Advances in Difference Equations
Volume 2010 (2010), Article ID 795145, 14 pages
doi:10.1155/2010/795145
Research Article

Nonlinear Delay Discrete Inequalities and Their Applications to Volterra Type Difference Equations

Yu Wu,1 Xiaopei Li,2 and Shengfu Deng3

1Yibin University, Yibin, Sichuan 644007, China
2Department of Mathematics, Zhanjiang Normal University, Zhanjiang, Guangdong 524048, China
3Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China

Received 7 September 2009; Accepted 14 January 2010

Academic Editor: Abdelkader Boucherif

Copyright © 2010 Yu Wu et al. 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

Delay discrete inequalities with more than one nonlinear term are discussed, which generalize some known results and can be used in the analysis of various problems in the theory of certain classes of discrete equations. Application examples to show boundedness and uniqueness of solutions of a Volterra type difference equation are also given.

1. Introduction

Gronwall-Bellman inequalities and their various linear and nonlinear generalizations play very important roles in the discussion of existence, uniqueness, continuation, boundedness, and stability properties of solutions of differential equations and difference equations. The literature on such inequalities and their applications is vast. For example, see [1–12] for continuous cases, and [13–20] for discrete cases. In particular, the book [21] written by Pachpatte considered three types of discrete inequalities:

𝑢 ( 𝑛 ) ≤ 𝑎 ( 𝑛 ) + 𝑛 − 1 𝑠 = 0 𝑢 𝑓 ( 𝑠 ) 𝑤 ( 𝑢 ( 𝑠 ) ) , 2 ( 𝑛 ) ≤ 𝑎 ( 𝑛 ) + 2 𝑛 − 1 𝑠 = 0 𝑢 𝑓 ( 𝑠 ) 𝑢 ( 𝑠 ) , 2 ( 𝑛 ) ≤ 𝑎 ( 𝑛 ) + 𝑛 − 1 𝑠 = 0 𝑓 ( 𝑠 ) 𝑤 ( 𝑢 ( 𝑠 ) ) . ( 1 . 1 ) In this paper, we consider a delay discrete inequality

𝑢 ( 𝑛 ) ≤ 𝑎 ( 𝑛 ) + 𝑚 𝑏 𝑖 = 1 𝑖 ( 𝑛 − 1 ) 𝑠 = 𝑏 𝑖 ( 0 ) 𝑓 𝑖 ( 𝑛 , 𝑠 ) 𝑤 𝑖 ( 𝑢 ( 𝑠 ) ) , 𝑛 ∈ 𝐍 0 ( 1 . 2 ) which has 𝑚 nonlinear terms where 𝐍 0 = { 0 , 1 , 2 , … } . We will show that many discrete inequalities like (1.1) can be reduced to this form. Our main result can be applied to analyze properties of solutions of discrete equations. We also give examples to show boundedness and uniqueness of solutions of a Volterra type difference equation.

2. Main Results

Assume that

( 𝐶 1 ) 𝑎 ( 𝑛 ) is nonnegative for 𝑛 ∈ 𝐍 0 and 𝑎 ( 0 ) > 0 ; ( 𝐶 2 ) 𝑏 𝑖 ( 𝑛 ) ( 𝑖 = 1 , … , 𝑚 ) are nondecreasing for 𝑛 ∈ 𝐍 0 , the range of each 𝑏 𝑖 belongs to 𝐍 0 , and 𝑏 𝑖 ( 𝑛 ) ≤ 𝑛 ; ( 𝐶 3 ) all 𝑓 𝑖 ( 𝑛 , 𝑗 ) ( 𝑖 = 1 , … , 𝑚 ) are nonnegative for 𝑛 , 𝑗 ∈ 𝐍 0 ; ( 𝐶 4 )all 𝑤 𝑖 ( 𝑖 = 1 , … , 𝑚 ) are continuous and nondecreasing functions on [ 0 , ∞ ) and are positive on ( 0 , ∞ ) . They satisfy the relationship 𝑤 1 ∝ 𝑤 2 ∝ ⋯ ∝ 𝑤 𝑚 where 𝑤 𝑖 ∝ 𝑤 𝑖 + 1 means that ( 𝑤 𝑖 + 1 ) / 𝑤 𝑖 is nondecreasing on ( 0 , ∞ ) (see [10]).

Let 𝑊 𝑖 ∫ ( 𝑢 ) = 𝑢 𝑢 𝑖 ( 𝑑 𝑧 / 𝑤 𝑖 ( 𝑧 ) ) for 𝑢 ≥ 𝑢 𝑖 where 𝑢 𝑖 > 0 is a given constant. Then, 𝑊 𝑖 is strictly increasing so its inverse 𝑊 𝑖 − 1 is well defined, continuous, and increasing in its corresponding domain. Define 𝑏 𝑖 ( − 1 ) = − 1 , Δ 𝑢 ( 𝑛 ) = 𝑢 ( 𝑛 + 1 ) − 𝑢 ( 𝑛 ) and Δ 2 𝑟 ( 𝑛 , 𝑗 ) = 𝑟 ( 𝑛 , 𝑗 + 1 ) − 𝑟 ( 𝑛 , 𝑗 ) .

Theorem 2.1. Suppose that ( 𝐶 1 )–( 𝐶 4 ) hold and 𝑢 ( 𝑛 ) is a nonnegative function for 𝑛 ∈ 𝐍 0 satisfying (1.2). Then 𝑢 ( 𝑛 ) ≤ 𝑊 𝑚 − 1 ⎡ ⎢ ⎢ ⎣ 𝑊 𝑚 ( ̃ 𝑎 ( 0 ) ) + 𝑏 𝑚 ( 𝑛 − 1 ) 𝑠 = 𝑏 𝑚 ( 0 ) 𝑓 𝑚 ( 𝑛 , 𝑠 ) + 𝑛 − 1 𝑠 = 0 Δ 2 𝑟 𝑚 ( 𝑛 , 𝑠 ) 𝜙 𝑚 𝑊 − 1 𝑚 − 1 𝑟 𝑚 ⎤ ⎥ ⎥ ⎦ ( 0 , 𝑠 ) , 𝑛 ≤ 𝑁 1 , ( 2 . 1 ) where ̃ 𝑎 ( 𝑛 ) = m a x 0 ≤ 𝜏 ≤ 𝑛 , 𝜏 ∈ 𝐍 0 𝑎 ( 𝜏 ) , 𝑓 𝑖 ( 𝑛 , 𝑗 ) = m a x 0 ≤ 𝜏 ≤ 𝑛 , 𝜏 ∈ 𝑁 0 𝑓 𝑖 ( 𝜏 , 𝑗 ) , 𝑟 𝑚 ( 𝑛 , 𝑗 ) is determined recursively by 𝑟 1 𝑟 ( 𝑛 , 𝑗 ) = ̃ 𝑎 ( 𝑗 ) , 𝑖 + 1 ( 𝑛 , 𝑗 ) = 𝑊 𝑖 𝑟 1 + ( 𝑛 , 0 ) 𝑏 𝑖 ( 𝑗 − 1 ) 𝑠 = 𝑏 𝑖 ( 0 ) 𝑓 𝑖 ( 𝑛 , 𝑠 ) + 𝑗 − 1 𝑠 = 0 Δ 2 𝑟 𝑖 ( 𝑛 , 𝑠 ) 𝜙 𝑖 𝑊 − 1 𝑖 − 1 𝑟 𝑖 ( 0 , 𝑠 ) , 𝑖 = 1 , … , 𝑚 − 1 , ( 2 . 2 ) 𝜙 𝑖 ( 𝑢 ) = 𝑤 𝑖 ( 𝑢 ) / 𝑤 𝑖 − 1 ( 𝑢 ) , 𝜙 1 ( 𝑢 ) = 𝑤 1 ( 𝑢 ) , 𝑊 0 = 𝐼 (Identity), and 𝑁 1 is the largest positive integer such that 𝑊 𝑖 ( ̃ 𝑎 ( 0 ) ) + 𝑏 𝑖 ( 𝑁 1 − 1 ) 𝑠 = 𝑏 𝑖 ( 0 ) 𝑓 𝑖 𝑁 1 + , 𝑠 𝑁 1 − 1 𝑠 = 0 Δ 2 𝑟 𝑖 𝑁 1 , 𝑠 𝜙 𝑖 𝑊 − 1 𝑖 − 1 𝑟 𝑖 ≤ ( 0 , 𝑠 ) ∞ 𝑢 𝑖 𝑑 𝑧 𝑤 i ( 𝑧 ) , 𝑖 = 1 , … , 𝑚 . ( 2 . 3 )

Remark 2.2. (1) 𝑁 1 is defined by (2.3) and 𝑁 1 = ∞ when all 𝑤 𝑖 ( 𝑖 = 1 , … , 𝑚 ) satisfy ∫ ∞ 𝑢 𝑖 ( 𝑑 𝑧 / 𝑤 𝑖 ( 𝑧 ) ) = ∞ . Different choices of 𝑢 𝑖 in 𝑊 𝑖 do not affect our results (see [2]).
(2) If 𝑏 𝑖 ( 𝑛 ) = 𝑛 for 𝑖 = 1 , … , 𝑚 , then (2.1) gives the estimate of the following inequality:
𝑢 ( 𝑛 ) ≤ 𝑎 ( 𝑛 ) + 𝑚 𝑖 = 1 𝑛 − 1 𝑠 = 0 𝑓 𝑖 ( 𝑛 , 𝑠 ) 𝑤 𝑖 ( 𝑢 ( 𝑠 ) ) , 𝑛 ∈ 𝐍 0 ( 2 . 4 ) by replacing 𝑏 𝑚 ( 𝑛 − 1 ) , 𝑏 𝑚 ( 0 ) , 𝑏 𝑖 ( 𝑗 − 1 ) , 𝑏 𝑖 ( 0 ) , and 𝑏 𝑖 ( 𝑁 1 − 1 ) with 𝑛 − 1 , 0 , 𝑗 − 1 , 0 and 𝑁 1 − 1 , respectively. Especially, if 𝑏 1 ( 𝑛 ) = 𝑛 and 𝑓 1 ( 𝑛 , 𝑠 ) = 𝑓 ( 𝑠 ) , then (1.2) for 𝑚 = 1 becomes the first inequality of (1.1). Equation (2.1) shows the same estimate given by ( 𝑏 1 ) of Theorem 4.2.3 in the book [21].

Lemma 2.3. Δ 2 𝑟 𝑖 ( 𝑛 , 𝑗 ) is nonnegative and nondecreasing in 𝑛 , and 𝑟 𝑖 ( 𝑛 , 𝑗 ) is nonnegative and nondecreasing in 𝑛 and 𝑗 for 𝑖 = 1 , … , 𝑚 .

Proof. By the definitions of ̃ 𝑎 ( 𝑛 ) and 𝑓 𝑖 ( 𝑛 , 𝑗 ) , it is easy to check that they are nonnegative and nondecreasing in 𝑛 , and ̃ 𝑎 ( 𝑛 ) ≥ 𝑎 ( 𝑛 ) and 𝑓 𝑖 ( 𝑛 , 𝑗 ) ≥ 𝑓 𝑖 ( 𝑛 , 𝑗 ) for each fixed 𝑗 where 𝑖 = 1 , … , 𝑚 . 𝑎 ( 0 ) > 0 in ( 𝐶 1 ) implies that ̃ 𝑎 ( 𝑛 ) > 0 for all 𝑛 ≤ 𝑁 1 . Clearly, Δ 2 𝑟 1 ( 𝑛 + 1 , 𝑗 ) − Δ 2 𝑟 1 Δ ( 𝑛 , 𝑗 ) = 0 , 2 𝑟 2 ( 𝑛 + 1 , 𝑗 ) − Δ 2 𝑟 2 𝑓 ( 𝑛 , 𝑗 ) = 1 𝑛 + 1 , 𝑏 1 − 𝑓 ( 𝑗 ) 1 𝑛 , 𝑏 1 + Δ ( 𝑗 ) 2 𝑟 1 ( 𝑛 + 1 , 𝑗 ) − Δ 2 𝑟 1 ( 𝑛 , 𝑗 ) 𝑤 1 𝑟 1 ( 0 , 𝑗 ) ≥ 0 , ( 2 . 5 ) where 𝑟 1 ( 0 , 𝑗 ) = ̃ 𝑎 ( 𝑗 ) > 0 is used, which yields that Δ 2 𝑟 1 ( 𝑛 , 𝑗 ) and Δ 2 𝑟 2 ( 𝑛 , 𝑗 ) are nondecreasing in 𝑛 . Assume that Δ 2 𝑟 𝑙 ( 𝑛 , 𝑗 ) is nondecreasing in 𝑛 . Then Δ 2 𝑟 𝑙 + 1 ( 𝑛 + 1 , 𝑗 ) − Δ 2 𝑟 𝑙 + 1 𝑓 ( 𝑛 , 𝑗 ) = 𝑙 𝑛 + 1 , 𝑏 𝑙 − 𝑓 ( 𝑗 ) 𝑙 𝑛 , 𝑏 𝑙 + Δ ( 𝑗 ) 2 𝑟 𝑙 ( 𝑛 + 1 , 𝑗 ) − Δ 2 𝑟 𝑙 ( 𝑛 , 𝑗 ) 𝜙 𝑙 𝑊 − 1 𝑙 − 1 𝑟 𝑙 ( 0 , 𝑗 ) ≥ 0 , ( 2 . 6 ) which implies that Δ 2 𝑟 𝑙 + 1 ( 𝑛 , 𝑗 ) is nondecreasing in 𝑛 . By induction, Δ 2 𝑟 𝑖 ( 𝑛 , 𝑗 ) ( 𝑖 = 1 , … , 𝑚 ) are nondecreasing in 𝑛 . Similarly, we can prove that they are nonnegative by induction again. Then 𝑟 𝑖 ( 𝑛 , 𝑗 ) ( 𝑖 = 1 , … , 𝑚 ) are nonnegative and nondecreasing in 𝑛 and 𝑗 .

Proof of Theorem 2.1. Take any arbitrary positive integer ̃ 𝑛 ≤ 𝑁 1 and consider the auxiliary inequality 𝑢 ( 𝑛 ) ≤ 𝑟 1 ( ̃ 𝑛 , 𝑛 ) + 𝑚 𝑏 𝑖 = 1 𝑖 ( 𝑛 − 1 ) 𝑠 = 𝑏 𝑖 ( 0 ) 𝑓 𝑖 ( ̃ 𝑛 , 𝑠 ) 𝑤 𝑖 ( 𝑢 ( 𝑠 ) ) , 𝑛 ≤ ̃ 𝑛 . ( 2 . 7 ) Claim that 𝑢 ( 𝑛 ) in (2.7) satisfies 𝑢 ( 𝑛 ) ≤ 𝑊 𝑚 − 1 ⎡ ⎢ ⎢ ⎣ 𝑊 𝑚 𝑟 1 + ( ̃ 𝑛 , 0 ) 𝑏 𝑚 ( 𝑛 − 1 ) 𝑠 = 𝑏 𝑚 ( 0 ) 𝑓 m ( ̃ 𝑛 , 𝑠 ) + 𝑛 − 1 𝑠 = 0 Δ 2 𝑟 𝑚 ( ̃ 𝑛 , 𝑠 ) 𝜙 𝑚 𝑊 − 1 𝑚 − 1 𝑟 𝑚 ⎤ ⎥ ⎥ ⎦ ( 0 , 𝑠 ) ( 2 . 8 ) for 𝑛 ≤ { ̃ 𝑛 , 𝑁 2 } where 𝑁 2 is the largest positive integer such that 𝑊 𝑖 𝑟 1 + ( ̃ 𝑛 , 0 ) 𝑏 𝑖 ( 𝑁 2 − 1 ) 𝑠 = 𝑏 𝑖 ( 0 ) 𝑓 𝑖 ( ̃ 𝑛 , 𝑠 ) + 𝑁 2 − 1 𝑠 = 0 Δ 2 𝑟 𝑖 ( ̃ 𝑛 , 𝑠 ) 𝜙 𝑖 𝑊 − 1 𝑖 − 1 𝑟 𝑖 ≤ ( 0 , 𝑠 ) ∞ 𝑢 𝑖 𝑑 𝑧 𝑤 𝑖 ( 𝑧 ) , ( 2 . 9 ) 𝑖 = 1 , … , 𝑚 .
Before we prove (2.8), notice that 𝑁 1 ≤ 𝑁 2 . In fact, 𝑟 𝑖 ( ̃ 𝑛 , 𝑛 ) , Δ 2 𝑟 𝑖 ( ̃ 𝑛 , 𝑛 ) , and 𝑓 𝑖 ( ̃ 𝑛 , 𝑛 ) are nondecreasing in ̃ 𝑛 by Lemma 2.3. Thus, 𝑁 2 satisfying (2.9) gets smaller as ̃ 𝑛 is chosen larger. In particular, 𝑁 2 satisfies the same (2.3) as 𝑁 1 for ̃ 𝑛 = 𝑁 1 if 𝑟 1 ( ̃ 𝑛 , 0 ) = ̃ 𝑎 ( 0 ) is applied.
We divide the proof of (2.8) into two steps by using induction.
Step 1 ( 𝑚 = 1 ). Let ∑ 𝑧 ( 𝑛 ) = 𝑏 1 ( 𝑛 − 1 ) 𝑠 = 𝑏 1 ( 0 ) 𝑓 1 ( ̃ 𝑛 , 𝑠 ) 𝑤 1 ( 𝑢 ( 𝑠 ) ) for 𝑛 ≤ ̃ 𝑛 and 𝑧 ( 0 ) = 0 . It is clear that 𝑧 ( 𝑛 ) is nonnegative and nondecreasing. Observe that (2.7) is equivalent to 𝑢 ( 𝑛 ) ≤ 𝑟 1 ( ̃ 𝑛 , 𝑛 ) + 𝑧 ( 𝑛 ) for 𝑛 ≤ ̃ 𝑛 and by assumptions ( 𝐶 2 ) and ( 𝐶 4 ) and Lemma 2.3, 𝑓 Δ 𝑧 ( 𝑛 ) = 1 ̃ 𝑛 , 𝑏 1 𝑤 ( 𝑛 ) 1 𝑢 𝑏 1 ≤ 𝑓 ( 𝑛 ) 1 ̃ 𝑛 , 𝑏 1 𝑤 ( 𝑛 ) 1 𝑟 1 ̃ 𝑛 , 𝑏 1 𝑏 ( 𝑛 ) + 𝑧 1 ≤ 𝑓 ( 𝑛 ) 1 ̃ 𝑛 , 𝑏 1 𝑤 ( 𝑛 ) 1 𝑟 1 . ( ̃ 𝑛 , 𝑛 ) + 𝑧 ( 𝑛 ) ( 2 . 1 0 ) Since 𝑤 1 is nondecreasing and 𝑟 1 ( ̃ 𝑛 , 𝑛 ) = ̃ 𝑎 ( 𝑛 ) > 0 , we have Δ 𝑧 ( 𝑛 ) + Δ 2 𝑟 1 ( ̃ 𝑛 , 𝑛 ) 𝑤 1 𝑟 1 ≤ 𝑓 ( ̃ 𝑛 , 𝑛 ) + 𝑧 ( 𝑛 ) 1 ̃ 𝑛 , 𝑏 1 + Δ ( 𝑛 ) 2 𝑟 1 ( ̃ 𝑛 , 𝑛 ) 𝑤 1 𝑟 1 ≤ 𝑓 ( ̃ 𝑛 , 𝑛 ) + 𝑧 ( 𝑛 ) 1 ̃ 𝑛 , 𝑏 1 + Δ ( 𝑛 ) 2 𝑟 1 ( ̃ 𝑛 , 𝑛 ) 𝑤 1 𝑟 1 . ( 0 , 𝑛 ) ( 2 . 1 1 ) Then 𝑧 ( 𝑛 + 1 ) + 𝑟 1 ( ̃ 𝑛 , 𝑛 + 1 ) 𝑧 ( 𝑛 ) + 𝑟 1 ( ̃ 𝑛 , 𝑛 ) 𝑑 𝜏 𝑤 1 ≤ ( 𝜏 ) 𝑧 ( 𝑛 + 1 ) + 𝑟 1 ( ̃ 𝑛 , 𝑛 + 1 ) 𝑧 ( 𝑛 ) + 𝑟 1 ( ̃ 𝑛 , 𝑛 ) 𝑑 𝜏 𝑤 1 𝑧 ( 𝑛 ) + 𝑟 1 ≤ ( ̃ 𝑛 , 𝑛 ) Δ 𝑧 ( 𝑛 ) + Δ 2 𝑟 1 ( ̃ 𝑛 , 𝑛 ) 𝑤 1 𝑧 ( 𝑛 ) + 𝑟 1 ≤ 𝑓 ( ̃ 𝑛 , 𝑛 ) 1 ̃ 𝑛 , 𝑏 1 + Δ ( 𝑛 ) 2 𝑟 1 ( ̃ 𝑛 , 𝑛 ) 𝑤 1 𝑟 1 , ( 0 , 𝑛 ) ( 2 . 1 2 ) and so 𝑧 ( 𝑛 ) + 𝑟 1 ( ̃ 𝑛 , 𝑛 ) 𝑧 ( 0 ) + 𝑟 1 ( ̃ 𝑛 , 0 ) 𝑑 𝜏 𝑤 1 = ( 𝜏 ) 𝑛 − 1 𝑠 = 0 𝑧 ( 𝑠 + 1 ) + 𝑟 1 ( ̃ 𝑛 , 𝑠 + 1 ) 𝑧 ( 𝑠 ) + 𝑟 1 ( ̃ 𝑛 , 𝑠 ) 𝑑 𝜏 𝑤 1 ≤ ( 𝜏 ) 𝑛 − 1 𝑠 = 0 𝑓 1 ̃ 𝑛 , 𝑏 1 + ( 𝑠 ) 𝑛 − 1 𝑠 = 0 Δ 2 𝑟 1 ( ̃ 𝑛 , 𝑠 ) 𝑤 1 𝑟 1 = ( 0 , 𝑠 ) 𝑏 1 ( 𝑛 − 1 ) 𝑠 = 𝑏 1 ( 0 ) 𝑓 1 ( ̃ 𝑛 , 𝑠 ) + 𝑛 − 1 𝑠 = 0 Δ 2 𝑟 1 ( ̃ 𝑛 , 𝑠 ) 𝑤 1 𝑟 1 ( . 0 , 𝑠 ) ( 2 . 1 3 ) The definition of 𝑊 1 in Theorem 2.1 and 𝑧 ( 0 ) = 0 show 𝑊 1 𝑧 ( 𝑛 ) + 𝑟 1 ( ̃ 𝑛 , 𝑛 ) ≤ 𝑊 1 𝑟 1 + ( ̃ 𝑛 , 0 ) 𝑏 1 ( 𝑛 − 1 ) 𝑠 = 𝑏 1 ( 0 ) 𝑓 1 ( ̃ 𝑛 , 𝑠 ) + 𝑛 − 1 𝑠 = 0 Δ 2 𝑟 1 ( ̃ 𝑛 , 𝑠 ) 𝑤 1 𝑟 1 ( 0 , 𝑠 ) , 𝑛 ≤ ̃ 𝑛 . ( 2 . 1 4 ) Equation (2.9) shows that the right side of (2.14) is in the domain of 𝑊 1 − 1 for all 𝑛 ≤ ̃ 𝑛 . Thus the monotonicity of 𝑊 1 − 1 implies 𝑢 ( 𝑛 ) ≤ 𝑧 ( 𝑛 ) + 𝑟 1 ( ̃ 𝑛 , 𝑛 ) ≤ 𝑊 1 − 1 ⎡ ⎢ ⎢ ⎣ 𝑊 1 𝑟 1 ( + ̃ 𝑛 , 0 ) 𝑏 1 ( 𝑛 − 1 ) 𝑠 = 𝑏 1 ( 0 ) 𝑓 1 ( ̃ 𝑛 , 𝑠 ) + 𝑛 − 1 𝑠 = 0 Δ 2 𝑟 1 ( ̃ 𝑛 , 𝑠 ) 𝑤 1 𝑟 1 ⎤ ⎥ ⎥ ⎦ ( 0 , 𝑠 ) ( 2 . 1 5 ) for 𝑛 ≤ ̃ 𝑛 ; that is, (2.8) is true for 𝑚 = 1 .Step 2 ( 𝑚 = 𝑘 + 1 ). Assume that (2.8) is true for 𝑚 = 𝑘 . Consider 𝑢 ( 𝑛 ) ≤ 𝑟 1 ( ̃ 𝑛 , 𝑛 ) + 𝑘 + 1 𝑏 𝑖 = 1 𝑖 ( 𝑛 − 1 ) 𝑠 = 𝑏 𝑖 ( 0 ) 𝑓 𝑖 ( ̃ 𝑛 , 𝑠 ) 𝑤 𝑖 ( 𝑢 ( 𝑠 ) ) , 𝑛 ≤ ̃ 𝑛 . ( 2 . 1 6 ) Let ∑ 𝑧 ( 𝑛 ) = 𝑘 + 1 𝑖 = 1 ∑ 𝑏 𝑖 ( 𝑛 − 1 ) 𝑠 = 𝑏 𝑖 ( 0 ) 𝑓 𝑖 ( ̃ 𝑛 , 𝑠 ) 𝑤 𝑖 ( 𝑢 ( 𝑠 ) ) and 𝑧 ( 0 ) = 0 . Then 𝑧 ( 𝑛 ) is nonnegative and nondecreasing and satisfies 𝑢 ( 𝑛 ) ≤ 𝑟 1 ( ̃ 𝑛 , 𝑛 ) + 𝑧 ( 𝑛 ) for 𝑛 ≤ ̃ 𝑛 . Moreover, we have Δ 𝑧 ( 𝑛 ) = 𝑘 + 1 𝑖 = 1 𝑓 𝑖 ̃ 𝑛 , 𝑏 𝑖 𝑤 ( 𝑛 ) 𝑖 𝑢 𝑏 𝑖 ≤ ( 𝑛 ) 𝑘 + 1 𝑖 = 1 𝑓 𝑖 ̃ 𝑛 , 𝑏 𝑖 𝑤 ( 𝑛 ) 𝑖 𝑟 1 ̃ 𝑛 , 𝑏 𝑖 𝑏 ( 𝑛 ) + 𝑧 𝑖 . ( 𝑛 ) ( 2 . 1 7 ) Since 𝑤 𝑖 and 𝑟 1 are nondecreasing in their arguments and 𝑟 1 ( ̃ 𝑛 , 𝑛 ) > 0 , we have by the assumption 𝑏 𝑖 ( 𝑛 ) ≤ 𝑛 Δ 𝑧 ( 𝑛 ) + Δ 2 𝑟 1 ( ̃ 𝑛 , 𝑛 ) 𝑤 1 𝑧 ( 𝑛 ) + 𝑟 1 ≤ ∑ ( ̃ 𝑛 , n ) 𝑘 + 1 𝑖 = 1 𝑓 𝑖 ̃ 𝑛 , 𝑏 𝑖 𝑤 ( 𝑛 ) 𝑖 𝑧 𝑏 𝑖 ( 𝑛 ) + 𝑟 1 ̃ 𝑛 , 𝑏 𝑖 ( 𝑛 ) 𝑤 1 𝑧 ( 𝑛 ) + 𝑟 1 + Δ ( ̃ 𝑛 , 𝑛 ) 2 𝑟 1 ( ̃ 𝑛 , 𝑛 ) 𝑤 1 𝑟 1 ≤ 𝑓 ( ̃ 𝑛 , 𝑛 ) 1 ̃ 𝑛 , 𝑏 1 + ( 𝑛 ) 𝑘 + 1 𝑖 = 2 𝑓 𝑖 ̃ 𝑛 , 𝑏 𝑖 𝑤 ( 𝑛 ) 𝑖 𝑧 𝑏 𝑖 ( 𝑛 ) + 𝑟 1 ̃ 𝑛 , 𝑏 𝑖 ( 𝑛 ) 𝑤 1 𝑧 𝑏 𝑖 ( 𝑛 ) + 𝑟 1 ̃ 𝑛 , 𝑏 𝑖 + Δ ( 𝑛 ) 2 𝑟 1 ( ̃ 𝑛 , 𝑛 ) 𝑤 1 𝑟 1 ≤ 𝑓 ( 0 , 𝑛 ) 1 ̃ 𝑛 , 𝑏 1 ( + 𝑛 ) 𝑘 𝑖 = 1 𝑓 𝑖 + 1 ̃ 𝑛 , 𝑏 𝑖 + 1 ( 𝜙 𝑛 ) 𝑖 + 1 𝑧 𝑏 𝑖 + 1 ( 𝑛 ) + 𝑟 1 ̃ 𝑛 , 𝑏 𝑖 + 1 ( + Δ 𝑛 ) 2 𝑟 1 ( ̃ 𝑛 , 𝑛 ) 𝑤 1 𝑟 1 ( 0 , 𝑛 ) ( 2 . 1 8 ) for 𝑛 ≤ ̃ 𝑛 where 𝜙 𝑖 + 1 ( 𝑢 ) = 𝑤 𝑖 + 1 ( 𝑢 ) / 𝑤 1 ( 𝑢 ) for 𝑖 = 1 , … , 𝑘 , which gives 𝑧 ( 𝑛 + 1 ) + 𝑟 1 ( ̃ 𝑛 , 𝑛 + 1 ) 𝑧 ( 𝑛 ) + 𝑟 1 ( ̃ 𝑛 , 𝑛 ) 𝑑 𝜏 𝑤 1 ≤ ( 𝜏 ) 𝑧 ( 𝑛 + 1 ) + 𝑟 1 ( ̃ 𝑛 , 𝑛 + 1 ) 𝑧 ( 𝑛 ) + 𝑟 1 ( ̃ 𝑛 , 𝑛 ) 𝑑 𝜏 𝑤 1 𝑧 ( 𝑛 ) + 𝑟 1 ≤ ( ̃ 𝑛 , 𝑛 ) Δ 𝑧 ( 𝑛 ) + Δ 2 𝑟 1 ( ̃ 𝑛 , 𝑛 ) 𝑤 1 𝑟 1 ≤ 𝑓 ( ̃ 𝑛 , 𝑛 ) + 𝑧 ( 𝑛 ) 1 ̃ 𝑛 , 𝑏 1 + Δ ( 𝑛 ) 2 𝑟 1 ( ̃ 𝑛 , 𝑛 ) 𝑤 1 𝑟 1 + ( 0 , 𝑛 ) 𝑘 𝑖 = 1 𝑓 𝑖 + 1 ̃ 𝑛 , 𝑏 𝑖 + 1 𝜙 ( 𝑛 ) 𝑖 + 1 𝑧 𝑏 𝑖 + 1 ( 𝑛 ) + 𝑟 1 ̃ 𝑛 , 𝑏 𝑖 + 1 . ( 𝑛 ) ( 2 . 1 9 ) Therefore, 𝑧 ( 𝑛 ) + 𝑟 1 ( ̃ 𝑛 , 𝑛 ) 𝑧 ( 0 ) + 𝑟 1 ( ̃ 𝑛 , 0 ) 𝑑 𝜏 𝑤 1 ≤ ( 𝜏 ) 𝑏 1 ( 𝑛 − 1 ) 𝑠 = 𝑏 1 ( 0 ) 𝑓 1 ( ̃ 𝑛 , 𝑠 ) + 𝑛 − 1 𝑠 = 0 Δ 2 𝑟 1 ( ̃ 𝑛 , 𝑠 ) 𝑤 1 𝑟 1 + ( 0 , 𝑠 ) 𝑘 𝑖 = 1 𝑛 − 1 𝑠 = 0 𝑓 𝑖 + 1 ̃ 𝑛 , 𝑏 𝑖 + 1 ( 𝜙 𝑠 ) 𝑖 + 1 𝑧 𝑏 𝑖 + 1 ( 𝑠 ) + 𝑟 1 ̃ 𝑛 , 𝑏 𝑖 + 1 ( , 𝑠 ) ( 2 . 2 0 ) that is, 𝑊 1 𝑧 ( 𝑛 ) + 𝑟 1 ( ̃ 𝑛 , 𝑛 ) ≤ 𝑊 1 𝑟 1 + ( ̃ 𝑛 , 0 ) 𝑏 1 ( 𝑛 − 1 ) 𝑠 = 𝑏 1 ( 0 ) 𝑓 1 ( ̃ 𝑛 , 𝑠 ) + 𝑛 − 1 𝑠 = 0 Δ 2 𝑟 1 ( ̃ 𝑛 , 𝑠 ) 𝑤 1 𝑟 1 + ( 0 , 𝑠 ) 𝑘 𝑏 𝑖 = 1 𝑖 + 1 ( 𝑛 − 1 ) 𝑠 = 𝑏 𝑖 + 1 ( 0 ) 𝑓 𝑖 + 1 𝜙 ( ̃ 𝑛 , 𝑠 ) 𝑖 + 1 𝑧 ( 𝑠 ) + 𝑟 1 , ( ̃ 𝑛 , 𝑠 ) ( 2 . 2 1 ) or equivalently 𝜉 ( 𝑛 ) ≤ 𝑐 1 ( ̃ 𝑛 , 𝑛 ) + 𝑘 𝑏 𝑖 = 1 𝑖 + 1 ( 𝑛 − 1 ) 𝑠 = 𝑏 𝑖 + 1 ( 0 ) 𝑓 𝑖 + 1 𝜙 ( ̃ 𝑛 , 𝑠 ) 𝑖 + 1 𝑊 1 − 1 ( 𝜉 ( 𝑠 ) ) , 𝑛 ≤ ̃ 𝑛 , ( 2 . 2 2 ) the same as (2.7) for 𝑚 = 𝑘 where 𝜉 ( 𝑛 ) = 𝑊 1 ( 𝑧 ( 𝑛 ) + 𝑟 1 ( ̃ 𝑛 , 𝑛 ) ) and 𝑐 1 ( ̃ 𝑛 , 𝑛 ) = 𝑊 1 𝑟 1 + ( ̃ 𝑛 , 0 ) 𝑏 1 ( 𝑛 − 1 ) 𝑠 = 𝑏 1 ( 0 ) 𝑓 1 ( ̃ 𝑛 , 𝑠 ) + 𝑛 − 1 𝑠 = 0 Δ 2 𝑟 1 ( ̃ 𝑛 , 𝑠 ) 𝑤 1 𝑟 1 ( 0 , 𝑠 ) . ( 2 . 2 3 ) From the assumption ( 𝐶 4 ), each 𝜙 𝑖 + 1 ( 𝑊 1 − 1 ) , 𝑖 = 1 , … , 𝑘 , is continuous and nondecreasing on [ 0 , ∞ ) and is positive on ( 0 , ∞ ) since 𝑊 1 − 1 is continuous and nondecreasing on [ 0 , ∞ ) . Moreover, 𝜙 2 ( 𝑊 1 − 1 𝜙 ) ∝ 3 ( 𝑊 1 − 1 𝜙 ) ∝ ⋯ ∝ 𝑘 + 1 ( 𝑊 1 − 1 ) . By the inductive assumption, we have 𝜉 ( 𝑛 ) ≤ Φ − 1 𝑘 + 1 ⎡ ⎢ ⎢ ⎣ Φ 𝑘 + 1 𝑐 1 + ( ̃ 𝑛 , 0 ) 𝑏 𝑘 + 1 ( 𝑛 − 1 ) 𝑠 = 𝑏 𝑘 + 1 ( 0 ) 𝑓 𝑘 + 1 ( ̃ 𝑛 , 𝑠 ) + 𝑛 − 1 𝑠 = 0 Δ 2 𝑐 𝑘 ( ̃ 𝑛 , 𝑠 ) 𝜓 𝑘 + 1 Φ 𝑘 − 1 𝑐 𝑘 ( ⎤ ⎥ ⎥ ⎦ 0 , 𝑠 ) ( 2 . 2 4 ) for 𝑛 ≤ m i n { ̃ 𝑛 , 𝑁 3 } where Φ 𝑖 + 1 ∫ ( 𝑢 ) = 𝑢 ̃ 𝑢 𝑖 + 1 𝜙 ( 𝑑 𝑧 / 𝑖 + 1 ( 𝑊 1 − 1 ( 𝑧 ) ) ) , 𝑢 > 0 , Φ 1 = 𝐼 (Identity), ̃ 𝑢 𝑖 + 1 = 𝑊 1 ( 𝑢 𝑖 + 1 ) , Φ − 1 𝑖 + 1 is the inverse of Φ 𝑖 + 1 , 𝜓 𝑖 + 1 𝜙 ( 𝑢 ) = 𝑖 + 1 ( 𝑊 1 − 1 𝜙 ( 𝑢 ) ) / 𝑖 ( 𝑊 1 − 1 ( 𝑢 ) ) = 𝑤 𝑖 + 1 ( 𝑊 1 − 1 ( 𝑢 ) ) / 𝑤 𝑖 ( 𝑊 1 − 1 ( 𝑢 ) ) , 𝑖 = 1 , … , 𝑘 , 𝑐 𝑖 + 1 ( ̃ 𝑛 , 𝑛 ) = Φ 𝑖 + 1 𝑐 1 + ( ̃ 𝑛 , 0 ) 𝑏 𝑖 + 1 ( 𝑛 − 1 ) 𝑠 = 𝑏 𝑖 + 1 ( 0 ) 𝑓 𝑖 + 1 ( ̃ 𝑛 , 𝑠 ) + 𝑛 − 1 𝑠 = 0 Δ 2 𝑐 𝑖 ( ̃ 𝑛 , 𝑠 ) 𝜓 𝑖 + 1 Φ 𝑖 − 1 𝑐 𝑖 ( 0 , 𝑠 ) , ( 2 . 2 5 ) i = 1 , … , 𝑘 − 1 , and 𝑁 3 is the largest positive integer such that Φ 𝑖 + 1 𝑐 1 + ( ̃ 𝑛 , 0 ) 𝑏 𝑖 + 1 ( 𝑁 3 − 1 ) 𝑠 = 𝑏 𝑖 + 1 ( 0 ) 𝑓 𝑖 + 1 ( ̃ 𝑛 , 𝑠 ) + 𝑁 3 − 1 𝑠 = 0 Δ 2 𝑐 𝑖 ( ̃ 𝑛 , 𝑠 ) 𝜓 𝑖 + 1 Φ 𝑖 − 1 𝑐 𝑖 ≤ ( 0 , 𝑠 ) 𝑊 1 ( ∞ ) ̃ 𝑢 𝑖 + 1 𝑑 𝑧 𝜙 𝑖 + 1 𝑊 1 − 1 ( 𝑧 ) , 𝑖 = 1 , … , 𝑘 . ( 2 . 2 6 ) Note that Φ 𝑖 ( 𝑢 ) = 𝑢 ̃ 𝑢 𝑖 𝑑 𝑧 𝜙 𝑖 𝑊 1 − 1 ( = 𝑧 ) 𝑢 𝑊 1 ( 𝑢 𝑖 ) 𝑤 1 𝑊 1 − 1 ( 𝑧 ) 𝑑 𝑧 𝑤 𝑖 𝑊 1 − 1 = ( 𝑧 ) 𝑊 1 − 1 𝑢 ( 𝑢 ) 𝑖 𝑑 𝑧 𝑤 𝑖 ( 𝑧 ) = 𝑊 𝑖 ∘ 𝑊 1 − 1 𝜓 ( 𝑢 ) , 𝑖 = 2 , … , 𝑘 + 1 , 𝑖 + 1 Φ 𝑖 − 1 ( = 𝑤 𝑢 ) 𝑖 + 1 𝑊 1 − 1 Φ 𝑖 − 1 ( 𝑢 ) 𝑤 𝑖 𝑊 1 − 1 Φ 𝑖 − 1 = 𝑤 ( 𝑢 ) 𝑖 + 1 𝑊 1 − 1 𝑊 1 𝑊 𝑖 − 1 ( 𝑢 ) 𝑤 𝑖 𝑊 1 − 1 𝑊 1 𝑊 𝑖 − 1 = 𝑤 ( 𝑢 )