)−a(s))≤RΓ(β+1),0≤I0+δ[y~(s)−b(s)]∗≤r2Γ(δ+1).$

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 $y~(t)=λ∫01G2(t,s)[f2(s,I0+β[x~(s)−a(s)]∗,I0+δ[y~(s)−b(s)]∗,[y~(s)−b(s)]∗)+q2(s)]ds≥λ∫01G2(t,s)q2(s)ds=b(t),t∈(0,1).$

• Case (iii)

If ∥∥ < $\begin{array}{}\frac{r}{2}\end{array}$ and ∥∥ > $\begin{array}{}\frac{r}{2}\end{array}$, similar to (ii), we have (t) ≥ a(t), (t) > b(t), t ∈ (0, 1).

So by Lemma 2.10 we know that $\begin{array}{}\left(\overline{u}\left(t\right),\overline{v}\left(t\right)\right)=\left({I}_{{0}^{+}}^{\beta }\left(\stackrel{~}{x\phantom{\rule{thinmathspace}{0ex}}}\left(t\right)-a\left(t\right)\right),{I}_{{0}^{+}}^{\delta }\left(\stackrel{~}{y\phantom{\rule{thinmathspace}{0ex}}}\left(t\right)-b\left(t\right)\right)\right)\end{array}$ is a positive solution of systems (1). □

## 4 An Example

#### Example 4.1

Consider the following problem: $D0+52(D0+14u(t))+λπt(1−t)(u(t)−90)2+(D0+14u(t)−3)4+v2(t)685t4−148=0,t∈(0,1),D0+94(D0+18v(t))+λΓ(94)1−t4u2(t)+(v(t)−30)2+(D0+18v(t)−3)230t−112=0,t∈(0,1),D0+14u(0)=D0+54u(0)=0,D0+54u(1)=9697D0+14u116,D0+18v(0)=D0+98v(0)=0,D0+98v(1)=4041D0+18v116.$(14)

Problem (14) can be regarded as a problem of the form (1) with $\begin{array}{}\alpha =\frac{11}{4},\beta =\frac{1}{4},\phantom{\rule{thinmathspace}{0ex}}\gamma =\frac{19}{8},\phantom{\rule{thinmathspace}{0ex}}\delta =\frac{1}{8},\end{array}$ $A(t)=0,t∈0,116,9697,t∈116,1,B(t)=0,t∈0,116,4041,t∈116,1,$ $f1(t,u1,u2,u3)=πt341−t(u1−90)2+(u2−3)4+u32685−π48t(1−t),f2(t,v1,v2,v3)=Γ(94)t1−t4v12+(v2−30)2+(v3−3)230−Γ(94)121−t4.$

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

By direct calculations, we get $k1(t)=t32,k2(t)=t54,γ1(t)=23t32,γ2(t)=45t54,ρ1=2,ρ2=2,ȷ1=∫01γ1(t)dA(t)=9697×13×11632=197,ℓ1(s)=9697G1∗116,s≥0,ȷ2=∫01γ2(t)dB(t)=4041×45×11654=141,ℓ2(s)=4041G2∗116,s≥0,$

so condition (H1) is satisfied. Let $q1(t)=π48t(1−t),p1(t)=πt341−t,g1(t,u2,u2,u3)=(u1−90)2+(u2−3)4+u32685,q2(t)=Γ(94)121−t4,p2(t)=Γ(94)t1−t4,g2(t,v2,v2,v3)=v12+(v2−30)2+(v3−3)230,$

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, $max2ρ12Γ(α−β)∫01q1(s)ds,2ρ22Γ(γ−δ)∫01q2(s)ds=89.$

Select r = 2, then $f1(t,u1,u2,u3)≥πt(1−t)(u1−90)2+(u2−3)4+u32685−148>0,(t,u1,u2,u3)∈(0,1)×0,1Γ(54)×[0,1]×[0,+∞),$ $f2(t,v1,v2,v3)≥Γ(94)1−t4v12+(v2−30)2+(v3−3)230−112>0,(t,v1,v2,v3)∈(0,1)×[0,+∞)×0,1Γ(98)×[0,1].$

Hence, condition (H3) is satisfied. In addition, for any [a, b] ⊂ (0, 1), $limu2→+∞mint∈[a,b]u1,u3≥0⁡f1(t,u1,u2,u3)u2=+∞,limv3→+∞mint∈[a,b]v1,v2≥0⁡f2(t,v1,v2,v3)v3=+∞,$

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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## Artikelinformationen

Erhalten: 19.11.2017

Angenommen: 19.04.2018

Online erschienen: 07.06.2018

Quellenangabe: Open Mathematics, Band 16, Heft 1, Seiten 581–596, ISSN (Online) 2391-5455,

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