Multiple Positive Solutions for Singular Periodic Boundary Value Problems of Impulsive Differential Equations in Banach Spaces

Abstract

By means of the fixed point theory of strict set contraction operators, we establish a new existence theorem on multiple positive solutions to a singular boundary value problem for second-order impulsive differential equations with periodic boundary conditions in a Banach space. Moreover, an application is given to illustrate the main result.

1. Introduction

The theory of impulsive differential equations describes processes that experience a sudden change of their state at certain moments. In recent years, a great deal of work has been done in the study of the existence of solutions for impulsive boundary value problems, by which a number of chemotherapy, population dynamics, optimal control, ecology, industrial robotics, and physics phenomena are described. For the general aspects of impulsive differential equations, we refer the reader to the classical monograph [1]. For some general and recent works on the theory of impulsive differential equations, we refer the reader to [2–14]. Meanwhile, the theory of ordinary differential equations in abstract spaces has become a new important branch (see [15–18]). So it is interesting and important to discuss the existence of positive solutions for impulsive boundary value problem in a Banach space.

Let (𝐸,‖⋅‖) be a real Banach space, 𝐽=[0,2𝜋], 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑚<𝑡_𝑚+1=2𝜋, 𝐽₀=[0,𝑡₁], and 𝐽_𝑖=(𝑡_𝑖,𝑡_𝑖+1], 𝑖=1,…,𝑚. Note that 𝑃𝐶[𝐽,𝐸]={𝑢∶𝑢 is a map from𝐽into𝐸such that 𝑢(𝑡)is continuous at 𝑡≠𝑡_𝑘 and left continuous at𝑡=𝑡_𝑘 and 𝑢(𝑡^+𝑘)exist, 𝑘=1,2,…,𝑚}, and it is also a Banach space with norm ‖𝑢‖_𝑃𝐶=sup_𝑡∈𝐽‖𝑢(𝑡)‖.(1.1)

Let the Banach space 𝐸 be partially ordered by a cone 𝑃 of 𝐸; that is, 𝑥≤𝑦 if and only if 𝑦−𝑥∈𝑃, and 𝑃𝐶[𝐽,𝐸] is partially ordered by 𝐾={𝑢∈𝑃𝐶[𝐽,𝐸]∶𝑢(𝑡)∈𝑃,𝑡∈𝐽}∶𝑢≤𝑣 if and only if 𝑣−𝑢∈𝐾; that is, 𝑢(𝑡)≤𝑣(𝑡) for all 𝑡∈𝐽.

In this paper, we consider the following singular periodic boundary value problem with impulsive effects in Banach 𝐸−𝑢^(𝑡)+𝑀²Δ𝑢𝑢(||𝑡)=_{𝑡=𝑡𝑘}𝑓(𝑡,𝑢(𝑡)),𝑡∈𝐽,𝑡≠𝑡_𝑘,Δ𝑢_=𝐼_𝑘𝑢𝑡_𝑘,𝑢||(0_{𝑡=𝑡𝑘})==−𝑢𝐼(2_𝑘𝜋)𝑢,𝑡_𝑘𝑢_,𝑘=1,2,…,𝑚,(0)=𝑢^(2𝜋),(1.2) where 𝑀>0 is constant, 𝑓(𝑡,𝑢) may be singular at 𝑡=0and/or𝑡=2𝜋, 𝑓∈𝐶[(0,2𝜋)×𝑃,𝑃], 𝐼_𝑘,𝐼_𝑘∈𝐶[𝑃,𝑃], Δ𝑢|_{𝑡=𝑡𝑘}=𝑢(𝑡^+𝑘)−𝑢(𝑡^−𝑘), Δ𝑢^|_{𝑡=𝑡𝑘}=𝑢^(𝑡^+𝑘)−𝑢^(𝑡^−𝑘), 𝑘=1,2,…,𝑚, and 𝑢^𝑖(𝑡^+𝑘) (resp., 𝑢^𝑖(𝑡^−𝑘)) denote the right limit (resp., left limit) of 𝑢^𝑖(𝑡) at 𝑡=𝑡_𝑘, 𝑖=0,1.

In the special case where 𝐸=ℝ⁺=[0,+∞), and 𝐼_𝑘=𝐼_𝑘=0, 𝑘=1,2,…,𝑚, problem (1.2) is reduced to the usual second-order periodic boundary value problem. For example, in [19], the periodic boundary value problem: −𝑢^(𝑢𝑡)(0+)𝑀=𝑢𝑢((2𝑡)𝜋=),𝑓(𝑡,𝑢𝑢_(𝑡)),𝑡∈(0,2𝜋),(0)=𝑢^(2𝜋),(1.3) was proved to have at least one positive solution, by Jiang [19].

In [20], the authors studied the multiplicity of positive solutions for IBVP(1.2) in 𝐸=ℝ⁺; the main tool is the theory of fixed point index.

In [21], the author considers the following periodic boundary value problem of second-order integrodifferential equations of mixed type in Banach space: −𝑢^=𝑓Δ(𝑢𝑡||,𝑢,_{𝑡=𝑡𝑘}𝑇𝑢,𝑆𝑢),𝑡∈(0,2𝜋),Δ𝑢_=𝐼_𝑘𝑢𝑡_𝑘,𝑢||(0_{𝑡=𝑡𝑘})==−𝑢𝐼(2_𝑘𝜋)𝑢,𝑡_𝑘𝑢_,𝑘=1,2,…,𝑚,(0)=𝑢^(2𝜋),(1.4) where 𝑓∈𝐶[𝐽×𝐸×𝐸×𝐸,𝐸], 𝐼_𝑘,𝐼_𝑘∈𝐶[𝐸,𝐸], and the operators 𝑇, 𝑆 are given by 𝑇𝑢(𝑡)=^𝑡0𝑘(𝑡,𝑠)𝑢(𝑠)𝑑𝑠,𝑆𝑢(𝑡)=^02𝜋𝑘₁(𝑡,𝑠)𝑢(𝑠)𝑑𝑠,(1.5) with 𝑘∈𝐶[𝐷,ℝ], 𝐷={(𝑡,𝑠)∈ℝ²∶0≤𝑠≤𝑡≤2𝜋}, 𝑘₁∈𝐶[𝐽×𝐽,ℝ]. By applying the monotone iterative technique and cone theory based on a comparison result, the author obtained an existence theorem of minimal and maximal solutions for the IBVP(1.4).

Motivated by the above facts, our aim is to study the multiplicity of positive solutions for IBVP(1.2) in a Banach space. By means of the fixed point index theory of strict set contraction operators, we establish a new existence theorem on multiple positive solutions for IBVP(1.2). Moreover, an application is given to illustrate the main result.

The rest of this paper is organized as follows. In Section 2, we present some basic lemmas and preliminary facts which will be needed in the sequel. Our main result and its proof are arranged in Section 3. An example is given to show the application of the result in Section 4.

2. Preliminaries

Let 𝑇_𝑟={𝑥∈𝐸∶‖𝑥‖≤𝑟}, 𝐵_𝑟={𝑢∈𝑃𝐶[𝐽,𝐸]∶‖𝑢‖_𝑃𝐶≤𝑟} (𝑟>0); for 𝐷⊂𝑃𝐶[𝐽,𝐸], we denote 𝐷(𝑡)={𝑢(𝑡)∶𝑢∈𝐷}⊂𝐸(𝑡∈𝐽).𝛼 denotes the Kuratowski measure of noncompactness.

Let 𝑃𝐶¹[𝐽,𝐸]={𝑢∣𝑢be a map from𝐽into𝐸such that 𝑢(𝑡)is continuously differentiable at 𝑡≠𝑡_𝑘 and left continuous at𝑡=𝑡_𝑘 and𝑢(𝑡^+𝑘), 𝑢^(𝑡^−𝑘), 𝑢^(𝑡^+𝑘)exist, 𝑘=1,2,…,𝑚}. Evidently, 𝑃𝐶¹[𝐽,𝐸] is a Banach space with norm ‖𝑢‖_𝑃𝐶¹=max‖𝑢‖_𝑃𝐶,‖𝑢′‖_𝑃𝐶.(2.1)

Let 𝐽^=𝐽⧵{𝑡₁,𝑡₂,…,𝑡_𝑚}; a map 𝑢∈𝑃𝐶¹[𝐽,𝐸]∩𝐶²[𝐽′,𝐸] is a solution of IBVP(1.2) if it satisfies (1.2).

Now, we first give the following lemmas in order to prove our main result.

Lemma 2.1 (see [17]). Let 𝐾 be a cone in real Banach space 𝐸, and let Ω be a nonempty bounded open convex subset of 𝐾. Suppose that 𝐴∶Ω→𝐾 is a strict set contraction and 𝐴(Ω)⊂𝐾. Then the fixed-point index 𝑖(𝐴,Ω,𝐾)=1.

Lemma 2.2 (see [21]). 𝑢∈𝑃𝐶¹[𝐽,𝐸]∩𝐶²[𝐽′,𝐸] is a solution of IBVP (1.2) if and only if 𝑢∈𝑃𝐶[𝐽,𝐸] is a solution of the impulsive integral equation: 𝑢(𝑡)=^02𝜋𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑚_𝑘=1𝐺𝑡,𝑡^𝑘𝐼_𝑘𝑢𝑡_𝑘+𝐻𝑡,𝑡_𝑘𝐼_𝑘𝑢𝑡_𝑘,(2.2) where 𝐺(𝑡,𝑠)=2𝑀𝑒^2𝜋𝑀−1⁻¹𝑒^{𝑀(2𝜋−𝑡+𝑠)}𝑒+𝑒^{𝑀(𝑡−𝑠)},0≤𝑠≤𝑡≤2𝜋,_{𝑀(2𝜋+𝑡−𝑠)}𝐻(𝑡,𝑠)=2𝑒_2𝜋𝑀+𝑒^{𝑀(𝑠−𝑡)},0≤𝑡≤𝑠≤2𝜋,−1⁻¹𝑒^{𝑀(2𝜋−𝑡+𝑠)}𝑒−𝑒^{𝑀(𝑡−𝑠)},0≤𝑠≤𝑡≤2𝜋,_{𝑀(𝑠−𝑡)}−𝑒^{𝑀(2𝜋+𝑡−𝑠)},0≤𝑡<𝑠≤2𝜋.(2.3) By simple calculations, we obtain that for (𝑡,𝑠)∈𝐽×𝐽, 𝑙₀∶=𝑒^𝜋𝑀𝑀𝑒^2𝜋𝑀−1≤𝐺(𝑡,𝑠)≤𝑒^2𝜋𝑀+12𝑀𝑒^2𝜋𝑀−1∶=𝑙¹||𝐻(𝑡,𝑠)||≤1,(2.4)2,𝑀𝐺(𝑡,𝑠)+𝐻(𝑡,𝑠)>0.(2.5)

To establish the existence of multiple positive solutions in 𝑃𝐶¹[𝐽,𝐸]∩𝐶²[𝐽′,𝐸] of IBVP(1.2), let us list the following assumptions:(A1)‖𝑓(𝑡,𝑥)‖≤𝑔(𝑡)‖ℎ(𝑥)‖, 𝑡∈(0,2𝜋), 𝑥∈𝑃, where 𝑔∶(0,2𝜋)→(0,∞) is continuous andℎ∶𝑃→𝑃 is bounded and continuous and satisfies ∫^02𝜋𝑔(𝑠)𝑑𝑠<+∞.(A2)ℎ(𝑥) in (A1) satisfies𝑐𝑙₁^02𝜋𝑔(𝑠)𝑑𝑠+𝑙_1𝑚_𝑘=1ℎ_𝑘+12^𝑚_𝑘=1𝑑𝑙₁𝑐_𝑘<1,^02𝜋𝑔(𝑠)𝑑𝑠+𝑙_1𝑚_𝑘=1𝑒_𝑘+12^𝑚_𝑘=1𝑑_𝑘<1,(2.6) where 𝑐=lim_{‖𝑥‖→0}‖ℎ(𝑥)‖‖𝑥‖,𝑑=lim_{‖𝑥‖→+∞}‖ℎ(𝑥)‖𝑐‖𝑥‖_𝑘,=lim_{‖𝑥‖→0}‖‖𝐼_𝑘‖‖‖𝑥‖,𝑑^𝑘=lim_{‖𝑥‖→+∞}‖‖𝐼_𝑘‖‖ℎ‖𝑥‖_𝑘,=lim_{‖𝑥‖→0}‖‖𝐼_𝑘‖‖‖𝑥‖,𝑒^𝑘=lim_{‖𝑥‖→+∞}‖‖𝐼_𝑘‖‖‖𝑥‖.(2.7)(A3)For any 𝑟>0 and [𝑎,𝑏]⊂(0,2𝜋), 𝑓 is uniformly continuous on [𝑎,𝑏]×𝑇_𝑟.(A4)There exist 𝐿,𝐿_𝑘,𝐻_𝑘≥0 such that 𝛼(𝑓(𝑡,𝐷))≤𝐿𝛼(𝐷), 𝛼(𝐼_𝑘(𝐷))≤𝐿_𝑘𝛼(𝐷), 𝛼(𝐼_𝑘(𝐷))≤𝐻_𝑘𝛼(𝐷) (𝑘=1,…,𝑚), and 4𝜋𝐿𝑙₁+𝑙₁∑^𝑚𝑘=1𝐻_𝑘+(1/2)∑^𝑚𝑘=1𝐿_𝑘<1,for 𝑡∈(0,2𝜋), and 𝐷⊂𝑃 is bounded.(A5)For any 𝑥∈𝑃, 𝐼_𝑘(𝑥)≥𝑀𝐼_𝑘(𝑥);(A6)𝑃 is a solid cone, and there exist 𝑢₀∈𝑃^∘, 𝐽^0=[𝑎′,𝑏′]⊂𝐽 such that 𝑡∈𝐽′₀, 𝑥≥𝑢₀

imply 𝑓(𝑡,𝑥)≥ℎ(𝑡)𝑢₀, ℎ∈𝐶(𝐽^0,[0,+∞)), and 𝑙∶=𝑙₀∫^𝑏′_𝑎′ℎ(𝑠)𝑑𝑠>1.

Define an operator 𝐴 as follows: (𝐴𝑢)(𝑡)=^02𝜋𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑚_𝑘=1𝐺𝑡,𝑡^𝑘𝐼_𝑘𝑢𝑡_𝑘+𝐻𝑡,𝑡_𝑘𝐼_𝑘𝑢𝑡_𝑘,𝑡∈𝐽.(2.8)

Lemma 2.3. Assuming (A1) and (A4) hold, then, for any 𝑟>0, 𝐴∶𝑃𝐶[𝐽,𝑃]∩𝐵_𝑟→𝑃𝐶[𝐽,𝑃] is bounded and continuous.

Proof. According to (A1) and (A4), we obtain that 𝐴 is a bounded operator. In the following, we will show that 𝐴 is continuous.
Let {𝑢_𝑛},{𝑢}⊂𝑃𝐶[𝐽,𝑃]∩𝐵_𝑟, and ‖𝑢_𝑛−𝑢‖_𝑃𝐶→0. Next we show that ‖𝐴𝑢_𝑛−𝐴𝑢‖_𝑃𝐶→0. By (A1), {(𝐴𝑢_𝑛)(𝑡)} is equicontinuous on each 𝐽_𝑖(𝑖=0,…,𝑚). By the Lebesgue dominated convergence theorem and (2.4), we have ‖‖𝐴𝑢≤_𝑛(𝑡‖‖‖)−𝐴𝑢(𝑡)_02𝜋‖‖𝐺(𝑡,𝑠)𝑓𝑠,𝑢_𝑛(𝑠)−𝑓(𝑠,𝑢(𝑠))𝑑𝑠‖‖‖+^𝑚_𝑘=1+𝐺𝑡,𝑡_𝑘‖‖𝐼_𝑘𝑢_𝑛𝑡_𝑘−𝐼_𝑘𝑢𝑡_𝑘_𝑚‖‖_𝑘=1≤𝑙₁||𝐻𝑡,𝑡_𝑘||‖‖𝐼_𝑘𝑢_𝑛𝑡_𝑘−𝐼_𝑘𝑢𝑡_𝑘‖‖^02𝜋‖‖𝑓𝑠,𝑢_𝑛(𝑠)−𝑓(𝑠,𝑢(𝑠))‖‖𝑑𝑠+𝑙_1𝑚_𝑘=1+1‖‖𝐼_𝑘𝑢_𝑛𝑡_𝑘−𝐼_𝑘𝑢𝑡_𝑘‖‖2^𝑚_𝑘=1‖‖𝐼_𝑘𝑢_𝑛𝑡_𝑘−𝐼_𝑘𝑢𝑡_𝑘‖‖⟶0(𝑛⟶∞).(2.9) In view of the Ascoli-Arzela theorem, {𝐴𝑢_𝑛} is a relatively compact set in 𝑃𝐶[𝐽,𝐸]. In the following we will verify that ‖𝐴𝑢_𝑛−𝐴𝑢‖_𝑃𝐶→0(𝑛→∞).
If this is not true, then there are 𝜀₀>0 and {𝑢_𝑛𝑖}⊂{𝑢_𝑛} such that ‖𝐴𝑢_𝑛𝑖−𝐴𝑢‖_𝑃𝐶≥𝜀₀(𝑖=1,2,…). Since {𝐴𝑢_𝑛} is a relatively compact set, there exists a subsequence of {𝐴𝑢_𝑛𝑖} which converges to 𝑣∈𝑃𝐶[𝐽,𝑃], without loss of generality, and we assume that lim_𝑖→∞𝐴𝑢_𝑛𝑖=𝑣, that is, lim_𝑖→∞‖𝐴𝑢_𝑛𝑖−𝑣‖_𝑃𝐶=0, so 𝑣=𝐴𝑢, which imply a contradiction. Therefore 𝐴 is continuous.

Lemma 2.4. Assuming (A1), (A3), and (A4) hold, then, for any 𝑅>0, 𝐴∶𝑃𝐶[𝐽,𝑃]∩𝐵_𝑅→𝑃𝐶[𝐽,𝑃] is a strict set contraction operator.

Proof. For any 𝑅>0, 𝑆⊂𝑃𝐶[𝐽,𝑃]∩𝐵_𝑅, by (A1), 𝐴𝑆 is bounded and equicontinuous on each 𝐽_𝑖, 𝑖=0,…,𝑚, and by [17], 𝛼_𝑃𝐶(𝐴𝑆)=sup_𝑡∈𝐽𝛼((𝐴𝑆)(𝑡)),(2.10) where (𝐴𝑆)(𝑡)={𝐴𝑢(𝑡)∶𝑢∈𝑆,𝑡∈𝐽}.
Let 𝐷𝐷=^02𝜋𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠∶𝑢∈𝑆_𝛿,=^{𝛿2𝜋−𝛿}𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠∶𝑢∈𝑆,0<𝛿<min𝜋,𝑡₁,2𝜋−𝑡_𝑚.(2.11) By (A1) and (2.4), for any 𝑢∈𝑆, ‖‖‖^{𝛿2𝜋−𝛿}≤𝑙₁𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠−^02𝜋𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠‖‖‖max_{𝑥∈𝑇𝑅}‖ℎ(𝑢)‖^𝛿0𝑔(𝑠)𝑑𝑠+𝑙₁max_{𝑢∈𝑇𝑅}‖ℎ(𝑥)‖^{2𝜋2𝜋−𝛿}𝑔(𝑠)𝑑𝑠.(2.12) In view of (2.12) and (A1), we have 𝑑_𝐻(𝐷_𝛿,𝐷)→0(𝛿→0⁺), where 𝑑_𝐻(𝐷_𝛿,𝐷) denotes the Hausdorff distance of 𝐷 and 𝐷_𝛿.
Therefore, lim_𝛿→0+𝛼𝐷_𝛿=𝛼(𝐷).(2.13) Next we will estimate 𝛼(𝐷_𝛿). Since ^{𝛿2𝜋−𝛿}𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠∈(2𝜋−2𝛿)co({𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))∶𝑠∈[𝛿,2𝜋−𝛿]}),(2.14) thus 𝛼𝐷_𝛿≤=𝛼2(𝜋−𝛿)𝛼^{𝛿2𝜋−𝛿}𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠∶𝑢∈𝑆≤≤22𝜋𝜋𝛼𝑙₁({𝐺(𝑡,𝑠c)o𝑓{(𝐺𝑠,𝑢(𝑡(,𝑠𝑠)))𝑓∶(𝑠𝑠,∈𝑢([𝑠𝛿)),2∶𝜋𝑠−∈𝛿][𝛿,,𝑢2∈𝜋𝑆−}𝛿)],𝑢∈𝑆}𝛼𝑓𝐼_𝛿×𝑆𝐼_𝛿,(2.15) where 𝐼_𝛿=[𝛿,2𝜋−𝛿], 𝑆(𝐼_𝛿)={𝑢(𝑡)∶𝑡∈𝐼_𝛿,𝑢∈𝑆}.
By (A3) and (A4), it is not difficult to prove that 𝛼𝑓𝐼_𝛿×𝑆𝐼_𝛿=max_{𝑡∈𝐼𝛿}𝛼𝑓𝑡,𝑆𝐼_𝛿≤𝐿𝛼𝑆𝐼_𝛿≤𝐿𝛼(𝑆(𝐽)).(2.16) By [17], we have 𝐿𝛼(𝑆(𝐽))≤2𝐿𝛼_𝑃𝐶(𝑆).(2.17) Let 𝛿→0⁺, and making use of the fact that lim_𝛿→0⁺𝛼(𝐷_𝛿)=𝛼(𝐷), we obtain 𝛼(𝐷)≤2𝜋𝑙₁2𝐿𝛼_𝑃𝐶(𝑆)=4𝜋𝐿𝑙₁𝛼_𝑃𝐶(𝑆).(2.18) It is clear that 𝛼^𝑚_𝑘=1𝐺𝑡,𝑡_𝑘𝐼_𝑘𝑢𝑡_𝑘∶𝑢∈𝑆≤𝑙_1𝑚_𝑘=1𝐻_𝑘𝛼^𝑃𝐶𝛼_𝑚(𝑆),(2.19)_𝑘=1𝐻𝑡,𝑡_𝑘𝐼_𝑘𝑢𝑡_𝑘∶𝑢∈𝑆≤12^𝑚_𝑘=1𝐿_𝑘𝛼^𝑃𝐶(𝑆).(2.20) Hence, according to (2.18)–(2.20), we have 𝛼_𝑃𝐶(𝐴𝑆)≤4𝜋𝐿𝑙₁+𝑙_1𝑚_𝑘=1𝐻_𝑘+12^𝑚_𝑘=1𝐿_𝑘𝛼^𝑃𝐶(𝑆).(2.21)
By (A4) and Lemma 2.3, 𝐴 is a strict set contraction operator from 𝑃𝐶[𝐽,𝑃] into 𝑃𝐶[𝐽,𝑃].

3. Main Result

Theorem 3.1. Assuming that (A1)–(A6) hold, then the IBVP (1.2) has at least two positive solutions 𝑢₁ and 𝑢₂ satisfying 𝑢₁(𝑡)≥𝑙𝑢₀(𝑡),for𝑡∈𝐽^0=𝑎^,𝑏^,𝑙>1,(3.1) where 𝑙 was specified in (A6).

Proof. First we verify that there exists 𝛿>0 such that ‖𝑣‖≥𝛿 for 𝑣≥𝑢₀. If this is not true, then there exists {𝑣_𝑛}⊂𝐸which satisfies 𝑣_𝑛≥𝑢₀ and ‖𝑣_𝑛‖<(1/𝑛)(𝑛=1,2,…), so we have 𝑢₀≤𝜃, which is a contradiction with 𝑢₀∈𝑃^∘.
By (A2), there exist 𝑐^>𝑐, 𝑐^𝑘>𝑐_𝑘, 𝑑^>𝑑, 𝑑^𝑘>𝑑_𝑘, ℎ^𝑘>ℎ_𝑘, and 𝑒^𝑘>𝑒_𝑘, and 0<𝑟₁<𝛿,𝑟₂>max𝛿,𝑙‖‖𝑢₀‖‖(3.2) satisfy 𝑐^𝑙₁^02𝜋𝑔(𝑠)𝑑𝑠+𝑙_1𝑚_𝑘=1ℎ^𝑘+12^𝑚_𝑘=1𝑏∶=𝑑_𝑐^𝑘<1,(3.3)𝑙₁^02𝜋𝑔(𝑠)𝑑𝑠+𝑙_1𝑚_𝑘=1𝑒^𝑘+12^𝑚_𝑘=1𝑑^𝑘<1.(3.4) For 𝑥∈𝑇_𝑟1∩𝑃, ‖ℎ(𝑥)‖≤𝑐^‖𝑥‖,‖‖𝐼_𝑘(𝑥)‖‖≤𝑐^𝑘‖𝑥‖,‖‖𝐼_𝑘(𝑥)‖‖≤ℎ^𝑘‖𝑥‖.(3.5) For ‖𝑥‖≥𝑟₂ and 𝑥∈𝑃, ‖ℎ(𝑥)‖≤𝑑^‖𝑥‖,‖‖𝐼_𝑘(𝑥)‖‖≤𝑑^𝑘‖𝑥‖,‖‖𝐼_𝑘(𝑥)‖‖≤𝑒^𝑘‖𝑥‖.(3.6) Therefore, for any 𝑥∈𝑃, we have ‖ℎ(𝑥)‖≤𝑑^‖𝑥‖+𝑀^,‖‖𝐼_𝑘(𝑥)‖‖≤𝑑^𝑘‖𝑥‖+𝑀^,‖‖𝐼_𝑘(𝑥)‖‖≤𝑒^𝑘‖𝑥‖+𝑀^,(3.7) where 𝑀𝑀^=max𝑀₀,𝑀₁,…,𝑀_𝑚,𝐾₁,…,𝐾_𝑚,𝑀₀=sup‖ℎ(𝑥)‖∶𝑥∈𝑇_𝑟2∩𝑃_𝑘,=sup‖‖𝐼_𝑘(𝑥)‖‖∶𝑥∈𝑇_𝑟2∩𝑃,𝐾_𝑘=sup‖‖𝐼_𝑘(𝑥)‖‖∶𝑥∈𝑇_𝑟2∩𝑃(𝑘=1,2,…,𝑚).(3.8)
Let 𝑟₃=𝑟₂+(1−𝑏)⁻¹𝐺, 𝐺=𝑀^[𝑙₁∫^02𝜋𝑔(𝑠)𝑑𝑠+𝑚𝑙₁+𝑚/2], 𝑈₁={𝑢∈𝑃𝐶[𝐽,𝑃]∶‖𝑢‖_𝑃𝐶<𝑟₁}, 𝑈₂={𝑢∈𝑃𝐶[𝐽,𝑃]∶‖𝑢‖_𝑃𝐶<𝑟₃}, 𝑈₃={𝑢∈𝑃𝐶[𝐽,𝑃]∶‖𝑢‖_𝑃𝐶<𝑟₃, 𝑢(𝑡)≥𝑙𝑢₀ for 𝑡∈𝐽^0 and 𝑙>1}. It is clear that 𝑈₁,𝑈₂,𝑈₃ are nonempty, bounded, and convex open sets in 𝑃𝐶[𝐽,𝑃], and 𝑈₁=𝑃𝐶[𝐽,𝑃]∩𝐵_𝑟1, 𝑈₂=𝑃𝐶[𝐽,𝑃]∩𝐵_𝑟3, and 𝑈₃={𝑢∈𝑈₂∶𝑢(𝑡)≥𝑙𝑢₀,𝑡∈𝐽^0}.
From (3.2), we obtain 𝑈₁⊂𝑈₂,𝑈₃⊂𝑈₂,𝑈₁∩𝑈₃=∅.(3.9) According to Lemma 2.4, 𝐴∶𝑈₂→𝑃𝐶[𝐽,𝑃] is a strict set contraction operator, and for 𝑢∈𝑈₂, by (2.4) and (3.7), we obtain ‖(𝐴𝑢)(𝑡)‖=‖‖‖‖^02𝜋𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑚_𝑘=1𝐺𝑡,𝑡^𝑘≤𝑙₁𝐼_𝑘𝑢𝑡_𝑘+𝐻𝑡,𝑡_𝑘𝐼_𝑘𝑢𝑡_𝑘‖‖‖‖^02𝜋𝑔(𝑠)𝑑𝑠‖ℎ(𝑢)‖+^𝑚_𝑘=1𝑙¹‖‖𝐼_𝑘‖‖+12‖‖𝐼^𝑘≤𝑙₁‖‖^02𝜋𝑔(𝑠)𝑑𝑠𝑑^‖𝑢‖+𝑀^+𝑙_1𝑚_𝑘=1𝑒^𝑘‖𝑢‖+𝑀^+12^𝑚_𝑘=1𝑑^𝑘=𝑑_‖𝑢‖+𝑀^𝑙₁^02𝜋𝑔(𝑠)𝑑𝑠+𝑙_1𝑚_𝑘=1𝑒^𝑘+12^𝑚_𝑘=1𝑑^𝑘‖𝑢‖+𝑀^𝑙¹^02𝜋𝑔(𝑠)𝑑𝑠+𝑚𝑙₁+𝑚=≤𝑏𝑏𝑟‖₃𝑢‖+𝐺2+𝐺<𝑟₃.(3.10) Hence 𝐴𝑈₂⊂𝑈².(3.11)
Similarly, 𝐴∶𝑈₁→𝑃𝐶[𝐽,𝑃] is a strict set contraction operator, and for 𝑢∈𝑈₁, by (3.3) and (3.5), we obtain ‖(𝐴𝑢)(𝑡)‖=‖‖‖‖^02𝜋𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑚_𝑘=1𝐺𝑡,𝑡^𝑘≤𝑐_𝐼_𝑘𝑢𝑡_𝑘+𝐻𝑡,𝑡_𝑘𝐼_𝑘𝑢𝑡_𝑘‖‖‖‖𝑙₁^02𝜋𝑔(𝑠)𝑑𝑠‖𝑢‖+^𝑚_𝑘=1𝑙¹ℎ^𝑘‖𝑢‖+12𝑐^𝑘=𝑐‖𝑢‖_𝑙₁^02𝜋𝑔(𝑠)𝑑𝑠+𝑙_1𝑚_𝑘=1ℎ^𝑘+12^𝑚_𝑘=1𝑐^𝑘<‖𝑢‖≤𝑟₁‖𝑢‖,(3.12) so 𝐴𝑈₁⊂𝑈¹.(3.13)
Let 𝑢∈𝑈₃, by (3.11), we have ‖𝐴𝑢‖_𝑃𝐶<𝑟₃.
By (2.5), (A5), and (A6), for 𝑡∈𝐽^0, (𝐴𝑢)(𝑡)=^02𝜋𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑚_𝑘=1𝐺𝑡,𝑡^𝑘≥𝐼_𝑘𝑢𝑡_𝑘+𝐻𝑡,𝑡_𝑘𝐼_𝑘𝑢𝑡_𝑘_02𝜋𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑚≥_𝑘=1𝑀𝐺𝑡,𝑡_𝑘+𝐻𝑡,𝑡_𝑘𝐼_𝑘𝑢𝑡_𝑘_02𝜋≥𝑙₀𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠^𝑏′_𝑎′≥𝑙₀𝑓(𝑠,𝑢(𝑠))𝑑𝑠^𝑏′_𝑎′ℎ(𝑠)𝑑𝑠𝑢₀=𝑙𝑢₀.(3.14) So 𝐴𝑢∈𝑈₃, and 𝐴𝑈₃⊂𝑈³.(3.15) According to (3.11)–(3.15) and Lemma 2.1, we have 𝑖𝐴,𝑈_𝑗,𝑃𝐶[𝐽,𝑃]=1(𝑗=1,2,3).(3.16) Hence 𝑖𝐴,=𝑈𝑖²⧵𝐴,𝑈𝑈₁₂∪𝑈₃,𝑃𝐶[𝐽,𝑃]=−1.,𝑃𝐶[𝐽,𝑃]−𝑖𝐴,𝑈₁,𝑃𝐶[𝐽,𝑃]−𝑖𝐴,𝑈₃,𝑃𝐶[𝐽,𝑃](3.17)
Thus, 𝐴 has two fixed points 𝑢₁ and 𝑢₂ in 𝑈₃ and 𝑈₂⧵(𝑈₁∪𝑈₃), respectively, which means 𝑢₁(𝑡) and 𝑢₂(𝑡) are positive solution of the IBVP (1.2), where 𝑢₁(𝑡)≥𝑙𝑢₀, for 𝑡∈𝐽^0 and 𝑙>1.

4. Example

To illustrate how our main result can be used in practice, we present an example.

Example 4.1. Consider the following problem: −𝑥^𝑛(𝑡)+4𝑥_𝑛=1√𝑡32+9𝜋99𝑡ln1+𝑥^2𝑛+1Δ𝑥_𝑛+𝑥3^𝑛,𝑡∈𝐽,||_𝑡=1/3=1Δ𝑥_𝑛8𝑥^𝑛31,||_𝑡=1/3=−1𝑥_𝑛4𝑥^𝑛31,(0)=𝑥_𝑛(2𝜋),𝑥^𝑛(0)=𝑥^𝑛(2𝜋),(4.1) where 𝑥_𝑚+𝑛=𝑥_𝑛 (𝑛=1,2,…,𝑚).

Conclusion
IBVP (4.1) has at least two positive solutions {𝑥_1𝑛(𝑡)} and {𝑥_2𝑛(𝑡)} such that 𝑥_1𝑛(𝑡)>1 for 𝑡∈[𝜋,2𝜋], 𝑛=1,2,…,𝑚.

Proof. Let 𝐽=[0,2𝜋], 𝐸=ℝ^𝑚={𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑚)∶𝑥_𝑛∈ℝ,𝑛=1,2,…,𝑚}; then, 𝐸 is a Banach space with norm ‖𝑥‖=max_{1≤𝑛≤𝑚}|𝑥_𝑛|. Let 𝑃={(𝑥₁,𝑥₂,…,𝑥_𝑚)∶𝑥_𝑛≥0,𝑛=1,2,…,𝑚}; then, 𝑃 is a solid cone in 𝐸. Compared to IBVP (1.2), 𝑓(𝑡,𝑥)=(𝑓₁,𝑓₂,…,𝑓_𝑚), 𝑓_𝑛(𝑡,𝑥)=(1/√𝑡)[3(2+(999/𝜋)𝑡)ln(1+𝑥^2𝑛+1)+𝑥_𝑛/3)] is singular at 𝑡=0. 𝐼(𝑥)=(𝐼₁(𝑥),𝐼₂(𝑥),…,𝐼_𝑚(𝑥)), and 𝐼_𝑛(𝑥)=(1/8)𝑥_𝑛(1/3). 𝐼(𝑥)=(𝐼₁(𝑥),…,𝐼_𝑚(𝑥)), and 𝐼_𝑛(𝑥)=(1/4)𝑥_𝑛(1/3), 𝑛=1,2,…,𝑚.
Next we will verify that the conditions in Theorem 3.1 are satisfied.
Let 𝑔(𝑡)=1/√𝑡, ℎ(𝑥)=(ℎ₁(𝑥),ℎ₂(𝑥),…,ℎ_𝑚(𝑥)), and ℎ_𝑛(𝑥)=6000ln(1+𝑥^2𝑛+1)+𝑥_𝑛/3. It is clear that ‖𝑓(𝑡,𝑥)‖≤𝑔(𝑡)‖ℎ(𝑥)‖, for 𝑡∈(0,2𝜋) and 𝑥∈𝐸, so (A1) is satisfied.
By simple calculations, we have 𝑀=2, 𝑐=𝑑=1/3, 𝑐₁=𝑑₁=1/8, ℎ₁=𝑒₁=1/4, ∫^02𝜋𝑔(𝑠)𝑑𝑠=5.01326, 𝑙₀=0.00093, and 𝑙₁=0.25. Hence, 𝑙₁∫^02𝜋𝑔(𝑠)𝑑𝑠𝑐+𝑙₁ℎ₁+(1/2)𝑐₁<1; that is, (A2) is satisfied.
Since 𝐸 is a finite-dimensional space, it is obvious that (A3) and (A4) are satisfied.
It is clear that 𝐼_𝑛(𝑥)=(1/4)𝑥_𝑛(1/3) and 𝑀𝐼_𝑛(𝑥)=2×(1/8)𝑥_𝑛(1/3)=(1/4)𝑥_𝑛(1/3), so 𝐼_𝑛(𝑥)=𝑀𝐼_𝑛(𝑥); that is, (A5) is satisfied.
Let 𝑢₀=(1,1,…,1)∈𝑃^∘ and 𝐽^0=[𝜋,2𝜋]⊂[0,2𝜋]; for 𝑡∈𝐽^0 and 𝑥≥𝑢₀, we have 𝑓_𝑛(𝑡,𝑥)=1√𝑡32+9𝜋99𝑡ln1+𝑥^2𝑛+1+𝑥3^𝑛>3000ln2√𝑡.(4.2) Let ℎ(𝑡)=3000ln2/√𝑡; then, for 𝑡∈𝐽^0 and 𝑥≥𝑢₀, we obtain that 𝑓(𝑡,𝑥)≥ℎ(𝑡)𝑢₀ and 𝑙₀∫^𝜋2𝜋ℎ(𝑠)𝑑𝑠>1. Therefore (A6) is satisfied.
By Theorem 3.1, IBVP (4.1) has at least two positive solutions {𝑥_1𝑛(𝑡)} and {𝑥_2𝑛(𝑡)} and satisfies 𝑥_1𝑛(𝑡)>1, 𝑛=1,2,…,𝑚.

Acknowledgments

This work is supported by the NNSF of China (no. 10871160) and project of NWNU-KJCXGC-3-47.