Multiple Positive Solutions to Multipoint Boundary Value Problem for a System of Second-Order Nonlinear Semipositone Differential Equations on Time Scales

Abstract

We study a system of second-order dynamic equations on time scales (𝑝₁𝑢^∇₁)^Δ(𝑡)−𝑞₁(𝑡)𝑢₁(𝑡)+𝜆𝑓₁(𝑡,𝑢₁(𝑡),𝑢₂(𝑡))=0,𝑡∈(𝑡₁,𝑡_𝑛),(𝑝₂𝑢^∇₂)^Δ(𝑡)−𝑞₂(𝑡)𝑢₂(𝑡)+𝜆𝑓₂(𝑡,𝑢₁(𝑡), 𝑢₂(𝑡))=0, satisfying four kinds of different multipoint boundary value conditions, 𝑓_𝑖 is continuous and semipositone. We derive an interval of 𝜆 such that any 𝜆 lying in this interval, the semipositone coupled boundary value problem has multiple positive solutions. The arguments are based upon fixed-point theorems in a cone.

1. Introduction

In this paper, we consider the following dynamic equations on time scales: 𝑝₁𝑢^∇₁^Δ(𝑡)−𝑞₁(𝑡)𝑢₁(𝑡)+𝜆𝑓₁𝑡,𝑢₁(𝑡),𝑢₂(𝑡)𝑡=0,𝑡∈₁,𝑡_𝑛𝑝,𝜆>0,₂𝑢^∇₂^Δ(𝑡)−𝑞₂(𝑡)𝑢₂(𝑡)+𝜆𝑓₂𝑡,𝑢₁(𝑡),𝑢₂(𝑡)=0, (1) satisfying one of the boundary value conditions 𝛼₁𝑢₁𝑡₁−𝛽₁𝑝₁𝑡₁𝑢^∇₁𝑡₁𝛾=0,₁𝑢₁𝑡_𝑛+𝛿₁𝑝₁𝑡_𝑛𝑢^∇₁𝑡_𝑛=^𝑛−2_𝑖=2𝑏_𝑖𝑢₁𝑡_𝑖,𝛼₂𝑢₂𝑡₁−𝛽₂𝑝₂𝑡₁𝑢^∇₂𝑡₁𝛾=0,₂𝑢₂𝑡_𝑛+𝛿₂𝑝₁𝑡_𝑛𝑢^∇₂𝑡_𝑛=^𝑛−2_𝑖=2𝑏_𝑖𝑢₂𝑡_𝑖,𝛼₁𝑢₁𝑡₁−𝛽₁𝑝₁𝑡₁𝑢^∇₁𝑡₁=^𝑛−2_𝑖=2𝑎_𝑖𝑢₁𝑡_𝑖,𝛾₁𝑢₁𝑡_𝑛+𝛿₁𝑝₁𝑡_𝑛𝑢^∇₁𝑡_𝑛𝛼=0,₂𝑢₂𝑡₁−𝛽₂𝑝₂𝑡₁𝑢^∇₂𝑡₁=^𝑛−2_𝑖=2𝑎_𝑖𝑢₂𝑡_𝑖,𝛾₂𝑢₂𝑡_𝑛+𝛿₂𝑝₁𝑡_𝑛𝑢^∇₂𝑡_𝑛=0,𝛼₁𝑢₁𝑡₁−𝛽₁𝑝₁𝑡₁𝑢^∇₁𝑡₁𝛾=0,₁𝑢₁𝑡_𝑛+𝛿₁𝑝₁𝑡_𝑛𝑢^∇₁𝑡_𝑛=^𝑛−2_𝑖=2𝑏_𝑖𝑢₁𝑡_𝑖,𝛼₂𝑢₂𝑡₁−𝛽₂𝑝₂𝑡₁𝑢^∇₂𝑡₁=^𝑛−2_𝑖=2𝑎_𝑖𝑢₂𝑡_𝑖,𝛾₂𝑢₂𝑡_𝑛+𝛿₂𝑝₁𝑡_𝑛𝑢^∇₂𝑡_𝑛=0,𝛼₁𝑢₁𝑡₁−𝛽₁𝑝₁𝑡₁𝑢^∇₁𝑡₁=^𝑛−2_𝑖=2𝑎_𝑖𝑢₁𝑡_𝑖,𝛾₁𝑢₁𝑡_𝑛+𝛿₁𝑝₁𝑡_𝑛𝑢^∇₁𝑡_𝑛𝛼=0,₂𝑢₂𝑡₁−𝛽₂𝑝₂𝑡₁𝑢^∇₂𝑡₁𝛾=0,₂𝑢₂𝑡_𝑛+𝛿₂𝑝₁𝑡_𝑛𝑢^∇₂𝑡_𝑛=^𝑛−2_𝑖=2𝑏_𝑖𝑢₂𝑡_𝑖, (2)(3)(4)(5) where 𝑝_𝑖,𝑞_𝑖∶𝑡₁,𝑡_𝑛→(0,+∞)with𝑝_𝑖∈𝐶^Δ𝑡₁,𝑡_𝑛,𝑞_𝑖∈𝐶𝑡₁,𝑡_𝑛𝛼for𝑖=1,2;_𝑖,𝛽_𝑖,𝛾_𝑖,𝛿_𝑖∈[0,+∞)with𝛼_𝑖𝛾_𝑖+𝛼_𝑖𝛿_𝑖+𝛽_𝑖𝛾_𝑖>0for𝑖=1,2, (6) and 𝑓_𝑖 is continuously and nonegative functionsquad, 𝑎_𝑖,𝑏_𝑖∈[0,+∞) for 𝑖∈{1,2,…,𝑛}; the points 𝑡_𝑖∈𝕋^𝜅_𝜅 for 𝑖∈{1,2,…,𝑛} with 𝑡₁<𝑡₂<⋯<𝑡_𝑛.

In the past few years, the boundary value problems of dynamic equations on time scales have been studied by many authors (see [1–19] and references). Recently, multipoint boundary value problems on time scale have been studied, for instance, see [1–12].

In 2006, Anderson and Ma [1] studied the second-order multiple time-scale eigenvalue problem: 𝑝𝑦^∇Δ(𝑡)(𝑡)𝑦(𝑡)(𝑡)𝑓𝑦𝑡−𝑞+𝜆ℎ=0,𝑡∈₁,𝑡_𝑛𝑡,𝜆>0,𝛼𝑦₁−𝛽𝑝𝑡₁𝑦^∇𝑡₁=^𝑛−2_𝑖=2𝑎_𝑖𝑦𝑡_𝑖,𝛾𝑦𝑡_𝑛+𝛿𝑝𝑡_𝑛𝑦^∇𝑡_𝑛=^𝑛−2_𝑖=2𝑏_𝑖𝑦𝑡_𝑖, (7) where the functions 𝑓∶[0,+∞)→[0,+∞) and ℎ∶[𝑡₁,𝑡_𝑛]→[0,+∞) are continuous. The authors discuss conditions for the existence of at least one positive solution to the second-order Sturm-Liouville-type multiple eigenvalue problem on time scales.

In 2009, Feng et al. [2] studied 𝑝𝑦^∇Δ(𝑡)(𝑡)𝑦(𝑡),𝑡−𝑞=𝑓𝑡,𝑦𝑡∈₁,𝑡_𝑛,𝛼𝑦𝑡₁−𝛽𝑝𝑡₁𝑦^∇𝑡₁=^𝑛−2_𝑖=2𝑎_𝑖𝑦𝑡_𝑖,𝛾𝑦𝑡_𝑛+𝛿𝑝𝑡_𝑛𝑦^∇𝑡_𝑛=^𝑛−2_𝑖=2𝑏_𝑖𝑦𝑡_𝑖, (8) where the functions 𝑓(𝑡,𝑦)=∑^𝑛_𝑗=1𝑐_𝑗(𝑡)𝑦^𝑣𝑗, 𝑐_𝑗∈𝐶([𝑡₁,𝑡_𝑛],[0,∞)), 𝑣_𝑖∈[0,∞), 𝑗=1,2,…,𝑛. This paper shows the existence of multiple positive solutions for the boundary value problem on time scales.

In 2009, Topal and Yantir [3] studied the second-order nonlinear 𝑚-point boundary value problems 𝑢^Δ∇(𝑡)+𝑎(𝑡)𝑢^Δ(𝑡)(𝑡)𝑢(𝑡)(𝑡)𝑓((𝑡()+𝑏+𝜆𝑞𝑡,𝑢))=0,𝑡∈0,1_𝕋,𝑢𝜌(0)=0,𝑢(𝜎(1))=^𝑚−2_𝑖=1𝛼_𝑖𝑢𝜂_𝑖, (9) where 𝛼_𝑖≥0,0<𝜂_𝑖<𝜂_𝑖+1<1; for  all 𝑖=1,2,…,𝑚−2; 𝑎∈𝐶([0,1],[0,+∞)), 𝑏∈𝐶([0,1],(−∞,0]), 𝑓,𝑞 are continuously and nonegative functions. The authors deal with determining the value of 𝜆, and the existences of multiple positive solutions of the equation are obtained. In 2010, Yuan and Liu [4] also study the second-order 𝑚-point boundary value problems; Yuan and Liu shows the existence of multiple positive solutions if 𝑓 is semipositone and superlinear.

Motivated by the above results mentioned, we study the second-order nonlinear 𝑚-point boundary value problem (1) with boundary condition (𝑘), and nonlinear term may be singularity and semipositone.

In this paper, the nonlinear term 𝑓_𝑖 of (1) is suit to and semipositone and the superlinear case, we shall prove our two existence results for the problem (1) with (𝑘) by using a nonlinear alternative of Leray-Schauder type and Krasnosel'skii fixed-point theorem. This paper is organized as follows. In Section 2, we start with some preliminary lemmas. In Section 3, we give the main result which state the sufficient conditions for (1) with 𝑚-point boundary value (𝑘) to have existence of positive solutions (𝑘=2,…,5).

2. Preliminaries

In this section, we state the preliminary information that we need to prove the main results.

In this paper, for our constructions, we shall consider the Banach space 𝐸=𝐶[𝜌(𝑡₁),𝑡_𝑛] equipped with standard norm ‖𝑥‖=max_{𝜌(𝑡1)≤𝑡≤𝑡𝑛}|𝑥(𝑡)|,𝑥∈𝐸; for each (𝑥,𝑦)∈𝐸×𝐸, we write ||(𝑥,𝑦)||₁=||𝑥||+||𝑦||. Clearly, (𝐸×𝐸,||⋅||₁) is a Banach space. Denote by 𝜙_𝑖1 and 𝜙_𝑖2 (𝑖=1,2), the solutions of the equation 𝑝_𝑖𝑢^∇_𝑖^Δ(𝑡)−𝑞_𝑖(𝑡)𝑢_𝑖(𝑡)=0,𝑡∈𝑡₁,𝑡_𝑛, (10) under the initial conditions 𝑢_𝑖𝑡₁=𝛽_𝑖,𝑝_𝑖𝑡₁𝑢^∇_𝑖𝑡₁=𝛼_𝑖,𝑢_𝑖𝑡_𝑛=𝛿_𝑖,𝑝_𝑖𝑡_𝑛𝑢^∇_𝑖𝑡_𝑛=−𝛾_𝑖, (11) respectively. So that 𝜙_𝑖1 and 𝜙_𝑖2 (𝑖=1,2) satisfy 𝑝_𝑖𝜙^∇_𝑖1^Δ(𝑡)−𝑞_𝑖(𝑡)𝜙_𝑖1(𝑡)𝑡=0,𝑡∈₁,𝑡_𝑛,𝜙_𝑖1𝑡₁=𝛽_𝑖,𝑝_𝑖𝑡₁𝜙^∇_𝑖1𝑡₁=𝛼_𝑖,𝑝_𝑖𝜙^∇_𝑖2^Δ(𝑡)−𝑞_𝑖(𝑡)𝜙_𝑖2(𝑡)𝑡=0,𝑡∈₁,𝑡_𝑛,𝜙_𝑖2𝑡_𝑛=𝛿_𝑖,𝑝_𝑖𝑡_𝑛𝜙^∇_𝑖2𝑡_𝑛=−𝛾_𝑖, (12) respectively. For 𝑖=1,2, set 𝑑_𝑖=𝛼_𝑖𝜙_𝑖2(𝑡₁)−𝛽_𝑖𝑝_𝑖(𝑡₁)𝜙^∇_𝑖2(𝑡₁)=𝛾_𝑖𝜙_𝑖1(𝑡_𝑛)+𝛿_𝑖𝑝_𝑖(𝑡_𝑛)𝜙^∇_𝑖1(𝑡_𝑛), the Green's function of the corresponding homogeneous boundary value problem is defined by 𝐺_𝑖()=1𝑡,𝑠𝑑_𝑖𝜙_𝑖2(𝑡)𝜙_𝑖1(𝑠),𝜌𝑡₁≤𝑠≤𝑡≤𝑡_𝑛,𝜙_𝑖1(𝑡)𝜙_𝑖2(𝑠),𝜌𝑡₁≤𝑡≤𝑠≤𝑡_𝑛,for𝑖=1,2. (13) From Lemmas 3.1 and 3.3 in [1], we have the following lemma.

Lemma 1. If 𝑑_𝑖≠0, (𝑢₁,𝑢₂) is a solution of (1) with boundary value condition (𝑘) if only and if 𝑢₁(𝑡)=𝜆^𝑡𝑛_𝑡1𝐻_1𝑘(𝑡,𝑠)𝑓₁𝑠,𝑢₁(𝑠),𝑢₂(𝑠)𝜌𝑡Δ𝑠,𝑡∈₁,𝑡_𝑛,𝑢₂(𝑡)=𝜆^𝑡𝑛_𝑡1𝐻_2𝑘(𝑡,𝑠)𝑓₂𝑠,𝑢₁(𝑠),𝑢₂(𝑠)Δ𝑠, (14)(15) where 𝑘=2,…,5, and 𝐻_𝑖2(𝑡,𝑠)=𝐺_𝑖()+1𝑡,𝑠𝑑_𝑖−∑^𝑛−1_𝑗=2𝑏_𝑗𝜙_𝑖1𝑡_𝑗^𝑛−1_𝑗=2𝑏_𝑗𝐺_𝑖𝑡_𝑗,𝑠𝜙_𝑖1(𝑡),(),𝐻𝑖=1,2_𝑖3(𝑡,𝑠)=𝐺_𝑖()+1𝑡,𝑠𝑑_𝑖−∑^𝑛−1_𝑗=2𝑎_𝑗𝜙_𝑖2𝑡_𝑗^𝑛−1_𝑗=2𝑎_𝑗𝐺_𝑖𝑡_𝑗,𝑠𝜙_𝑖2(𝑡),(),𝐻𝑖=1,2₁₄(𝑡,𝑠)=𝐺₁()+1𝑡,𝑠𝑑₁−∑^𝑛−1_𝑗=2𝑏_𝑖𝜙₁₁𝑡_𝑖^𝑛−1_𝑗=2𝑏_𝑖𝐺₁𝑡_𝑖,𝑠𝜙₁₁(𝑡)=𝐻₁₂(),𝐻𝑡,𝑠₂₄(𝑡,𝑠)=𝐺₂()+1𝑡,𝑠𝑑₂−∑^𝑛−1_𝑗=2𝑎_𝑖𝜙₂₂𝑡_𝑖^𝑛−1_𝑗=2𝑎_𝑖𝐺₂𝑡_𝑖,𝑠𝜙₂₂(𝑡)=𝐻₂₃(),𝐻𝑡,𝑠₁₅(𝑡,𝑠)=𝐺₁()+1𝑡,𝑠𝑑₁−∑^𝑛−1_𝑗=2𝑎_𝑖𝜙₁₂𝑡_𝑖^𝑛−1_𝑗=2𝑎_𝑖𝐺₁𝑡_𝑖,𝑠𝜙₁₂(𝑡)=𝐻₁₃(),𝐻𝑡,𝑠₂₅(𝑡,𝑠)=𝐺₂()+1𝑡,𝑠𝑑₂−∑^𝑛−1_𝑗=2𝑏_𝑖𝜙₂₁𝑡_𝑖^𝑛−1_𝑗=2𝑏_𝑖𝐺₂𝑡_𝑖,𝑠𝜙₂₁(𝑡)=𝐻₂₂(𝑡,𝑠). (16)

For the rest of the paper, we need the following assumption: 0<^𝑛−1_𝑗=2𝑏_𝑗𝜙_𝑖1𝑡_𝑗,^𝑛−1_𝑗=2𝑎_𝑗𝜙_𝑖2𝑡_𝑗<𝑑_𝑖,for𝑖=1,2. (C)

From 𝜙_𝑖1 is nondecreasing on [𝜌(𝑡₁),𝑡_𝑛], 𝜙_𝑖2 is nonincreasing on [𝜌(𝑡₁),𝑡_𝑛] (see [2, Proposition 2.3]), it is easy to verify the following inequalities: 𝑑_𝑖𝐺_𝑖(𝑡,𝑠)≤𝜙_𝑖1(𝑡)𝜙_𝑖2(𝑡),𝑑_𝑖𝐺_𝑖(𝑡,𝑠)≤𝜙_𝑖1(𝑠)𝜙_𝑖2(𝑠),𝑑_𝑖𝐺_𝑖()≥1𝑡,𝑠𝜙_𝑖1𝜙_𝑖2𝜙_𝑖1(𝑡)𝜙_𝑖2(𝑡)𝜙_𝑖1(𝑠)𝜙_𝑖2(𝑠). (17)

Lemma 2. The Green's function 𝐺_𝑖(𝑡,𝑠) has properties 𝐺_𝑖(𝑡,𝑠)≤𝐺_𝑖(),𝑑𝑡,𝑡_𝑖𝜙_𝑖1𝜙_𝑖2𝐺_𝑖(𝑡,𝑡)𝐺_𝑖(𝑠,𝑠)≤𝐺_𝑖(𝑡,𝑠)≤𝐺_𝑖(𝑠,𝑠). (18)

Lemma 3. For 𝐻_𝑖𝑘(𝑡,𝑠), 𝑘=2,…,5 and 𝑖=1,2, one has the conclusions 𝐻_𝑖𝑘(𝑡,𝑠)≤𝐶^∗𝐺_𝑖(𝑠,𝑠) and 𝑐_∗𝜙_𝑖1(𝑡)𝐺_𝑖(𝑠,𝑠)≤𝐻_𝑖2(𝑡,𝑠)≤𝐶^∗𝜙_𝑖1(𝑡),(),𝑐𝑖=1,2_∗𝜙_𝑖2(𝑡)𝐺_𝑖(𝑠,𝑠)≤𝐻_𝑖3(𝑡,𝑠)≤𝐶^∗𝜙_𝑖2(𝑡),(),𝑐𝑖=1,2_∗𝜙_𝑖𝑖(𝑡)𝐺₁(𝑠,𝑠)≤𝐻_𝑖4(𝑡,𝑠)≤𝐶^∗𝜙_𝑖𝑖(𝑡),(),𝑐𝑖=1,2_∗𝜙₁₂(𝑡)𝐺₁(𝑠,𝑠)≤𝐻₁₅(𝑡,𝑠)≤𝐶^∗𝜙₁₂(𝑡),𝑐_∗𝜙₂₁(𝑡)𝐺₁(𝑠,𝑠)≤𝐻₂₅(𝑡,𝑠)≤𝐶^∗𝜙₂₁(𝑡), (19) where 𝐶^∗=𝐶₁+𝐶₂ and 𝐶₁=max_𝑖=1,2𝜙1+^𝑖1𝑑_𝑖−∑^𝑛−1_𝑗=2𝑏_𝑗𝜙_𝑖1𝑡_𝑗^𝑛−1_𝑖=1𝑏_𝑗+𝜙_𝑖2𝑑_𝑖−∑^𝑛−1_𝑗=2𝑎_𝑗𝜙_𝑖2𝑡_𝑗^𝑛−1_𝑗=2𝑎_𝑗,𝐶₂=max_𝑖=1,2𝜙_𝑖1+𝜙_𝑖2𝑑_𝑖+1𝑑_𝑖−∑^𝑛−1_𝑗=2𝑏_𝑗𝜙_𝑖1𝑡_𝑗×^𝑛−1_𝑗=2𝑏_𝑗𝐺_𝑖𝑡_𝑗,𝑡_𝑗+1𝑑_𝑖−∑^𝑛−1_𝑗=2𝑏_𝑗𝜙_𝑖2𝑡_𝑗×^𝑛−1_𝑗=2𝑎_𝑗𝐺₂𝑡_𝑗,𝑡_𝑗,𝑐_∗𝑑=min^𝑖𝜙_𝑖1𝜙_𝑖21𝑑_𝑖−∑^𝑛−1_𝑗=2𝑏_𝑗𝜙_𝑖1𝑡_𝑗×^𝑛−1_𝑗=2𝑏_𝑗𝐺_𝑖𝑡_𝑗,𝑡_𝑗,𝑑_𝑖𝜙_𝑖1𝜙_𝑖21𝑑_𝑖−∑^𝑛−1_𝑗=2𝑎_𝑗𝜙_𝑖2𝑡_𝑗×^𝑛−1_𝑗=2𝑎_𝑗𝐺_𝑖𝑡_𝑗,𝑡_𝑗;𝑖=1,2. (20)

Proof. From Lemma 2 and 1𝑑_𝑖−∑^𝑛−1_𝑗=2𝑏_𝑗𝜙_𝑖1𝑡_𝑗^𝑛−1_𝑗=2𝑏_𝑗𝐺_𝑖𝑡_𝑗,𝑠𝜙_𝑖1(𝑡)≤1𝑑_𝑖−∑^𝑛−1_𝑗=2𝑏_𝑗𝜙_𝑖1𝑡_𝑗^𝑛−1_𝑗=2𝑏_𝑗𝐺_𝑖(𝑠,𝑠)𝜙_𝑖1,1𝑑_𝑖−∑^𝑛−1_𝑗=2𝑎_𝑗𝜙_𝑖2𝑡_𝑗^𝑛−1_𝑗=2𝑎_𝑗𝐺_𝑖𝑡_𝑗,𝑠𝜙_𝑖2(𝑡)≤1𝑑_𝑖−∑^𝑛−1_𝑗=2𝑎_𝑗𝜙_𝑖2𝑡_𝑗^𝑛−1_𝑗=2𝑎_𝑗𝐺_𝑖(𝑠,𝑠)𝜙_𝑖2, (21) we have 𝐻_𝑖𝑘(𝑡,𝑠)≤𝐶₁𝐺_𝑖(𝑠,𝑠)≤𝐶^∗𝐺_𝑖(𝑠,𝑠). (22)
For 𝑘=2 or 3, we have 𝐻_𝑖2()≤𝜙𝑡,𝑠^𝑖2(𝑡)𝑑_𝑖𝜙_𝑖1(𝑡)+1𝑑_𝑖−∑^𝑛−1_𝑗=2𝑏_𝑖𝜙_𝑖1𝑡_𝑖×^𝑛−1_𝑗=2𝑏_𝑗𝐺_𝑖𝑡_𝑗,𝑡_𝑗𝜙_𝑖1(𝑡)≤𝐶^∗𝜙_𝑖1(𝑡),𝐻_𝑖3()≤𝜙𝑡,𝑠^𝑖1(𝑡)𝑑_𝑖𝜙_𝑖2(𝑡)+1𝑑_𝑖−∑^𝑛−1_𝑗=2𝑎_𝑖𝜙_𝑖2𝑡_𝑖×^𝑛−1_𝑗=2𝑎_𝑗𝐺_𝑖𝑡_𝑗,𝑡_𝑗𝜙_𝑖2(𝑡)≤𝐶^∗𝜙_𝑖2(𝑡),𝐻_𝑖2()≥1𝑡,𝑠𝑑_𝑖−∑^𝑛−1_𝑗=2𝑏_𝑗𝜙_𝑖1𝑡_𝑗×^𝑛−1_𝑗=2𝑏_𝑗𝐺_𝑖𝑡_𝑗,𝑠𝜙_𝑖1(𝑡)≥1𝑑_𝑖−∑^𝑛−1_𝑗=2𝑏_𝑗𝜙_𝑖1𝑡_𝑗×^𝑛−1_𝑗=2𝑏_𝑗𝑑_𝑖𝜙₁𝜙₂𝐺_𝑖𝑡_𝑗,𝑡_𝑗𝐺_𝑖(𝑠,𝑠)𝜙_𝑖1(𝑡)≥𝑐_∗𝜙_𝑖1(𝑡)𝐺_𝑖(),𝐻𝑠,𝑠_𝑖3()≥1𝑡,𝑠𝑑_𝑖−∑^𝑛−1_𝑗=2𝑎_𝑗𝜙_𝑖2𝑡_𝑗×^𝑛−1_𝑗=2𝑎_𝑗𝐺_𝑖𝑡_𝑗,𝑠𝜙_𝑖2(𝑡)≥1𝑑_𝑖−∑^𝑛−1_𝑗=2𝑎_𝑗𝜙_𝑖1𝑡_𝑗×^𝑛−1_𝑗=2𝑎_𝑗𝑑_𝑖𝜙₁𝜙₂𝐺_𝑖𝑡_𝑗,𝑡_𝑗𝐺_𝑖(𝑠,𝑠)𝜙_𝑖2(𝑡)≥𝑐_∗𝜙_𝑖2(𝑡)𝐺_𝑖(𝑠,𝑠). (23) So, we have 𝑐_∗𝜙_𝑖1(𝑡)𝐺_𝑖(𝑠,𝑠)≤𝐻_𝑖2(𝑡,𝑠)≤𝐶^∗𝜙_𝑖1(𝑡),(),𝑐𝑖=1,2_∗𝜙_𝑖2(𝑡)𝐺_𝑖(𝑠,𝑠)≤𝐻_𝑖3(𝑡,𝑠)≤𝐶^∗𝜙_𝑖2(𝑡),(𝑖=1,2). (24)
Since 𝐻₁₄(𝑡,𝑠)=𝐻₁₂(𝑡,𝑠),𝐻₂₄(𝑡,𝑠)=𝐻₂₃(𝑡,𝑠),𝐻₁₅(𝑡,𝑠)=𝐻₁₃(𝑡,𝑠),𝐻₂₅(𝑡,𝑠)=𝐻₂₂(𝑡,𝑠), then we also have 𝑐_∗𝜙₁₁(𝑡)𝐺₁(𝑠,𝑠)≤𝐻₁₄(𝑡,𝑠)=𝐻₁₂(𝑡,𝑠)≤𝐶^∗𝜙₁₁(𝑡),𝑐_∗𝜙₂₂(𝑡)𝐺₁(𝑠,𝑠)≤𝐻₂₄(𝑡,𝑠)=𝐻₂₃(𝑡,𝑠)≤𝐶^∗𝜙₂₂(𝑡),𝑐_∗𝜙₁₂(𝑡)𝐺₁(𝑠,𝑠)≤𝐻₁₅(𝑡,𝑠)=𝐻₁₃(𝑡,𝑠)≤𝐶^∗𝜙₁₂(𝑡),𝑐_∗𝜙₂₁(𝑡)𝐺₁(𝑠,𝑠)≤𝐻₂₅(𝑡,𝑠)=𝐻₂₂(𝑡,𝑠)≤𝐶^∗𝜙₂₁(𝑡). (25) The proof is complete.

The following theorems will play a major role in our next analysis.

Theorem 4 (see [20]). Let 𝑋 be a Banach space, and Ω⊂𝑋 closed and convex. Assume 𝑈 is a relatively open subset of Ω with 0∈𝑈, and let 𝑆∶𝑈→Ω be a compact, continuous map. Then either (1)𝑆has a fixed point in 𝑈, or (2)there exists 𝑢∈𝜕𝑈 and 𝜈∈(0,1), with 𝑢=𝜈𝑆𝑢.

Theorem 5 (see [21]). Let 𝑋 be a Banach space, and let 𝑃⊂𝑋 be a cone in 𝑋. Let Ω₁,Ω₂ be bounded open subsets of 𝑋 with 0∈Ω₁⊂Ω₁⊂Ω₂, and let 𝑆∶𝑃→𝑃 be a completely continuous operator such that, either (1)‖𝑆𝑤‖≤‖𝑤‖, 𝑤∈𝑃∩𝜕Ω₁, ‖𝑆𝑤‖≥‖𝑤‖,𝑤∈𝑃∩𝜕Ω₂, or (2)‖𝑆𝑤‖≥‖𝑤‖, 𝑤∈𝑃∩𝜕Ω₁, ‖𝑆𝑤‖≤‖𝑤‖,𝑤∈𝑃∩𝜕Ω₂. Then 𝑆 has a fixed point in 𝑃∩Ω₂⧵Ω₁.

3. Main Results

We make the following assumptions:(H₁)𝑓_𝑖(𝑡,𝑢₁,𝑢₂)∈𝐶([𝑡₁,𝑡_𝑛]×[0,+∞)²,(−∞,+∞)), moreover there exists a function 𝑔(𝑡)∈𝐿¹([𝑡₁,𝑡_𝑛],(0,+∞)) such that 𝑓_𝑖(𝑡,𝑢₁,𝑢₂)≥−𝑔(𝑡), for any 𝑡∈(𝑡₁,𝑡_𝑛), 𝑢_𝑖∈[0,+∞), 𝑖=1,2.(H^∗₁)𝑓_𝑖(𝑡,𝑢₁,𝑢₂)∈𝐶((𝑡₁,𝑡_𝑛)×[0,+∞)²,(−∞,+∞)) may be singular at 𝑡=𝑡₁,𝑡_𝑛; moreover, there exists a function 𝑔(𝑡)∈𝐿¹((𝑡₁,𝑡_𝑛),(0,+∞)) such that 𝑓_𝑖(𝑡,𝑢₁,𝑢₂)≥−𝑔(𝑡), for any 𝑡∈(𝑡₁,𝑡_𝑛), 𝑢_𝑖∈[0,+∞).(H₂)𝑓_𝑖(𝑡,0,0)>0, for 𝑡∈[𝑡₁,𝑡_𝑛]  (𝑖=1,2).(H₃) There exists [𝜃₁,𝜃₂]⊂(𝑡₁,𝑡_𝑛) such that lim_{𝑢1+𝑢2↑+∞}min_{𝑡∈[𝜃1,𝜃2]}(𝑓_𝑖(𝑡,𝑢₁,𝑢₂)/(𝑢₁+𝑢₂))=+∞(𝑖=1,2).(H₄)∫^𝑡𝑛_𝑡1𝐺_𝑖(𝑠,𝑠)𝑔(𝑠)∇𝑠<+∞ and ∫^𝑡𝑛_𝑡1𝐺_𝑖(𝑠,𝑠)𝑓_𝑖(𝑠,𝑧₁,𝑧₂)∇𝑠<+∞ for any 𝑧_𝑖∈[0,𝑚], 𝑚>0 is any constant (𝑖=1,2).In fact, we only consider the system 𝑝₁𝑥^∇₁^Δ(𝑡)−𝑞₁(𝑡)𝑥₁(𝑡)+𝜆𝑓₁𝑡,𝑥₁(𝑡)−𝑣_1𝑘(𝑡)^∗,𝑥₂(𝑡)−𝑣_2𝑘(𝑡)^∗(𝑡)𝑝+𝑔=0,𝜆>0,₂𝑥^∇₂^Δ(𝑡)−𝑞₂(𝑡)𝑥₂(𝑡)+𝜆𝑓₂𝑡,𝑥₁(𝑡)−𝑣_1𝑘(𝑡)^∗,𝑥₂(𝑡)−𝑣_2𝑘(𝑡)^∗+𝑔(𝑡)=0,𝜆>0, (26) with one of the boundary value conditions 𝛼₁𝑥₁𝑡₁−𝛽₁𝑝₁𝑡₁𝑥^∇₁𝑡₁𝛾=0,₁𝑥₁𝑡_𝑛+𝛿₁𝑝₁𝑡_𝑛𝑥^∇₁𝑡_𝑛=^𝑛−2_𝑖=2𝑏_𝑖𝑥₁𝜂_𝑖,𝛼₂𝑥₂𝑡₁−𝛽₂𝑝₂𝑡₁𝑥^∇₂𝑡₁𝛾=0,₂𝑥₂𝑡_𝑛+𝛿₂𝑝₁𝑡_𝑛𝑥^∇₂𝑡_𝑛=^𝑛−2_𝑖=2𝑏_𝑖𝑥₂𝜂_𝑖,𝛼₁𝑥₁𝑡₁−𝛽₁𝑝₁𝑡₁𝑥^∇₁𝑡₁=^𝑛−2_𝑖=2𝑎_𝑖𝑥₁𝜂_𝑖,𝛾₁𝑥₁𝑡_𝑛+𝛿₁𝑝₁𝑡_𝑛𝑥^∇₁𝑡_𝑛𝛼=0,₂𝑥₂𝑡₁−𝛽₂𝑝₂𝑡₁𝑥^∇₂𝑡₁=^𝑛−2_𝑖=2𝑎_𝑖𝑥₂𝜂_𝑖,𝛾₂𝑥₂𝑡_𝑛+𝛿₂𝑝₁𝑡_𝑛𝑥^∇₂𝑡_𝑛𝛼=0,₁𝑥₁𝑡₁−𝛽₁𝑝₁𝑡₁𝑥^∇₁𝑡₁𝛾=0,₁𝑥₁𝑡_𝑛+𝛿₁𝑝₁𝑡_𝑛𝑥^∇₁𝑡_𝑛=^𝑛−2_𝑖=2𝑏_𝑖𝑥₁𝜂_𝑖,𝛼₂𝑥₂𝑡₁−𝛽₂𝑝₂𝑡₁𝑥^∇₂𝑡₁=^𝑛−2_𝑖=2𝑎_𝑖𝑥₂𝜂_𝑖,𝛾₂𝑥₂𝑡_𝑛+𝛿₂𝑝₁𝑡_𝑛𝑥^∇₂𝑡_𝑛𝛼=0,₁𝑥₁𝑡₁−𝛽₁𝑝₁𝑡₁𝑥^∇₁𝑡₁=^𝑛−2_𝑖=2𝑎_𝑖𝑥₁𝜂_𝑖,𝛾₁𝑥₁𝑡_𝑛+𝛿₁𝑝₁𝑡_𝑛𝑥^∇₁𝑡_𝑛𝛼=0,₂𝑥₂𝑡₁−𝛽₂𝑝₂𝑡₁𝑥^∇₂𝑡₁𝛾=0,₂𝑥₂𝑡_𝑛+𝛿₂𝑝₁𝑡_𝑛𝑥^∇₂𝑡_𝑛=^𝑛−2_𝑖=2𝑏_𝑖𝑥₂𝜂_𝑖, (27) where 𝑦(𝑡)^∗=𝑦(𝑡)(𝑡)(𝑡),𝑦≥0,0,𝑦<0, (28) and 𝑣_𝑖𝑘(𝑡)=𝜆∫^𝑡𝑛_𝑡1𝐻_𝑖𝑘(𝑡,𝑠)𝑔(𝑠)Δ𝑠. For 𝑘=2,…,5, from Lemma 1, (𝑣_1𝑘(𝑡),𝑣_2𝑘(𝑡)) is the solution of the equation 𝑝₁𝑣^∇₁^Δ(𝑡)−𝑞₁(𝑡)𝑣₁(𝑡)+𝜆𝑔(𝑡)=0,𝜆>0,𝑡₁<𝑡<𝑡_𝑛,𝑝₂𝑣^∇₂^Δ(𝑡)−𝑞₂(𝑡)𝑣₂(𝑡)+𝜆𝑔(𝑡)=0,𝜆>0, (29) respectively, satisfying the following boundary value conditions: 𝛼₁𝑣₁𝑡₁−𝛽₁𝑝₁𝑡₁𝑣^∇₁𝑡₁𝛾=0,₁𝑣₁𝑡_𝑛+𝛿₁𝑝₁𝑡_𝑛𝑣^∇₁𝑡_𝑛=^𝑛−2_𝑖=2𝑏_𝑖𝑣₁𝜂_𝑖,𝛼₂𝑣₂𝑡₁−𝛽₂𝑝₂𝑡₁𝑣^∇₂𝑡₁𝛾=0,₂𝑣₂𝑡_𝑛+𝛿₂𝑝₁𝑡_𝑛𝑣^∇₂𝑡_𝑛=^𝑛−2_𝑖=2𝑏_𝑖𝑣₂𝜂_𝑖,𝛼₁𝑣₁𝑡₁−𝛽₁𝑝₁𝑡₁𝑣^∇₁𝑡₁=^𝑛−2_𝑖=2𝑎_𝑖𝑣₁𝜂_𝑖,𝛾₁𝑣₁𝑡_𝑛+𝛿₁𝑝₁𝑡_𝑛𝑣^∇₁𝑡_𝑛𝛼=0,₂𝑣₂𝑡₁−𝛽₂𝑝₂𝑡₁𝑣^∇₂𝑡₁=^𝑛−2_𝑖=2𝑎_𝑖𝑣₂𝜂_𝑖,𝛾₂𝑣₂𝑡_𝑛+𝛿₂𝑝₁𝑡_𝑛𝑣^∇₂𝑡_𝑛𝛼=0,₁𝑣₁𝑡₁−𝛽₁𝑝₁𝑡₁𝑣^∇₁𝑡₁𝛾=0,₁𝑣₁𝑡_𝑛+𝛿₁𝑝₁𝑡_𝑛𝑣^∇₁𝑡_𝑛=^𝑛−2_𝑖=2𝑏_𝑖𝑣₁𝜂_𝑖,𝛼₂𝑣₂𝑡₁−𝛽₂𝑝₂𝑡₁𝑣^∇₂𝑡₁=^𝑛−2_𝑖=2𝑎_𝑖𝑣₂𝜂_𝑖,𝛾₂𝑣₂𝑡_𝑛+𝛿₂𝑝₁𝑡_𝑛𝑣^∇₂𝑡_𝑛𝛼=0,₁𝑣₁𝑡₁−𝛽₁𝑝₁𝑡₁𝑥^∇₁𝑡₁=^𝑛−2_𝑖=2𝑎_𝑖𝑣₁𝜂_𝑖,𝛾₁𝑣₁𝑡_𝑛+𝛿₁𝑝₁𝑡_𝑛𝑥^∇₁𝑡_𝑛𝛼=0,₂𝑣₂𝑡₁−𝛽₂𝑝₂𝑡₁𝑥^∇₂𝑡₁𝛾=0,₂𝑣₂𝑡_𝑛+𝛿₂𝑝₁𝑡_𝑛𝑥^∇₂𝑡_𝑛=^𝑛−2_𝑖=2𝑏_𝑖𝑣₂𝜂_𝑖. (30)

We will show that there exists a solution (𝑥_1𝑘,𝑥_2𝑘) to the boundary value problem (𝑘) of the system (26) with 𝑥_𝑖𝑘(𝑡)≥𝑣_𝑖𝑘(𝑡),𝑡∈[𝜌(𝑡₁),𝑡_𝑛]. If this is true, then 𝑢_𝑖𝑘(𝑡)=𝑥_𝑖𝑘(𝑡)−𝑣_𝑖𝑘(𝑡) is a nonnegative solution (positive on (𝜌(𝑡₁),𝑡_𝑛)) of the system (1) with the boundary value problem (𝑘), (where𝑖=1,2;𝑘=2,…,5,𝑘=𝑘+11). Since for any 𝑡∈(𝑡₁,𝑡_𝑛), from 𝑝_𝑖𝑥^∇_𝑖𝑘^Δ(𝑡)−𝑞_𝑖(𝑡)𝑥^∇_𝑖𝑘(𝑡)=𝑝_𝑖𝑢_𝑖𝑘+𝑣_𝑖𝑘^∇Δ(𝑡)−𝑞_𝑖(𝑡)𝑢_𝑖𝑘+𝑣_𝑖𝑘^∇(𝑡)=−𝜆𝑓_𝑖𝑡,𝑥_1𝑘(𝑡)−𝑣_1𝑘(𝑡)^∗,𝑥_2𝑘(𝑡)−𝑣_2𝑘(𝑡)^∗(𝑡)𝑓+𝑔=−𝜆_𝑖𝑡,𝑢_1𝑘(𝑡),𝑢_2𝑘(𝑡)+𝑔(𝑡), (31) we have 𝑝_𝑖𝑢^∇_𝑖𝑘^Δ(𝑡)−𝑞_𝑖(𝑡)𝑢_𝑖𝑘(𝑡)=−𝜆𝑓_𝑖𝑡,𝑢_1𝑘(𝑡),𝑢_2𝑘(𝑡). (32) As a result, we will concentrate our study on (26) with the boundary value problem (𝑘).

Employing Lemma 1, we note that (𝑥_1𝑘(𝑡),𝑥_2𝑘(𝑡)) is a solution of the system (26) with boundary value (𝑘) if and only if 𝑥_1𝑘(𝑡)=𝜆^𝑡𝑛_𝑡1𝐻_1𝑘(𝑡,𝑠)𝑓₁𝑠,𝑥_1𝑘(𝑠)−𝑣_1𝑘(𝑠)^∗,𝑥_2𝑘(𝑠)−𝑣_2𝑘(𝑠)^∗(𝑠)𝜌𝑡+𝑔Δ𝑠,𝑡∈₁,𝑡_𝑛,𝑥_2𝑘(𝑡)=𝜆^𝑡𝑛_𝑡1𝐻_2𝑘(𝑡,𝑠)𝑓₂𝑠,𝑥_1𝑘(𝑠)−𝑣_1𝑘(𝑠)^∗,𝑥_2𝑘(𝑠)−𝑣_2𝑘(𝑠)^∗+𝑔(𝑠)Δ𝑠. (33)

We define a cone 𝑃_𝑖𝑗  (𝑖,𝑗=1,2) by 𝑃_𝑖𝑗=(𝑡)≥𝑐𝑥∈𝐸∣𝑥^∗𝐶^∗𝜙_𝑖𝑗(𝑡)‖𝑥‖,𝑡∈𝜌𝑡₁,𝑡_𝑛. (34) It is clearly that 𝑃_𝑖𝑗×𝑃_𝑚𝑛 is a cone of 𝐸×𝐸, (𝑖,𝑗,𝑚,𝑛=1,2). Define the integral operator 𝑇₂∶𝑃₁₁×𝑃₂₁→𝐸×𝐸, 𝑇₃∶𝑃₁₂×𝑃₂₂→𝐸×𝐸, 𝑇₄∶𝑃₁₂×𝑃₂₁→𝐸×𝐸, 𝑇₅∶𝑃₂₁×𝑃₁₂→𝐸×𝐸, by 𝑇_𝑘𝑥_1𝑘,𝑥_2𝑘=𝑇_1𝑘𝑥_1𝑘,𝑥_2𝑘,𝑇_2𝑘𝑥_1𝑘,𝑥_2𝑘,(𝑘=2,…,5), (35) where operators 𝑇_𝑖𝑘 are defined by 𝑇_𝑖𝑘𝑥_1𝑘,𝑥_2𝑘(𝑡)=𝜆^𝑡𝑛_𝑡1𝐻_𝑖𝑘(𝑡,𝑠)𝑓_𝑖𝑠,𝑥_1𝑘(𝑠)−𝑣_1𝑘(𝑠)^∗,𝑥_2𝑘(𝑠)−𝑣_2𝑘(𝑠)^∗(𝑠)𝜌𝑡+𝑔Δ𝑠,𝑡∈₁,𝑡_𝑛, (36) where 𝑖=1,2. Clearly, if (𝑥_1𝑘,𝑥_2𝑘) is a fixed point of 𝑇_𝑘, then (𝑥_1𝑘,𝑥_2𝑘) is a solution of system (26) with (𝑘)  (𝑘=2,…,5,𝑘=𝑘+11).

For 𝑘=2,…,5, from (35) and Lemma 3, we have 𝑇_𝑘(𝑥_1𝑘,𝑥_2𝑘)(𝑡)≥0 on [0,1], for (𝑥_1𝑘,𝑥_2𝑘)∈𝑃_𝑖𝑗×𝑃_𝑚𝑛, we have 𝑇_𝑖𝑘𝑥_1𝑘,𝑥_2𝑘(𝑡)=𝜆^𝑡𝑛_𝑡1𝐻_𝑖𝑘(𝑡,𝑠)𝑓_𝑖𝑠,𝑥_1𝑘(𝑠)−𝑣_1𝑘(𝑠)^∗,𝑥_2𝑘(𝑠)−𝑣_2𝑘(𝑠)^∗+𝑔(𝑠)Δ𝑠≤𝐶^∗𝜆×^{𝜎(𝑡𝑛)}_𝜌𝑡1𝐺_𝑖(𝑠,𝑠)𝑓_𝑖𝑠,[𝑥(𝑠)−𝑣(𝑠)]^∗+𝑔(𝑠)Δ𝑠, (37) then ‖𝑇_𝑖𝑘(𝑥_1𝑘,𝑥_2𝑘)‖≤𝐶^∗𝜆∫^{𝜎(𝑡𝑛)}_𝜌(𝑡1)𝐺_𝑖(𝑠,𝑠)(𝑓(𝑡,[𝑥(𝑡)−𝑣(𝑡)]^∗)+𝑔(𝑡))Δ𝑠.

On the other hand, when 𝑘=2, we have 𝑇_𝑖2𝑥_1𝑘,𝑥_2𝑘(𝑡)=𝜆^𝑡𝑛_𝑡1𝐻_𝑖2(𝑡,𝑠)𝑓_𝑖𝑠,𝑥_1𝑘(𝑠)−𝑣_1𝑘(𝑠)^∗,𝑥_2𝑘(𝑠)−𝑣_2𝑘(𝑠)^∗+𝑔(𝑠)Δ𝑠≥𝜆^𝑡𝑛_𝑡1𝑐_∗𝜙_𝑖1(𝑡)𝐺_𝑖(𝑠,𝑠)𝑓_𝑖𝑠,𝑥_1𝑘(𝑠)−𝑣_1𝑘(𝑠)^∗,𝑥_2𝑘(𝑠)−𝑣_2𝑘(𝑠)^∗(𝑠)≥𝑐+𝑔Δ𝑠_∗𝐶^∗𝜙_𝑖1(𝑡)𝜆×^{𝜎(𝑡𝑛)}_𝜌𝑡1𝐶^∗𝐺_𝑖(𝑠,𝑠)𝑓_𝑖𝑠,[𝑥(𝑠)−𝑣(𝑠)]^∗(𝑠)≥𝑐+𝑔Δ𝑠_∗𝐶^∗𝜙_𝑖1(𝑡)𝑇_𝑖2𝑥_1𝑘,𝑥_2𝑘. (38) Thus, 𝑇_𝑖2(𝑃₁₁×𝑃₂₁)⊂𝑃_𝑖1. Hence 𝑇₂(𝑃₁₁×𝑃₂₁)⊂𝑃₁₁×𝑃₂₁.

When 𝑘=3, we have 𝑇_𝑖3𝑥_1𝑘,𝑥_2𝑘(𝑡)=𝜆^𝑡𝑛_𝑡1𝐻_𝑖3(𝑡,𝑠)𝑓_𝑖𝑠,𝑥_1𝑘(𝑠)−𝑣_1𝑘(𝑠)^∗,𝑥_2𝑘(𝑠)−𝑣_2𝑘(𝑠)^∗+𝑔(𝑠)Δ𝑠≥𝜆^𝑡𝑛_𝑡1𝑐_∗𝜙_𝑖2(𝑡)𝐺_𝑖(𝑠,𝑠)𝑓_𝑖𝑠,𝑥_1𝑘(𝑠)−𝑣_1𝑘(𝑠)^∗,𝑥_2𝑘(𝑠)−𝑣_2𝑘(𝑠)^∗(𝑠)≥𝑐+𝑔Δ𝑠_∗𝐶^∗𝜙_𝑖2(𝑡)𝜆×^{𝜎(𝑡𝑛)}_𝜌𝑡1𝐶^∗𝐺_𝑖(𝑠,𝑠)𝑓_𝑖𝑠,[𝑥(𝑠)−𝑣(𝑠)]^∗(𝑠)≥𝑐+𝑔Δ𝑠_∗𝐶^∗𝜙_𝑖2(𝑡)𝑇_𝑖3𝑥_1𝑘,𝑥_2𝑘. (39) Thus, 𝑇_𝑖3(𝑃₁₂×𝑃₂₂)⊂𝑃_𝑖2. Hence 𝑇₃(𝑃₁₂×𝑃₂₂)⊂𝑃₁₂×𝑃₂₂.

Similarly discussion, we also have 𝑇₄(𝑃₁₂×𝑃₂₁)⊂𝑃₁₂×𝑃₂₁, 𝑇₅(𝑃₂₁×𝑃₁₂)⊂𝑃₂₁×𝑃₁₂. In addition, standard arguments show that 𝑇_𝑘 is a completely continuous operator.

For simplicity, we adopt the notation: 𝑃₁₄=𝑃₂₅∶=𝑃₁₂ and 𝑃₂₄=𝑃₁₅∶=𝑃₂₁, then, we can write 𝑇_𝑘(𝑃_1(𝑘−1)×𝑃_2(𝑘−1))⊂𝑃_1(𝑘−1)×𝑃_2(𝑘−1), that is, 𝑇_𝑖𝑘(𝑃_1(𝑘−1)×𝑃_2(𝑘−1))⊂𝑃_{𝑖(𝑘−1)}, (𝑖=1,2,𝑘=2,…,5).

Theorem 6. Suppose that (𝐻₁)- (𝐻₂) hold. Then there exists a constant 𝜆>0 such that, for any 0<𝜆≤𝜆, (1) with boundary value condition (𝑘) has at least one positive solution (𝑘=2,…,5).

Proof. Fix 𝛿∈(0,1) and 𝑘 (𝑘=2,…,5). From (H₂) let 0<𝜀<1 be such that 𝑓_𝑖𝑡,𝑧₁,𝑧₂≥𝛿𝑓_𝑖(𝑡,0,0),for𝑡₁≤𝑡≤𝑡_𝑛,0≤𝑧_𝑖≤𝜀,𝑖=1,2. (40)
Let 𝑓(𝜀)=max_{𝑡1≤𝑡≤𝑡𝑛,0≤𝑧1,𝑧2≤𝜀}{max_𝑖=1,2{𝑓_𝑖(𝑡,𝑧₁,𝑧₂)}+𝑔(𝑡)}, and 𝑐=∫^𝑡𝑛_𝑡1𝐶^∗𝐺_𝑖(𝑠,𝑠)Δ𝑠. We have lim_𝑧↓0𝑓(𝑧)𝑧=+∞. (41)
Set 𝜆=𝜀/4𝑐𝑓(𝜀), since for any 0<𝜆≤𝜆, fix the 𝜆∈(0,𝜆], we always have lim_𝑧↓0𝑓(𝑧)𝑧=+∞,𝑓(𝜀)𝜀<1,4𝑐𝜆 (42) Then there exists a 𝑅₀∈(0,𝜀] such that 𝑓𝑅₀𝑅₀=1.4𝑐𝜆 (43)
Let 𝑈_𝑘={(𝑥_1𝑘,𝑥_2𝑘)∈𝑃_1(𝑘−1)×𝑃_2(𝑘−1)∶‖(𝑥_1𝑘,𝑥_2𝑘)‖₁<𝑅₀}, (𝑥_1𝑘,𝑥_2𝑘)∈𝜕𝑈_𝑘 and 𝜈∈(0,1) be such that (𝑥_1𝑘,𝑥_2𝑘)=𝜈𝑇_𝑘(𝑥_1𝑘,𝑥_2𝑘), that is, 𝑥_𝑖𝑘=𝜈𝑇_𝑖𝑘(𝑥_1𝑘,𝑥_2𝑘)  (𝑖=1,2). We claim that ‖(𝑥_1𝑘,𝑥_2𝑘)‖₁≠𝑅₀. In fact for (𝑥_1𝑘,𝑥_2𝑘)∈𝜕𝑈_𝑘 and ‖(𝑥_1𝑘,𝑥_2𝑘)‖₁=𝑅₀, we have 𝑥_𝑖𝑘=𝜈𝑇_𝑖𝑘𝑥_1𝑘,𝑥_2𝑘≤𝜆^𝑡𝑛_𝑡1𝐻_𝑖𝑘(𝑡,𝑠)𝑓_𝑖𝑠,𝑥_1𝑘(𝑠)−𝑣_1𝑘(𝑠)^∗,𝑥_2𝑘(𝑠)−𝑣_2𝑘(𝑠)^∗+𝑔(𝑠)Δ𝑠≤𝜆^𝑡𝑛_𝑡1𝐶^∗𝐺_𝑖(𝑠,𝑠)𝑓_𝑖𝑠,𝑥_1𝑘(𝑠)−𝑣_1𝑘(𝑠)^∗,𝑥_2𝑘(𝑠)−𝑣_2𝑘(𝑠)^∗+𝑔(𝑠)Δ𝑠≤𝜆^𝑡𝑛_𝑡1𝐶^∗𝐺_𝑖(𝑠,𝑠)𝑓𝑅₀Δ𝑠≤𝜆𝑐𝑓𝑅₀. (44) It follows that 𝑅₀=𝑥_1𝑘,𝑥_2𝑘1≤2𝜆𝑐𝑓𝑅₀, (45) that is, 𝑓𝑅₀𝑅₀≥1>12𝑐𝜆=4𝑐𝜆𝑓𝑅₀𝑅₀, (46) which implies that ‖(𝑥_1𝑘,𝑥_2𝑘)‖₁≠𝑅₀. By the nonlinear alternative of Leray-Schauder type, 𝑇_𝑘 has a fixed point (𝑥_𝑖1,𝑥_𝑖2)∈𝑈_𝑘. Moreover combining (40) and the fact that 𝑅₀<𝜀, we obtain 𝑥_𝑖𝑘=𝜆^𝑡𝑛_𝑡1𝐻_𝑖𝑘(𝑡,𝑠)𝑓_𝑖𝑡,𝑥_1𝑘(𝑡)−𝑣_1𝑘(𝑡)^∗,𝑥_2𝑘(𝑡)−𝑣_2𝑘(𝑡)^∗+𝑔(𝑡)Δ𝑠≥𝜆^𝑡𝑛_𝑡1𝐻_𝑖𝑘(𝑡,𝑠)𝛿𝑓(𝑠,0,0)+𝑔(𝑡)Δ𝑠≥𝜆^𝑡𝑛_𝑡1𝐻_𝑖𝑘(𝑡,𝑠)𝑔(𝑡)Δ𝑠=𝑣_𝑖𝑘(𝑡)for𝑡∈𝜌𝑡₁,𝑡_𝑛. (47)
Then 𝑇_𝑘 has a positive fixed point (𝑥_𝑖1,𝑥_𝑖2) and ‖(𝑥_𝑖1,𝑥_𝑖2)‖₁≤𝑅₀<1; that is, (𝑥_𝑖1,𝑥_𝑖2) is a positive solution of the boundary value problem (26) with 𝑥_𝑖𝑘>𝑣_𝑖𝑘(𝑡) for 𝑡∈(𝑡₁,𝑡_𝑛).
Let 𝑢_𝑖𝑘(𝑡)=𝑥_𝑖𝑘(𝑡)−𝑣_𝑖𝑘(𝑡