Existence of Positive Solutions for 𝑚-Point Boundary Value Problems on Time Scales

Abstract

We study the one-dimensional 𝑝-Laplacian 𝑚-point boundary value problem (𝜑_𝑝(𝑢^Δ(𝑡)))^Δ+𝑎(𝑡)𝑓(𝑡,𝑢(𝑡))=0, 𝑡∈[0,1]_𝑇, 𝑢(0)=0, 𝑢(1)=∑^𝑚−2_𝑖=1𝑎_𝑖𝑢(𝜉_𝑖), where 𝑇 is a time scale, 𝜑_𝑝(𝑠)=|𝑠|^𝑝−2𝑠, 𝑝>1, some new results are obtained for the existence of at least one, two, and three positive solution/solutions of the above problem by using Krasnosel^′skll^′s fixed point theorem, new fixed point theorem due to Avery and Henderson, as well as Leggett-Williams fixed point theorem. This is probably the first time the existence of positive solutions of one-dimensional 𝑝-Laplacian 𝑚-point boundary value problem on time scales has been studied.

1. Introduction

With the development of p-Laplacian dynamic equations and theory of time scales, a few authors focused their interest on the study of boundary value problems for p-Laplacian dynamic equations on time scales. The readers are referred to the paper [1–7].

In 2005, He [1] considered the following boundary value problems:(𝜑_𝑝(𝑢^Δ(𝑡)))^∇+𝑎(𝑡)𝑓(𝑢(𝑡))=0,𝑡∈[0,𝑇]_𝑇,𝑢(0)−𝐵₀𝑢^Δ(𝜂)=0,𝑢^Δ(𝑇)=0or𝑢^Δ(0)=0,𝑢(𝑇)+𝐵₁𝑢^Δ(𝜂)=0, (1.1)where 𝑇 is a time scales, 𝜑_𝑝(𝑠)=|𝑠|^𝑝−2𝑠,𝑝>1,𝜂∈(0,𝜌(𝑡))_𝑇. The author showed the existence of at least two positive solutions by way of a new double fixed point theorem.

In 2004, Anderson et al. [2] used the virtue of the fixed point theorem of cone and obtained the existence of at least one solution of the boundary value problem:(𝑔(𝑢^Δ(𝑡)))^∇+𝑐(𝑡)𝑓(𝑢)=0,𝑎<𝑡<𝑏,𝑢(𝑎)−𝐵₀𝑢^Δ(𝛾)=0,𝑢^Δ(𝑏)=0. (1.2)

In 2007, Geng and Zhu [3] used the Avery-Peterson and another fixed theorem of cone and obtained the existence of three positive solutions of the boundary value problem:(𝜑_𝑝(𝑢^Δ(𝑡)))^∇+𝑎(𝑡)𝑓(𝑢(𝑡))=0,𝑡∈[0,𝑇]_𝑇,𝑢(0)−𝐵₀𝑢^Δ(𝜂)=0,𝑢^Δ(𝑇)=0. (1.3)Also, in 2007, Sun and Li [4] discussed the existence of at least one, two or three positive solutions of the following boundary value problem:(𝜑_𝑝(𝑢^Δ(𝑡)))^Δ+ℎ(𝑡)𝑓(𝑢^𝜎(𝑡))=0,𝑡∈[𝑎,𝑏]_𝑇,𝑢(𝑎)−𝐵₀𝑢^Δ(𝑎)=0,𝑢^Δ(𝜎(𝑏))=0. (1.4)

In this paper, we are concerned with the existence of multiple positive solutions to the m-point boundary value problem for the one dimension p-Laplcaian dynamic equation on time scale 𝑇(𝜑_𝑝(𝑢^Δ(𝑡)))^Δ+𝑎(𝑡)𝑓(𝑡,𝑢(𝑡))=0,𝑡∈[0,1]_𝑇,𝑢(0)=0,𝑢(1)=∑^𝑚−2_𝑖=1𝑎_𝑖𝑢(𝜉_𝑖), (1.5) where 𝑇 is a time scale, 𝜑_𝑝(𝑠)=|𝑠|^𝑝−2𝑠,𝑝>1,0<𝜉₁<𝜉₂<⋯<𝜉_𝑚−2<1,0≤𝑎_𝑖,𝑖=1,2,…,𝑚−3,𝑎_𝑚−2>0, and

(H₁)∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖<1;(H₂)𝑓∈𝐶_rd([0,1]_𝑇×[0,∞),[0,∞));(H₃)𝑎∈𝐶_rd([0,1]_𝑇,[0,∞)) and there exists 𝑡₀∈(𝜉_𝑚−2,1) such that 𝑎(𝑡₀)>0.

In this paper, we have organized the paper as follows. In Section 2, we give some lemmas which are needed later. In Section 3, we apply the Krassnoselskiifs [8] fixed point theorem to prove the existence of at least one positive solution to the MBVP(1.5). In Section 4, conditions for the existence of at least two positive solutions to the MBVP (1.5) are discussed by using Avery and Henderson [9] fixed point theorem. In Section 5, to prove the existence of at least three positive solutions to the MBVP (1.5) are discussed by using Leggett and Williams [10] fixed point theorem.

For completeness, we introduce the following concepts and properties on time scales.

A time scale 𝑇 is a nonempty closed subset of 𝑅, assume that 𝑇 has the topology that it inherits from the standards topology on 𝑅.

Definition 1.1. Let 𝑇 be a time scale, for 𝑡∈𝑇, one defines the forward jump operator 𝜎∶𝑇→𝑇 by 𝜎(𝑡)=inf{𝑠∈𝑇∶𝑠>𝑡}, and the backward jump operator 𝜌∶𝑇→𝑇 by 𝜌(𝑡)=sup{𝑠∈𝑇∶𝑠<𝑡}, while the graininess function 𝜇∶𝑇→[0,∞) is defined by 𝜇(𝑡)=𝜎(𝑡)−𝑡. If 𝜎(𝑡)>𝑡, one says that is right-scattered, while if 𝜌(𝑡)<𝑡, one says that tis left-scattered. Also, if 𝑡<sup𝑇 and 𝜎(𝑡)=𝑡, then 𝑡 is called right-dense, and if 𝑡>inf𝑇 and 𝜌(𝑡)=𝑡, then 𝑡 is called left-dense. One also needs below the set 𝑇^𝑘 as follows: if 𝑇 has a left-scattered maximum 𝑚, then 𝑇^𝑘=𝑇−𝑚, otherwise 𝑇^𝑘=𝑇. For instance, if sup𝑇=∞, then 𝑇^𝑘=𝑇.

Definition 1.2. Assume 𝑓∶𝑇→𝑅 is a function and let 𝑡∈𝑇. Then , one defines 𝑓^Δ(𝑡) to be the number (provided it exists) with the property that any given 𝜀>0, there is a neighborhood 𝑈 of 𝑡 such that|[𝑓(𝜎(𝑡))−𝑓(𝑠)]−𝑓^Δ(𝑡)[𝜎(𝑡)−𝑠]|≤𝜀|𝜎(𝑡)−𝑠|, (1.6)for all 𝑠∈𝑈. One says that 𝑓 is delta differentiable (or in short: differentiable) on 𝑇 provided 𝑓^Δ(𝑡) exist for all 𝑡∈𝑇.

If 𝑇=𝑅, then 𝑓^Δ(𝑡)=𝑓′(𝑡), if 𝑇=ℤ, then 𝑓^Δ(𝑡)=Δ𝑓(𝑡).

A function 𝑓∶𝑇→𝑅.

(i)If 𝑓 is continuous , then 𝑓 is rd-continuous.(ii)The jump operator 𝜎 is rd-continuous.(iii)If 𝑓 is rd-continuous, then so is 𝑓^𝜎.

A function 𝐹∶𝑇→𝑅 is called an antidervative of 𝑓∶𝑇→𝐑, provided 𝐹^Δ(𝑡)=𝑓(𝑡) holds for all 𝑡∈𝑇^𝑘. One defines the definite integral by^𝑏_𝑎𝑓(𝑡)Δ𝑡=𝐹(𝑏)−𝐹(𝑎). (1.7)For all 𝑎,𝑏∈𝑇. If 𝑓^Δ(𝑡)≥0, then 𝑓 is nondecreasing.

2. The Preliminary Lemmas

Lemma 2.1 (see [5, 6]). Assume that (H₁)–(H₃) hold. Then 𝑢(𝑡) is a solution of the MBVP (1.5) on [0,1]_𝑇 if and only if𝑢(𝑡)=−^𝑡₀𝜑_𝑞(^𝑠₀∑𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠−𝑡⋅^{𝑚−2𝑖=1}𝑎_𝑖∫^𝜉𝑖₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖∫+𝑡⋅¹⁰𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖, (2.1)where 𝜑_𝑞(𝑠)=|𝑠|^𝑞−2𝑠,(1/𝑝)+(1/𝑞)=1, and 𝑞>1.

Lemma 2.2. Assume that conditions (H₁)–(H₃) are satisfied, then the solution of the MBVP (1.5) on [0,1]_𝑇 satisfies𝑢(𝑡)≥0,𝑡∈[0,1]_𝑇. (2.2)

Lemma 2.3 (see [5]). If the conditions (H₁)–(H₃) are satisfied, then𝑢(𝑡)≥𝛾‖𝑢‖,𝑡∈[𝜉_𝑚−2,1], (2.3)where ‖𝑢‖=sup_{𝑡∈[0,1]𝑇}𝑎|𝑢(𝑡)|,𝛾=min{^𝑚−2(1−𝜉_𝑚−2)1−𝑎_𝑚−2𝜉_𝑚−2,𝑎_𝑚−2𝜉_𝑚−2,𝜉₁}. (2.4)

Lemma 2.4 (see [6]). min_{𝑡∈[𝜉𝑚−2,1]}𝐴𝑢(𝑡)=min{𝐴𝑢(1),𝐴𝑢(𝜉_𝑚−2)}.

Let 𝐸 denote the Banach space 𝐶_rd[0,1]_𝑇 with the norm ‖𝑢‖=sup_{𝑡∈[0,1]𝑇}|𝑢(𝑡)|.

Define the cone 𝑃⊂𝐸, by𝑃={𝑢∈𝐸∣𝑢(𝑡)≥0,𝑡∈[𝜉_𝑚−2,1]min𝑢(𝑡)≥𝛾‖𝑢‖,𝑢isconcave}. (2.5)

The solutions of MBVP (1.5) are the points of the operator 𝐴 defined by𝐴𝑢(𝑡)=−^𝑡₀𝜑_𝑞(^𝑠₀∑𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠−𝑡⋅^{𝑚−2𝑖=1}𝑎_𝑖∫^𝜉𝑖₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖∫+𝑡⋅¹⁰𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖=𝑢(𝑡). (2.6)So, 𝐴𝑃⊂𝑃. It is easy to check that 𝐴∶𝑃→𝑃 is completely continuous.

3. Existence of at least One Positive Solutions

Theorem 3.1 (see [8]). Let 𝐸 be a Banach space, and let 𝑃⊂𝐸 be a cone. Assume Ω₁ and Ω₂ are open boundary subsets of 𝐸 with 0∈Ω₁,Ω₁⊂Ω₂, and let 𝐴∶𝑃∩(Ω₂⧵Ω₁)→𝑃 be a completely continuous operator such that either
(i)‖𝐴𝑢‖≤‖𝑢‖ for 𝑢∈𝑃∩𝜕Ω₁,‖𝐴𝑢‖≥‖𝑢‖ for 𝑢∈𝑃∩𝜕Ω₂; or(ii)‖𝐴𝑢‖≥‖𝑢‖ for 𝑢∈𝑃∩𝜕Ω₁,‖𝐴𝑢‖≤‖𝑢‖ for 𝑢∈𝑃∩𝜕Ω₂ hold. Then A has a fixed point in 𝑃∩(Ω₂⧵Ω₁).

Theorem 3.2. Assume conditions (H₁)–(H₃) are satisfied. In addition, suppose there exist numbers 0<𝑟<𝑅<∞ such that 𝑓(𝑡,𝑢)≤𝜑_𝑝(𝑚)𝜑_𝑝(𝑟), if 𝑡∈[0,𝜎(1)],0≤𝑢≤𝑟, and 𝑓(𝑡,𝑢)≥𝜑_𝑝(𝑀𝛾)𝜑_𝑝(𝑅), if 𝑡∈[𝜉_𝑚−2,1],𝑅≤𝑢<∞, where𝑀=1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖𝛾𝜉_𝑚−2∫¹_𝜉𝑚−2𝜑_𝑞(∫^𝑠_𝜉𝑚−2,𝑎(𝜏)Δ𝜏)Δ𝑠𝑚=1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖∫¹₀𝜑_𝑞(∫^𝑠₀.𝑎(𝜏)Δ𝜏)Δ𝑠 (3.1)

Then the MBVP (1.5) has at least one positive solution.

Proof. Define the cone 𝑃 as in (2.5), define a completely continuous integral operator 𝐴∶𝑃→𝑃 by𝐴𝑢(𝑡)=−^𝑡₀𝜑_𝑞(^𝑠₀∑𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠−𝑡⋅^{𝑚−2𝑖=1}𝑎_𝑖∫^𝜉𝑖₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖∫+𝑡⋅¹⁰𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖. (3.2)From (H₁)–(H₃), Lemmas 2.1 and 2.2, 𝐴𝑃⊂𝑃. If 𝑢∈𝑃 with ‖𝑢‖=𝑟, then we get𝐴𝑢(𝑡)=−^𝑡₀𝜑_𝑞(^𝑠₀∑𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠−𝑡⋅^{𝑚−2𝑖=1}𝑎_𝑖∫^𝜉𝑖₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖∫+𝑡⋅¹⁰𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖∫≤𝑡⋅¹⁰𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖≤𝜑_𝑞(𝜑_𝑝(𝑚)𝜑_𝑝∫(𝑟))⋅¹⁰𝜑_𝑞(∫^𝑠₀𝑎(𝜏)Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖∫≤𝑟𝑚⋅¹⁰𝜑_𝑞(∫^𝑠₀𝑎(𝜏)Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖=𝑟=‖𝑢‖. (3.3) This implies that ‖𝐴𝑢‖≤‖𝑢‖. So, if we set Ω₁={𝑢∈𝐶_rd([0,1])∣‖𝑢‖<𝑟}, then ‖𝐴𝑢‖≤‖𝑢‖, for 𝑢∈𝑃∩𝜕Ω₁.
Let us now set Ω₂={𝑢∈𝐶_rd([0,1])∣‖𝑢‖<𝑅}.
Then for 𝑢∈𝑃 with ‖𝑢‖<𝑅, by Lemma 2.4 we have 𝑢(𝑡)≥𝛾‖𝑢‖,𝑡∈[𝜉_𝑚−2,1]. Therefore, we have‖𝐴𝑢(𝑡)‖≥𝐴𝑢(𝜉_𝑚−2)=−^𝜉𝑚−2₀𝜑_𝑞(^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠−𝜉_𝑚−2∑^𝑚−2_𝑖=1𝑎_𝑖∫^𝜉𝑖₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖+𝜉_𝑚−2⋅∫¹₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖=𝜉_𝑚−2∫¹₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠−∫^𝜉𝑚−2₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖+11−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖^𝑚−2_𝑖=1𝑎_𝑖(𝜉_𝑖^𝜉𝑚−2₀𝜑_𝑞(^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠−𝜉_𝑚−2^𝜉𝑖₀𝜑_𝑞(^𝑠₀≥𝜉𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠)_𝑚−2∫¹₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖−∫^𝜉𝑚−2₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖≥𝜉_𝑚−2∫¹_𝜉𝑚−2𝜑_𝑞(∫^𝑠_𝜉𝑚−2𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖≥𝜑_𝑞(𝜑_𝑝(𝑀𝛾)𝜑_𝑝𝜉(𝑅))^𝑚−2∫¹_𝜉𝑚−2𝜑_𝑞(∫^𝑠_𝜉𝑚−2𝑎(𝜏)Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖≥𝑀𝛾𝑅1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖⋅𝜉_𝑚−2¹_𝜉𝑚−2𝜑_𝑞(^𝑠_𝜉𝑚−2𝑎(𝜏)Δ𝜏)Δ𝑠=‖𝑢‖. (3.4)Hence, ‖𝐴𝑢‖≥‖𝑢‖ for 𝑢∈𝑃∩𝜕Ω₂. Thus by the Theorem 3.1, 𝐴 has a fixed point 𝑢 in 𝑃∩(Ω₂⧵Ω₁). Therefore, the MBVP (1.5) has at least one positive solution.

4. Existence of at least Two Positive Solutions

In this section, we apply the Avery-Henderson fixed point theorem [9] to prove the existence of at least two positive solutions to the nonlinear MBVP (1.5) .

Theorem 4.1 (see Avery and Henderson [9]). Let 𝑃 be a cone in a real Banach space 𝐸. Set𝑃(Φ,𝜌₃)={𝑢∈𝑃∣Φ(𝑢)<𝜌₃}. (4.1)

If 𝜈 and Φ are increasing, nonnegative continuous functionals on 𝑃, let 𝜃 be a nonnegative continuous functional on 𝑃 with 𝜃(0)=0 such that, for some positive constants 𝜌₃ and 𝑀>0,Φ(𝑢)≤𝜃(𝑢)≤𝜈(𝑢) and ‖𝑢‖≤𝑀Φ(𝑢), for all 𝑢∈𝑃(Φ,𝜌₃). Suppose that there exist positive numbers 𝜌₁<𝜌₂<𝜌₃ such that 𝜃(𝜆𝑢)=𝜆𝜃(𝑢) for all 0≤𝜆≤1 and 𝑢∈𝜕𝑃(𝜃,𝜌₂).

If 𝐴∶𝑃(Φ,𝜌₃)→𝑃 is a completely continuous operator satisfying

(i)Φ(𝐴𝑢)>𝜌₃ for all 𝑢∈𝜕𝑃(Φ,𝜌₃);(ii)𝜃(𝐴𝑢)<𝜌₂ for all 𝑢∈𝜕𝑃(𝜃,𝜌₂);(iii)𝑃(𝜈,𝜌₁)≠𝜙 and 𝜈(𝐴𝑢)>𝜌₁ for all 𝑢∈𝜕𝑃(𝜈,𝜌₁), then 𝐴 has at least two fixed points 𝑢₁ and 𝑢₂ such that 𝜌₁<𝜈(𝑢₁) with 𝜃(𝑢₁)<𝜌₂ and 𝜌₃<(𝑢₂) with Φ(𝑢₂)<𝜌₃.

Let 𝑙∈(0,1)_𝐓 and 0<𝜉_𝑚−2<𝑙<1. Define the increasing, nonnegative and continuous functionals Φ,𝜃, and 𝜈 on 𝑃, by Φ(𝑢)=𝑢(𝜉_𝑚−2),𝜃(𝑢)=𝑢(𝜉_𝑚−2), and 𝜈(𝑢)=𝑢(𝑙).

From Lemma 2.4, for each 𝑢∈𝑃,Φ(𝑢)=𝜃(𝑢)≤𝜈(𝑢).

In addition, for each 𝑢∈𝑃, Lemma 2.3 implies Φ(𝑢)=𝑢(𝜉_𝑚−2)≥𝛾‖𝑢‖.

Thus,1‖𝑢‖<𝛾Φ(𝑢),∀𝑢∈𝑃. (4.2)

We also see that 𝜃(0)=0 and 𝜃(𝜆𝑢)=𝜆𝜃(𝑢) for all 0≤𝜆≤1 and 𝑢∈𝜕𝑃(𝜃,𝑞).

Theorem 4.2. Assume (H₁)–(H₃) hold, suppose there exist positive numbers 𝜌₁<𝜌₂<𝜌₃, such that the function 𝑓 satisfies the following conditions:
(B₁)𝑓(𝑡,𝑢)>𝜑_𝑝(𝑚𝛾)𝜑_𝑝(𝜌₁), for 𝑡∈[𝜉_𝑚−2,𝑙] and 𝑢∈[𝛾𝜌₁,𝜌₁];(B₂)𝑓(𝑡,𝑢)<𝜑_𝑝(𝑚)𝜑_𝑝(𝜌₂), for 𝑡∈[𝜉_𝑚−2,1] and 𝑢∈[0,𝜌₂];(B₃)𝑓(𝑡,𝑢)>𝜑_𝑝(𝑀𝛾)𝜑_𝑝(𝜌₃), for 𝑡∈[𝜉_𝑚−2,𝑙] and 𝑢∈[𝜌₃,(1/𝛾)𝜌₃].

Then the MBVP (1.5) has at least two positive solutions 𝑢₁ and 𝑢₂ such that 𝑢₁(𝑡)>𝜌₁ with 𝑢₁(𝑙)<𝜌₂ and 𝑢₂(𝑙)>𝜌₂ with 𝑢₂(𝑙)<𝜌₃.

Proof. We now verify that all of the conditions of Theorem 4.1 are satisfied.
Define the cone 𝑃 as (2.5), define a completely continuous integral operator 𝐴∶𝑃→𝑃 by 𝐴𝑢(𝑡)=−^𝑡₀𝜑_𝑞(^𝑠₀∑𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠−𝑡⋅^{𝑚−2𝑖=1}𝑎_𝑖∫^𝜉𝑖₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖∫+𝑡⋅¹⁰𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖. (4.3)
𝑀 and 𝑚 as in (3.1). To verify that condition (i) of Theorem 4.1 holds, we choose 𝑢∈𝜕𝑃(Φ,𝜌₃), then Φ(𝑢)=𝜌₃. This implies 𝜌₃≤‖𝑢‖≤(1/𝛾)Φ(𝑢). Note that ‖𝑢‖≤(1/𝛾)Φ(𝑢)=(1/𝛾)𝜌₃. We have 𝜌₃≤𝑢(𝑡)≤(1/𝛾)𝜌₃, for 𝑡∈[𝜉_𝑚−2,1]_𝑇. As a consequence of (B₃), 𝑓(𝑡,𝑢)>𝜑_𝑝(𝑀𝛾)𝜑_𝑝(𝜌₃), for 𝑡∈[𝜉_𝑚−2,𝑙]_𝑇. Since 𝐴𝑢∈𝑃, we have from Lemma 2.2, Φ(𝐴𝑢)=(𝐴𝑢)(𝜉_𝑚−2)=−^𝜉𝑚−2₀𝜑_𝑞(^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠−𝜉_𝑚−2⋅∑^𝑚−2_𝑖=1𝑎_𝑖∫^𝜉𝑖₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖+𝜉_𝑚−2⋅∫¹₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖=𝜉_𝑚−2∫¹₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠−∫^𝜉𝑚−2₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖+11−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖^{𝑚−2𝑖=1}𝑎_𝑖(𝜉_𝑖^𝜉𝑚−2₀𝜑_𝑞(^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠−𝜉_𝑚−2^𝜉𝑖₀𝜑_𝑞(^𝑠₀≥𝜉𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠)_𝑚−2∫¹₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠−∫^𝜉𝑚−2₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖≥𝜉_𝑚−2∫¹_𝜉𝑚−2𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖≥𝜑_𝑞(𝜑_𝑝(𝑀𝛾)𝜑_𝑝(𝜌₃𝜉))^𝑚−2∫¹_𝜉𝑚−2𝜑_𝑞(∫^𝑠₀𝑎(𝜏)Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖≥𝑀𝛾𝜌₃𝜉_𝑚−21−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖^1𝜉_𝑚−2𝜑_𝑞(^𝑠₀𝑎(𝜏)Δ𝜏)Δ𝑠≥𝜌₃. (4.4)Then condition (i) of Theorem 4.1 holds.
Let 𝑢∈𝜕𝑃(𝜃,𝜌₂). Then 𝜃(𝑢)=𝜌₂. This implies 0≤𝑢(𝑡)≤‖𝑢‖≤(1/𝛾)𝜌₂, for 𝑡∈[𝜉_𝑚−2,1]. From (B₂), we have𝜃(𝐴𝑢)=(𝐴𝑢)(𝜉_𝑚−2)≤𝜉_𝑚−2⋅∫¹₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖≤𝜉_𝑚−2⋅∫¹₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖≤𝑚𝜌₂⋅∫¹₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖=𝜌₂=‖𝑢‖. (4.5)Hence condition (ii) of Theorem 4.1 holds.
If we first define 𝑢(𝑡)=𝜌₁/2, for 𝑡∈[0,1]_𝑇, then 𝜈(𝑢)=𝜌₁/2<𝜌₁. So 𝑃(𝜈,𝜌₁)≠𝜙.
Now, let 𝑢∈𝜕𝑃(𝜈,𝜌₁), then 𝜈(𝑢)=𝑢(𝑙)=𝜌₁. This mean that 𝜌₁/𝛾≤𝑢(𝑡)≤‖𝑢‖≤𝜌₁. From (B₁) and Lemma 2.4, we get𝜈(𝐴𝑢)=(𝐴𝑢)(𝑙)≥(𝐴𝑢)(𝜉_𝑚−2)=−^𝜉𝑚−2₀𝜑_𝑞(^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠−𝜉_𝑚−2⋅∑^𝑚−2_𝑖=1𝑎_𝑖∫^𝜉𝑖₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖+𝜉_𝑚−2⋅∫¹₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖=𝜉_𝑚−2∫¹₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠−∫^𝜉𝑚−2₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖+11−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖^𝑚−2_𝑖=1𝑎_𝑖(𝜉_𝑖^𝜉𝑚−2₀𝜑_𝑞(^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠−𝜉_𝑚−2^𝜉𝑖₀𝜑_𝑞(^𝑠₀≥𝜉𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠)_𝑚−2∫¹₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠−∫^𝜉𝑚−2₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖≥𝜉_𝑚−2∫¹_𝜉𝑚−2𝜑_𝑞(∫^𝑠_𝜉𝑚−2𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖≥𝜑_𝑞(𝜑_𝑝(𝑚𝛾)𝜑_𝑝(𝜌₁𝜉))^𝑚−2∫¹_𝜉𝑚−2𝜑_𝑞(∫^𝑠_𝜉𝑚−2𝑎(𝜏)Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖=𝑚𝛾𝜌11−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖⋅𝜉_𝑚−2¹_𝜉𝑚−2𝜑_𝑞(^𝑠_𝜉𝑚−2𝑎(𝜏)Δ𝜏)Δ𝑠≥𝜌₁. (4.6)Then condition (iii) of Theorem 4.1 holds.
Since all conditions of Theorem 4.1 are satisfied, the MBVP (1.5) has at least two positive solutions 𝑢₁ and 𝑢₂ such that 𝑢₁(𝑡)>𝜌₁ with 𝑢₁(𝑙)<𝜌₂ and 𝑢₂(𝑙)>𝜌₂ with 𝑢₂(𝑙)<𝜌₃.

5. Existence of at least Three Positive Solutions

We will use the Leggett-Williams fixed point theorem [10] to prove the existence of at least three positive solutions to the nonlinear MBVP (1.5).

Theorem 5.1 (see Leggett and Williams [10]). Let 𝑃 be a cone in the real Banach space 𝐸. Set𝑃𝑟={𝑥∈𝑃∣‖𝑥‖<𝑟},𝑃(Ψ,𝑎,𝑏)={𝑥∈𝑃∣𝑎≤Ψ(𝑥),‖𝑥‖≤𝑏}. (5.1)

Suppose 𝐴∶𝑃𝑟→𝑃𝑟 be a completely continuous operator and be a nonnegative continuous concave functional on 𝑃 with Ψ(𝑢)≤‖𝑢‖ for all 𝑢∈𝑃𝑟. If there exists 0<𝜌₁<𝜌₂<(1/𝛾)𝜌₂≤𝜌₃ such that the following condition hold:

(i){𝑢∈𝑃(Ψ,𝜌₂,(1/𝛾)𝜌₂)∣Ψ(𝑢)>𝜌₂}≠𝜙 and Ψ(𝐴𝑢)>𝜌₂ for all 𝑢∈𝑃(Ψ,𝜌₂,(1/𝛾)𝜌₂);(ii)‖𝐴𝑢‖<𝜌₁ for ‖𝑢‖≤𝜌₁;(iii)Ψ(𝐴𝑢)>𝜌₂ for 𝑢∈𝑃(Ψ,𝜌₂,(1/𝛾)𝜌₂) with ‖𝐴𝑢‖>(1/𝛾)𝜌₂, then 𝐴 has at least three fixed points 𝑢₁,𝑢₂ and 𝑢₃ in 𝑃𝑟 satisfying ‖𝑢₁‖<𝜌₁,Ψ(𝑢₂)>𝜌₂,𝜌₁<‖𝑢₃‖ with Ψ(𝑢₂)<𝜌₂.

Theorem 5.2. Assume (H₁)–(H₃) hold . Suppose that there exist constants 0<𝜌₁<𝜌₂<(1/𝛾)𝜌₂≤𝜌₃ such that
(C₁)𝑓(𝑡,𝑢)≤𝜑_𝑝(𝑚)𝜑_𝑝(𝜌₃), for 𝑡∈[𝜉_𝑚−2,𝑙] and 𝑢∈[0,𝜌₃];(C₂)𝑓(𝑡,𝑢)>𝜑_𝑝(𝑀𝛾)𝜑_𝑝(𝜌₂), for 𝑡∈[𝜉_𝑚−2,𝑙] and 𝑢∈[𝜌₂,(1/𝛾)𝜌₂];(C₃)𝑓(𝑡,𝑢)<𝜑_𝑝(𝑚)𝜑_𝑝(𝜌₁), for 𝑡∈[𝜉_𝑚−2,1] and 𝑢∈[0,𝜌₁].

Then the MBVP (1.5) has at least three positive solutions 𝑢₁,𝑢₂, and 𝑢₃ such that 𝑢₁(𝜉)<𝜌₁,𝑢₂(𝑙)>𝜌₂,𝑢₃(𝜉)>𝜌₁ with 𝑢₃(𝑙)<𝜌₂.

Proof. The conditions of Theorem 5.1 will be shown to be satisfied. Define the nonnegative continuous concave functional Ψ∶𝑃→[0,∞) to be Ψ(𝑢)=𝑢(𝜉_𝑚−2), the cone 𝑃 as in (2.5), 𝑀 and 𝑚 as in (3.1). We have Ψ(𝑢)≤‖𝑢‖ for all 𝑢∈𝑃. If 𝑢∈𝑃_𝜌3, then ‖𝑢‖≤𝜌₃, and from assumption (C₁) , then we have(𝐴𝑢)(𝑡)=−^𝑡₀𝜑_𝑞(^𝑠₀∑𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠−𝑡⋅^{𝑚−2𝑖=1}𝑎_𝑖∫^𝜉𝑖₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖∫+𝑡⋅¹⁰𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖∫≤𝑡⋅¹⁰𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖≤𝜑_𝑞(𝜑_𝑝(𝑚)𝜑_𝑝(𝜌₃∫))⋅¹⁰𝜑_𝑞(∫^𝑠₀𝑎(𝜏)Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖≤𝑚𝜌₃⋅∫¹₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖=𝜌₃. (5.2)
This implies that ‖𝐴𝑢‖≤𝜌₃. Thus, we have 𝐴∶𝑃_𝜌3→𝑃_𝜌3. Since (1/𝛾)𝜌₂∈𝑃(Ψ,𝜌₂,(1/𝛾)𝜌₂) and Ψ((1/𝛾)𝜌₂)=(1/𝛾)𝜌₂>𝜌₂,{𝑢∈𝑃(Ψ,𝜌₂,(1/𝛾)𝜌₂)|Ψ(𝑢)>𝜌₂}≠𝜙. For 𝑢∈𝑃(Ψ,𝜌₂,(1/𝛾)𝜌₂) we have 𝜌₂≤𝑢(𝜉_𝑚−2)≤‖𝑢‖≤(1/𝛾)𝜌₂. Using assumption (C₂), 𝑓(𝑡,𝑢)>𝜑_𝑝(𝑀𝛾)𝜑_𝑝(𝜌₂), we obtainΨ(𝐴𝑢)=(𝐴𝑢)(𝜉_𝑚−2)=−^𝜉𝑚−2₀𝜑_𝑞(^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠−𝜉_𝑚−2⋅∑^𝑚−2_𝑖=1𝑎_𝑖∫^𝜉𝑖₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖+𝜉_𝑚−2⋅∫¹₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖=𝜉_𝑚−2∫¹₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠−∫^𝜉𝑚−2₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖+11−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖^𝑚−2_𝑖=1𝑎_𝑖(𝜉_𝑖^𝜉𝑚−2₀𝜑_𝑞(^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠−𝜉_𝑚−2^𝜉𝑖₀𝜑_𝑞(^𝑠₀≥𝜉𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠)_𝑚−2∫¹₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠−∫^𝜉𝑚−2₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖≥𝜉_𝑚−2∫¹_𝜉𝑚−2𝜑_𝑞(∫^𝑠_𝜉𝑚−2𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖≥𝜑_𝑞(𝜑_𝑝(𝑀𝛾)𝜑_𝑝(𝜌₂𝜉))^𝑚−2∫¹_𝜉𝑚−2𝜑_𝑞(∫^𝑠_𝜉𝑚−2𝑎(𝜏)Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖≥𝑀𝛾𝜌₂1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖⋅𝜉_𝑚−2¹_𝜉𝑚−2𝜑_𝑞(^𝑠_𝜉𝑚−2𝑎(𝜏)Δ𝜏)Δ𝑠≥𝜌₂. (5.3)Hence, condition (i) of Theorem 5.1 holds.
If ‖𝑢‖≤𝜌₁, from assumption (C₃), we obtain(𝐴𝑢)(𝑡)≤𝜑_𝑞(𝜑_𝑝(𝑚)𝜑_𝑝(𝜌₁∫))⋅¹⁰𝜑_𝑞(∫^𝑠₀𝑎(𝜏)Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖≤𝑚𝜌₁⋅∫¹₀𝜑_𝑞(∫^𝑠₀𝑎(𝜏)Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖=𝜌₁. (5.4)This implies that ‖𝐴𝑢‖≤𝜌₁.
Consequently, condition (ii) of Theorem 5.1 holds.
We suppose that 𝑢∈𝑃(Ψ,𝜌₂,𝜌₃), with ‖𝐴𝑢‖>(1/𝛾)𝜌₂. Then we getΨ(𝐴𝑢)=(𝐴𝑢)(𝜉_𝑚−2)≥𝜉_𝑚−2∫¹_𝜉𝑚−2𝜑_𝑞(∫^𝑠_𝜉𝑚−2𝑎(𝜏)𝑓(𝜏,𝑢(𝜏))Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖≥𝜑_𝑞(𝜑_𝑝(𝑀𝛾)𝜑_𝑝(𝜌₂𝜉))^𝑚−2∫¹_𝜉𝑚−2𝜑_𝑞(∫^𝑠_𝜉𝑚−2𝑎(𝜏)Δ𝜏)Δ𝑠1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖≥𝑀𝛾𝜌₂1−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖⋅𝜉_𝑚−2¹_𝜉𝑚−2𝜑_𝑞(^𝑠_𝜉𝑚−2𝑎(𝜏)Δ𝜏)Δ𝑠≥𝜌₂. (5.5)Hence, condition (iii) of Theorem 5.1 holds.
Because all of the hypotheses of the Leggett-Williams fixed point theorem are satisfied, the nonlinear MBVP (1.5) has at least three positive solutions 𝑢₁,𝑢₂, and 𝑢₃ such that 𝑢₁(𝜉)<𝜌₁,𝑢₂(𝑙)>𝜌₂, and 𝑢₃(𝜉)>𝜌₁ with 𝑢₃(𝑙)<𝜌₂.

Acknowledgment

This work is supported by the Research and Development Foundation of College of Shanxi Province (no. 200811043).