Abstract and Applied Analysis

Volume 2013 (2013), Article ID 164851, 8 pages

http://dx.doi.org/10.1155/2013/164851

Research Article

Convexity of Solutions for an Iterative Equation in Banach Spaces

Xiaobing Gong1,2

¹Key Laboratory of Numerical Simulation of Sichuan Province, Neijiang, Sichuan 641100, China

²College of Mathematics and Information Science, Neijiang Normal University, Neijiang, Sichuan 641100, China

Received 18 March 2013; Accepted 2 August 2013

Academic Editor: Abdelghani Bellouquid

Copyright © 2013 Xiaobing Gong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

By applying Schauder's fixed point theorem we investigate the existence of increasing (decreasing) solutions of the iterative equation and further give conditions under which those solutions are convex or concave. As corollaries we obtain results on iterative equation in Banach spaces, where .

1. Introduction

Iterative root problem [1, 2], being a weak version of the problem of embedding flows, plays an important role in the theory of dynamical systems. As a natural generalization of the iterative root problem, the polynomial-like iterative equation

where is a subset of a linear space over , is a given function, s are real constants, is the unknown function, and is the th iterate of , that is, and for all , is one of the important iterative functional equations [3, 4] and was studied in many papers. For , while some works (e.g., [5–11]) are contributed to the case of linear , there are many results given to the case of nonlinear , for example, [12, 13] for , [14] for general , [15] for smoothness, and [16] for analyticity. Some efforts were also devoted to (1) in high-dimensional spaces such as in [17, 18]; radially monotonic solutions were discussed in high-dimensional Euclidean spaces by properties of orthogonal group in [18], and the existence of convex solutions was proved by introducing a partial order in Banach spaces in [17]. A general iterative equation can be presented as

where and . In 1995, solutions of (2) were discussed in [19], and as continuations of   [19], and solutions were studied in [20, 21], respectively, for . In 2007, by lifting maps on the unit circle and maps on the torus , the existence, uniqueness, and stability of continuous solutions for (2) were proved on the unit circle in [22]. A more general iterative functional equation

was studied in [23, 24] in high-dimensional spaces, where is a operator. Equation (3) is a generalization of iterative equation (2). In fact, if , then (3) becomes (2). In [23], the existence of Lipschitzian solutions for (3) was proved on a compact convex subset of , and by using this result, the existence of Lipschitzian solutions for equation

was investigated on a compact interval of and a compact convex subset of . Later, the results were partially generalized to an arbitrary closed (not necessarily convex) subset of a Banach space and the existence of solutions for iterative functional equations

was proved in [24], where s are bounded linear operators on the Banach space.

Convexity is an important property of functions and the study of convexity for iterative equations can be traced to 1968 when Kuczma and Smajdor [25] investigated the convexity of iterative roots. Some recent results can be found from [17, 26–28]. In [27, 28], convexity of solutions for (1) was discussed on a compact interval, and in [26], nondecreasing convex solutions for (1) on open intervals were discussed. In [17], convexity of solutions for (1) was studied in Banach spaces. Up to now, there are no further results on monotonicity and convexity of solutions for (2) and (3) in Banach spaces. In fact, there are much more difficulties on monotonicity and convexity of solutions for these two equations in Banach spaces.

In this paper we study monotonicity and convexity of solutions for (2) and (3) in Banach spaces and generalize the results in [17]. Using Schauder’s fixed point theorem, we discuss increasing (decreasing) solutions for (3) and further give conditions under which those solutions are convex or concave. As corollaries, we obtain results on (2). The uniqueness and continuous dependence of those solutions are also discussed.

2. Preliminaries

As in [17], in order to discuss monotonicity and convexity of solutions in Banach spaces, we need to introduce a partial order. For convenience, we use the conventions of [17]. As in [29], a nonempty subset of a real vector space is called a cone if implies that for all . A nonempty and nontrivial (, where denotes the zero element of ) subset is called an order cone in if is a convex cone and satisfies . Having chosen such an order cone in , we can define a partial order in , simply called the -order, if

A real vector space equipped with a -order is called an ordered vector space, abbreviated by OVS and denoted by . A real Banach space associated with a -order is called an ordered real Banach space, abbreviated by OBS and denoted by , if is closed. One can define increasing (decreasing) operators as in [30] in an ordered real vector space . An operator is said to be increasing (resp., decreasing) in the sense of the -order if implies (resp., ). An operator , where is a convex subset, is said to be convex (resp., concave) in the sense of the -order if (resp., ) for all and for every pair of distinct comparable points (i.e., either or ).

Let be a compact convex subset of an ordered real Banach space with nonempty interior, and let consist of all continuous functions . is a Banach space equipped with the norm . For , define

Similar to Lemma  2.2 in [17], , , , and are compact convex subsets of .

As shown in [29, 30], an order cone in an ordered real Banach space is said to be normal if there exists a constant such that if in . The smallest constant , denoted by , is called the normal constant of .

3. Main Result

We first discuss monotonicity and convexity of solutions for iterative functional equation (3) in the ordered real Banach space such that is normal and . Consider (3) with the following hypothesis:

(H1).

3.1. Increasing and Decreasing Solutions

Theorem 1. Suppose that (H1) holds and , where is a constant. Let such that

where is an increasing function and for any . If there exists such that

and is continuous, then (3) has a solution .

Proof. Define a mapping by

We first prove that is a self-mapping on . Obviously, is well defined and . By for any and the definition of , . Further, when are not comparable, that is, and , by (8) and , we have

where the monotonicity of the function is employed, which implies that

because of (9). When are comparable, suppose that . By the definition of , ; thus . Hence, by (8), we get

Consequently, we have

where the monotonicity of the function is employed, which implies that

because of (9). Thus, (12) and (15) imply that is a self-mapping on . The continuity of implies that is continuous on . Since is a compact convex subset, by Schauder’s fixed point theorem, we see that has a fixed point . Thus, is an increasing solution of (3). The proof is completed.

The following is devoted to decreasing solutions.

Theorem 2. Suppose that (H1) holds and . Let such that

where is an increasing function and for any . If condition (9) holds for a constant and is continuous, then (3) has a solution .

The proof is almost the same as the proof of Theorem 1; we omit it here.

3.2. Convex and Concave Solutions

On the basis of the last subsection, we can discuss convexity of solutions for (3).

Theorem 3. Suppose that (H1) holds and , where is a constant. Let such that

where is an increasing function and for any . If there exists such that

and is continuous, then (3) has a solution .

In order to prove Theorem 3, we need the following lemma.

Lemma 4 (see [17, Lemma 3.1]). Let be an ordered real Banach space. Then composition is convex (resp., concave) if both and are convex (resp., concave) and increasing. In particular, for increasing convex (resp., concave) operator , the iterate is also convex (resp., concave).

Proof of Theorem 3. Define a mapping as in Theorem 1. In order to prove that is a self-mapping on , it suffices to prove that is convex in the sense of -order on . In fact, by (17), we know is increasing and convex on . Hence, by Lemma 4 and , is convex in the sense of -order on . Consequently, by ,

for every pair of distinct comparable points and . So is an self-mapping on . The remaining part of the proof is the same as the proof of Theorem 1. This completes the proof.

Similarly, we can prove the following results for concavity of solutions.

Theorem 5. Suppose that (H1) holds and , where is a constant. Let such that

where is a increasing function and for any . If there exists such that

and is continuous, then (3) has a solution .

4. Iterative Equation in Banach Spaces

In this section, we are going to discuss monotonicity and convexity of solutions for (2) in the ordered real Banach space such that is normal and . Consider (2) with the following hypotheses:

(H2), .

Before discussing convexity, we prove the existence of increasing and decreasing solutions of (2).

4.1. Increasing and Decreasing Solutions

We first study increasing solutions. Consider (2) with the following hypothesis:

(H3) such that

Theorem 6. Suppose that (H2) and (H3) hold and , where is a constant. If for any and

for a constant , then (2) has a solution . Additionally, if

then the solution is unique in and depends continuously on .

In order to prove Theorem 6, we need the following lemmas.

Lemma 7 (see [17, Lemma 3.2]). Let be an ordered real Banach space such that is normal, and let (resp., , , and ), where . Then

Lemma 8 (see [17, Lemma 3.3]). Let be an ordered real Banach space, and let , where . Then

for all in .

Proof of Theorem 6. We apply Theorem 1. Define

By (H2), . Next we prove that is an operator from to such that

By the definition of , for , we have

if , and

if are not comparable, that is, and . Note that

if , and

if are not comparable, that is, and . By (H3), we get

if , and

if are not comparable, that is, and , where is employed. Hence, . Let . Obviously, function is increasing on . By the definition of , we have for any . Hence, .

Let , and let ; by (23),

Next, we prove that is continuous. By (H3), for any , by Lemma 7, we have

By Theorem 1, there exists an such that

By (H2), we have

This completes the proof of existence. Under the additional hypothesis (24), we see from (36) that the mapping defined as (10) is a contraction mapping on the closed subset . The uniqueness of solution follows. Let be solutions of (2) with the given functions respectively. By the uniqueness, and . Hence,

By (24), we have , implying that the solution of (2) depends continuously on . The proof is completed.

Similarly, we can prove the following result for decreasing solutions. We need the following hypothesis:

(H4) such that

Theorem 9. Suppose that (H2) and (H4) hold and all even order iterates in (2) vanish. Let , where is a constant. If , for all and

for a constant , then (2) has a solution . Additionally, if

then the solution is unique in and depends continuously on .

By using Lemma 8, the proof is almost the same as the proof of Theorem 6. We omit it here.

4.2. Convex and Concave Solutions

On the basis of the last subsection we can discuss convexity of solutions for (2). Consider (2) with the following hypothesis:

(H5) if  , then

Theorem 10. Suppose that (H2), (H3), and (H5) hold and , where is a constant. If for any and

for a constant , then (2) has a solution . Additionally, if

then the solution is unique in and depends continuously on .

Proof. Similar to Theorem 6, by applying Theorem 3, it suffices to prove that is convex in the sense of -order. In fact, each is convex in the sense of -order because is increasing and convex by Lemma 4. Hence, for every distinct comparable point , suppose ; by (H3) and (H5), we have

The proof is completed.

Similarly, we can prove the following results for concavity of solutions. We need the following hypothesis:

(H6)if  , then

Theorem 11. Suppose that (H2), (H3), and (H6) hold and , where is a constant. If for any and

for a constant , then (2) has a solution . Additionally, if

then the solution is unique in and depends continuously on .

Example 12. Let equipped with the norm for . Let

a subset of . Then, the equation

is an iterative equation of the form (2) in the infinite-dimensional setting, where

and . Note that

is a normal order cone in and . Then, is a compact convex subset of the ordered real Banach space . Clearly, (H2) is satisfied and

If , then

For any ,

where . Hence, (H3) is satisfied.

If or in , then

Hence, (H5) is satisfied. Similar to Example  4.2 in [17], . for any . In this case,

for all ; that is, inequality (44) holds. By Theorem 10, (51) has an increasing solution .

Additionally, if , we have

Hence, the solution is unique and depends continuously on .

It is difficult to discuss the convexity of solutions for (2) without hypothesis (H2) and (3) without hypothesis (H1) because of the difficulties in discussing the inverse of a mapping in Banach spaces. The reason also leads to difficulties in discussing the convexity of solutions for (2) and (3) by applying the method in [23].

Acknowledgments

The author is grateful to the editor and the referee for their valuable comments and encouragement. This work is supported by Key Project of Sichuan Provincial Department of Education (12ZA086).

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