The Solutions of Mixed Monotone Fredholm-Type Integral Equations in Banach Spaces

Abstract

By introducing new definitions of 𝜙 convex and −𝜑 concave quasioperator and 𝑣₀ quasilower and 𝑢₀ quasiupper, by means of the monotone iterative techniques without any compactness conditions, we obtain the iterative unique solution of nonlinear mixed monotone Fredholm-type integral equations in Banach spaces. Our results are even new to 𝜙 convex and −𝜑 concave quasi operator, and then we apply these results to the two-point boundary value problem of second-order nonlinear ordinary differential equations in the ordered Banach spaces.

1. Introduction

In this paper, we will consider the following nonlinear Fredholm integral equation:𝑢(𝑡)=_𝐼𝐻(𝑡,𝑠,𝑢(𝑠))𝑑𝑠,𝑡∈𝐼, (1) where 𝐼=[𝑎,𝑏] and 𝐻∈𝐶[𝐼×𝐼×𝐸,𝐸], 𝐸 is a real Banach space with the norm ‖⋅‖, and there exists a function 𝐺∈𝐶[𝐼×𝐼×𝐸×𝐸,𝐸] such that for any (𝑡,𝑠,𝑥)∈𝐼×𝐼×𝐸𝐻(𝑡,𝑠,𝑥)=𝐺(𝑡,𝑠,𝑥,𝑥). (2)

Guo and Lakshmikantham [1] introduced the definition of mixed monotone operator and coupled fixed point; there are many good results (see [1–13]). In the special case where 𝐻(𝑡,𝑠,𝑥) is nondecreasing in 𝑥 for fixed 𝑡,𝑠∈𝐼, Guo [2] established an existence theorem of the maximal and minimal solutions for (1) in the ordered Banach spaces by means of monotone iterative techniques. Recently, Jingxian and Lishan [3] and Lishan [4] obtained iterative sequences that converge uniformly to solutions and coupled minimal and maximal quasisolutions of the nonlinear Fredholm integral equations in ordered Banach spaces by using the Möuch fixed point theorem and establishing new comparison results. But these all required the compactness conditions and the monotone conditions in the above papers, and furthermore they did not obtain the unique solutions. In addition, extensive studies have also been carried out to study the global or iterative solutions of initial value problems [8–13].

In this paper, by introducing new definitions of 𝜙 convex and −𝜑 concave quasioperator and 𝑣₀ quasilower and 𝑢₀ quasiupper, by means of the monotone iterative techniques without any compactness conditions which are of the essence in [2–4, 7, 8, 14], we obtain the iterative unique solution of nonlinear mixed monotone Fredholm-type integral equations in Banach spaces and then apply these results to the two-point boundary value problem of second-order nonlinear ordinary differential equations.

2. Preliminaries and Definitions

Let 𝑃 be a cone in 𝐸, that is, a closed convex subset such that 𝜆𝑃⊂𝑃 for any 𝜆≥0 and 𝑃∩{−𝑃}={𝜃}. By means of 𝑃, a partial order ≤ is defined as 𝑥≤𝑦 if and only if 𝑦−𝑥∈𝑃. A cone 𝑃 is said to be normal if there exists a constant 𝑁>0 such that 𝑥,𝑦∈𝐸, 𝜃≤𝑥≤𝑦 implies ‖𝑥‖≤𝑁‖𝑦‖, where 𝜃 denotes the zero element of 𝐸 (see [2, 14]), and we call the smallest number 𝑁 the normal constant of 𝑃 and denote 𝑁_𝑃. The cone 𝑃 is normal if and only if every ordered interval [𝑥,𝑦]={𝑧∈𝐸∶𝑥≤𝑧≤𝑦} is bounded.

Let 𝑃_𝐼={𝑢∈𝐶[𝐼,𝐸]∶𝑢(𝑡)≥𝜃forall𝑡∈𝐼}, where 𝐶[𝐼,𝐸] denotes the Banach space of all the continuous mapping 𝑢∶𝐼→𝐸 with the norm ‖𝑢‖_𝐶=max_𝑡∈𝐼|𝑢(𝑡)|. It is clear that 𝑃_𝐼 is a cone of space 𝐶[𝐼,𝐸], and so it defines a partial ordering in 𝐶[𝐼,𝐸]. Obviously, the normality of 𝑃 implies the normality of 𝑃_𝐼 and the normal constants of 𝑃_𝐼, and 𝑃 are the same.

Let 𝑢₀,𝑣₀∈𝐶[𝐼,𝐸]. Then, 𝑢₀,𝑣₀ are said to be coupled lower and upper quasi-solutions of (1) if 𝑢₀(𝑡)≤_𝐼𝐺𝑡,𝑠,𝑢₀(𝑠),𝑣₀(𝑠)𝑣𝑑𝑠,𝑡∈𝐼,₀(𝑡)≥_𝐼𝐺𝑡,𝑠,𝑣₀(𝑠),𝑢₀(𝑠)𝑑𝑠,𝑡∈𝐼. (3) If the equality in (3) holds, then 𝑢₀,𝑣₀ are said to be coupled quasi-solutions of (1).

We will always assume in this paper that 𝑃 is a normal cone of 𝐸. For any 𝑢₀,𝑣₀∈𝐶[𝐼,𝐸] such that 𝑣₀≤𝑤₀, we define the ordered interval 𝐷=[𝑢₀,𝑣₀]={𝑢∈𝐶[𝐼,𝐸]∶𝑢₀≤𝑢≤𝑣₀}.

Next, we will give the new definition of 𝜙 convex and −𝜑 concave quasi operator and 𝑣₀ quasi-lower and 𝑢₀ quasi-upper.

Definition 1. Suppose that, 𝐺∈𝐶[𝐼×𝐼×𝐸×𝐸,𝐸]. Then 𝐺 is called 𝜙 convex and −𝜑 concave quasi operator, if there exist functions ()×()⟶(),()×()⟶(),𝜙∶0,∞0,∞0,∞𝜑∶0,∞0,∞0,∞ (4) such that(1)𝐺(𝑡,𝑠,𝛼𝑢,𝛽𝑣)≥𝜙(𝛼,𝛽)𝐺(𝑡,𝑠,𝑢,𝑣),𝛼<𝛽,𝛼,𝛽∈(0,∞), for all 𝑢,𝑣∈𝐸, (2)𝐺(𝑡,𝑠,𝛼𝑢,𝛽𝑣)≤𝜑(𝛼,𝛽)𝐺(𝑡,𝑠,𝑢,𝑣),𝛼≥𝛽,𝛼,𝛽∈(0,∞), for all 𝑢,𝑣∈𝐸.

Definition 2. Suppose that 𝐺∈𝐶[𝐼×𝐼×𝐸×𝐸,𝐸], 𝑢₀∈𝑃. Then, 𝐺 is called 𝑢₀ quasi-upper, if for any 𝑢,𝑣∈𝐸, 𝑢,𝑣<𝑢₀ such that ∫_𝐼𝐺(𝑡,𝑠,𝑢,𝑣)𝑑𝑠<𝑢₀.

Definition 3. Suppose that 𝐺∈𝐶[𝐼×𝐼×𝐸×𝐸,𝐸], 𝑣₀∈𝐸. Then, 𝐺 is called 𝑣₀ quasi-lower, if for any 𝑢,𝑣∈𝐸, 𝑢,𝑣>𝑣₀ such that ∫_𝐼𝐺(𝑡,𝑠,𝑢,𝑣)𝑑𝑠>𝑣₀.

Let us list the following assumption for convenience. (𝐻₁)𝐺 is uniformly continuous on 𝐼×𝐼×𝐸×𝐸, and 𝐺 is 𝜙 convex and −𝜑 concave quasi operator.(𝐻₂)𝐺(𝑡,𝑠,𝑥,𝑦) is nondecreasing in 𝑥∈𝐸 for fixed (𝑡,𝑠,𝑦)∈𝐼×𝐼×𝐸. 𝐺(𝑡,𝑠,𝑥,𝑦) is nonincreasing in 𝑦∈𝐸 for fixed (𝑡,𝑠,𝑥)∈𝐼×𝐼×𝐸.(𝐻₃)𝜙(𝛼,𝛽),𝜑(𝛼,𝛽) are all increasing in 𝛼, decreasing in 𝛽, and 𝜙(𝛼₀,𝛽₀)≥𝛼₀, 𝜑(𝛽₀,𝛼₀)≤𝛽₀ and for 𝛼,𝛽∈[𝛼₀,𝛽₀], 𝛼<𝛽, 𝜑𝛽,𝛼−𝜙𝛼,𝛽≤𝑙𝛽−𝛼,0<𝑙<1. (5)

3. The Main Result

The main results of this paper are the following three theorems.

Theorem 4. Let 𝑃 be a normal cone of 𝐸, let 𝑢₀,𝑣₀∈𝑃_𝐼 be coupled lower and upper quasi-solutions of (1). Assume that conditions (𝐻₁),(𝐻₂), and (𝐻₃) hold and(𝐻₄) There exists 𝑤₀∈𝑃_𝐼 such that 𝑢₀≤𝑤₀≤𝑣₀, and for 𝛼₀,𝛽₀∈(0,∞) of (𝐻₃) such that 𝑢₀≥𝛼₀𝑤₀, 𝛽₀𝑤₀≥𝑣₀.
Then, (1) has a unique solution 𝑥^∗(𝑡)∈𝐷=[𝑢₀,𝑣₀], and for any initial 𝑥₀,𝑦₀∈[𝑢₀,𝑣₀], one has 𝑥_𝑛(𝑡)⟶𝑥^∗(𝑡),𝑦_𝑛(𝑡)⟶𝑥^∗(𝑡),uniformlyon𝑡∈𝐼as𝑛⟶∞, (6) where {𝑥_𝑛(𝑡)},{𝑦_𝑛(𝑡)} are defined as 𝑥_𝑛(𝑡)=_𝐼𝐺𝑡,𝑠,𝑥_𝑛−1(𝑠),𝑦_𝑛−1(𝑠)𝑦𝑑𝑠,_𝑛(𝑡)=_𝐼𝐺𝑡,𝑠,𝑦_𝑛−1(𝑠),𝑥_𝑛−1(𝑠)𝑑𝑠,𝑡∈𝐼. (7)

Proof. We first define the operator 𝐴∶[𝑢₀,𝑣₀]×[𝑢₀,𝑣₀]→𝐶[𝐼,𝐸] by the formula 𝐴(𝑢,𝑣)=_𝐼𝐺(𝑡,𝑠,𝑢(𝑠),𝑣(𝑠))𝑑𝑠. (8) It follows from the assumption (𝐻₂) that 𝐴 is a mixed monotone operator, that is, 𝐴(𝑢,𝑣) is nondecreasing in 𝑢∈[𝑢₀,𝑣₀] and nonincreasing in 𝑣∈[𝑢₀,𝑣₀], and 𝑢₀≤𝐴(𝑢₀,𝑣₀), 𝐴(𝑣₀,𝑢₀)≤𝑣₀.
By (7), we have 𝑥_𝑛(𝑡)=𝐴(𝑥_𝑛−1(𝑡),𝑦_𝑛−1(𝑡)), 𝑦_𝑛(𝑡)=𝐴(𝑦_𝑛−1(𝑡),𝑥_𝑛−1(𝑡)) and set 𝑤_𝑛(𝑡)=𝐴(𝑤_𝑛−1(𝑡),𝑤_𝑛−1(𝑡)) for initial 𝑤₀ in (𝐻₄), and we also define that 𝑢_𝑛(𝑡)=𝐴𝑢_𝑛−1(𝑡),𝑣_𝑛−1(𝑡),𝑣_𝑛(𝑡)=𝐴𝑣_𝑛−1(𝑡),𝑢_𝑛−1(𝑡). (9) Since 𝐴 is a mixed monotone operator, it is easy to see that 𝑢₀≤𝑢₁≤⋯≤𝑢_𝑛≤⋯≤𝑣_𝑛≤⋯≤𝑣₁≤𝑣₀,𝑢_𝑛≤𝑤_𝑛≤𝑣_𝑛. (10)
Obviously, by induction, it is easy to see that 𝑢_𝑛≥𝛼_𝑛𝑤_𝑛,𝑣_𝑛≤𝛽_𝑛𝑤_𝑛,𝑛=0,1,…,𝛼₀≤𝛼₁≤⋯≤𝛼_𝑛≤⋯≤1≤⋯≤𝛽_𝑛≤⋯≤𝛽₁≤𝛽₀, (11)(12) where 𝛼_𝑛=𝜙(𝛼_𝑛−1,𝛽_𝑛−1), 𝛽_𝑛=𝜑(𝛽_𝑛−1,𝛼_𝑛−1),𝑛=1,2,….
In fact, by the assumption (𝐻₄), we have that inequality (11) holds as 𝑛=0. Suppose that inequality (11) holds as 𝑛=𝑘, that is, 𝑢_𝑘≥𝛼_𝑘𝑤_𝑘, 𝑣_𝑘≤𝛽_𝑘𝑤_𝑘. Then, as 𝑛=𝑘+1, by the assumption (𝐻₃), we have 𝑢_𝑘+1=𝐴𝑢_𝑘,𝑣_𝑘=_𝐼𝐺𝑡,𝑠,𝑢_𝑘(𝑠),𝑣_𝑘(𝑠)≥𝑑𝑠_𝐼𝐺𝑡,𝑠,𝛼_𝑘𝑤_𝑘,𝛽_𝑘𝑤_𝑘𝛼𝑑𝑠≥𝜙_𝑘,𝛽_𝑘𝐼𝐺𝑡,𝑠,𝑤_𝑘(𝑠),𝑤_𝑘(𝑠)𝑑𝑠=𝛼_𝑘+1𝑤_𝑘+1,𝑣_𝑘+1=𝐴𝑣_𝑘,𝑢_𝑘=_𝐼𝐺𝑡,𝑠,𝑣_𝑘(𝑠),𝑢_𝑘(𝑠)≤𝑑𝑠_𝐼𝐺𝑡,𝑠,𝛽_𝑘𝑤_𝑘,𝛼_𝑘𝑤_𝑘𝛽𝑑𝑠≤𝜑_𝑘,𝛼_𝑘𝐼𝐺𝑡,𝑠,𝑤_𝑘(𝑠),𝑤_𝑘(𝑠)𝑑𝑠=𝛽_𝑘+1𝑤_𝑘+1. (13) Then, it is easy to show by induction that inequality (11) holds.
For inequality (12), by 𝑢_𝑘+1≤𝑤_𝑘+1≤𝑣_𝑘+1 and the above discussion, we have 0<𝛼_𝑘+1≤1≤𝛽_𝑘+1. Obviously, it follows from the assumption (𝐻₄) that 𝛼₀≤𝛼₁, 𝛽₁≤𝛽₀. Suppose that 𝛼_𝑘−1≤𝛼_𝑘, 𝛽_𝑘≤𝛽_𝑘−1, so it is easy to show by (𝐻₃) that 𝜙𝛼_𝑘−1,𝛽_𝑘−1≤𝜙𝛼_𝑘,𝛽_𝑘,𝜑𝛽_𝑘,𝛼_𝑘≤𝜑𝛽_𝑘−1,𝛼_𝑘−1, (14) that is, 𝛼_𝑘≤𝛼_𝑘+1, 𝛽_𝑘+1≤𝛽_𝑘. Then, it is easy to show by induction that inequality (12) holds.
Then, it follows from the inequality (12) that there exist limits of the sequences {𝛼_𝑛},{𝛽_𝑛}. Suppose that there exist 𝛼,𝛽 such that 𝛼_𝑛→𝛼, 𝛽_𝑛→𝛽, and  𝑛→∞, and by (𝐻₃), we also have 0≤𝛽_𝑛−𝛼_𝑛=𝜑𝛽_𝑛−1,𝛼_𝑛−1−𝜙𝛼_𝑛−1,𝛽_𝑛−1≤𝑙𝛽_𝑛−1−𝛼_𝑛−1≤⋯≤𝑙^𝑛𝛽₀−𝛼₀, (15) they 0<𝑙<1, and taking limits in the above inequality as 𝑛→∞, we have 𝛼=𝛽.
Next, we will show that the sequences {𝑢_𝑛},{𝑣_𝑛} are all Cauchy sequences on 𝐷.
In fact, by (10) and (11), for any natural number 𝑝, we know that 𝜃≤𝑢_𝑛+𝑝−𝑢_𝑛≤𝑣_𝑛−𝑢_𝑛≤𝛽_𝑛−𝛼_𝑛𝑢₀,𝜃≤𝑣_𝑛−𝑣_𝑛+𝑝≤𝑣_𝑛−𝑢_𝑛≤𝛽_𝑛−𝛼_𝑛𝑢₀. (16) By the normality of 𝑃_𝐼 and (15), we have 𝑢_𝑛+𝑝−𝑢_𝑛𝐶≤𝑁_𝑃𝑙^𝑛𝛽₀−𝛼₀𝑢_0𝐶,𝑣_𝑛−𝑣_{𝑛+𝑝𝐶}≤𝑁_𝑃𝑙^𝑛𝛽₀−𝛼₀𝑢_0𝐶, (17) where 𝑁_𝑃 is a normal constant. So {𝑢_𝑛},{𝑣_𝑛} are all Cauchy sequences on 𝐷, and then there exists 𝑢^∗,𝑣^∗∈[𝑢₀,𝑣₀] such that lim_𝑛→∞𝑢_𝑛=𝑢^∗, lim_𝑛→∞𝑣_𝑛=𝑣^∗.
It is easy to know by (10) and (11) that 𝜃≤𝑣_𝑛−𝑢_𝑛≤𝛽_𝑛𝑤_𝑛−𝛼_𝑛𝑤_𝑛≤𝛽_𝑛−𝛼_𝑛𝑢₀≤𝑙^𝑛𝛽₀−𝛼₀𝑢₀, (18) so by the normality of 𝑃_𝐼, we have 𝑣_𝑛−𝑢_𝑛𝐶≤𝑁_𝑃𝑙^𝑛𝛽₀−𝛼₀𝑢_0𝐶, (19) and taking limits in the above inequality as 𝑛→∞, we have 𝑥^∗=𝑢^∗=𝑣^∗∈[𝑢₀,𝑣₀], and for any natural number 𝑛, we also have 𝑢_𝑛≤𝑥^∗≤𝑣_𝑛,  𝑡∈𝐼.
Then, by the mixed monotone quality of 𝐴 we have 𝑢_𝑛+1=𝐴𝑢_𝑛,𝑣_𝑛≤𝐴𝑥^∗,𝑥^∗≤𝐴𝑣_𝑛,𝑢_𝑛=𝑣_𝑛+1, (20) and taking limits in above inequality as 𝑛→∞, we know that 𝑥^∗=𝐴𝑥^∗,𝑥^∗, (21) that is, 𝑥^∗∈[𝑢₀,𝑣₀] is the fixed point of 𝐴; thus, 𝑥^∗ is the solution of (1) on 𝐷=[𝑢₀,𝑣₀].
Furthermore, we will show that the solution is unique. Suppose that 𝑦^∗∈[𝑢₀,𝑣₀] satisfy 𝑦^∗=𝐴(𝑦^∗,𝑦^∗). Then, by the mixed monotone quality of 𝐴 and induction, for any natural number 𝑛, it is easy to have that 𝑢_𝑛≤𝑦^∗≤𝑣_𝑛. Then, by the normality of 𝑃_𝐼 and taking limits in the above inequality as 𝑛→∞ and the above discussion, we have 𝑦^∗=𝑥^∗.
For any initial 𝑥₀,𝑦₀∈[𝑢₀,𝑣₀], by (7) and (8), the mixed monotone quality of 𝐴 and induction, for any natural number 𝑛, we have 𝑢_𝑛(𝑡)≤𝑥_𝑛(𝑡)≤𝑣_𝑛(𝑡), 𝑢_𝑛(𝑡)≤𝑦_𝑛(𝑡)≤𝑣_𝑛(𝑡), 𝑡∈𝐼. Then, the normality of 𝑃_𝐼 and (19) imply that 𝑥_𝑛−𝑢_𝑛𝐶≤𝑁_𝑃𝑙^𝑛𝛽₀−𝛼₀𝑢_0𝐶,𝑦_𝑛−𝑢_𝑛𝐶≤𝑁_𝑃𝑁_𝑃𝑙^𝑛𝛽₀−𝛼₀𝑢_0𝐶. (22) Thus, the sequence {𝑥_𝑛(𝑡)},{𝑦_𝑛(𝑡)} all converges uniformly to 𝑥^∗(𝑡) on 𝑡∈𝐼. This completes the proof of Theorem 4.

Theorem 5. Let 𝑃 be a normal cone of 𝐸, let 𝑢₀,𝑣₀∈𝑃_𝐼 be coupled lower and upper quasi-solutions of (1). Assume that conditions (𝐻₁), (𝐻₂), and (𝐻₃) hold.(𝐻^′₄)𝐺 is 𝑢₀ quasi-upper, and there exists 𝑤₀∈𝑃_𝐼 such that 𝑤₀<𝑢₀<𝑣₀, and there exist 𝛼₀=sup{𝛼>0∶𝑢₀≥𝛼𝑤₀}, 𝛽₀=inf{𝛽>0∶𝑣₀≤𝛽𝑤₀}.
Then, (1) has a unique solution 𝑥^∗(𝑡)∈𝐷=[𝑢₀,𝑣₀], and for any initial 𝑥₀,𝑦₀∈[𝑢₀,𝑣₀], one has 𝑥_𝑛(𝑡)⟶𝑥^∗(𝑡),𝑦_𝑛(𝑡)⟶𝑥^∗(𝑡),uniformlyon𝑡∈𝐼as𝑛⟶∞, (23) where {𝑥_𝑛(𝑡)},{𝑦_𝑛(𝑡)} are defined as 𝑥_𝑛(𝑡)=_𝐼𝐺𝑡,𝑠,𝑥_𝑛−1(𝑠),𝑦_𝑛−1(𝑠)𝑦𝑑𝑠,_𝑛(𝑡)=_𝐼𝐺𝑡,𝑠,𝑦_𝑛−1(𝑠),𝑥_𝑛−1(𝑠)𝑑𝑠,𝑡∈𝐼. (24)

Proof. We first define the operator 𝐴∶[𝑢₀,𝑣₀]×[𝑢₀,𝑣₀]→𝐶[𝐼,𝐸] by the formula 𝐴(𝑢,𝑣)=_𝐼𝐺(𝑡,𝑠,𝑢(𝑠),𝑣(𝑠))𝑑𝑠. 8^′ It follows from the assumption (𝐻₂) that 𝐴 is a mixed monotone operator, that is, 𝐴(𝑢,𝑣) is nondecreasing in 𝑢∈[𝑢₀,𝑣₀] and nonincreasing in 𝑣∈[𝑢₀,𝑣₀] and 𝑢₀≤𝐴(𝑢₀,𝑣₀), 𝐴(𝑣₀,𝑢₀)≤𝑣₀. By (7), we have 𝑥_𝑛(𝑡)=𝐴(𝑥_𝑛−1(𝑡), 𝑦_𝑛−1(𝑡)),𝑦_𝑛(𝑡)=𝐴(𝑦_𝑛−1(𝑡), 𝑥_𝑛−1(𝑡)) and set 𝑤_𝑛(𝑡)=𝐴(𝑤_𝑛−1(𝑡), 𝑤_𝑛−1(𝑡)), and we also define 𝑢_𝑛(𝑡)=𝐴𝑢_𝑛−1(𝑡),𝑣_𝑛−1(𝑡),𝑣_𝑛(𝑡)=𝐴𝑣_𝑛−1(𝑡),𝑢_𝑛−1(𝑡). (25) Since 𝐴 is a mixed monotone operator, it is easy to see that 𝑢₀≤𝑢₁≤⋯≤𝑢_𝑛≤⋯≤𝑣_𝑛≤⋯≤𝑣₁≤𝑣₀. 10^′
Because 𝐺 is 𝑢₀ quasi-upper and 𝑤₀<𝑢₀, we have 𝑤₁(𝑡)=𝐴𝑤₀(𝑡),𝑤₀(𝑡)=_𝐼𝐺𝑡,𝑠,𝑤₀(𝑠),𝑤₀(𝑠)𝑑𝑠<𝑢₀. (26) So for any natural number 𝑛, by induction, we know that 𝑤_𝑛(𝑡)=𝐴(𝑤_𝑛−1(𝑡),𝑤_𝑛−1(𝑡))<𝑢₀.
It is easy to see by induction that 𝑢_𝑘≥𝛼_𝑘𝑤_𝑘,𝑣_𝑘≤𝛽_𝑘𝑤_𝑘,𝑎₀≤𝑎₁≤⋯≤𝑎_𝑘≤⋯≤𝑏_𝑘≤⋯≤𝑏₁≤𝑏₀, 11^′12^′ where 𝛼_𝑘=𝜙(𝛼_𝑘−1,𝛽_𝑘−1), 𝛽_𝑘=𝜑(𝛽_𝑘−1,𝛼_𝑘−1),𝑘=1,2,….
In fact, by the assumptions (𝐻₁) and (𝐻₃) and the above discussion, as 𝑛=0, we have𝑢₁=𝐴𝑢₀,𝑣₀=_𝐼𝐺𝑡,𝑠,𝑢₀(𝑠),𝑣₀(𝑠)≥𝑑𝑠_𝐼𝐺𝑡,𝑠,𝛼₀𝑤₀,𝛽₀𝑤₀𝛼𝑑𝑠≥𝜙₀,𝛽_0𝐼𝐺𝑡,𝑠,𝑤₀(𝑠),𝑤₀(𝑠)𝑑𝑠=𝛼₁𝑤₁,𝑣₁=𝐴𝑣₀,𝑢₀=_𝐼𝐺𝑡,𝑠,𝑣₀(𝑠),𝑢₀(𝑠)≤𝑑𝑠_𝐼𝐺𝑡,𝑠,𝛽₀𝑤₀,𝛼₀𝑤₀𝛽𝑑𝑠≤𝜑₀,𝛼_0𝐼𝐺𝑡,𝑠,𝑤₀(𝑠),𝑤₀(𝑠)𝑑𝑠=𝛽₁𝑤₁. (27)(28) By the above two inequalities and assumption (𝐻₃), we have 𝑎₀≤𝛼₁=𝜙𝛼₀,𝛽₀≤𝜑𝛽₀,𝛼₀=𝛽₁≤𝑏₀. (29) Suppose that for 𝑘−1 we have 𝑢_𝑘−1≥𝛼_𝑘−1𝑤_𝑘−1, 𝑣_𝑘−1≤𝛽_𝑘−1𝑤_𝑘−1, and 𝛼_𝑘−2≤𝛼_𝑘−1≤𝛽_𝑘−1≤𝛽_𝑘−2. Then, for 𝑘+1, by the assumption (𝐻₃), we have 𝑢_𝑘=𝐴𝑢_𝑘−1,𝑣_𝑘−1=_𝐼𝐺𝑡,𝑠,𝑢_𝑘−1(𝑠),𝑣_𝑘−1(𝑠)≥𝑑𝑠_𝐼𝐺𝑡,𝑠,𝛼_𝑘−1𝑤_𝑘−1,𝛽_𝑘−1𝑤_𝑘−1𝛼𝑑𝑠≥𝜙_𝑘−1,𝛽_𝑘−1𝐼𝐺𝑡,𝑠,𝑤_𝑘−1(𝑠),𝑤_𝑘−1(𝑠)𝑑𝑠=𝛼_𝑘𝑤_𝑘,𝑣_𝑘=𝐴𝑣_𝑘−1,𝑢_𝑘−1=_𝐼𝐺𝑡,𝑠,𝑣_𝑘−1(𝑠),𝑢_𝑘−1(𝑠)≤𝑑𝑠_𝐼𝐺𝑡,𝑠,𝛽_𝑘−1𝑤_𝑘−1,𝛼_𝑘−1𝑤_𝑘−1𝛽𝑑𝑠≤𝜑_𝑘−1,𝛼_𝑘−1𝐼𝐺𝑡,𝑠,𝑤_𝑘−1(𝑠),𝑤_𝑘−1(𝑠)𝑑𝑠=𝛽_𝑘𝑤_𝑘. (30) By the above two inequalities and assumption (𝐻₃), we have 𝛼_𝑘−1=𝜙𝛼_𝑘−2,𝛽_𝑘−2≤𝜙𝛼_𝑘−1,𝛽_𝑘−1=𝛼_𝑘≤𝛽_𝑘=𝜑𝛽_𝑘−1,𝛼_𝑘−1≤𝜑𝛽_𝑘−2,𝛼_𝑘−2=𝛽_𝑘−1. (31) Then, it is easy to show by induction that inequalities 11^′ and 12^′ hold.
The following proof is similar to that of Theorem 4. This completes the proof of Theorem 5.

By a similar argument to that of Theorem 5, we obtain the following results.

Theorem 6. Let 𝑃 be a normal cone of 𝐸, and let 𝑢₀,𝑣₀∈𝑃_𝐼 be coupled lower and upper quasi-solutions of (1). Assume that condition (𝐻₁),(𝐻₂), and (𝐻₃) hold.(𝐻₄^′′)𝐺 is 𝑣₀ quasi-lower, and there exists 𝑤₀∈𝑃_𝐼 such that 𝑢₀<𝑣₀<𝑤₀, and there exist 𝛼₀=sup{𝛼>0∶𝑢₀≥𝛼𝑤₀}, 𝛽₀=inf{𝛽>0∶𝑣₀≤𝛽𝑤₀}.
Then, (1) has a unique solution 𝑥^∗(𝑡)∈𝐷=[𝑢₀,𝑣₀], and for any initial 𝑥₀,𝑦₀∈[𝑢₀,𝑣₀], one has 𝑥_𝑛(𝑡)⟶𝑥^∗(𝑡),𝑦_𝑛(𝑡)⟶𝑥^∗(𝑡),uniformlyon𝑡∈𝐼as𝑛⟶∞, (32) where {𝑥_𝑛(𝑡)},{𝑦_𝑛(𝑡)} are defined as 𝑥_𝑛(𝑡)=_𝐼𝐺𝑡,𝑠,𝑥_𝑛−1(𝑠),𝑦_𝑛−1(𝑠)𝑦𝑑𝑠,_𝑛(𝑡)=_𝐼𝐺𝑡,𝑠,𝑦_𝑛−1(𝑠),𝑥_𝑛−1(𝑠)𝑑𝑠,𝑡∈𝐼. (33)

4. Applications

Consider the following two-point BVP in the Banach space: −𝑢^′′()[],𝑢(0)(1)=𝑓𝑡,𝑢,𝑡∈𝐽=0,1=𝑢=0, (34) where 𝑓∈𝐶[𝐽×𝑃,𝑃], 𝑃 is a cone in a real Banach space 𝐸. Suppose that there exists a mapping 𝑔∈𝐶[𝐽×𝑃×𝑃,𝑃] such that 𝑓(𝑡,𝑥)=𝑔(𝑡,𝑥,𝑥), and that 𝑔 satisfies the following conditions:(𝐶₁)𝑔 is uniformly continuous on 𝐽×𝑃×𝑃, and 𝐺 is 𝜙 convex and −𝜑 concave quasi operator,(𝐶₂)𝑔(𝑡,𝑥,𝑦) is nondecreasing in 𝑥∈𝑃 for fixed (𝑡,𝑦)∈𝐽×𝑃, and 𝑔(𝑡,𝑥,𝑦) is nonincreasing in 𝑦∈𝑃, for fixed (𝑡,𝑥)∈𝐽×𝑃,(𝐶₃) there exist the bounded nonnegative Lebesgue integrable functions 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡), and 𝑑(𝑡) satisfying ∫_𝐽𝑎(𝑠)𝑑𝑠<8, ∫_𝐽𝑐(𝑠)𝑑𝑠<8 such that 𝑎(𝑡)𝑥+𝑏(𝑡)≤𝑔𝑡,𝑥,𝑦≤𝑐(𝑡)𝑥+𝑑(𝑡),𝑡∈𝐽,𝑥,𝑦∈𝑃. (35)

It is well known that 𝑢∈𝐶²[𝐽,𝑃] is a solution of BVP(34) in 𝐶²[𝐽,𝑃] if and only if 𝑢∈𝐶[𝐽,𝑃] is a solution of the following integral equation: 𝑢(𝑡)=_𝐽ℎ(𝑡,𝑠)𝑔(𝑠,𝑢(𝑠),𝑢(𝑠))𝑑𝑠,𝑡∈𝐽, (36) where ℎ()=𝑡()𝑠()𝑡,𝑠1−𝑠,𝑡≤𝑠,1−𝑡,𝑡>𝑠. (37)

Lemma 7. If assumption (𝐶₃) holds, then there exists 𝑢₀,𝑣₀∈𝐶[𝐽,𝑃] such that 𝑢₀(𝑡)≤_𝐽ℎ(𝑡,𝑠)𝑔𝑠,𝑢₀(𝑠),𝑣₀(𝑠)𝑣𝑑𝑠,𝑡∈𝐽,₀(𝑡)≥_𝐽ℎ(𝑡,𝑠)𝑔𝑠,𝑣₀(𝑠),𝑢₀(𝑠)𝑑𝑠,𝑡∈𝐽. (38)

Proof. In fact, let 𝐿₁𝑢(𝑡)=_𝐽ℎ()𝑎(𝑠)𝑢(𝑠)𝑥𝑡,𝑠𝑑𝑠,₀(𝑡)=_𝐽ℎ()𝑏(𝑠)𝐿𝑡,𝑠𝑑𝑠,𝑡∈𝐽,₂𝑣(𝑡)=_𝐽ℎ()𝑐(𝑠)𝑣(𝑠)𝑦𝑡,𝑠𝑑𝑠,₀(𝑡)=_𝐽ℎ(𝑡,𝑠)𝑑(𝑠)𝑑𝑠,𝑡∈𝐽. (39) Obviously, by assumption (𝐶₃), we can get that ‖𝐿₁‖≤max_𝑡∈𝐼(𝑡(1−𝑡)/2)∫_𝐽𝑎(𝑠)𝑑𝑠=(1/8)∫_𝐽𝑎(𝑠)𝑑𝑠<1, then the equation (𝐼−𝐿₁)𝑢=𝑥₀ has a unique solution 𝑢₀(𝑡)=𝐼−𝐿₁⁻¹𝑥₀=^∞_𝑛=0𝐿^𝑛₁𝑥₀∈𝑃_𝐼. (40)
Similarly, the equation (𝐼−𝐿₂)𝑣=𝑦₀ has a unique solution 𝑣₀(𝑡)=𝐼−𝐿₂⁻¹𝑦₀=^∞_𝑛=0𝐿^𝑛₂𝑦₀∈𝑃_𝐼. (41) Thus, by assumption (𝐶₃), for any 𝑡∈𝐽, we have _𝐽ℎ(𝑡,𝑠)𝑔𝑠,𝑢₀(𝑠),𝑣₀(𝑠)≥𝑑𝑠_𝐽ℎ(𝑡,𝑠)(𝑎(𝑠)𝑥+𝑏(𝑠))𝑑𝑠=𝐿₁𝑢₀(𝑡)+𝑥₀(𝑡)=𝑢₀(𝑡),_𝐽ℎ(𝑡,𝑠)𝑔𝑠,𝑣₀(s),𝑢₀(𝑠)≤𝑑𝑠_𝐽ℎ(𝑡,𝑠)𝑐(𝑠)𝑣₀(𝑠)+𝑏(𝑠)𝑑𝑠=𝐿₂𝑣₀(𝑡)+𝑦₀(𝑡)=𝑣₀(𝑡), (42) that is, (38) holds.

Theorem 8. Let 𝑃 be a normal cone of 𝐸. Assume that (𝐶₁) and (𝐶₃) hold,(𝐶₄) there exists 𝑤₀∈𝑃_𝐼 and 𝑢₀,𝑣₀ in (38) of Lemma 7 such that 𝑢₀<𝑤₀<𝑣₀, and also there exists 𝛼₀,𝛽₀∈(0,∞) such that 𝑢₀≥𝛼₀𝑤₀, 𝛽₀𝑤₀≥𝑣₀,(𝐶₅)𝜙(𝛼,𝛽),𝜑(𝛼,𝛽) are all increasing in 𝛼, decreasing in 𝛽 and 𝜙(𝛼₀,𝛽₀)≥𝛼₀, 𝜑(𝛽₀,𝛼₀)≤𝛽₀, for 𝛼,𝛽∈[𝛼₀,𝛽₀], 𝛼<𝛽, 𝜑𝛽,𝛼−𝜙𝛼,𝛽≤𝑙𝛽−𝛼,0<𝑙<1. (43)
Then, (34) has a unique solution 𝑥^∗(𝑡)∈𝐷=[𝑢₀,𝑣₀], and for any initial 𝑥₀,𝑦₀∈[𝑢₀,𝑣₀], one has 𝑥_𝑛(𝑡)⟶𝑥^∗(𝑡),𝑦_𝑛(𝑡)⟶𝑥^∗(𝑡),uniformlyon𝑡∈𝐼as𝑛⟶∞, (44) where {𝑥_𝑛(𝑡)},{𝑦_𝑛(𝑡)} are defined as 𝑥_𝑛(𝑡)=_𝐽ℎ(𝑡,𝑠)𝑔𝑠,𝑥_𝑛−1(𝑠),𝑦_𝑛−1(𝑠)𝑦𝑑𝑠,_𝑛(𝑡)=_𝐽ℎ(𝑡,𝑠)𝑔𝑠,𝑦_𝑛−1(𝑠),𝑥_𝑛−1(𝑠)𝑑𝑠,𝑡∈𝐽. (45)

Proof. It is easy to see by conditions (𝐶₁) and (𝐶₂) that 𝐺(𝑡,𝑠,𝑥,𝑦)=ℎ(𝑡,𝑠)𝑔(𝑠,𝑥,𝑦) satisfy the conditions (𝐻₁) and (𝐻₂) of Theorem 4. By (𝐶₃) and (38), we have 𝑢₀,𝑣₀∈𝐶[𝐼,𝑃] as coupled lower and upper quasi-solutions of (34).
Thus, the assumption (𝐻₁)--(𝐻₄) of Theorem 4 is satisfied from the assumption (𝐶₁)--(𝐶₅) of Theorem 8. The conclusion of Theorem 8 follows from Theorem 4.

Example 9. In fact, we can construct the function 𝑓(𝑡,𝑥) in Theorem 8.

Let 𝑓()1𝑡,𝑥=𝑔𝑡,𝑥,𝑦=𝑥+𝑦[],𝜙1,𝑡∈0,1𝛼,𝛽=sin𝛼+,𝜋2𝛽𝛼∈0,2,𝜑𝛼,𝛽=3𝛼−5𝛽, (46) then 𝐺()𝑔()1𝑡,𝑠,𝛼𝑥,𝛽𝑦=ℎ𝑡,𝑠𝑠,𝛼𝑥,𝛽𝑦=ℎ𝑡,𝑠𝛼𝑥+()1𝛽𝑦≥ℎ𝑡,𝑠sin𝛼+12𝛽𝑥+𝑦=𝜙𝛼,𝛽𝐺𝑡,𝑠,𝑥,𝑦,𝐺()𝑔()1𝑡,𝑠,𝛼𝑥,𝛽𝑦=ℎ𝑡,𝑠𝑠,𝛼𝑥,𝛽𝑦=ℎ𝑡,𝑠𝛼𝑥+(1𝛽𝑦≤ℎ𝑡,𝑠)3𝛼−5𝛽𝑥+𝑦=𝜑𝛼,𝛽𝐺𝑡,𝑠,𝑥,𝑦. (47)(48) Thus, 𝐺 is 𝜙 convex and −𝜑 concave quasi operator and thus satisfies (𝐶₁).

It is easy to check that 𝑔(𝑡,𝑥,𝑦) is nondecreasing in 𝑥 for fixed (𝑡,𝑦) and is nonincreasing in 𝑦 for fixed (𝑡,𝑥) and thus satisfies (𝐶₂).

There exist 𝑎(𝑡)=𝑡/2, 𝑏(𝑡)=𝑡/100, 𝑐(𝑡)=2𝑡, and 𝑑(𝑡)=1000𝑡 satisfying ¹₀𝑎(𝑠)1𝑑𝑠=2¹₀1𝑡𝑑𝑡=4<8,₁₀𝑐(𝑠)𝑑𝑠=2¹₀𝑠𝑑𝑠=1<8, (49) such that 𝑎(𝑡)𝑥+𝑏(𝑡)≤𝑔𝑡,𝑥,𝑦≤𝑐(𝑡)𝑥+𝑑(𝑡). (50) Thus, (𝐶₃) holds.

There exist 𝑢₀=¹₀ℎ(𝑡,𝑠)𝑢₀(𝑠)+1𝑣₀(𝑠)𝑣𝑑𝑠,₀=2𝑢₀=¹₀ℎ(𝑡,𝑠)𝑣₀(𝑠)+1𝑢₀(𝑠)𝑑𝑠. (51) Choose 𝑤₀=(3/2)𝑢₀ such that 𝑢₀<𝑤₀<𝑣₀, and also there exist 𝛼₀=2/3, 𝛽₀=4/3 such that 𝑢₀=𝛼₀32𝑢₀=𝛼₀𝑤₀,𝛽₀32𝑢₀=2𝑢₀=𝑣₀. (52) Thus, (𝐶₄) is satisfied.

𝜙(𝛼,𝛽),𝜑(𝛼,𝛽) are all increasing in 𝛼 and nondecreasing in 𝛽, 𝜙23,432=sin3+1()≥22×4/33=𝛼₀,𝜑43,234=3×32−5×3=23≤43=𝛽₀, (53) and for 𝛼,𝛽∈[2/3,4/3], 𝛼<𝛽, we have 𝜑1𝛽,𝛼−𝜙𝛼,𝛽=3𝛽−5𝛼−sin𝛼−≤2𝛽99.100𝛽−𝛼 (54) Thus, (𝐶₅) also holds.

Acknowledgment

The author was supported financially by Natural Science Foundation of China (11071141) and the Project of Shandong Province Higher Educational Science and Technology Program (J13LI12, J10LA62).