Smooth Solutions of a Class of Iterative Functional Differential Equations

Abstract

By Faà di Bruno’s formula, using the fixed-point theorems of Schauder and Banach, we study the existence and uniqueness of smooth solutions of an iterative functional differential equation 𝑥^′(𝑡)=1/(𝑐₀𝑥^[0](𝑡)+𝑐₁𝑥^[1](𝑡)+⋯+𝑐_𝑚𝑥^[𝑚](𝑡)).

1. Introduction

There has been a lot of monographs and research articles to discuss the kinds of solutions of functional differential equations since the publication of Jack Hale's paper [1]. Several papers discussed the iterative functional differential equations of the form 𝑥^′(𝑡)=𝐻𝑥^[0](𝑡),𝑥^[1](𝑡),…,𝑥^[𝑚](𝑡), (1.1) where 𝑥^[0](𝑡)=𝑡,𝑥^[1](𝑡)=𝑥(𝑡),𝑥^[𝑘](𝑡)=𝑥(𝑥^[𝑘−1](𝑡)),𝑘=2,…,𝑚. More specifically, Eder [2] considered the functional differential equation𝑥′(𝑡)=𝑥^[2](𝑡) (1.2) and proved that every solution either vanishes identically or is strictly monotonic. Furthermore, Fečkan [3] and Wang [4] studied the equation 𝑥^′(𝑡)=𝑓𝑥^[2](𝑡) (1.3) with different conditions. Staněk [5] considered the equation 𝑥^′(𝑡)=𝑥(𝑡)+𝑥^[2](𝑡) (1.4) and obtained every solution either vanishes identically or is strictly monotonic. Si and his coauthors [6, 7] studied the following equations: 𝑥^′(𝑡)=𝑥^[𝑚](𝑡),(𝑡)=1𝑥′𝑥^[𝑚](𝑡),𝑥^′(𝑡)=1𝑐₀𝑥^[0](𝑡)+𝑐₁𝑥^[1](𝑡)+⋯+𝑐_𝑚𝑥^[𝑚](𝑡) (1.5)(1.6) and established sufficient conditions for the existence of analytic solutions. Especially in [8, 9], the smooth solutions of the following equations: 𝑥^′(𝑡)=^𝑚_𝑗=1𝑎_𝑗𝑥^[𝑗](𝑡)(𝑡),𝑥+𝐹_′(𝑡)=^𝑚_𝑗=1𝑎_𝑗(𝑡)𝑥^[𝑗](𝑡)+𝐹(𝑡), (1.7) have been studied by the fixed-point theorems of Schauder and Banach.

A smooth function is taken to mean one that has a number of continuous derivatives and for which the highest continuous derivative is also Lipschitz. Let 𝑥∈𝐶^𝑛 if 𝑥^′,…,𝑥^(𝑛) are continuous, 𝐶^𝑛(𝐼,𝐼) is the set in which 𝑥∈𝐶^𝑛 and maps a closed interval I into I. As in [9], we, using the same symbols, denote the norm ‖𝑥‖_𝑛=^𝑛_𝑘=0𝑥^(𝑘),‖𝑥‖=max_𝑡∈𝐼{|𝑥(𝑡)|}, (1.8) then 𝐶^𝑛(𝐼,𝑅) with ‖⋅‖_𝑛 is a Banach space, and 𝐶^𝑛(𝐼,𝐼) is a subset of 𝐶^𝑛(𝐼,𝑅). For given 𝑀_𝑖>0(𝑖=1,2,…,𝑛+1), let Ω𝑀₁,…,𝑀_𝑛+1;𝐼=𝑥∈𝐶^𝑛(𝐼,𝐼)∶𝑥^(𝑖)(𝑡)≤𝑀_𝑖𝑥,𝑖=1,2,…,𝑛;_(𝑛)𝑡₁−𝑥^(𝑛)𝑡₂≤𝑀_𝑛+1𝑡₁−𝑡₂,𝑡,𝑡₁,𝑡₂∈𝐼. (1.9)

For convenience, we will make use of the notation 𝑥_𝑖𝑗(𝑡)=𝑥^(𝑖)𝑥^[𝑗](𝑡),𝑥_∗𝑗𝑘(𝑡)=𝑥^[𝑗](𝑡)^(𝑘), (1.10) where 𝑖,𝑗, and 𝑘 are nonnegative integers. Let 𝐼 be a closed interval in 𝑅. By induction, we may prove that𝑥_∗𝑗𝑘(𝑡)=𝑃_𝑗𝑘𝑥₁₀(𝑡),…,𝑥_1,𝑗−1(𝑡);…;𝑥_𝑘0(𝑡),…,𝑥_{𝑘,𝑗−1}(𝑡𝛽),_𝑗𝑘=𝑃𝑥^′(𝜉),…,𝑥_𝑗𝑘^{𝑗terms0𝑥0𝑓𝑒37′}(𝜉);…;𝑥^(𝑘)(𝜉),…,𝑥^{𝑗terms0𝑥0𝑓𝑒37(𝑘)}(𝜉𝐻),_𝑗𝑘=𝑃1,…,_𝑗𝑘^{𝑗terms0𝑥0𝑓𝑒37}1;𝑀₂,…,^{𝑗terms0𝑥0𝑓𝑒37}𝑀₂;…;𝑀_𝑘,…,^{𝑗terms0𝑥0𝑓𝑒37}𝑀_𝑘, (1.11)(1.12)(1.13) where 𝑃_𝑗𝑘 is a uniquely defined multivariate polynomial with nonnegative coefficients. The proof can be found in [8].

In order to seek a solution 𝑥(𝑡) of (1.6), in 𝐶^𝑛(𝐼,𝐼) such that 𝜉 is a fixed point of the function 𝑥(𝑡), that is, 𝑥(𝜉)=𝜉, it is natural to seek an interval 𝐼 of the form [𝜉−𝛿,𝜉+𝛿] with 𝛿>0.

Let us define 𝑋𝜉;𝜉₀,…,𝜉_𝑛;1,𝑀₂,…,𝑀_𝑛+1=;𝐼𝑥∈Ω1,𝑀₂,…,𝑀_𝑛+1;𝐼∶𝑥(𝜉)=𝜉₀=𝜉,𝑥^(𝑖)(𝜉)=𝜉_𝑖,𝑖=1,2,…,𝑛. (1.14)

2. Smooth Solutions of (1.6)

In this section, we will prove the existence theorem of smooth solutions for (1.6). First of all, we have the inequalities in the following for all 𝑥(𝑡),𝑦(𝑡)∈𝑋:𝑥^[𝑗]𝑡₁−𝑥^[𝑗]𝑡₂≤𝑡₁−𝑡₂,𝑡₁,𝑡₂𝑥∈𝐼,𝑗=0,1,…,𝑚,_[𝑗]−𝑥^[𝑗]≤𝑗𝑥−𝑦,𝑗=1,…,𝑚,𝑥−𝑦≤𝛿^𝑛𝑥^(𝑛)−𝑦^(𝑛), (2.1)(2.2)(2.3) and the proof can be found in [9].

Theorem 2.1. Let 𝐼=[𝜉−𝛿,𝜉+𝛿], where 𝜉 and 𝛿 satisfy 1𝜉≥𝑐₀−∑^𝑚_𝑖=1𝑐_𝑖1,0<𝛿≤𝜉−𝑐₀−∑^𝑚_𝑖=1𝑐_𝑖, (2.4) where |𝑐₀|>∑^𝑚_𝑖=0|𝑐_𝑖|, then (1.6) has a solution in 𝑋𝜉;𝜉₀,…,𝜉_𝑛;1,M₂,…,𝑀_𝑛+1;𝐼, (2.5) provided the following conditions hold:
(i)𝜉₁=𝜉^−1𝑚_𝑖=0𝑐_𝑖⁻¹,𝜉_𝑘()=−1^𝑠(𝑘−1)!𝑠!𝑠₁!𝑠₂!⋯𝑠_𝑘−1!𝜉^𝑚_𝑖=0𝑐_𝑖^−𝑠−1∑^𝑚_𝑖=0𝑐_𝑖𝛽_𝑖11!^𝑠1×∑^𝑚_𝑖=0𝑐_𝑖𝛽_𝑖22!^𝑠2⋯∑^𝑚_𝑖=0𝑐_𝑖𝛽_𝑖𝑘−1(𝑘−1)!^𝑠𝑘−1, (2.6)(2.7) where 𝑘=2,…,𝑛, and the sum is over all nonnegative integer solutions of the Diophantine equation 𝑠₁+2𝑠₂+⋯+(𝑘−1)𝑠_𝑘−1=𝑘−1 and 𝑠=𝑠₁+𝑠₂+⋯+𝑠_𝑘−1,
(ii)(𝑘−1)!𝑠!𝑠₁!𝑠₂!⋯𝑠_𝑘−1!(𝜉−𝛿)^−s−1𝑐₀−^𝑚_𝑖=1𝑐_𝑖^−𝑠−1∑^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖11!^𝑠1×∑^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖22!^𝑠2⋯∑^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖𝑘−1(𝑘−1)!^𝑠𝑘−1≤𝑀_𝑘,𝑘=2,…,𝑛, (2.8) where 𝑠₁+2𝑠₂+⋯+(𝑘−1)𝑠_𝑘−1=𝑘−1 and 𝑠=𝑠₁+𝑠₂+⋯+𝑠_𝑘−1,
(iii)(𝑛−1)!𝑠!𝑠₁!𝑠₂!⋯𝑠_𝑛−1!1!^𝑠12!^𝑠2⋯(𝑛−1)!^𝑠𝑛−1×(𝑠+1)(𝜉−𝛿)^−𝑠−2𝑐₀−^𝑚_𝑖=1𝑐_𝑖^{−𝑠−2𝑚}_𝑖=0𝑐_𝑖𝐻_𝑖1^𝑠1+1𝑚_𝑖=0𝑐_𝑖𝐻_𝑖2^𝑠2×⋯×^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖𝑛−1^𝑠𝑛−1+𝑠₁(𝜉−𝛿)^−𝑠−1𝑐₀−^𝑚_𝑖=1𝑐_𝑖^{−𝑠−1𝑚}_𝑖=0𝑐_𝑖𝐻_𝑖1^𝑠1−1×^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖2^𝑠2+1𝑚_𝑖=0𝑐_𝑖𝐻_𝑖3^𝑠3⋯^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖𝑛−1^𝑠𝑛−1+⋯+𝑠_𝑛−1(𝜉−𝛿)^−𝑠−1𝑐₀−^𝑚_𝑖=1𝑐_𝑖^{−𝑠−1𝑚}_𝑖=0𝑐_𝑖𝐻_𝑖1^𝑠1⋯^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖𝑛−2^𝑠𝑛−2×^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖𝑛−1^{𝑠𝑛−1−1𝑚}_𝑖=0𝑐_𝑖𝐻_𝑖𝑛≤𝑀_𝑛+1, (2.9) where 𝑠₁+2𝑠₂+⋯+(𝑛−1)𝑠_𝑛−1=𝑛−1 and 𝑠=𝑠₁+𝑠₂+⋯+𝑠_𝑛−1,

Proof. Define an operator 𝑇 from 𝑋 into 𝐶^𝑛(𝐼,𝐼) by (𝑇𝑥)(𝑡)=𝜉+^𝑡_𝜉1𝑐₀𝑥^[0](𝑠)+𝑐₁𝑥^[1](𝑠)+⋯+𝑐_𝑚𝑥^[𝑚](𝑠)𝑑𝑠. (2.10) We will prove that for any 𝑥∈𝑋,𝑇𝑥∈𝑋, (𝑇𝑥)(𝑡)−𝜉=^𝑡_𝜉1∑^𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖](𝑠)𝑑𝑠≤(𝜉−𝛿)𝑐₀−^𝑚_𝑖=1𝑐_𝑖⁻¹𝑡−𝜉≤𝛿, (2.11) where the second inequality is from (2.4) and 𝑥(𝐼)⊆𝐼. Thus, (𝑇𝑥)(𝐼)⊆𝐼.
It is easy to see that (𝑇𝑥)^′(𝑡)=1∑^𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖](𝑡), (2.12) and by Faà di Bruno's formula, for 𝑘=2,…,𝑛, we have (𝑇𝑥)^(𝑘)(𝑡)=1∑^𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖](𝑡)^(𝑘−1)()=−1^𝑠(𝑘−1)!𝑠!𝑠₁!𝑠₂!⋯𝑠_𝑘−1!1∑^𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖](𝑡)^𝑠+1∑^𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖](𝑡)^′1!^𝑠1×∑^𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖](𝑡)^′′2!^𝑠2⋯∑^𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖](𝑡)^(𝑘−1)(𝑘−1)!^𝑠𝑘−1()=−1^𝑠(𝑘−1)!𝑠!𝑠₁!𝑠₂!⋯𝑠_𝑘−1!1∑^𝑚_𝑖=0c_𝑖𝑥^[𝑖](𝑡)^𝑠+1∑^𝑚_𝑖=0𝑐_𝑖𝑥_∗𝑖1(𝑡)1!^𝑠1×∑^𝑚_𝑖=0𝑐_𝑖𝑥_∗𝑖2(𝑡)2!^𝑠2⋯∑^𝑚_𝑖=0𝑐_𝑖𝑥_{∗𝑖𝑘−1}(𝑡)(𝑘−1)!^𝑠𝑘−1, (2.13) where the sum is over all nonnegative integer solutions of the Diophantine equation 𝑠₁+2𝑠₂+⋯+(𝑘−1)𝑠_𝑘−1=𝑘−1 and 𝑠=𝑠₁+𝑠₂+⋯+𝑠_𝑘−1.
Furthermore, note (𝑇𝑥)(𝜉)=𝜉, by (2.6) and (2.7), (𝑇𝑥)^′(𝜉)=1∑^𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖](𝜉)=1𝜉∑^𝑚_𝑖=0𝑐_𝑖=𝜉₁,(𝑇𝑥)^(𝑘)(𝜉)()=−1^𝑠(𝑘−1)!𝑠!𝑠₁!𝑠₂!⋯𝑠_𝑘−1!1𝜉∑^𝑚_𝑖=0𝑐_𝑖^𝑠+1∑^𝑚_𝑖=0𝑐_𝑖𝑥_∗𝑖1(𝜉)1!^𝑠1×∑^𝑚_𝑖=0𝑐_𝑖𝑥_∗𝑖2(𝜉)2!^𝑠2⋯∑^𝑚_𝑖=0𝑐_𝑖𝑥_{∗𝑖𝑘−1}(𝜉)(𝑘−1)!^𝑠𝑘−1=(−1)^𝑠(𝑘−1)!𝑠!𝑠₁!𝑠₂!⋯𝑠_𝑘−1!1𝜉∑^𝑚_𝑖=0𝑐_𝑖^𝑠+1∑^𝑚_𝑖=0𝑐_𝑖𝛽_𝑖11!^𝑠1×∑^𝑚_𝑖=0𝑐_𝑖𝛽_𝑖22!^𝑠2⋯∑^𝑚_𝑖=0𝑐_𝑖𝛽_𝑖𝑘−1(𝑘−1)!^𝑠𝑘−1=𝜉_𝑘,𝑘=2,…,𝑛, (2.14) where 𝑠₁+2𝑠₂+⋯+(𝑘−1)𝑠_𝑘−1=𝑘−1 and 𝑠=𝑠₁+𝑠₂+⋯+𝑠_𝑘−1. Thus, (𝑇𝑥)^(𝑘)(𝜉)=𝜉_𝑘 for 𝑘=0,1,…,𝑛, (𝑇𝑥)^′(𝑡=1)∑^𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖](𝑡)≤(𝜉−𝛿)𝑐₀−^𝑚_𝑖=1𝑐_𝑖⁻¹≤1=𝑀₁, (2.15) By (2.8), we have (𝑇𝑥)^(𝑘)(𝑡())≤𝑘−1!𝑠!𝑠₁!𝑠₂!⋯𝑠_𝑘−1!1∑^𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖](𝑡)^𝑠+1∑^𝑚_𝑖=0𝑐_𝑖𝑥_∗𝑖1(𝑡)1!^𝑠1×∑^𝑚_𝑖=0𝑐_𝑖𝑥_∗𝑖2(𝑡)2!^𝑠2⋯∑^𝑚_𝑖=0𝑐_𝑖𝑥_{∗𝑖𝑘−1}(𝑡)(𝑘−1)!^𝑠𝑘−1()≤𝑘−1!𝑠!𝑠₁!𝑠₂!⋯𝑠_𝑘−1!(𝜉−𝛿)^−𝑠−1𝑐₀−^𝑚_𝑖=1𝑐_𝑖^−𝑠−1∑^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖11!^𝑠1×∑^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖22!^𝑠2⋯∑^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖𝑘−1(𝑘−1)!^𝑠𝑘−1≤𝑀_𝑘,𝑘=2,…,𝑛, (2.16) where 𝑠₁+2𝑠₂+⋯+(𝑘−1)𝑠_𝑘−1=𝑘−1 and 𝑠=𝑠₁+𝑠₂+⋯+𝑠_𝑘−1.
Finally, (𝑇𝑥)^(𝑛)𝑡₁−(𝑇𝑥)^(𝑛)𝑡₂()≤𝑛−1!𝑠!s₁!𝑠₂!⋯𝑠_𝑛−1!1!^𝑠12!^𝑠2⋯(𝑛−1)!^𝑠𝑛−1×^𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖]𝑡₁^{−𝑠−1𝑚}_𝑖=0𝑐_𝑖𝑥_∗𝑖1𝑡₁^𝑠1𝑚_𝑖=0𝑐_𝑖𝑥_∗𝑖2𝑡₁^𝑠2⋯^𝑚_𝑖=0𝑐_𝑖𝑥_{∗𝑖𝑛−1}𝑡₁^𝑠𝑛−1−^𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖]𝑡₂^{−𝑠−1𝑚}_𝑖=0𝑐_𝑖𝑥_∗𝑖1𝑡₂^𝑠1𝑚_𝑖=0𝑐_𝑖𝑥_∗𝑖2𝑡₂^𝑠2⋯^𝑚_𝑖=0𝑐_𝑖𝑥_{∗𝑖𝑛−1}𝑡₂^𝑠𝑛−1()≤𝑛−1!𝑠!𝑠₁!𝑠₂!⋯𝑠_𝑛−1!1!^𝑠12!^𝑠2⋯(𝑛−1)!^𝑠𝑛−1×1∑^𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖]𝑡₁^𝑠+1−1∑^𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖]𝑡₂^𝑠+1𝑚_𝑖=0𝑐_𝑖𝑥_∗𝑖1𝑡₁^𝑠1×⋯^𝑚_𝑖=0𝑐_𝑖𝑥_{∗𝑖𝑛−1}𝑡₁^𝑠𝑛−1+^𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖]𝑡₂^−𝑠−1×^𝑚_𝑖=0𝑐_𝑖𝑥_∗𝑖1𝑡₁^𝑠1−^𝑚_𝑖=0𝑐_𝑖𝑥_∗𝑖1𝑡₂^𝑠1𝑚_𝑖=0𝑐_𝑖𝑥_∗𝑖2𝑡₁^𝑠2⋯×^𝑚_𝑖=0𝑐_𝑖𝑥_{∗𝑖𝑛−1}𝑡₁^𝑠𝑛−1+⋯+^𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖]𝑡₂^{−𝑠−1𝑚}_𝑖=0𝑐_𝑖𝑥_∗𝑖1𝑡₂^𝑠1×^𝑚_𝑖=0𝑐_𝑖𝑥_∗𝑖2𝑡₂^𝑠2⋯^𝑚_𝑖=0𝑐_𝑖𝑥_{∗𝑖𝑛−2}𝑡₂^𝑠𝑛−2×^𝑚_𝑖=0𝑐_𝑖𝑥_{∗𝑖𝑛−1}𝑡₁^𝑠𝑛−1−^𝑚_𝑖=0𝑐_𝑖𝑥_{∗𝑖𝑛−1}𝑡₂^𝑠𝑛−1()≤𝑛−1!𝑠!𝑠₁!𝑠₂!⋯𝑠_𝑛−1!1!^𝑠12!^𝑠2⋯(𝑛−1)!^𝑠𝑛−1×(𝑠+1)(𝜉−𝛿)^−𝑠−2𝑐₀−^𝑚_𝑖=1𝑐_𝑖^{−𝑠−2𝑚}_𝑖=0𝑐_𝑖𝐻_𝑖1^𝑠1+1𝑚_𝑖=0𝑐_𝑖𝐻_𝑖2^𝑠2×⋯×^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖𝑛−1^𝑠𝑛−1+𝑠₁(𝜉−𝛿)^−𝑠−1𝑐₀−^𝑚_𝑖=1𝑐_𝑖^{−𝑠−1𝑚}_𝑖=0𝑐_𝑖𝐻_𝑖1^𝑠1−1×^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖2^𝑠2+1𝑚_𝑖=0𝑐_𝑖𝐻_𝑖3^𝑠3⋯^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖𝑛−1^𝑠𝑛−1+⋯+𝑠_𝑛−1(𝜉−𝛿)^−𝑠−1𝑐₀−^𝑚_𝑖=1𝑐_𝑖^{−𝑠−1𝑚}_𝑖=0𝑐_𝑖𝐻_𝑖1^𝑠1⋯^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖𝑛−2^𝑠𝑛−2×^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖𝑛−1^{𝑠𝑛−1−1𝑚}_𝑖=0𝑐_𝑖𝐻_𝑖𝑛𝑡₁−𝑡₂, (2.17) where 𝑠₁+2𝑠₂+⋯+(𝑘−1)𝑠_𝑘−1=𝑘−1 and 𝑠=𝑠₁+𝑠₂+⋯+𝑠_𝑘−1. By (2.9), we see that (𝑇𝑥)^(𝑛)𝑡₁−(𝑇𝑥)^(𝑛)𝑡₂≤𝑀_𝑛+1𝑡₁−𝑡₂. (2.18) Now, we can say that 𝑇 is an operator from 𝑋 into itself.
Next, we will show that 𝑇 is continuous. Let 𝑥,𝑦∈𝑋, then 𝑇𝑥−𝑇𝑦_𝑛=𝑇𝑥−𝑇𝑦+(𝑇𝑥)^′−𝑇𝑦^′+^𝑛_𝑘=2(𝑇𝑥)^(𝑘)−𝑇𝑦^(𝑘)=max_𝑡∈𝐼^𝑡_𝜉1∑^𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖](𝑠)−1∑^𝑚_𝑖=0𝑐_𝑖𝑦^[𝑖](𝑠)𝑑𝑠+max_𝑡∈𝐼1∑^𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖](𝑡)−1∑^𝑚_𝑖=0𝑐_𝑖𝑦^[𝑖](𝑡)+^𝑛_𝑘=2max_𝑡∈𝐼(𝑘−1)!𝑠!𝑠₁!𝑠₂!⋯𝑠_𝑘−1!1!^𝑠12!^𝑠2⋯(𝑘−1)!^𝑠𝑘−1×1∑^𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖](𝑡)^𝑠+1𝑚_𝑖=0𝑐_𝑖𝑥_∗𝑖1(𝑡)^𝑠1𝑚_𝑖=0𝑐_𝑖𝑥_∗𝑖2(𝑡)^𝑠2⋯×^𝑚_𝑖=0𝑐_𝑖𝑥_{∗𝑖𝑘−1}(𝑡)^𝑠𝑘−1−1∑^𝑚_𝑖=0𝑐_𝑖𝑦^[𝑖](𝑡)^𝑠+1𝑚_𝑖=0𝑐_𝑖𝑦_∗𝑖1(𝑡)^𝑠1×^𝑚_𝑖=0𝑐_𝑖𝑦_∗𝑖2(𝑡)^𝑠2⋯^𝑚_𝑖=0𝑐_𝑖𝑦_{∗𝑖𝑘−1}(𝑡)^𝑠𝑘−1≤𝛿(𝜉−𝛿)⁻²𝑐₀−^𝑚_𝑖=1𝑐_𝑖^−2𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖]−𝑦^[𝑖]+(𝜉−𝛿)⁻²𝑐₀−^𝑚_𝑖=1𝑐_𝑖^−2𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖]−𝑦^[𝑖]+^𝑛_𝑘=2max_𝑡∈𝐼(𝑘−1)!𝑠!𝑠₁!𝑠₂!⋯𝑠_𝑘−1!1!^𝑠12!^𝑠2⋯(𝑘−1)!^𝑠𝑘−1×^𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖](𝑡)^−𝑠−1−^𝑚_𝑖=0𝑐_𝑖𝑦^[𝑖](𝑡)^{−𝑠−1𝑚}_𝑖=0𝑐_𝑖𝑥_∗𝑖1(𝑡)^s1⋯×^𝑚_𝑖=0𝑐_𝑖𝑥_{∗𝑖𝑘−1}(𝑡)^𝑠𝑘−1+^𝑚_𝑖=0𝑐_𝑖𝑦^[𝑖](𝑡)^−𝑠−1×^𝑚_𝑖=0𝑐_𝑖𝑥_∗𝑖1(𝑡)^𝑠1−^𝑚_𝑖=0𝑐_𝑖𝑦_∗𝑖1(𝑡)^𝑠1×^𝑚_𝑖=0𝑐_𝑖𝑥_∗𝑖2(𝑡)^𝑠2⋯^𝑚_𝑖=0𝑐_𝑖𝑥_{∗𝑖𝑘−1}(𝑡)^𝑠𝑘−1+⋯+^𝑚_𝑖=0𝑐_𝑖𝑦^[𝑖](𝑡)^{−𝑠−1𝑚}_𝑖=0𝑐_𝑖𝑦_∗𝑖1(𝑡)^𝑠1⋯^𝑚_𝑖=0𝑐_𝑖𝑦_{∗𝑖𝑘−2}(𝑡)^𝑠𝑘−2×^𝑚_𝑖=0𝑐_𝑖𝑥_{∗𝑖𝑘−1}(𝑡)^𝑠𝑘−1−^𝑚_𝑖=0𝑐_𝑖𝑦_{∗𝑖𝑘−1}(𝑡)^𝑠𝑘−1≤(𝜉−𝛿)⁻²𝑐₀−^𝑚_𝑖=1𝑐_𝑖⁻²𝑐₀+^𝑚_𝑖=1𝑖𝑐_𝑖+𝑥−𝑦_𝑛𝑘=2(𝑘−1)!𝑠!𝑠₁!𝑠₂!⋯𝑠_𝑘−1!1!^𝑠12!^𝑠2⋯(𝑘−1)!^𝑠𝑘−1×(𝑠+1)(𝜉−𝛿)^−𝑠−2𝑐₀−^𝑚_𝑖=1𝑐_𝑖^{−𝑠−2𝑚}_𝑖=0𝑐_𝑖𝑥^[𝑖]−𝑦^[𝑖]×^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖1^𝑠1⋯^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖𝑘−1^𝑠𝑘−1+𝑠₁(𝜉−𝛿)^−𝑠−1𝑐₀−^𝑚_𝑖=1𝑐_𝑖^{−𝑠−1𝑚}_𝑖=0𝑐_𝑖𝐻_𝑖1^𝑠1−1×^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖2^𝑠2+1⋯^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖𝑘−1^{𝑠𝑘−1𝑚}_𝑖=0𝑐_𝑖𝑥^[𝑖]−𝑦^[𝑖]+⋯+𝑠_𝑘−1(𝜉−𝛿)^−𝑠−1𝑐₀−^𝑚_𝑖=1𝑐_𝑖^{−𝑠−1𝑚}_𝑖=0𝑐_𝑖𝐻_𝑖1^𝑠1⋯×^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖𝑘−2^{𝑠𝑘−2𝑚}_𝑖=0𝑐_𝑖𝐻_𝑖𝑘−1^{𝑠𝑘−1−1𝑚}_𝑖=0𝑐_𝑖𝐻_𝑖𝑘×^𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖]−𝑦^[𝑖]+𝛿^𝑛+1(𝜉−𝛿)⁻²𝑐₀−^𝑚_𝑖=1𝑐_𝑖⁻²𝑐₀+^𝑚_𝑖=1𝑖c_𝑖𝑥^(𝑛)−𝑦^(𝑛), (2.19) where 𝑠₁+2𝑠₂+⋯+(𝑘−1)𝑠_𝑘−1=𝑘−1 and 𝑠=𝑠₁+𝑠₂+⋯+𝑠_𝑘−1.
Moreover, we can find some constants 𝑃_𝑘 such that ^𝑛_𝑘=2(𝑘−1)!𝑠!𝑠₁!𝑠₂!⋯𝑠_𝑘−1!1!^𝑠12!^𝑠2⋯(𝑘−1)!^𝑠𝑘−1×(𝑠+1)(𝜉−𝛿)^−𝑠−2𝑐₀−^𝑚_𝑖=1𝑐_𝑖^{−𝑠−2𝑚}_𝑖=0𝑐_𝑖𝑥^[𝑖]−𝑦^[𝑖]××^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖1^𝑠1⋯^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖𝑘−1^𝑠𝑘−1+𝑠₁(𝜉−𝛿)^−𝑠−1𝑐₀−^𝑚_𝑖=1𝑐_𝑖^{−𝑠−1𝑚}_𝑖=0𝑐_𝑖𝐻_𝑖1^𝑠1−1×^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖2^𝑠2+1⋯^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖𝑘−1^{𝑠𝑘−1𝑚}_𝑖=0𝑐_i𝑥^[𝑖]−𝑦^[𝑖]+⋯+𝑠_𝑘−1(𝜉−𝛿)^−𝑠−1𝑐₀−^𝑚_𝑖=1𝑐_𝑖^{−𝑠−1𝑚}_𝑖=0𝑐_𝑖𝐻_𝑖1^𝑠1⋯×^𝑚_𝑖=0𝑐_𝑖𝐻_𝑖𝑘−2^{𝑠𝑘−2𝑚}_𝑖=0𝑐_𝑖𝐻_𝑖𝑘−1^{𝑠𝑘−1−1𝑚}_𝑖=0𝑐_𝑖𝐻_𝑖𝑘×^𝑚_𝑖=0𝑐_𝑖𝑥^[𝑖]−𝑦^[𝑖]≤^𝑛−1_𝑘=1𝑃_𝑘𝜉,𝛿,𝑐_𝑖,𝐻_𝑖𝑗𝑥^(𝑘)−𝑦^(𝑘), (2.20) where 𝑃_𝑘𝜉,𝛿,𝑐_𝑖,𝐻_𝑖𝑗=𝑃𝜉,𝛿;𝑐₁,…,𝑐_𝑚;𝐻₁₁,…,𝐻_1𝑘+1;…;𝐻_𝑚1,…,𝐻_𝑚𝑘+1; (2.21) are the positive constants depend on 𝜉,𝛿,𝑐_𝑖, and 𝐻_𝑖𝑗,𝑖=1,…,𝑚;𝑗=1,…,𝑘+1. Then 𝑇𝑥−𝑇𝑦_𝑛≤(𝜉−𝛿)⁻²𝑐₀−^𝑚_𝑖=1𝑐_𝑖⁻²𝑐₀+^𝑚_𝑖=1𝑖𝑐_𝑖+𝑥−𝑦_{𝑛−1𝑘=1}𝑃_𝑘𝜉,𝛿,𝑐_𝑖,𝐻_𝑖𝑗𝑥^(𝑘)−𝑦^(𝑘)+𝛿^𝑛+1(𝜉−𝛿)⁻²𝑐₀−^𝑚_𝑖=1𝑐_𝑖⁻²𝑐₀+^𝑚_𝑖=1𝑖𝑐_𝑖𝑥^(𝑛)−𝑦^(𝑛)≤Γ𝑥−𝑦_𝑛. (2.22) Here, Γ=max(𝜉−𝛿)⁻²𝑐₀−^𝑚_𝑖=1𝑐_𝑖⁻²𝑐₀+^𝑚_𝑖=1𝑖𝑐_𝑖;max_{1≤𝑘≤𝑛−1}𝑃_𝑘𝜉,𝛿,𝑐_𝑖,𝐻_𝑖𝑗;𝛿^𝑛+1(𝜉−𝛿)⁻²𝑐₀−^𝑚_𝑖=1𝑐_𝑖⁻²𝑐₀+^𝑚_𝑖=1𝑖𝑐_𝑖, (2.23)𝑘=1,…,𝑛−1. So we can say that 𝑇 is continuous.
It is easy to see that 𝑋 is closed and convex. We now show that 𝑋 is a relatively compact subset of 𝐶^𝑛(𝐼,𝐼). For any 𝑥=𝑥(𝑡)∈𝑋, ‖𝑥‖_𝑛≤‖𝑥‖+^𝑛_𝑘=1𝑥^(𝑘)≤𝜉+𝛿+1+^𝑛_𝑘=2𝑀_𝑘. (2.24) Next, for any 𝑡₁,𝑡₂ in 𝐼, we have 𝑥𝑡₁−𝑥𝑡₂≤𝑡₁−𝑡₂. (2.25) Hence, 𝑋 is bounded in 𝐶^𝑛(𝐼,𝐼) and equicontinuous on 𝐼, and by the Arzela-Ascoli theorem, we know 𝑋 is relatively compact in 𝐶^𝑛(𝐼,𝐼), since 𝐶^𝑛(𝐼,𝐼) is the subset of 𝐶^𝑛(𝐼,𝑅), and we can say that 𝑋 is relatively compact in 𝐶^𝑛(𝐼,𝑅).
From Schauder's fixed-point theorem, we conclude that 𝑥(𝑡)=𝜉+^𝑡_𝜉1𝑐₀𝑥^[0](𝑠)+𝑐₁𝑥^[1](𝑠)+⋯+𝑐_𝑚𝑥^[𝑚](𝑠)𝑑𝑠, (2.26) for some 𝑥=𝑥(𝑡) in 𝑋. By differentiating both sides of the above equality, we see that 𝑥 is the desired solution of (1.6). This completes the proof.

Theorem 2.2. Let 𝐼=[𝜉−𝛿,𝜉+𝛿], where 𝜉 and 𝛿 satisfy (2.4), then (1.6) has a unique solution in 𝑋𝜉;𝜉₀,…,𝜉_𝑛;1,𝑀₂,…,𝑀_𝑛+1;𝐼, (2.27) provided the conditions (2.6)–(2.9) hold and Γ<1 in (2.23).

Proof. Since Γ<1, we see that 𝑇 defined by (2.10) is contraction mapping on the close subset 𝑋 of 𝐶^𝑛(𝐼,𝐼). Thus, the fixed point 𝑥 in the proof of Theorem 2.1 must be unique. This completes the proof.

Remark 2.3. By Theorem 2.1 or Theorem 2.2, the existence and uniqueness of smooth solutions of an iterative functional differential equation of the form (1.6) can be obtained. If 𝑛→+∞, we can also find that the solution is 𝐶^∞-smooth.
Now, we will show that the conditions in Theorem 2.1 do not self-contradict. Consider the following equation: (𝑡)=1𝑥′()𝑥(𝑡)+()𝑥(𝑥(𝑡,𝑡+1/21/4)) (2.28) where 𝑐₀=1,𝑐₁=(1/2),𝑐₂=(1/4), and 𝜉≥4. Moreover, we take 0<𝛿≤𝜉−4. Then, (2.4) is satisfied, and 𝜉,𝛿 define the interval 𝐼=[𝜉−𝛿,𝜉+𝛿]. Now, take 𝜉₀=𝜉, 𝜉₁=47𝜉⁻¹,𝜉₂=−16𝜉49⁻²11+2𝜉₁+14𝜉²₁,𝜉₃=4𝜉343⁻³4+2𝜉₁+𝜉²₁²−4𝜉49⁻²𝜉₂2+𝜉₁+𝜉²₁, (2.29) then (2.6) and (2.7) are satisfied.
Finally, if we take 𝑀₁=1,𝑀₂=28(𝜉−𝛿)⁻²,𝑀₃=392(𝜉−𝛿)⁻³+16(𝜉−𝛿)⁻²𝑀₂ (2.30) as positive, and 𝑀₄=8232(𝜉−𝛿)⁻⁴+576(𝜉−𝛿)⁻³𝑀₂+8(𝜉−𝛿)⁻²6𝑀²₂+5𝑀₃, (2.31) then (2.8) and (2.9) are satisfied.
Thus, we have shown that when 𝜉₀,…,𝜉₃ and 𝑀₁,…,𝑀₄ are defined as above, then there will be a solution for (2.28) in 𝑋(𝜉;𝜉₀,…,𝜉₃;1,…,𝑀₄;𝐼).

Acknowledgment

This work was partially supported by the Natural Science Foundation of Chongqing Normal University (Grant no. 12XLB003).