Positive Solutions of a Singular Positone and Semipositone Boundary Value Problems for Fourth-Order Difference Equations

Abstract

This paper studies the boundary value problems for the fourth-order nonlinear singular difference equations Δ⁴𝑢(𝑖−2)=𝜆𝛼(𝑖)𝑓(𝑖,𝑢(𝑖)), 𝑖∈[2,𝑇+2], 𝑢(0)=𝑢(1)=0, 𝑢(𝑇+3)=𝑢(𝑇+4)=0. We show the existence of positive solutions for positone and semipositone type. The nonlinear term may be singular. Two examples are also given to illustrate the main results. The arguments are based upon fixed point theorems in a cone.

1. Introduction

In this paper, we consider the following boundary value problems of difference equations:Δ⁴𝑢(𝑖−2)=𝜆𝑎(𝑖)𝑓(𝑖,𝑢(𝑖)),𝑖∈[2,𝑇+2],𝑢(0)=𝑢(1)=0,𝑢(𝑇+3)=𝑢(𝑇+4)=0. (1.1) Here [2,𝑇+2]={2,3,…,𝑇+2} and 𝑢∶[0,𝑇+4]→𝑅. We will let [𝑎,𝑏] denote the discrete integer set [𝑎,𝑏]={𝑎,𝑎+1,…,𝑏}, and 𝐶([𝑎,𝑏]) denotes the set of continuous function on [𝑎,𝑏] (discrete topology) with norm ‖⋅‖=max_{𝑘∈[𝑎,𝑏]}|⋅|.

Due to the wide applications in many fields such as computer science, economics, neural network, ecology, and cybernetics, the theory of nonlinear difference equations has been widely studied since the 70's of last century. Recently, many literatures on the boundary value of difference equations have appeared. We refer the reader to [1–13] and the references therein, which include work on Agarwal, Elaydi, Eloe, Erber, O'Regan, Henderson, Merdivenci, Yu, and Ma et al., concerning the existence of positive solutions and the corresponding eigenvalue problems. Recently, the existence of positive solutions of fourth-order discrete boundary value problems has been studied by several authors; for example, see [14–16] and the references therein.

On the other hand, fourth-order boundary value problems of ordinary value problems have important application in various branches of pure and applied science. They arise in the mathematical modeling of viscoelastic and inelastic flows, deformation of beams and plate deflection theory [17–19]. For example, the deformations of an elastic beam can be described by the boundary value problems of the fourth-order ordinary differential equations. There have been extensive studies on fourth-order boundary value problems with diverse boundary conditions via many methods, for example, [20–26] and the references therein. We also find that the differential equations on time scales is due to its unification of the theory of differential and difference equations, see [27–30] and the references therein.

In this paper, the boundary value problem (1.1) can be viewed as the discrete analogue of the following boundary value problems for ordinary differential equation:𝑢⁽⁴⁾(𝑡)=𝜆𝑎(𝑡)𝑓(𝑡,𝑢(𝑡)),𝑡∈(0,1),𝑢(0)=𝑢^′(1)=0,𝑢(1)=𝑢^′(1)=0. (1.2) Equation (1.2) describes an elastic beam in an equilibrium state whose both ends are simply supported. However, very little is known about the existence of solutions of the discrete boundary value problems (1.1). This motivates us to study (1.1).

In this paper, we discuss separately the cases when 𝑓 is positone and when 𝑓 is semipositone; the nonlinear term 𝑓 is singularity at 𝑢=0, and we will prove our two existence results for the problem (1.1) by using Krasnosel'skii fixed point theorem. This paper is organized as follows. In Section 2, starting with some preliminary lemmas, we state the Krasnosel'skii fixed point theorem. In Section 3, we give the sufficient conditions which state the existence of multiple positive solutions to the positone boundary value problem (1.1). In Section 4, we give the sufficient conditions which state the existence of at least one positive solution to the semipositone boundary value problem (1.1).

2. Preliminaries

In this section, we state the preliminary information that we need to prove the main results. From [28, Definition 2.1], we have the following lemma.

Lemma 2.1. 𝑢(𝑖) is a solution of (1.1) if only and if 𝑢(𝑖)=^𝑇+2_𝑗=2G(𝑖,𝑗)𝑎(𝑗)𝑓(𝑗,𝑢(𝑗)),𝑖∈[0,𝑇+4], (2.1) where 𝐺(𝑖,𝑗)=(𝑇+4−𝑖)²(𝑗−1)²2(𝑖(𝑇+3)²−(𝑇+4+2𝑖)(𝑗−1)3(𝑇+4)³𝑖),2≤𝑗≤𝑖+1,₂(𝑇+4−𝑗)²6((𝑇+3−𝑖)(𝑇+4+2𝑗)(𝑇+4)³−𝑇+4−𝑗(𝑇+3)²),𝑖+1<𝑗≤𝑇+2. (2.2)

Lemma 2.2. Green's function 𝐺(𝑡,𝑠) defined by (2.2) has the following properties: 𝐶₀𝑖²(𝑇+4−𝑖)²(𝑗−1)²(𝑇+4−𝑗)²≤𝐺(𝑖,𝑗)≤(𝑗−1)²(𝑇+4−𝑗)²,𝐺(𝑖,𝑗)≤𝑖²(𝑇+4−𝑖)², (2.3) where 𝐶₀=1/3(𝑇+4)⁷.

Proof. For 2≤𝑗≤𝑖+1, we have 𝐺(𝑖,𝑗)=(𝑇+4−𝑖)²(𝑗−1)²2(𝑖(𝑇+3)²−(𝑇+4+2𝑖)(𝑗−1)3(𝑇+4)³)≤(𝑇+4−𝑖)²(𝑗−1)²2𝑖(𝑇+3)²≤(𝑇+4−𝑗)²(𝑗−1)²𝑖2(𝑇+3)²≤(𝑗−1)²(𝑇+4−𝑗)²,𝐺(𝑖,𝑗)=(𝑇+4−𝑖)²(𝑗−1)²2(𝑖(𝑇+3)²−(𝑇+4+2𝑖)(𝑗−1)3(𝑇+4)³)≤(𝑇+4−𝑖)²(𝑗−1)²2𝑖(𝑇+3)²≤𝑖²(𝑇+4−𝑖)²(𝑗−1)²2(𝑇+3)²≤𝑖²(𝑇+4−𝑖)². (2.4)
On the other hand, 𝐺(𝑖,𝑗)=(𝑇+4−𝑖)²(𝑗−1)²2(𝑖(𝑇+3)²−(𝑇+4+2𝑖)(𝑗−1)3(𝑇+4)³)≥(𝑇+4−𝑖)²(𝑗−1)²2(𝑖(𝑇+4)²−(𝑇+4+2𝑖)(𝑗−1)3(𝑇+4)³)=(𝑇+4−𝑖)²(𝑗−1)²6(𝑇+4)³≥((𝑇+4)((𝑖+1)−𝑗)+2𝑖((𝑇+5)−𝑗))2𝑖(𝑇+4−𝑖)²(𝑗−1)²((𝑇+5)−𝑗)6(𝑇+4)³≥𝑖(𝑇+4−𝑖)²(𝑗−1)²((𝑇+4)−𝑗)3(𝑇+4)³≥𝑖²(𝑇+4−𝑖)²(𝑗−1)²((𝑇+4)−𝑗)²3(𝑇+4)⁵≥𝑖²(𝑇+4−𝑖)²(𝑗−1)²((𝑇+4)−𝑗)²3(𝑇+4)⁷. (2.5)
Then, for 2≤𝑗≤𝑖+1, we have 𝐶₀𝑖²(𝑇+4−𝑖)²(𝑗−1)²(𝑇+4−𝑗)²≤𝐺(𝑖,𝑗)≤(𝑗−1)²(𝑇+4−𝑗)²,𝐺(𝑖,𝑗)≤𝑖²(𝑇+4−𝑖)². (2.6)
For 𝑖+1<𝑗≤𝑇+2, we have 𝑖𝐺(𝑖,𝑗)=²(𝑇+4−𝑗)²6((𝑇+3−𝑖)(𝑇+4+2𝑗)(𝑇+4)³−𝑇+4−𝑗(𝑇+3)²)≤𝑖²(𝑇+4−𝑗)²6(𝑇+3−𝑖)(𝑇+4+2𝑗)(𝑇+4)³≤𝑖²(𝑇+4−𝑖)²6(𝑇+3−𝑖)(𝑇+4+2𝑗)(𝑇+4)³≤𝑖²(𝑇+4−𝑖)²(𝑇+4+2𝑗)6(𝑇+4)³≤3𝑖²(𝑇+4−𝑖)²(𝑇+4)6(𝑇+4)³≤𝑖²(𝑇+4−𝑖)²,𝑖𝐺(𝑖,𝑗)=²(𝑇+4−𝑗)²6((𝑇+3−𝑖)(𝑇+4+2𝑗)(𝑇+4)³−𝑇+4−𝑗(𝑇+3)²)≤𝑖²(𝑇+4−𝑗)²6(𝑇+3−𝑖)(𝑇+4+2𝑗)(𝑇+4)³≤(𝑗−1)²(𝑇+4−𝑗)²6(𝑇+3−𝑖)(𝑇+4+2𝑗)(𝑇+4)³≤3(𝑗−1)²(𝑇+4−𝑗)²(𝑇+4)²6(𝑇+4)³≤(𝑗−1)²(𝑇+4−𝑗)². (2.7)
On the other hand, 𝑖𝐺(𝑖,𝑗)=²(𝑇+4−𝑗)²6((𝑇+3−𝑖)(𝑇+4+2𝑗)(𝑇+4)³−𝑇+4−𝑗(𝑇+3)²)≥𝑖²(𝑇+4−𝑗)²6((𝑇+3−𝑖)(𝑇+4+2𝑗)(𝑇+4)³−𝑇+3−𝑖(𝑇+3)²)≥𝑖²(𝑇+4−𝑗)²(𝑇+3−𝑖)6((𝑇+8)(𝑇+4)³−1(𝑇+3)²)≥𝑖²(𝑇+4−𝑗)²(𝑇+3−𝑖)62𝑇²+9𝑇+8(𝑇+4)³(𝑇+3)²≥𝑖²(𝑇+4−𝑗)²(𝑇+3−𝑖)31(𝑇+4)³≥𝑖²(𝑇+4−𝑖)²(𝑗−1)²((𝑇+4−𝑗)²3(𝑇+4)⁷. (2.8)
Then, for 𝑖+1<𝑗≤𝑇+2, we have also 𝐶₀𝑖²(𝑇+4−𝑖)²(𝑗−1)²(𝑇+4−𝑗)²≤𝐺(𝑖,𝑗),𝐺(𝑖,𝑗)≤(𝑗−1)²(𝑇+4−𝑗)²,𝐺(𝑖,𝑗)≤𝑖²(𝑇+4−𝑖)². (2.9)

We note that 𝑢(𝑡) is a solution of (1.1) if and only if𝑢(𝑖)=𝜆^𝑇+2_𝑗=2𝐺(𝑖,𝑗)𝑎(𝑗)𝑓(𝑗,𝑢(𝑗)),𝑖∈[0,𝑇+4]. (2.10)

For our constructions, we will consider the Banach space 𝐸=𝐶([0,𝑇+4]) equipped with the standard norm ‖𝑢‖=max_{0≤𝑖≤𝑇+4}|𝑢(𝑖)|, 𝑢∈𝐸. We define a cone 𝑃 by 𝑃={𝑢∈𝐸∣𝑢(𝑖)≥𝐶₀𝑖²(𝑇+4−𝑖)²‖𝑢‖,𝑖∈[0,𝑇+4]}. (2.11)

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

Theorem 2.3 (see [1]). Let 𝑋 be a Banach space, and let 𝑃⊂𝑋 be a cone in 𝑋. Let Ω₁,Ω₂ be 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. Singular Positone Problems

Theorem 3.1. Assume that the following conditions are satisfied: (H1)𝑓∈𝐶([2,𝑇+2]×(0,+∞),[0,+∞));(H2)𝑓(𝑖,𝑢)≤𝐾(𝑖)(𝑔(𝑢)+ℎ(𝑢)) on [2,𝑇+2]×(0,∞) with 𝑔>0 continuous and nonincreasing on (0,∞), ℎ≥0 continuous on [0,∞), and ℎ/𝑔 nondecreasing on (0,∞), ∃𝐾₀ with 𝑔(𝑥𝑦)≤𝐾₀𝑔(𝑥)𝑔(𝑦) for all 𝑥>0, 𝑦>0;(H3) there exists [𝛼,𝛽]⊂[2,𝑇+2] such that lim_𝑢→+∞inf(𝑓(𝑖,𝑢)/𝑢)=+∞ for 𝑖∈[𝛼,𝛽]; (H4)there exists [𝛼₁,𝛽₁]⊂[2,𝑇+2] such that lim_𝑢→0+inf(𝑓(𝑖,𝑢)/𝑢)=+∞ for 𝑖∈[𝛼₁,𝛽₁]. Then, for each 𝑟>0, there exists a positive number 𝜆^∗ such that the positone problem (1.1) has at least two positive solutions 𝑢₁ and 𝑢₂ with ‖𝑢₁‖<𝑟≤‖𝑢₂‖ for 0<𝜆<𝜆^∗.

Proof. Now, we define the integral operator 𝑇∶𝑃→𝐸 by 𝑇𝑢(𝑖)=𝜆^𝑇+2_𝑗=2𝐺(𝑖,𝑗)𝑎(𝑗)𝑓(𝑗,𝑢(𝑗)), (3.1) where 𝑃={𝑢∈𝑋∣𝑢(𝑖)≥𝐶₀𝑖²(𝑇+4−𝑖)²‖𝑢‖,𝑖∈[0,𝑇+4]}.
It is easy to check that 𝑇(𝑃)⊂𝑃. In fact, for each 𝑢∈𝑃, we have by Lemma 2.2 that 𝑇𝑢(𝑖)≤𝜆^𝑇+2_𝑗=2(𝑗−1)²(𝑇+4−𝑗)²𝑎(𝑗)𝑓(𝑗,𝑢(𝑗)). (3.2) This implies ‖𝑇𝑢‖≤𝜆∑^𝑇+2_𝑗=2(𝑗−1)²(𝑇+4−𝑗)²𝑎(𝑗)𝑓(𝑗,𝑢(𝑗)). On the other hand, we have 𝑇𝑢(𝑖)≥𝐶₀𝑖²(𝑇+4−𝑖)²𝜆^𝑇+2_𝑗=2(𝑗−1)²(𝑇+4−𝑗)²𝑎(𝑗)𝑓(𝑗,𝑢(𝑗)). (3.3) Thus, we have 𝑇𝑢(𝑖)≥𝐶₀𝑖²(𝑇+4−𝑖)²‖𝑇𝑢‖. In addition, standard argument shows that 𝑇 is completely continuous.
For any given 𝑟>0, we fix it, and take Ω_𝑟={𝑢∈𝐸∣‖𝑢‖<𝑟}. Choose 𝜆^∗=𝑟𝐾²₀𝑔(𝐶₀)∑^𝑇+2_𝑗=2(𝑗−1)²(𝑇+4−𝑗)²𝑎(𝑗)𝐾(𝑗)𝑔(𝑗²(𝑇+4−𝑗)².)(𝑔(𝑟)+ℎ(𝑟)) (3.4)
For 𝑢∈𝑃∩𝜕Ω_𝑟, from (H2) and (3.4), we have 𝑇𝑢(𝑡)=𝜆^𝑇+2_𝑗=2𝐺(𝑖,𝑗)𝑎(𝑗)𝑓(𝑗,𝑢(𝑗))≤𝜆^𝑇+2_𝑗=2(𝑗−1)²(𝑇+4−𝑗)²𝑎(𝑗)𝐾(𝑗)(𝑔(𝑢)+ℎ(𝑢))≤𝜆^𝑇+2_𝑗=2(𝑗−1)²(𝑇+4−𝑗)²𝑎(𝑗)𝐾(𝑗)𝑔(𝑢)(1+ℎ(𝑢))𝑔(𝑢)≤𝜆^𝑇+2_𝑗=2(𝑗−1)²(𝑇+4−𝑗)²𝑎(𝑗)𝐾(𝑗)𝑔(𝐶₀𝑗²(𝑇+4−𝑗)²𝑟)(1+ℎ(𝑟))𝑔(𝑟)≤𝜆𝐾²₀𝑔(𝐶₀)^𝑇+2_𝑗=2(𝑗−1)²(𝑇+4−𝑗)²𝑎(𝑗)𝐾(𝑗)𝑔(𝑗²(𝑇+4−𝑗)²)(𝑔(𝑟)+ℎ(𝑟))<𝑟. (3.5) Thus, ‖𝑇𝑢‖<‖𝑢‖,for𝑢∈𝑃∩𝜕Ω_𝑟. (3.6) Further, choose a constant 𝑀^∗>0 satisfying that 𝜆𝑀^∗𝐶₀𝜎max_0≤𝑡≤1{^𝛽_𝑗=𝛼𝐺(𝑖,𝑗)𝑎(𝑗)}>1, (3.7) where 𝜎=min_{𝛼≤𝑖≤𝛽}{𝑖²(𝑇+4−𝑖)²}.
By (H3), there is a constant 𝐿>0 such that 𝑓(𝑖,𝑢)≥𝑀^∗𝑢,∀𝑢≥𝐿,𝑖∈[𝛼,𝛽]. (3.8) Let 𝑅=𝑟+𝐿/𝐶₀𝜎 and Ω_𝑅={𝑢∈𝐸∣‖𝑢‖<𝑅}. For 𝑢∈𝑃∩𝜕Ω_𝑅, we have that 𝑢(𝑖)≥𝐶₀𝑖²(𝑇+4−𝑖)²‖𝑢‖≥𝐶₀𝑅𝑖²(𝑇+4−𝑖)²≥𝐶₀𝑅𝜎≥𝐿,𝑖∈[𝛼,𝛽]. (3.9) It follows that 𝑓(𝑖,𝑢(𝑖))≥𝑀^∗𝑢(𝑖)≥𝑀^∗𝐶₀𝑅𝜎,𝑖∈[𝛼,𝛽]. (3.10)
Then, for 𝑢∈𝑃∩𝜕Ω_𝑅, we have ‖𝑇𝑢‖=𝜆max_0≤𝑡≤1{^𝑇+2_𝑗=2𝐺(𝑖,𝑗)𝑎(𝑗)𝑓(𝑗,𝑢(𝑗))}≥𝜆max_0≤𝑡≤1{^𝛽_𝑗=𝛼𝐺(𝑖,𝑗)𝑎(𝑗)𝑓(𝑗,𝑢(𝑗))}≥𝜆max_0≤𝑡≤1{^𝛽_𝑗=𝛼𝐺(𝑖,𝑗)𝑎(𝑗)𝑀^∗𝐶₀𝑅𝜎}≥𝜆𝑀^∗𝐶₀𝑅𝜎max_0≤𝑡≤1{^𝛽_𝑗=𝛼𝐺(𝑖,𝑗)𝑎(𝑗)}≥𝑅. (3.11)
Therefore, by the first part of the Fixed Point Theorem 2.3, 𝑇 has a fixed point 𝑦 with 𝑟≤‖𝑢₂‖≤𝑅.
Finally, choose a constant 𝑀_∗>0 satisfying that 𝜆𝑀^∗𝐶₀max_0≤𝑡≤1{^𝛽1_𝑗=𝛼1𝐺(𝑖,𝑗)𝑎(𝑗)𝑗²(𝑇+4−𝑗)²}>1. (3.12)
By (𝐻4), there is a constant 𝛿>0 and 𝛿<𝑟 such that 𝑓(𝑖,𝑢)≥𝑀_∗𝑢,∀𝑢≤𝛿,𝑖∈[𝛼₁,𝛽₁]. (3.13) Let 𝑟_∗=𝛿/2 and Ω_𝑟∗={𝑢∈𝐸∣‖𝑢‖<𝑟_∗}. For 𝑢∈𝑃∩𝜕Ω_𝑟∗, we have 𝑢(𝑖)≥𝐶₀𝑖²(𝑇+4−𝑖)²‖𝑢‖≥𝐶₀𝑟_∗𝑖²(𝑇+4−𝑖)². (3.14) It follows that 𝑓(𝑖,𝑢(𝑖))≥𝑀^∗𝑢(𝑖)≥𝑀^∗𝐶₀𝑟_∗𝑖²(𝑇+4−𝑖)²,𝑖∈[𝛼₁,𝛽₁]. (3.15)
Then, for 𝑢∈𝑃∩𝜕Ω_𝑟∗, we have ‖𝑇𝑢‖=𝜆max_0≤𝑡≤1{^𝑇+2_𝑗=2𝐺(𝑖,𝑗)𝑎(𝑗)𝑓(𝑗,𝑢(𝑗))}≥𝜆max_0≤𝑡≤1{^𝛽1_𝑗=𝛼1𝐺(𝑖,𝑗)𝑎(𝑗)𝑓(𝑗,𝑢(𝑗))}≥𝜆max_0≤𝑡≤1{^𝛽1_𝑗=𝛼1𝐺(𝑖,𝑗)𝑎(𝑗)𝑀^∗𝐶₀𝑟_∗𝑗²(𝑇+4−𝑗)²}≥𝜆𝑀^∗𝐶₀𝑟_∗max_0≤𝑡≤1{^𝛽1_𝑗=𝛼1𝐺(𝑖,𝑗)𝑎(𝑗)𝑗²(𝑇+4−𝑗)²}≥𝑟_∗. (3.16)
Therefore, by the first part of the Fixed Point Theorem 2.3, 𝑇 has a fixed point 𝑢₁ with 𝑟_∗≤‖𝑢₁‖≤r. It follows from (3.6) that ‖𝑢₁‖≠𝑟.
Then, for each 𝑟>0, there exists a positive number 𝜆^∗ such that the positone problem (1.1) has at least two positive solutions 𝑢_𝑖(𝑖=1,2) with 𝑟_∗≤‖𝑢₁‖<𝑟≤‖𝑢₂‖≤𝑅 for 0<𝜆<𝜆^∗.

From the proof of Theorem 3.1, we have the following result.

Theorem 3.2. Assume that (H1)–(H3) are satisfied. Then, for each 𝑟>0, there exists a positive number 𝜆^∗ such that the positone problem (1.1) has at least one positive solution 𝑢₂ with 𝑟≤‖𝑢₂‖ for 0<𝜆<𝜆^∗.

Theorem 3.3. Assume that (H1), (H2), and (H4) are satisfied. Then, for each 𝑟>0, there exists a positive number 𝜆^∗ such that the positone problem (1.1) has at least one positive solution 𝑢₁ with ‖𝑢₁‖<𝑟 for 0<𝜆<𝜆^∗.

Example 3.4. Consider the boundary value problem Δ⁴𝑢(𝑖−2)=𝜆𝑎(𝑖)(𝑢^−𝛼+𝑢^𝛽(sin²𝑢+1)),𝑖∈[2,𝑇+2],𝑢(0)=𝑢(1)=0,𝑢(𝑇+3)=𝑢(𝑇+4)=0, (3.17) where 0<𝛼<1<𝛽 are constants. Then, for each 𝑟>0, there exists a positive number 𝜆^∗ such that the problem (3.17) has at least two positive solutions for 0<𝜆<𝜆^∗.
In fact, it is clear that 𝑓(𝑖,𝑢)=𝑢^−𝛼+𝑢^𝛽(sin²𝑢+1),lim_𝑢→0+𝑓(𝑖,𝑢)𝑢=+∞,lim_𝑢→+∞𝑓(𝑖,𝑢)𝑢=+∞. (3.18)
Letting 𝐾(𝑖)=1, 𝑔(𝑢)=𝑢^−𝛼, and ℎ(𝑢)=2𝑢^𝛽, we have 𝑓(𝑖,𝑢)≤𝐾(𝑖)(𝑔(𝑢)+ℎ(𝑢)),𝐾₀=1 (3.19) with 𝑔>0 continuous and nonincreasing on (0,∞), ℎ≥0 continuous on [0,∞), and ℎ/𝑔=2𝑢^𝛼+𝛽 nondecreasing on (0,∞); 𝐾₀=1 with 𝑔(𝑥𝑦)=𝑔(𝑥)𝑔(𝑦)≤𝐾₀𝑔(𝑥)𝑔(𝑦) for ∀𝑥>0, 𝑦>0.
Then, by Theorem 3.1, for each given 𝑟>0, we choose 𝜆^∗=𝐶^𝛼₀𝑟^1+𝛼(1+𝑟^𝛼+𝛽)∑^𝑇+2_𝑗=2𝑗^−2𝛼(𝑗−1)²(𝑇+4−𝑗)^2(1−𝛼),𝑎(𝑗) (3.20) such that the problem (3.17) has at least two positive solutions for 0<𝜆<𝜆^∗.

4. Singular Semipositone Problems

Before we prove our next main result, we first state a result.

Lemma 4.1. The boundary value problem Δ⁴𝑤(𝑖−2)=𝜆𝑎(𝑖)𝑒(𝑖),𝑖∈[2,𝑇+2],𝑤(0)=𝑤(1)=0,𝑤(𝑇+3)=𝑤(𝑇+4)=0 (4.1) has a solution 𝑤 with 𝑤(𝑡)≤𝑐₀𝑖²(𝑇+4−𝑖)², where 𝑐₀=∑^𝑇+2_𝑗=2𝑎(𝑗)𝑒(𝑗).

In fact, from Lemma 2.1, (4.1) has solution 𝑤(𝑡)=^𝑇+2_𝑗=2𝐺(𝑖,𝑗)𝑎(𝑗)𝑒(𝑗). (4.2) According to Lemma 2.2, we have 𝑤(𝑡)≤𝑖²(𝑇+4−𝑖)^2𝑇+2_𝑗=2𝑎(𝑗)𝑒(𝑗)=𝑐₀𝑖²(𝑇+4−𝑖)². (4.3)

Theorem 4.2. Assume that the following conditions are satisfied: (B1)𝑓∶[2,𝑇+2]×(0,∞)→𝐑 is continuous and there exists a function 𝑒∈𝐶([2,𝑇+2],(0,+∞)) with 𝑓(𝑖,𝑢)+𝑒(𝑖)≥0 for (𝑖,𝑢)∈[2,𝑇+2]×(0,∞); (B2)𝑓^∗(𝑖,𝑢)=𝑓(𝑖,𝑢)+𝑒(𝑖)≤𝐾(𝑖)(𝑔(𝑢)+ℎ(𝑢)) on [2,𝑇+2]×(0,∞) with 𝑔>0 continuous and nonincreasing on (0,∞), ℎ≥0 continuous on [0,∞), and ℎ/𝑔 nondecreasing on (0,∞);(B3)∃𝐾₀ with 𝑔(𝑥𝑦)≤𝐾₀𝑔(𝑥)𝑔(𝑦) for all 𝑥>0,𝑦>0;(B4) there exists [𝛼,𝛽]⊂[2,𝑇+2] such that lim_𝑢→+∞inf(𝑓(𝑖,𝑢)/𝑢)=+∞ for 𝑖∈[𝛼,𝛽]. Then, for each 𝑟>0, there exists a positive number 𝜆^∗ such that the semipositone problem (1.1) has at least one positive solution for 0<𝜆<𝜆^∗.

Proof. To show that (1.1) has a nonnegative solution, we will look at the boundary value problem Δ⁴𝑦(𝑖−2)=𝜆𝑎(𝑖)𝑓^∗(𝑖,𝑦(𝑖)−𝜑(𝑖)),𝑖∈[2,𝑇+2],𝑦(0)=𝑦(1)=0,𝑦(𝑇+3)=𝑦(𝑇+4)=0, (4.4) where 𝜑(𝑖)=𝜆𝑤(𝑖) and 𝑤 is as in Lemma 4.1.
We will show, using Theorem 2.3, that there exists a solution 𝑦 to (4.4) with 𝑦(𝑖)>𝜑(𝑖) for 𝑖∈[2,𝑇+2]. If this is true, then 𝑢(𝑖)=𝑦(𝑖)−𝜑(𝑖)(0≤𝑖≤𝑇+4) is a nonnegative solution (positive on [2,𝑇+2]) of (1.1), since Δ⁴𝑢(𝑖−2)=Δ⁴(𝑦(𝑖−2)−𝜑(𝑖−2))=𝜆𝑎(𝑖)𝑓^∗(𝑖,𝑦(𝑖)−𝜑(𝑖))−𝜆𝑎(𝑖)𝑒(𝑖)=𝜆𝑎(𝑖)[𝑓(𝑖,𝑦(𝑖)−𝜑(𝑖))+𝑒(𝑖)]−𝜆𝑎(𝑖)𝑒(𝑖)=𝜆𝑎(𝑖)𝑓(𝑖,𝑦(𝑖)−𝜑(𝑖))=𝜆𝑎(𝑖)𝑓(𝑖,𝑢(𝑖)),𝑖∈[0,𝑇+4]. (4.5)
Next, let 𝑇∶𝐾→𝐸 be defined by (𝑇𝑦)(𝑖)=𝜆^𝑇+2_𝑗=2𝐺(𝑖,𝑗)𝑎(𝑗)𝑓^∗(𝑗,𝑦(𝑗)−𝜑(𝑗)),0≤𝑖≤𝑇+4. (4.6) In addition, standard argument shows that 𝑇(𝑃)⊂𝑃 and 𝑇 is completely continuous.
For any given 𝑟>0, fix it. We choose 𝜆^∗C=min{⁰𝑟2𝑐₀,𝑟𝐾²₀𝑎₀(𝑔(𝑟)+ℎ(𝑟))}, (4.7) where 𝑎₀=𝑔(𝐶₀/2)∑^𝑇+2_𝑗=2(𝑗−1)²(𝑇+4−𝑗)²𝑎(𝑗)𝐾(𝑗)𝑔(𝑗²(𝑇+4−𝑗)²).
Now, let Ω_𝑟={𝑦∈𝐸∣‖𝑦‖<𝑟}. (4.8) We show that ‖𝑇𝑦‖≤‖𝑦‖for𝑦∈𝑃∩𝜕Ω_𝑟. (4.9) To see this, let 𝑦∈𝑃∩𝜕Ω_𝑟. Then, ‖𝑦‖=𝑟 and 𝑦(𝑡)≥𝐶₀𝑖²(𝑇+4−𝑖)²𝑟 for 𝑖∈[0,𝑇+4]. Now, for 𝑖∈[0,𝑇+4], the Lemma 4.1 implies 𝑦(𝑖)−𝜑(𝑖)≥𝐶₀𝑟𝑖²(𝑇+4−𝑖)²−𝜆𝑐₀𝑖²(𝑇+4−𝑖)²≥(𝐶₀𝑟−𝜆𝑐₀)𝑖²(𝑇+4−𝑖)²≥𝐶₀𝑟2𝑖²(𝑇+4−𝑖)²>0, (4.10) so for 𝑖∈[0,𝑇+4], we have (𝑇𝑦)(𝑖)=𝜆^𝑇+2_𝑗=2𝐺(𝑖,𝑗)𝑎(𝑗)𝑓^∗(𝑗,𝑦(𝑗)−𝜑(𝑗))≤𝜆^𝑇+2_𝑗=2(𝑗−1)²(𝑇+4−𝑗)²𝑎(𝑗)𝐾(𝑗)[𝑔(𝑦(𝑗)−𝜑(𝑗))+ℎ(𝑦(𝑗)−𝜑(𝑗))]=𝜆^𝑇+2_𝑗=2(𝑗−1)²(𝑇+4−𝑗)²𝑎(𝑗)𝐾(𝑗)𝑔(𝑦(𝑗)−𝜑(𝑗)){1+ℎ(𝑦(𝑗)−𝜑(𝑗))}𝑔(𝑦(𝑗)−𝜑(𝑗))≤𝜆^𝑇+2_𝑗=2(𝑗−1)²(𝑇+4−𝑗)²𝐶𝑎(𝑗)𝐾(𝑗)𝑔(⁰𝑟2𝑗²(𝑇+4−𝑗)²){1+ℎ(𝑟)𝑔(𝑟)}𝑑𝑠≤𝜆𝐾²₀𝐶𝑔(⁰2)(𝑔(𝑟)+ℎ(𝑟))^𝑇+2_𝑗=2(𝑗−1)²(𝑇+4−𝑗)²𝑎(𝑗)𝐾(𝑗)𝑔(𝑗²(𝑇+4−𝑗)²)=𝜆𝐾²₀𝑎₀(𝑔(𝑟)+ℎ(𝑟))<𝑟. (4.11) This yields ‖𝑇𝑦‖≤𝑟=‖𝑦‖, so (4.9) is satisfied.
Further, choose a constant 𝑀^∗>0 satisfying that 𝜆𝑀^∗𝐶₀2𝜎max_0≤𝑡≤1{^𝛽_𝑗=𝛼𝐺(𝑖,𝑗)𝑎(𝑗)}>1, (4.12) where 𝜎=min_{𝛼≤𝑖≤𝛽}{𝑖²(𝑇+4−𝑖)²}.
By (B4), there is a constant 𝐿>0 such that 𝑓^∗(𝑖,𝑥)≥𝑀^∗𝑥,∀𝑥≥𝐿,𝑖∈[𝛼,𝛽]. (4.13) Let 𝑅=𝑟+max{2𝜆𝑐₀𝐶₁/𝐶₂,2𝐶₁𝐿/𝐶₂𝜎} and Ω_𝑅={𝑦∈𝐸∣‖𝑦‖<𝑅}.
Next, we show that ‖𝑇𝑦‖≥‖𝑦‖,for𝑦∈𝑃∩𝜕Ω_𝑅. (4.14) To see this, let 𝑦∈𝑃∩𝜕Ω_𝑅. We have 𝑦(𝑡)−𝜑(𝑡)≥𝐶₀𝑖²(𝑇+4−𝑖)²‖𝑦‖−𝜆𝑐₀𝑖²(𝑇+4−𝑖)²≥𝐶₀2𝑅𝑖²(𝑇+4−𝑖)²≥𝐶₀2𝑅𝜎≥𝐿,𝑖∈[𝛼,𝛽]. (4.15) It follows that, for 𝑦∈𝑃∩𝜕Ω_𝑅, we have 𝑓^∗(𝑖,𝑦(𝑖)−𝜑(𝑖))≥𝑀^∗(𝑦(𝑖)−𝜑(𝑖))≥𝑀^∗𝐶₀2𝑅𝜎,𝑖∈[𝛼,𝛽]. (4.16)
Then, we have ‖𝑇𝑦‖=𝜆max_{0≤𝑖≤𝑇+4}{^𝑇+2_𝑗=2𝐺(𝑖,𝑗)𝑎(𝑗)𝑓^∗(𝑗,𝑦(𝑗)−𝜑(𝑗))}≥𝜆max_{0≤𝑖≤𝑇+4}{^𝛽_𝑗=𝛼𝐺(𝑖,𝑗)𝑎(𝑗)𝑓^∗(𝑗,𝑦(𝑗)−𝜑(𝑗))}≥𝜆max_{0≤𝑖≤𝑇+4}{^𝛽_𝑗=𝛼𝐺(𝑖,𝑗)𝑎(𝑗)𝑀^∗𝐶₀2𝑅𝜎}≥𝜆𝑀^∗𝐶₀2𝑅𝜎max_{0≤𝑖≤𝑇+4}{^𝛽_𝑗=𝛼𝐺(𝑖,𝑗)𝑎(𝑗)}≥𝑅. (4.17) This yields ‖𝑇𝑦‖≥‖𝑦‖, so (4.14) holds.
Therefore, by the first part of the Fixed Point Theorem 2.3, 𝑇 has a fixed point 𝑦 with 𝑟≤‖y‖≤𝑅, since 𝑦(𝑖)−𝜑(𝑖)≥𝐶₀𝑖²(𝑇+4−𝑖)²𝑟−𝜆𝑐₀𝑖²(𝑇+4−𝑖)²≥(𝐶₀𝑟−𝜆𝑐₀)𝑖²(𝑇+4−𝑖)²>0,𝑖∈[0,𝑇+4]. (4.18) Namely, 𝑢=𝑦−𝜑 is a positive solution of the semipositone problem (1.1).
Then, for each 𝑟>0, there exists a positive number 𝜆^∗ such that the semipositone problem (1.1) has at least one positive solution for 0<𝜆<𝜆^∗.

Example 4.3. Consider the boundary value problem Δ⁴𝑦(𝑖−2)=𝜆𝑎(𝑖)(𝑢^−𝛼+𝑢^𝛽−sin(𝑖𝑢+𝑖^1/2))=0,𝑖∈[2,𝑇+2],𝑦(0)=𝑦(1)=0,𝑦(𝑇+3)=𝑦(𝑇+4)=0, (4.19) where 0<𝛼<1<𝛽 are constants. Then, for each 𝑟>0, there exists a positive number 𝜆^∗ such that the problem (4.19) has at least one positive solution for 0<𝜆<𝜆^∗.
To see this, we will apply Theorem 4.2 (here 𝜆^∗>0 will be chosen later). From 𝑓(𝑡,𝑢)=𝑢^−𝛼+𝑢^𝛽−sin(𝑖𝑢+𝑖^1/2), (4.20) we let 𝑔(𝑢)=𝑢^−𝛼,ℎ(𝑢)=𝑢^𝛽+2,𝐾(𝑖)=1,𝑒(𝑡)=1,𝐾₀=1. (4.21)
It is clear that 0≤𝑓(𝑖,𝑢)+𝑒(𝑖)≤𝐾(𝑖)(𝑔(𝑢)+ℎ(𝑢)), 𝑔(𝑥𝑦)≤𝐾₀𝑔(𝑥)𝑔(𝑦), and lim_𝑢→+∞inf(𝑓(𝑖,𝑢)/𝑢)=+∞, 𝑖∈[𝛼,𝛽]⊂[2,𝑇+2] hold.
Then, the (B1)–(B4) of Theorem 4.2 hold. Now, we have 𝑐₀=^𝑇+2_𝑗=2𝑎(𝑗),𝑎₀=2^𝛼𝐶₀^{−𝛼𝑇+2}_𝑗=2𝑗^−2𝛼(𝑗−1)²(𝑇+4−𝑗)²(1−𝛼))𝑎(𝑗). (4.22)
For each 𝑟>0, we can choose 𝜆^∗𝐶=min{²𝑟2𝑐₀𝐶₁,𝑟^1+𝛼𝐾₀𝑎₀(1+(𝑟+1)^𝛼+𝛽)}. (4.23) Thus, all the conditions of Theorem 4.2 are satisfied, so the existence of positive solution is guaranteed for 0<𝜆<𝜆^∗.

Acknowledgments

This work was supported by Scientific Research Fund of Heilongjiang Provincial Education Department (no. 11544032) and NNSF of China (no. 10971021).