Existence and Monotone Iteration of Positive Pseudosymmetric Solutions for a Third-Order Four-Point BVP with 𝑝-Laplacian

Abstract

We study the existence and monotone iteration of solutions for a third-order four-point boundary value problem with 𝑝-Laplacian. An existence result of positive, concave, and pseudosymmetric solutions and its monotone iterative scheme are established by using the monotone iterative technique. Meanwhile, as an application of our result, an example is given.

1. Introduction

The third-order equations arise in many areas of applied mathematics and physics [1] and thus have been discussed by many authors and many excellent results were obtained; see [1–31] and the references therein. Recently, wide attention has been paid to the third-order boundary value problems with the 𝑝-Laplace operator. In fact, the third-order equations involving the 𝑝-Laplace operator can be seen as a generalized model for various physical, natural or physiological phenomena such as the flow of a thin film of viscous fluid over a solid surface, the solitary wave solution of the Korteweg-de Vries equation or a thyroid-pituitary interaction [17].

In 2005, Cabada et al. [7] studied the one-dimensional nonlinear third-order 𝜙-Laplacian equation −𝜙𝑢^′′(𝑡)^′=𝑓(𝑡,𝑢(𝑡)),𝑡∈[𝑎,𝑏] (1) with the boundary conditions 𝑢(𝑎)=𝐴,𝑢^′′(𝑎)=𝐵,𝑢^′′(𝑏)=𝐶, (2) where 𝜙∶ℝ→ℝ is an increasing homeomorphism with 𝜙(0)=0. By applying the monotone iterative technique based on suitable antimaximum principles, they obtained the existence of extremal solutions for the problem.

In 2006, using the monotone iterative technique, Zhou and Ma [30] obtained the existence of positive solutions and established a corresponding iterative scheme for the following third-order 𝑝-Laplacian problem of the form: 𝜙_𝑝𝑢^′′(𝑡)^′(𝑡)𝑓((𝑡[],𝑢(0)==𝑞𝑡,𝑢)),𝑡∈0,1_𝑚𝑖=1𝛼_𝑖𝑢𝜉_𝑖,𝑢^′𝜂𝑢=0,_′′(1)=^𝑛_𝑖=1𝛽_𝑖𝑢^′′𝜃_𝑖. (3) In 2007, Wang and Ge [26] considered third-order differential equation 𝜙𝑢^′′(𝑡)^′+𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡),𝑢^′′(𝑡)=0,𝑡∈(0,1) (4) subject to the following integral boundary conditions: 𝑢(0)𝑢=0,_′(0)−𝑘₁𝑢^′′(0)=¹₀ℎ₁(𝑢(𝑠))d𝑢𝑠,_′(1)+𝑘₂𝑢^′′(1)=¹₀ℎ₂(𝑢(𝑠))d𝑠. (5) The existence result to the problem is obtained by applying the method of upper and lower solutions and Leray-Schauder degree theory.

In 2009, Sun et al. [24] studied the existence of positive solutions for the following third-order 𝑝-Laplacian problem: 𝜙_𝑝𝑢^′′(𝑡)^′=𝑞(𝑡)𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡),𝑢^′′(𝑡)[],𝑢(0)=,𝑡∈0,1_𝑚𝑖=1𝛼_𝑖𝑢𝜉_𝑖,𝑢^′𝜂=0,𝑢^′′(1)=^𝑛_𝑖=1𝛽_𝑖𝑢^′′𝜃_𝑖. (6) By applying a monotone iterative method, the authors obtained the existence of positive solutions for the problem and established iterative schemes for approximating the solutions.

In 2010, Jin and Lu [17] considered the following third-order 𝑝-Laplacian resonant problem of the form: 𝜙_𝑝𝑥^′′(𝑡)^′=𝑓𝑡,𝑥(𝑡),𝑥^′(𝑥),𝑥^′′(𝑡)(),𝑥(0),𝑡∈0,1=0,𝑥^′(1)=^𝑚−2_𝑖=1𝑎_𝑖𝑥^′𝜉_𝑖,𝑥^′′(0)=0. (7) The authors obtained the existence of solutions for the problem by using Mawhin's continuation theorem.

In 2010, by using the fixed point index method, Yang and Yan [31] established the existence of at least one or at least two positive solutions for the following third-order 𝑝-Laplacian problem: 𝜙_𝑝𝑢^′′(𝑡)^′((𝑡(),(0)+𝑓𝑡,𝑢))=0,𝑡∈0,1𝛼𝑢−𝛽𝑢^′(0)=0,𝛾𝑢(1)+𝛿𝑢^′(1)𝑢=0,_′′(0)=0. (8) Motivated by the above works and [32, 33], in this paper, we consider the existence and monotone iteration of positive, pseudosymmetric solutions of the following third-order four-point 𝑝-Laplacian boundary value problem: 𝜙_𝑝𝑢^′′(𝑡)^′+𝑞(𝑡)𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡)=0,𝑡∈(0,1) (9) subject to boundary conditions 𝑢(0)=0,𝑢(1)=𝑢𝜂,𝑢^′′1+𝜂2=0, (10) where 𝜙_𝑝(𝑠)=|𝑠|^𝑝−2𝑠,𝑝>1, and 𝜂∈(0,1) be constant. Here 𝑢^∗(𝑡)  is said to be a positive solution of BVP (9), (10) if and only if 𝑢^∗(𝑡) is the solution of BVP (9), (10) and satisfies 𝑢^∗(𝑡)>0 for 𝑡∈(0,1]. BVP (9), (10) can model the static deflection of an elastic beam with linear supports at both endpoints.

To the best of our knowledge, the existence results of the pseudosymmetric solutions for the third-order boundary value problem has not been considered.

This work is organized as follows. In Section 2, some notations and preliminaries are introduced. The main results are discussed in Section 3. As applications of our results, an example is given in the last section.

2. Preliminary

In this section, we give some definitions and lemmas which help to simplify the presentation of our main result.

Definition 1 (see [32]). Let 𝑢∈𝐶[0,1], 𝜂∈(0,1). One says that 𝑢 is pseudosymmetric about 𝜂 on [0,1], if 𝑢 is symmetric on [𝜂,1], that is, 𝑢(𝑡)=𝑢1+𝜂−𝑡,∀𝑡∈𝜂,1. (11)

Definition 2. Let 𝑢∈𝐶[0,1], 𝜂∈(0,1). One says that 𝑢 is pseudo-antisymmetric about 𝜂 on [0,1], if 𝑢 is antisymmetric on [𝜂,1], that is, 𝑢(𝑡)+𝑢1+𝜂−𝑡=0,∀𝑡∈𝜂,1. (12) Let the Banach space 𝐸=𝐶¹[0,1] be endowed with the norm ‖𝑢‖=max_0≤𝑡≤1𝑢²(𝑡)+𝑢^′2(𝑡)^1/2, (13) and define the cone 𝑃⊂𝐸 by 𝑃=𝑢∈𝐸∶𝑢isnonnegative,concaveandpseudosymmetricabout𝜂on[0,1], (14) and let 𝑃_𝑎={𝑢∈𝑃∶‖𝑢‖≤𝑎}. (15) For convenience, we consider the following. (H₀)𝑞(𝑡) is a nonnegative continuous function defined on (0,1), 𝑞(𝑡)≢0 on any subinterval of (0,1). In addition, ∫¹₀𝑞(𝑡)d𝑡<+∞ and 𝑞(𝑡) is pseudosymmetric about 𝜂 on [0,1]. (H₁)𝑓(𝑡,𝑢,𝑣)∶[0,1]×[0,+∞)×ℝ→ℝ is continuous and 𝑓(𝑡,𝑢,𝑣)≥0,∀(𝑡,𝑢,𝑣)∈0,1+𝜂2×[0,+∞)×ℝ. (16) (H₂)𝑓(𝑡,𝑢,𝑣)+𝑓(1+𝜂−𝑡,𝑢,−𝑣)=0 for all (𝑡,𝑢,𝑣)∈[𝜂,1]×[0,+∞)×ℝ. (H₃)𝑓(𝑡,0,0)≢0 on [0,1].
Now, we define an operator 𝑇∶𝐶¹[0,1]→𝐶¹[0,1] as follows: for 𝑢∈𝐶¹[0,1], (𝑡)=∫𝑇𝑢)(^𝑡0∫_𝜏^(1+𝜂)/2𝜙_𝑝⁻¹∫_𝑠^(1+𝜂)/2𝑞(𝑟)×𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)d𝑟d𝑠d𝜏,0≤𝑡≤1+𝜂2,∫^𝜂₀∫_𝜏^(1+𝜂)/2𝜙_𝑝⁻¹∫_𝑠^(1+𝜂)/2𝑞(𝑟)×𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)d𝑟d𝑠d𝜏−∫¹_𝑡∫^𝜏_1+𝜂/2𝜙_𝑝⁻¹∫^𝑠_1+𝜂/2𝑞(𝑟)×𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)d𝑟d𝑠d𝜏,1+𝜂2≤𝑡≤1. (17)
Obviously under assumptions (H₀) and (H₁), the operator 𝑇 is well defined and it is easy to verify that BVP (9), (10) has a solution if and only if 𝑇∶𝐶¹[0,1]→𝐶¹[0,1] has a fixed point.
The next lemmas are some properties of the operator 𝑇.

Lemma 3. Assume that (H₀), (H₁), and (H₂) hold. Then 𝑇𝑃⊂𝑃.

Proof. From the definition of 𝑇, it is easy to check that 𝑇𝑢 is nonnegative on [0,1] and satisfies (10) for all 𝑢∈𝑃. Furthermore, since (𝑇𝑢)^′′(𝑡)=−𝜙_𝑝⁻¹∫_𝑡^(1+𝜂)/2𝑞(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)d𝑟≤0,0≤𝑡≤1+𝜂2,𝜙_𝑝⁻¹∫^𝑡_1+𝜂/2𝑞(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)d𝑟≤0,1+𝜂2≤𝑡≤1, (18) it follows that 𝑇𝑢 is concave on [0,1].
Next we prove that 𝑇𝑢 is pseudosymmetric about 𝜂 on [0,1]. In fact, if 𝑡∈[𝜂,(1+𝜂)/2], then 1+𝜂−𝑡∈[(1+𝜂)/2,1], and it follows that (𝑇𝑢)1+𝜂−𝑡=^𝜂_0𝜏^(1+𝜂)/2𝜙_𝑝⁻¹×_𝑠^(1+𝜂)/2𝑞(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)d𝑟d𝑠d𝜏−¹_{1+𝜂−𝑡}^𝜏_1+𝜂/2𝜙_𝑝⁻¹×^𝑠_1+𝜂/2𝑞(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)d𝑟d𝑠d𝜏=^𝜂_0𝜏^(1+𝜂)/2𝜙_𝑝⁻¹×_𝑠^(1+𝜂)/2𝑞(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)d𝑟d𝑠d𝜏+^𝜂₁^𝜏_1+𝜂/2𝜙_𝑝⁻¹×^𝑠_1+𝜂/2𝑞(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)d𝑟d𝑠d𝜏+^𝑡_𝜂^𝜏_1+𝜂/2𝜙_𝑝⁻¹×^𝑠_1+𝜂/2𝑞(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)d𝑟d𝑠d𝜏+_𝑡^{1+𝜂−𝑡𝜏}_1+𝜂/2𝜙_𝑝⁻¹×^𝑠_1+𝜂/2𝑞(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)d𝑟d𝑠d𝜏. (19) Also since 𝑢 is pseudosymmetric about 𝜂 on [0,1], that is, 𝑢(𝑡)=𝑢(1+𝜂−𝑡) for 𝑡∈[𝜂,1], then 𝑢^′(𝑡)=−𝑢^′1+𝜂−𝑡,𝑡∈𝜂,1. (20) Thus, for all 𝑡∈[𝜂,1], from (H₂), we have 𝑞(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)=−𝑞1+𝜂−𝑟𝑓1+𝜂−𝑟,𝑢(𝑟),−𝑢^′(𝑟)=−𝑞1+𝜂−𝑟×𝑓1+𝜂−𝑟,𝑢1+𝜂−𝑟,𝑢^′1+𝜂−𝑟. (21) Hence 𝑞(𝑟)𝑓(𝑟,𝑢(𝑟),𝑢^′(𝑟)) is pseudo-antisymmetric about 𝜂 on [0,1], and thus ∫^𝑠_(1+𝜂)/2𝑞(𝑟)𝑓(𝑟,𝑢(𝑟),𝑢^′(𝑟))d𝑟 is pseudosymmetric about 𝜂 on [0,1]. Furthermore 𝜙_𝑝⁻¹(∫^𝑠_(1+𝜂)/2𝑞(𝑟)𝑓(𝑟,𝑢(𝑟),𝑢^′(𝑟))d𝑟) is pseudosymmetric about 𝜂 on [0,1]. Thus the function ∫^𝜏_(1+𝜂)/2𝜙_𝑝⁻¹(∫^𝑠_(1+𝜂)/2𝑞(𝑟)𝑓(𝑟,𝑢(𝑟),𝑢^′(𝑟))d𝑟)d𝑠 is pseudo-antisymmetric about 𝜂 on [0,1], and hence ^𝜂₁^𝜏_1+𝜂/2𝜙_𝑝^−1𝑠_1+𝜂/2𝑞(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)d𝑟d𝑠d𝜏=0. (22) Using the similar technique, we can get _𝑡^{1+𝜂−𝑡𝜏}_1+𝜂/2𝜙_𝑝^−1𝑠_1+𝜂/2𝑞(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)d𝑟d𝑠d𝜏=0. (23) From (19), (22), and (23), it follows that (𝑇𝑢)1+𝜂−𝑡=^𝜂_0𝜏^(1+𝜂)/2𝜙_𝑝⁻¹×_𝑠^(1+𝜂)/2𝑞(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)d𝑟d𝑠d𝜏+^𝑡_𝜂^𝜏_1+𝜂/2𝜙_𝑝⁻¹×^𝑠_1+𝜂/2𝑞(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)d𝑟d𝑠d𝜏=^𝑡₀^𝜏_1+𝜂/2𝜙_𝑝⁻¹×^𝑠_1+𝜂/2𝑞(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)d𝑟d𝑠d𝜏=(𝑇𝑢)(𝑡),𝑡∈𝜂,1+𝜂2. (24) If 𝑡∈[(1+𝜂)/2,1], then 1+𝜂−𝑡∈[𝜂,(1+𝜂)/2]. From (24), it follows that ()=()=(𝑡)𝑇𝑢1+𝜂−𝑡𝑇𝑢1+𝜂−1+𝜂−𝑡𝑇𝑢)(,𝑡∈1+𝜂2,1. (25) This together with (24) implies that (𝑇𝑢)(𝑡)=(𝑇𝑢)1+𝜂−𝑡,𝑡∈𝜂,1. (26) In summary, 𝑇𝑢∈𝑃, and then 𝑇𝑃⊂𝑃.

The following lemma can be easily verified by a standard argument.

Lemma 4. Assume that (H₀), (H₁), and (H₂) hold. Then 𝑇∶𝑃→𝑃 is completely continuous.

Lemma 5. Assume that (H₀), (H₁), and (H₂) hold. Suppose also that there exists 𝑎>0 such that for 0≤𝑡≤(1+𝜂)/2, 0≤𝑢₁≤𝑢₂≤𝑎, 0≤|𝑣₁|≤|𝑣₂|≤𝑎, 𝑓𝑡,𝑢₁,𝑣₁≤𝑓𝑡,𝑢₂,𝑣₂. (27) Then for 𝑢₁, 𝑢₂∈𝑃_𝑎 with 𝑢₁(𝑡)≤𝑢₂(𝑡),𝑢^′₁(𝑡)≤𝑢^′₂(𝑡),𝑡∈[0,1], (28) we have 𝑇𝑢₁(𝑡)≤𝑇𝑢₂(𝑡),𝑇𝑢₁^′(𝑡)≤𝑇𝑢₂^′(𝑡,[].)𝑡∈0,1 (29)

Proof. First we prove that, for all 𝑡∈[0,(1+𝜂)/2], 𝑇𝑢₁(𝑡)≤𝑇𝑢₂(𝑡),𝑇𝑢₁^′(𝑡)≤𝑇𝑢₂^′(𝑡). (30) From assumptions, we have 𝑓𝑟,𝑢₁(𝑟),𝑢^′₁(𝑟)≤𝑓𝑟,𝑢₂(𝑟),𝑢^′₂(𝑟),𝑟∈0,1+𝜂2, (31) and hence _𝑠^(1+𝜂)/2𝑓𝑟,𝑢₁(𝑟),𝑢^′₁(𝑟)d𝑟≤_𝑠^(1+𝜂)/2𝑓𝑟,𝑢₂(𝑟),𝑢^′₂(𝑟)d𝑟,𝑠∈0,1+𝜂2. (32) Since 𝜙_𝑝⁻¹ is strictly increasing on ℝ, then for all 𝑠∈[0,(1+𝜂)/2], we have 𝜙_𝑝⁻¹_𝑠^(1+𝜂)/2𝑓𝑟,𝑢₁(𝑟),𝑢^′₁(𝑟)d𝑟≤𝜙_𝑝⁻¹_𝑠^(1+𝜂)/2𝑓𝑟,𝑢₂(𝑟),𝑢^′₂(𝑟)d𝑟. (33) Thus for 𝑡∈[0,(1+𝜂)/2], 𝑇𝑢₁(𝑡)−𝑇𝑢₂(𝑡)=^𝑡_0𝜏^(1+𝜂)/2𝜙_𝑝⁻¹_𝑠^(1+𝜂)/2𝑞(𝑟)𝑓𝑟,𝑢₁(𝑟),𝑢^′₁(𝑟)d𝑟−𝜙_𝑝⁻¹_𝑠^(1+𝜂)/2𝑞(𝑟)𝑓𝑟,𝑢₂(𝑟),𝑢^′₂(𝑟)d𝑟d𝑠d𝜏≤0,𝑇𝑢₁^′(𝑡)−𝑇𝑢₂^′(𝑡=_{𝑡(1+𝜂)/2})𝜙_𝑝⁻¹_𝑠^(1+𝜂)/2𝑞(𝑟)𝑓𝑟,𝑢₁(𝑟),𝑢^′₁(𝑟)d𝑟−𝜙_𝑝⁻¹_𝑠^(1+𝜂)/2𝑞(𝑟)𝑓𝑟,𝑢₂(𝑟),𝑢^′₂(𝑟)d𝑟d𝑠≤0. (34) Therefore, (30) holds for 𝑡∈[0,(1+𝜂)/2].
Next we prove that (30) holds for 𝑡∈[(1+𝜂)/2,1]. In fact, if 𝑡∈[(1+𝜂)/2,1], then 1+𝜂−𝑡∈[0,(1+𝜂)/2], and hence from the fact that 𝑇𝑢₁ and 𝑇𝑢₂ are pseudosymmetric about 𝜂 on [0,1], it follows that, for 𝑡∈[(1+𝜂)/2,1], 𝑇𝑢₁(𝑡)−𝑇𝑢₂(𝑡)=𝑇𝑢₁−1+𝜂−𝑡𝑇𝑢₂1+𝜂−𝑡≤0,𝑇𝑢₁^′(𝑡)−𝑇𝑢₂^′(𝑡)=𝑇𝑢₁^′−1+𝜂−𝑡𝑇𝑢₂^′1+𝜂−𝑡≤0. (35) In summary, 𝑇𝑢₁(𝑡)≤𝑇𝑢₂(𝑡),𝑇𝑢₁^′(𝑡)≤𝑇𝑢₂^′(𝑡,[].)𝑡∈0,1 (36)

Now, we introduce some notations as follows: 𝐴₁=₀^(1+𝜂)/2𝜙_𝑝⁻¹_𝑠^(1+𝜂)/2𝑞(𝑟)d𝑟d𝐴𝑠,₂=𝜙_𝑝⁻¹₀^(1+𝜂)/2𝑞(𝑟)d𝑟,𝐴=max√2𝐴₁,√2𝐴₂=√2𝐴₂. (37)

Lemma 6. Assume that (H₀), (H₁), and (H₂) hold. Suppose also that there exists 𝑎>0 such that for 0≤𝑡≤(1+𝜂)/2, 0≤𝑢₁≤𝑢₂≤𝑎, 0≤|𝑣₁|≤|𝑣₂|≤𝑎, 𝑓𝑡,𝑢₁,𝑣₁≤𝑓𝑡,𝑢₂,𝑣₂,max_{0≤𝑡≤(1+𝜂)/2}𝑓(𝑡,𝑎,𝑎)≤𝜙_𝑝𝑎𝐴. (38) Then 𝑇∶𝑃_𝑎→𝑃_𝑎.

Proof. Define two functionals on 𝐸 as follows: 𝛼(𝑢)∶=max_0≤𝑡≤1|𝑢(𝑡)|,𝛽(𝑢)∶=max_0≤𝑡≤1𝑢^′(𝑡). (39) Then ‖𝑢‖≤√2max𝛼(𝑢),𝛽(𝑢). (40) If 𝑢∈𝑃_𝑎, then 0≤𝑢(𝑡)≤max_0≤𝑡≤1|𝑢(𝑡≤‖𝑢‖[],𝑢)|≤𝑎,𝑡∈0,10≤^′(𝑡)≤max_0≤𝑡≤1𝑢^′(𝑡)≤‖𝑢‖≤𝑎,𝑡∈[0,1]. (41) From the assumptions, for all 𝑡∈[0,(1+𝜂)/2], 0≤𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡)≤𝑓(𝑡,𝑎,𝑎)≤max_{0≤𝑡≤1+𝜂/2}𝑓(𝑡,𝑎,𝑎)≤𝜙_𝑝𝑎𝐴. (42) Then, 𝛼(𝑇𝑢)=max_0≤𝑡≤1|(𝑇𝑢)(𝑡)|=(𝑇𝑢)1+𝜂2=₀^(1+𝜂)/2_𝜏^(1+𝜂)/2𝜙_𝑝⁻¹×_𝑠^(1+𝜂)/2𝑞(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)d𝑟d𝑠d𝜏≤₀^(1+𝜂)/2_𝜏^(1+𝜂)/2𝜙_𝑝⁻¹×_𝑠^(1+𝜂)/2𝑞(𝑟)𝜙_𝑝𝑎𝐴d𝑟d𝑠d𝜏≤𝐴₁⋅𝑎𝐴≤√22𝛽()𝑎,𝑇𝑢=max_0≤𝑡≤1(𝑇𝑢)^′(𝑡)=(𝑇𝑢)^′(0)=₀^(1+𝜂)/2𝜙_𝑝⁻¹_𝑠^(1+𝜂)/2𝑞(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)d𝑟d𝑠≤₀^(1+𝜂)/2𝜙_𝑝⁻¹_𝑠^(1+𝜂)/2𝑞(𝑟)𝜙_𝑝𝑎𝐴d𝑟d𝑠≤𝐴₂⋅𝑎𝐴≤√22𝑎. (43) So we have ‖𝑇𝑢‖≤√2max𝛼(𝑇𝑢),𝛽(𝑇𝑢)≤√√2⋅22𝑎=𝑎. (44) Thus 𝑇∶𝑃_𝑎→𝑃_𝑎.

3. Main Result

Now we establish existence result of positive, concave, and pseudosymmetric solutions and its monotone iterative scheme for BVP (9), (10).

Theorem 7. Assume that (H₀), (H₁), (H₂), and (H₃) hold. Suppose also that there exists 𝑎>0 such that for 0≤𝑡≤(1+𝜂)/2, 0≤𝑢₁≤𝑢₂≤𝑎, 0≤|𝑣₁|≤|𝑣₂|≤𝑎, 𝑓𝑡,𝑢₁,𝑣₁≤𝑓𝑡,𝑢₂,𝑣₂,max_{0≤𝑡≤1+𝜂/2}𝑓(𝑡,𝑎,𝑎)≤𝜙_𝑝𝑎𝐴. (45) Then BVP (9), (10) has two positive, concave, and pseudosymmetric solutions 𝑤^∗ and 𝑣^∗ with 0<𝑤^∗(𝑡)≤√22𝑎,𝑡∈(0,1],lim_𝑛→∞𝑤_𝑛=lim_𝑛→∞𝑇^𝑛𝑤₀=𝑤^∗𝑖𝑛𝐶¹norm,0<𝑣^∗(𝑡)≤𝑎,𝑡∈(0,1],lim_𝑛→∞𝑣_𝑛=lim_𝑛→∞𝑇^𝑛𝑣₀=𝑣^∗𝑖𝑛𝐶¹norm, (46) where 𝑤₀(𝑡)=√22𝑎𝑡,0≤𝑡≤1+𝜂2,√22𝑎1+𝜂−𝑡,1+𝜂2𝑣≤𝑡≤1;₀(𝑡)(𝑡)=∫≡0,0≤𝑡≤1,𝑇𝑢)(^𝑡0∫_𝜏^(1+𝜂)/2𝜙_𝑝⁻¹∫_𝑠^(1+𝜂)/2𝑞(𝑟)×𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)d𝑟d𝑠d𝜏,0≤𝑡≤1+𝜂2,∫^𝜂₀∫_𝜏^(1+𝜂)/2𝜙_𝑝⁻¹∫_𝑠^(1+𝜂)/2𝑞(𝑟)×𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)d𝑟d𝑠d𝜏−∫¹_𝑡∫^𝜏_1+𝜂/2𝜙_𝑝⁻¹∫^𝑠_1+𝜂/2𝑞(𝑟)×𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)d𝑟d𝑠d𝜏,1+𝜂2≤𝑡≤1. (47)

Proof. Let 𝑤₀(𝑡)=√22𝑎𝑡,0≤𝑡≤1+𝜂2,√22𝑎1+𝜂−𝑡,1+𝜂2𝑤≤𝑡≤1,₁(𝑡)=∫^𝑡₀∫_𝜏^(1+𝜂)/2𝜙_𝑝⁻¹∫_𝑠^(1+𝜂)/2𝑞(𝑟)×𝑓𝑟,𝑤₀(𝑟),𝑤^′₀(𝑟)d𝑟d𝑠d𝜏,0≤𝑡≤1+𝜂2,∫^𝜂₀∫_𝜏^(1+𝜂)/2𝜙_𝑝⁻¹∫_𝑠^(1+𝜂)/2𝑞(𝑟)×𝑓𝑟,𝑤₀(𝑟),𝑤^′₀(𝑟)d𝑟d𝑠d𝜏,−∫¹_𝑡∫^𝜏_1+𝜂/2𝜙_𝑝⁻¹∫^𝑠_1+𝜂/2𝑞(𝑟)×𝑓𝑟,𝑤₀(𝑟),𝑤^′₀(𝑟)d𝑟d𝑠d𝜏,1+𝜂2≤𝑡≤1. (48) Then 𝑤₁(𝑡)∈𝐶¹[0,(1+𝜂)/2]∩𝐶¹[(1+𝜂)/2,1].
Next we prove that lim_{𝑡→1+𝜂/2−}𝑤₁(𝑡)=lim_{𝑡→1+𝜂/2+}𝑤₁(𝑡),lim_{𝑡→1+𝜂/2−}𝑤^′₁(𝑡)=lim_{𝑡→1+𝜂/2+}𝑤^′₁(𝑡). (49)(50) In fact, from (H₂) it follows that lim_{𝑡→(1+𝜂/2)+}𝑤₁(𝑡)=^𝜂_0𝜏^(1+𝜂)/2𝜙_𝑝⁻¹×_𝑠^(1+𝜂)/2𝑞(𝑟)𝑓√𝑟,22√𝑎𝑟,22𝑎d𝑟d𝑠d𝜏−¹_1+𝜂/2^𝜏_1+𝜂/2𝜙_𝑝⁻¹×^𝑠_1+𝜂/2𝑞(𝑟)𝑓√𝑟,22𝑎√1+𝜂−𝑟,−22𝑎d𝑟d𝑠d𝜏=₀^(1+𝜂)/2+^𝜂_1+𝜂/2×_𝜏^(1+𝜂)/2𝜙_𝑝⁻¹_𝑠^(1+𝜂)/2𝑞(𝑟)𝑓√𝑟,22√𝑎𝑟,22𝑎d𝑟d𝑠d𝜏−¹_1+𝜂/2^𝜏_1+𝜂/2𝜙_𝑝⁻¹×^𝑠_1+𝜂/2𝑞(𝑟)√×𝑓𝑟,22𝑎√1+𝜂−𝑟,−22𝑎d𝑟d𝑠d𝜏=₀^(1+𝜂)/2_𝜏^(1+𝜂)/2𝜙_𝑝⁻¹×_𝑠^(1+𝜂)/2𝑞(𝑟)𝑓√𝑟,22√𝑎𝑟,22𝑎d𝑟d𝑠d𝜏+^𝜂_1+𝜂/2𝜏^(1+𝜂)/2𝜙_𝑝⁻¹×_𝑠^(1+𝜂)/2𝑞(𝑟)𝑓√𝑟,22√𝑎𝑟,22𝑎d𝑟d𝑠d𝜏−¹_1+𝜂/2^𝜏_1+𝜂/2𝜙_𝑝⁻¹×^𝑠_1+𝜂/2𝑞(𝑟)√×𝑓𝑟,22𝑎√1+𝜂−𝑟,−22𝑎d𝑟d𝑠d𝜏=₀^(1+𝜂)/2_𝜏^(1+𝜂)/2𝜙_𝑝⁻¹×_𝑠^(1+𝜂)/2𝑞(𝑟)𝑓√𝑟,22√𝑎𝑟,22𝑎d𝑟d𝑠d𝜏=lim_{𝑡→1+𝜂/2−}𝑤₁(𝑡). (51) Then (49) holds. Equation (50) can be obtained in a similar way. Thus from (49) and (50), it follows that 𝑤₁(𝑡)∈𝐶¹[0,1]. (52) We note that for 𝑡∈[0,(1+𝜂)/2], 0≤𝑤₁(𝑡)=^𝑡_0𝜏^(1+𝜂)/2𝜙_𝑝⁻¹×_𝑠^(1+𝜂)/2𝑞(𝑟)𝑓√𝑟,22√𝑎𝑟,22𝑎d𝑟d𝑠d𝜏≤^𝑡_0𝜏^(1+𝜂)/2𝜙_𝑝⁻¹_𝑠^(1+𝜂)/2𝑞(𝑟)𝜙_𝑝𝑎𝐴d𝑟d𝑠d𝜏≤𝑎𝐴^𝑡_0𝜏^(1+𝜂)/2𝜙_𝑝⁻¹₀^(1+𝜂)/2𝑞(𝑟)d𝑟d𝑠d𝜏≤𝑎𝐴𝐴₂𝑡=𝑤₀(𝑡), (53) and for 𝑡∈[(1+𝜂)/2,1], 𝑤₁(𝑡)=^𝜂_0𝜏^(1+𝜂)/2𝜙_𝑝⁻¹×_𝑠^(1+𝜂)/2𝑞(𝑟)𝑓√𝑟,22√𝑎𝑟,22𝑎d𝑟d𝑠d𝜏−¹_𝑡^𝜏_1+𝜂/2𝜙_𝑝⁻¹×^𝑠_1+𝜂/2𝑞(𝑟)√×𝑓𝑟,22𝑎√1+𝜂−𝑟,−22𝑎d𝑟d𝑠d𝜏≤^𝜂_0𝜏^(1+𝜂)/2𝜙_𝑝⁻¹_𝑠^(1+𝜂)/2𝑞(𝑟)𝜙_𝑝𝑎𝐴d𝑟d𝑠d𝜏+¹_𝑡^𝜏_1+𝜂/2𝜙_𝑝^−1𝑠_1+𝜂/2𝑞(𝑟)𝜙_𝑝𝑎𝐴d𝑟d𝑠d𝜏≤𝑎𝐴^𝜂_0𝜏^(1+𝜂)/2𝜙_𝑝⁻¹₀^(1+𝜂)/2𝑞(𝑟)d𝑟d𝑠d𝜏+𝑎𝐴¹_𝑡^𝜏_1+𝜂/2𝜙_𝑝⁻¹¹_1+𝜂/2𝑞(𝑟)d𝑟d𝑠d𝜏≤𝑎𝐴^𝜂_0𝜏^(1+𝜂)/2𝐴₂d𝑠d𝜏+𝑎𝐴¹_𝑡^𝜏_1+𝜂/2𝜙_𝑝⁻¹₀^(1+𝜂)/2𝑞(𝑟)d𝑟d𝑠d𝜏≤𝑎𝐴^𝜂₀𝐴₂d𝑎𝜏+𝐴¹_𝑡^𝜏_1+𝜂/2𝐴₂d𝑠d𝜏≤𝑎𝐴𝐴₂𝑎𝜂+𝐴𝐴₂(1−𝑡)=𝑤₀(𝑡). (54) So, 𝑤₁(𝑡)≤𝑤₀(𝑡)≤√22𝑎,𝑡∈[0,1]. (55) Thus, 𝛼𝑤₁∶=max_0≤𝑡≤1𝑤₁(𝑡≤√)22𝑎. (56) From assumptions, for 𝑡∈[0,(1+𝜂)/2], 𝑤^′₁(𝑡)=_𝑡^(1+𝜂)/2𝜙_𝑝⁻¹×_𝑠^(1+𝜂)/2𝑞(𝑟)𝑓√𝑟,22√𝑎𝑟,22𝑎d𝑟d𝑠≤_𝑡^(1+𝜂)/2𝜙_𝑝⁻¹_𝑠^(1+𝜂)/2𝑞(𝑟)𝜙_𝑝𝑎𝐴d𝑟d𝑠≤𝑎𝐴_𝑡^(1+𝜂)/2𝜙_𝑝⁻¹₀^(1+𝜂)/2𝑞(𝑟)d𝑟d𝑠≤𝑎𝐴𝐴₂≤√22𝑎=𝑤^′₀(𝑡), (57) and for 𝑡∈[(1+𝜂)/2,1], 𝑤^′₁(𝑡=_𝑡1)_+𝜂/2𝜙_𝑝^−1𝑠_1+𝜂/2𝑞(𝑟)√×𝑓𝑟,22𝑎√1+𝜂−𝑟,−22𝑎d𝑟d𝑠≤^𝑡_1+𝜂/2𝜙_𝑝⁻¹¹_1+𝜂/2𝑞(𝑟)𝜙_𝑝𝑎𝐴d𝑟d𝑠≤𝑎𝐴^𝑡_1+𝜂/2𝜙_𝑝⁻¹₀^(1+𝜂)/2𝑞(𝑟)d𝑟d𝑠≤𝑎𝐴𝐴₂≤√22𝑎=𝑤^′₀(𝑡). (58) Hence from (57) and (58), we have 𝛽𝑤₁∶=max_0≤𝑡≤1𝑤^′₁(𝑡≤√)22𝑎. (59) Consequently, from (56) and (59), it follows that 𝑤₁≤√2max𝛼𝑤₁,𝛽𝑤₁≤𝑎. (60) From the proof of Lemma 3, we see that 𝑤₁ is nonnegative, concave, and pseudosymmetric about 𝜂 on [0,1], and hence 𝑤₁∈𝑃_𝑎. (61) Define {𝑤_𝑛} as follows: 𝑤_𝑛+1=𝑇𝑤_𝑛=𝑇^𝑛𝑤₁=𝑇^𝑛+1𝑤₀,𝑛=0,1,…. (62) Then {𝑤_𝑛} is well defined and for 𝑛=1, 2, …, 𝑤_𝑛+1(𝑡)≤𝑤_𝑛(𝑡),𝑤^′_𝑛+1(𝑡)≤𝑤^′_𝑛(𝑡),𝑡∈[0,1]. (63) In fact, for 𝑡∈[0,(1+𝜂)/2], 𝑤₂(𝑡)=𝑇𝑤₁(𝑡)=^𝑡_0𝜏^(1+𝜂)/2𝜙_𝑝⁻¹×_𝑠^(1+𝜂)/2𝑞(𝑟)×𝑓𝑟,𝑤₁(𝑟),𝑤^′₁(𝑟)d𝑟d𝑠d𝜏≤^𝑡_0𝜏^(1+𝜂)/2𝜙_𝑝⁻¹×_𝑠^(1+𝜂)/2𝑞(𝑟)√×𝑓𝑟,22√𝑎𝑟,22𝑎d𝑟d𝑠d𝜏=𝑤₁(𝑡),𝑤^′₂(𝑡)=𝑇𝑤^′₁(𝑡=_{𝑡(1+𝜂)/2})𝜙_𝑝⁻¹×_𝑠^(1+𝜂)/2𝑞(𝑟)×𝑓𝑟,𝑤₁(𝑟),𝑤^′₁(𝑟)d𝑟d𝑠≤_𝑡^(1+𝜂)/2𝜙_𝑝⁻¹×_𝑠^(1+𝜂)/2𝑞(𝑟)√×𝑓𝑟,22√𝑎𝑟,22𝑎d𝑟d𝑠=𝑤^′₁(𝑡). (64)(65) For 𝑡∈[(1+𝜂)/2,1], since 𝑤₁, 𝑤₂∈𝑃_𝑎, it follows from (64) and (65) that 𝑤₂(𝑡)=𝑤₂1+𝜂−𝑡≤𝑤₁1+𝜂−𝑡=𝑤₁(𝑡),𝑤^′₂(𝑡)=𝑤^′₂1+𝜂−𝑡≤𝑤^′₁1+𝜂−𝑡=𝑤^′₁(𝑡). (66)(67) So from (64)–(67), we have 𝑤₂(𝑡)≤𝑤₁(𝑡),𝑤^′₂(𝑡)≤𝑤^′₁(𝑡),𝑡∈[0,1], (68) that is, (63) holds when 𝑛=1. Assume that (63) holds when 𝑛=𝑘, that is, 𝑤_𝑘+1(𝑡)≤𝑤_𝑘(𝑡),𝑤^′_𝑘+1(𝑡)≤𝑤^′_𝑘(𝑡),𝑡∈[0,1]. (69) Then from Lemma 5, we obtain 𝑤_𝑘+2(𝑡)=𝑇𝑤_𝑘+1(𝑡)≤𝑇𝑤_𝑘(𝑡)=𝑤_𝑘+1(𝑡)[],𝑤,𝑡∈0,1_′𝑘+2(𝑡)=𝑇𝑤_𝑘+1^′(𝑡)≤𝑇𝑤_𝑘^′(𝑡=𝑤_′𝑘+1)(𝑡),𝑡∈[0,1]. (70) So by induction (63) holds.
Since 𝑇∶𝑃_𝑎→𝑃_𝑎 is completely continuous, it follows that {𝑤_𝑛}^∞_𝑛=1 is relative compact. Then {𝑤_𝑛} has a convergent subsequence {𝑤_𝑛𝑘} and 𝑤^∗∈𝑃_𝑎 such that 𝑤_𝑛𝑘⟶𝑤^∗(𝑘⟶∞), (71) that is, 𝑤_𝑛𝑘(𝑡)⇉𝑤^∗(𝑡),𝑤)(𝑘⟶∞_′𝑛𝑘(𝑡)⇉𝑤^∗′(𝑡)(𝑘⟶∞)on[0,1]. (72) While from (63) and the fact for each 𝑛=1,2,…, 𝑤^′_𝑛((1+𝜂)/2)=0 and 𝑤_𝑛^′′(𝑡)≤0 on [0,1], it follows that 𝑤₁(𝑡)≥𝑤₂(𝑡)≥⋯≥𝑤_𝑛(𝑡)≥𝑤_𝑛+1(𝑡)[],𝑤≥⋯,𝑛=1,2,…,on0,1_′1(𝑡)≥𝑤^′₂(𝑡)≥⋯≥𝑤^′_𝑛(𝑡)≥𝑤^′_𝑛+1(𝑡)≥⋯,𝑛=1,2,…,on0,1+𝜂2,𝑤^′₁(𝑡)≤𝑤^′₂(𝑡)≤⋯≤𝑤^′_𝑛(𝑡)≤𝑤^′_𝑛+1(𝑡)≤⋯,𝑛=1,2,…,on1+𝜂2,1. (73) Hence, 𝑤_𝑛(𝑡)⇉𝑤^∗(𝑡),𝑤)(𝑛⟶∞_′𝑛(𝑡)⇉𝑤^∗′(𝑡)(𝑛⟶∞)on[0,1], (74) that is, 𝑤_𝑛⟶𝑤^∗(𝑛⟶∞). (75) This together with the continuity of 𝑇 and 𝑤_𝑛+1=𝑇𝑤_𝑛, implies that 𝑇𝑤^∗=𝑤^∗. (76) Also, since 0≤𝑤_𝑛(𝑡)≤𝑤₀(𝑡)=√22𝑎𝑡,0≤𝑡≤1+𝜂2,√22𝑎1+𝜂−𝑡,1+𝜂2𝑤≤𝑡≤1,0≤^′_𝑛(𝑡)≤𝑤^′₁(𝑡≤√)22𝑎,𝑡∈[0,1], (77) we have 0≤𝑤^∗(𝑡)≤√22𝑎,0≤𝑤^∗′(𝑡≤√)22𝑎,𝑡∈[0,1]. (78) Furthermore, we have 𝑤^∗(𝑡)>0,𝑡∈(0,1]. (79) In fact, from (H₃) and 𝑤^∗(𝑡)≢0 on [0,1], we have 𝑤^∗((1+𝜂)/2)>0. Since 𝑤^∗(𝑡) is concave on [0,1], then 𝑤^∗(𝑡)≥𝑤^∗1+𝜂/2−0𝑡=21+𝜂/2−0𝑤1+𝜂^∗1+𝜂2𝑡>0,𝑡∈0,1+𝜂2. (80) Consequently from the fact 𝑤^∗ is pseudosymmetric on [0,1], we have 𝑤^∗(𝑡)>0,𝑡∈(0,1]. (81) Let 𝑣₀(𝑡)≡0 on [0,1], then 𝑣₀∈𝑃_𝑎. Set 𝑣_𝑛+1=𝑇𝑣_𝑛, 𝑛=0, 1, 2, …. Then from Lemma 6, it follows that 𝑣_𝑛∈𝑃_𝑎,𝑛=1,2,…. (82) From Lemma 4, we see that {𝑣_𝑛}^∞_𝑛=1 is relative compact, and hence there exists a convergent subsequence {𝑣_𝑛𝑘}⊂{𝑣_𝑛} and 𝑣^∗∈𝑃_𝑎 such that 𝑣_𝑛𝑘⟶𝑣^∗(𝑘⟶∞), (83) that is, 𝑣_𝑛𝑘(𝑡)⇉𝑣^∗(𝑡)(𝑘⟶∞)on[0,1],𝑣^′_𝑛𝑘(𝑡)⇉𝑣^∗′(𝑡)(𝑘⟶∞)on[0,1]. (84)(85) Since 𝑣₁=𝑇𝑣₀=𝑇0∈𝑃_𝑎, then 𝑣₁(𝑡)=𝑇𝑣₀(𝑡)=(𝑡)[],𝑣𝑇0)(≥0,𝑡∈0,1_′1(𝑡)=𝑇𝑣₀^′(𝑡)=(𝑇0)^′(𝑡)≥0,𝑡∈[0,1]. (86) Thus from Lemma 5, 𝑣₂(𝑡)=𝑇𝑣₁(𝑡)≥𝑇𝑣₀(𝑡)=𝑣₁(𝑡)[],𝑣,𝑡∈0,1_′2(𝑡)=𝑇𝑣₁^′(𝑡)≥𝑇𝑣₀^′(𝑡)=𝑣^′₁(𝑡),𝑡∈[0,1]. (87) By induction, it is easy to see that for 𝑛=1, 2, …, 𝑣_𝑛+1(𝑡)≥𝑣_𝑛(𝑡),𝑡∈[0,1],𝑣^′_𝑛+1(𝑡)≥𝑣^′_𝑛(𝑡),𝑡∈[0,1]. (88)(89) From (84)–(89), we see that 𝑣_𝑛(𝑡)⇉𝑣^∗(𝑡),𝑣)(𝑛⟶∞_′𝑛(𝑡)⇉𝑣^∗′(𝑡)(𝑛⟶∞)on[0,1]. (90) Therefore, 𝑣_𝑛→𝑣^∗(𝑛→∞), 𝑣^∗∈𝑃_𝑎. By the continuity of 𝑇 and 𝑣_𝑛+1=𝑇𝑣_𝑛, we have 𝑇𝑣^∗=𝑣^∗. (91) Again from (H₃), we have 𝑣^∗(𝑡)>0 on (0,1].
Since every fixed point of 𝑇 in 𝑃 is the solution of BVP (9), (10), then 𝑤^∗ and 𝑣^∗ are two positive, concave and pseudosymmetric solutions of BVP (9), (10). This completes the proof of the theorem.

4. An Example

Consider the following third-order four-point boundary value problem: 𝑢^′′′(𝑡)+𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡)(),𝑢(0)(1)1=0,𝑡∈0,1=0,𝑢=𝑢2,𝑢^′′34=0, (92) where 𝑓()=√𝑡,𝑢,𝑣23844−𝑡𝑢𝑣²+16√2,(𝑡,𝑢,𝑣)∈[0,1]×[0,+∞)×ℝ. (93) It is easy to see that BVP (92) corresponds to BVP (9), (10) when 𝑝=2, 𝑞(𝑡)≡1, and 𝜂=1/2. Take 𝑎=6√2, and then 𝐴=3√2/4.

Next we verify that all conditions of Theorem 7 are satisfied. In fact, obviously the conditions (H₀), (H₁), (H₂), and (H₃) hold. In addition, for 0≤𝑡≤3/4, 0≤𝑢₁≤𝑢₂≤6√2, 0≤|𝑣₁|≤|𝑣₂|≤6√2, 𝑓𝑡,𝑢₁,𝑣₁≤𝑓𝑡,𝑢₂,𝑣₂,max_{0≤𝑡≤3/4}𝑓(𝑡,𝑎,𝑎)=𝑓0,6√2,6√2=8=𝜙₂𝑎𝐴. (94) Hence, from Theorem 7, BVP (92) has two positive, concave, and pseudosymmetric solutions 𝑤^∗ and 𝑣^∗ such that 0<𝑤^∗(𝑡)≤6,0<𝑤^∗′(𝑡)≤6,𝑡∈[0,1],lim_𝑛→∞𝑤_𝑛=lim_𝑛→∞𝑇^𝑛𝑤₀=𝑤^∗,lim_𝑛→∞𝑤^′_𝑛=lim_𝑛→∞𝑇^𝑛𝑤₀^′=𝑤^∗′, (95) where 𝑤₀(𝑡)=36𝑡,0≤𝑡≤4,39−6𝑡,4≤𝑡≤1, (96) for 𝑛=0,1,2,…, 𝑤_𝑛+1(𝑡)=∫^𝑡₀∫_𝜏^3/4∫_𝑠^3/4√23844×𝑤−𝑟_𝑛(𝑟)𝑤_𝑛^′2(𝑟)+16√2d𝑟d𝑠d3𝜏,0≤𝑡≤4,∫₀^1/2∫_𝜏^3/4∫_𝑠^3/4√23844×𝑤−𝑟_𝑛(𝑟)𝑤_𝑛^′2(𝑟)+16√2d𝑟d𝑠d𝜏−∫¹_𝑡∫^𝜏_3/4∫^𝑠_3/4√23844×𝑤−𝑟_𝑛(𝑟)𝑤_𝑛^′2(𝑟)+16√2d𝑟d𝑠d3𝜏,4≤𝑡≤1,0<𝑣^∗(𝑡)≤6√2,0<𝑣^∗′(𝑡)≤6√2,𝑡∈[0,1],lim_𝑛→∞𝑣_𝑛=lim_𝑛→∞𝑇^𝑛𝑣₀=𝑣^∗,lim_𝑛→∞𝑣^′_𝑛=lim_𝑛→∞𝑇^𝑛𝑣₀^′=𝑣^∗′, (97) where 𝑣₀(𝑡)≡0 on [0,1] and for 𝑛=0,1,2,…, 𝑣_𝑛+1(𝑡)=∫^𝑡₀∫_𝜏^3/4∫_𝑠^3/4√23844×𝑣−𝑟_𝑛(𝑟)𝑣_𝑛^′2(𝑟)+16√2d𝑟d𝑠d3𝜏,0≤𝑡≤4,∫₀^1/2∫_𝜏^3/4∫_𝑠^3/4√23844×𝑣−𝑟_𝑛(𝑟)𝑣_𝑛^′2(𝑟)+16√2d𝑟d𝑠d𝜏−∫¹_𝑡∫^𝜏_3/4∫^𝑠_3/4√23844×𝑣−𝑟_𝑛(𝑟)𝑣_𝑛^′2(𝑟)+16√2d𝑟d𝑠d3𝜏,4≤𝑡≤1. (98) The first two terms of {𝑤_𝑛(𝑡)} and three terms of {𝑣_𝑛(𝑡)}, respectively, are as follows: 𝑤₀(𝑡)=36𝑡,0≤𝑡≤4,39−6𝑡,4≤𝑡≤1;𝑤₁(𝑡)=−3√2𝑡70⁵+9√2−1112𝑡63⁴+1𝑡21³−81√2+3896𝑡56²+243√2+335843112𝑡,0≤𝑡≤4,3√2𝑡70⁵−27√2+1112𝑡63⁴+27√2+156𝑡21³−405√2+3896𝑡56²+729√2+33584𝑡−√1127292,3358404𝑣≤𝑡≤1;₀(𝑡)[],𝑣≡0,𝑡∈0,1₁(𝑡)1=−𝑡63⁴+1𝑡21³−3𝑡56²+3[],𝑣112𝑡,𝑡∈0,1₂(𝑡)=√2𝑡2867038902¹⁴−√2𝑡273051324¹³+√2𝑡56010528¹²−√2𝑡18670176¹¹+47√2𝑡426746880¹⁰−65√2𝑡398297088⁹+√2𝑡5619712⁸−59√2𝑡413048832⁷+13√2𝑡157351936⁶−51√2𝑡1573519360⁵+9√2−11258815488𝑡63⁴+1𝑡21³−81√2+3161128382464𝑡56²+81√2+3598476849152[].112𝑡,𝑡∈0,1 (99)(100)

Acknowledgment

This work was supported by the NSFC (11126339 and 11201008).