Existence and Uniqueness of Solution to Nonlinear Boundary Value Problems with Sign-Changing Green’s Function

Abstract

By using the cone theory and the Banach contraction mapping principle, the existence and uniqueness results are established for nonlinear higher-order differential equation boundary value problems with sign-changing Green’s function. The theorems obtained are very general and complement previous known results.

1. Introduction

Boundary value problems (BVPs for short) for nonlinear differential equations arise in a variety of areas of applied mathematics, physics, and variational problems of control theory. The study of multipoint BVPs for second-order differential equations was initiated by Bicadze and Samarskiĭ [1] and later continued by II'in and Moiseev [2, 3] and Gupta [4]. Since then, great efforts have been devoted to nonlinear multipoint BVPs due to their theoretical challenge and great application potential. Many results on the existence of solutions for multipoint BVPs have been obtained; the methods used therein mainly depend on the fixed point theorems, degree theory, upper and lower techniques, and monotone iteration. The existence results are available in the literature [5–25] and the references therein.

Recently, by applying the fixed point theorems on cones, the authors of papers [5–7] established the existence and multiplicity of positive solutions for the 𝑛th-order three-point BVP: 𝑢^(𝑛)(𝑡)(𝑡)𝑓((𝑡(),𝑢(0)+𝑎𝑡,𝑢))=0,𝑡∈0,1=𝑢^′(0)=…=𝑢^(𝑛−2)(0)=0,𝑢(1)=𝛼𝑢𝜂, (1) where 𝑛≥2,0<𝜂<1 and 0<𝛼𝜂^𝑛−1<1. The 𝑛th-order 𝑚-point BVP 𝑢^(𝑛)(𝑡)(𝑡)𝑓((𝑡(),𝑢(0)+𝑎𝑡,𝑢))=0,𝑡∈0,1=𝑢^′(0)=…=𝑢^(𝑛−2)(0)=0,𝑢(1)=^𝑚−2_𝑖=1𝛼_𝑖𝑢𝜂_𝑖 (2) has been studied in [8–10], where 𝑛≥2,   0<𝜂₁<𝜂₂<⋯𝜂_𝑚−2<1 and 𝛼_𝑖>0(𝑖=1,2,⋯,𝑚−2) with 0<∑^𝑚−2_𝑖=1𝛼_𝑖𝜂_𝑖^𝑛−1<1. The existence and multiplicity results of solutions were shown by using various fixed point theorems and fixed point index theory.

By using the cone theory and the Banach contraction mapping principle, the author [26] established the existence and uniqueness for singular third-order three-point boundary value problems.

The purpose of this paper is to investigate the existence and uniqueness of solution of the following higher-order differential equation boundary value problem: 𝑢^(𝑛)(𝑡)+𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡),…,𝑢^(𝑛−1)(𝑡)𝑢(0)=0,𝑡∈𝐽,=𝑢^′(0)=…=𝑢^(𝑛−2)(0)=0,𝑢(1)=^𝑚−2_𝑖=1𝛼_𝑖𝑢𝜂_𝑖, (3) where 𝑛≥2,  𝑓∈𝐶(𝐽×ℝ^𝑛,ℝ),𝐽=(0,1), ∑^𝑚−2_𝑖=1𝛼_𝑖𝜂_𝑖^𝑛−1≠1,and0<𝜂₁<⋯<𝜂_𝑚−2<1.

Here, we give the unique solution of BVP (3) under the conditions that 𝑓 is mixed nonmonotone. The methods used in this paper are motivated by [26], and the arguments are based upon the cone theory and the Banach contraction mapping principle.

2. The Preliminary Lemmas

Lemma 1. For any 𝑓∈𝐿(𝐼), the BVP 𝑢^′(𝑡)+𝑓(𝑡)=0,𝑡∈𝐽,₁₀(1−𝑡)^𝑛−2𝑢(𝑡)𝑑𝑡=^𝑚−2_𝑖=1𝛼_𝑖^𝜂𝑖₀𝜂_𝑖−𝑡^𝑛−2𝑢(𝑡)𝑑𝑡 (4)(5) has a unique solution 𝑢(𝑡)=∫¹₀𝐺(𝑡,𝑠)𝑓(𝑠)𝑑𝑠, where 𝐺()=1𝑡,𝑠−1+𝜎(1−𝑠)^𝑛−1−∑^𝑚−2_𝑖=1𝛼_𝑖𝜂_𝑖−𝑠^𝑛−1,0≤𝑠≤𝜂₁1,𝑠≤𝑡,𝜎(1−𝑠)^𝑛−1−∑^𝑚−2_𝑖=1𝛼_𝑖𝜂_𝑖−𝑠^𝑛−1,0≤𝑡≤𝑠≤𝜂₁,1−1+𝜎(1−𝑠)^𝑛−1−∑^𝑚−2_𝑖=𝑗+1𝛼_𝑖𝜂_𝑖−𝑠^𝑛−1,𝜂_𝑗≤𝑠≤𝜂_𝑗+11,𝑠≤𝑡,𝜎(1−𝑠)^𝑛−1−∑^𝑚−2_𝑖=𝑗+1𝛼_𝑖𝜂_𝑖−𝑠^𝑛−1,𝜂_𝑗≤𝑠≤𝜂_𝑗+1(),𝑡≤𝑠,−1+1−𝑠^𝑛−1𝜎,𝜂_𝑚−2()≤𝑠≤𝑡≤1,1−𝑠^𝑛−1𝜎,𝜂_𝑚−2≤𝑠≤1,𝑡≤𝑠,𝜎=1−^𝑚−2_𝑖=1𝛼_𝑖𝜂_𝑖^𝑛−1,𝐼=[0,1]. (6)

Proof. First, suppose that 𝑢∈𝐶(𝐼) is a solution to problem (4) and (5). It is easy to see by integration of (4) that 𝑢(𝑡)=𝑢(0)−^𝑡₀𝑓(𝑠)𝑑𝑠. (7) Substituting (7) into (5), we obtain ¹₀(1−𝑡)^𝑛−2𝑢(0)−^𝑡₀𝑓(𝑠)=𝑑𝑠𝑑𝑡_{𝑚−2𝑖=1}𝛼_𝑖^𝜂𝑖₀𝜂_𝑖−𝑡^𝑛−2𝑢(0)−^𝑡₀𝑓(𝑠)𝑑𝑠𝑑𝑡, (8) and so 𝑢(0)=¹₀(1−𝑡)^𝑛−2𝑑𝑡−^𝑚−2_𝑖=1𝛼_𝑖^𝜂𝑖₀𝜂_𝑖−𝑡^𝑛−2𝑑𝑡⁻¹×¹₀(1−𝑡)^𝑛−2𝑡₀𝑓(𝑠)−𝑑𝑠𝑑𝑡_{𝑚−2𝑖=1}𝛼_𝑖^𝜂𝑖₀𝜂_𝑖−𝑡^𝑛−2𝑡₀𝑓(𝑠)=𝑑𝑠𝑑𝑡1−∑^𝑚−2_𝑖=1𝛼_𝑖𝜂_𝑖^𝑛−1𝑛−1⁻¹×¹₀(1−𝑡)^𝑛−2𝑡₀𝑓(𝑠)−𝑑𝑠𝑑𝑡_{𝑚−2𝑖=1}𝛼_𝑖^𝜂𝑖₀𝜂_𝑖−𝑡^𝑛−2𝑡₀𝑓(𝑠)=𝑑𝑠𝑑𝑡𝑛−1𝜎¹₀(1−𝑡)^𝑛−2𝑡₀𝑓(𝑠)−𝑑𝑠𝑑𝑡_{𝑚−2𝑖=1}𝛼_𝑖^𝜂𝑖₀𝜂_𝑖−𝑡^𝑛−2𝑡₀𝑓(𝑠)=𝑑𝑠𝑑𝑡𝑛−1𝜎¹₀𝑓(𝑠)¹_𝑠(1−𝑡)^𝑛−2−𝑑𝑡𝑑𝑠_{𝑚−2𝑖=1}𝛼_𝑖^𝜂𝑖₀𝑓(𝑠)^𝜂𝑖_𝑠𝜂_𝑖−𝑡^𝑛−2=1𝑑𝑡𝑑𝑠𝜎¹₀𝑓(𝑠)(1−𝑠)^𝑛−1−𝑑𝑠_{𝑚−2𝑖=1}𝛼_𝑖^𝜂𝑖₀𝑓(𝑠)𝜂_𝑖−𝑠^𝑛−1𝑑𝑠. (9) Substituting (9) into (7), we have 𝑢(𝑡)=−^𝑡₀𝑓(𝑠)+1𝑑𝑠𝜎¹₀𝑓(𝑠)(1−𝑠)^𝑛−1−𝑑𝑠_{𝑚−2𝑖=1}𝛼_𝑖^𝜂𝑖₀𝑓(𝑠)𝜂_𝑖−𝑠^𝑛−1𝑑𝑠=¹₀𝐺(𝑡,𝑠)𝑓(𝑠)𝑑𝑠. (10) Conversely, suppose that 𝑢(𝑡)=∫¹₀𝐺(𝑡,𝑠)𝑓(𝑠)𝑑𝑠; then it is easy to verify that (4) and (5) are satisfied. The lemma is proved.

For any 𝑢∈𝐶(𝐼), let 𝐼_𝑖𝑢(𝑡)=^𝑡₀(𝑡−𝑠)^𝑖−1()!𝑢(𝑠)(𝑡)=𝑖−1𝑑𝑠,𝑖=1,2,…,𝑛−1,𝐹𝑢)(¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝐼_𝑛−1𝑢(𝑠)𝐼,…,₁𝑢(𝑠),𝑢(𝑠)𝑑𝑠,𝑡∈𝐼. (11)

Lemma 2. (i) If 𝑢∈𝐶^𝑛−1(𝐼) is a solution to problem (3), then 𝑣(𝑡)=𝑢^(𝑛−1)(𝑡)∈𝐶(𝐼) is a fixed point of 𝐹.
(ii) If 𝑣∈𝐶(𝐼) is a fixed point of 𝐹, then 𝑢(𝑡)=(𝐼_𝑛−1𝑣)(𝑡)=∫^𝑡₀((𝑡−𝑠)^𝑛−2/(𝑛−2)!)𝑣(𝑠)𝑑𝑠∈𝐶^𝑛−1(𝐼) is a solution to problem (3).

By Lemma 1, the proof follows by routine calculations.

Let ℎ₁(𝑡)=max¹₀|𝐺(𝑡,𝑠)|𝑑𝑠,^𝑡₀¹₀|𝐺(,ℎ𝑠,𝑥)|𝑑𝑥𝑑𝑠_𝑘(𝑡)=max¹₀|𝐺(𝑡,𝑠)|ℎ_𝑘−1(𝑠)𝑑𝑠,_𝑡010|𝐺(𝑠,𝑥)|ℎ_𝑘−1(𝑥),𝜌(𝐺)𝑑𝑥𝑑𝑠𝑘=2,3,…,=lim_𝑘→∞sup_𝑡∈𝐽ℎ_𝑘(𝑡)^−1/𝑘. (12)

It is easy to see that 𝜌(𝐺)≥(sup_𝑡∈𝐽ℎ_𝑘(𝑡))^−1/𝑘≥(sup_{𝑡,𝑠∈𝐽}|𝐺(𝑡,𝑠)|)⁻¹>0.

Lemma 3 (see [27, 28]). 𝑃 is a generating cone in Banach space (𝐸,‖⋅‖) if and only if there exists a constant 𝜏>0 such that every element 𝑢∈𝐸 can be represented in the form 𝑢=𝑣−𝑤, where 𝑣,𝑤∈𝑃 and ‖𝑣‖≤𝜏‖𝑢‖,‖𝑤‖≤𝜏‖𝑢‖.

3. Main Results

This section discusses the solution of nonlinear higher-order differential equation BVP (3).

Let 𝑃={𝑢∈𝐶(𝐼)|𝑢(𝑡)≥0,forall𝑡∈[0,1]}. Obviously, 𝑃 is a normal solid cone of Banach space 𝐶(𝐼), by Lemma 2.1.2 in [29], and we have that 𝑃 is a generating cone in 𝐶(𝐼).

Theorem 4. Suppose that 𝑔∈𝐶(𝐽×ℝ^2𝑛,ℝ), 𝑓(𝑡,𝑥₀,𝑥₁,…,𝑥_𝑛−1)=𝑔(𝑡,𝑥₀,𝑥₀,𝑥₁,𝑥₁,…,𝑥_𝑛−1,𝑥_𝑛−1), and there exist positive constants 𝐾₀,𝑀₀,𝐾₁,𝑀₁,…,𝐾_𝑛−1,𝑀_𝑛−1 with 𝐾₀+𝑀₀()!+𝐾𝑛−1¹+𝑀₁()!𝐾𝑛−2+…+^𝑛−3+𝑀_𝑛−32!+𝐾_𝑛−2+𝑀_𝑛−2+𝐾_𝑛−1+𝑀_𝑛−1<𝜌(𝐺), (13) such that for any 𝑡∈𝐼,𝑠₀₁,𝑡₀₁,𝑠₀₂, 𝑡₀₂,𝑠₁₁,𝑡₁₁,𝑠₁₂,𝑡₁₂,…,𝑠_1,𝑛−1,𝑡_1,𝑛−1,𝑠_2,𝑛−1,𝑡_2,𝑛−1∈ℝ with 𝑠₀₁≤𝑡₀₁,𝑠₀₂≥𝑡₀₂,𝑠₁₁≤𝑡₁₁,𝑠₁₂≥𝑡₁₂,…,𝑠_𝑛−1,1≤𝑡_𝑛−1,1,𝑠_𝑛−1,2≥𝑡_𝑛−1,2, one has −𝐾₀𝑡₀₁−𝑠₀₁−𝑀₀𝑠₀₂−𝑡₀₂−𝐾₁𝑡₁₁−𝑠₁₁−𝑀₁𝑠₁₂−𝑡₁₂−…−𝐾_𝑛−1𝑡_𝑛−1,1−𝑠_𝑛−1,1−𝑀_𝑛−1𝑠_𝑛−1,2−𝑡_𝑛−1,2≤𝑔𝑡,𝑠₀₁,𝑠₀₂,𝑠₁₁,𝑠₁₂,…,𝑠_𝑛−1,1,𝑠_𝑛−1,2−𝑔𝑡,𝑡₀₁,𝑡₀₂,𝑡₁₁,𝑡₁₂,…,𝑡_𝑛−1,1,𝑡_𝑛−1,2≤−𝐾₀𝑡₀₁−𝑠₀₁−𝑀₀𝑠₀₂−𝑡₀₂−𝐾₁𝑡₁₁−𝑠₁₁−𝑀₁𝑠₁₂−𝑡₁₂−…−𝐾_𝑛−1𝑡_𝑛−1,1−𝑠_𝑛−1,1−𝑀_𝑛−1𝑠_𝑛−1,2−𝑡_𝑛−1,2, (14) and there exist 𝑢₀,𝑣₀∈𝐶^𝑛−1(𝐼), such that ¹₀𝑔𝑡,𝑢₀(𝑡),𝑣₀(𝑡),𝑢^′₀(𝑡),𝑣^′₀(𝑡)𝑢,…,_0(𝑛−1)(𝑡),𝑣₀^(𝑛−1)(𝑡)𝑑𝑡 (15) converges. Then, BVP (3) has a unique solution 𝐼_𝑛−1𝑢^∗ in 𝐶(𝐼), and moreover,  for any 𝑢₀∈𝐶(𝐼), the iterative sequence 𝑢_𝑚(𝑡)=¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝐼_𝑛−1𝑢_𝑚−1(𝑠)𝐼,…,₁𝑢_𝑚−1(𝑠),𝑢_𝑚−1(𝑠)𝑑𝑠,𝑚=1,2,…, (16) converges to 𝑢^∗ in 𝐶(𝐼)(𝑚→∞).

Remark 5. Recently, in the study of BVP (3), almost all the papers have supposed that Green’s function 𝐺(𝑡,𝑠) is nonnegative. However, the scope of 𝛼_𝑖 is not limited to ∑^𝑚−2_𝑖=1𝛼_𝑖𝜂_𝑖<1 in Theorem 4, so, we do not need to suppose that 𝐺(𝑡,𝑠) is nonnegative.

Remark 6. The function 𝑓 in Theorem 4 is not monotone or convex; the conclusions and the proof used in this paper are different from the known papers in essence.

Proof of Theorem 4. It is easy to see that, for any 𝑡∈𝐽,𝐺(𝑡,𝑠) can be divided into finite partitioned monotone and bounded function on (0,1), and then, by (15), we have that ¹₀𝐺(𝑡,𝑠)𝑔𝑠,𝑢₀(𝑠),𝑣₀(𝑠),𝑢^′₀(𝑠),𝑣^′₀(𝑠)𝑢,…,_0(𝑛−1)(𝑠),𝑣₀^(𝑛−1)(𝑠)𝑑𝑠 (17) converges. Let 𝑝(𝑡)=𝑢₀^(𝑛−1)(𝑡),𝑞(𝑡)=𝑣₀^(𝑛−1)(𝑡); then ¹₀𝐺(𝑡,𝑠)𝑔𝑠,𝐼_𝑛−1𝑝(𝑠),𝐼_𝑛−1𝑞(𝑠)𝐼,…,₁𝑝(𝑠),𝐼₁𝑞(𝑠),𝑝(𝑡),𝑞(𝑠)𝑑𝑠 (18) converges.
For any 𝑢,𝑣∈𝐶(𝐼), let 𝑥(𝑡)=|𝑝(𝑡)|+|𝑢(𝑡)|,𝑦(𝑡)=−|𝑞(𝑡)|−|𝑣(𝑡)| and then 𝑥≥𝑝,𝑦≤𝑞. By (14), we have −𝐾₀𝐼_𝑛−1𝑥−𝐼_𝑛−1𝑝(𝑡)−𝑀₀𝐼_𝑛−1𝑞−𝐼_𝑛−1𝑦(𝑡)−𝐾₁𝐼_𝑛−2𝑥−𝐼_𝑛−2𝑝(𝑡)−𝑀₁𝐼_𝑛−2𝑞−𝐼_𝑛−2𝑦(𝑡)−…−𝐾_𝑛−2𝐼₁𝑥−𝐼₁𝑝(𝑡)−𝑀_𝑛−2𝐼₁𝑞−𝐼₁𝑦(𝑡)−𝐾_𝑛−1𝑥−𝑝(𝑡)−𝑀_𝑛−1(𝑡)𝐼𝑞−𝑦≤𝑔𝑡,_𝑛−1𝑥(𝑡),𝐼_𝑛−1𝑦𝑡)𝐼(,…,₁𝑥(𝑡),𝐼₁𝑦(𝑡)(𝑡)(𝑡)𝐼,𝑥,𝑦−𝑔𝑡,_𝑛−1𝑝(𝑡),𝐼_𝑛−1𝑞(𝑡)𝐼,…,₁𝑝(𝑡),𝐼₁𝑞(𝑡),𝑝(𝑡),𝑞(𝑡)≤𝐾₀𝐼_𝑛−1𝑥−𝐼_𝑛−1𝑝(𝑡)+𝑀₀𝐼_𝑛−1𝑞−𝐼_𝑛−1𝑦(𝑡)+𝐾₁𝐼_𝑛−2𝑥−𝐼_𝑛−2𝑝(𝑡)+𝑀₁𝐼_𝑛−2𝑞−𝐼_𝑛−2𝑦(𝑡)+…+𝐾_𝑛−2𝐼₁𝑥−𝐼₁𝑝(𝑡)+𝑀_𝑛−2𝐼₁𝑞−𝐼₁𝑦(𝑡)+𝐾_𝑛−1𝑥−𝑝(𝑡)+𝑀_𝑛−1𝑞−𝑦(𝑡). (19) Hence, 𝐺(𝑡,𝑠)𝑔𝑡,𝐼_𝑛−1𝑥(𝑡),𝐼_𝑛−1𝑦(𝑡)𝐼,…,₁𝑥(𝑡),𝐼₁𝑦𝑡)(𝑡)(𝑡)()𝑔𝐼(,𝑥,𝑦−𝐺𝑡,𝑠𝑡,_𝑛−1𝑝(𝑡),𝐼_𝑛−1𝑞(𝑡)𝐼,…,₁𝑝(𝑡),𝐼₁𝑞(𝑡)(𝑡)(𝑡)≤|𝐺(𝐾,𝑝,𝑞𝑡,𝑠)|₀𝐼_𝑛−1𝑥(𝑡)−𝐼_𝑛−1𝑝(𝑡)+𝑀₀𝐼_𝑛−1𝑞(𝑡)−𝐼_𝑛−1𝑦(𝑡)+𝐾₁𝐼_𝑛−2𝑥(𝑡)−𝐼_𝑛−2𝑝(𝑡)+𝑀₁𝐼_𝑛−2𝑞(𝑡)−𝐼_𝑛−2𝑦(𝑡)+…+𝐾_𝑛−2𝐼₁𝑥(𝑡)−𝐼₁𝑝(𝑡)+𝑀_𝑛−2𝐼₁𝑞(𝑡)−𝐼₁𝑦(𝑡)+𝐾_𝑛−1𝑥(𝑡)−𝑝(𝑡)+𝑀_𝑛−1𝑞(𝑡)(𝑡≤|𝐺(𝐾−𝑦)𝑡,𝑠)|₀+𝐾₁+…+𝐾_𝑛−1+𝑀𝑥−𝑝₀+𝑀₁+…+𝑀_𝑛−1𝑞−𝑦. (20) Following the former inequality, we can easily have that ¹₀𝐺(𝑡,𝑠)𝑔𝑠,𝐼_𝑛−1𝑥(𝑠),𝐼_𝑛−1𝑦(𝑠)𝐼,…,₁𝑥(𝑠),𝐼₁𝑦(𝑠)(𝑠)(𝑠)𝐼,𝑥,𝑦−𝑔𝑠,_𝑛−1𝑝(𝑠),𝐼_𝑛−1𝑞(𝑠)𝐼,…,₁𝑝(𝑠),𝐼₁𝑞(𝑠),𝑝(𝑠),𝑞(𝑠)𝑑𝑠 (21) converges, thus, ¹₀𝐺(𝑡,𝑠)𝑔𝑠,𝐼_𝑛−1𝑥(𝑠),𝐼_𝑛−1𝑦(𝑠)𝐼,…,₁𝑥(𝑠),𝐼₁𝑦(𝑠)(𝑠)(𝑠)=,𝑥,𝑦𝑑𝑠₁₀𝐺(𝑡,𝑠)𝑔𝑠,𝐼_𝑛−1𝑝(𝑠),𝐼_𝑛−1𝑞(𝑠)𝐼,…,₁𝑝(𝑠),𝐼₁𝑞(𝑠)(𝑠)(𝑠)+,𝑝,𝑞𝑑𝑠₁₀𝐺(𝑡,𝑠)𝑔𝑠,𝐼_𝑛−1𝑥(𝑠),𝐼_𝑛−1𝑦(𝑠)𝐼,…,₁𝑥(𝑠),𝐼₁𝑦(𝑠)(𝑠)(𝑠)𝐼,𝑥,𝑦−𝑔𝑠,_𝑛−1𝑝(𝑠),𝐼_𝑛−1𝑞𝑠)𝐼(,…,₁𝑝(𝑠),𝐼₁𝑞(𝑠),𝑝(𝑠),𝑞(𝑠)𝑑𝑠 (22) is converged.
Similarly, by 𝑥≥𝑢,𝑦≤𝑣, ¹₀𝐺(𝑡,𝑠)𝑔𝑠,𝐼_𝑛−1𝑥(𝑠),𝐼_𝑛−1𝑦(𝑠)𝐼,…,₁𝑥(𝑠),𝐼₁𝑦(𝑠),𝑥(𝑠),𝑦(𝑠)𝑑𝑠 (23) is converged, and we have that ¹₀𝐺(𝑡,𝑠)𝑔𝑠,𝐼_𝑛−1𝑢(𝑠),𝐼_𝑛−1𝑣(𝑠)𝐼,…,₁𝑢(𝑠),𝐼₁𝑣(𝑠),𝑢(𝑠),𝑣(𝑠)𝑑𝑠 (24) converges.
Define the operator 𝐹∶𝐶(𝐼)×𝐶(𝐼)→𝐶(𝐼) by 𝐹(𝑢,𝑣)(𝑡)=¹₀𝐺()𝐼𝑡,𝑠×𝑔𝑠,_𝑛−1𝑢(𝑠),𝐼_𝑛−1𝑣(𝑠)𝐼,…,₁𝑢(𝑠),𝐼₁𝑣(𝑠),𝑢(𝑠),𝑣(𝑠)𝑑𝑠,∀𝑡∈𝐼. (25) Let 𝐴₀𝑢(𝑡)=¹₀|𝐺(𝑡,𝑠)|𝐾₀𝑢(𝑠)𝐵𝑑𝑠,₀𝑣(𝑡)=¹₀|𝐺(𝑡,𝑠)|𝑀₀𝑣(𝑠)𝐴𝑑𝑠,_𝑖𝑢(𝑡)=¹₀|𝐺(𝑡,𝑠)|𝐾_𝑖𝐼_𝑖𝑢𝑠)𝐵(𝑑𝑠,𝑖=1,2,…,𝑛−1,_𝑖𝑣(𝑡)=¹₀|𝐺(𝑡,𝑠)|𝑀_𝑖𝐼_𝑖𝑣(𝑠)(𝑡)=𝐴𝑑𝑠,𝑖=1,2,…,𝑛−1,𝐴𝑢)(₀𝑢+𝐴₁𝑢+…+𝐴_𝑛−1𝑢(𝑡),(𝐵𝑣)(𝑡)=𝐵₀𝑣+𝐵₁𝑣+…+𝐵_𝑛−1𝑣(𝑡). (26) By (14) and (25), for any 𝑢₁,𝑢₂,𝑣₁,𝑣₂∈𝐶(𝐼),𝑢₁≤𝑢₂,𝑣₁≥𝑣₂, we have −𝐴𝑢₂−𝑢₁−𝐵𝑣₁−𝑣₂≤𝐹𝑢₁,𝑣₁−𝐹𝑢₂,𝑣₂≤𝐴𝑢₂−𝑢₁+𝐵𝑣₁−𝑣₂,)𝑢𝑡)=((𝐴+𝐵)(₁₀|𝐺(𝑡,𝑠)|𝐾₀𝑢+𝑀₀𝑢+𝐾₁𝐼₁𝑢+𝑀₁𝐼₁𝑢+…+𝐾_𝑛−1𝐼_𝑛−1𝑢+𝑀_𝑛−1𝐼_𝑛−1𝑢𝑠)≤𝐾(𝑑𝑠₀+𝑀₀()!+𝐾𝑛−2¹+𝑀₁()!𝐾𝑛−3+…+^𝑛−3+𝑀_𝑛−31!+𝐾_𝑛−2+𝑀_𝑛−2+𝐾_𝑛−1+𝑀_𝑛−1⋅‖𝑢‖ℎ₁(𝑡),(𝐴+𝐵)^𝑚𝑢(𝑡)=¹₀|𝐺(𝑡,𝑠)|(𝐴+𝐵)(𝐴+𝐵)^𝑚−1(𝑢𝑠)≤𝐾)(𝑑𝑠₀+𝑀₀()!+𝐾𝑛−2¹+𝑀₁()!𝐾𝑛−3+…+^𝑛−3+𝑀_𝑛−31!+𝐾_𝑛−2+𝑀_𝑛−2+𝐾_𝑛−1+𝑀_𝑛−1^𝑚‖𝑢‖ℎ_𝑚(𝑡))𝑚=2,3,…,(𝐴+𝐵^𝑚≤𝐾₀+𝑀₀()!+𝐾𝑛−2¹+𝑀₁()!+𝐾𝑛−3+…_𝑛−3+𝑀_𝑛−31!+𝐾_𝑛−2+𝑀_𝑛−2+𝐾_𝑛−1+𝑀_𝑛−1^𝑚⋅sup_𝑡∈𝐽𝑒_𝑚(𝑡),𝑟()≤𝐾𝐴+𝐵⁰+𝑀₀()!+𝐾𝑛−2¹+𝑀₁()!+𝐾𝑛−3+…_𝑛−3+𝑀_𝑛−31!+𝐾_𝑛−2+𝑀_𝑛−2+𝐾_𝑛−1+𝑀_𝑛−1×𝜌(𝐺)⁻¹<1. (27)(28) So we can choose 𝛽∈(0,1), which satisfies lim_𝑘→∞‖(𝐴+𝐵)^𝑘‖^1/𝑘=𝑟(𝐴+𝐵)<𝛽<1, and so there exists a positive integer 𝑘₀ such that (𝐴+𝐵)^𝑘<𝛽^𝑘<1,𝑘≥𝑘₀. (29)
Since 𝑃 is a generating cone in 𝐶(𝐼), from Lemma 3, there exists 𝜏>0 such that every element 𝑢∈𝐶(𝐼) can be represented in ‖𝑣‖‖𝑢‖,‖𝑤‖‖𝑢‖;𝑢=𝑣−𝑤,𝑣,𝑤∈𝑃,≤𝜏≤𝜏 (30) this implies −(𝑣+𝑤)≤𝑢≤𝑣+𝑤. (31) Let ‖𝑢‖₀=inf{‖ℎ‖∣ℎ∈𝑃,−ℎ≤𝑢≤ℎ}. (32) By (31), we know that ‖𝑢‖₀ is well defined for any 𝑢∈𝐶(𝐼). It is easy to verify that ‖⋅‖₀ is a norm in 𝐶(𝐼). By (30)–(32), we get ‖𝑢‖₀≤‖𝑣+𝑤‖≤2𝜏‖𝑢‖,∀𝑢∈𝐶(𝐼). (33)
On the other hand, for any ℎ∈𝑃 which satisfies −ℎ≤𝑢≤ℎ, we have 0≤𝑢+ℎ≤2ℎ; thus, ‖𝑢‖≤‖𝑢+ℎ‖+‖−ℎ‖≤(2𝑁+1)‖ℎ‖, where 𝑁 denotes the normal constant of 𝑃. Since ℎ is arbitrary, we have ‖𝑢‖≤(2𝑁+1)‖𝑢‖₀,∀𝑢∈𝐶(𝐼). (34) It follows from (33) and (34) that the norms ‖⋅‖₀ and ‖⋅‖ are equivalent. Now, for any 𝑢,𝑣∈𝐶(𝐼) and ℎ∈𝑃 which satisfies −ℎ≤𝑢−𝑣≤ℎ, let 𝑢₁=12(),𝑢𝑢+𝑣−ℎ₂=12(),𝑢𝑢−𝑣+ℎ₃=12(−𝑢+𝑣+ℎ), (35) then 𝑢≥𝑢₁,𝑣≥𝑢₁,𝑢−𝑢₁=𝑢₂,𝑣−𝑢₁=𝑢₃,𝑢₂+𝑢₃=ℎ.
It follows from (27) that −𝐴𝑢₂≤𝐹(𝑢,𝑢)−𝐹𝑢₁,𝑢≤𝐴𝑢₂,−𝐴𝑢₃−𝐵𝑢₂≤𝐹𝑣,𝑢₁−𝐹𝑢₁,𝑢≤𝐴𝑢₂+𝐵𝑢₃,−𝐵𝑢₃≤𝐹𝑣,𝑢₁−𝐹(𝑣,𝑣)≤𝐹𝑢₃; (36)(37)(38) subtracting (37) from (36) + (38), we obtain −(𝐴+𝐵)ℎ≤𝐹(𝑢,𝑢)−𝐹(𝑣,𝑣)≤(𝐴+𝐵)ℎ. (39) Let 𝐺(𝑢)=𝐹(𝑢,𝑢); then we have −(𝐴+𝐵)ℎ≤𝐺(𝑢)−𝐺(𝑣)≤(𝐴+𝐵)ℎ. (40)
As 𝐴 and 𝐵 are both positive linear bounded operators, so 𝐴+𝐵 is a positive linear bounded operator, and therefore, (𝐴+𝐵)ℎ∈𝑃. Hence, by mathematical induction, it is easy to know that for natural number 𝑘₀ in (29), we have −(𝐴+𝐵)^𝑘0ℎ≤𝐺^𝑘0(𝑢)−𝐺^𝑘0(𝑣)≤(𝐴+𝐵)^𝑘0ℎ,(𝐴+𝐵)^𝑘0ℎ∈𝑃; (41) since (𝐴+𝐵)^𝑘0ℎ∈𝑃, we see that 𝐺^𝑘0(𝑢)−𝐺^𝑘0(𝑣)₀≤(𝐴+𝐵)^𝑘0‖ℎ‖, (42) which implies by virtue of the arbitrariness of ℎ that 𝐺^𝑘0𝑢−𝐺^𝑘0𝑣₀≤(𝐴+𝐵)^𝑘0‖𝑢−𝑣‖₀≤𝛽^𝑘0‖𝑢−𝑣‖₀. (43) By 0<𝛽<1, we have 0<𝛽^𝑘0<1. Thus, the Banach contraction mapping principle implies that 𝐺^𝑘0 has a unique fixed point 𝑢^∗ in 𝐶(𝐼), and so 𝐺 has a unique fixed point 𝑢^∗ in 𝐶(𝐼); by the definition of 𝐺,𝐹 has a unique fixed point 𝑢^∗ in 𝐶(𝐼); then, by Lemma 2, 𝐼_𝑛−1𝑢^∗ is the unique solution of (3). And, for any 𝑢₀∈𝐶(𝐼), let 𝑢_𝑚=𝐹(𝑢_𝑚−1,𝑢_𝑚−1)(𝑚=1,2,…); we have ‖𝑢_𝑚−𝑢^∗‖₀→0(𝑘→∞). By the equivalence of ‖⋅‖₀ and ‖⋅‖ again, we get ‖𝑢_𝑚−𝑢^∗‖→0(𝑚→∞). This completes the proof.

4. Example

In this paper, the results apply to a very wide range of functions, and we are following only one example to illustrate.

Consider the following 𝑛th-order three-point boundary value problem: 𝑢^(𝑛)(𝑡)+𝑆₀𝑢(𝑡)+𝑆₁𝑢^′(𝑡)(𝑡)(|𝑥(𝑡(),𝑢(0)+𝑘ln3+)|),𝑡∈0,1=𝑢^′(0)=…=𝑢^(𝑛−2)(0)𝑢(1)1=0,=2𝑢2, (44) where (𝑆_𝑖𝑢^(𝑖))(𝑡)=∫¹₀ℎ_𝑖(𝑡,𝑠)𝑢^(𝑖)(𝑠)𝑑𝑠,ℎ_𝑖,𝑘∈𝐶(𝐼×𝐼,ℝ), 𝑖=0,1.

Applying Theorem 4, we can find that (44) has a unique solution 𝐼_𝑛𝑥^∗(𝑡)∈𝐶^(𝑛)(𝐼) provided sup_{𝑡,𝑠∈𝐼}|(ℎ₀(𝑡,𝑠)/(𝑛−2)!)+(ℎ₁(𝑡,𝑠)/(𝑛−3)!)+(𝑘(𝑡)/3(𝑛−2)!)|<1, and moreover, for any 𝑢₀∈𝐶(𝐼), the iterative sequence 𝑥_𝑚(𝑡)=¹₀𝐺(𝑡,𝑠)𝑆₀𝐼_𝑛−1𝑥_𝑚−1(𝑠)+𝑆₁𝐼_𝑛−2𝑥_𝑚−1(𝑠)+𝑘(𝑠)ln3+𝑥_𝑚−1(𝑠)𝑑𝑠 (45)(𝑚=1,2,…) converges to 𝑥^∗ uniformly for all 𝑡 in 𝐼(𝑚→∞).

To see that, let 𝐺₁()=2𝑡,𝑠−1+^𝑛−22^𝑛−2)−1(1−𝑠^𝑛−11−22−𝑠^𝑛−1,10≤𝑠≤22,𝑠≤𝑡,_𝑛−22^𝑛−2)−1(1−𝑠^𝑛−11−22−𝑠^𝑛−1,10≤𝑡≤𝑠≤2,2−1+^𝑛−22^𝑛−2()−11−𝑠^𝑛−1,122≤𝑠≤𝑡,_𝑛−22^𝑛−2()−11−𝑠^𝑛−1,12𝑒≤𝑠,𝑡≤𝑠,_∗1(𝑡)=max¹₀𝐺₁(𝑡,𝑠)𝑑𝑠,^𝑡₀¹₀𝐺₁(𝑠,𝑥)𝑑𝑥𝑑𝑠; (46) then 𝐺₁(𝑡,𝑠) is Green’s function of (44). It is easy to verify that |𝐺₁(𝑡,𝑠)|≤1, and so 𝜌(𝐺₁)≥(sup_{𝑡,𝑠∈𝐼}𝑒^∗₁(𝑡))⁻¹≥1.

Let 𝑔𝑡,𝑢(𝑡),𝑣(𝑡),𝑢^′(𝑡),𝑣^′(𝑡),…,𝑢^(𝑛−1)(𝑡),𝑣^(𝑛−1)(𝑡)=𝑆₀𝑢(𝑡)+𝑆₁𝑢^′(𝑡)(𝑡)(|𝑣(𝑡,𝐾+𝑘ln3+)|)_𝑖𝑢(𝑡)=𝐻^∗_𝑖¹₀𝑢(𝑠)𝑀𝑑𝑠,𝑖=0,1,₀𝑣𝑡)=𝐾(^∗3^𝑡₀𝑣(𝑠)𝑀𝑑𝑠,_𝑖𝑢𝑡)𝑢(=0,𝑖=1,…,𝑛−1,₀=𝑣₀=0, (47) where 𝐻^∗_𝑖=sup_{𝑡,𝑠∈𝐼}|ℎ_𝑖(𝑡,𝑠)|(𝑖=0,1),𝐾^∗=sup_𝑡∈𝐼|𝑘(𝑡)|; then it is easy to verify that all conditions in Theorem 4 are satisfied.

Acknowledgments

Peiguo Zhang and Lishan Liu were supported financially by the National Natural Science Foundation of China (11071141, 11371221), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001), the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province, and the Project of Shandong Province Higher Educational Science and Technology Program (J11LA06, J13LI02). Yonghong Wu was supported financially by the Australian Research Council through an ARC Discovery Project grant.