Positive Solutions for Nonlinear Integro-Differential Equations of Mixed Type in Banach Spaces

Abstract

We establish some new existence theorems on the positive solutions for nonlinear integro-differential equations which do not possess any monotone properties in ordered Banach spaces by means of Banach contraction mapping principle and cone theory based on some new comparison results.

1. Introduction

In this paper, we consider the existence of the unique positive solution and at least one positive solution for the following initial value problem (IVP) of the nonlinear integro-differential equation of mixed type in ordered Banach spaces 𝐸: 𝑢^′′(𝑡)((𝑡),(𝑡),(𝑡𝑢(0)=𝑔𝑡,𝑢𝑇𝑢)(𝑆𝑢)()),∀𝑡∈𝐼,=𝑢₀,𝑢^′(0)=𝑢₁, (1) where 𝐼=[0,𝑎](𝑎>0), 𝑔(𝑡,𝑢(𝑡),(𝑇𝑢)(𝑡), (𝑆𝑢)(𝑡))=𝑓(𝑡,𝑢(𝑡),𝑢(𝑡),(𝑇𝑢)(𝑡),(𝑆𝑢)(𝑡)), 𝑢₀,𝑢₁∈𝐸, 𝑓∈𝐶[𝐼×𝑃×𝑃×𝑃×𝑃,𝑃], and 𝑓 is nonincreasing with the second variable 𝑢(𝑡), with 𝑓 is nondecreasing with the third variable 𝑢(𝑡). 𝑃 is a positive cone in ordered Banach spaces 𝐸, and (𝑇𝑢)(𝑡)=^𝑡₀𝑘()𝑢(𝑠)(𝑡)=𝑡,𝑠𝑑𝑠,𝑆𝑢)(^𝑎₀ℎ(𝑡,𝑠)𝑢(𝑠)𝑑𝑠,𝑡∈𝐼. (2) In (2), 𝑘∈𝐶[𝐷,𝑅⁺] and ℎ∈𝐶[𝐷₀,𝑅⁺], where 𝐷={(𝑡,𝑠)∈𝑅×𝑅∶0≤𝑠≤𝑡≤𝑎}, 𝐷₀={(𝑡,𝑠)∈𝑅×𝑅∶(𝑡,𝑠)∈𝐼×𝐼}, 𝑅⁺ denotes the set of nonnegative real numbers, and 𝑅 denotes the set of real numbers. In the special case when 𝑓 does not contain the third argument, Guo [1] proved the minimal and maximal solution of the following initial value problem: 𝑢^′=𝑓(𝑡,𝑢,𝑇𝑢),𝑡∈𝐼,𝑢𝑡₀=𝑥₀, (3) under some stronger conditions. He used the topological degree theory and the monotone iterative technique. When 𝑓 does not possess any monotone assumption, the problem of proving the existence results is an interesting and important question. The aim of the paper is to study this kind of problem. By means of Banach contraction mapping principle and cone theory based on some new comparison results, we obtain some new existence theorems of the solutions for the initial value problems of the nonlinear integro-differential equations of mixed type which does not possess any monotone properties in ordered Banach spaces. The new results obtained are quite general and are used to generalize, improve, and unify many recent results of [2–8]. Their results assert the existence of one solution, the minimal and maximal solution using monotone iterative technique with the stronger conditions. Our method is different from the method of mixed monotony, and the results obtained in the paper are new even if the space 𝐸 is finite dimensional.

Inspired and motivated greatly by the above work, the aim of the paper is to consider the existence of positive solutions for the boundary value problem (1) under simpler conditions. The main results of problem (1) are obtained by making use of new fixed point theorem. More specifically, in the proof of these theorems, we establish new comparison theorem and construct a special cone for strict set contraction operator. Our main results in essence improve and generalize the corresponding results of [1, 9–12]. Moreover, our method is different from those in [5, 10, 13].

The rest of the paper is organized as follows. In Section 2, we present some known results and introduce conditions to be used in the next section. The main theorems are formulated and proved in Section 3.

2. Preliminaries and Lemmas

In this section, we will state some necessary definitions and preliminary results.

Definition 1. Let 𝐸 be a real Banach space. A nonempty closed set 𝑃⊂𝐸 is called a cone if it satisfies the following two conditions:(1)𝑧∈𝑃, 𝜆>0 implies 𝜆𝑧∈𝑃;(2)𝑧∈𝑃, −𝑧∈𝑃 implies 𝑧=𝜃, where 𝜃 denotes the zero element of 𝐸.
A cone is said to be solid if it contains interior points, 𝑃^∘≠𝜃. The positive cone 𝑃 is said to be generating if 𝐸=𝑃−𝑃; that is, every element 𝑦∈𝐸 can be represented in the form 𝑦=𝑥−𝑧, where 𝑥,𝑧∈𝑃. Every cone with nonempty interior is generating. A cone 𝑃 induces a partial ordering in 𝐸 given by 𝑢≤𝑣 if 𝑣−𝑢∈𝑃. If 𝑢≤𝑣 and 𝑢≠𝑣, we write 𝑢<𝑣; if cone 𝑃 is solid and 𝑣−𝑢∈𝑃^∘, we write 𝑢≪𝑣.

Definition 2. A cone 𝑃⊂𝐸 is said to be normal if there exists a positive constant 𝜐 such that ‖𝑥+𝑦‖≥𝜐, ∀𝑥,𝑦∈𝑃, ‖𝑥‖=1, ‖𝑦‖=1.

Proposition 3. A positive cone 𝑃 is called normal if and only if there exists 𝑁>0 such that 0≤𝑥≤𝑦 implies ‖𝑥‖≤𝑁‖𝑦‖. The norm on 𝐸 is called monotone if 0≤𝑢≤𝑣 implies ‖𝑢‖≤‖𝑣‖. The cone 𝑃 is said to be normal if and only if there exists an equivalent monotone norm on 𝐸.
For further details on cone theory, one can refer to [2, 14].
Let 𝑃={𝑢∈𝐶[𝐼,𝐸]∶𝑢(𝑡)≥𝜃𝑓𝑜𝑟𝑡∈𝐼}, where 𝐶[𝐼,𝐸] denotes the Banach space of all continuous mapping 𝑢∶𝐼→𝐸 with norm ‖𝑢‖=max{𝑢∶𝑡∈𝐼}. It is easy to see that 𝑃 is a cone in 𝐶[𝐼,𝐸], and so it defines a partial order in 𝐶[𝐼,𝐸] by 𝑢≤𝑣 if and only if 𝑢(𝑡)≤𝑣(𝑡) for all 𝑡∈𝐼. Given a cone 𝑃, the dual of 𝑃 is defined as 𝑃^∗={𝜑∈𝐸^∗∶𝜑(𝑢)≥0𝑓𝑜𝑟𝑢∈𝑃}, where 𝐸^∗ is the dual of 𝐸.
Let 𝑘₀=max{𝑘(𝑡,𝑠)∶(𝑡,𝑠)∈𝐷}, ℎ₀=max{ℎ(𝑡,𝑠)∶(𝑡,𝑠)∈𝐷₀}. For any 𝐻⊂𝐶[𝐼,𝐸] and 𝑡∈𝐼, let 𝐻(𝑡)={𝑢(𝑡)},(𝑡)=𝑡)},(𝑡)=𝑡)}.∶𝑢∈𝐻𝑇𝐻)({(𝑇𝑢)(∶𝑢∈𝐻𝑆𝐻)({(𝑆𝑢)(∶𝑢∈𝐻 (4)

Definition 4. Let 𝐸 be a metric space and 𝑆 be a bounded subset of 𝐸. The measure of noncompactness Υ(𝑆) of 𝑆 is defined by Υ(𝑆)}.=inf𝛿>0∶𝑆admitsafinitecoverbysetsofdiameter≤𝛿 (5)

Definition 5. An operator 𝐵∶𝐷→𝐸 is said to be completely continuous if it is continuous and compact. 𝐵 is called a 𝑘-set-contraction (𝑘≥0) if it is continuous, bounded, and Υ(𝐵(𝑆))≤𝑘Υ(𝑆) for any bounded set 𝑆⊂𝐷, where Υ(𝑆) denotes the measure of noncompactness of 𝑆.

A 𝑘-set-contraction is called a strict-set-contraction if  𝑘<1. An operator 𝐵 is said to be condensing if it is continuous, bounded, and Υ(𝐵(𝑆))<Υ(𝑆) for any bounded set 𝑆⊂𝐷 with Υ(𝑆)>0.

Obviously, if 𝐵 is a strict-set-contraction, then 𝐵 is a condensing mapping, and if operator 𝐵 is completely continuous, then 𝐵 is a strict-set-contraction.

Definition 6 (see [15]). Let 𝐸 be a Banach space, and let 𝐴∶𝐸×𝐸→𝐸. The mapping 𝐴 is said to have the mixed monotone property if 𝐴(𝑥,𝑦) is monotone nonincreasing in 𝑢 and is monotone nondecreasing in 𝑣; that is, forany𝑢₁,𝑢₂∈𝐸,𝑢₁≤𝑢₂⟹𝐴𝑢₁,𝑣≥𝐴𝑢₂,𝑣,forany𝑣₁,𝑣₂∈𝐸,𝑣₁≤𝑢₂⟹𝐴𝑢,𝑣₁≤𝐴𝑢,𝑣₂. (6)

Definition 7 (see [15]). A couple point (𝑥,𝑦)∈𝐸×𝐸 is said to be a coupled fixed point of the mapping 𝐴∶𝐸×𝐸→𝐸 if 𝑢=𝐴(𝑢,𝑣) and 𝑣=𝐴(𝑣,𝑢).

Lemma 8 (see [2, 14]). A positive cone 𝑃 is normal and generating if and only if there exists a positive constant 𝜏>0 and 𝑦,𝑧∈𝑃 such that 𝑥=𝑦−𝑧 for all 𝑥∈𝑃 and ‖𝑦‖≤𝜏‖𝑥‖,‖𝑧‖≤𝜏‖𝑥‖.

The following lemma plays a key role for improving the main results.

Lemma 9. Let 𝑃 be a normal generate cone, and let 𝐴∶𝑃×𝑃→𝑃 be a nonlinear operator. If there exists a positive bounded linear operator 𝐿∶𝑃→𝑃 such that 𝑟(𝐿)<1 and −𝐿𝑥−𝑥+𝑦−𝑦≤𝐴𝑥,𝑦−𝐴𝑥,𝑦≤𝐿𝑥−𝑥+𝑦−𝑦,∀𝑥,𝑥,𝑦,𝑦∈𝑃,𝑥≥𝑥,𝑦≤𝑦, (7) then 𝐴 has a unique positive fixed point 𝑥^∗, and for any 𝑥₀∈𝑃, let 𝑥_𝑛=𝐴(𝑥_𝑛−1,𝑥_𝑛−1)(𝑛=1,2,…), 𝑥_𝑛→𝑥^∗(𝑛→∞).

Proof. Since 𝑟(𝐿)<1, then 𝑟(𝐿)=lim_𝑛→∞‖𝐿^𝑛‖^1/𝑛<1. Thus, there exists a natural number 𝑛₀ and 0<𝛼<1 such that 𝐿^𝑛0≤𝛼^𝑛0. (8) Since 𝑃 is generate, it follows from Lemma 8 that there exist 𝜏>0,  𝑦,𝑧∈𝑃 such that 𝑥=𝑦−𝑧 and 𝑦≤𝜏‖𝑥‖,‖𝑧‖≤𝜏‖𝑥‖,∀𝑥∈𝑃. (9) Then ‖𝑦+𝑧‖≤2𝜏‖𝑥‖,∀𝑥∈𝑃. Suppose that 𝑥,𝑦,𝑧∈𝑃 are not in the same line for any 0≤𝑥−𝑦≤𝑧. Let 𝑢₁=(1/2)(𝑥+𝑦−𝑧), 𝑢₂=𝑥−𝑦+𝑧, 𝑢₃=−𝑥+𝑦+𝑧. Then 𝑢₁,𝑢₂,𝑢₃∈𝑃 and 𝑥≥𝑦≥𝑢₁,  𝑥−𝑢₁=(1/2)𝑢₂,  𝑦−𝑢₁=(1/2)𝑢₃,  𝑢₂+𝑢₃=2𝑧. It follows from (7) that −𝐿2𝑢₂≤𝐴(𝑥,𝑥)−𝐴𝑢₁≤𝐿,𝑥2𝑢₂,−𝐿2𝑢₂+𝑢₃≤𝐴𝑢₁,𝑥−𝐴𝑦,𝑢₁≤𝐿2𝑢₂+𝑢₃,−𝐿2𝑢₃≤𝐴𝑦,𝑢₁≤𝐿−𝐴𝑦,𝑦2𝑢₃. (10)(11)(12) By virtue of (10) + (12) − (11), we have −𝐿2𝑢₂+𝑢₃()≤𝐿≤𝐴𝑥,𝑥−𝐴𝑦,𝑦2𝑢₂+𝑢₃,−𝐿⋅2𝑧≤𝐴(𝑥,𝑥)−𝐴𝑦,𝑦≤𝐿⋅2𝑧. (13) We first define the operator 𝐴∶𝑃→𝑃 as the following 𝐴𝑥=𝐴(𝑥,𝑥), 𝑥∈𝑃. Let 𝑤=2𝑧. Then −𝐿𝑤≤𝐴𝑥−𝐴𝑦≤𝐿𝑤. Since 𝐿 is a positive bounded linear operator, then 𝐿𝑢∈𝑃. By induction, it is easy to see that −𝐿^𝑛𝑤≤𝐴^𝑛𝑥−𝐴^𝑛𝑦≤𝐿^𝑛𝑤,𝐿^𝑛𝑤∈𝑃. (14) Since 𝑤∈𝑃 is arbitrary and 𝑛₀ is taken as in (8), we get 𝐴^𝑛0𝑥−𝐴^𝑛0𝑦≤𝐿^𝑛0𝑥−𝑦≤𝛼^𝑛0𝑥−𝑦. (15) From 0<𝛼<1 and Banach contract principle, we know that 𝐴^𝑛0 has the unique positive fixed point 𝑥^∗ in 𝑃; that is, 𝑥^∗ is the unique positive solution of operator equation 𝑥=𝐴(𝑥,𝑥). And for any 𝑥₀∈𝐸, let 𝑥_𝑛=𝐴(𝑥_𝑛−1,𝑥_𝑛−1)(𝑛=1,2,…), we have ‖𝑥_𝑛−𝑥^∗‖→0(𝑛→∞).

Remark 10. The results of Lemma 9 obtained are quite general and are used to improve, generalize, and unify many results of Chen [13], Guo [9], Krasnosel’skii and Zabreiko [16], Zhang [11], Su et al. [10], and Liu [12]. Not only do we obtain the existence and uniqueness of fixed points for mixed nonmonotone binary operators in ordered Banach spaces, but we also solve the difficult open problem that nonmonotone binary operator has unique fixed points under some weaker conditions.

Lemma 11 (see [17]). Let 𝐸 be a Banach space and 𝐻⊂𝐶[𝐼,𝐸] if 𝐻 is a countable set of strongly measurable functions 𝑦∶𝐼→𝐸 such that there exists 𝜑∈𝐿[𝐼,𝑅⁺] such that ‖𝑦(𝑡)‖≤𝜑(𝑡),  𝑡∈𝐼, 𝑦∈𝐻. Then Υ(𝐻(𝑡)) is Lebesgue integrable on 𝐼, and Υ_𝐼𝑦(𝑠)𝑑𝑠∶𝑦∈𝐻≤2_𝐼Υ(𝐻(𝑠))𝑑𝑠. (16)

Lemma 12 (see [1, 18]). Let 𝐻⊂𝐶[𝐽,𝐸] be bounded and equicontinuous; then Υ(𝐻(𝑡)) is continuous on 𝐼, and Υ_𝐼𝐻(𝑠)𝑑𝑠≤_𝐼Υ(𝐻(𝑠))𝑑𝑠. (17)

Lemma 13 (see [2, 19]). Let 𝑋 be a Banach space, 𝐾⊂𝑋 closed and convex, and 𝐹∶𝐾→𝐾 continuous with the further property that for some 𝑥∈𝐾 we have 𝐵⊂𝐾 countable. 𝐵=co({𝛼}∩𝐹(𝐵)) implies that 𝐵 is relatively compact. Then 𝐹 has a fixed point in 𝐾.

Lemma 14. Assume that 𝑚∈𝐶[𝐼,𝑅⁺] satisfies 𝑚(𝑡)≤𝑀^𝑡_𝑡0𝑚(𝑠)𝑑𝑠+𝑀^∗_𝑡^𝑡+𝑎₀𝑚(𝑠)𝑑𝑠,𝑡∈𝐼, (18) where 𝑀>0 and 𝑀^∗≥0 are constants. Then 𝑚(𝑡)≡0 for 𝑡∈𝐼 provided one of the following two conditions hold:(i)𝑀>𝑀^∗(𝑒^{𝑀(𝑡−𝑡0)}−1);(ii)𝑎(𝑀+𝑀^∗)<1.

Proof. Suppose that (i) holds. Let 𝑝(𝑡)=∫^𝑡_𝑡0𝑚(𝑠)𝑑𝑠; then 𝑝(𝑡₀)=0, 𝑝^′(𝑡)=𝑚(𝑡)≥0, for 𝑡∈𝐼. From (18), we know that 𝑝^′(𝑡)≤𝑀𝑝(𝑡)+𝑀^∗𝑝𝑡₀+𝑎,𝑡∈𝐼. (19) Since 𝑒^{−𝑀(𝑡−𝑡0)}>0, we have 𝑝(𝑡)𝑒^{−𝑀(𝑡−𝑡0)′}≤𝑀^∗𝑝𝑡₀+𝑎𝑒^{−𝑀(𝑡−𝑡0)},𝑡∈𝐼. (20) Integrate from 𝑡₀ to 𝑡; noticing 𝑝(𝑡₀)=0, we have 0≤𝑝(𝑡)𝑒^{−𝑀(𝑡−𝑡0)}≤^𝑡_𝑡0𝑀^∗𝑝𝑡₀+𝑎𝑒^{−𝑀(𝑠−𝑡0)}=𝑀𝑑𝑠_∗𝑀𝑝𝑡₀+𝑎1−𝑒^{𝑀(𝑡−𝑡0)},𝑡∈𝐼. (21) Hence, (𝑡)≤𝑀0≤𝑝^∗𝑀𝑝𝑡₀+𝑎𝑒^{𝑀(𝑡−𝑡0)}−1,𝑡∈𝐼. (22) We claim that 𝑝𝑡₀+𝑎=0. (23) Otherwise, 𝑝(𝑡₀+𝑎)>0; then we take 𝑡=𝑡₀+𝑎 in (22), we obtain 𝑀≤𝑀^∗(𝑒^{𝑀(𝑡−𝑡0)}−1), which contradicts assumption (i).
Therefore, from (22) and (23), we know that 𝑝(𝑡)=∫^𝑡_𝑡0𝑚(𝑠)𝑑𝑠=0. Consequently, 𝑚(𝑠)≡0 a.e. on 𝑡∈𝐼.
Let us suppose now that (ii) holds. Then 𝑚(𝑡)≤𝑀+𝑀^∗𝑡0_𝑡^+𝑎₀𝑚(𝑠)𝑑𝑠,𝑡∈𝐼. (24) It follows by integrating the above inequality that ^𝑡0_𝑡^+𝑎₀𝑚(𝑠)𝑑𝑠≤𝑎𝑀+𝑀^∗𝑡0_𝑡^+𝑎₀𝑚(𝑠)𝑑𝑠, (25) which, by assumption (ii), implies that ∫^𝑡0_𝑡^+𝑎₀𝑚(𝑠)𝑑𝑠=0 and so 𝑚(𝑡)≡0 a.e. on 𝐼. The proof of Lemma 14 is therefore complete.

Lemma 15 (see [2, 19]). 𝐻⊂𝐶¹[𝐼,𝐸] is relatively compact if and only if each element 𝑦(𝑡)∈𝐻 and 𝑦^′(𝑡)∈𝐻 are uniformly bounded and equicontinuous on 𝐼.

3. Main Results

We are now in a position to prove our main results concerning the unique positive solution and at least one positive solution.

Theorem 16. Suppose that 𝑃 is a normal solid cone whenever 𝑡∈𝐼; there exists 𝑀>0, 𝑦₁,𝑦₁,𝑦₂,𝑦₂∈𝑃, 𝑦₁≥𝑦₁,𝑦₂≤𝑦₂ such that 𝑓𝑡,𝑦₁,𝑦₂,𝑇𝑦₁,𝑆𝑦₁−𝑓𝑡,𝑦₁,𝑦₂,𝑇𝑦₁,𝑆𝑦₁≥−𝑀𝑦₁−𝑦₁+𝑦₂−𝑦₂+𝑇𝑦₁−𝑇𝑦₁,𝑓𝑡,𝑦₁,𝑦₂,𝑇𝑦₁,𝑆𝑦₁−𝑓𝑡,𝑦₁,𝑦₂,𝑇𝑦₁,𝑆𝑦₁≤𝑀𝑦₁−𝑦₁+𝑦₂−𝑦₂+𝑇𝑦₁−𝑇𝑦₁. (26)(27) Then IVP (1) has the unique positive solution 𝑥^∗ in 𝐶²[𝐼,𝑃].

Proof. Define operator 𝐴∶𝐶[𝐼,𝑃]×𝐶[𝐼,𝑃]→𝐶[𝐼,𝑃] as the following: 𝐴(𝑢,𝑣)=𝑢₀+𝑡𝑢₁+^𝑡₀()𝑓((𝑠)(𝑠),(𝑠),(𝑠𝑡−𝑠𝑠,𝑢,𝑣𝑇𝑢)(𝑆𝑢)())𝑑𝑠,𝑡∈𝐼. (28) Then 𝑢∈𝐶²[𝐼,𝑃] is a solution of IVP (1) if and only if 𝑢∈𝐶[𝐼,𝑃] is a fixed point of 𝐴; that is, 𝑢=𝐴(𝑢,𝑢) for any 𝑢,𝑣,𝑢,𝑣∈𝐶[𝐼,𝑃], 𝑢≥𝑢,𝑣≤𝑣. From (26) and (28), we know that 𝐴(𝑢,𝑣)−𝐴(𝑢,𝑣)≥−𝑀𝑎^𝑡₀[(𝑢−𝑢)+(𝑣−𝑣)+𝑇(𝑢−𝑢))]𝑑𝑠≥−𝑀𝑎(1+𝑎^𝑡₀(𝑢−𝑢)𝑑𝑠+^𝑡₀()()𝑣−𝑣𝑑𝑠≥−𝑀𝑎1+𝑎^𝑡₀(𝑢−𝑢+)𝐴()(𝑣−𝑣𝑑𝑠,𝑢,𝑣−𝐴𝑢,𝑣)≤𝑀𝑎^𝑡₀[(𝑢−𝑢)+(𝑣−𝑣)+𝑇(𝑢−𝑢≤))]𝑑𝑠𝑀𝑎(1+𝑎^𝑡₀(𝑢−𝑢)𝑑𝑠+^𝑡₀()≤()𝑣−𝑣𝑑𝑠𝑀𝑎1+𝑎^𝑡₀(𝑢−𝑢+𝑣−𝑣)𝑑𝑠. (29)(30) From (29) and (30), for any 𝑢,𝑣,𝑢,𝑣∈𝐶[𝐼,𝑃], 𝑢≥𝑢,𝑣≤𝑣 we know that −𝐿(𝑢−𝑢+𝑣−𝑣)≤𝐴(𝑢,𝑣)−𝐴(𝑢,𝑣)≤𝐿(𝑢−𝑢+𝑣−𝑣), (31) where (𝐿𝑤)(𝑡)=𝑀𝑎(1+𝑎)^𝑡₀𝑤(𝑠)𝑑𝑠. (32) Now we prove that 𝑟(𝐿)<1, for 𝑡∈𝐼. From (32), we know that ‖(𝐿𝑢)(𝑡)‖=𝑀𝑎(1+𝑎)𝑡‖𝑢‖. (33) Denote 𝜂=𝑀𝑎(1+𝑎). From (32) and (33), we get 𝐿²𝑢(𝑡)≤𝜂^𝑡₀(𝑠)‖𝑢‖𝐿𝑢)(𝑑𝑠≤𝜂^𝑡₀𝜂𝑠𝑑𝑠=²𝑡²‖𝑢‖2!,∀𝑡∈𝐼. (34) By induction, for any natural number 𝑛, we have 𝐿^𝑛𝑢(𝑡≤𝜂)^𝑛𝑡^𝑛‖𝑢‖𝐿𝑛!,∀𝑡∈𝐼,_𝑛𝑢=max_𝑡∈𝐼𝐿^𝑛𝑢(𝑡≤𝜂)^𝑛𝑡^𝑛‖𝑢‖.𝑛! (35) Therefore, 𝑟(𝐿)=lim_𝑛→∞‖𝐿^𝑛‖^1/𝑛=0<1.
Since 𝑃 is normal solid cone in 𝐶[𝐼,𝐸] and 𝑃 is generate, by virtue of Lemma 9, we know that 𝐴 has a unique positive fixed point 𝑥^∗ in 𝑃; that is, IVP (1) has a unique positive solution 𝑥^∗ in 𝐶²[𝐼,𝑃]⊂𝐶²[𝐼,𝐸].

Remark 17. The conditions of Theorem 16 cannot be obtained by Chen and Zhuang [3], Lakshmikantham et al. [8], and Sun and Liu [20]. We obtain a unique positive solution of IVP (1). The conditions imposed on nonlinear term 𝑓 are sharper, and the result is new.
Denote 𝑀^′(𝑅)=sup{‖𝑓(𝑡,𝑢,𝑣,𝑤,𝑧)‖∶(𝑡,𝑢,𝑣,𝑤,𝑧)∈𝐼×𝐵_𝑅×𝐵_𝑅×𝐵_𝑅×𝐵_𝑅},  𝐵_𝑅={𝑥∈𝑃∶‖𝑥‖≤𝑅}.

Theorem 18. Assume that 𝑓∶𝐼×𝑃×𝑃×𝑃×𝑃→𝑃 satisfying the following conditions.(H₁) For any 𝑅>0, 𝑓 is uniformly continuous on 𝐼×𝐵_𝑅×𝐵_𝑅×𝐵_𝑅×𝐵_𝑅 and lim_𝑅→∞𝑀^′(𝑅)𝑅<𝑎𝑎₀⁻¹,𝑤ℎ𝑒𝑟𝑒𝑎₀=max1,𝑎𝑘₀,𝑎ℎ₀. (36)(H₂) There exists 𝐿>0 such that Υ𝑓(𝑡,𝐵(𝑡),𝐵(𝑡),(𝑇𝐵)(𝑡),(𝑆𝐵)(𝑡))≤𝜆₁Υ(𝐵(𝑡))+𝜆₂Υ((𝑇𝐵)(𝑡))+𝜆₃Υ((𝑆𝐵)(𝑡)) (37)for any bounded set 𝑆⊂𝐶[𝐼,𝑃]⊂𝐶[𝐼,𝐸], 𝑡∈𝐼, and 2(𝜆₁+𝜆₂𝑎𝑘₀)>0, 2𝜆₃ℎ₀𝑎≥0 with 2𝑎(𝜆₁+𝜆₂𝑎𝑘₀+𝜆₃𝑎ℎ₀)<1.
Then IVP (1) has at least one positive solution in 𝐶²[𝐼,𝑃]⊂𝐶²[𝐼,𝐸].

Proof. We first define the operator 𝐹∶𝐶[𝐼,𝑃]→𝐶[𝐼,𝑃] by the formula (𝐹𝑢)(𝑡)=𝑢₀+𝑡𝑢₁+^𝑡₀(𝑡−𝑠)𝑓(𝑠,𝑢(𝑠),𝑢(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠))𝑑𝑠,𝑡∈𝐼. (38) It is easy to know that 𝑢 is a solution of IVP (1) if and only if 𝑢 is a fixed point of 𝐹.
It follows from (H₁) that there exist 0<𝑟<(𝑎𝑎₀)⁻¹ and 𝑅₁>0 such that 𝑀^′(𝑅)𝑅<𝑟,for𝑅≥𝑎₀𝑅₁. (39)
Let 𝑅^∗=max𝑅₁,𝑢₀+𝑎𝑢₁1−𝑎𝑎₀𝑟⁻¹,𝐵_𝐼𝑅^∗=𝑢∈𝐶[𝐼,𝑃]∶‖𝑢‖≤𝑅^∗. (40) Then for any 𝑢∈𝐵_𝐼(𝑅^∗), we have ‖𝑢‖=𝑅^∗≤𝑎₀𝑅^∗,‖𝑇𝑢‖≤𝑎𝑘₀‖𝑢‖≤𝑎𝑘₀𝑅^∗≤𝑎₀𝑅^∗,‖𝑆𝑢‖≤𝑎ℎ₀‖𝑢‖≤𝑎ℎ₀𝑅^∗≤𝑎₀𝑅^∗. (41) It follows from (38) and (39) that ‖𝐹𝑢‖≤𝑢₀+𝑎𝑢₁+𝑎𝑀^′𝑎₀𝑅^∗≤𝑢₀+𝑎𝑢₁+𝑎𝑎₀𝑅^∗𝑟≤𝑅^∗. (42) Set 𝐾=co(𝐵_𝐼(𝑅^∗)). Then 𝐹 is a continuous operator from 𝐾 into 𝐾. It is easy to see from (38), (H₁), and the normality of 𝑃 that 𝐾⊂𝐵_𝐼(𝑅^∗) is uniformly bounded and equicontinuous on 𝐼.
Let 𝐵⊂𝐾⊂𝐶[𝐼,𝑃] be any countable subset satisfying 𝐵=co({𝑢(𝑡)}∩(𝐹𝐵)(𝑡)) for any 𝑡∈𝐼. By applying Lemma 12, we get Υ(𝐵(𝑡))=Υ(𝐵(𝑡))=Υ((𝐹𝐵)(𝑡))≤2^𝑡₀𝜆₁Υ(𝐵(𝑠))+𝜆₂Υ((𝑇𝐵)(𝑠))+𝜆₃Υ((𝑆𝐵)(𝑠))𝑑𝑠≤2^𝑡₀𝜆₁Υ(𝐵(𝑠))+𝜆₂𝑘₀^𝑠₀Υ(𝐵(𝜏))𝑑𝜏+𝜆₃ℎ₀^𝑡0₀^+𝑎Υ(𝐵(𝜏))𝑑𝜏𝑑𝑠=2𝜆₁^𝑡₀Υ(𝐵(𝑠))𝑑𝑠+2𝜆₂𝑘₀^𝑡₀(𝑡−𝑠)Υ(𝐵(𝑠))𝑑𝑠+2𝜆₃ℎ₀𝑡^𝑡0₀^+𝑎Υ(𝐵(𝑠≤))𝑑𝑠2𝜆₁+2𝜆₂𝑎𝑘₀×^𝑡₀Υ(𝐵(𝑠))𝑑𝑠+2𝜆₃𝑎ℎ₀^𝑡0₀^+𝑎Υ(𝐵(𝑠))𝑑𝑠=𝑀^𝑡₀Υ(𝐵(𝑠))𝑑𝑠+𝑀^∗𝑡0₀^+𝑎Υ(𝐵(𝑠))𝑑𝑠=0, (43) where 𝑀=(2𝜆₁+2𝜆₂𝑎𝑘₀),𝑀^∗=2𝜆₃𝑎ℎ₀. Thus, by Ascoli-Arzela theorem, 𝐵 is relative compact in 𝑋=𝐶[𝐼,𝐸]. It follows from Lemma 14 that 𝐹 has a fixed point 𝑥^∗∈𝐾=co(𝐹(𝐵_𝐼(𝑅^∗))) which is a solution of IVP (1). This completes the proof of Theorem 18.

Remark 19. The conditions imposed on nonlinear term do not possess any monotone properties. The results of the paper cannot be obtained by making use of the fixed point theorems on decreasing and increasing operators or mixed monotone operators.

Conflict of Interests

The author declares no conflict of interests.

Acknowledgments

The author is very grateful to Professor Lishan Liu, Professor Tijun Xiao, and Professor R. P. Agarwal for their many valuable comments. The work is supported financially by the Foundation of Shanghai Natural Science (13ZR1430100) and the Foundation of Shanghai Municipal Education Commission (DYL201105).