Differentiable Solutions of Equations Involving Iterated Functional Series

Abstract

The nonmonotonic differentiable solutions of equations involving iterated functional series are investigated. Conditions for the existence, uniqueness, and stability of such solutions are given. These extend earlier results due to Murugan and Subrahmanyam.

1. Introduction

Let 𝑓 be a self-mapping on a topological space 𝑋 and 𝑓^𝑚 denote the mth iterate of 𝑓, that is, 𝑓^𝑚=𝑓∘𝑓^𝑚−1,𝑓⁰=𝑖𝑑,𝑚=1,2,…. Let 𝐶(𝑋,𝑋) be the set of all continuous self-mappings on 𝑋. Equations having iteration as their main operation, that is, including iterates of the unknown mapping, are called iterative equations. It is one of the most interesting classes of functional equations [1–4], because it concludes the problem of iterative roots [1, 5, 6], that is, finding 𝑓∈𝐶(𝑋,𝑋) such that 𝑓^𝑛 is identical to a given 𝐹∈𝐶(𝑋,𝑋). As a natural generalization of the problem of iterative roots, a class of iterative equations named as polynomial-like iterative equation𝜆₁𝑓(𝑥)+𝜆₂𝑓²(𝑥)+⋯+𝜆_𝑛𝑓^𝑛(𝑥)=𝐹(𝑥),𝑥∈𝐼=[𝑎,𝑏],(1.1)had fascinated many scholars, such as Dhombres [7], Zhao [8], Mukherjea and Ratti [9]. Despite their nice constructive proofs, the classical methods prevented them from obtaining more fruitful results. In 1986, Zhang [10] constructed an interesting operator called “structural operator” for (1.1) and used the fixed point theory in Banach space to get the solutions of (1.1). Hence he overcame the difficulties encountered by the formers. By means of this method, Zhang and Si made a series of work concerning these qualitative problems, such as [11–15]. In 2002, Kulczycki and Tabor [16] improved Zhang's method and investigated the existence of Lipschitzian solutions of the iterative functional equation^∞_𝑛=1𝜆_𝑛𝑓^𝑛(𝑥)=𝐹(𝑥),𝑥∈𝐵,(1.2)where 𝐵 is a compact convex subset of ℝ^𝑛 and 𝐹∶𝐵→𝐵 are a given Lipschitz function. It is easy to see that (1.1) is the special case of (1.2) with 𝜆_𝑖=0, 𝑖=𝑛+1,… and 𝐵=[𝑎,𝑏].

Recently Zhang et al. [17] and Xu et al. [18] developed this method and they have got the nonmonotonic, convex, and decreasing continuous solutions of (1.1). In fact they have answered the open problem 2 which was proposed by J. Zhang et al. [19].

The problem of differentiable solutions of iterative equation had also fascinated many scholars' attentions. In Zhang [12] and Si [15], the 𝐶¹ and 𝐶² solutions of (1.1) are considered. In Wang and Si [20] the differentiable solutions of the below equation𝐻𝑥,𝜙^𝑛1(𝑥),…,𝜙^𝑛𝑖(𝑥)=𝐹(𝑥),𝑥∈𝐼=[𝑎,𝑏](1.3) are considered. Murugan and Subrahmanyam [21, 22] discussed the existence and uniqueness of 𝐶¹ solutions of the more general equations^∞_𝑖=1𝜆_𝑖𝐻_𝑖𝑓^𝑖(𝑥)=𝐹(𝑥),𝑥∈𝐼=[𝑎,𝑏],(1.4)_∞𝑖=1𝜆_𝑖𝐻_𝑖𝑥,𝜙^𝑎𝑖1(𝑥),…,𝜙^{𝑎𝑖𝑖𝑛}(𝑥)=𝐹(𝑥),𝑥∈𝐼=[𝑎,𝑏],(1.5)which involve iterated functional series. All the above references only got the increasing differentiable solutions for the above equations because they only considered the case that 𝐹 is increasing. Li and Deng [23] considered the 𝐶¹ solutions of the (1.2). In [24] 𝐶¹ solutions of the equation^∞_𝑛=1𝜆_𝑛(𝑥)𝑓^𝑛(𝑥)=𝐹(𝑥),𝑥∈𝐵,(1.6)where 𝐵 is a compact convex subset of ℝ^𝑛 and 𝜆_𝑛(𝑥)∶𝐵→𝑅 are discussed. Li and Deng [23] and Li [24] work in higher dimensional case, they do not require monotonicity. It should be pointed out that Mai and Liu [25] made an important contribution to 𝐶^𝑚 solutions of iterative equations. Mai and Liu proved the existence, uniqueness of 𝐶^𝑚 solutions of a relatively general kind of iterative equations𝐺𝑥,𝑓(𝑥),…,𝑓^𝑛(𝑥)=0,𝑥∈𝐽,(1.7)where 𝐽 is a connected closed subset of ℝ and 𝐺∈𝐶^𝑚(𝐽^𝑛+1,ℝ),𝑛≥2. Here 𝐶^𝑚(𝐽^𝑛+1,ℝ) denotes the set of all 𝐶^𝑚 mappings from 𝐽^𝑛+1 to ℝ.

Inspired by the above work, we will investigate (1.4) and extend earlier results due to Murugan and Subrahmanyam in two directions. In [21] the authors only get increasing solutions of (1.4), so the present paper will investigate the nonmonotonic differentiable solutions of (1.4) and give conditions for the existence, uniqueness, and stability of such solutions. In [21] the authors require not only all coefficients are nonnegative but also 𝐻_𝑖,𝑖=2,3,… are all increasing, but we will find that those conditions are not necessary.

2. Preliminaries

Let 𝐼=[𝑎,𝑏] and 𝐽=[𝑐,𝑑] be two compact intervals. Let 𝐶¹(𝐼,𝐽) be the set of all continuously differentiable functions from 𝐼 to 𝐽. Then 𝐶¹(𝐼,𝐽) is a closed subset of the Banach space 𝐶¹(𝐼,ℝ) consisting of all continuously differentiable functions from 𝐼 to ℝ with norm ‖⋅‖_𝑐1. Here the norm ‖⋅‖_𝑐1 is defined by ‖𝜑‖_𝑐1=‖𝜑‖_𝑐0+‖𝜑′‖_𝑐0,𝜑∈𝐶¹(𝐼,ℝ), where ‖𝜑‖_𝑐0=max_𝑥∈𝐼|𝜑(𝑥)| and 𝜑^′ is the derivative of 𝜑. Following Zhang [12], we define the families of functions𝔄(𝐼,𝐽,𝑚,𝑀,𝑁)=𝜑∈𝐶¹(𝐼,𝐽)∶𝜑(𝑎)=𝑐,𝜑(𝑏)=𝑑,𝑚≤𝜑^′𝜑(𝑥)≤𝑀,_′(𝑥₁)−𝜑^′(𝑥₂)≤𝑁𝑥₁−𝑥₂,∀𝑥,𝑥₁,𝑥₂,𝔄∈𝐼_′(𝐼,𝐽,𝑚,𝑀,𝑁)=𝜑∈𝐶¹(𝐼,𝐽)∶𝜑(𝑎)=𝑑,𝜑(𝑏)=𝑐,𝑚≤𝜑^′𝜑(𝑥)≤𝑀,_′(𝑥₁)−𝜑^′(𝑥₂)≤𝑁𝑥₁−𝑥₂,∀𝑥,𝑥₁,𝑥₂∈𝐼,(2.1)where 0≤𝑚<𝑀, 𝑁>0 are all constants.

Lemma 2.1. Both 𝔄(𝐼,𝐽,𝑚,𝑀,𝑁) and 𝔄^′(𝐼,𝐽,𝑚,𝑀,𝑁) are compact convex subsets of 𝐶¹(𝐼,𝐽).

The Lemma above can be proved by a method which is contained in the proof of Theorem 3.1 in [12].

Lemma 2.2 (see [12]). Suppose that 𝜑,𝜙∈𝔄(𝐼,𝐽,𝑚,𝑀,𝑁) (or 𝜑,𝜙∈𝔄^′(𝐼,𝐽,𝑚,𝑀,𝑁)). Then for 𝑛=1,2,…,(𝜑^𝑛)^′(𝑥)≤𝑀^𝑛,∀𝑥∈𝐼,(𝜑^𝑛)^′(𝑥₁)−(𝜑^𝑛)^′(𝑥₂)≤𝑁^2𝑛−2_{𝑖=𝑛−1}𝑀^𝑖𝑥₁−𝑥₂,∀𝑥₁,𝑥₂𝜑∈𝐼,_𝑛−𝜙^𝑛_𝑐0≤^𝑛_𝑖=1𝑀^𝑖−1𝜑−𝜙_𝑐0,(𝜑^𝑛)^′−(𝜙^𝑛)^′_𝑐0≤𝑛𝑀^𝑛−1𝜑^′−𝜙^′_𝑐0+𝑄(𝑛)𝑁^𝑛−1_𝑖=1(𝑛−𝑖)𝑀^{𝑛+𝑖−2}𝜑−𝜙_𝑐0,(2.2)where 𝑄(1)=0,𝑄(𝑚)=1 as 𝑚=2,3,… and (𝜑^𝑛)^′ denotes 𝑑𝜑^𝑛/𝑑𝑥.

We can get the following Lemma from [12]. In [12] the author proved that the lemma is valid for 𝐶¹(𝐼,𝐼), but we find it is also valid for 𝐶¹(𝐼,𝐽).

Lemma 2.3 (see [12]). Suppose that 𝑓∈𝐶¹(𝐼,𝐽) satisfies that0<𝛿≤𝑓^′𝑓(𝑥),∀𝑥∈𝐼,_′(𝑥₁)−𝑓^′(𝑥₂)≤𝑀^∗𝑥₁−𝑥₂,∀𝑥₁,𝑥₂∈𝐼,(2.3)where 𝛿, 𝑀^∗ are positive constants. Then(𝑓⁻¹)^′(𝑦₁)−(𝑓⁻¹)^′(𝑦₂)≤𝑀^∗𝛿³𝑦₁−𝑦₂,∀𝑦₁,𝑦₂∈𝐽.(2.4)

Lemma 2.4 (see [18]). If both 𝑓_𝑖∶𝐼→𝐽,𝑖=1,2 are homeomorphisms from 𝐼 to 𝐽 such that𝑓_𝑖(𝑥₁)−𝑓_𝑖(𝑥₂)≤𝐾𝑥₁−𝑥₂,∀𝑥₁,𝑥₂∈𝐼,(2.5)where 𝐾 is a positive constant. Then𝑓₁−𝑓_2𝑐0≤𝐾𝑓₁⁻¹−𝑓₂⁻¹_𝑐0.(2.6)

3. Differentiable Solutions of (1.4)

3.1. Existence of Solutions

Let {𝜆_𝑖}^∞_𝑖=1 be coefficients of (1.4) and 𝐼=[𝑎,𝑏]. For any 𝑓∈𝔄(𝐼,𝐼,0,𝑀,𝑁)(or𝔄′(𝐼,𝐼,0,𝑀,𝑁)) define 𝐴=∑^∞_𝑖=1𝜆_𝑖𝐻_𝑖(𝑓^𝑖−1(𝑎)) and 𝐵=∑^∞_𝑖=1𝜆_𝑖𝐻_𝑖(𝑓^𝑖−1(𝑏)). It is easy to see that both the convergence and the value of 𝐴,𝐵 have nothing to do with the choice of 𝑓.

Theorem 3.1. Suppose 𝑀>1, 𝐿 are positive constants and 𝐻₁∈𝔄(𝐼,𝐼,𝑚₁,𝑀₁,𝑁₁),𝐻_𝑖∈𝔄(𝐼,𝐼,0,𝑀_𝑖,𝑁_𝑖)or𝐻_𝑖∈𝔄^′(𝐼,𝐼,0,𝑀_𝑖,𝑁_𝑖) for 𝑖=2,3,…, where 𝑚₁>0 and 𝑀_𝑖≥1,𝑁_𝑖 are positive constants for 𝑖=1,2,…. Assume further the following conditions:𝐾₀=𝜆₁𝑚₁−^∞_𝑖=2𝜆_𝑖𝑀_𝑖𝑀^𝑖−1𝐾>0,₁=1(𝑀−1)^∞_𝑖=1𝜆_𝑖+1𝑀_𝑖+1𝑀^𝑖−1(𝑀^𝑖𝐾−1)<∞,₀−𝐾₁𝑀²𝐾>0,₂=^∞_𝑖=1𝜆_𝑖𝑁_𝑖𝑀^𝑖−1(𝑀^𝑖−1)<∞,−∞<𝐴<𝐵<∞,(3.1) hold. Then for any given 𝐹∈𝔄(𝐼,𝐽,0,𝐾₀𝑀,𝐿)(or𝔄′(𝐼,𝐽,0,𝐾₀𝑀,𝐿)), (1.4) has a solution 𝑓∈𝔄(𝐼,𝐼,0,𝑀,𝑀^∗)(or𝔄′(𝐼,𝐼,0,𝑀,𝑀^∗)), where 𝑀^∗≥(𝐿+𝐾₄𝑀²)/(𝐾₀−𝐾₁𝑀²), 𝐾₄=∑^∞_𝑖=1|𝜆_𝑖|𝑁_𝑖𝑀^2(𝑖−1) and 𝐼=[𝑎,𝑏], 𝐽=[𝐴,𝐵].

As in [22], firstly we give the following three Lemmas which lead directly to the proof of Theorem 3.1. In the sequel we denote𝑁=𝐿+𝐾₄𝑀²𝐾₀−𝐾₁𝑀².(3.2)

Lemma 3.2. Under the assumptions of Theorem 3.1 the following series: 𝐾₃=𝜆₁𝑀₁+^∞_𝑖=2𝜆_𝑖𝑀_𝑖𝑀^𝑖−1,𝐾₄=^∞_𝑖=1𝜆_𝑖𝑁_𝑖𝑀^2(𝑖−1),𝐾₅=^∞_𝑘=1𝜆_𝑘+1𝑀_𝑘+1^𝑘_𝑖=1𝑀^𝑖−1,𝐾₆=^∞_𝑘=1𝜆_𝑘+1𝑀_𝑘+1𝑘𝑀^𝑘−1,𝐾₇=^∞_𝑘=1𝜆_𝑘+1𝑁_𝑘+1𝑀^𝑘𝑘_𝑖=1𝑀^𝑖−1,𝐾₈=^∞_𝐾=2𝜆_𝑘+1𝑀_𝑘+1^𝑘−1_𝑗=1(𝑘−𝑗)𝑀^{𝑘+𝑗−2},(3.3) are all convergent.

Proof. The convergence of 𝐾₃, 𝐾₄, 𝐾₅, 𝐾₆, and 𝐾₇ is easy to be verified. As mentioned in [21], the equality^𝑛−1_𝑖=1(𝑛−𝑖)𝑥^{𝑛+𝑖−2}=𝑥^𝑛−1𝑥^𝑛−1(𝑥−1)²−𝑛𝑥−1,𝑥≠1(3.4)holds. We get that𝐾₈=^∞_𝐾=2|𝜆_𝑘+1|𝑀_𝑘+1𝑀^𝑘−1𝑀^𝑘−1(𝑀−1)²−𝑘.𝑀−1(3.5)By the convergence of 𝐾₁ and 𝐾₆, 𝐾₈ is also convergent.

Lemma 3.3. Under the assumptions of Theorem 3.1, for each 𝑓∈𝔄(𝐼,𝐼,0,𝑀,𝑁)(or𝔄′(𝐼,𝐼,0,𝑀,𝑁)) the mapping 𝐿_𝑓∶𝐼→ℝ defined by𝐿_𝑓(𝑥)=^∞_𝑖=1𝜆_𝑖𝐻_𝑖𝑓^𝑖−1(𝑥),𝑥∈𝐼(3.6)has the following properties: (i)𝐿_𝑓∈𝔄(𝐼,𝐽,𝐾₀,𝐾₃,𝐾₁𝑁+𝐾₄);(ii)𝐿_𝑓⁻¹∈𝔄(𝐽,𝐼,1/𝐾₃,1/𝐾₀,(𝐾₁𝑁+𝐾₄)/(𝐾₀)³), where 𝐼=[𝑎,𝑏] and 𝐽=[𝐴,𝐵].

Proof. For any 𝑓∈𝔄(𝐼,𝐼,0,𝑀,𝑁)(or𝔄′(𝐼,𝐼,0,𝑀,𝑁)), we have𝐿_𝑓(𝑎)=^∞_𝑖=1𝜆_𝑖𝐻_𝑖𝑓^𝑖−1(𝑎)<𝐿_𝑓(𝑏)=^∞_𝑖=1𝜆_𝑖𝐻_𝑖𝑓^𝑖−1(𝑏).(3.7)It is easy to see that for any 𝑥∈𝐼0<𝜆₁𝑚₁−^∞_𝑖=2𝜆_𝑖𝑀_𝑖𝑀^𝑖−1≤^∞_𝑖=1𝜆_𝑖𝐻^′_𝑖𝑓^𝑖−1(𝑥)𝑓^𝑖−1′(𝑥)≤𝜆₁𝑀₁+^∞_𝑖=2𝜆_𝑖𝑀_𝑖𝑀^𝑖−1.(3.8)We have for any 𝑥∈𝐼0<𝐾₀≤𝐿^′_𝑓(𝑥)≤𝐾₃,(3.9)and for any 𝑦∈𝐽10<𝐾₃≤𝐿_𝑓^−1′1(𝑦)≤𝐾₀.(3.10)Thus 𝐿_𝑓∶𝐼→𝐽 is an orientation-preserving diffeomorphism.

By Lemma 2.2 we can see that for any 𝑥₁,𝑥₂∈𝐼, 𝐿^′_𝑓(𝑥₁)−𝐿^′_𝑓(𝑥₂)=^∞_𝑖=1𝜆_𝑖𝐻^′_𝑖𝑓^𝑖−1(𝑥₁)𝑓^𝑖−1′(𝑥₁)−^∞_𝑖=1𝜆_𝑖𝐻^′_𝑖𝑓^𝑖−1(𝑥₂)𝑓^𝑖−1′(𝑥₂)≤^∞_𝑖=1𝜆_𝑖𝐻^′_𝑖𝑓^𝑖−1(𝑥₁)⋅𝑓^𝑖−1′(𝑥₁)−𝑓^𝑖−1′(𝑥₂)+𝐻^′_𝑖𝑓^𝑖−1(𝑥₁)−𝐻^′_𝑖𝑓^𝑖−1(𝑥₂)⋅𝑓^𝑖−1′(𝑥₂)≤^∞_𝑖=2𝜆_𝑖𝑀_𝑖𝑁^2(𝑖−2)_{𝑗=𝑖−2}𝑀^𝑗+^∞_𝑖=1𝜆_𝑖𝑁_𝑖𝑀^2(𝑖−1)𝑥₁−𝑥₂=𝐾₁𝑁+𝐾₄𝑥₁−𝑥₂.(3.11)By (3.9), (3.11), and Lemma 2.3 we get for any 𝑦₁,𝑦₂∈𝐽:𝐿_𝑓^−1′(𝑦₁)−𝐿_𝑓^−1′𝑦₂≤𝐾₁𝑁+𝐾₄(𝐾₀)³𝑦₁−𝑦₂.(3.12)

Lemma 3.4. Under the assumptions of Theorem 3.1, for each 𝑓₁,𝑓₂∈𝔄(𝐼,𝐼,0,𝑀,𝑁)(or𝔄′(𝐼,𝐼,0,𝑀,𝑁)) the mappings 𝐿_𝑓1,𝐿_𝑓2∶𝐼→𝐽 satisfy the following inequalities: (i)‖𝐿_𝑓⁻¹₁−𝐿_𝑓⁻¹₂‖_𝑐0≤𝐾₅/𝐾₀‖𝑓₁−𝑓₂‖_𝑐0;(ii)‖(𝐿_𝑓⁻¹₁)′−(𝐿_𝑓⁻¹₂)′‖_𝑐0≤(((𝐾₁𝑁+𝐾₄)𝐾₅+(𝐾₇+𝑁𝐾₈)𝐾₀)/(𝐾₀)³)‖𝑓₁−𝑓₂‖_𝑐0 +𝐾₆/(𝐾₀)²‖𝑓^′₁−𝑓^′₂‖_𝑐0.

Proof. Firstly we have 𝐿_𝑓1−𝐿_𝑓2𝑐0≤^∞_𝑘=1𝜆_𝑘+1⋅𝐻_𝑘+1∘𝑓^𝑘₁−𝐻_𝑘+1∘𝑓^𝑘_2𝑐0(2.2)≤^∞_𝑘=1𝜆_𝑘+1𝑀_𝑘+1^𝑘_𝑖=1𝑀^𝑖−1⋅𝑓₁−𝑓_2𝑐0,(3.13)and thus𝐿_𝑓⁻¹₁−𝐿≤_𝑓⁻¹_2𝑐0^(Lemma(2.4))1𝐾₀𝐿_𝑓1−𝐿_𝑓2𝑐0≤𝐾₅𝐾₀𝑓₁−𝑓_2𝑐0.(3.14)Secondly we get𝐿_𝑓⁻¹₁^′−𝐿_𝑓⁻¹₂^′_𝑐0=max_𝑦∈𝐽1𝐿^′_𝑓1(𝐿_𝑓⁻¹₁−1(𝑦))𝐿^′_𝑓2(𝐿_𝑓⁻¹₂(𝑦))_(3.9)≤max_𝑦∈𝐽𝐿^′_𝑓2𝐿_𝑓⁻¹₂(𝑦)−𝐿^′_𝑓1𝐿_𝑓⁻¹₁(𝑦)(𝐾₀)²≤1(𝐾₀)²⋅max_𝑦∈𝐽𝐿^′_𝑓2𝐿_𝑓⁻¹₂(𝑦)−𝐿^′_𝑓2𝐿_𝑓⁻¹₁+1(𝑦)(𝐾₀)²⋅max_𝑦∈𝐽𝐿^′_𝑓2𝐿_𝑓⁻¹₁(𝑦)−𝐿^′_𝑓1𝐿_𝑓⁻¹₁(𝑦)_(3.11)≤𝐾₁𝑁+𝐾₄(𝐾₀)²⋅max_𝑦∈𝐽𝐿_𝑓⁻¹₂(𝑦)−𝐿_𝑓⁻¹₁+1(𝑦)(𝐾₀)²⋅max_𝑥∈𝐼𝐿^′_𝑓2(𝑥)−𝐿^′_𝑓1=𝐾(𝑥)₁𝑁+𝐾₄(𝐾₀)²⋅𝐿_𝑓⁻¹₂−𝐿_𝑓⁻¹_1𝑐0+1(𝐾₀)²⋅𝐿^′_𝑓2−𝐿^′_𝑓1𝑐0.(3.15)Notice that𝐿^′_𝑓2−𝐿^′_𝑓1𝑐0≤∑^∞_𝑘=1𝜆_𝑘+1⋅(𝐻_𝑘+1∘𝑓^𝑘₂)^′−(𝐻_𝑘+1∘𝑓^𝑘₁)^′_𝑐0(3.16)and for 𝑘=1,2,…,𝐻_𝑘+1∘𝑓^𝑘₂^′−𝐻_𝑘+1∘𝑓^𝑘₁^′_𝑐0=𝐻^′_𝑘+1∘𝑓^𝑘₂⋅𝑓^𝑘₂^′−𝐻^′_𝑘+1∘𝑓^𝑘₁⋅𝑓^𝑘₁^′_𝑐0≤𝐻^′_𝑘+1∘𝑓^𝑘₂−𝐻^′_𝑘+1∘𝑓^𝑘_1𝑐0⋅𝑓^𝑘₂^′_𝑐0+𝐻^′_𝑘+1∘𝑓^𝑘_1𝑐0𝑓^𝑘₂^′−𝑓^𝑘₁^′_𝑐0≤𝑀^𝑘𝑁^𝑘_𝑘+1𝑖=1𝑀^𝑖−1𝑓₂−𝑓_1𝑐0+𝑀_𝑘+1𝑘𝑀^𝑘−1𝑓^′₂−𝑓^′_1𝑐0+𝑀_𝑘+1𝑄(𝑘)𝑁^𝑘−1_𝑖=1(𝑘−𝑖)𝑀^{𝑘+𝑖−2}𝑓₂−𝑓_1𝑐0,(3.17)then𝐿^′_𝑓2−𝐿^′_𝑓1𝑐0≤𝐾₇+𝑁𝐾₈𝑓₂−𝑓_1𝑐0+𝐾₆𝑓^′₂−𝑓^′_1𝑐0.(3.18)Finally we get𝐿_𝑓⁻¹₁^′−𝐿_𝑓⁻¹₂^′_𝑐0≤𝐾₁𝑁+𝐾₄𝐾₅+𝐾₇+𝑁𝐾₈𝐾₀𝐾₀³𝑓₁−𝑓_2𝑐0+𝐾₆𝐾₀²𝑓^′₁−𝑓^′_2𝑐0.(3.19)

Proof of Theorem 3.1.. For any 𝑓∈𝔄(𝐼,𝐼,0,𝑀,𝑁)(or𝔄′(𝐼,𝐼,0,𝑀,𝑁)) we define Θ(𝑓) as follows:Θ(𝑓)=𝐿_𝑓⁻¹∘𝐹,(3.20)and denote Θ(𝑓)=𝑔 for convenience. Clearly 𝑔∈𝐶¹(𝐼,𝐼), 𝑔(𝑎)=𝑎, 𝑔(𝑏)=𝑏, (or 𝑔(𝑎)=𝑏,𝑔(𝑏)=𝑎), and (3.10) yields that for any 𝑥∈𝐼, 𝑔^′(𝑥)=𝐿_𝑓^−1′(𝐹(𝑥))⋅𝐹^′≤𝐾(𝑥)⁰𝑀𝐾₀=𝑀.(3.21)Furthermore by (3.12) we get that for any 𝑥₁,𝑥₂∈𝐼, 𝑔^′(𝑥₁)−𝑔^′(𝑥₂)≤𝐿_𝑓^−1′𝐹𝑥₁⋅𝐹^′𝑥₁−𝐹^′𝑥₂+𝐿_𝑓^−1′𝐹𝑥₁−𝐿_𝑓^−1′𝐹𝑥₂⋅𝐹^′𝑥₂≤1𝐾₀𝐿𝑥₁−𝑥₂+𝐾₀𝐾𝑀⋅¹𝑁+𝐾₄𝐾₀³⋅𝐾₀𝑀𝑥₁−𝑥₂=𝑁𝑥₁−𝑥₂.(3.22)So 𝑔∈𝔄(𝐼,𝐼,0,𝑀,𝑁)(or𝔄′(𝐼,𝐼,0,𝑀,𝑁)), which means that Θ(𝔄(𝐼,𝐼,0,𝑀,𝑁))⊂𝔄(𝐼,𝐼,0,𝑀,𝑁) (orΘ(𝔄′(𝐼,𝐼,0,𝑀,𝑁))⊂𝔄′(𝐼,𝐼,0,𝑀,𝑁)).
Secondly we prove thatΘ∶𝔄(𝐼,𝐼,0,𝑀,𝑁)⟶𝔄(𝐼,𝐼,0,𝑀,𝑁)(orΘ∶𝔄^′(𝐼,𝐼,0,𝑀,𝑁)⟶𝔄^′(𝐼,𝐼,0,𝑀,𝑁))(3.23) is continuous. For any 𝑓₁,𝑓₂∈𝔄(𝐼,𝐼,0,𝑀,𝑁)(or𝔄′(𝐼,𝐼,0,𝑀,𝑁)) we denote 𝑔_𝑖=Θ(𝑓_𝑖),𝑖=1,2. It is easy to see that𝑔₁−𝑔_2𝑐0=𝐿_𝑓⁻¹₁∘𝐹−𝐿_𝑓⁻¹₂∘𝐹_𝑐0≤𝐿_𝑓⁻¹₁−𝐿_𝑓⁻¹_2𝑐0≤𝐾₅𝐾₀𝑓₁−𝑓_2𝑐0.(3.24)By Lemma 3.4 we get𝑔^′₁−𝑔^′_2𝑐0=𝐿_𝑓⁻¹₁^′∘𝐹⋅𝐹^′−𝐿_𝑓⁻¹₂^′∘𝐹⋅𝐹^′_𝑐0≤𝐾₀𝑀⋅𝐿_𝑓⁻¹₁^′−𝐿_𝑓⁻¹₂^′_𝑐0≤𝐾₁𝑁+𝐾₄𝐾₅𝑀+𝐾₇+𝑁𝐾₈𝐾₀𝑀𝐾₀²𝑓₁−𝑓_2𝑐0+𝑀𝐾₆𝐾₀𝑓^′₁−𝑓^′_2𝑐0.(3.25)By the discussion above we get‖𝑔₁−𝑔₂‖_𝑐1=𝑔₁−𝑔_2𝑐0+𝑔^′₁−𝑔^′_2𝑐0≤𝐾₅𝐾₀+𝐾₁𝑁+𝐾₄𝐾₅𝑀+𝐾₇+𝑁𝐾₈𝐾₀𝑀𝐾₀²𝑓₂−𝑓_1𝑐0+𝑀𝐾₆𝐾₀𝑓^′₂−𝑓^′_1𝑐0≤𝐸𝑓₁−𝑓_2𝑐1,(3.26)where𝐾𝐸=max⁵𝐾₀+𝐾₁𝑁+𝐾₄𝐾₅𝑀+𝐾₇+𝑁𝐾₈𝐾₀𝑀𝐾₀²,𝑀𝐾₆𝐾₀.(3.27)

Hence Θ∶𝔄(𝐼,𝐼,0,𝑀,𝑁)→𝔄(𝐼,𝐼,0,𝑀,𝑁)(orΘ∶𝔄′(𝐼,𝐼,0,𝑀,𝑁)→𝔄′(𝐼,𝐼,0,𝑀,𝑁)) is continuous. By Schauder fixed point theorem, there exists a function 𝑓∈𝔄(𝐼,𝐼,0,𝑀,𝑁)(or𝔄′(𝐼,𝐼,0,𝑀,𝑁)) such that𝑓=Θ(𝑓)=𝐿_𝑓⁻¹∘𝐹.(3.28)That means 𝑓 is a solution of (1.4) in 𝔄(𝐼,𝐼,0,𝑀,𝑁)(or𝔄′(𝐼,𝐼,0,𝑀,𝑁)).

4. Uniqueness and Stability of Solutions

Theorem 4.1. Let {𝜆_𝑖}^∞_𝑖=1 be coefficient of (1.4) and 𝑀>1, 𝑁>0 positive constants. Suppose that the conditions in (3.1) are valid. Further one assumes that𝐾𝐸=max⁵𝐾₀+𝐾₁𝑁+𝐾₄𝐾₅𝑀+𝐾₇+𝑁𝐾₈𝐾₀𝑀𝐾₀²,𝑀𝐾₆𝐾₀<1.(4.1)Then for any 𝐹∈𝔄(𝐼,𝐽,0,𝐾₀𝑀,𝐿)(or𝔄′(𝐼,𝐽,0,𝐾₀𝑀,𝐿)), there exists a unique function 𝑓∈𝔄(𝐼,𝐼,0,𝑀,𝑀^∗)(or𝔄′(𝐼,𝐼,0,𝑀,𝑀^∗)) satisfying (1.4), where 𝑀^∗≥(𝐿+𝐾₄𝑀²)/(𝐾₀−𝐾₁𝑀²). Furthermore the solution 𝑓 depends continuously on the given function 𝐹.

Proof. If 𝐸<1, then by (3.26) the map Θ defined in Theorem 3.1 becomes a strict contraction. The fix point of Θ, which is a solution of (1.4), is unique by Banach's contraction principle.
Let 𝑓₁,𝑓₂ be the solutions of (1.4) for the corresponding functions 𝐹₁,𝐹₂. First, since𝑓_𝑖=𝐿_𝑓⁻¹_𝑖∘𝐹_𝑖,𝑖=1,2,(4.2)we get𝑓₁−𝑓_2𝑐0=𝐿_𝑓⁻¹₁∘𝐹₁−𝐿_𝑓⁻¹₂∘𝐹_2𝑐0≤𝐿_𝑓⁻¹₁∘𝐹₁−𝐿_𝑓⁻¹₁∘𝐹_2𝑐0+𝐿_𝑓⁻¹₁∘𝐹₂−𝐿_𝑓⁻¹₂∘𝐹_{2𝑐0(Lemma(3.4))}≤1𝐾₀𝐹₁−𝐹_2𝑐0+𝐾₅𝐾₀𝑓₁−𝑓_2𝑐0.(4.3)Second, we have𝑓^′₁−𝑓^′_2𝑐0=𝐿_𝑓⁻¹₁^′∘𝐹₁⋅𝐹^′₁−𝐿_𝑓⁻¹₂^′∘𝐹₂⋅𝐹^′_2𝑐0≤𝐿_𝑓⁻¹₁^′∘𝐹₁⋅𝐹^′₁−𝐿_𝑓⁻¹₁^′∘𝐹₁⋅𝐹^′_2𝑐0+𝐿_𝑓⁻¹₁^′∘𝐹₁⋅𝐹^′₂−𝐿_𝑓⁻¹₁^′∘𝐹₂⋅𝐹^′_2𝑐0+𝐿_𝑓⁻¹₁^′∘𝐹₂⋅𝐹^′₂−𝐿_𝑓⁻¹₂^′∘𝐹₂⋅𝐹^′_2𝑐0(3.12)≤1𝐾₀𝐹^′₁−𝐹^′_2𝑐0+𝑀𝐾₁𝑁+𝐾₄𝐾₀²𝐹₁−𝐹_2𝑐0+𝐾₀𝑀𝐿_𝑓⁻¹₁^′−𝐿_𝑓⁻¹₂^′_{𝑐0(Lemma(3.4))}≤1𝐾₀𝐹^′₁−𝐹^′_2𝑐0+𝑀𝐾₁𝑁+𝐾₄𝐾₀²𝐹₁−𝐹_2𝑐0+𝐾₁𝑁+𝐾₄𝐾₅𝑀+𝐾₇+𝑁𝐾₈𝐾₀𝑀𝐾₀²𝑓₁−𝑓_2𝑐0+𝑀𝐾₆𝐾₀𝑓^′₂−𝑓^′_1𝑐0.(4.4)
By the above discussion we get𝑓₁−𝑓_2𝑐1=𝑓₁−𝑓_2𝑐0+𝑓^′₁−𝑓^′_2𝑐0≤𝐸𝑓₁−𝑓_2𝑐1+𝑀𝐾₁𝑁+𝐾₄𝐾₀²+1𝐾₀𝐹₁−𝐹_2𝑐1,(4.5)which means that𝑓₁−𝑓_2𝑐1≤1𝑀𝐾1−𝐸¹𝑁+𝐾₄𝐾₀²+1𝐾₀𝐹₁−𝐹_2𝑐1.(4.6)So the solution 𝑓 depends continuously on the given function 𝐹.

4.1. Examples

Example 4.2. Let 𝑀=10,𝐿=20 and 𝐼=[0,1]. The equation1001𝑒^𝑓(𝑥)−1−𝑓1000(𝑒−1)²(𝑥)²11000=𝑥+2sin(2𝜋𝑥),(4.7)where 𝑥∈[0,1], has a unique solution 𝑓∈𝔄([0,1],[0,1],0,10,900).

Proof. It is easy to see that 𝐻₁(𝑥)=(𝑒^𝑥−1)/(𝑒−1)∈𝔄(𝐼,𝐼,1/2,2,2),𝐻₂(𝑥)=𝑥²∈𝔄(𝐼,𝐼,0,2,2) and 𝐹(𝑥)=𝑥+1/2sin(2𝜋𝑥)∈𝔄(𝐼,𝐼,0,4.5,20). By simple calculation we get that𝐾₀=9612000,𝐾₁=1500,𝐾₂=9999500,𝐾₃=20221000,𝐾₄=1101,𝐾500₅=𝐾₆=1500,𝐾₇=150,𝐾₈=0,𝐾₀𝑀=961,𝐾200₀−𝐾₁𝑀²=5612000,𝐴=0<𝐵=1,𝑁=(𝐿+𝐾₄𝑀²)(𝐾₀−𝐾₁𝑀²)=480400561<900,(4.8) by Theorem 3.1 the equation has a solution 𝑓∈𝔄([0,1],[0,1],0,10,900). Further we get that𝐾₅𝐾₀+𝐾₁𝑁+𝐾₄𝐾₅𝑀+𝐾₇+𝑁𝐾₈𝐾₀𝑀𝐾₀²<804,961𝑀𝐾₆𝐾₀=40,961(4.9)this means 𝐸<1. By Theorem 4.1 the solution is unique.

By similar discussion we have the following example.

Example 4.3. Let 𝑀=10,𝐿=20, and 𝐼=[0,1]. The equation1001𝑒^𝑓(𝑥)−1−𝑓1000(𝑒−1)1−²(𝑥)²11000=1−𝑥−2sin(2𝜋𝑥),(4.10)where 𝑥∈[0,1] has a unique solution 𝑓∈𝔄^′([0,1],[0,1],0,10,900).

Acknowledgements

The author would like to thank the referees for their detailed comments and helpful suggestions. He is supported by Project HITC200706 supported by Science Research Foundation in Harbin Institute of Technology.