Smooth Solutions of a Class of Iterative Functional Differential Equations

and Applied Analysis 3 where i, j, and k are nonnegative integers. Let I be a closed interval in R. By induction, we may prove that x∗jk t Pjk ( x10 t , . . . , x1,j−1 t ; . . . ;xk0 t , . . . , xk,j−1 t ) , 1.11 βjk Pjk ⎛ ⎜⎝ j terms { }} { x′ ξ , . . . , x′ ξ ; . . . ; j terms { }} { x k ξ , . . . , x k ξ ⎞ ⎟⎠, 1.12 Hjk Pjk ⎛ ⎜⎝ j terms { }} { 1, . . . , 1; j terms { }} { M2, . . . ,M2; . . . ; j terms { }} { Mk, . . . ,Mk ⎞ ⎟⎠, 1.13 where Pjk 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 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 X ξ; ξ0, . . . , ξn; 1,M2, . . . ,Mn 1; I { x ∈ Ω 1,M2, . . . ,Mn 1; I : x ξ ξ0 ξ, x i ξ ξi, 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, t2 ∈ I, j 0, 1, . . . , m, 2.1 ∥∥x j − x j ∥∥ ≤ j∥∥x − y∥∥, j 1, . . . , m, 2.2 ∥∥x − y∥∥ ≤ δ ∥∥x n − y n ∥∥, 2.3 and the proof can be found in 9 . Theorem 2.1. Let I ξ − δ, ξ δ , where ξ and δ satisfy ξ ≥ 1 |c0| − ∑m i 1|ci| , 0 < δ ≤ ξ − 1 |c0| − ∑m i 1|ci| , 2.4 where |c0| > ∑m i 0 |ci|, then 1.6 has a solution in X ξ; ξ0, . . . , ξn; 1,M2, . . . ,Mn 1; I , 2.5 4 Abstract and Applied Analysis provided the following conditions hold: i ξ1 ξ−1 ( m ∑ i 0 ci )−1 , 2.6 ξk ∑ −1 s k − 1 !s! s1!s2! · · · sk−1! ( ξ m ∑ i 0 ci )−s−1(∑m i 0 ciβi1 1! )s1 × (∑m i 0 ciβi2 2! )s2 · · · (∑m i 0 ciβik−1 k − 1 ! )sk−1 , 2.7 where k 2, . . . , n, and the sum is over all nonnegative integer solutions of the Diophantine equation s1 2s2 · · · k − 1 sk−1 k − 1 and s s1 s2 · · · sk−1, ii ∑ k − 1 !s! s1!s2! · · · sk−1! ξ − δ −s−1 ( |c0| − m ∑ i 1 |ci| )−s−1(∑m i 0|ci|Hi1 1! )s1 × (∑m i 0|ci|Hi2 2! )s2 · · · (∑m i 0|ci|Hik−1 k − 1 ! )sk−1 ≤ Mk, k 2, . . . , n, 2.8 where s1 2s2 · · · k − 1 sk−1 k − 1 and s s1 s2 · · · sk−1, iii ∑ n − 1 !s! s1!s2! · · · sn−1!1!2! · · · n − 1 !n−1 × ⎡ ⎣ s 1 ξ − δ −s−2 ( |c0| − m ∑ i 1 |ci| )−s−2( m ∑ i 0 |ci|Hi1 )s1 1( m ∑ i 0 |ci|Hi2 )s2 × · · · × ( m ∑ i 0 |ci|Hin−1 )sn−1 s1 ξ − δ −s−1 ( |c0| − m ∑ i 1 |ci| )−s−1( m ∑ i 0 |ci|Hi1 )s1−1 × ( m ∑ i 0 |ci|Hi2 )s2 1( m ∑ i 0 |ci|Hi3 )s3 · · · ( m ∑ i 0 |ci|Hin−1 )sn−1 · · · sn−1 ξ − δ −s−1 ( |c0| − m ∑ i 1 |ci| )−s−1( m ∑ i 0 |ci|Hi1 )s1 · · · ( m ∑ i 0 |ci|Hin−2 )sn−2 × ( m ∑ i 0 |ci|Hin−1 )sn−1−1( m ∑ i 0 |ci|Hin )⎤ ⎦ ≤ Mn 1, 2.9 where s1 2s2 · · · n − 1 sn−1 n − 1 and s s1 s2 · · · sn−1, Abstract and Applied Analysis 5 Proof. Define an operator T from X into C I, I byand Applied Analysis 5 Proof. Define an operator T from X into C I, I by