The Existence and Multiplicity of Positive Solutions for Second-Order Periodic Boundary Value Problem

Abstract

The existence and multiplicity of positive solutions are established for second-order periodic boundary value problem. Our results are based on the theory of a fixed point index for A-proper semilinear operators defined on cones due to Cremins. Our approach is different in essence from other papers and the main results of this paper are also new.

1. Introduction

In the present paper, we discuss the existence of positive solutions of the periodic boundary value problem (PBVP) for second-order differential equation𝑥(𝑥0^)(=𝑡)𝑥=(1𝑓)(,𝑡,𝑥),𝑥_0<𝑡<1,(0)=𝑥^(1),(1.1) where 𝑓∶[0,1]×ℝ→ℝ is a continuous function. Our purpose here is to provide sufficient conditions for the existence of multiple positive solutions to the periodic boundary value problem (1.1). This will be done by applying the theory of a fixed point index for A-proper semilinear operators defined on cones obtained by Cremins [1].

We are interested in positive solutions of (1.1), because we have been motivated by a problem from the Theory of Nonlinear Elasticity modelling radial oscillations of an elastic spherical membrane made up of a neo-Hookean material and subjected to an internal pressure. Because of wide interests in physics and engineering, second-order periodic boundary value problems have been studied widely in the literature; we refer the reader to [2–30] and references therein. In [6, 7], by using Krasnoselskii’s fixed point theorem, the existence and multiplicity of positive solutions are established to the periodic boundary value problem on𝑥−𝑥^𝑥+(0𝜌)²=𝑥=𝑥(𝑓2𝜋(𝑡),,𝑥),𝑥0′(<0)𝑡=<𝑥2′𝜋(2,𝜋𝜌),_>0,+𝜌²𝑥=𝑓(𝑡,𝑥),0<𝑡<2𝜋,0<𝜌<1𝑥(0)=𝑥(2𝜋),𝑥_2,(0)=𝑥^(2𝜋).(1.2)

Agarwal et al. [8] discussed the existence of positive solutions for the second-order differential equation−𝑥^(𝑡)+𝑥(𝑏0()𝑡)=𝑥𝑥(𝑡()𝜔=),𝑔(𝑡)𝑓𝑥(_𝑡,𝑥(𝑡)),0<𝑡<𝜔,(0)=𝑥^(𝜔),(1.3) where 𝑏(𝑡) and 𝑔(𝑡) are continuous 𝜔-periodic positive functions and 𝑓∈𝐶(ℝ×[0,∞),[0,∞)). By employing fixed point index theory in cones, they found sufficient conditions for the existence of at least one positive solution. Recently, Torres [9] and Yao [10] obtained some results on the existence of positive solutions of a general periodic boundary value problem𝑥𝑥(0^)(𝑡=)=𝑥(𝑓2𝜋(𝑡),,𝑥(𝑡)),𝑥_0<𝑡<2𝜋,(0)=𝑥^(2𝜋).(1.4) In this case, the problem (1.4) has no Green function. In order to overcome this difficulty, their main technique is to rewrite the original PBVP (1.4) as an equivalent one, so that the Krasnoselskii fixed point theorem on compression and expansion of cones can be applied. Inspired by the above work, the aim of this paper is to consider the existence and multiplicity of positive solutions for the periodic boundary value problem (1.1). The method we used here is different in essence from other papers and the main results of this paper are also new.

This paper is organized as follows. In Section 2, we give some preliminaries and establish several lemmas, and the main theorems are formulated and proved in Section 3. Finally, in Section 4, we give two examples to illustrate our results.

2. Notation and Preliminaries

We start by introducing some basic notation relative to theory of the fixed point index for A-proper semilinear operators defined on cones established by Cremins (see [1]).

Let 𝑋 and 𝑌 be Banach spaces, 𝐷 a linear subspace of 𝑋, {𝑋_𝑛}⊂𝐷, and {𝑌_𝑛}⊂𝑌 sequences of oriented finite-dimensional subspaces such that 𝑄_𝑛𝑦→𝑦 in 𝑌 for every 𝑦 and dist(𝑥,𝑋_𝑛)→0 for every 𝑥∈𝐷, where 𝑄_𝑛∶𝑌→𝑌_𝑛 and 𝑃_𝑛∶𝑋→𝑋_𝑛 are sequences of continuous linear projections. The projection scheme Γ={𝑋_𝑛,𝑌_𝑛,𝑃_𝑛,𝑄_𝑛} is then said to be admissible for maps from 𝐷⊂𝑋 to 𝑌.

Definition 2.1 (see [1]). A map 𝑇∶𝐷⊂𝑋→𝑌 is called approximation-proper (A-proper) at a point 𝑦∈𝑌 with respect to Γ if 𝑇_𝑛≡𝑃_𝑛𝑇|_{𝐷∩𝑋𝑛} is continuous for each 𝑛∈ℕ and whenever {𝑥_𝑛𝑗∶𝑥_𝑛𝑗∈𝐷∩𝑋_𝑛𝑗} is bounded with 𝑇_𝑛𝑗𝑥_𝑛𝑗→𝑦, then there exists a subsequence {𝑥_𝑛𝑗𝑘} such that 𝑥_𝑛𝑗𝑘→𝑥∈𝐷, and 𝑇𝑥=𝑦. 𝑇 is said to be A-proper on a set Ω if it is A-proper at all points of Ω.
Let 𝐾 be a cone in a finite-dimensional Banach space 𝑋, and let Ω⊂𝑋 be open and bounded with Ω∩𝐾=Ω_𝐾≠∅. Let 𝑇∶Ω_𝐾→𝐾 be continuous such that 𝑇𝑥≠𝑥 on 𝜕Ω_𝐾=𝜕Ω∩𝐾, where Ω_𝐾 and 𝜕Ω_𝐾 denote the closure and boundary, respectively, of Ω_𝐾 relative 𝐾. Let 𝜌∶𝑋→𝐾 be an arbitrary retraction.
The following definition of finite-dimensional index forms the basis of generalized index for A-proper maps 𝐼−𝑇.

Definition 2.2 (see [1]). One defines 𝑖_𝐾(𝑇,Ω)=deg_𝐵𝐼−𝑇𝜌,𝜌⁻¹(Ω)∩𝐵_𝑅,0,(2.1) where the degree is the Brouwer degree and 𝐵_𝑅 is a ball containing Ω_𝐾.
Now let 𝐾 be a cone in an infinite-dimensional Banach space 𝑋 with projection scheme Γ such that 𝑄_𝑛(𝐾)⊆𝐾 for every 𝑛∈ℕ. Let 𝜌∶𝑋→𝐾 be an arbitrary retraction and Ω⊂𝑋 an open bounded set such that Ω_𝐾=Ω∩𝐾≠∅. Let 𝑇∶Ω_𝐾→𝐾 be such that 𝐼−𝑇 is A-proper at 0. Write 𝐾_𝑛=𝐾∩𝑋_𝑛=𝑄_𝑛𝐾 and Ω_𝑛=Ω_𝐾∩𝑋_𝑛. Then 𝑄_𝑛𝜌∶𝑋_𝑛→𝐾_𝑛 is a finite dimensional retraction.

Definition 2.3 (see [1]). If 𝑇𝑥≠𝑥 on 𝜕Ω_𝐾, then one defines ind_𝐾(𝑇,Ω)=𝑘∈ℤ∪{±∞}∶𝑖_𝐾𝑛𝑗𝑄^𝑛_𝑗𝑇,Ω_𝑛𝑗⟶𝑘forsome𝑛^𝑗⟶∞,(2.2) that is, the index is the set of limit points of 𝑖_𝐾𝑛𝑗(𝑄_𝑛𝑗𝑇,Ω_𝑛𝑗), where the finite dimensional index is that defined above.
Let 𝐿∶dom𝐿⊂𝑋→𝑌 be a Fredholm map of index zero, and let 𝑃∶𝑋→𝑋,𝑄∶𝑌→𝑌 be continuous projectors such that Im𝑃=Ker𝐿, Ker𝑄=Im𝐿 and 𝑋=Ker𝐿⊕ Ker𝑃, 𝑌=Im𝐿⊕Im𝑄. The restriction of 𝐿 to dom𝐿∩Ker𝑃, denoted 𝐿₁, is a bijection onto Im𝐿 with continuous inverse 𝐿¹⁻¹∶ Im𝐿→dom𝐿∩Ker𝑃. Since dim Im𝑄=dim Ker𝐿, there exists a continuous bijection 𝐽∶ Im𝑄→ Ker𝐿. Let 𝐾 be a cone in an infinite-dimensional Banach space 𝑋 with projection scheme Γ. If we let 𝐻=𝐿+𝐽⁻¹𝑃, then 𝐻∶ dom𝐿⊂𝑋→𝑌 is a linear bijection with bounded inverse. Thus 𝐾₁=𝐻(𝐾∩dom𝐿) is a cone in the Banach space 𝑌.
Let Ω⊂𝑋 be open and bounded with Ω_𝐾∩dom𝐿≠∅,𝐿∶dom𝐿⊂𝑋→𝑌 a bounded Fredholm operator of index zero, and 𝑁∶Ω_𝐾∩ dom𝐿→𝑌 a bounded continuous nonlinear operator such that 𝐿−𝑁 is A-proper at 0.
We can now extend the definition of the index to A-proper maps of the form 𝐿−𝑁 acting on cones.

Definition 2.4 (see [1]). Let 𝜌₁ be a retraction from 𝑌 to 𝐾₁, and assume 𝑄_𝑛𝐾₁⊂𝐾₁,𝑃+𝐽𝑄𝑁+𝐿¹⁻¹(𝐼−𝑄)𝑁 maps 𝐾∩dom𝐿 to 𝐾∩dom𝐿 and 𝐿𝑥≠𝑁𝑥 on 𝜕Ω_𝐾. One defines the fixed point index of 𝐿−𝑁 over Ω_𝐾 as ind_𝐾([𝐿,𝑁],Ω)=ind_𝐾1(𝑇,𝑈),(2.3) where 𝑈=𝐻(Ω_𝐾),𝑇∶𝑌→𝑌 is defined as 𝑇𝑦=(𝑁+𝐽⁻¹𝑃)𝐻⁻¹𝑦 for each 𝑦∈𝑌, and the index on the right is that of Definition 2.3.
For convenience, we recall some properties of ind_𝐾.

Proposition 2.5 (see [1]). Let 𝐿∶dom𝐿→𝑌 be Fredholm of index zero, and let Ω⊂𝑋 be open and bounded. Assume that 𝑃+𝐽𝑄𝑁+𝐿¹⁻¹(𝐼−𝑄)𝑁 maps 𝐾 to 𝐾, and 𝐿𝑥≠𝑁𝑥 on 𝜕Ω_𝐾. Then one has
(𝑃₁)(existence property) if ind_𝐾([𝐿,𝑁],Ω)≠{0}, then there exists 𝑥∈Ω_𝐾 such that 𝐿𝑥=𝑁𝑥;(𝑃₂)(normality property) if 𝑥₀∈Ω_𝐾, then ind_𝐾([𝐿,−𝐽⁻¹𝑃+𝑦₀̂],Ω)={1}, where 𝑦₀̂=𝐻𝑥₀ and 𝑦₀̂(𝑦)=𝑦₀ for every 𝑦∈𝐻(Ω_𝐾);(𝑃₃)(additivity property) if 𝐿𝑥≠𝑁𝑥 for 𝑥∈Ω_𝐾⧵(Ω₁∪Ω₂), where Ω₁ and Ω₂ are disjoint relatively open subsets of Ω_𝐾, thenind_𝐾([𝐿,𝑁],Ω)⊆ind_𝐾[𝐿,𝑁],Ω₁+ind_𝐾[𝐿,𝑁],Ω₂(2.4) with equality if either of indices on the right is a singleton;(𝑃₄)(homotopy invariance property) if 𝐿−𝑁(𝜆,𝑥) is an A-proper homotopy on Ω_𝐾 for 𝜆∈[0,1] and (𝑁(𝜆,𝑥)+𝐽⁻¹𝑃)𝐻⁻¹∶𝐾₁→𝐾₁ and 𝜃∉(𝐿−𝑁(𝜆,𝑥))(𝜕Ω_𝐾) for 𝜆∈[0,1], then ind_𝐾([𝐿,𝑁(𝜆,𝑥)],Ω)=ind_𝐾1(𝑇_𝜆,𝑈) is independent of 𝜆∈[0,1], where 𝑇_𝜆=(𝑁(𝜆,𝑥)+𝐽⁻¹𝑃)𝐻⁻¹.

The following two lemmas will be used in this paper.

Lemma 2.6. If 𝐿∶dom𝐿→𝑌 is Fredholm of index zero, Ω is an open bounded set, and Ω_𝐾∩dom𝐿≠∅, and let 𝐿−𝜆𝑁 be A-proper for 𝜆∈[0,1]. Assume that 𝑁 is bounded and 𝑃+𝐽𝑄𝑁+𝐿¹⁻¹(𝐼−𝑄)𝑁 maps 𝐾 to 𝐾. If there exists 𝑒∈𝐾₁⧵{𝜃}, such that 𝐿𝑥−𝑁𝑥≠𝜇𝑒,(2.5) for every 𝑥∈𝜕Ω_𝐾 and all 𝜇≥0, then ind_𝐾([𝐿,𝑁],Ω)={0}.

Proof. Choose a real number 𝑙 such that 𝑙>sup_𝑥∈Ω‖𝐿𝑥−𝑁𝑥‖‖𝑒‖,(2.6) and define 𝑁(𝜇,𝑥)∶[0,1]×Ω_𝐾→𝑌 by 𝑁(𝜇,𝑥)=𝑁𝑥+𝑙𝜇𝑒.(2.7) Trivially, (𝑁(𝜇,𝑥)+𝐽⁻¹𝑃)𝐻⁻¹∶𝐾₁→𝐾₁ and from (2.5) we obtain 𝑁𝑥+𝑙𝜇𝑒≠𝐿𝑥,forany(𝜇,𝑥)∈[0,1]×𝜕Ω_𝐾.(2.8) Again, by homotopy invariance property in Proposition 2.5, we have ind_𝐾([𝐿,𝑁(0,𝑥)],Ω)=ind_𝐾([𝐿,𝑁],Ω)=ind_𝐾([𝐿,𝑁(1,𝑥)],Ω).(2.9) However, ind_𝐾([𝐿,𝑁(1,𝑥)],Ω)={0}.(2.10) In fact, if ind_𝐾([𝐿,𝑁(1,𝑥)],Ω)≠{0}, the existence property in Proposition 2.5 implies that there exists 𝑥₀∈Ω_𝐾 such that 𝐿𝑥₀=𝑁𝑥₀+𝑙𝑒.(2.11) Then 𝑙=‖‖𝐿𝑥⁰−𝑁𝑥₀‖‖‖𝑒‖,(2.12) which contradicts (2.6). So ind_𝐾([𝐿,𝑁],Ω)={0}.(2.13)

Remark 2.7. The original condition of [1, Theorem  5] was given with 𝜃≠𝑒∈𝐿(𝐾∩dom𝐿) instead of 𝑒∈𝐾₁⧵{𝜃}. The modification is necessary since otherwise it cannot guarantee that (𝑁+𝜇𝑒+𝐽⁻¹𝑃)𝐻⁻¹∶𝐾₁→𝐾₁.
We assume that there is a continuous bilinear form [𝑦,𝑥] on 𝑌×𝑋 such that 𝑦∈Im𝐿 if and only if [𝑦,𝑥]=0 for each 𝑥∈Ker𝐿. This condition implies that if {𝑥₁,𝑥₂,…,𝑥_𝑛} is a basis in Ker𝐿, then the linear map 𝐽∶Im𝑄→Ker𝐿 defined by 𝐽𝑦=𝛽∑^𝑛𝑖=1[𝑦,𝑥_𝑖]𝑥_𝑖, 𝛽∈ℝ⁺ is an isomorphism and that if 𝑦=∑^𝑛𝑖=1𝑦_𝑖𝑥_𝑖, then [𝐽⁻¹𝑦,𝑥_𝑖]=𝑦_𝑖/𝛽 for 1≤𝑖≤𝑛 and [𝐽⁻¹𝑥₀,𝑥₀]>0 for 𝑥₀∈Ker𝐿.
In [1], Cremins extended a continuation theorem related to that of Mawhin [31] and Petryshyn [32] for semilinear equations to cones; refer to [1, Corollary  1] for the details. By Lemma 2.6 and [1, Corollary  1], we obtain the following existence theorem of positive solutions to a semilinear equation in cones. It is worth mentioning that the positive or nonnegative solutions of an operator equation 𝐿𝑥=𝑁𝑥 were also discussed by a recent paper of O’Regan and Zima [33] and the earlier papers [34–38].

Lemma 2.8. If 𝐿∶dom𝐿→𝑌 is Fredholm of index zero, 𝐾⊂𝑋 is a cone, Ω₁ and Ω₂ are open bounded sets such that 𝜃∈Ω₁⊂Ω₁⊂Ω₂ and Ω₂∩𝐾∩dom𝐿≠∅. Suppose that 𝐿−𝜆𝑁 is A-proper for 𝜆∈[0,1] with 𝑁∶Ω₂∩𝐾→𝑌 bounded. Assume that
(𝐶₁)(𝑃+𝐽𝑄𝑁)(𝐾)⊂𝐾 and (𝑃+𝐽𝑄𝑁+𝐿¹⁻¹(𝐼−𝑄)𝑁)(𝐾)⊂𝐾,
(𝐶₂)𝐿𝑥≠𝜆𝑁𝑥 for 𝑥∈𝜕Ω₂∩𝐾,𝜆∈(0,1],
(𝐶₃)𝑄𝑁𝑥≠0 for 𝑥∈𝜕Ω₂∩𝐾∩Ker𝐿,
(𝐶₄)[𝑄𝑁𝑥,𝑥]≤0,𝑓𝑜𝑟𝑎𝑙𝑙𝑥∈𝜕Ω₂∩𝐾∩Ker𝐿,
(𝐶₅) there exists 𝑒∈𝐾₁⧵{𝜃}, such that 𝐿𝑥−𝑁𝑥≠𝜇𝑒,𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦𝜇≥0,𝑥∈𝜕Ω₁∩𝐾.(2.14) Then there exists 𝑥∈dom𝐿∩𝐾∩(Ω₂⧵Ω₁) such that 𝐿𝑥=𝑁𝑥.

Corollary 2.9. Assume all conditions of Lemma 2.8 hold except (𝐶₂) and assume (𝐶₂)′‖𝐿𝑥−𝑁𝑥‖²≥‖𝑁𝑥‖²−‖𝐿𝑥‖² for each 𝑥∈𝜕Ω₂∩𝐾. Then the same conclusion holds.

Proof. We show that (𝐶₂)^ implies (𝐶₂), that is, 𝐿𝑥≠𝜆𝑁𝑥, for each 𝑥∈𝜕Ω₂∩𝐾,𝜆∈(0,1]. Here 𝜆∈[0,1). Otherwise, the proof is finished. If 𝑥∈Ker𝐿∩𝜕Ω₂∩𝐾, then it follows from 𝐿𝑥=𝜆𝑁𝑥=𝜃 that 𝐿𝑥=𝑁𝑥 has a solution in dom𝐿∩𝐾∩(Ω₂⧵Ω₁), and Corollary 2.9 is proved. If 𝑥∈dom𝐿⧵Ker𝐿∩𝜕Ω₂∩𝐾 and 𝐿𝑥=𝜆𝑁𝑥 for some 𝜆∈(0,1), then 𝑁𝑥=𝜆⁻¹𝐿𝑥 and (𝜆−1)²‖𝑁𝑥‖²=‖𝐿𝑥−𝑁𝑥‖²≥‖𝑁𝑥‖²−‖𝐿𝑥‖²=1−𝜆²‖𝑁𝑥‖²,(2.15) by condition (𝐶₂)′; that is, (𝜆−1)²≥1−𝜆², contradicting the fact that 𝜆∈(0,1). This completes the proof of Corollary 2.9.

The following lemma can be found by (𝑎) of [32, Lemma  2].

Lemma 2.10. Suppose either 𝑁 or 𝐿¹⁻¹ is compact, then 𝐿−𝜆𝑁 is A-proper for 𝜆∈[0,1].

3. Main Results

The goal of this section is to apply Lemma 2.8 to discuss the existence and multiplicity of positive solutions for the PBVP (1.1).

Let 𝑋={𝑥∈𝐶[0,1]∶𝑥^∈𝐶[0,1],𝑥(0)=𝑥(1),𝑥^(0)=𝑥^(1)} endowed with the norm ‖𝑥‖_𝑋=max_𝑡∈[0,1]|𝑥(𝑡)|, and let 𝑌=𝐶[0,1] with the norm ‖𝑦‖_𝑌=max_𝑡∈[0,1]|𝑦(𝑡)| and 𝐾={𝑥∈𝑋∶𝑥(𝑡)≥0,𝑡∈[0,1]}, then 𝐾 is a cone of 𝑋.

We define𝐿∶dom𝐿⟶do𝑌m,𝐿=𝐿𝑋𝑥,(𝑡)=−𝑥_𝑁∶𝑋⟶𝑌,𝑁𝑥(𝑡)=−𝑓(𝑡,𝑥(𝑡()𝑡)),,(3.1) then PBVP (1.1) can be written as𝐿𝑥=𝑁𝑥,𝑥∈𝐾.(3.2)

It is easy to check thatKIemr𝐿𝐿=={𝑥𝑦∈∈d𝑌om∶𝐿₁₀∶𝑥(𝑡)≡𝑐on[0,1],𝑐∈ℝ},dimKer𝐿=codimIm𝐿=𝑦(1𝑠,)𝑑𝑠=0,(3.3) so that 𝐿 is a Fredholm operator of index zero.

Next, define the projections 𝑃∶𝑋→𝑋 by𝑃𝑥=¹⁰𝑥(𝑠)𝑑𝑠(3.4) and 𝑄∶𝑌→𝑌 by𝑄𝑦=¹⁰𝑦(𝑠)𝑑𝑠.(3.5)

Furthermore, we define the isomorphism 𝐽∶Im𝑄→Im𝑃 as 𝐽𝑦=𝛽𝑦, where 𝛽=1/24. It is easy to verify that the inverse operator 𝐿¹⁻¹∶Im𝐿→dom𝐿∩Ker𝑃 of 𝐿|_{dom𝐿∩Ker𝑃}∶dom𝐿∩Ker𝑃→Im𝐿 as (𝐿¹⁻¹𝑦)(𝑡)=∫¹⁰𝐺(𝑠,𝑡)𝑦(𝑠)𝑑𝑠, where𝐺(𝑠,𝑡)=⎧⎪⎨⎪⎩𝑠21(1−2𝑡+𝑠),0≤𝑠<𝑡≤1,2(1−𝑠)(2𝑡−𝑠),0≤𝑡≤𝑠≤1.(3.6)

For notational convenience, we set 𝐻(𝑠,𝑡)=1/24+𝐺(𝑠,𝑡)−∫¹⁰𝐺(𝑠,𝑡)𝑑𝑠 or𝐻(𝑠,𝑡)=⎧⎪⎨⎪⎩1214+2𝑠(1−2𝑡+𝑠)+𝑡2²−2𝑡+112,0≤𝑠<𝑡≤1,24+12(1−𝑠)(2𝑡−𝑠)+𝑡2²+2𝑡+112,0≤𝑡≤𝑠≤1.(3.7) By routine methods of advanced calculus, we get max_{𝑠,𝑡∈[0,1]}𝐻(𝑠,𝑡)=1/8.

Now we can state and prove our main results.

Theorem 3.1. Assume that there exist two positive numbers 𝑎,𝑏 such that
(𝐻₁)𝑓(𝑡,𝑥)≤𝑥,𝑓𝑜𝑟𝑎𝑙𝑙𝑡∈[0,1],𝑥∈[min{𝑎,𝑏},max{𝑎,𝑏}](𝐻₂)if one of the two conditions(i)max_𝑡∈[0,1]𝑓(𝑡,𝑎)<0,min_𝑡∈[0,1]𝑓(𝑡,𝑏)>0,(ii)min_𝑡∈[0,1]𝑓(𝑡,𝑎)>0,max_𝑡∈[0,1]𝑓(𝑡,𝑏)<0 is satisfied, then the PBVP (1.1) has at least one positive solution 𝑥^∗∈𝐾 satisfying min{𝑎,𝑏}≤‖𝑥^∗‖_𝑋≤max{𝑎,𝑏}.

Proof. It is easy to see 𝑎≠𝑏. Without loss of generality, let 𝑎<𝑏.
First, we note that 𝐿, as so defined, is Fredholm of index zero, 𝐿¹⁻¹ is compact by Arzela-Ascoli theorem, and thus 𝐿−𝜆𝑁 is A-proper for 𝜆∈[0,1] by (𝑎) of Lemma 2.10.
For each 𝑥∈𝐾, then by condition (𝐻₁)𝑃𝑥+𝐽𝑄𝑁𝑥=¹⁰𝑥(𝑠)𝑑𝑠−124¹⁰𝑓(𝑠,𝑥(𝑠))𝑑𝑠≥2324¹⁰𝑃𝑥+𝐽𝑄𝑁𝑥+𝐿₁₋₁𝑥(𝑠)𝑑𝑠≥0,(𝐼−𝑄)𝑁𝑥=¹⁰𝑥(𝑠)𝑑𝑠−124¹⁰+𝑓(𝑠,𝑥(𝑠))𝑑𝑠₁₀=𝐺(𝑠,𝑡)−𝑓(𝑠,𝑥(𝑠))+¹⁰𝑓(𝑠,𝑥(𝑠))𝑑𝑠𝑑𝑠₁₀≥𝑥(𝑠)𝑑𝑠−¹⁰𝐻(𝑠,𝑡)𝑓(𝑠,𝑥(𝑠))𝑑𝑠₁₀(1−𝐻(𝑠,𝑡))𝑥(𝑠)𝑑𝑠≥0.(3.8)
This implies that condition (𝐶₁) of Lemma 2.8 is satisfied. To apply Lemma 2.8, we should define two open bounded subsets Ω₁,Ω₂ of 𝑋 so that (𝐶₂)–(𝐶₅) of Lemma 2.8 hold.
We prove only Case (𝐻₂)(i). In the same way, we can prove Case (𝐻₂)(ii).
Let Ω₁=𝑥∈𝑋∶‖𝑥‖_𝑋<𝑎,Ω₂=𝑥∈𝑋∶‖𝑥‖_𝑋<𝑏.(3.9) Clearly, Ω₁ and Ω₂ are bounded and open sets and 𝜃∈Ω₁⊂Ω₁⊂Ω₂.(3.10) Next we show that (𝐻₂)(i) implies (𝐶₂). For this purpose, suppose that there exist 𝑥₀∈𝐾∩𝜕Ω₂ and 𝜆₀∈(0,1] such that 𝐿𝑥₀=𝜆₀𝑁𝑥₀ then 𝑥^0(𝑡)=𝜆₀𝑓(𝑡,𝑥₀(𝑡)) for all 𝑡∈[0,1]. Let 𝑡₀∈[0,1], such that 𝑥₀(𝑡₀)=max_𝑡∈[0,1]𝑥₀(𝑡)=𝑏. From boundary conditions, we have 𝑡₀∈[0,1). Then we have the following two cases. Case 1 (𝑡₀=0). In this case, 𝑥^0(0)≤0,𝑥^0(1)≥0. Since boundary condition 𝑥^0(0)=𝑥^0(1), we have 𝑥^0(0)=𝑥^0(1)=0. So we have 𝑥^0(0)=𝜆₀𝑓(0,𝑥₀(0))=𝜆₀𝑓(0,𝑏)>0. It follows from 𝑥^0(𝑡) being continuous in [0,1] that there exists 𝛿∈(0,1), such that 𝑥^0(𝑡)>0 when 𝑡∈(0,𝛿]. Thus, 𝑥^0(𝑡)=𝑥^0(0)+∫^𝑡0𝑥^0(𝑠)𝑑𝑠>0. Hence, 𝑥₀(𝑡)=𝑥₀(0)+^𝑡0𝑥^0(𝑠)𝑑𝑠>𝑥₀(0),𝑡∈(0,𝛿],(3.11) and 𝑥₀(0) is not the maximum on [0,1], a contradiction.Case 2 (𝑡₀∈(0,1)). In this case, 𝑥^0(𝑡₀)=0,𝑥^0(𝑡₀)≤0. This gives 0≥𝑥^0𝑡₀=𝜆₀𝑓𝑡₀,𝑥𝑡₀=𝜆₀𝑓𝑡₀,𝑏>0,(3.12) which contradicts (𝐻₂)(i). So for each 𝑥∈𝜕Ω₂∩𝐾 and 𝜆∈(0,1], we have 𝐿𝑥≠𝜆𝑁𝑥. Thus, (𝐶₂) of Lemma 2.8 is satisfied.
To prove (𝐶₄) of Lemma 2.8, we define the bilinear form [⋅,⋅]∶𝑌×𝑋→ℝ as [𝑦,𝑥]=¹⁰𝑦(𝑡)𝑥(𝑡)𝑑𝑡.(3.13) It is clear that [⋅,⋅] is continuous and satisfies [𝑦,𝑥]=0 for every 𝑥∈Ker𝐿,𝑦∈Im𝐿. In fact, for any 𝑥∈Ker𝐿 and 𝑦∈Im𝐿, we have 𝑥≡𝑐, a constant, and there exists 𝑥∈𝑋 such that 𝑦(𝑡)=−𝑥^(𝑡) for each 𝑡∈[0,1]. By 𝑥′(0)=𝑥′(1), we get [𝑦,𝑥]=¹⁰𝑦(𝑡)𝑥(𝑡)𝑑𝑡=−𝑐¹⁰𝑥^(𝑡)𝑑𝑡=0.(3.14) Let 𝑥∈Ker𝐿∩𝜕Ω₂∩𝐾, then 𝑥(𝑡)≡𝑏, so we have by condition (𝐻₂)(i) [𝑄𝑁𝑥𝑄,𝑁𝑥]𝑥==−−¹⁰₁₀𝑓(𝑡,𝑏)𝑑𝑡≠0,𝑓(𝑡,𝑏)𝑑𝑡⋅𝑏𝑑𝑠<0.(3.15) Thus, (𝐶₃) and (𝐶₄) of Lemma 2.8 are verified.
Finally, we prove (𝐶₅) of Lemma 2.8 is satisfied. We may suppose that 𝐿𝑥≠𝑁𝑥,forall𝑥∈𝜕Ω₁∩𝐾. Otherwise, the proof is completed.
Let 𝑒≡1∈𝐾₁⧵{𝜃}. We claim that 𝐿𝑥−𝑁𝑥≠𝜇𝑒,∀𝑥∈𝜕Ω₁∩𝐾,𝜇≥0.(3.16) In fact, if not, there exist 𝑥₂∈𝜕Ω₁∩𝐾,𝜇₁>0, such that 𝐿𝑥₂−𝑁𝑥₂=𝜇₁.(3.17) Since 𝑄𝐿=𝜃, operating on both sides of the latter equation by 𝑄, we obtain𝑄𝑁𝑥₂+𝑄𝜇₁=0,(3.18) that is, ¹⁰−𝑓𝑡,𝑥₂+𝜇₁𝑑𝑡=0.(3.19) For any 𝑥₂∈𝜕Ω₁∩𝐾, we have ‖𝑥₂‖_𝑋=𝑎. Then there exists 𝑡₁∈[0,1], such that 𝑥₂(𝑡₁)=𝑎. By condition (𝐻₂)(i) and 𝜇₁>0, ¹⁰−𝑓𝑡,𝑥₂𝑡₁+𝜇₁𝑑𝑡=¹⁰−𝑓(𝑡,𝑎)+𝜇₁𝑑𝑡>0,(3.20) in contradiction to (3.19). So (3.16) holds; that is, (𝐶₅) of Lemma 2.8 is verified.
Thus, all conditions of Lemma 2.8 are satisfied and there exists 𝑥∈𝐾∩(Ω₂⧵Ω₁) such that 𝐿𝑥=𝑁𝑥 and the assertion follows. Thus, 𝑥^∗∈𝐾 and 𝑎≤‖𝑥^∗‖_𝑋≤𝑏.

Let [𝑐] be the integer part of 𝑐. The following result concerns the existence of 𝑛 positive solutions.

Theorem 3.2. Assume that there exist 𝑛+1 positive numbers 𝑎₁<𝑎₂<⋯<𝑎_𝑛+1 such that
(𝐻₁)^𝑓(𝑡,𝑥)≤𝑥,𝑓𝑜𝑟𝑎𝑙𝑙𝑡∈[0,1],𝑥∈[𝑎₁,𝑎_𝑛+1], (𝐻₂)′if one of the two conditions(i)max_𝑡∈[0,1]𝑓(𝑡,𝑎_2𝑖−1)<0,𝑖=1,2,…,[(𝑛+2)/2], min_𝑡∈[0,1]𝑓(𝑡,𝑎_2𝑖)>0,𝑖=1,2,…,[(𝑛+1)/2](ii)min_𝑡∈[0,1]𝑓(𝑡,𝑎_2𝑖−1)>0,𝑖=1,2,…,[(𝑛+2)/2], min_𝑡∈[0,1]𝑓(𝑡,𝑎_2𝑖)<0,𝑖=1,2,…,[(𝑛+1)/2] is satisfied, then the PBVP (1.1) has at least 𝑛 positive solutions 𝑥^∗𝑖∈𝐾,𝑖=1,2,…,𝑛 satisfying 𝑎_𝑖<‖𝑥^∗𝑖‖_𝑋<𝑎_𝑖+1.

Proof. Modeling the proof of Theorem 3.1, we can prove that if there exist two positive numbers 𝑎,𝑏 such that max_𝑡∈[0,1]𝑓(𝑡,𝑎)<0,min_𝑡∈[0,1]𝑓(𝑡,𝑏)>0, then PBVP (1.1) has at least one positive solution 𝑥^∗∈𝐾 satisfying min{𝑎,𝑏}<‖𝑥^∗‖_𝑋<max{𝑎,𝑏}.
By the claim, for every pair of positive numbers {𝑎_𝑖,𝑎_𝑖+1},𝑖=1,2,…,𝑛, (1.1) has at least 𝑛 positive solutions 𝑥^∗𝑖∈𝐾 satisfying 𝑎_𝑖<‖𝑥^∗𝑖‖_𝑋<𝑎_𝑖+1.

We have the following existence result for two positive solutions.

Corollary 3.3. Assume that there exist three positive numbers 𝑎₁<𝑎₂<𝑎₃ such that (𝐻₁)^𝑓(𝑡,𝑥)≤𝑥,𝑓𝑜𝑟𝑎𝑙𝑙𝑡∈[0,1],𝑥∈[𝑎₁,𝑎₃], (𝐻₂)^if one of the two conditions(i)max_𝑡∈[0,1]𝑓(𝑡,𝑎₁)<0,min_𝑡∈[0,1]𝑓(𝑡,𝑎₂)>0, max_𝑡∈[0,1]𝑓(𝑡,𝑎₃)<0,(ii)min_𝑡∈[0,1]𝑓(𝑡,𝑎₁)>0,max_𝑡∈[0,1]𝑓(𝑡,𝑎₂)<0,min_𝑡∈[0,1]𝑓(𝑡,𝑎₃)>0is satisfied, then the PBVP (1.1) has at least two positive solutions 𝑥^∗1,𝑥^∗2∈𝐾 satisfying 𝑎₁≤‖𝑥^∗1‖_𝑋<𝑎₂<‖𝑥^∗2‖_𝑋≤𝑎₃.

We also have the following existence result for three positive solutions.

Corollary 3.4. Assume that there exist four positive numbers 𝑎₁<𝑎₂<𝑎₃<𝑎₄ such that(𝐻₁)^𝑓(𝑡,𝑥)≤𝑥,𝑓𝑜𝑟𝑎𝑙𝑙𝑡∈[0,1],𝑥∈[𝑎₁,𝑎₄], (𝐻₂)^if one of the two conditions(i)max_𝑡∈[0,1]𝑓(𝑡,𝑎₁)<0,min_𝑡∈[0,1]𝑓(𝑡,𝑎₂)>0,max_𝑡∈[0,1]𝑓(𝑡,𝑎₃)<0, min_𝑡∈[0,1]𝑓(𝑡,𝑎₄)>0, and (ii)min_𝑡∈[0,1]𝑓(𝑡,𝑎₁)>0, max_𝑡∈[0,1]𝑓(𝑡,𝑎₂)<0,min_𝑡∈[0,1]𝑓(𝑡,𝑎₃)>0, max_𝑡∈[0,1]𝑓(𝑡,𝑎₄)<0,is satisfied, then the PBVP (1.1) has at least three positive solutions 𝑥^∗1,𝑥^∗2,𝑥^∗3∈𝐾 satisfying 𝑎₁≤‖𝑥^∗1‖_𝑋<𝑎₂<‖𝑥^∗2‖_𝑋<𝑎₃<‖𝑥^∗3‖_𝑋≤𝑎₄.

Remark 3.5. From similar arguments and techniques, we can also deal with the following periodic boundary value problem (PBVP) 𝑥−(0𝑥)^=(𝑡)𝑥=(1𝑓),(𝑡,𝑥),𝑥_0<𝑡<1,(0)=𝑥^(1).(3.21) We can also verify that the similar results presented in this paper are valid for PBVP (3.21); we omit the details here.

4. Some Examples

In this section, we give some examples to illustrate the main results of the paper.

Example 4.1. Consider the following second-order periodic boundary value problem (PBVP): 𝑥^(𝑡)=45𝑡²𝑥−(𝑡0−)=1𝑥(21𝑥)³,+3𝑥𝑥²_−12𝑥+6𝑥,0<𝑡<1,(0)=𝑥^(1),(4.1) where 𝑓(𝑡,𝑥(𝑡))=(𝑡²−𝑡−1)(2𝑥³+3𝑥²−12𝑥+6)𝑥. In this case, 𝑓(𝑡,𝑥)≤𝑥,𝑥≥0,0≤𝑡≤1.
Corresponding to the assumptions of Corollary 3.3, we set 𝑎₁=1/2,𝑎₂=1, and 𝑎₃=2. It is easy to check that the other conditions of Corollary 3.3 are satisfied; hence, PBVP (4.1) has at least two positive solutions 𝑥^∗1,𝑥^∗2 satisfying 1/2≤‖𝑥^∗1‖_𝑋<1<‖𝑥^∗2‖_𝑋≤2.

Example 4.2. Consider the periodic boundary value problem (PBVP) 𝑥^(𝑡)=5𝑡+2𝑥(0)=𝑥(110),sin𝑥𝑥,_0<𝑡<1,(0)=𝑥^(1).(4.2) Now, let 𝑓(𝑡,𝑥)=((5𝑡+2)/10)sin𝑥; thus, 𝑓(𝑡,𝑥)≤𝑥,𝑥≥0,0≤𝑡≤1. Set 𝑎₁=𝜋/2,𝑎₂=3𝜋/2,𝑎₃=5𝜋/2,𝑎₄=7𝜋/2. Then Corollary 3.4 ensures that there exist at least three positive solutions 𝑥^∗1,𝑥^∗2,𝑥^∗3 satisfying 𝜋/2≤‖𝑥^∗1‖_𝑋<3𝜋/2<‖𝑥^∗2‖_𝑋<5𝜋/2<‖𝑥^∗3‖_𝑋≤7𝜋/2.

Acknowledgments

The project is supported financially by the National Science Foundation of china (10971179) and the Natural Science Foundation of Changzhou university (JS201008).