Multiple Positive Symmetric Solutions to 𝑝-Laplacian Dynamic Equations on Time Scales

Abstract

This paper makes a study on the existence of positive solution to 𝑝-Laplacian dynamic equations on time scales 𝕋. Some new sufficient conditions are obtained for the existence of at least single or twin positive solutions by using Krasnosel'skii's fixed point theorem and new sufficient conditions are also obtained for the existence of at least triple or arbitrary odd number positive solutions by using generalized Avery-Henderson fixed point theorem and Avery-Peterson fixed point theorem. As applications, two examples are given to illustrate the main results and their differences. These results are even new for the special cases of continuous and discrete equations, as well as in the general time-scale setting.

1. Introduction

Initiated by Hilger in his Ph.D. thesis [1] in 1988, the theory of time scales has been improved ever since, especially in the unification of the theory of differential equations in the continuous case and the theory of difference equations in the discrete case. For the time being, it remains a field of vitality and attracts attention of many distinguished scholars worldwide. In particular, the theory is also widely applied to biology, heat transfer, stock market, wound healing, epidemic models [2–5], and so forth.

Recent research results indicate that considerable achievement has been made in the existence problems of positive solutions to dynamic equations on time scales. For details, please see [6–13] and the references therein. Symmetry and pseudosymmetry have been widely used in science and engineering [14]. The reason is that symmetry and pseudosymmetry are not only of its theoretical value in studying the metric manifolds [15] and symmetric graph [16, 17], and so forth, but also of its practical value, for example, we can apply this characteristic to study graph structure [18, 19] and chemistry structure [20]. Yet, few literature resource [21, 22] is available concerning the characteristics of positive solutions to 𝑝-Laplacian dynamic equations on time scales.

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 convenience, we think of the blanket as an assumption that 𝑎,𝑏 are points in 𝕋, for an interval (𝑎,𝑏)_𝕋 we always mean (𝑎,𝑏)∩𝕋. Other type of intervals is defined similarly.

We would like to mention the results of Sun and Li [11, 12]. In [12], Sun and Li considered the two-point BVP

(𝜑_𝑝(𝑢^Δ(𝑡)))^Δ+ℎ(𝑡)𝑓(𝑢^𝜎(𝑡))=0,𝑡∈[𝑎,𝑏]_𝕋,𝑢(𝑎)−𝐵₀(𝑢^Δ(𝑎))=0,𝑢^Δ(𝜎(𝑏))=0, (1.1) and established the existence theory for positive solutions of the above problem. They [11] also considered the 𝑚-point boundary value problem with 𝑝-Laplacian

(𝜑_𝑝(𝑢^Δ(𝑡)))^∇+ℎ(𝑡)𝑓(𝑡,𝑢(𝑡))=0,𝑡∈(0,𝑇)_𝕋,𝑢^Δ(0)=0,𝑢(𝑇)=^𝑚−2_𝑖=1𝑐_𝑖𝑢(𝜉_𝑖), (1.2) and gave the existence of single or multiple positive solutions to the above problem. The main tools used in these two papers are some fixed-point theorems [23–25].

It is also noted that the researchers mentioned above [11, 12] only considered the existence of positive solutions. As a results, they failed to further provide characteristics of solutions, such as, symmetry. Naturally, it is quite necessary to consider the characteristics of solutions to 𝑝-Laplacian dynamic equations on time scales.

Let 𝕋 be a symmetric time scale such that 0,𝑇∈𝕋. we consider the following 𝑝-Laplacian boundary value problem on time scales 𝕋 of the form:

(𝜑_𝑝(𝑢^Δ(𝑡)))^∇+ℎ(𝑡)𝑓(𝑢(𝑡))=0,𝑡∈[0,𝑇]_𝕋,𝑢(0)=𝑢(𝑇)=0,𝑢^Δ(0)=−𝑢^Δ(𝑇). (1.3) By using symmetric technique, the Krasnosel'skii's fixed point theorem [24], the generalized Avery-Henderson fixed point theorem [26], and Avery-Peterson fixed point theorem [27], we obtain the existence of at least single, twin, triple, or arbitrary odd positive symmetric solutions of problem (1.3). As applications, two examples are given to illustrate the main results and their differences. These results are even new for the special cases of continuous and discrete equations as well as in the general time-scale setting.

The rest of the paper is organized as follows. In Section 2, we present several fixed point results. In Section 3, by using Krasnosel'skii's fixed point theorem, we obtain the existence of at least single or twin positive symmetric solutions to problem (1.3). In Section 4, the existence criteria for at least triple positive or arbitrary odd positive symmetric solutions to problem (1.3) are established. In Section 5, we present two simple examples to illustrate our results.

For convenience, we now give some symmetric definitions.

Definition 1.1. The interval [0,𝑇]_𝕋 is said to be symmetric if any given 𝑡∈[0,𝑇]_𝕋, we have 𝑇−𝑡∈[0,𝑇]_𝕋.

We note that such a symmetric time scale 𝕋 exists. For example, let

𝕋={0,0.05,0.1,0.15}∪[0.22,0.44]∪{0.5,0.85,0.9,0.95,1}∪[0.56,0.78]. (1.4) It is obvious that 𝕋 is a symmetric time scale.

Definition 1.2. A function 𝑢∶[0,𝑇]_𝕋→ℝ is said to be symmetric if 𝑢 is symmetric over the interval [0,𝑇]_𝕋. That is, 𝑢(𝑡)=𝑢(𝑇−𝑡), for any given 𝑡∈[0,𝑇]_𝕋.

Definition 1.3. We say 𝑢 is a symmetric solution to problem (1.3) on [0,𝑇]_𝕋 provided that 𝑢 is a solution to boundary value problem (1.3) and is symmetric over the interval [0,𝑇]_𝕋.

Basic definitions on time scale can be found in [6, 7, 28]. Another excellent sources on dynamical systems on measure chains are the book in [29].

Throughout this paper, it is assumed that

(H1)𝑓∶[0,∞)→[0,∞) is continuous, and does not vanish identically;(H2)ℎ∈𝐶_ld([0,𝑇]_𝕋,[0,∞)) is symmetric over the interval [0,𝑇]_𝕋 and does not vanish identically on any closed subinterval of [0,𝑇]_𝕋, where 𝐶_ld([0,𝑇]_𝕋,[0,∞)) denotes the set of all left dense continuous functions from [0,𝑇]_𝕋 to [0,∞).

2. Preliminaries

Let 𝐸=𝐶_ld([0,𝑇]_𝕋,ℝ) and equip norm

‖𝑢‖=sup_{𝑡∈[0,𝑇]𝕋}|𝑢(𝑡)|, (2.1) then 𝐸 is a Banach space. Define a cone 𝑃⊂𝐸 by

𝑃={𝑢∈𝐸∣𝑢(0)=𝑢(𝑇)=0,𝑢issymmetric,nonnegative,andconcaveontheinterval[0,𝑇]_𝕋}. (2.2)

Assume that 𝑟,𝜂∈(0,𝑇/2)_𝕋 with 𝜂<𝑟. By using the symmetric and concave characters of 𝑢∈𝑃 and 𝑢(0)=𝑢(𝑇)=0, it is easy to obtain the following results.

Lemma 2.1. Assume that 𝑟,𝜂∈(0,𝑇/2)_𝕋 with 𝜂<𝑟. If 𝑢∈𝑃, then (i)𝑢(𝜂)≥(𝜂/𝑟)𝑢(𝑟);(ii)(𝑇/2)𝑢(𝑟)≥𝑟𝑢(𝑇/2).

From the previous lemma we know that ‖𝑢‖=𝑢(𝑇/2) for 𝑢∈𝑃.

The operator 𝐴∶𝑃→𝐸 is defined by

𝐴𝑢(𝑡)=^𝑡0𝜑_𝑞(_𝑠^𝑇/2𝑇ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠,𝑡∈[0,2]_𝕋,_𝑇𝑡𝜑_𝑞(^𝑠_𝑇/2𝑇ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠,𝑡∈[2,𝑇]_𝕋. (2.3) It is obvious that 𝐴 is completely continuous operator and all the fixed points of 𝐴 are the solutions to the boundary value problem (1.3).

In addition, it is easy to see that the operator 𝐴 is symmetric. In fact, for 𝑡∈[0,𝑇/2]_𝕋, we have 𝑇−𝑡∈[𝑇/2,𝑇]_𝕋, by using the integral transform, we have

𝐴𝑢(𝑇−𝑡)=^𝑇_𝑇−𝑡𝜑_𝑞(^𝑠_𝑇/2ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠_{𝑠→𝑇−𝑠}=^𝑡₀𝜑_𝑞(^𝑇−𝑠_𝑇/2ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠_{𝑟→𝑇−𝑟}=^𝑡₀𝜑_𝑞(_𝑠^𝑇/2=ℎ(𝑇−𝑟)𝑓(𝑢(𝑇−𝑟))∇𝑟)Δ𝑠_𝑡0𝜑_𝑞(_𝑠^𝑇/2ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠=𝐴𝑢(𝑡). (2.4) Hence, 𝐴 is symmetric.

Now, we provide some background material from the theory of cones in Banach spaces [24, 26, 27, 30], and then state several fixed point theorems needed later.

Firstly, we list the Krasnosel'skii's fixed point theorem [24].

Lemma 2.2 (see [24]). Let 𝑃 be a cone in a Banach space 𝐸. Assume that Ω₁, Ω₂ are open 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‖, forall𝑥∈𝑃∩𝜕Ω₁ and ‖𝐴𝑥‖≥‖𝑥‖, forall𝑥∈𝑃∩𝜕Ω₂ or(ii)‖𝐴𝑥‖≥‖𝑥‖, forall𝑥∈𝑃∩𝜕Ω₁ and ‖𝐴𝑥‖≤‖𝑥‖, forall𝑥∈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}.

Secondly, we state the generalized Avery-Henderson fixed point theorem [26].

Lemma 2.3 (see [26]). Let 𝑃 be a cone in a real Banach space 𝐸. Let 𝛼, 𝛽, and 𝛾 be increasing, nonnegative continuous functional on 𝑃 such that for some 𝑐>0 and 𝐻>0,𝛾(𝑥)≤𝛽(𝑥)≤𝛼(𝑥) and ‖𝑥‖≤𝐻𝛾(𝑥) for all 𝑥∈𝑃(𝛾,𝑐). Suppose that there exist positive numbers 𝑎 and 𝑏 with 𝑎<𝑏<𝑐 and 𝐴∶𝑃(𝛾,𝑐)→𝑃 is a completely continuous operator such that (i)𝛾(𝐴𝑥)<𝑐 for all 𝑥∈𝜕𝑃(𝛾,𝑐);(ii)𝛽(𝐴𝑥)>𝑏 for all 𝑥∈𝜕𝑃(𝛽,𝑏);(iii)𝑃(𝛼,𝑎)≠∅ and 𝛼(𝐴𝑥)<𝑎 for 𝑥∈𝜕𝑃(𝛼,𝑎),then 𝐴 has at least three fixed points 𝑥₁, 𝑥₂, and 𝑥₃ belonging to 𝑃(𝛾,𝑐) such that 0≤𝛼(𝑥₁)<𝑎<𝛼(𝑥₂)with𝛽(𝑥₂)<𝑏<𝛽(𝑥₃),𝛾(𝑥₃)<𝑐. (2.5)

The following lemma can be found in [21].

Lemma 2.4 (see [21]). Let 𝑃 be a cone in a real Banach space 𝐸. Let 𝛼, 𝛽, and 𝛾 be increasing, nonnegative continuous functional on 𝑃 such that for some 𝑐>0 and 𝐻>0,𝛾(𝑥)≤𝛽(𝑥)≤𝛼(𝑥) and ‖𝑥‖≤𝐻𝛾(𝑥) for all 𝑥∈𝑃(𝛾,𝑐). Suppose that there exist positive numbers 𝑎 and 𝑏 with 𝑎<𝑏<𝑐 and 𝐴∶𝑃(𝛾,𝑐)→𝑃 is a completely continuous operator such that:(i)𝛾(𝐴𝑥)>𝑐 for all 𝑥∈𝜕𝑃(𝛾,𝑐);(ii)𝛽(𝐴𝑥)<𝑏 for all 𝑥∈𝜕𝑃(𝛽,𝑏);(iii)𝑃(𝛼,𝑎)≠∅ and 𝛼(𝐴𝑥)>𝑎 for 𝑥∈𝜕𝑃(𝛼,𝑎),then 𝐴 has at least three fixed points 𝑥₁,𝑥₂, and 𝑥₃ belonging to 𝑃(𝛾,𝑐) such that 0≤𝛼(𝑥₁)<𝑎<𝛼(𝑥₂)with𝛽(𝑥₂)<𝑏<𝛽(𝑥₃),𝛾(𝑥₃)<𝑐. (2.6)

Let 𝛽 and 𝜙 be nonnegative continuous convex functionals on 𝑃, 𝜆 is a nonnegative continuous concave functional on 𝑃, and 𝜑 is a nonnegative continuous functional, respectively on 𝑃. We define the following convex sets:

𝑃(𝜙,𝜆,𝑏,𝑑)={𝑥∈𝑃∶𝑏≤𝜆(𝑥),𝜙(𝑥)≤𝑑},𝑃(𝜙,𝛽,𝜆,𝑏,𝑐,𝑑)={𝑥∈𝑃∶𝑏≤𝜆(𝑥),𝛽(𝑥)≤𝑐,𝜙(𝑥)≤𝑑}, (2.7) and a closed set 𝑅(𝜙,𝜑,𝑎,𝑑)={𝑥∈𝑃∶𝑎≤𝜑(𝑥),𝜙(𝑥)≤𝑑}.

Finally, we list the fixed point theorem due to Avery-Peterson [27].

Lemma 2.5 (see [27]). Let 𝑃 be a cone in a real Banach space 𝐸 and 𝛽,𝜙,𝜆,𝜑 defined as above, moreover, 𝜑 satisfies 𝜑(𝜆^′𝑥)≤𝜆^′𝜑(𝑥) for 0≤𝜆^′≤1 such that, for some positive numbers ℎ and 𝑑, 𝜆(𝑥)≤𝜑(𝑥),‖𝑥‖≤ℎ𝜙(𝑥) (2.8) for all 𝑥∈𝑃(𝜙,𝑑). Suppose that 𝐴∶𝑃(𝜙,𝑑)→𝑃(𝜙,𝑑) is completely continuous and there exist positive real numbers 𝑎, 𝑏, 𝑐, with 𝑎<𝑏 such that (i){𝑥∈𝑃(𝜙,𝛽,𝜆,𝑏,𝑐,𝑑)∶𝜆(𝑥)>𝑏}≠∅ and 𝜆(𝐴(𝑥))>𝑏 for 𝑥∈𝑃(𝜙,𝛽,𝜆,𝑏,𝑐,𝑑);(ii)𝜆(𝐴(𝑥))>𝑏 for 𝑥∈𝑃(𝜙,𝜆,𝑏,𝑑) with 𝛽(𝐴(𝑥))>𝑐;(iii)0∉𝑅(𝜙,𝜑,𝑎,𝑑) and 𝜆(𝐴(𝑥))<𝑎 for all 𝑥∈𝑅(𝜙,𝜑,𝑎,𝑑) with 𝜑(𝑥)=𝑎,then 𝐴 has at least three fixed points 𝑥₁,𝑥₂,𝑥₃∈𝑃(𝜙,𝑑) such that 𝜙(𝑥_𝑖)≤𝑑for𝑖=1,2,3,𝑏<𝜆(𝑥₁),𝑎<𝜑(𝑥₂),𝜆(𝑥₂)<𝑏with𝜑(𝑥₃)<𝑎. (2.9)

3. Single or Twin Solutions

Let

𝑓₀=lim_𝑢→0+𝑓(𝑢)𝜑_𝑝(𝑢),𝑓_∞=lim_𝑢→∞𝑓(𝑢)𝜑_𝑝.(𝑢) (3.1) We define 𝑖₀=number of zeros in the set {𝑓₀,𝑓_∞} and 𝑖_∞=number of infinities in the set {𝑓₀,𝑓_∞}. Clearly, 𝑖₀, 𝑖_∞=0,1, or 2 and there exist six possible cases: (i) 𝑖₀=1 and 𝑖_∞=1; (ii) 𝑖₀=0 and 𝑖_∞=0; (iii) 𝑖₀=0 and 𝑖_∞=1; (iv) 𝑖₀=0 and 𝑖_∞=2; (v) 𝑖₀=1 and 𝑖_∞=0; (vi) 𝑖₀=2 and 𝑖_∞=0. In the following, by using Krasnosel'skii's fixed point theorem in a cone, we study the existence of positive symmetric solutions to problem (1.3) under the above six possible cases.

3.1. For the Case 𝑖₀=1 and 𝑖_∞=1

In this subsection, we discuss the existence of single positive symmetric solution of the problem (1.3) under 𝑖₀=1 and 𝑖_∞=1.

Theorem 3.1. Problem (1.3) has at least one positive symmetric solution in the case 𝑖₀=1 and 𝑖_∞=1.

Proof. We divide the proof into two cases.
Case 1 (𝑓₀=0 and 𝑓_∞=∞). In view of 𝑓₀=0, there exists an 𝐻₁>0 such that 𝑓(𝑢)≤𝜑_𝑝(𝜀)𝜑_𝑝(𝑢)=𝜑_𝑝(𝜀𝑢) for 𝑢∈(0,𝐻₁], where 𝜀 arbitrary small and satisfies 0<(𝜀𝑇/2)𝜑_𝑞(∫₀^𝑇/2ℎ(𝑟)∇𝑟)≤1.
If 𝑢∈𝑃 with ‖𝑢‖=𝐻₁, then ‖𝐴𝑢‖=sup_{𝑡∈[0,𝑇]𝕋}𝑇|𝐴𝑢|=𝐴𝑢(2)=₀^𝑇/2𝜑_𝑞(_𝑠^𝑇/2ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠≤𝜀‖𝑢‖₀^𝑇/2𝜑_𝑞(₀^𝑇/2ℎ(𝑟)∇𝑟)Δ𝑠≤‖𝑢‖𝜀𝑇2𝜑_𝑞(₀^𝑇/2ℎ(𝑟)∇𝑟)≤‖𝑢‖. (3.2) We let Ω_𝐻1={𝑢∈𝐸∶‖𝑢‖<𝐻₁}, then ‖𝐴𝑢‖≤‖𝑢‖ for 𝑢∈𝑃∩𝜕Ω_𝐻1.
From 𝑓_∞=∞, there exists an 𝐻^′₂>0 such that 𝑓(𝑢)≥𝜑_𝑝(𝑘)𝜑_𝑝(𝑢)=𝜑_𝑝(𝑘𝑢) for 𝑢∈[𝐻^′₂,∞), where 𝑘>0, and satisfies the following inequality: 2𝑘𝜂²𝑇𝜑_𝑞(_𝜂^𝑇/2ℎ(𝑟)∇𝑟)≥1. (3.3) Set 𝐻₂=max{2𝐻₁,𝐻^′₂𝑇2𝜂},Ω_𝐻2={𝑢∈𝐸∶‖𝑢‖<𝐻₂}. (3.4) If 𝑢∈𝑃 with ‖𝑢‖=𝐻₂, then, by Lemma 2.1, one has min_{𝑡∈[𝜂,𝑇/2]𝕋}𝑢(𝑡)≥2𝜂𝑇𝑇𝑢(2)≥𝐻^′₂. (3.5) For 𝑢∈𝑃∩𝜕Ω_𝐻2, in terms of (3.3) and (3.5), we get ‖𝐴𝑢‖=sup_{𝑡∈[0,𝑇]𝕋}=|𝐴𝑢|≥𝐴𝑢(𝜂)_𝜂0𝜑_𝑞(_𝑠^𝑇/2≥ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠_𝜂0𝜑_𝑞(_𝜂^𝑇/2≥ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠_𝜂0𝜑_𝑞(_𝜂^𝑇/2ℎ(𝑟)𝜑_𝑝≥(𝑘𝑢(𝑟))∇𝑟)Δ𝑠2𝑘𝜂²𝑇‖𝑢‖𝜑_𝑞(_𝜂^𝑇/2ℎ(𝑟)∇𝑟)≥‖𝑢‖. (3.6) Thus, by (i) of Lemma 2.2, problem (1.3) has at least single positive symmetric solution 𝑢 in 𝑃∩(Ω_𝐻2⧵Ω_𝐻1) with 𝐻₁≤‖𝑢‖≤𝐻₂.Case 2 (𝑓₀=∞ and 𝑓_∞=0). Since 𝑓₀=∞, there exists an 𝐻₃>0 such that 𝑓(𝑢)≥𝜑_𝑝(𝑚)𝜑_𝑝(𝑢)=𝜑_𝑝(𝑚𝑢) for 𝑢∈(0,𝐻₃], where 𝑚 is such that 2𝑚𝜂²𝑇𝜑_𝑞(_𝜂^𝑇/2ℎ(𝑟)∇𝑟)≥1. (3.7) If 𝑢∈𝑃 with ‖𝑢‖=𝐻₃, then, by (3.7), one has ‖𝐴𝑢‖=sup_{𝑡∈[0,𝑇]𝕋}≥|𝐴𝑢|_𝜂0𝜑_𝑞(_𝜂^𝑇/2ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠≥𝜂𝜑_𝑞(_𝜂^𝑇/2ℎ(𝑟)𝜑_𝑝(𝑚𝑢(𝑟))∇𝑟)≥𝜂𝑚2𝜂𝑇‖𝑢‖𝜑_𝑞(_𝜂^𝑇/2ℎ(𝑟)∇𝑟)≥‖𝑢‖. (3.8) If we let Ω_𝐻3={𝑢∈𝐸∶‖𝑢‖<𝐻₃}, then ‖𝐴𝑢‖≥‖𝑢‖ for 𝑢∈𝑃∩𝜕Ω_𝐻3.
Now, we consider 𝑓_∞=0. By definition, there exists 𝐻^′₄>0 such that 𝑓(𝑢)≤𝜑_𝑝(𝛿)𝜑_𝑝(𝑢)=𝜑_𝑝(𝛿𝑢)for𝑢∈[𝐻^′₄,∞), (3.9) where 𝛿>0 satisfies 𝛿𝑇2𝜑_𝑞(₀^𝑇/2ℎ(𝑟)∇𝑟)≤1. (3.10)
Suppose that 𝑓 is bounded, then 𝑓(𝑢)≤𝜑_𝑝(𝐾) for all 𝑢∈[0,∞) and some constant 𝐾>0. Pick 𝐻₄=max{2𝐻₃,𝐾𝑇2𝜑_𝑞(₀^𝑇/2ℎ(𝑠)Δ𝑠)}. (3.11) If 𝑢∈𝑃 with ‖𝑢‖=𝐻₄, then 𝑇‖𝐴𝑢‖=𝐴𝑢(2)≤₀^𝑇/2𝜑_𝑞(₀^𝑇/2𝑇ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠≤𝐾2𝜑_𝑞(₀^𝑇/2ℎ(𝑠)Δ𝑠)≤𝐻₄=‖𝑢‖. (3.12)
Suppose that 𝑓 is unbounded. From 𝑓∈𝐶([0,+∞),[0,+∞)), we have 𝑓(𝑢)≤𝐶₃ for arbitrary 𝑢∈[0,𝐶₄], here 𝐶₃ and 𝐶₄ are arbitrary positive constants. This implies that 𝑓(𝑢)→+∞ if 𝑢→+∞. Hence, it is easy to know that there exists 𝐻₄≥max{2𝐻₃,(𝑇/2𝜂)𝐻^′₄} such that 𝑓(𝑢)≤𝑓(𝐻₄) for 𝑢∈[0,𝐻₄]. If 𝑢∈𝑃 with ‖𝑢‖=𝐻₄, then by using (3.9) and (3.10), we have 𝑇‖𝐴𝑢‖=𝐴𝑢(2)=₀^𝑇/2𝜑_𝑞(_𝑠^𝑇/2≤ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠_0𝑇/2𝜑_𝑞(₀^𝑇/2ℎ(𝑟)𝑓(𝐻₄)∇𝑟)Δ𝑠≤𝛿𝐻₄𝑇2𝜑_𝑞(₀^𝑇/2ℎ(𝑟)∇𝑟)≤‖𝑢‖. (3.13) Consequently, in either case, if we take Ω_𝐻4={𝑢∈𝐸∶‖𝑢‖<𝐻₄}, then, for 𝑢∈𝑃∩𝜕Ω_𝐻4, we have ‖𝐴𝑢‖≤‖𝑢‖. Thus, condition (ii) of Lemma 2.2 is satisfied. Consequently, problem (1.3) has at least single positive symmetric solution 𝑢 in 𝑃∩(Ω_𝐻4⧵Ω_𝐻3) with 𝐻₃≤‖𝑢‖≤𝐻₄. The proof is complete.

3.2. For the Case 𝑖₀=0 and 𝑖_∞=0

In this subsection, we discuss the existence of positive symmetric solutions to problems (1.3) under 𝑖₀=0 and 𝑖_∞=0 .

First, we will state and prove the following main result of problem (1.3).

Theorem 3.2. Suppose that the following conditions hold:(i)there exists constant 𝑝^′>0 such that 𝑓(𝑢)≤𝜑_𝑝(𝑝^′Λ₁) for 𝑢∈[0,𝑝^′], where Λ₁={(𝑇/2)𝜑_𝑞(∫₀^𝑇/2ℎ(𝑟)∇𝑟)}⁻¹;(ii)there exists constant 𝑞^′>0 such that 𝑓(𝑢)≥𝜑_𝑝(𝑞^′Λ₂) for 𝑢∈[(2𝜂/𝑇)𝑞^′,𝑞^′], where Λ₂={𝜂𝜑_𝑞(∫_𝜂^𝑇/2ℎ(𝑟)∇𝑟)}⁻¹, furthermore, 𝑝^′≠𝑞^′,then problem (1.3) has at least one positive symmetric solution 𝑢 such that ‖𝑢‖ lies between 𝑝^′ and 𝑞^′.

Proof. Without loss of generality, we may assume that 𝑝^′<𝑞^′.
Let Ω_𝑝′={𝑢∈𝐸∶‖𝑢‖<𝑝^′}. For any 𝑢∈𝑃∩𝜕Ω_𝑝′, in view of condition (i), we have 𝑇‖𝐴𝑢‖=𝐴𝑢(2)=₀^𝑇/2𝜑_𝑞(_𝑠^𝑇/2ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠≤𝑝^′Λ₁𝑇2𝜑_𝑞(₀^𝑇/2ℎ(𝑟)∇𝑟)=𝑝^′, (3.14) which yields ‖𝐴𝑢‖≤‖𝑢‖for𝑢∈𝑃∩𝜕Ω_𝑝′. (3.15)
Now, set Ω_𝑞′={𝑢∈𝐸∶‖𝑢‖<𝑞^′}. For 𝑢∈𝑃∩𝜕Ω_𝑞′, Lemma 2.1 implies that 2𝜂𝑇𝑞^′≤𝑢(𝑡)≤𝑞^′for𝑇𝑡∈[𝜂,2]_𝕋. (3.16) Hence, by condition (ii) we get 𝑇‖𝐴𝑢‖=𝐴𝑢(2)≥^𝜂₀𝜑_𝑞(_𝜂^𝑇/2ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠≥𝑞^′Λ₂𝜂𝜑_𝑞(_𝜂^𝑇/2ℎ(𝑟)∇𝑟)=𝑞^′. (3.17) So, if we take Ω_𝑞′={𝑢∈𝐸∶‖𝑢‖<𝑞^′}, then ‖𝐴𝑢‖≥‖𝑢‖,𝑢∈𝑃∩𝜕Ω_𝑞′. (3.18) Consequently, in view of 𝑝^′<𝑞^′, (3.15) and (3.18), it follows from Lemma 2.2 that problem (1.3) has a positive symmetric solution 𝑢 in 𝑃∩(Ω_𝑞′⧵Ω_𝑝′). The proof is complete.

3.3. For the Case 𝑖₀=1 and 𝑖_∞=0 or 𝑖₀=0 and 𝑖_∞=1

In this subsection, under the conditions 𝑖₀=1 and 𝑖_∞=0 or 𝑖₀=0 and 𝑖_∞=1, we discuss the existence of positive symmetric solutions to problem (1.3).

Theorem 3.3. Suppose that 𝑓₀∈[0,𝜑_𝑝(Λ₁)) and 𝑓_∞∈(𝜑_𝑝((𝑇/2𝜂)Λ₂),∞) hold. Then problem (1.3) has at least one positive symmetric solution.

Proof. It is easy to see that under the assumptions, conditions (i) and (ii) in Theorem 3.2 are satisfied. So the proof is easy and we omit it here.

Theorem 3.4. Suppose that 𝑓₀∈(𝜑_𝑝((𝑇/2𝜂)Λ₂),∞) and 𝑓_∞∈[0,𝜑_𝑝(Λ₁)) hold, then problem (1.3) has at least one positive symmetric solution.

Proof. Firstly, let 𝜀₁=𝑓₀−𝜑_𝑝((𝑇/2𝜂)Λ₂)>0, there exists a sufficiently small 𝑞^′>0 that satisfies 𝑓(𝑢)𝜑_𝑝(𝑢)≥𝑓₀−𝜀₁=𝜑_𝑝(𝑇Λ2𝜂²)for𝑢∈(0,𝑞^′]. (3.19) Thus, 𝑢∈[(2𝜂/𝑇)𝑞^′,𝑞^′], we have 𝑓(𝑢)≥𝜑_𝑝(𝑇Λ2𝜂²)𝜑_𝑝(𝑢)≥𝜑_𝑝(Λ₂𝑞^′), (3.20) which implies that condition (ii) in Theorem 3.2 holds.
Nextly, for 𝜀₂=𝜑_𝑝(Λ₁)−𝑓_∞>0, there exists a sufficiently large 𝑝^′′(>𝑞^′) such that 𝑓(𝑢)𝜑_𝑝(𝑢)≤𝑓_∞+𝜀₂=𝜑_𝑝(Λ₁)for𝑢∈[𝑝^′′,∞). (3.21) We consider two cases.
Case 1. Assume that 𝑓 is bounded, that is, 𝑓(𝑢)≤𝜑_𝑝(𝐾₁)for𝑢∈[0,∞), (3.22) here 𝐾₁>0 some constant. If we take sufficiently large 𝑝^′ such that 𝑝^′≥max{𝐾₁/Λ₁,𝑝^′′}, then 𝑓(𝑢)≤𝜑_𝑝(𝐾₁)≤𝜑_𝑝(Λ₁𝑝^′)for𝑢∈[0,𝑝^′]. (3.23) Consequently, from the above inequality, condition (i) of Theorem 3.2 is true.Case 2. Assume that 𝑓 is unbounded.
From 𝑓∈𝐶([0,∞),[0,∞)), there exists 𝑝^′>𝑝^′′ such that 𝑓(𝑢)≤𝑓(𝑝^′)for𝑢∈[0,𝑝^′]. (3.24) Since 𝑝^′>𝑝^′′, by (3.21), we get 𝑓(𝑝^′)≤𝜑_𝑝(Λ₁𝑝^′), hence 𝑓(𝑢)≤𝑓(𝑝^′)≤𝜑_𝑝(Λ₁𝑝^′)for𝑢∈[0,𝑝^′]. (3.25) Thus, condition (i) of Theorem 3.2 is fulfilled.
Consequently, Theorem 3.2 implies that the conclusion of this theorem holds.

From the proof of Theorems 3.1 and 3.2, respectively, we have the following two results.

Corollary 3.5. Suppose that 𝑓₀=0 and condition (ii) in Theorem 3.2 hold, then problem (1.3) has at least one positive symmetric solution.

Corollary 3.6. Suppose that 𝑓_∞=0 and condition (ii) in Theorem 3.2 hold, then problem (1.3) has at least one positive symmetric solution.

Theorem 3.7. Suppose that 𝑓₀∈(0,𝜑_𝑝(Λ₁)) and 𝑓_∞=∞ hold, then problem (1.3) has at least one positive symmetric solution.

Proof. First, in view of 𝑓_∞=∞, then by inequality (3.7), we have ‖𝐴𝑢‖≥‖𝑢‖ for 𝑢∈𝑃∩𝜕Ω_𝐻2. Next, by 𝑓₀∈(0,𝜑_𝑝(Λ₁)), for 𝜀₃=𝜑_𝑝(Λ₁)−𝑓₀>0, there exists a sufficiently small 𝑝^′∈(0,𝐻₂) such that 𝑓(𝑢)≤(𝑓₀+𝜀₃)𝜑_𝑝(𝑢)=𝜑_𝑝(Λ₁𝑢)≤𝜑_𝑝(Λ₁𝑝^′)for𝑢∈[0,𝑝^′], (3.26) which implies that (i) of Theorem 3.2 holds, that is, (3.14) is true. Hence, we obtain ‖𝐴𝑢‖≤‖𝑢‖ for 𝑢∈𝑃∩𝜕Ω_𝑝′. The result is obtained and the proof is complete.

Theorem 3.8. Suppose that 𝑓₀=∞ and 𝑓_∞∈(0,𝜑_𝑝(Λ₁)) hold, then problem (1.3) has at least one positive symmetric solution.

Proof. On one hand, since 𝑓₀=∞, by inequality (3.9), one gets ‖𝐴𝑢‖≥‖𝑢‖, 𝑢∈𝑃∩𝜕Ω_𝐻3. On the other hand, since 𝑓_∞∈(0,𝜑_𝑝(Λ₁)), from the technique similar to the second part proof in Theorem 3.4, one obtains that condition (i) of Theorem 3.2 is satisfied, that is, inequality (3.14) holds, one has ‖𝐴𝑢‖≤‖𝑢‖, 𝑢∈𝑃∩𝜕Ω_𝑝′, where 𝑝^′>𝐻₃. Hence, problem (1.3) has at least one positive symmetric solution. The proof is complete.

From Theorems 3.7 and 3.8, respectively, it is easy to obtain the following two corollaries.

Corollary 3.9. Assume that 𝑓_∞=∞ and condition (i) in Theorem 3.2 hold, then problem (1.3) has at least one positive symmetric solution.

Corollary 3.10. Assume that 𝑓₀=∞ and condition (i) in Theorem 3.2 hold, then problem (1.3) has at least one positive symmetric solution.

3.4. For the Case 𝑖₀=0 and 𝑖_∞=2 or 𝑖₀=2 and 𝑖_∞=0

In this subsection, under 𝑖₀=0 and 𝑖_∞=2 or 𝑖₀=2 and 𝑖_∞=0, we study the existence of multiple positive solutions to problems (1.3).

Combining the proofs of Theorems 3.1 and 3.2, it is easy to prove the following two theorems.

Theorem 3.11. Suppose that 𝑖₀=0 and 𝑖_∞=2 and condition (i) of Theorem 3.2 hold, then problem (1.3) has at least two positive solutions 𝑢₁,𝑢₂∈𝑃 such that 0<‖𝑢₁‖<𝑝^′<‖𝑢₂‖.

Theorem 3.12. Suppose that 𝑖₀=2 and 𝑖_∞=0 and condition (ii) of Theorem 3.2 hold, then problem (1.3) has at least two positive solutions 𝑢₁,𝑢₂∈𝑃 such that 0<‖𝑢₁‖<𝑞^′<‖𝑢₂‖.

4. Triple Solutions

In the previous section, we have obtained some results on the existence of at least single or twin positive symmetric solutions to problem (1.3). In this section, we will further discuss the existence of positive symmetric solutions to problem (1.3) by using two different methods. And the conclusions we will arrive at are different with their own distinctive advantages.

Based on the obtained symmetric solution position and local properties, we can only get some local properties of solutions by using method one; however, the position of solutions is not determined. In contrast, by means of method two, we cannot only get some local properties of solutions but also give the position of all solutions, with regard to some subsets of the cone, which has to meet some conditions which are stronger than those of method one. Obviously, the local properties of obtained solutions are different by using the two different methods. Hence, it is convenient for us to comprehensively comprehend the solutions of the models by using the two different techniques.

In Section 5, two examples are given to illustrate the differences of the results obtained by the two different methods.

For the notational convenience, we denote

𝑀_𝜉=𝜂𝜑_𝑞(₀^𝑇/2ℎ(𝑟)∇𝑟),𝑁_𝜉=𝜂𝜑_𝑞(_𝜂^𝑇/2𝐿ℎ(𝑟)∇𝑟),_𝜉=𝑟𝜑_𝑞(₀^𝑇/2ℎ(𝑟)∇𝑟),𝐿_𝜃=𝑟𝜑_𝑞(_𝑟^𝑇/2ℎ(𝑟)∇𝑟),𝑊_𝜉=𝑇2𝜑_𝑞(₀^𝑇/2ℎ(𝑟)∇𝑟). (4.1)

4.1. Result 1

In this subsection, in view of the generalized Avery-Henderson fixed-point theorem [26], the existence criteria for at least triple and arbitrary odd positive symmetric solutions to problems (1.3) are established.

For 𝑢∈𝑃, we define the nonnegative, increasing, continuous functionals 𝛾,𝛽, and 𝛼 by

𝛾(𝑢)=max_{𝑡∈[0,𝜂]𝕋}𝑢(𝑡)=𝑢(𝜂),𝛽(𝑢)=min_{𝑡∈[𝜂,𝑇/2]𝕋}𝑢(𝑡)=𝑢(𝜂),𝛼(𝑢)=max_{𝑡∈[0,𝑟]𝕋}𝑢(𝑡)=𝑢(𝑟). (4.2) It is obvious that 𝛾(𝑢)≤𝛽(𝑢)≤𝛼(𝑢) for each 𝑢∈𝑃. By Lemma 2.1, one obtains ‖𝑢‖≤𝐶^∗𝛾(𝑢) for all 𝑢∈𝑃, here 𝐶^∗=𝑇/2𝜂.

We now present the results in this subsection.

Theorem 4.1. If there are positive numbers 𝑎^′, 𝑏^′, 𝑐^′ such that 𝑎^′<(2𝑟/𝑇)𝑏^′<(2𝑟/𝑇)(𝑐^′𝑁_𝜉/𝑀_𝜉). In addition, 𝑓(𝑢) satisfies the following conditions:(i)𝑓(𝑢)<𝜑_𝑝(𝑐^′/𝑀_𝜉) for 𝑢∈[0,(𝑇/2𝜂)𝑐^′];(ii)𝑓(𝑢)>𝜑_𝑝(𝑏^′/𝑁_𝜉) for 𝑢∈[𝑏^′,(𝑇/2𝜂)𝑏^′];(iii)𝑓(𝑢)<𝜑_𝑝(𝑎^′/𝐿_𝜉) for 𝑢∈[0,(𝑇/2𝑟)𝑎^′].Then problem (1.3) has at least three positive symmetric solutions 𝑢₁, 𝑢₂, and 𝑢₃ such that 0<max_{𝑡∈[0,𝑟]𝕋}𝑢₁(𝑡)<𝑎^′<max_{𝑡∈[0,𝑟]𝕋}𝑢₂(𝑡),min_{𝑡∈[𝜂,𝑇/2]𝕋}𝑢₂(𝑡)<𝑏^′<min_{𝑡∈[𝜂,𝑇/2]𝕋}𝑢₃(𝑡),max_{𝑡∈[0,𝜂]𝕋}𝑢₃(𝑡)<𝑐^′. (4.3)

Proof. By the definition of completely continuous operator 𝐴 and its properties, it has to be demonstrated that all the conditions of Lemma 2.3 hold with respect to 𝐴. It is easy to obtain that 𝐴∶𝑃(𝛾,𝑐)→𝑃.
Firstly, we verify that if 𝑢∈𝜕𝑃(𝛾,𝑐^′), then 𝛾(𝐴𝑢)<𝑐^′.
If 𝑢∈𝜕𝑃(𝛾,𝑐^′), then 𝛾(𝑢)=max_{𝑡∈[0,𝜂]𝕋}𝑢(𝑡)=𝑢(𝜂)=𝑐^′. (4.4) Lemma 2.1 implies that 𝑇‖𝑢‖≤𝑇2𝜂𝑢(𝜂)=𝑐2𝜂^′, (4.5) we have 𝑇0≤𝑢(𝑡)≤𝑐2𝜂^′𝑇,𝑡∈[0,2]_𝕋. (4.6) Thus, by condition (i), one has 𝛾(𝐴𝑢)=max_{𝑡∈[0,𝜂]𝕋}=𝐴𝑢(𝑡)=𝐴𝑢(𝜂)_𝜂0𝜑_𝑞(_𝑠^𝑇/2ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠≤^𝜂₀𝜑_𝑞(₀^𝑇/2ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠<𝜂𝜑_𝑞(₀^𝑇/2ℎ(𝑟)𝜑_𝑝(𝑐^′𝑀_𝜉=𝑐)∇𝑟)_′𝑀_𝜉𝜂𝜑_𝑞(₀^𝑇/2ℎ(𝑟)∇𝑟)=𝑐^′. (4.7)
Secondly, we show that 𝛽(𝐴𝑢)>𝑏^′ for 𝑢∈𝜕𝑃(𝛽,𝑏^′).
If we choose 𝑢∈𝜕𝑃(𝛽,𝑏^′), then 𝛽(𝑢)=min_{𝑡∈[𝜂,𝑇/2]𝕋}𝑢(𝑡)=𝑏^′. In view of Lemma 2.1, we have 𝑇‖𝑢‖≤𝑇2𝜂𝑢(𝜂)=𝑏2𝜂^′. (4.8) So 𝑏^′𝑇≤𝑢(𝑡)≤𝑏2𝜂^′𝑇,𝑡∈[𝜂,2]_𝕋. (4.9) Using condition (ii), we get ≥𝛽(𝐴𝑢)=𝐴𝑢(𝜂)_𝜂0𝜑_𝑞(_𝜂^𝑇/2ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠>𝜂𝜑_𝑞(_𝜂^𝑇/2ℎ(𝑟)𝜑_𝑝(𝑏^′𝑁_𝜉>𝑏)∇𝑟)_′𝑁_𝜉𝜂𝜑_𝑞(_𝜂^𝑇/2ℎ(𝑟)∇𝑟)=𝑏^′. (4.10)
Finally, we prove that 𝑃(𝛼,𝑎^′)≠∅ and 𝛼(𝐴𝑢)<𝑎^′ for all 𝑢∈𝜕𝑃(𝛼,𝑎^′).
In fact, the constant function 𝑎^′/2∈𝑃(𝛼,𝑎^′). Moreover, for 𝑢∈𝜕𝑃(𝛼,𝑎^′), we have 𝛼(𝑢)=max_{𝑡∈[0,𝑟]𝕋}𝑢(𝑡)=𝑎^′, which implies 0≤𝑢(𝑡)≤𝑎^′ for 𝑡∈[0,𝑟]_𝕋. Hence, ‖𝑢‖≤(𝑇/2𝑟)𝑢(𝑟). Therefore 𝑇0≤𝑢(𝑡)≤𝑎2𝑟^′𝑇,𝑡∈[0,2]_𝕋. (4.11) By using assumption (iii), one has =𝛼(𝐴𝑢)=(𝐴𝑢)(𝑟)_𝑟0𝜑_𝑞(_𝑠^𝑇/2≤ℎ(𝑟)𝑓(𝑢(𝑟)))∇𝑟Δ𝑠_𝑟0𝜑_𝑞(₀^𝑇/2≤ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠_𝑟0𝜑_𝑞(₀^𝑇/2ℎ(𝑟)𝜑_𝑝(𝑎^′𝐿_𝜉=𝑎)∇𝑟)Δ𝑠_′𝐿_𝜉𝑟𝜑_𝑞(₀^𝑇/2ℎ(𝑟)∇𝑟)=𝑎^′. (4.12) Thus, all the conditions in Lemma 2.3 are satisfied. From (H1) and (H2), we have that the solutions to problem (1.3) do not vanish identically on any closed subinterval of [0,𝑇]_𝕋. Consequently, problem (1.3) has at least three positive symmetric solutions 𝑢₁, 𝑢₂, and 𝑢₃ belonging to 𝑃(𝛾,𝑐^′), and satisfying (4.3). The proof is complete.

From Theorem 4.1, we see that, when assumptions as (i), (ii), and (iii) are imposed appropriately on 𝑓, we can establish the existence of an arbitrary odd number of positive symmetric solutions to problem (1.3).

Theorem 4.2. Let 𝑙=1,2,…,𝑛. Suppose that there exist positive numbers 𝑎^′_𝑠𝑙, 𝑏^′_𝑠𝑙, 𝑐^′_𝑠𝑙 such that 𝑎^′_𝑠1<2𝑟𝑇𝑏^′_𝑠1<2𝑟𝑇𝑁_𝜉𝑀_𝜉𝑐^′_𝑠1<𝑎^′_𝑠2<2𝑇𝑟𝑏^′_𝑠2<2𝑟𝑇𝑁_𝜉𝑀_𝜉𝑐^′_𝑠2<𝑎^′_𝑠3<⋯<𝑎^′_𝑠𝑙. (4.13) In addition, 𝑓(𝑢) satisfies the following conditions:(i)𝑓(𝑢)<𝜑_𝑝(𝑐^′_𝑠𝑙/𝑀_𝜉) for 𝑢∈[0,(𝑇/2𝜂)𝑐^′_𝑠𝑙];(ii)𝑓(𝑢)>𝜑_𝑝(𝑏^′_𝑠𝑙/𝑁_𝜉) for 𝑢∈[𝑏^′_𝑠𝑙,(𝑇/2𝜂)𝑏^′_𝑠𝑙];(iii)𝑓(𝑢)<𝜑_𝑝(𝑎^′_𝑠𝑙/𝐿_𝜉) for 𝑢∈[0,(𝑇/2𝑟)𝑎^′_𝑠𝑙].Then problem (1.3) has at least 2𝑙+1 positive symmetric solutions.

Proof. When 𝑙=1, it is clear that Theorem 4.1 holds. Then we can obtain at least three positive symmetric solutions 𝑢₁,𝑢₂, and 𝑢₃ satisfying 0<max_{𝑡∈[0,𝑟]𝕋}𝑢₁(𝑡)<𝑎_𝑠1<max_{𝑡∈[0,𝑟]𝕋}𝑢₂(𝑡),min_{𝑡𝑖∈[𝜂,𝑇/2]𝕋}𝑢₂(𝑡)<𝑏_𝑠1<min_{𝑡𝑖∈[𝜂,𝑇/2]𝕋}𝑢₃(𝑡),max_{𝑡∈[0,𝜂]𝕋}𝑢₃(𝑡)<𝑐_𝑠1. (4.14) Following this way, we finish the proof by induction. The proof is complete.

Using Lemma 2.4, it is easy to have the following results.

Theorem 4.3. Suppose that there are positive numbers 𝑎^′, 𝑏^′, 𝑐^′ such that 𝑎^′<(𝐿_𝜃/𝑀_𝜉)𝑏^′<(2𝜂/𝑇)(𝐿_𝜃/𝑀_𝜉)𝑐^′. In addition, 𝑓(𝑢) satisfies the following conditions:(i)𝑓(𝑢)>𝜑_𝑝(𝑐^′/𝑁_𝜉) for 𝑢∈[𝑐^′,(𝑇/2𝜂)𝑐^′];(ii)𝑓(𝑢)<𝜑_𝑝(𝑏^′/𝑀_𝜉) for 𝑢∈[0,(𝑇/2𝜂)𝑏^′];(iii)𝑓(𝑢)>𝜑_𝑝(𝑎^′/𝐿_𝜃) for 𝑢∈[𝑎^′,(𝑇/2𝑟)𝑎^′].Then problem (1.3) has at least three positive symmetric solutions 𝑢₁, 𝑢₂, and 𝑢₃ such that 0<max_{𝑡∈[0,𝑟]𝕋}𝑢₁(𝑡)<𝑎^′<max_{𝑡∈[0,𝑟]𝕋}𝑢₂(𝑡),min_{𝑡∈[𝜂,𝑇/2]𝕋}𝑢₂(𝑡)<𝑏^′<min_{𝑡∈[𝜂,𝑇/2]𝕋}𝑢₃(𝑡),max_{𝑡∈[0,𝜂]𝕋}𝑢₃(𝑡)<𝑐^′. (4.15)

From Theorem 4.3, we can obtain Theorem 4.4 and Corollary 4.5.

Theorem 4.4. Let 𝑙=1,2,…,𝑛. Suppose that there existence positive numbers 𝑎^′_𝜆𝑙, 𝑏^′_𝜆𝑙, 𝑐^′_𝜆𝑙 such that 𝑎^′_𝜆1<𝐿_𝜃𝑀_𝜉𝑏^′_𝜆1<2𝜂𝑇𝐿_𝜃𝑐^′_𝜆1𝑁_𝜉<𝑎^′_𝜆2<𝐿_𝜃𝑀_𝜉𝑏^′_𝜆2<2𝜂𝑇𝐿_𝜃𝑐^′_𝜆2𝑁_𝜉<𝑎^′_𝜆3<⋯<𝑎^′_𝜆𝑙. (4.16) In addition, 𝑓(𝑢) satisfies the following conditions:(i)𝑓(𝑢)>𝜑_𝑝(𝑐^′_𝜆𝑙/𝑁_𝜉) for 𝑢∈[𝑐^′_𝜆𝑙,(𝑇/2𝜂)𝑐^′_𝜆𝑙];(ii)𝑓(𝑢)<𝜑_𝑝(𝑏^′_𝜆𝑙/𝑀_𝜉) for 𝑢∈[0,(𝑇/2𝜂)𝑏^′_𝜆𝑙];(iii)𝑓(𝑢)>𝜑_𝑝(𝑎^′_𝜆𝑙/𝐿_𝜃) for 𝑢∈[𝑎^′_𝜆𝑙,(𝑇/2𝑟)𝑎^′_𝜆𝑙].Then problem (1.3) has at least 2𝑙+1 positive symmetric solutions.

Corollary 4.5. Assume that 𝑓 satisfies the following conditions:(i)𝑓₀=∞,𝑓_∞=∞,(ii)there exists 𝑐₀>0 such that 𝑓(𝑢)<𝜑_𝑝((2𝜂/𝑇)(𝑐₀/𝑀_𝜉)) for 𝑢∈[0,𝑐₀],then problem (1.3) has at least three positive symmetric solutions.

Proof. First, by condition (ii), let 𝑏^′=(2𝜂/𝑇)𝑐₀, one gets 𝑓(𝑢)<𝜑_𝑝(𝑏^′𝑀_𝜉)for𝑇𝑢∈[0,𝑏2𝜂^′], (4.17) which implies that (ii) of Theorem 4.3 holds.
Second, choose 𝐾₃ sufficiently large to satisfy 𝐾₃𝐿_𝜃=𝐾₃𝑟𝜑_𝑞(_𝑟^𝑇/2ℎ(𝑟)∇𝑟)>1. (4.18) Since 𝑓₀=∞, there exists 𝑟^′₁>0 sufficiently small such that 𝑓(𝑢)≥𝜑_𝑝(𝐾₃)𝜑_𝑝(𝑢)=𝜑_𝑝(𝐾₃𝑢)for𝑢∈[0,𝑟^′₁]. (4.19) Without loss of generality, suppose 𝑟^′₁≤(𝐿_𝜃𝑇/2𝑟𝑀_𝜉)𝑏^′. Choose 𝑎^′>0 such that 𝑎^′<(2𝑟/𝑇)𝑟^′₁. For 𝑎^′≤𝑢≤(𝑇/2𝑟)𝑎^′, we have 𝑢≤𝑟^′₁ and 𝑎^′<(𝐿_𝜃/𝑀_𝜉)𝑏^′. Thus, by (4.18) and (4.19), we have 𝑓(𝑢)≥𝜑_𝑝(𝐾₃𝑢)≥𝜑_𝑝(𝐾₃𝑎^′)>𝜑_𝑝(𝑎^′𝐿_𝜃)for𝑢∈[𝑎^′,𝑇𝑎2𝑟^′], (4.20) this implies that (iii) of Theorem 4.3 is true.
Third, choose 𝐾₂ sufficiently large such that 𝐾₂𝑁_𝜉=𝐾₂𝜂𝜑_𝑞(_𝜂^𝑇/2ℎ(𝑟)∇𝑟)>1. (4.21) Since 𝑓_∞=∞, there exists 𝑟^′₂>0 sufficiently large such that 𝑓(𝑢)≥𝜑_𝑝(𝐾₂)𝜑_𝑝(𝑢)=𝜑_𝑝(𝐾₂𝑢)for𝑢≥𝑟^′₂. (4.22) Without loss of generality, suppose 𝑟^′₂>(𝑇/2𝜂)𝑏^′. Choose 𝑐^′=𝑟^′₂. Then 𝑓(𝑢)≥𝜑_𝑝(𝐾₂𝑢)≥𝜑_𝑝(𝐾₂𝑐^′)>𝜑_𝑝(𝑐^′𝑁_𝜉)for𝑢∈[𝑐^′,𝑇𝑐2𝜂^′], (4.23) which means that (i) of Theorem 4.3 holds.
From above analysis, we get 0<𝑎^′<𝐿_𝜃𝑀_𝜉𝑏^′<2𝜂𝑇𝐿_𝜃𝑐^′𝑀_𝜉, (4.24) then, all conditions in Theorem 4.3 are satisfied. Hence, problem (1.3) has at least three positive symmetric solutions.

In terms of Theorem 4.1, we also have the following corollary.

Corollary 4.6. Assume that 𝑓 satisfies conditions(i)𝑓₀=0,𝑓_∞=0;(ii)there exists 𝑐₀>0 such that 𝑓(𝑢)>𝜑_𝑝((2𝜂/𝑇)(𝑐₀/𝑁_𝜉)) for 𝑢∈[(2𝜂/𝑇)𝑐₀,𝑐₀],then problem (1.3) has at least three positive symmetric solutions.

4.2. Result 2

In this subsection, the existence criteria for at least triple positive or arbitrary odd positive symmetric solutions to problems (1.3) are established by using the Avery-Peterson fixed point theorem [27].

Define the nonnegative continuous convex functionals 𝜙 and 𝛽, nonnegative continuous concave functional 𝜆, and nonnegative continuous functional 𝜑, respectively, on 𝑃 by

𝜙(𝑢)=max_{𝑡∈[0,𝑇/2]𝕋}𝑇𝑢(𝑡)=𝑢(2),𝛽(𝑢)=max_{𝑡∈[𝑟,𝑇/2]𝕋𝜅}|𝑢^Δ(𝑡)|=|𝑢^Δ(𝑟)|,𝜆(𝑢)=𝜑(𝑢)=min_{𝑡∈[𝜂,𝑇/2]𝕋}𝑢(𝑡)=𝑢(𝜂). (4.25)

Now, we list and prove the results in this subsection.

Theorem 4.7. Suppose that there exist constants 𝑎^∗, 𝑏^∗, 𝑑^∗ such that 0<𝑎^∗<(2𝜂/𝑇)𝑏^∗<(2𝜂/𝑇)(𝑁_𝜉𝑑^∗/𝑊_𝜉). In addition, suppose that 𝑊_𝜉>𝜑_𝑞(∫_𝜂^𝑇/2ℎ(𝑠)∇𝑠) holds, 𝑓 satisfies the following conditions:(i)𝑓(𝑢)≤𝜑_𝑝(𝑑^∗/𝑊_𝜉) for 𝑢∈[0,𝑑^∗];(ii)𝑓(𝑢)>𝜑_𝑝(𝑏^∗/𝑁_𝜉) for 𝑢∈[𝑏^∗,𝑑^∗];(iii)𝑓(𝑢)<𝜑_𝑝(𝑎^∗/𝑀_𝜉) for 𝑢∈[0,(𝑇/2𝜂)𝑎^∗],then problem (1.3) has at least three positive symmetric solutions 𝑢₁,𝑢₂,𝑢₃ such that ‖𝑢_𝑖‖≤𝑑^∗,𝑖=1,2,3,𝑏^∗<𝑢₁(𝜂),𝑢₂(𝜂)<𝑏^∗,𝑢₃(𝜂)<𝑎^∗. (4.26)

Proof. By the definition of completely continuous operator 𝐴 and its properties, it suffices to show that all the conditions of Lemma 2.5 hold with respect to 𝐴.
For all 𝑢∈𝑃,𝜆(𝑢)=𝜑(𝑢) and ‖𝑢‖=𝑢(𝑇/2)=𝜙(𝑢). Hence, condition (2.8) is satisfied.
Firstly, we show that 𝐴∶𝑃(𝜙,𝑑^∗)→𝑃(𝜙,𝑑^∗).
For any 𝑢∈𝑃(𝜙,𝑑^∗), in view of 𝜙(𝑢)=‖𝑢‖≤𝑑^∗ and assumption (i), one has 𝑇‖𝐴𝑢‖=𝐴𝑢(2)=₀^𝑇/2𝜑_𝑞(_𝑠^𝑇/2≤ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠_0𝑇/2𝜑_𝑞(₀^𝑇/2≤𝑑ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠_∗𝑊_𝜉𝑇2𝜑_𝑞(₀^𝑇/2ℎ(𝑟)∇𝑟)=𝑑^∗. (4.27) From the above analysis, it remains to show that (i)–(iii) of Lemma 2.5 hold.
Secondly, we verify that condition (i) of Lemma 2.5 holds, let 𝑢(𝑡)≡𝑘𝑏^∗ with 𝑘=𝑊_𝜉/𝑁_𝜉>1. From the definitions of 𝑁_𝜉,𝑊_𝜉, and 𝛽(𝑢), respectively, it is easy to see that 𝑢(𝑡)=𝑘𝑏^∗>𝑏^∗ and 𝛽(𝑢)=0<𝑘𝑏^∗. In addition, in view of 𝑏^∗<(𝑁_𝜉/𝑊_𝜉)𝑑^∗, we have 𝜙(𝑢)=𝑘𝑏^∗<𝑑^∗. Thus {𝑢∈𝑃(𝜙,𝛽,𝜆,𝑏^∗,𝑘𝑏^∗,𝑑^∗)∶𝜆(𝑥)>𝑏^∗}≠∅. (4.28) For any 𝑢∈𝑃(𝜙,𝛽,𝜆,𝑏^∗,𝑘𝑏^∗,𝑑^∗), then we get 𝑏^∗≤𝑢(𝑡)≤𝑑^∗ for all 𝑡∈[𝜂,𝑇/2]_𝕋. Hence, by assumption (ii), we have =𝜆(𝐴𝑢)=𝐴𝑢(𝜂)_𝜂0𝜑_𝑞(_𝑠^𝑇/2≥ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠_𝜂0𝜑_𝑞(_𝜂^𝑇/2>𝑏ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠_∗𝑁_𝜉𝜂𝜑_𝑞(_𝜂^𝑇/2ℎ(𝑟)∇𝑟)=𝑏^∗. (4.29)
Thirdly, we prove that condition (ii) of Lemma 2.5 holds. For any 𝑢∈𝑃(𝜙,𝜆,𝑏^∗,𝑑^∗) with 𝛽(𝐴𝑢)>𝑘𝑏^∗, that is, 𝛽(𝐴𝑢)=|(𝐴𝑢)^Δ(𝑟)|=𝜑_𝑞(_𝑟^𝑇/2ℎ(𝑠)𝑓(𝑢(𝑠))∇𝑠)>𝑘𝑏^∗. (4.30) So, in view of 𝑘=𝑊_𝜉/𝑁_𝜉, 𝑊_𝜉>𝜑_𝑞(∫_𝜂^𝑇/2ℎ(𝑠)∇𝑠) and (4.30), one has =𝜆(𝐴𝑢)=𝐴𝑢(𝜂)_𝜂0𝜑_𝑞(_𝑠^𝑇/2≥ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠_𝜂0𝜑_𝑞(_𝜂^𝑇/2>ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠_𝜂0𝜑_𝑞(_𝑟^𝑇/2ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠>𝜂𝑘𝑏^∗>𝑏^∗. (4.31)
Finally, we check condition (iii) of Lemma 2.5. Clearly, since 𝜑(0)=0<𝑎^∗, we have 0∉𝑅(𝜙,𝜑,𝑎^∗,𝑑^∗). If 𝑢∈𝑅(𝜙,𝜑,𝑎^∗,𝑑^∗) with 𝜑(𝑢)=min_{𝑡∈[𝜂,𝑇/2]𝕋}𝑢(𝑡)=𝑎^∗, then Lemma 2.1 implies that 𝑇‖𝑢‖≤𝑇2𝜂𝑢(𝜂)=𝑎2𝜂^∗. (4.32) This yields 0≤𝑢(𝑡)≤(𝑇/2𝜂)𝑎^∗ for all 𝑡∈[0,𝑇/2]_𝕋. Hence, by assumption (iii), we have 𝜆(𝐴𝑢)=𝐴𝑢(𝜂)≤^𝜂₀𝜑_𝑞(₀^𝑇/2𝑎ℎ(𝑟)𝑓(𝑢(𝑟))∇𝑟)Δ𝑠<^∗𝑀_𝜉𝜂𝜑_𝑞(₀^𝑇/2ℎ(𝑟)∇𝑟)=𝑎^∗. (4.33) Consequently, all conditions of Lemma 2.5 are satisfied. The proof is completed.

We remark that condition (i) in Theorem 4.7 can be replaced by the following condition (i^′):

lim_𝑢→∞𝑓(𝑢)𝜑_𝑝(𝑢)≤𝜑_𝑝(1𝑊_𝜉), (4.34) which is a special case of (i).

Corollary 4.8. If condition (i) in Theorem 4.7 is replaced by (i^′), then the conclusion of Theorem 4.7 also holds.

Proof. By Theorem 4.7, we only need to prove that (i') implies that (i) holds, that is, if (i') holds, then there is a number 𝑑^∗≥max{(𝑇/2𝜂)𝑎^∗,(𝑊_𝜉/𝑁_𝜉)𝑏^∗} such that 𝑓(𝑢)≤𝜑_𝑝(𝑑^∗/𝑊_𝜉) for 𝑢∈[0,𝑑^∗].
Suppose on the contrary that for any 𝑑^∗≥max{(𝑇/2𝜂)𝑎^∗,(𝑊_𝜉/𝑁_𝜉)𝑏^∗}, there exists 𝑢_𝑐∈[0,𝑑^∗] such that 𝑓(𝑢_𝑐)>𝜑_𝑝(𝑑^∗/𝑊_𝜉). Hence, if we choose 𝑐^′_𝑛𝑇>max{𝑎2𝜂^∗,𝑊_𝜉𝑁_𝜉𝑏^∗}(𝑛=1,2,…) (4.35) with 𝑐^′_𝑛→∞, then there exist 𝑢_𝑛∈[0,𝑐^′_𝑛] such that 𝑓(𝑢_𝑛)>𝜑_𝑝(𝑐^′_𝑛𝑊_𝜉), (4.36) and so lim_𝑛→∞𝑓(𝑢_𝑛)=∞. (4.37) Since condition (i^′) holds, then there exists 𝜏>0 such that 𝑓(𝑢)≤𝜑_𝑝(𝑢𝑊_𝜉)for𝑢∈(𝜏,∞). (4.38) Hence, we have 𝑢_𝑛≤𝜏. Otherwise, if 𝑢_𝑛>𝜏, then it follows from (4.38) that 𝑓(𝑢_𝑛)≤𝜑_𝑝(𝑢_𝑛𝑊_𝜉)≤𝜑_𝑝(𝑐^′_𝑛𝑊_𝜉), (4.39) which contradicts (4.36).
Let 𝑊=max_{𝑢∈[0,𝜏]}𝑓(𝑢), then 𝑓(𝑢_𝑛)≤𝑊(𝑛=1,2,…), which also contradicts (4.37). The proof is complete.

Theorem 4.9. Let 𝑙=1,2,…,𝑛. Suppose that there exist constants 𝑎^∗_𝑙, 𝑏^∗_𝑙, 𝑑^∗_𝑙 such that 0<𝑎^∗₁<2𝜂𝑇𝑏^∗₁<2𝜂𝑇𝑑^∗₁𝑁_𝜉𝑊_𝜉<𝑎^∗₂<2𝜂_𝑖𝑇𝑏^∗₂<2𝜂𝑇𝑑^∗₂𝑁_𝜉𝑊_𝜉<𝑎^∗₃<⋯<𝑎^∗_𝑙. (4.40) In addition, suppose that 𝑊_𝜉>𝜑_𝑞(∫_𝜂^𝑇/2ℎ(𝑠)∇𝑠) holds, then 𝑓 satisfies the following conditions:(i)𝑓(𝑢)<𝜑_𝑝(𝑑^∗_𝑙/𝑊_𝜉) for 𝑢∈[0,𝑑^∗_𝑙];(ii)𝑓(𝑢)>𝜑_𝑝(𝑏^∗_𝑙/𝑁_𝜉) for 𝑢∈[𝑏^∗_𝑙,𝑑^∗_𝑙];(iii)𝑓(𝑢)<𝜑_𝑝(𝑎^∗_𝑙/𝑀_𝜉) for 𝑢∈[0,(𝑇/2𝜂)𝑎^∗_𝑙],then problem (1.3) has at least 2𝑙+1 positive symmetric solutions.

Proof. Similar to the proof of Theorem 4.2, we omit it here.

5. Examples

In this section, we give two simple examples to illustrate that the conclusions we will arrive at are different with their own distinctive advantages.

Example 5.1. Let 𝕋={0,0.05,0.1,0.15,0.45,0.48,0.5,0.55,0.52,0.85,0.9,0.95,1}∪[0.22,0.44]∪[0.56,0.78]. (5.1)
Consider the following boundary value problem with 𝑝=3: (𝜑_𝑝(𝑢^Δ(𝑡)))^∇+ℎ(𝑡)𝑓(𝑢(𝑡))=0,𝑡∈(0,1)_𝕋,𝑢(0)=𝑢(1)=0,𝑢^Δ(0)=−𝑢^Δ(1), (5.2) where ℎ(𝑡)=𝑡+𝜌(𝑡),𝑡∈[0,0.5]_𝕋,1−𝑡+𝜌(1−𝑡),𝑡∈[0.5,1]_𝕋,𝑓(𝑢)=0.16,𝑢∈[0,0.6],1049.6𝑢−629.6,𝑢∈[0.6,1],420,𝑢∈[1,5],2.8364𝑢+405.82,𝑢∈[5,60],576,𝑢∈[60,∞). (5.3) Note that 𝑇=1. If we choose 𝜂=0.1, 𝑟=0.25, then direct calculation shows that 𝑀_𝜉=𝜂𝜑_𝑞(₀^1/2ℎ(𝑟)∇𝑟)=0.05, (5.4) and 𝑁_𝜉≈0.0489,𝐿_𝜉=0.125,𝑊_𝜉=0.2500. If we take 𝑎^′=0.3, 𝑏^′=1, 𝑐^′=12, then 𝑎^′=0.3<2𝑟𝑇𝑏^′≈0.9714<2𝑟𝑇𝑐^′𝑁_𝜉𝑀_𝜉≈9.173. (5.5) Furthermore, 𝑓(𝑢)=0.16<5.76=𝜑_𝑝(𝑎^′𝐿_𝜉),𝑢∈[0,0.6],𝑓(𝑢)=420>418.2≈𝜑_𝑝(𝑏^′𝑁_𝜉),𝑢∈[1,5],𝑓(𝑢)≤576<57600=𝜑_𝑝(𝑐^′𝑀_𝜉),𝑢∈[0,60]. (5.6) By Theorem 4.1 we see that the boundary value problem (5.2) has at least three positive symmetric solutions 𝑢₁,𝑢₂, and 𝑢₃ such that 0<𝑢₁(0.25)<0.3<𝑢₂(0.25),𝑢₂(0.1)<1<𝑢₃(0.1),𝑢₃(0.1)<12. (5.7) Yet, 𝜑_𝑞(∫^1/2_0.1ℎ(𝑠)∇𝑠)≈0.489>𝑊_𝜉≈0.25, hence, the existence of positive solutions of boundary value problem (5.2) is not obtained by using Theorem 4.7.

Example 5.2. Let 𝕋={0,0.05,0.1,0.15,0.45,0.46,0.48,0.5,0.52,0.54,0.55,0.85,0.9,0.95,1}∪[0.22,0.44]∪[0.56,0.78]. (5.8)
Consider the following boundary value problem: (𝜑_𝑝(𝑢^Δ(𝑡)))^∇+ℎ(𝑡)𝑓(𝑢(𝑡))=0,𝑡∈(0,1)_𝕋,𝑢(0)=𝑢(1)=0,𝑢^Δ(0)=−𝑢^Δ(1), (5.9) where ℎ(𝑡)=𝑡+𝜌(𝑡),𝑡∈[0,0.5]_𝕋,1−𝑡+𝜌(1−𝑡),𝑡∈[0.5,1]_𝕋. (5.10) Note that 𝑇=1. If we choose 𝜂=0.45, 𝑟=0.48, then a direct calculation shows that 𝑀_𝜉=𝜂𝜑_𝑞(₀^1/2𝑁ℎ(𝑟)∇𝑟)=0.225,_𝜉≈0.0981,𝐿_𝜉≈0.24,𝑊_𝜉≈0.25,𝑊_𝜉≈0.25>0.218≈𝜑_𝑞(^0.5_0.45ℎ(𝑟)∇𝑟). (5.11) Let 𝜀 be an arbitrary small positive number, 𝑎^∗,𝑏^∗, and 𝑑^∗ are arbitrary positive numbers with 0<𝑎^∗<2𝜂𝑏^∗𝑁<2𝜂^𝜉𝑑^∗𝑊_𝜉. (5.12) That is 0<𝑎^∗<0.9𝑏^∗<0.3924𝑑^∗,𝜑𝑓(𝑢)=^𝑝(𝑎^∗0.225)−𝜀,0≤𝑢≤𝑎^∗,𝑙(𝑢),𝑎^∗≤𝑢≤𝑏^∗,𝜑_𝑝(𝑏^∗0.0981)+𝜀,𝑏^∗≤𝑢≤𝑑^∗, (5.13) where 𝑙(𝑢) satisfy 𝑙(𝑎^∗)=𝜑_𝑝(𝑎^∗/0.225)−𝜀, 𝑙(𝑏^∗)=𝜑_𝑝(𝑏^∗/0.0981)+𝜀, (𝑙^Δ(𝑢))^Δ=0, 𝑢∈[𝑎^∗,𝑏^∗].
It is obvious that (i), (ii), and (iii) in Theorem 4.7 are satisfied. By Theorem 4.7, we see that the boundary value problem (5.9) has at least three positive solutions 𝑢₁,𝑢₂, and 𝑢₃ such that ‖𝑢_𝑖‖≤𝑑^∗,𝑖=1,2,3,𝑏^∗<𝑢₁(0.45),𝑎^∗<𝑢₂(0.45),𝑢₃(0.45)<𝑎^∗. (5.14)
However, for arbitrary positive numbers 𝑎^∗, 𝑏^∗, 𝑑^∗, condition (iii) of Theorem 4.1 is not satisfied. Therefore, Theorem 4.1 is not fit to study the boundary value problem (5.9).

Acknowledgments

This work was supported by the Grant of Department of Education Jiangsu Province (09KJD110006) and Science Foundation of Lanzhou University of Technology (BS10200903).