Existence Theory for Pseudo-Symmetric Solution to 𝑝-Laplacian Differential Equations Involving Derivative

Abstract

We all-sidedly consider a three-point boundary value problem for 𝑝-Laplacian differential equation with nonlinear term involving derivative. Some new sufficient conditions are obtained for the existence of at least one, triple, or arbitrary odd positive pseudosymmetric solutions by using pseudosymmetric technique and fixed-point theory in cone. As an application, two examples are given to illustrate the main results.

1. Introduction

Recent research results indicate that considerable achievement was made in the existence of positive solutions to dynamic equations; for details, please see [1–6] and the references therein. In particular, the existence of positive pseudosymmetric solutions to 𝑝-Laplacian difference and differential equations attract many researchers' attention, such as [7–11]. The reason is that the pseudosymmetry problem not only has theoretical value, such as in the study of metric manifolds [12], but also has practical value itself; for example, we can apply this characteristic into studying the chemistry structure [13]. On another hand, there are much attention paid to the positive solutions of boundary value problems (BVPs) for differential equation with the nonlinear term involved with the derivative explicitly [14–18]. Hence, it is natural to continue study pseudosymmetric solutions to 𝑝-Laplacian differential equations with the nonlinear term involved with the first-order derivative explicitly.

First, let us recall some relevant results about BVPs with 𝑝-Laplacian, We would like to mention the results of Avery and Henderson [7, 8], Ma and Ge [11] and Sun and Ge [16]. Throughout this paper, we denote the 𝑝-Laplacian operator by 𝜑_𝑝(𝑢); that is, 𝜑_𝑝(𝑢)=|𝑢|^𝑝−2𝑢 for 𝑝>1 with (𝜑_𝑝)⁻¹=𝜑_𝑞 and 1/𝑝+1/𝑞=1.

For the three-point BVPs with 𝑝-Laplacian 𝜑_𝑝𝑢^′(𝑡)^′+ℎ(𝑡)𝑓(𝑡,𝑢(𝑡))=0for[],𝑢(0)𝜂(1),𝑡∈0,1=0,𝑢=𝑢 (1.1) here, 𝜂∈(0,1) is constant, by using the five functionals fixed point theorem in a cone [19], Avery and Henderson [8] established the existence of at least three positive pseudosymmetric solutions to BVPs (1.1). The authors also obtained the similar results in their paper [7] for the discrete case. In addition, Ma and Ge [11] developed the existence of at least two positive pseudosymmetric solutions to BVPs (1.1) by using the monotone iterative technique.

For the three-point 𝑝-Laplacian BVPs with dependence on the first-order derivative 𝜑_𝑝𝑢^′(𝑡)^′+ℎ(𝑡)𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡)=0for[],𝑢(0)𝜂(1),𝑡∈0,1=0,𝑢=𝑢 (1.2) Sun and Ge [16] obtained the existence of at least two positive pseudosymmetric solutions to BVPs (1.2) via the monotone iterative technique again. However, it is worth mentioning that the above-mentioned papers [7, 8, 10, 11, 16], the authors only considered results on the existence of positive pseudosymmetric solutions partly, they failed to further provide comprehensive results on the existence of positive pseudosymmetric solutions to 𝑝-Laplacian. Naturally, in this paper, we consider the existence of positive pseudosymmetric solutions for 𝑝-Laplacian differential equations in all respects.

Motivated by the references [7, 8, 10, 11, 16, 18], in present paper, we consider all-sidedly 𝑝-Laplacian BVPs (1.2), using the compression and expansion fixed point theorem [20] and Avery-Peterson fixed point theorem [21]. We obtain that there exist at least one, triple or arbitrary odd positive pseudosymmetric solutions to problem (1.2). In particular, we not only get some local properties of pseudosymmetric solutions, but also obtain that the position of pseudosymmetric solutions is determined under some conditions, which is much better than the results in papers [8, 11, 16]. Correspondingly, we generalize and improve the results in papers Avery and Henderson [8]. From the view of applications, two examples are given to illustrate the main results.

Throughout this paper, we assume that(S1)𝑓(𝑡,𝑢,𝑢′)∶[0,1]×[0,∞)×(−∞,+∞)→[0,∞) is continuous, does not vanish identically on interval [0,1], and 𝑓(𝑡,𝑢,𝑢^′) is pseudosymmetric about 𝜂 on [0,1],(S2)ℎ(𝑡)∈𝐿([0,1],[0,∞)) is pseudosymmetric about 𝜂 on [0,1], and does not vanish identically on any closed subinterval of [0,1]. Furthermore, 0<∫¹₀ℎ(𝑡)𝑑𝑡<∞.

2. Preliminaries

In the preceding of this section, we state the definition of cone and several fixed point theorems needed later [20, 22]. In the rest of this section, we will prove that solving BVPs (1.2) is equivalent to finding the fixed points of a completely continuous operator.

We first list the definition of cone and the compression and expansion fixed point theorem [20, 22].

Definition 2.1. Let 𝐸 be a real Banach space. A nonempty, closed, convex set 𝑃⊂𝐸 is said to be a cone provided the following conditions are satisfied:(i)if 𝑥∈𝑃 and 𝜆≥0, then 𝜆𝑥∈𝑃,(ii)if 𝑥∈𝑃 and −𝑥∈𝑃, then 𝑥=0.

Lemma 2.2 (see [20, 22]). Let 𝑃 be a cone in a Banach space 𝐸. Assume that Ω₁,Ω₂ are open bounded subsets of 𝐸 with 0∈Ω₁,Ω₁⊂Ω₂. If 𝐴∶𝑃∩(Ω₂⧵Ω₁)→𝑃 is a completely continuous operator such that either (i)‖𝐴𝑥‖≤‖𝑥‖,∀𝑥∈𝑃∩𝜕Ω₁and‖𝐴𝑥‖≥‖𝑥‖,∀𝑥∈𝑃∩𝜕Ω₂, or(ii)‖𝐴𝑥‖≥‖𝑥‖,∀𝑥∈𝑃∩𝜕Ω₁and‖𝐴𝑥‖≤‖𝑥‖,∀𝑥∈𝑃∩𝜕Ω₂. Then, 𝐴 has a fixed point in 𝑃∩(Ω₂⧵Ω₁).

Given a nonnegative continuous functional 𝛾 on a cone 𝑃 of a real Banach space 𝐸, we define, for each 𝑑>0, the set 𝑃(𝛾,𝑑)={𝑥∈𝑃∶𝛾(𝑥)<𝑑}.

Let 𝛾 and 𝜃 be nonnegative continuous convex functionals on 𝑃, 𝛼 a nonnegative continuous concave functional on 𝑃, and 𝜓 a nonnegative continuous functional on 𝑃 respectively. We define the following convex sets: 𝑃=(𝑥)(𝑥),𝑃=(𝑥)(𝑥)(𝑥),𝛾,𝛼,𝑏,𝑑𝑥∈𝑃∶𝑏≤𝛼,𝛾≤𝑑𝛾,𝜃,𝛼,𝑏,𝑐,𝑑𝑥∈𝑃∶𝑏≤𝛼,𝜃≤𝑐,𝛾≤𝑑 (2.1) and a closed set 𝑅(𝛾,𝜓,𝑎,𝑑)={𝑥∈𝑃∶𝑎≤𝜓(𝑥),𝛾(𝑥)≤𝑑}.

Next, we list the fixed point theorem due to Avery-Peterson [21].

Lemma 2.3 (see [21]). Let 𝑃 be a cone in a real Banach space 𝐸 and 𝛾,𝜃,𝛼,𝜓 defined as above; moreover, 𝜓 satisfies 𝜓(𝜆^′𝑥)≤𝜆^′𝜓(𝑥) for 0≤𝜆^′≤1 such that for some positive numbers ℎ and 𝑑, 𝛼(𝑥)≤𝜓(𝑥),‖𝑥‖≤ℎ𝛾(𝑥), (2.2) for all 𝑥∈𝑃(𝛾,𝑑). Suppose that 𝐴∶𝑃(𝛾,𝑑)→𝑃(𝛾,𝑑) is completely continuous and there exist positive real numbers 𝑎,𝑏,𝑐 with 𝑎<𝑏 such that(i){𝑥∈𝑃(𝛾,𝜃,𝛼,𝑏,𝑐,𝑑)∶𝛼(𝑥)>𝑏}≠∅ and 𝛼(𝐴(𝑥))>𝑏 for 𝑥∈𝑃(𝛾,𝜃,𝛼,𝑏,𝑐,𝑑),(ii)𝛼(𝐴(𝑥))>𝑏 for 𝑥∈𝑃(𝛾,𝛼,𝑏,𝑑) with 𝜃(𝐴(𝑥))>𝑐,(iii)0∉𝑅(𝛾,𝜓,𝑎,𝑑) and 𝜓(𝐴(𝑥))<𝑎 for all 𝑥∈𝑅(𝛾,𝜓,𝑎,𝑑) with 𝜓(𝑥)=𝑎.Then, 𝐴 has at least three fixed points 𝑥₁,𝑥₂,𝑥₃∈𝑃(𝛾,𝑑) such that 𝛾𝑥_𝑖≤𝑑for𝑖=1,2,3,𝑏<𝛼𝑥₁,𝑎<𝜓𝑥₂,𝛼𝑥₂<𝑏with𝜓𝑥₃<𝑎. (2.3)

Now, let 𝐸=𝐶¹([0,1],ℝ). Then, 𝐸 is a Banach space with norm ‖𝑢‖=maxmax_[]𝑡∈0,1|𝑢(𝑡)|,max_[]𝑡∈0,1𝑢^′(𝑡). (2.4) Define a cone 𝑃⊂𝐸 by 𝑃=𝑢∈𝐸∣𝑢(0)=0,𝑢isconcave,nonnegativeon[0,1]and𝑢issymmetricon𝜂,1. (2.5)

The following lemma can be founded in [11], which is necessary to prove our result.

Lemma 2.4 (see [11]). If 𝑢∈𝑃, then the following statements are true: (i)𝑢(𝑡)≥(𝑢(𝜔₁)/𝜔₁)min{𝑡,1+𝜂−𝑡} for 𝑡∈[0,1], here 𝜔₁=(𝜂+1)/2, (ii)𝑢(𝑡)≥(𝜂/𝜔₁)𝑢(𝜔₁) for 𝑡∈[𝜂,𝜔₁],(iii)max_𝑡∈[0,1]𝑢(𝑡)=𝑢(𝜔₁).

Lemma 2.5. If 𝑢∈𝑃, then the following statements are true: (i)𝑢(𝑡)≤max_𝑡∈[0,1]|𝑢′(𝑡)|, (ii)‖𝑢(𝑡)‖=max_𝑡∈[0,1]|𝑢′(𝑡)|=max{|𝑢′(0)|,|𝑢′(1)|},(iii)min_{𝑡∈[0,𝜔1]}𝑢(𝑡)=𝑢(0)andmin_{𝑡∈[𝜔1,1]}𝑢(𝑡)=𝑢(1).

Proof. (i) Since 𝑢(𝑡)=𝑢(0)+^𝑡₀𝑢^′(𝑡)𝑑𝑡for𝑡∈[0,1], (2.6) which reduces to 𝑢(𝑡)≤^𝑡₀𝑢^′(𝑡)𝑑𝑡≤max_[]𝑡∈0,1𝑢^′(𝑡). (2.7)  
(ii) By using 𝑢^′′(𝑡)≤0 for 𝑡∈[0,1], we have 𝑢′(𝑡) is monotone decreasing function on [0,1]. Moreover, max_[]𝑡∈0,1𝑢(𝑡)=𝑢𝜂+12=𝑢𝜔₁, (2.8) which implies that 𝑢′(𝜔₁)=0, so, 𝑢′(𝑡)≥0 for 𝑡∈[0,𝜔₁] and 𝑢′(𝑡)≤0 for 𝑡∈[𝜔₁,1].

Now, we define the operator 𝐴∶𝑃→𝐸 by (𝐴𝑢)(𝑡)=^𝑡0𝜑_𝑞^𝜔1_𝑠ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝑑𝑟𝑑𝑠for𝑡∈0,𝜔₁,𝑤𝜂+¹_𝑡𝜑_𝑞^𝑠_𝜔1ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝑑𝑟𝑑𝑠for𝑡∈𝜔₁,1, (2.9) here, 𝑤(𝜂)=(𝐴𝑢)(𝜂).

Lemma 2.6. 𝐴∶𝑃→𝑃 is a completely continuous operator.

Proof. In fact, (𝐴𝑢)(𝑡)≥0 for 𝑡∈[0,1], (𝐴𝑢)(𝜂)=(𝐴𝑢)(1) and (𝐴𝑢)(0)=0.
It is easy to see that the operator 𝐴 is pseudosymmetric about 𝜔₁ on [0,1].
In fact, for 𝑡∈[𝜂,𝜔₁], we have 1−𝑡+𝜂∈[𝜔₁,1], and according to the integral transform, one has ¹_{1−𝑡+𝜂}𝜑_𝑞^𝑠_𝜔1ℎ(𝑟)𝑓(𝑟)(𝑟)=𝑟,𝑢,𝑢′𝑑𝑟𝑑𝑠_𝑡𝜂𝜑_𝑞^𝜔1_𝑠1ℎ𝑟₁𝑓𝑟₁,𝑢𝑟₁,𝑢′𝑟₁𝑑𝑟₁𝑑𝑠₁, (2.10) here, 𝑠=1−𝑠₁+𝜂,𝑟=1−𝑟₁+𝜂. Hence, (𝐴𝑢)1−𝑡+𝜂=𝑤𝜂+¹_{1−𝑡+𝜂}𝜑_𝑞^𝑠_𝜔1ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝜂+𝑑𝑟𝑑𝑠=𝑤^𝑡_𝜂𝜑_𝑞^𝜔1_𝑠1ℎ𝑟₁𝑓𝑟₁,𝑢𝑟₁,𝑢′𝑟₁𝑑𝑟₁𝑑𝑠₁=^𝜂₀𝜑_𝑞^𝜔1_𝑠ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝑑𝑟𝑑𝑠+^𝑡_𝜂𝜑_𝑞^𝜔1_𝑠ℎ(𝑟)𝑓(𝑟)(𝑟)=𝑟,𝑢,𝑢′𝑑𝑟𝑑𝑠_𝑡0𝜑_𝑞^𝜔1_𝑠ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢′(𝑟)𝑑𝑟𝑑𝑠=(𝐴𝑢)(𝑡). (2.11)
For 𝑡∈[𝜔₁,1], we note that 1−𝑡+𝜂∈[𝜂,𝜔₁], by using the integral transform, one has _𝜂^{1−𝑡+𝜂}𝜑_𝑞^𝜔1_𝑠ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)=𝑑𝑟𝑑𝑠_1𝑡𝜑_𝑞^𝑠1_𝜔1ℎ𝑟₁𝑓𝑟₁,𝑢𝑟₁,𝑢^′𝑟₁𝑑𝑟₁𝑑𝑠₁, (2.12) where 𝑠=1−𝑠₁+𝜂,𝑟=1−𝑟₁+𝜂. Thus, (𝐴𝑢)1−𝑡+𝜂=₀^{1−𝑡+𝜂}𝜑_𝑞^𝜔1_𝑠ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝜂+𝑑𝑟𝑑𝑠=𝑤_𝜂^{1−𝑡+𝜂}𝜑_𝑞^𝜔1_𝑠ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝜂+𝑑𝑟𝑑𝑠=𝑤¹_𝑡𝜑_𝑞^𝑠1_𝜔1ℎ𝑟₁𝑓𝑟₁,𝑢𝑟₁,𝑢^′𝑟₁𝑑𝑟₁𝑑𝑠₁=𝑤𝜂+¹_𝑡𝜑_𝑞^𝑠_𝜔1ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝑑𝑟𝑑𝑠=(𝐴𝑢)(𝑡). (2.13)
Hence, 𝐴 is pseudosymmetric about 𝜂 on [0,1].
In addition, (𝐴𝑢)^′(𝑡)=𝜑_𝑞^𝜔1_𝑡ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝑑𝑠≥0,𝑡∈0,𝜔₁ (2.14) is continuous and nonincreasing in [0,𝜔₁]; moreover, 𝜑_𝑞(𝑥) is a monotone increasing continuously differentiable function ^𝜔1_𝑡ℎ(𝑠)𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠)𝑑𝑠^′=−ℎ(𝑡)𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡)≤0,𝑡∈0,𝜔₁, (2.15) it is easy to obtain (𝐴𝑢)^′′(𝑡)≤0for𝑡∈[0,𝜔₁]. By using the similar way, we can deduce (𝐴𝑢)^′′(𝑡)≤0for𝑡∈[𝜔₁,1]. So, 𝐴∶𝑃→𝑃. It is easy to obtain that 𝐴∶𝑃→𝑃 is completely continuous.

Hence, the solutions of BVPs (1.2) are fixed points of the completely continuous operator 𝐴.

3. One Solutions

In this section, we will study the existence of one positive pseudosymmetric solution to problem (1.2) by Krasnosel'skii's fixed point theorem in a cone.

Motivated by the notations in reference [23], for 𝑢∈𝑃, let 𝑓⁰=sup_[]𝑡∈0,1lim_{𝑢,𝑢′→()0,0}𝑓𝑡,𝑢,𝑢^′𝜑_𝑝𝑢^′,𝑓₀=inf_[]𝑡∈0,1lim_{𝑢,𝑢′→()0,0}𝑓𝑡,𝑢,𝑢^′𝜑_𝑝𝑢^′,𝑓^∞=sup_[]𝑡∈0,1lim_{𝑢,𝑢′→()∞,∞}𝑓𝑡,𝑢,𝑢^′𝜑_𝑝𝑢^′,𝑓_∞=inf_[]𝑡∈0,1lim_{𝑢,𝑢′→()∞,∞}𝑓𝑡,𝑢,𝑢^′𝜑_𝑝𝑢^′. (3.1)

In the following, we discuss the problem (1.2) under the following four possible cases.

Theorem 3.1. If 𝑓⁰=0 and 𝑓_∞=∞, problem (1.2) has at least one positive pseudosymmetric solution 𝑢.

Proof. In view of 𝑓⁰=0, there exists an 𝐻₁>0 such that 𝑓𝑡,𝑢,𝑢^′≤𝜑_𝑝(𝜀)𝜑_𝑝𝑢^′=𝜑_𝑝𝜀𝑢^′for𝑡,𝑢,𝑢^′∈[0,1]×0,𝐻₁×−𝐻₁,𝐻₁, (3.2) here, 𝜀>0 and satisfies 𝜀𝜑_𝑞^𝜔1₀ℎ(𝑠)𝑑𝑠≤1. (3.3)
If 𝑢∈𝑃 with ‖𝑢‖=𝐻₁, by Lemma 2.5, we have 𝑢(𝑡)≤max_[]𝑡∈0,1𝑢^′(𝑡)≤‖𝑢‖=𝐻₁for𝑡∈[0,1], (3.4) hence, ‖𝐴𝑢‖=max(𝐴𝑢)^′(0),(𝐴𝑢)^′(1𝜑)=max_𝑞^𝜔1₀ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝑑𝑟,𝜑_𝑞¹_𝜔1ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢′(𝑟)𝑑𝑟≤𝜀max_[]𝑡∈0,1𝑢^′(𝑡)𝜑_𝑞^𝜔1₀ℎ(𝑠)𝑑𝑠≤‖𝑢‖. (3.5) If set Ω_𝐻1={𝑢∈𝐸∶‖𝑢‖<𝐻₁}, one has ‖𝐴𝑢‖≤‖𝑢‖for𝑢∈𝑃∩𝜕Ω_𝐻1.
According to 𝑓_∞=∞, there exists an 𝐻^′₂>0 such that 𝑓𝑡,𝑢,𝑢^′≥max_[]𝑡∈0,1𝜑_𝑝(𝑘)𝜑_𝑝𝑢^′=max_[]𝑡∈0,1𝜑_𝑝𝑘𝑢^′, (3.6) where (𝑡,𝑢,𝑢′)∈[0,1]×[𝐻^′₂,∞)×(−∞,𝐻^′₂]∪[𝐻^′₂,∞), 𝑘>0 and satisfies 𝑘𝜑_𝑞¹_𝜔1ℎ(𝑟)𝑑𝑟≥1. (3.7) Set 𝐻₂=max2𝐻₁,𝜔₁𝜂𝐻^′₂,Ω_𝐻^∗₂=𝑢∈𝐸∶‖𝑢‖<5𝐻₂,Ω_𝐻2=𝑢∈Ω_𝐻^∗₂∶𝑢𝜔₁<𝐻₂. (3.8)
For 𝑢∈𝑃∩𝜕Ω_𝐻2, we have 𝑢(𝜔₁)=𝐻₂ since 𝑢(𝑡)≤|𝑢^′(𝑡)|for𝑢∈𝑃.
If 𝑢∈𝑃 with 𝑢(𝜔₁)=𝐻₂, Lemmas 2.4 and 2.5 reduce to min_{𝜔𝑡∈1,1}𝑢^′(𝑡)≥min_{𝜔𝑡∈1,1}𝑢(𝑡)(1)≥𝜔=𝑢𝜂𝑢₁𝜔₁≥𝐻^′₂. (3.9) For 𝑢∈𝑃∩𝜕Ω_𝐻2, according to (3.6), (3.7) and (3.9), we get ‖𝐴𝑢‖=max𝜑_𝑞^𝜔1₀ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝑑𝑟,𝜑_𝑞¹_𝜔1ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝑑𝑟≥𝜑_𝑞¹_𝜔1ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝑑𝑟≥𝑘max_[]𝑡∈0,1𝑢^′(𝑡)𝜑_𝑞^𝜔1₁ℎ(𝑟)𝑑𝑟=‖𝑢‖. (3.10) Thus, by (i) of Lemma 2.2, the problem (1.2) has at least one positive pseudosymmetric solution 𝑢 in 𝑃∩(Ω_𝐻2⧵Ω_𝐻1).

Theorem 3.2. If 𝑓₀=∞ and 𝑓^∞=0, problem (1.2) has at least one positive pseudosymmetric solution 𝑢.

Proof. Since 𝑓₀=∞, there exists an 𝐻₃>0 such that 𝑓𝑡,𝑢,𝑢^′≥max_[]𝑡∈0,1𝜑_𝑝(𝑚)𝜑_𝑝𝑢^′=max_[]𝑡∈0,1𝜑_𝑝𝑚𝑢^′, (3.11) here, (𝑡,𝑢,𝑢′)∈[0,1]×(0,𝐻₃]×[−𝐻₃,𝐻₃] and 𝑚 is such that 𝑚𝜑_𝑞¹_𝜔1ℎ(𝑟)𝑑𝑟≥1. (3.12) If 𝑢∈𝑃 with ‖𝑢‖=𝐻₃, Lemma 2.5 implies that 𝑢(𝑡)≤max_[]𝑡∈0,1𝑢^′(𝑡)≤‖𝑢‖=𝐻₃for𝑡∈[0,1], (3.13) now, by (3.11), (3.12), and (3.13), we have ‖𝐴𝑢‖=max𝜑_𝑞^𝜔1₀ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝑑𝑟,𝜑_𝑞¹_𝜔1ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝑑𝑟≥𝜑_𝑞¹_𝜔1ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝑑𝑟≥𝑚max_[]𝑡∈0,1𝑢^′(𝑡)𝜑_𝑞¹_𝜔1ℎ(𝑟)𝑑𝑟=‖𝑢‖. (3.14) If let Ω_𝐻3={𝑢∈𝐸∶‖𝑢‖<𝐻₃}, one has ‖𝐴𝑢‖≥‖𝑢‖for𝑢∈𝑃∩𝜕Ω_𝐻3.
Now, we consider 𝑓^∞=0.
Suppose that 𝑓 is bounded, for some constant 𝐾>0, then 𝑓𝑡,𝑢,𝑢^′≤𝜑_𝑝(𝐾)∀𝑡,𝑢,𝑢^′∈[0,1]×[0,∞)×(−∞,∞). (3.15) Pick 𝐻₄≥max𝐻^′₄,2𝐻₃,𝐾𝜑_𝑞^𝜔1₀ℎ(𝑠),𝐶𝑑𝑠𝛿, (3.16) here, 𝐶 is an arbitrary positive constant and satisfy the (3.21). Let Ω_𝐻4=𝑢∈𝐸∶‖𝑢‖<𝐻₄. (3.17)
If 𝑢∈𝑃∩𝜕Ω_𝐻4, one has ‖𝑢‖=𝐻₄, then (3.15) and (3.16) imply that ‖𝐴𝑢‖=max𝜑_𝑞^𝜔1₀ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝑑𝑟,𝜑_𝑞¹_𝜔1ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝑑𝑟≤𝐾𝜑_𝑞^𝜔1₀ℎ(𝑠)𝑑𝑠≤𝐻₄=‖𝑢‖. (3.18)
Suppose that 𝑓 is unbounded.
By definition of 𝑓^∞=0, there exists 𝐻^′₄>0 such that 𝑓𝑡,𝑢,𝑢^′≤𝜑_𝑝(𝛿)𝜑_𝑝𝑢^′=𝜑_𝑝𝛿𝑢^′, (3.19) where (𝑡,𝑢,𝑢′)∈[0,𝜔₁]×[𝐻^′₄,∞)×(−∞,𝐻^′₄]∪[𝐻^′₄,∞) and 𝛿>0 satisfies 𝛿𝜑_𝑞^𝜔1₀ℎ(𝑠)𝑑𝑠≤1. (3.20)
From 𝑓∈𝐶([0,1]×[0,+∞)×(−∞,∞),[0,+∞)), we have 𝑓𝑡,𝑢,𝑢^′≤𝜑_𝑝(𝐶)for𝑡,𝑢,𝑢^′∈[0,1]×0,𝐻^′₄×−𝐻^′₄,𝐻^′₄, (3.21) here, 𝐶 is an arbitrary positive constant.
Then, for (𝑡,𝑢,𝑢′)∈[0,1]×[0,∞)×(−∞,∞), we have 𝑓𝑡,𝑢,𝑢′≤max𝜑_𝑝(𝐶),𝜑_𝑝(𝛿)𝜑_𝑝𝑢^′. (3.22)
If 𝑢∈𝑃∩𝜕Ω_𝐻4, one has ‖𝑢‖=𝐻₄, which reduces to ‖𝐴𝑢‖=max𝜑_𝑞^𝜔1₀ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝑑𝑟,𝜑_𝑞¹_𝜔1ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝑢𝑑𝑟≤max𝐶,𝛿^′𝜑_𝑞^𝜔1₀ℎ(𝑟)𝑑𝑟≤𝐻₄=‖𝑢‖. (3.23) Consequently, for any cases, if we take Ω_𝐻4={𝑢∈𝐸∶‖𝑢‖<𝐻₄}, we have ‖𝐴𝑢‖≤‖𝑢‖for𝑢∈𝑃∩𝜕Ω_𝐻4. Thus, the condition (ii) of Lemma 2.2 is satisfied.
Consequently, the problem (1.2) has at least one positive pseudosymmetric solution 𝑢∈𝑃∩Ω_𝐻4⧵Ω_𝐻3with𝐻₃≤‖𝑢‖≤𝐻₄. (3.24)

Theorem 3.3. Suppose that the following conditions hold: (i)there exist nonzero finite constants 𝑐₁ and 𝑐₂ such that 𝑓⁰=𝑐₁ and 𝑓_∞=𝑐₂,(ii)there exist nonzero finite constants 𝑐₃ and 𝑐₄ such that 𝑓₀=𝑐₃ and 𝑓^∞=𝑐₄.Then, problem (1.2) has at least one positive pseudosymmetric solution 𝑢.

Proof. (i) In view of 𝑓⁰=𝑐₁, there exists an 𝐻₅>0 such that 𝑓𝑡,𝑢,𝑢^′≤𝜑_𝑝𝜀+𝑐₁₁𝜑_𝑝𝑢^′=𝜑_𝑝𝜀+𝑐₁₁𝑢^′for𝑡,𝑢,𝑢^′∈[0,1]×0,𝐻₅×−𝐻₅,𝐻₅, (3.25) here, 𝑐₁=𝜑_𝑝(𝑐₁₁+𝜀),𝜀>0 and satisfies 𝜀+𝑐₁₁𝜑_𝑞^𝜔1₀ℎ(𝑠)𝑑𝑠≤1. (3.26)
If 𝑢∈𝑃 with ‖𝑢‖=𝐻₅, by Lemma 2.5, we have 𝑢(𝑡)≤𝑢^′(𝑡)≤‖𝑢‖=𝐻₅for𝑡∈[0,1], (3.27) hence, ‖𝐴𝑢‖=max(𝐴𝑢)^′(0),(𝐴𝑢)^′(1𝜑)=max_𝑞^𝜔1₀ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝑑𝑟,𝜑_𝑞¹_𝜔1ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)≤𝑑𝑟𝜀+𝑐₁₁max_[]𝑡∈0,1𝑢^′(𝑡)𝜑_𝑞^𝜔1₀ℎ(𝑠)𝑑𝑠≤‖𝑢‖. (3.28) If set Ω_𝐻5={𝑢∈𝐸∶‖𝑢‖<𝐻₅}, one has ‖𝐴𝑢‖≤‖𝑢‖for𝑢∈𝑃∩𝜕Ω_𝐻5.
According to 𝑓_∞=𝑐₂, there exists an 𝐻^′₆>0 such that 𝑓𝑡,𝑢,𝑢^′≥max_[]𝑡∈0,1𝜑_𝑝𝑐₂₂−𝜀𝜑_𝑝𝑢^′=max_[]𝑡∈0,1𝜑_𝑝𝑐₂₂−𝜀𝑢^′, (3.29) where (𝑡,𝑢,𝑢′)∈[0,1]×[𝐻^′₆,∞)×(−∞,𝐻^′₆]∪[𝐻^′₆,∞), 𝑐₂=𝜑_𝑝(𝑐₂₂−𝜀),𝜀>0 and satisfies 𝑐₂₂−𝜀𝜑_𝑞¹_𝜔1ℎ(𝑟)𝑑𝑟≥1. (3.30) Set 𝐻₆=max2𝐻₅,𝜔₁𝜂𝐻^′₆,Ω_𝐻^∗₆=𝑢∈𝐸∶‖𝑢‖<5𝐻₆,Ω_𝐻6=𝑢∈Ω_𝐻^∗₆∶𝑢𝜔₁<𝐻₆. (3.31) If 𝑢∈𝑃 with 𝑢(𝜔₁)=𝐻₆, Lemmas 2.4 and 2.5 reduce to min_{𝜔𝑡∈1,1}𝑢^′(𝑡)≥min_{𝜔𝑡∈1,1}𝑢(𝑡)(1)≥𝜔=𝑢𝜂𝑢₁𝜔₁≥𝐻^′₆. (3.32) For 𝑢∈𝑃∩𝜕Ω_𝐻6, according to (3.29), (3.30) and (3.32), we get ‖𝐴𝑢‖=max𝜑_𝑞^𝜔1₀ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝑑𝑟,𝜑_𝑞¹_𝜔1ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝑑𝑟≥𝜑_𝑞¹_𝜔1ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟≥𝑐)𝑑𝑟₂₂−𝜀max_[]𝑡∈0,1𝑢^′(𝑡)𝜑_𝑞^𝜔1₁ℎ(𝑟)𝑑𝑟=‖𝑢‖. (3.33) Thus, by (i) of Lemma 2.2, the problem (1.2) has at least one positive pseudosymmetric solution 𝑢 in 𝑃∩(Ω_𝐻6⧵Ω_𝐻5).
(ii) By using the similar way as to Theorem 3.2, we can prove to it.

4. Triple Solutions

In the previous section, some results on the existence of at least one positive pseudosymmetric solutions to problem (1.2) are obtained. In this section, we will further discuss the existence criteria for at least three and arbitrary odd positive pseudosymmetric solutions of problems (1.2) by using the Avery-Peterson fixed point theorem [21].

Choose a 𝑟∈(𝜂,𝜔₁), for the notational convenience, we denote 𝑀=𝜔₁𝜑_𝑞^𝜔1₀ℎ(𝑟)𝑑𝑟,𝑁=𝜂𝜑_𝑞^𝜔1_𝜂ℎ(𝑟)𝑑𝑟,𝑊=𝜑_𝑞^𝜔1₀ℎ(𝑟)𝑑𝑟. (4.1)

Define the nonnegative continuous convex functionals 𝜃 and 𝛾, nonnegative continuous concave functional 𝛼, and nonnegative continuous functional 𝜑, respectively, on 𝑃 by 𝛾(𝑢)=max_[]𝑡∈0,1𝑢^′(𝑡)=max𝑢^′(0),𝑢^′(1=‖𝑢‖,𝜓(𝑢)(𝑢))=𝜃=max_{𝑡∈0,𝜔1}𝑢(𝑡)=𝑢𝜔₁≤‖𝑢‖,𝛼(𝑢)=min_{𝑡∈𝜂,𝜔1}𝑢(𝑡)=𝑢𝜂. (4.2)

Now, we state and prove the results in this section.

Theorem 4.1. Suppose that there exist constants 𝑎^∗,𝑏^∗, and 𝑑^∗ such that 0<𝑎^∗<𝑏^∗<(𝑁/𝑊)𝑑^∗. In addition, 𝑓 satisfies the following conditions: (i)𝑓(𝑡,𝑢,𝑢′)≤𝜑_𝑝(𝑑^∗/𝑊) for (𝑡,𝑢,𝑢^′)∈[0,1]×[0,𝑑^∗]×[−𝑑^∗,𝑑^∗],(ii)𝑓(𝑡,𝑢,𝑢′)>𝜑_𝑝(𝑏^∗/𝑁) for (𝑡,𝑢,𝑢^′)∈[𝜂,𝜔₁]×[𝑏^∗,𝑑^∗]×[−𝑑^∗,𝑑^∗],(iii)𝑓(𝑡,𝑢,𝑢′)<𝜑_𝑝(𝑎^∗/𝑀) for (𝑡,𝑢,𝑢′)∈[0,𝜔₁]×[0,𝑎^∗]×[−𝑑^∗,𝑑^∗].
Then, problem (1.2) has at least three positive pseudosymmetric solutions 𝑢₁,𝑢₂,and𝑢₃ such that 𝑥_𝑖≤𝑑^∗for𝑖=1,2,3,𝑏^∗<min_{𝑡∈𝜂,𝜔1}𝑢₁(𝑡),𝑎^∗<max_[]𝑡∈0,1𝑢₂(𝑡),min_{𝑡∈𝜂,𝜔1}𝑢₂(𝑡)<𝑏^∗withmax_[]𝑡∈0,1𝑢₃(𝑡)<𝑎^∗. (4.3)

Proof. According to the definition of completely continuous operator 𝐴 and its properties, we need to show that all the conditions of Lemma 2.3 hold with respect to 𝐴.
It is obvious that 𝜓𝜆^′𝑢=𝜆^′𝑢𝜔₁=𝜆^′𝜓(𝑢)for𝛼(𝑢)(𝑢)‖𝑢‖(𝑢)0<𝜆′<1,≤𝜓∀𝑢∈𝑃,=𝛾∀𝑢∈𝑃. (4.4)
Firstly, we show that 𝐴∶𝑃(𝛾,𝑑^∗)→𝑃(𝛾,𝑑^∗).
For any 𝑢∈𝑃(𝛾,𝑑^∗), we have 𝑢(𝑡)≤max_[]𝑡∈0,1𝑢^′(𝑡)≤‖𝑢‖=𝛾(𝑢)≤𝑑^∗for𝑡∈[0,1], (4.5) hence, the assumption (i) implies that ‖𝐴𝑢‖=max𝜑_𝑞^𝜔1₀ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝑑𝑟,𝜑_𝑞¹_𝜔1ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)≤𝑑𝑑𝑟_∗𝑊𝜑_𝑞^𝜔1₀ℎ(𝑟)𝑑𝑟=𝑑^∗. (4.6) From the above analysis, it remains to show that (i)–(iii) of Lemma 2.3 hold.
Secondly, we verify that condition (i) of Lemma 2.3 holds; let 𝑢(𝑡)≡(𝑡𝑏^∗/𝜂)+𝑏^∗,𝑡∈[0,1], and it is easy to see that 𝛼(𝑢)=𝑢𝜂=2𝑏^∗>𝑏^∗,𝜃(𝑢)=𝑢𝜔₁=𝜔₁𝑏^∗𝜂+𝑏^∗≤𝜔₁𝑏^∗𝜂+𝑏^∗, (4.7) in addition, we have 𝛾(𝑢)=(𝑏^∗/𝜂)<𝑑^∗, since 𝑏^∗<(𝑁/𝑊)𝑑^∗. Thus 𝑢∈𝑃𝛾,𝜃,𝛼,𝑏^∗,𝜔₁𝑏^∗𝜂+𝑏^∗,𝑑^∗∶𝛼(𝑥)>𝑏^∗≠∅. (4.8) For any 𝑢∈𝑃𝛾,𝜃,𝛼,𝑏^∗,𝜔₁𝑏^∗𝜂+𝑏^∗,𝑑^∗, (4.9) one has 𝑏^∗≤𝑢(𝑡)≤‖𝑢‖≤𝑑^∗∀𝑡∈𝜂,𝜔₁, (4.10) it follows from the assumption (ii) that 𝛼(𝐴𝑢)=(𝐴𝑢)𝜂=^𝜂₀𝜑_𝑞^𝜔1_𝑠ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)𝑑𝑟𝑑𝑠≥^𝜂₀𝜑_𝑞^𝜔1_𝜂ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)>𝑏𝑑𝑟𝑑𝑠_∗𝑁𝜂𝜑_𝑞^𝜔1_𝜂ℎ(𝑟)𝑑𝑟=𝑏^∗. (4.11)
Thirdly, we prove that the condition (ii) of Lemma 2.3 holds. In fact, 𝛼()𝜂,𝜃()𝐴𝑢=𝐴𝑢𝐴𝑢=max_{𝑡∈0,𝜔1}𝐴(𝑢)=𝐴𝑢𝜔₁. (4.12) For any 𝑢∈𝑃(𝛾,𝛼,𝑏^∗,𝑑^∗)with𝜃(𝐴𝑢)>(𝜔₁𝑏^∗/𝜂)+𝑏^∗, we have 𝛼()𝜂≥𝜂𝐴𝑢=𝐴𝑢𝜔₁𝐴𝑢𝜔₁≥𝜂𝜔₁𝜃(𝐴𝑢)=𝑏^∗+𝜔₁𝑏^∗𝜂>𝑏^∗. (4.13)
Finally, we check condition (iii) of Lemma 2.3.
Clearly, since 𝜓(0)=0<𝑎^∗, we have 0∉𝑅(𝛾,𝜓,𝑎^∗,𝑑^∗). If 𝑢∈𝑅𝛾,𝜓,𝑎^∗,𝑑^∗with𝜓(𝑢)=max_{𝑡∈0,𝜔1}𝑢(𝑡)=𝑢𝜔₁=𝑎^∗, (4.14) then 0≤𝑢(𝑡)≤𝑎^∗∀𝑡∈0,𝜔₁,max_[]𝑡∈0,1𝑢^′(𝑡)=‖𝑢‖=𝛾(𝑢)≤𝑑^∗. (4.15) Hence, by assumption (iii), we have 𝜓(𝐴𝑢)=(𝐴𝑢)𝜔₁≤^𝜔1₀𝜑_𝑞^𝜔1₀ℎ(𝑟)𝑓𝑟,𝑢(𝑟),𝑢^′(𝑟)<𝑎𝑑𝑟𝑑𝑠_∗𝑀𝜔₁𝜑_𝑞^𝜔1₀ℎ(𝑟)𝑑𝑟=𝑎^∗. (4.16) Consequently, from above, all the conditions of Lemma 2.3 are satisfied. The proof is completed.

Corollary 4.2. If the condition (𝑖) in Theorem 4.1 is replaced by the following condition (i′):(i′)lim_{(𝑢,𝑢′)→(∞,∞)}(𝑓(𝑡,𝑢,𝑢^′)/(𝜑_𝑝(|𝑢^′|)))≤𝜑_𝑝(1/𝑊),  then the conclusion of Theorem 4.1 also holds.

Proof. From Theorem 4.1, we only need to prove that (i′) implies that (i) holds. That is, assume that (i′) holds, then there exists a number 𝑑^∗≥(𝑊/𝑁)𝑏^∗ such that 𝑓𝑡,𝑢,𝑢^′≤𝜑_𝑝𝑑^∗𝑊for𝑡,𝑢,𝑢^′∈[0,1]×0,𝑑^∗×−𝑑^∗,𝑑^∗. (4.17)
Suppose on the contrary that for any 𝑑^∗≥(𝑊/𝑁)𝑏^∗, there exists (𝑢_𝑐,𝑢′_𝑐)∈[0,𝑑^∗]×[−𝑑^∗,𝑑^∗] such that 𝑓𝑡,𝑢_𝑐,𝑢^′_𝑐>𝜑_𝑝𝑑^∗𝑊for𝑡∈[0,1]. (4.18) Hence, if we choose 𝑐^∗_𝑛>(𝑊/𝑁)𝑏^∗(𝑛=1,2,…)with𝑐^∗_𝑛→∞, then there exist (𝑢_𝑛,𝑢^′_𝑛)∈[0,𝑐^∗_𝑛]×[−𝑐^∗_𝑛,𝑐^∗_𝑛] such that 𝑓𝑡,𝑢_𝑛,𝑢^′_𝑛>𝜑_𝑝𝑐^∗_𝑛𝑊for𝑡∈[0,1], (4.19) and so lim_𝑛→∞𝑓𝑡,𝑢_𝑛,𝑢^′_𝑛=∞for𝑡∈[0,1]. (4.20) Since the condition (i′) holds, there exists 𝜏>0 satisfying 𝑓𝑡,𝑢,𝑢^′≤𝜑_𝑝𝑢^′𝑊for𝑡,𝑢,𝑢^′∈[0,1]×[𝜏,∞)×(−∞,𝜏]∪[𝜏,∞). (4.21) Hence, we have 𝑢_𝑛<𝑢^′_𝑛≤𝜏. (4.22) Otherwise, if 𝑢^′_𝑛>𝑢_𝑛>𝜏for𝑡∈[0,1], (4.23) it follows from (4.21) that 𝑓𝑡,𝑢_𝑛,𝑢^′_𝑛≤𝜑_𝑝𝑢_𝑛𝑊≤𝜑_𝑝𝑐^∗_𝑛𝑊for𝑡∈[0,1], (4.24) which contradicts (4.19).
Let 𝑊=max_{𝑡,𝑢,𝑢′∈[]×[]×[]0,10,𝜏−𝜏,𝜏}𝑓𝑡,𝑢,𝑢^′, (4.25) then 𝑓𝑡,𝑢_𝑛,𝑢^′_𝑛≤𝑊(𝑛=1,2,…), (4.26) which also contradicts (4.20).

Theorem 4.3. Suppose that there exist constants 𝑎^∗_𝑖,𝑏^∗_𝑖,and𝑑^∗_𝑖 such that 0<𝑎^∗₁<𝑏^∗₁<𝑁𝑊𝑑^∗₂<𝑎^∗₂<𝑏^∗₂<𝑁𝑊𝑑^∗₃<⋯<𝑎^∗_𝑛<𝑏^∗_𝑛<𝑁𝑊𝑑^∗_𝑛+1, (4.27) here, 𝑛∈ℕ and 𝑖=1,2,…,𝑛. In addition, suppose that 𝑓 satisfies the following conditions:(i)𝑓(𝑡,𝑢,𝑢^′)≤𝜑_𝑝(𝑑^∗_𝑖/𝑊) for (𝑡,𝑢,𝑢^′)∈[0,1]×[0,𝑑^∗_𝑖]×[−𝑑^∗_𝑖,𝑑^∗_𝑖],(ii)𝑓(𝑡,𝑢,𝑢^′)>𝜑_𝑝(𝑏^∗_𝑖/𝑁) for (𝑡,𝑢,𝑢^′)∈[𝜂,𝜔₁]×[𝑏^∗_𝑖,𝑑^∗_𝑖]×[−𝑑^∗_𝑖,𝑑^∗_𝑖],(iii)𝑓(𝑡,𝑢,𝑢^′)<𝜑_𝑝(𝑎^∗_𝑖/𝑀) for (𝑡,𝑢,𝑢^′)∈[0,𝜔₁]×[0,𝑎^∗_𝑖]×[−𝑑^∗_𝑖,𝑑^∗_𝑖].Then, problem (1.2) has at least 2𝑛−1 positive pseudosymmetric solutions.

Proof. When 𝑛=1, it is immediate from condition (i) that 𝐴∶𝑃_𝑎^∗₁⟶𝑃_𝑎^∗₁⊂𝑃_𝑎^∗₁. (4.28) It follows from the Schauder fixed point theorem that 𝐴 has at least one fixed point 𝑢₁∈𝑃_𝑎^∗₁, (4.29) which means that 𝑢₁≤𝑎^∗₁. (4.30)
When 𝑛=2, it is clear that Theorem 4.1 holds (with 𝑎^∗=𝑎^∗₁,𝑏^∗=𝑏^∗₁,and𝑑^∗=𝑑^∗₂). Then, there exists at least three positive pseudosymmetric solutions 𝑢₁,𝑢₂, and 𝑢₃ such that 𝑥₁≤𝑑^∗₂,𝑥₂≤𝑑^∗₂,𝑥₃≤𝑑^∗₂,𝑏^∗<min_{𝑡∈𝜂,𝜔1}𝑢₁(𝑡),𝑎^∗₁<max_[]𝑡∈0,1𝑢₂(𝑡),min_{𝑡∈𝜂,𝜔1}𝑢₂(𝑡)<𝑏^∗₁withmax_[]𝑡∈0,1𝑢₃(𝑡)<𝑎^∗₁. (4.31) Following this way, we finish the proof by induction. The proof is complete.

5. Examples

In this section, we present two simple examples to illustrate our results.

Example 5.1. Consider the following BVPs: 𝜑_𝑝𝑢^′(𝑡)^′+𝑡𝑡+1+𝑢^′(𝑡)^𝑝−2[],𝑢(0)()(1).=0,𝑡∈0,1=0,𝑢0.2=𝑢 (5.1)
Note that 𝑓₀=inf_[]𝑡∈0,1lim_{𝑢,𝑢′→()0,0}𝑡+1+𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)𝑓=∞,_∞=sup_[]𝑡∈0,1lim_{𝑢,𝑢′→()∞,∞}𝑡+1+𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)=0. (5.2) Hence, Theorem 3.2 implies that the BVPs in (5.1) have at least one pseudosymmetric solution 𝑢.

Example 5.2. Consider the following BVPs with 𝑝=3: 𝜑_𝑝𝑢^′(𝑡)^′+ℎ(𝑡)𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡)[],𝑢(0)()(1),=0,𝑡∈0,1=0,𝑢0.2=𝑢 (5.3) where ℎ(𝑡)=2𝑡 and 𝑓𝑡,𝑢,𝑢^′=𝑢𝑡+4+^′5.5²[],𝑢,𝑢∈0,0.9𝑡+750𝑢−671+^′5.5²[],𝑢,𝑢∈0.9,1𝑡+79+^′5.5²[],𝑢,𝑢∈1,5.5𝑡+14.364𝑢+^′5.5²,𝑢∈[5.5,+∞). (5.4)
Note that 𝜂=0.2,𝜔₁=0.6, then a direct calculation shows that 𝑀=𝜔₁𝜑_𝑞^𝜔1₀ℎ(𝑟)𝑑𝑟=0.6×0.6=0.36,𝑁≈0.1131,𝑊=0.6. (5.5) If we take 𝑎^′=0.9,𝑏^′=1,𝑑^′=5.5, then 𝑎^′<𝑏^′<(𝑁/𝑊)𝑑^′ holds; furthermore, 𝑓𝑡,𝑢,𝑢^′<82<84.028≈𝜑_𝑝𝑑^′𝑊for𝑡,𝑢,𝑢^′∈[]×[]×[],𝑓0,0.60,5.5−5.5,5.5𝑡,𝑢,𝑢^′>79>78.176≈𝜑_𝑝𝑏^′𝑁for𝑡,𝑢,𝑢^′∈[]×[]×[],𝑓0.6,11,5.5−5.5,5.5𝑡,𝑢,𝑢^′<6.2<6.25=𝜑_𝑝𝑎^′𝑀for𝑡,𝑢,𝑢^′∈[0,0.6]×[0,0.9]×[−5.5,5.5]. (5.6) By Theorem 4.1, we see that the BVPs in (5.3) have at least three positive pseudosymmetric solutions 𝑢₁,𝑢₂ and 𝑢₃ such that 𝑥_𝑖≤5.5for𝑖=1,2,3,1<min_{[]𝑡∈0.2,0.6}𝑢₁(𝑡),0.9<max_[]𝑡∈0,1𝑢₂(𝑡),min_{[]𝑡∈0.2,0.6}𝑢₂(𝑡)<1withmax_[]𝑡∈0,1𝑢₃(𝑡)<0.9. (5.7)

Acknowledgments

Y. H. Su Sponsored by Qing Lan Project and XZIT (no. XKY2010101). X. Yan Supported by Fundamental Research Funds for the Central Universities Grant (no. 2010QNA40).