Existence and Iterative Algorithms of Positive Solutions for a Higher Order Nonlinear Neutral Delay Differential Equation

Abstract

This paper is concerned with the higher order nonlinear neutral delay differential equation [𝑎(𝑡)(𝑥(𝑡)+𝑏(𝑡)𝑥(𝑡−𝜏))^(𝑚)]^{(𝑛−𝑚)}+[ℎ(𝑡,𝑥(ℎ₁(𝑡)),…,𝑥(ℎ_𝑙(𝑡)))]^(𝑖)+𝑓(𝑡,𝑥(𝑓₁(𝑡)),…,𝑥(𝑓_𝑙(𝑡)))=𝑔(𝑡), for all 𝑡≥𝑡₀. Using the Banach fixed point theorem, we establish the existence results of uncountably many positive solutions for the equation, construct Mann iterative sequences for approximating these positive solutions, and discuss error estimates between the approximate solutions and the positive solutions. Nine examples are included to dwell upon the importance and advantages of our results.

1. Introduction and Preliminaries

In recent years, the existence problems of nonoscillatory solutions for neutral delay differential equations of first, second, third, and higher order have been studied intensively by using fixed point theorems; see, for example, [1–12] and the references therein.

Using the Banach, Schauder, and Krasnoselskii fixed point theorems, Zhang et al. [9] and Liu et al. [7] considered the existence of nonoscillatory solutions for the following first order neutral delay differential equations: [𝑥(𝑡)+𝑃(𝑡)𝑥(𝑡−𝜏)]^′+𝑄₁(𝑡)𝑥𝑡−𝜏₁−𝑄₂(𝑡)𝑥𝑡−𝜏₂=0,∀𝑡≥𝑡₀,[𝑥(𝑡)+𝑐(𝑡)𝑥(𝑡−𝜏)]^′+ℎ(𝑡)𝑓𝑥𝑡−𝜎₁,𝑥𝑡−𝜎₂,…,𝑥𝑡−𝜎_𝑘=𝑔(𝑡),∀𝑡≥𝑡₀,(1) where 𝑃∈𝐶([𝑡₀,+∞),ℝ⧵{±1}) and 𝑐∈𝐶([𝑡₀,+∞),ℝ). Making use of the Banach and Krasnoselskii fixed point theorems, Kulenović and Hadžiomerspahić [2] and Zhou [10] studied the existence of a nonoscillatory solution for the following second order neutral differential equations: [𝑥(𝑡)+𝑐𝑥(𝑡−𝜏)]^′′+𝑄₁(𝑡)𝑥𝑡−𝜎₁−𝑄₂(𝑡)𝑥𝑡−𝜎₂=0,∀𝑡≥𝑡₀,𝑟(𝑡)(𝑥(𝑡)+𝑃(𝑡)𝑥(𝑡−𝜏))^′′+^𝑚_𝑖=1𝑄_𝑖(𝑡)𝑓_𝑖𝑥𝑡−𝜎_𝑖=0,∀𝑡≥𝑡₀,(2) where 𝑐∈ℝ⧵{±1} and 𝑃∈𝐶([𝑡₀,∞),ℝ). Zhou and Zhang [11], Zhou et al. [12], and Liu et al. [4], respectively, investigated the existence of nonoscillatory solutions for the following higher order neutral delay differential equations: [𝑥(𝑡)+𝑐𝑥(𝑡−𝜏)]^(𝑛)+(−1)^𝑛+1[𝑃(𝑡)𝑥(𝑡−𝜎)−𝑄(𝑡)𝑥(𝑡−𝛿)]=0,∀𝑡≥𝑡₀,[𝑥(𝑡)+𝑃(𝑡)𝑥(𝑡−𝜏)]^(𝑛)+^𝑚_𝑖=1𝑄_𝑖(𝑡)𝑓_𝑖𝑥𝑡−𝜎_𝑖=𝑔(𝑡),∀𝑡≥𝑡₀,[𝑥(𝑡)+𝑎𝑥(𝑡−𝜏)]^(𝑛)+(−1)^𝑛+1𝑓𝑡,𝑥𝑡−𝜎₁,𝑥𝑡−𝜎₂,…,𝑥𝑡−𝜎_𝑘=𝑔(𝑡),∀𝑡≥𝑡₀,(3) where 𝑐∈ℝ⧵{±1}, 𝑃∈𝐶([𝑡₀,∞),ℝ) and 𝑎∈ℝ⧵{−1}. Candan [1] proved the existence of a bounded nonoscillatory solution for the higher order nonlinear neutral differential equation: 𝑟(𝑡)(𝑥(𝑡)+𝑃(𝑡)𝑥(𝑡−𝜏))^{(𝑛−1)′}+(−1)^𝑛𝑄₁(𝑡)𝑔₁𝑥𝑡−𝜎₁−𝑄₂(𝑡)𝑔₂𝑥𝑡−𝜎₂−𝑓(𝑡)=0,∀𝑡≥𝑡₀,(4) where 𝑃∈𝐶([𝑡₀,∞),ℝ⧵{±1}).

Motivated by the results in [1–12], in this paper we consider the following higher order nonlinear neutral delay differential equation: 𝑎(𝑡)(𝑥(𝑡)+𝑏(𝑡)𝑥(𝑡−𝜏))^{(𝑚)(𝑛−𝑚)}=+ℎ𝑡,𝑥ℎ₁(𝑡),…,𝑥ℎ_𝑙(𝑡)^(𝑖)=+𝑓𝑡,𝑥𝑓₁(𝑡),…,𝑥𝑓_𝑙(𝑡)=𝑔(𝑡),∀𝑡≥𝑡₀,(5) where 𝑚,𝑛∈ℕ and 𝑖∈ℕ₀ with 𝑖≤𝑛−𝑚−1, 𝜏>0, 𝑎∈𝐶([𝑡₀,+∞),ℝ⧵{0}), 𝑏,𝑔,𝑓_𝑗,ℎ_𝑗∈𝐶([𝑡₀,+∞),ℝ), ℎ∈𝐶^𝑖([𝑡₀,+∞)×ℝ^𝑙,ℝ) and 𝑓∈𝐶([𝑡₀,+∞)×ℝ^𝑙,ℝ) with lim_𝑡→+∞ℎ_𝑗(𝑡)=lim_𝑡→+∞𝑓_𝑗(𝑡)=+∞,𝑗∈{1,2,…,𝑙}.(6) It is clear that (5) includes (1)–(4) as special cases. Utilizing the Banach fixed point theorem, we prove several existence results of uncountably many positive solutions for (5), construct a few Mann iterative schemes, and discuss error estimates between the sequences generated by the Mann iterative schemes and the positive solutions. Nine examples are given to show that the results presented in this paper extend substantially the existing ones in [1, 2, 4, 5, 8, 9, 11].

Throughout this paper, we assume that ℝ=(−∞,+∞),ℝ⁺=[0,+∞), ℕ denotes the set of all positive integers, ℕ₀=ℕ∪{0}, 𝐻_𝑗=1()!!{},𝑡𝑚−1𝑛−𝑚−𝑗−1,𝑗∈0,𝑖𝛾=min₀−𝜏,inf_𝑡≥𝑡0ℎ_𝑗(𝑡),inf_𝑡≥𝑡0𝑓_𝑗(𝑡)∶𝑗∈{1,2,…,𝑙},(7)CB([𝛾,+∞),ℝ) stands for the Banach space of all continuous and bounded functions in [𝛾,+∞) with norm ‖𝑥‖=sup_𝑡≥𝛾|𝑥(𝑡)|, and for any 𝑀>𝑁>0Ω₁()=∶(𝑡),Ω𝑁,𝑀𝑥∈CB𝛾,+∞,ℝ𝑁≤𝑥≤𝑀,∀𝑡≥𝛾₂()=∶𝑁𝑁,𝑀𝑥∈CB𝛾,+∞,ℝ𝑏()(𝑡)≤𝑀𝑡+𝜏≤𝑥𝑏()𝑁𝑡+𝜏,∀𝑡≥𝑇;𝑏()(𝑡)≤𝑀𝑇+𝜏≤𝑥𝑏(),Ω𝑇+𝜏,∀𝑡∈𝛾,𝑇₃()=𝑁𝑁,𝑀𝑥∈CB𝛾,+∞,ℝ∶−𝑏()(𝑡)𝑀𝑡+𝜏≤𝑥≤−𝑏()𝑁𝑡+𝜏,∀𝑡≥𝑇;−𝑏()(𝑡)𝑀𝑇+𝜏≤𝑥≤−𝑏().𝑇+𝜏,∀𝑡∈𝛾,𝑇(8) It is easy to check that Ω₁(𝑁,𝑀), Ω₂(𝑁,𝑀) and Ω₃(𝑁,𝑀) are closed subsets of CB([𝛾,+∞),ℝ).

By a solution of (5), we mean a function 𝑥∈𝐶([𝛾,+∞),ℝ) for some 𝑇>1+|𝑡₀|+𝜏+|𝛾|, such that 𝑎(𝑡)(𝑥(𝑡)+𝑏(𝑡)𝑥(𝑡−𝜏))^(𝑚) are 𝑛−𝑚 times continuously differentiable in [𝑇,+∞) and such that (5) is satisfied for 𝑡≥𝑇.

Lemma 1. Let 𝜏>0,𝑐≥0, 𝐹∈𝐶([𝑐,+∞)³,ℝ⁺) and 𝐺∈𝐶([𝑐,+∞)²,ℝ⁺). Then(a)∫_𝑐^+∞∫_𝑟^+∞∫_𝑢^+∞𝑟𝐹(𝑠,𝑢,𝑟)𝑑𝑠𝑑𝑢𝑑𝑟<+∞⇔∑^∞_𝑗=0∫^+∞_{𝑐+𝑗𝜏}∫_𝑟^+∞∫_𝑢^+∞𝐹(𝑠,𝑢,𝑟)𝑑𝑠𝑑𝑢𝑑𝑟<+∞; (b)∫_𝑐^+∞∫_𝑢^+∞𝑢𝐺(𝑠,𝑢)𝑑𝑠𝑑𝑢<+∞⇔∑^∞_𝑗=0∫^+∞_{𝑐+𝑗𝜏}∫_𝑢^+∞𝐺(𝑠,𝑢)𝑑𝑠𝑑𝑢<+∞; (c)if ∫_𝑐^+∞∫_𝑟^+∞∫_𝑢^+∞𝑟𝐹(𝑠,𝑢,𝑟)𝑑𝑠𝑑𝑢𝑑𝑟<+∞, then ^∞_𝑗=1^+∞_{𝑡+𝑗𝜏𝑟}^+∞_𝑢^+∞𝐹()≤1𝑠,𝑢,𝑟𝑑𝑠𝑑𝑢𝑑𝑟𝜏^+∞_{𝑡+𝜏𝑟}^+∞_𝑢^+∞𝑟𝐹(𝑠,𝑢,𝑟)𝑑𝑠𝑑𝑢𝑑𝑟<+∞,∀𝑡≥𝑐;(9)(d)if ∫_𝑐^+∞∫_𝑢^+∞𝑢𝐺(𝑠,𝑢)𝑑𝑠𝑑𝑢<+∞, then ^∞_𝑗=1^+∞_{𝑡+𝑗𝜏𝑢}^+∞𝐺()≤1𝑠,𝑢𝑑𝑠𝑑𝑢𝜏^+∞_{𝑡+𝜏𝑢}^+∞𝑢𝐺(𝑠,𝑢)𝑑𝑠𝑑𝑢<+∞,∀𝑡≥𝑐.(10)

Proof. Let [𝑡] denote the largest integral number not exceeding 𝑡∈ℝ. Note that lim_𝑟→+∞[(𝑟−𝑐)/𝜏]+1𝑟=1𝜏,()(11)𝑐+𝑛𝜏≤𝑟<𝑐+𝑛+1𝜏⟺𝑛≤𝑟−𝑐𝜏<𝑛+1,∀𝑛∈ℕ₀.(12) Clearly (12) means that ^∞_𝑗=0^+∞_{𝑐+𝑗𝜏𝑟}^+∞_𝑢^+∞𝐹()=𝑠,𝑢,𝑟𝑑𝑠𝑑𝑢𝑑𝑟_{𝑐+∞𝑟+∞𝑢+∞}𝐹()+𝑠,𝑢,𝑟𝑑𝑠𝑑𝑢𝑑𝑟_{+∞𝑐+𝜏𝑟+∞𝑢+∞}𝐹()+𝑠,𝑢,𝑟𝑑𝑠𝑑𝑢𝑑𝑟_{+∞𝑐+2𝜏𝑟+∞𝑢+∞}𝐹()+𝑠,𝑢,𝑟𝑑𝑠𝑑𝑢𝑑𝑟_{+∞𝑐+3𝜏𝑟+∞𝑢+∞}𝐹()=𝑠,𝑢,𝑟𝑑𝑠𝑑𝑢𝑑𝑟+⋯_{𝑐𝑐+𝜏𝑟+∞𝑢+∞}𝐹(𝑠,𝑢,𝑟)𝑑𝑠𝑑𝑢𝑑𝑟+2^𝑐+2𝜏_{𝑐+𝜏𝑟}^+∞_𝑢^+∞𝐹(𝑠,𝑢,𝑟)𝑑𝑠𝑑𝑢𝑑𝑟+3^𝑐+3𝜏_{𝑐+2𝜏𝑟}^+∞_𝑢^+∞𝐹(𝑠,𝑢,𝑟)𝑑𝑠𝑑𝑢𝑑𝑟+4^𝑐+4𝜏_{𝑐+3𝜏𝑟}^+∞_𝑢^+∞𝐹()=𝑠,𝑢,𝑟𝑑𝑠𝑑𝑢𝑑𝑟+⋯_{∞𝑛=0𝑐+(𝑛+1)𝜏𝑐+𝑛𝜏𝑟+∞𝑢+∞}()𝐹()=𝑛+1𝑠,𝑢,𝑟𝑑𝑠𝑑𝑢𝑑𝑟_{∞𝑛=0𝑐+(𝑛+1)𝜏𝑐+𝑛𝜏𝑟+∞𝑢+∞}𝑟−𝑐𝜏()=+1×𝐹𝑠,𝑢,𝑟𝑑𝑠𝑑𝑢𝑑𝑟_{𝑐+∞𝑟+∞𝑢+∞}𝑟−𝑐𝜏+1𝐹(𝑠,𝑢,𝑟)𝑑𝑠𝑑𝑢𝑑𝑟.(13) Thus (a) follows from (11) and (13).
Assume that ∫_𝑐^+∞∫_𝑟^+∞∫_𝑢^+∞𝑟𝐹(𝑠,𝑢,𝑟)𝑑𝑠𝑑𝑢𝑑𝑟<+∞. As in the proof of (a), we infer that ^∞_𝑗=1^+∞_{𝑡+𝑗𝜏𝑟}^+∞_𝑢^+∞𝐹()=𝑠,𝑢,𝑟𝑑𝑠𝑑𝑢𝑑𝑟_{+∞𝑡+𝜏𝑟+∞𝑢+∞}𝑟−𝑡𝜏𝐹()≤1𝑠,𝑢,𝑟𝑑𝑠𝑑𝑢𝑑𝑟𝜏^+∞_{𝑡+𝜏𝑟}^+∞_𝑢^+∞()≤1𝑟𝐹𝑠,𝑢,𝑟𝑑𝑠𝑑𝑢𝑑𝑟𝜏_𝑐^+∞_𝑟^+∞_𝑢^+∞𝑟𝐹(𝑠,𝑢,𝑟)𝑑𝑠𝑑𝑢𝑑𝑟<+∞,∀𝑡≥𝑐,(14) that is, (c) holds.
Similar to the proofs of (a) and (c), we conclude that (b) and (d) hold. This completes the proof.

2. Existence of Uncountably Many Positive Solutions and Mann Iterative Schemes

Now we show the existence of uncountably many positive solutions for (5) and discuss the convergence of the Mann iterative sequences to these positive solutions.

Theorem 2. Assume that there exist three constants 𝑀, 𝑁, and 𝑏₀ and four functions 𝑃,𝑄,𝑅,𝑊∈𝐶([𝑡₀,+∞),ℝ⁺) satisfying 0<𝑁<𝑀,𝑏₀<𝑀−𝑁,|𝑏(𝑡2𝑀)|≤𝑏₀𝑓𝑒𝑣𝑒𝑛𝑡𝑢𝑎𝑙𝑙𝑦;(15)𝑡,𝑢₁,…,𝑢_𝑙−𝑓𝑡,𝑢₁,…,𝑢_𝑙≤𝑃(𝑡)max𝑢_𝑗−𝑢_𝑗,ℎ∶1≤𝑗≤𝑙𝑡,𝑢₁,…,𝑢_𝑙−ℎ𝑡,𝑢₁,…,𝑢_𝑙≤𝑅(𝑡)max𝑢_𝑗−𝑢_𝑗,∀∶1≤𝑗≤𝑙𝑡,𝑢₁,…,𝑢_𝑙,𝑢₁,…,𝑢_𝑙∈𝑡₀,+∞×[𝑁,𝑀]^2𝑙;𝑓(16)𝑡,𝑢₁,…,𝑢_𝑙≤𝑄(𝑡),ℎ𝑡,𝑢₁,…,𝑢_𝑙(𝑡),∀≤𝑊𝑡,𝑢₁,…,𝑢_𝑙∈𝑡₀,+∞×[𝑁,𝑀]^𝑙;(17)_{𝑡+∞0𝑢+∞}|𝑢|^𝑚−1|𝑎(𝑢𝑠|)||^{𝑛−𝑚−1}𝑃(𝑠)(𝑠),𝑔(𝑠+|𝑠|max,𝑄)_{𝑛−𝑚−𝑖−1}max{𝑅(𝑠),𝑊(𝑠)}𝑑𝑠𝑑𝑢<+∞.(18) Then
(a) for any 𝐿∈(𝑏₀𝑀+𝑁,(1−𝑏₀)𝑀), there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| such that for each 𝑥₀∈Ω₁(𝑁,𝑀), the Mann iterative sequence {𝑥_𝑘}_𝑘∈ℕ0 generated by the following scheme 𝑥_𝑘+1(𝑡)=1−𝛼_𝑘𝑥_𝑘(𝑡)+𝛼_𝑘𝐿−𝑏(𝑡)𝑥_𝑘(𝑡−𝜏)+(−1)^𝑛𝐻₀×∫_𝑡^+∞∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑡)^𝑚−1𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥_𝑘𝑓₁(𝑠),…,𝑥_𝑘𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢−1^{𝑛−𝑖−1}𝐻_𝑖×∫_𝑡^+∞∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑡)^𝑚−1𝑎(𝑢)×ℎ𝑥_𝑘ℎ₁(𝑠)𝑥,…,_𝑘ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢,𝑡≥𝑇,𝑘∈ℕ₀,1−𝛼_𝑘𝑥_𝑘(𝑇)+𝛼_𝑘𝐿−𝑏(𝑇)𝑥_𝑘()+()𝑇−𝜏−1^𝑛𝐻₀×∫_𝑇^+∞∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑇)^𝑚−1𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥_𝑘𝑓₁(𝑠),…,𝑥_𝑘𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢−1^{𝑛−𝑖−1}𝐻_𝑖×∫_𝑇^+∞∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑇)^𝑚−1𝑎(𝑢)×ℎ𝑠,𝑥_𝑘ℎ₁(𝑠)𝑥,…,_𝑘ℎ_𝑙(𝑠),𝑡𝑑𝑠𝑑𝑢₀≤𝑡<𝑇,𝑘∈ℕ₀(19) converges to a positive solution 𝑥∈Ω₁(𝑁,𝑀) of (5) and has the following error estimate: 𝑥_𝑘+1−𝑥≤𝑒^{−(1−𝜃)∑𝑘𝑝=0𝛼𝑝}𝑥₀−𝑥,∀𝑘∈ℕ₀,(20) where {𝛼_𝑘}_𝑘∈ℕ0 is an arbitrary sequence in [0,1] such that ^∞_𝑘=0𝛼_𝑘=+∞;(21)
(b) Equation (5) has uncountably many positive solutions in Ω₁(𝑁,𝑀).

Proof. Firstly, we prove that (a) holds. Set 𝐿∈(𝑏₀𝑀+𝑁,(1−𝑏₀)𝑀). From (15) and (18), we know that there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| satisfying |𝑏(𝑡)|≤𝑏₀,∀𝑡≥𝑇;(22)𝜃=𝑏₀+_𝑇^+∞_𝑢^+∞𝑢^𝑚−1|𝑎(𝑢𝐻)|⁰𝑠^{𝑛−𝑚−1}𝑃(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑅(𝑠)𝑑𝑠𝑑𝑢;(23)_{𝑇+∞𝑢+∞}𝑢^𝑚−1|𝑎(𝑢𝐻)|⁰𝑠^{𝑛−𝑚−1}𝑔(𝑠)+𝑄(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑊(𝑠)𝑑𝑠𝑑𝑢<min1−𝑏₀𝑀−𝐿,𝐿−𝑏₀𝑀−𝑁.(24) Define a mapping 𝑆_𝐿∶Ω₁(𝑁,𝑀)→CB([𝛾,+∞),ℝ) by 𝑆_𝐿𝑥(𝑡)=𝐿−𝑏(𝑡)𝑥(𝑡−𝜏)+(−1)^𝑛𝐻₀×∫_𝑡^+∞∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑡)^𝑚−1𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥𝑓₁(𝑠𝑥𝑓),…,_𝑙(𝑠+())𝑑𝑠𝑑𝑢−1^{𝑛−𝑖−1}𝐻_𝑖×∫_𝑡^+∞∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑡)^𝑚−1𝑎(𝑢)×ℎ𝑠,𝑥ℎ₁(𝑠),…,𝑥ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢,𝑡≥𝑇,𝑥∈Ω₁(),𝑆𝑁,𝑀_𝐿𝑥(𝑇),𝛾≤𝑡<𝑇,𝑥∈Ω₁(𝑁,𝑀).(25) It is obvious that 𝑆_𝐿𝑥 is continuous for each 𝑥∈Ω₁(𝑁,𝑀). By means of (16), (22), (23), and (25), we deduce that for any 𝑥,𝑦∈Ω₁(𝑁,𝑀) and 𝑡≥𝑇𝑆_𝐿𝑥(𝑡)−𝑆_𝐿𝑦(𝑡≤|𝑏(𝑡𝑥()())|𝑡−𝜏−𝑦𝑡−𝜏)+𝐻_0𝑡^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑡)^𝑚−1|𝑎(𝑢×𝑓𝑓)|𝑠,𝑥₁(𝑠),…,𝑥𝑓_𝑙(𝑠𝑓)−𝑓𝑠,𝑦₁(𝑠),…,𝑦𝑓_𝑙(𝑠)𝑑𝑠𝑑𝑢+𝐻_𝑖𝑡^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑡)^𝑚−1|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠),…,𝑥ℎ_𝑙(𝑠)−ℎ𝑠,𝑦ℎ₁(𝑠),…,𝑦ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢≤𝑏₀+×𝑥−𝑦𝑥−𝑦_{𝑇+∞𝑢+∞}𝑢^𝑚−1|𝑎(𝑢𝐻)|⁰𝑠^{𝑛−𝑚−1}𝑃(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑅(𝑠,)𝑑𝑠𝑑𝑢=𝜃𝑥−𝑦(26) which yields that 𝑆_𝐿𝑥−𝑆_𝐿𝑦≤𝜃𝑥−𝑦,∀𝑥,𝑦∈Ω₁(𝑁,𝑀).(27) On the basis of (17), (22), (24), and (25), we acquire that for any 𝑥∈Ω₁(𝑁,𝑀) and 𝑡≥𝑇𝑆_𝐿𝑥(𝑡)≤𝐿+|𝑏(𝑡)|𝑥(𝑡−𝜏)+𝐻_0𝑡^+∞_𝑢^+∞𝑠^{𝑛−𝑚−1}𝑢^𝑚−1|𝑎(𝑢×𝑔(𝑠+𝑓𝑓)|)𝑠,𝑥₁(𝑠)𝑥𝑓,…,_𝑙(𝑠)𝑑𝑠𝑑𝑢+𝐻_𝑖𝑡^+∞_𝑢^+∞𝑠^{𝑛−𝑚−𝑖−1}𝑢^𝑚−1|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠),…,𝑥ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢≤𝐿+𝑏₀𝑀+_𝑇^+∞_𝑢^+∞𝑢^𝑚−1|𝑎(𝑢×𝐻)|₀𝑠^{𝑛−𝑚−1}𝑔(𝑠)+𝑄(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑊(𝑠)𝑑𝑠𝑑𝑢<𝐿+𝑏₀𝑀+min1−𝑏₀𝑀−𝐿,𝐿−𝑏₀𝑆𝑀−𝑁≤𝑀,_𝐿𝑥(𝑡)≥𝐿−|𝑏(𝑡)|𝑥(𝑡−𝜏)−𝐻_0𝑡^+∞_𝑢^+∞𝑠^{𝑛−𝑚−1}𝑢^𝑚−1|𝑎(𝑢×𝑔(𝑠+𝑓𝑓)|)𝑠,𝑥₁(𝑠)𝑥𝑓,…,_𝑙(𝑠)𝑑𝑠𝑑𝑢−𝐻_𝑖𝑡^+∞_𝑢^+∞𝑠^{𝑛−𝑚−𝑖−1}𝑢^𝑚−1|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠),…,𝑥ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢≥𝐿−𝑏₀𝑀−_𝑇^+∞_𝑢^+∞𝑢^𝑚−1|𝑎(𝑢×𝐻)|₀𝑠^{𝑛−𝑚−1}𝑔(𝑠)+𝑄(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑊(𝑠)𝑑𝑠𝑑𝑢>𝐿−𝑏₀𝑀−min1−𝑏₀𝑀−𝐿,𝐿−𝑏₀𝑀−𝑁≥𝑁,(28) which guarantee that 𝑆_𝐿(Ω₁(𝑁,𝑀))⊆Ω₁(𝑁,𝑀). Consequently, (27) gives that 𝑆_𝐿 is a contraction mapping in Ω₁(𝑁,𝑀) and it has a unique fixed point 𝑥∈Ω₁(𝑁,𝑀). It is easy to see that 𝑥∈Ω₁(𝑁,𝑀) is a positive solution of (5).
It follows from (19), (25), and (27) that 𝑥_𝑘+1(𝑡)(𝑡=−𝑥)1−𝛼_𝑘𝑥_𝑘(𝑡)+𝛼_𝑘𝐿−𝑏(𝑡)𝑥_𝑘(𝑡−𝜏)+(−1)^𝑛𝐻₀×_𝑡^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑡)^𝑚−1𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥_𝑘𝑓₁(𝑠)𝑥,…,_𝑘𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢−1^{𝑛−𝑖−1}𝐻_𝑖×_𝑡^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑡)^𝑚−1𝑎(𝑢)×ℎ𝑠,𝑥_𝑘ℎ₁(𝑠),…,𝑥_𝑘ℎ_𝑙(𝑠)(𝑡≤𝑑𝑠𝑑𝑢−𝑥)1−𝛼_𝑘𝑥_𝑘(𝑡)−𝑥(𝑡)+𝛼_𝑘𝑆_𝐿𝑥_𝑘(𝑡)−𝑆_𝐿𝑥(𝑡≤)1−𝛼_𝑘𝑥_𝑘(𝑡)−𝑥(𝑡)+𝛼_𝑘𝜃𝑥_𝑘(𝑡)(𝑡=()𝛼−𝑥)1−1−𝜃_𝑘𝑥_𝑘(𝑡)−𝑥(𝑡)≤𝑒^{−(1−𝜃)𝛼𝑘}𝑥_𝑘−𝑥≤𝑒^{−(1−𝜃)∑𝑘𝑝=0𝛼𝑝}𝑥₀−𝑥,∀𝑘∈ℕ₀,𝑡≥𝑇,(29) which yields that 𝑥_𝑘+1−𝑥≤𝑒^{−(1−𝜃)∑𝑘𝑝=0𝛼𝑝}𝑥₀−𝑥,∀𝑘∈ℕ₀.(30) That is, (20) holds. Thus (20) and (21) ensure that lim_𝑘→∞𝑥_𝑘=𝑥.
Secondly, we show that (b) holds. Let 𝐿₁,𝐿₂∈(𝑏₀𝑀+𝑁,(1−𝑏₀)𝑀) with 𝐿₁≠𝐿₂. In light of (15) and (18), we know that for each 𝑝∈{1,2}, there exist 𝜃_𝑝∈(0,1), 𝑇_𝑝 and 𝑇^∗ with 𝑇_𝑝>1+|𝑡₀|+𝜏+|𝛾| and 𝑇^∗>max{𝑇₁,𝑇₂} satisfying (22)–(24) and _𝑇^+∞_∗𝑢^+∞𝑢^𝑚−1|𝑎(𝑢𝐻)|⁰𝑠^{𝑛−𝑚−1}𝑃(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑅(𝑠<𝐿)𝑑𝑠𝑑𝑢₁−𝐿₂,(31) where 𝜃 and 𝑇 are replaced by 𝜃_𝑝 and 𝑇_𝑝, respectively. Let the mapping 𝑆_𝐿𝑝 be defined by (25) with 𝐿 and 𝑇 replaced by 𝐿_𝑝 and 𝑇_𝑝, respectively. As in the proof of (a), we deduce easily that the mapping 𝑆_𝐿𝑝 possesses a unique fixed point 𝑧_𝑝∈Ω₁(𝑁,𝑀), that is, 𝑧_𝑝 is a positive solution of (5) in Ω₁(𝑁,𝑀). In order to prove (b), we need only to show that 𝑧₁≠𝑧₂. In fact, (25) means that for each 𝑡≥𝑇^∗ and 𝑝∈{1,2}𝑧_𝑝(𝑡)=𝐿_𝑝−𝑏(𝑡)𝑧_𝑝(𝑡−𝜏)+(−1)^𝑛𝐻₀×_𝑡^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑡)^𝑚−1𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑧_𝑝𝑓₁(𝑠),…,𝑧_𝑝𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢−1^{𝑛−𝑖−1}𝐻_𝑖𝑡^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑡)^𝑚−1𝑎(𝑢)×ℎ𝑠,𝑧_𝑝ℎ₁(𝑠),…,𝑧_𝑝ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢.(32) It follows from (16), (22), (31), and (32) that for each 𝑡≥𝑇^∗𝑧₁(𝑡)−𝑧₂(𝑡≥𝐿₁)−𝐿₂−|𝑏(𝑡)|𝑧₁(𝑡−𝜏)−𝑧₂(𝑡−𝜏)−𝐻_0𝑡^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑡)^𝑚−1|𝑎(𝑢×𝑓)|𝑠,𝑧₁𝑓₁(𝑠),…,𝑧₁𝑓_𝑙(𝑠)−𝑓𝑠,𝑧₂𝑓₁(𝑠),…,𝑧₂𝑓_𝑙(𝑠)𝑑𝑠𝑑𝑢−𝐻_𝑖𝑡^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑡)^𝑚−1|𝑎(𝑢×ℎ)|𝑠,𝑧₁ℎ₁(𝑠),…,𝑧₁ℎ_𝑙(𝑠)−ℎ𝑠,𝑧₂ℎ₁(𝑠),…,𝑧₂ℎ_𝑙(𝑠)≥𝐿𝑑𝑠𝑑𝑢₁−𝐿₂−𝑏₀𝑧₁−𝑧₂−𝑧₁−𝑧_2𝑇^+∞_∗𝑢^+∞𝑢^𝑚−1|𝑎(𝑢×𝐻)|₀𝑠^{𝑛−𝑚−1}𝑃(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑅(𝑠≥𝐿)𝑑𝑠𝑑𝑢₁−𝐿₂−𝑏₀+𝐿₁−𝐿₂𝑧₁−𝑧₂,(33) which implies that 𝑧₁−𝑧₂≥𝐿₁−𝐿₂1+𝑏₀+𝐿₁−𝐿₂>0,(34) that is, 𝑧₁≠𝑧₂. This completes the proof.

Theorem 3. Assume that there exist three constants 𝑀, 𝑁, and 𝑏₀ and four functions 𝑃,𝑄,𝑅,𝑊∈𝐶([𝑡₀,+∞),ℝ⁺) satisfying (16)–(18) and 0<𝑁<𝑀,𝑏₀<𝑀−𝑁𝑀,0≤𝑏(𝑡)≤𝑏₀𝑒𝑣𝑒𝑛𝑡𝑢𝑎𝑙𝑙𝑦.(35) Then(a)for any 𝐿∈(𝑏₀𝑀+𝑁,𝑀), there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| such that for each 𝑥₀∈Ω₁(𝑁,𝑀), the Mann iterative sequence {𝑥_𝑘}_𝑘∈ℕ0 generated by (19) converges to a positive solution 𝑥∈Ω₁(𝑁,𝑀) of (5) and has the error estimate (20), where {𝛼_𝑘}_𝑘∈ℕ0 is an arbitrary sequence in [0,1] satisfying (21);(b)Equation (5) has uncountably many positive solutions in Ω₁(𝑁,𝑀).

Proof. Let 𝐿∈(𝑏₀𝑀+𝑁,𝑀). Equations (18) and (36) ensure that there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| satisfying (23), 0≤𝑏(𝑡)≤𝑏₀,∀𝑡≥𝑇;(36)_{𝑇+∞𝑢+∞}𝑢^𝑚−1|𝑎(𝑢𝐻)|⁰𝑠^{𝑛−𝑚−1}𝑔(𝑠)+𝑄(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑊(𝑠)𝑑𝑠𝑑𝑢<min𝑀−𝐿,𝐿−𝑏₀𝑀−𝑁.(37) Define a mapping 𝑆_𝐿∶Ω₁(𝑁,𝑀)→CB([𝛾,+∞),ℝ) by (25). Obviously, 𝑆_𝐿𝑥 is continuous for every 𝑥∈Ω₁(𝑁,𝑀). Using (16), (23), (25), and (36), we conclude that for any 𝑥,𝑦∈Ω₁(𝑁,𝑀) and 𝑡≥𝑇𝑆_𝐿𝑥(𝑡)−𝑆_𝐿𝑦(𝑡(𝑡𝑥()()≤𝑏)𝑡−𝜏−𝑦𝑡−𝜏)+𝐻_0𝑡^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑡)^𝑚−1|𝑎(𝑢×𝑓𝑓)|𝑠,𝑥₁(𝑠),…,𝑥𝑓_𝑙(𝑠𝑓)−𝑓𝑠,𝑦₁(𝑠),…,𝑦𝑓_𝑙(𝑠)𝑑𝑠𝑑𝑢+𝐻_𝑖𝑡^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑡)^𝑚−1|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠),…,𝑥ℎ_𝑙(𝑠)−ℎ𝑠,𝑦ℎ₁(𝑠),…,𝑦ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢≤𝑏₀+×𝑥−𝑦𝑥−𝑦_{𝑇+∞𝑢+∞}𝑢^𝑚−1|𝑎(𝑢𝐻)|⁰𝑠^{𝑛−𝑚−1}𝑃(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑅(𝑠,)𝑑𝑠𝑑𝑢=𝜃𝑥−𝑦(38) which implies that (27) holds. In light of (17), (25), (36), and (37), we know that for any 𝑥∈Ω₁(𝑁,𝑀) and 𝑡≥𝑇𝑆_𝐿𝑥(𝑡)≤𝐿+𝐻_0𝑡^+∞_𝑢^+∞𝑠^{𝑛−𝑚−1}𝑢^𝑚−1|𝑎(𝑢×𝑔(𝑠+𝑓𝑓)|)𝑠,𝑥₁(𝑠𝑥𝑓),…,_𝑙(𝑠)𝑑𝑠𝑑𝑢+𝐻_𝑖𝑡^+∞_𝑢^+∞𝑠^{𝑛−𝑚−𝑖−1}𝑢^𝑚−1|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠),…,𝑥ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢≤𝐿+_𝑇^+∞_𝑢^+∞𝑢^𝑚−1|𝑎(𝑢𝐻)|⁰𝑠^{𝑛−𝑚−1}𝑔(𝑠)+𝑄(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑊(𝑠)𝑑𝑠𝑑𝑢<𝐿+min𝑀−𝐿,𝐿−𝑏₀𝑆𝑀−𝑁≤𝑀,_𝐿𝑥(𝑡)≥𝐿−|𝑏(𝑡)|𝑥(𝑡−𝜏)−𝐻_0𝑡^+∞_𝑢^+∞𝑠^{𝑛−𝑚−1}𝑢^𝑚−1|𝑎(𝑢×𝑔(𝑠+𝑓𝑓)|)𝑠,𝑥₁(𝑠𝑥𝑓),…,_𝑙(𝑠)𝑑𝑠𝑑𝑢−𝐻_𝑖𝑡^+∞_𝑢^+∞𝑠^{𝑛−𝑚−𝑖−1}𝑢^𝑚−1|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠),…,𝑥ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢≥𝐿−𝑏₀𝑀−_𝑇^+∞_𝑢^+∞𝑢^𝑚−1|𝑎(𝑢×𝐻)|₀𝑠^{𝑛−𝑚−1}𝑔(𝑠)+𝑄(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑊(𝑠)𝑑𝑠𝑑𝑢>𝐿−𝑏₀𝑀−min𝑀−𝐿,𝐿−𝑏₀𝑀−𝑁≥𝑁,(39) which mean that 𝑆_𝐿(Ω₁(𝑁,𝑀))⊆Ω₁(𝑁,𝑀). Equation (27) guarantees that 𝑆_𝐿 is a contraction mapping in Ω₁(𝑁,𝑀) and it possesses a unique fixed point 𝑥∈Ω₁(𝑁,𝑀). As in the proof of Theorem 2, we infer that 𝑥∈Ω₁(𝑁,𝑀) is a positive solution of (5). The rest of the proof is similar to that of Theorem 2 and is omitted. This completes the proof.

Theorem 4. Assume that there exist three constants 𝑀, 𝑁, and 𝑏₀ and four functions 𝑃,𝑄,𝑅,𝑊∈𝐶([𝑡₀,+∞),ℝ⁺) satisfying (16)–(18) and 0<𝑁<𝑀,𝑏₀<𝑀−𝑁𝑀,−𝑏₀≤𝑏(𝑡)≤0𝑒𝑣𝑒𝑛𝑡𝑢𝑎𝑙𝑙𝑦.(40) Then(a)for any 𝐿∈(𝑁,(1−𝑏₀)𝑀), there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| such that for each 𝑥₀∈Ω₁(𝑁,𝑀), the Mann iterative sequence {𝑥_𝑘}_𝑘∈ℕ0 generated by (19) converges to a positive solution 𝑥∈Ω₁(𝑁,𝑀) of (5) and has the error estimate (20), where {𝛼_𝑘}_𝑘∈ℕ0 is an arbitrary sequence in [0,1] satisfying (21);(b)Equation (5) has uncountably many positive solutions in Ω₁(𝑁,𝑀).

Proof. Set 𝐿∈(𝑁,(1−𝑏₀)𝑀). It follows from (18) and (40) that there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| satisfying (23), −𝑏₀≤𝑏(𝑡)≤0,∀𝑡≥𝑇;(41)_{𝑇+∞𝑢+∞}𝑢^𝑚−1|𝑎(𝑢𝐻)|⁰𝑠^{𝑛−𝑚−1}𝑔(𝑠)+𝑄(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑊(𝑠)𝑑𝑠𝑑𝑢<min𝐿−𝑁,1−𝑏₀𝑀−𝐿.(42) Define a mapping 𝑆_𝐿∶Ω₁(𝑁,𝑀)→CB([𝛾,+∞),ℝ) by (25). Distinctly, 𝑆_𝐿𝑥 is continuous for each 𝑥∈Ω₁(𝑁,𝑀). In terms of (16), (23), (25), and (41), we reason that for any 𝑥,𝑦∈Ω₁(𝑁,𝑀) and 𝑡≥𝑇𝑆_𝐿𝑥(𝑡)−𝑆_𝐿𝑦(𝑡≤|𝑏(𝑡𝑥()())|𝑡−𝜏−𝑦𝑡−𝜏)+𝐻_0𝑡^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑡)^𝑚−1|𝑎(𝑢×𝑓𝑓)|𝑠,𝑥₁(𝑠),…,𝑥𝑓_𝑙(𝑠𝑓)−𝑓𝑠,𝑦₁(𝑠),…,𝑦𝑓_𝑙(𝑠)𝑑𝑠𝑑𝑢+𝐻_𝑖𝑡^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑡)^𝑚−1|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠),…,𝑥ℎ_𝑙(𝑠)−ℎ𝑠,𝑦ℎ₁(𝑠),…,𝑦ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢≤𝑏₀+×𝑥−𝑦𝑥−𝑦_{𝑇+∞𝑢+∞}𝑢^𝑚−1|𝑎(𝑢𝐻)|⁰𝑠^{𝑛−𝑚−1}𝑃(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑅(𝑠,)𝑑𝑠𝑑𝑢=𝜃𝑥−𝑦(43) which means that (27) holds. Owing to (17), (25), (41), and (42), we earn that for any 𝑥∈Ω₁(𝑁,𝑀) and 𝑡≥𝑇𝑆_𝐿𝑥(𝑡)≤𝐿+|𝑏(𝑡)|𝑥(𝑡−𝜏)+𝐻_0𝑡^+∞_𝑢^+∞𝑠^{𝑛−𝑚−1}𝑢^𝑚−1|𝑎(𝑢×𝑔(𝑠+𝑓𝑓)|)𝑠,𝑥₁(𝑠)𝑥𝑓,…,_𝑙(𝑠)𝑑𝑠𝑑𝑢+𝐻_𝑖𝑡^+∞_𝑢^+∞𝑠^{𝑛−𝑚−𝑖−1}𝑢^𝑚−1|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠)𝑥ℎ,…,_𝑙(𝑠)𝑑𝑠𝑑𝑢≤𝐿+𝑏₀𝑀+_𝑇^+∞_𝑢^+∞𝑢^𝑚−1|𝑎(𝑢×𝐻)|₀𝑠^{𝑛−𝑚−1}𝑔(𝑠)+𝑄(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑊(𝑠)𝑑𝑠𝑑𝑢<𝐿+𝑏₀𝑀+min𝐿−𝑁,1−𝑏₀𝑆𝑀−𝐿≤𝑀,_𝐿𝑥(𝑡)≥𝐿−𝐻_0𝑡^+∞_𝑢^+∞𝑠^{𝑛−𝑚−1}𝑢^𝑚−1|𝑎(𝑢×𝑔(𝑠+𝑓𝑓)|)𝑠,𝑥₁(𝑠)𝑥𝑓,…,_𝑙(𝑠)𝑑𝑠𝑑𝑢−𝐻_𝑖𝑡^+∞_𝑢^+∞𝑠^{𝑛−𝑚−𝑖−1}𝑢^𝑚−1|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠)𝑥ℎ,…,_𝑙(𝑠)𝑑𝑠𝑑𝑢≥𝐿−_𝑇^+∞_𝑢^+∞𝑢^𝑚−1|𝑎(𝑢×𝐻)|₀𝑠^{𝑛−𝑚−1}𝑔(𝑠)+𝑄(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑊(𝑠)𝑑𝑠𝑑𝑢>𝐿−min𝐿−𝑁,1−𝑏₀𝑀−𝐿≥𝑁,(44) which yield that 𝑆_𝐿(Ω₁(𝑁,𝑀))⊆Ω₁(𝑁,𝑀). Thus (27) ensures that 𝑆_𝐿 is a contraction mapping in Ω₁(𝑁,𝑀) and it owns a unique fixed point 𝑥∈Ω₁(𝑁,𝑀). As in the proof of Theorem 2, we infer that 𝑥∈Ω₁(𝑁,𝑀) is a positive solution of (5). The rest of the proof is parallel to that of Theorem 2, and hence is elided. This completes the proof.

Theorem 5. Assume that there exist three constants 𝑀, 𝑁, and 𝑏₀ and four functions 𝑃,𝑄,𝑅,𝑊∈𝐶([𝑡₀,+∞),ℝ⁺) satisfying (18) and 𝑀>𝑁>0,𝑏₀>𝑀(𝑡)𝑀−𝑁,𝑏≥𝑏₀𝑓𝑒𝑣𝑒𝑛𝑡𝑢𝑎𝑙𝑙𝑦;(45)𝑡,𝑢₁,…,𝑢_𝑙−𝑓𝑡,𝑢₁,…,𝑢_𝑙≤𝑃(𝑡)max𝑢_𝑗−𝑢_𝑗,ℎ∶1≤𝑗≤𝑙𝑡,𝑢₁,…,𝑢_𝑙−ℎ𝑡,𝑢₁,…,𝑢_𝑙≤𝑅(𝑡)max𝑢_𝑗−𝑢_𝑗,∀∶1≤𝑗≤𝑙𝑡,𝑢₁,…,𝑢_𝑙,𝑢₁,…,𝑢_𝑙∈𝑡₀×𝑀,+∞0,𝑏₀^2𝑙;𝑓(46)𝑡,𝑢₁,…,𝑢_𝑙≤𝑄(𝑡),ℎ𝑡,𝑢₁,…,𝑢_𝑙(𝑡),∀≤𝑊𝑡,𝑢₁,…,𝑢_𝑙∈𝑡₀×𝑀,+∞0,𝑏₀^𝑙.(47) Then
(a) for any 𝐿∈(𝑁+𝑀/𝑏₀,𝑀), there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| such that for each 𝑥₀∈Ω₂(𝑁,𝑀), the Mann iterative sequence {𝑥_𝑘}_𝑘∈ℕ0 generated by the following scheme 𝑥_𝑘+1(𝑡)=1−𝛼_𝑘𝑥_𝑘(𝑡)+𝛼_𝑘𝑏()×𝑡+𝜏𝐿−𝑥_𝑘(𝑡+𝜏)+(−1)^𝑛𝐻₀×∫^+∞_𝑡+𝜏∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑡−𝜏)^𝑚−1𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥_𝑘𝑓₁(𝑠),…,𝑥_𝑘𝑓_𝑙(𝑠+())𝑑𝑠𝑑𝑢−1^{𝑛−𝑖−1}𝐻_𝑖×∫^+∞_𝑡+𝜏∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑡−𝜏)^𝑚−1𝑎(𝑢)×ℎ𝑠,𝑥_𝑘ℎ₁(𝑠𝑥),…,_𝑘ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢,𝑡≥𝑇,𝑘∈ℕ₀,1−𝛼_𝑘𝑥_𝑘(𝑇)+𝛼_𝑘𝑏()×𝑡+𝜏𝐿−𝑥_𝑘(𝑇+𝜏)+(−1)^𝑛𝐻₀×∫^+∞_𝑇+𝜏∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑇−𝜏)^𝑚−1𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥_𝑘𝑓₁(𝑠),…,𝑥_𝑘𝑓_𝑙(𝑠+())𝑑𝑠𝑑𝑢−1^{𝑛−𝑖−1}𝐻_𝑖×∫^+∞_𝑇+𝜏∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑇−𝜏)^𝑚−1𝑎(𝑢)×ℎ𝑠,𝑥_𝑘ℎ₁(𝑠)𝑥,…,_𝑘ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢,𝛾≤𝑡<𝑇,𝑘∈ℕ₀(48) converges to a positive solution 𝑥∈Ω₂(𝑁,𝑀) of (5) and has the error estimate (20), where {𝛼_𝑘}_𝑘∈ℕ0 is an arbitrary sequence in [0,1] with (21);
(b) Equation (5) has uncountably many positive solutions in Ω₂(𝑁,𝑀).

Proof. First of all, we show that (a) holds. Set 𝐿∈(𝑁+𝑀/𝑏₀,𝑀). It follows from (18) and (45) that there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| such that 𝑏(𝑡)≥𝑏₀1,∀𝑡≥𝑇;(49)𝜃=𝑏₀+1𝑏_0𝑇^+∞_𝑢^+∞𝑢^𝑚−1|𝑎(𝑢×𝐻)|₀𝑠^{𝑛−𝑚−1}𝑃(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑅(𝑠)𝑑𝑠𝑑𝑢;(50)_{𝑇+∞𝑢+∞}𝑢^𝑚−1|𝑎(𝑢𝐻)|⁰𝑠^{𝑛−𝑚−1}𝑔(𝑠)+𝑄(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑊(𝑠𝑀)𝑑𝑠𝑑𝑢<min𝑀−𝐿,𝐿−𝑏₀−𝑁.(51) Define a mapping 𝑆_𝐿∶Ω₂(𝑁,𝑀)→CB([𝛾,+∞),ℝ) by 𝑆_𝐿𝑥(𝑡)=𝐿𝑏()−𝑥()𝑡+𝜏𝑡+𝜏𝑏()+()𝑡+𝜏−1^𝑛𝐻₀𝑏(𝑡+𝜏)×∫^+∞_𝑡+𝜏∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑡−𝜏)^𝑚−1𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥𝑓₁(𝑠),…,𝑥𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢−1^{𝑛−𝑖−1}𝐻_𝑖𝑏(𝑡+𝜏)×∫^+∞_𝑡+𝜏∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑡−𝜏)^𝑚−1𝑎(𝑢)×ℎ𝑠,𝑥ℎ₁(𝑠),…,𝑥ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢,𝑡≥𝑇,𝑥∈Ω₂(),𝑆𝑁,𝑀_𝐿𝑥(𝑇),𝛾≤𝑡<𝑇,𝑥∈Ω₂(𝑁,𝑀).(52) In light of (46), (49), (50), and (52), we conclude that for 𝑥,𝑦∈Ω₂(𝑁,𝑀) and 𝑡≥𝑇𝑆_𝐿𝑥(𝑡)−𝑆_𝐿𝑦(𝑡≤1)𝑏()𝑥()(+𝐻𝑡+𝜏𝑡+𝜏−𝑦𝑡+𝜏)⁰𝑏()×𝑡+𝜏_{+∞𝑡+𝜏𝑢+∞}𝑠^{𝑛−𝑚−1}𝑢^𝑚−1|𝑎(𝑢×𝑓𝑓)|𝑠,𝑥₁(𝑠),…,𝑥𝑓_𝑙(𝑠)−𝑓𝑠,𝑦𝑓₁(𝑠),…,𝑦𝑓_𝑙(𝑠+𝐻)𝑑𝑠𝑑𝑢_𝑖𝑏()×𝑡+𝜏_{+∞𝑡+𝜏𝑢+∞}𝑠^{𝑛−𝑚−𝑖−1}𝑢^𝑚−1𝑎(𝑢)×ℎ𝑠,𝑥ℎ₁(𝑠),…,𝑥ℎ_𝑙(𝑠)−ℎ𝑠,𝑦ℎ₁(𝑠),…,𝑦ℎ_𝑙(𝑠)≤1𝑑𝑠𝑑𝑢𝑏₀+1𝑥−𝑦𝑏₀×𝑥−𝑦_{𝑇+∞𝑢+∞}𝑢^𝑚−1|𝑎(𝑢𝐻)|⁰𝑠^{𝑛−𝑚−1}𝑃(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑅(𝑠,)𝑑𝑠𝑑𝑢=𝜃𝑥−𝑦(53) which yields that 𝑆_𝐿𝑥−𝑆_𝐿𝑦≤𝜃𝑥−𝑦,∀𝑥,𝑦∈Ω₂(𝑁,𝑀).(54) In view of (47), (49), (51), and (52), we obtain that for any 𝑥∈Ω₂(𝑁,𝑀) and 𝑡≥𝑇𝑆_𝐿𝑥(𝑡)≤1𝑏()×()𝑡+𝜏𝐿−𝑥𝑡+𝜏+𝐻₀^+∞_{𝑡+𝜏𝑢}^+∞𝑠^{𝑛−𝑚−1}𝑢^𝑚−1|𝑎(𝑢×𝑔(𝑠+𝑓𝑓)|)𝑠,𝑥₁(𝑠),…,𝑥𝑓_𝑙(𝑠)𝑑𝑠𝑑𝑢+𝐻_𝑖^+∞_{𝑡+𝜏𝑢}^+∞𝑠^{𝑛−𝑚−𝑖−1}𝑢^𝑚−1|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠)𝑥ℎ,…,_𝑙(𝑠)≤1𝑑𝑠𝑑𝑢𝑏()×𝑁𝑡+𝜏𝐿−𝑏()+𝑡+𝜏_{𝑇+∞𝑢+∞}𝑢^𝑚−1|𝑎(𝑢×𝐻)|₀𝑠^{𝑛−𝑚−1}𝑔(𝑠)+𝑄(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑊(𝑠<1)𝑑𝑠𝑑𝑢𝑏()×𝑁𝑡+𝜏𝐿−𝑏()𝑀𝑡+𝜏+min𝑀−𝐿,𝐿−𝑏₀≤𝑀−𝑁𝑏(),𝑆𝑡+𝜏_𝐿𝑥(𝑡)≥1𝑏()×()𝑡+𝜏𝐿−𝑥𝑡+𝜏−𝐻₀^+∞_{𝑡+𝜏𝑢}^+∞𝑠^{𝑛−𝑚−1}𝑢^𝑚−1|𝑎(𝑢×𝑔(𝑠+𝑓𝑓)|)𝑠,𝑥₁(𝑠𝑥𝑓),…,_𝑙(𝑠)𝑑𝑠𝑑𝑢−𝐻_𝑖^+∞_{𝑡+𝜏𝑢}^+∞𝑠^{𝑛−𝑚−𝑖−1}𝑢^𝑚−1|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠)𝑥ℎ,…,_𝑙(𝑠)≥1𝑑𝑠𝑑𝑢𝑏()×𝑀𝑡+𝜏𝐿−𝑏()−𝑡+𝜏_{𝑇+∞𝑢+∞}𝑢^𝑚−1|𝑎(𝑢×𝐻)|₀𝑠^{𝑛−𝑚−1}𝑔(𝑠)+𝑄(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑊(𝑠>1)𝑑𝑠𝑑𝑢𝑏()×𝑀𝑡+𝜏𝐿−𝑏()𝑀𝑡+𝜏−min𝑀−𝐿,𝐿−𝑏₀≥𝑁−𝑁𝑏(),𝑡+𝜏(55) which imply that 𝑆_𝐿(Ω₂(𝑁,𝑀))⊆Ω₂(𝑁,𝑀). It follows from (50) and (54) that 𝑆_𝐿 is a contraction mapping in Ω₂(𝑁,𝑀) and it has a unique fixed point 𝑥∈Ω₂(𝑁,𝑀). It is clear that 𝑥∈Ω₂(𝑁,𝑀) is a positive solution of (5).
Note that (48), (52), and (54) undertake that 𝑥_𝑘+1(𝑡)(𝑡=−𝑥)1−𝛼_𝑘𝑥_𝑘(𝑡)+𝛼_𝑘𝑏()×𝑡+𝜏𝐿−𝑥_𝑘(𝑡+𝜏)+(−1)^𝑛𝐻₀×^+∞_{𝑡+𝜏𝑢}^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑡−𝜏)^𝑚−1𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥_𝑘𝑓₁(𝑠),…,𝑠𝑥_𝑘𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢−1^{𝑛−𝑖−1}𝐻_𝑖×^+∞_{𝑡+𝜏𝑢}^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑡−𝜏)^𝑚−1𝑎(𝑢)×ℎ𝑠,𝑥_𝑘ℎ₁(𝑠)𝑥,…,_𝑘ℎ_𝑙(𝑠)(𝑡≤𝑑𝑠𝑑𝑢−𝑥)1−𝛼_𝑘𝑥_𝑘(𝑡)−𝑥(𝑡)+𝛼_𝑘𝑆_𝐿𝑥_𝑘(𝑡)−𝑆_𝐿𝑥(𝑡≤)1−𝛼_𝑘𝑥_𝑘(𝑡)−𝑥(𝑡)+𝛼_𝑘𝜃𝑥_𝑘(𝑡)(𝑡=()𝛼−𝑥)1−1−𝜃_𝑘𝑥_𝑘(𝑡)−𝑥(𝑡)≤𝑒^{−(1−𝜃)𝛼𝑘}𝑥_𝑘−𝑥≤𝑒^{−(1−𝜃)∑𝑘𝑝=0𝛼𝑝}𝑥₀−𝑥,∀𝑘∈ℕ₀,𝑡≥𝑇,(56) which indicates that (20) holds. Thus (20) and (21) assure that lim_𝑘→∞𝑥_𝑘=𝑥.
Next we prove that (b) holds. Let 𝐿₁,𝐿₂∈(𝑁+𝑀/𝑏₀,𝑀) with 𝐿₁≠𝐿₂. As in the proof of (a) we infer that for each 𝑝∈{1,2} there exist 𝜃_𝑝∈(0,1), 𝑇_𝑝>1+|𝑡₀|+𝜏+|𝛾| and 𝑆_𝐿𝑝 satisfying (49)–(52), where 𝐿, 𝜃, 𝑇, and 𝑆_𝐿 are replaced by 𝐿_𝑝, 𝜃_𝑝, 𝑇_𝑝, and 𝑆_𝐿𝑝, respectively, and 𝑆_𝐿𝑝 has a unique fixed point 𝑧_𝑝∈Ω₂(𝑁,𝑀), which is a positive solution of (5) in Ω₂(𝑁,𝑀). It follows that for each 𝑡≥𝑇_𝑝 and 𝑝∈{1,2}𝑧_𝑝(𝑡)=𝐿_𝑝𝑏()−𝑧𝑡+𝜏^𝑝(𝑡+𝜏)𝑏()+()𝑡+𝜏−1^𝑛𝐻₀𝑏()×𝑡+𝜏_{+∞𝑡+𝜏𝑢+∞}(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑡−𝜏)^𝑚−1𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑧_𝑝𝑓₁(𝑠),…,𝑧_𝑝𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢−1^{𝑛−𝑖−1}𝐻_𝑖𝑏()×𝑡+𝜏_{+∞𝑡+𝜏𝑢+∞}(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑡−𝜏)^𝑚−1𝑎(𝑢)×ℎ𝑠,𝑧_𝑝ℎ₁(𝑠),…,𝑧_𝑝ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢.(57) On behalf of proving (b), we need only to show that 𝑧₁≠𝑧₂. Notice that (18) guarantees that there exits 𝑇₃>max{𝑇₁,𝑇₂} satisfying _𝑇^+∞_3𝑢^+∞𝑢^𝑚−1|𝑎(𝑢𝐻)|⁰𝑠^{𝑛−𝑚−1}𝑃(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑅(𝑠<𝐿)𝑑𝑠𝑑𝑢₁−𝐿₂1+2𝑧₁−𝑧₂.(58) Due to (46), (51), (57), and (58), we conclude that for each 𝑡≥𝑇₃𝑧₁(𝑡)−𝑧₂(𝑡)+𝑧₁(𝑡+𝜏)𝑏()−𝑧𝑡+𝜏²(𝑡+𝜏)𝑏()≥1𝑡+𝜏𝑏()×𝐿𝑡+𝜏₁−𝐿₂−𝐻_0𝑇^+∞_3𝑢^+∞𝑠^{𝑛−𝑚−1}𝑢^𝑚−1|𝑎(𝑢×𝑓)|𝑠,𝑧₁𝑓₁(𝑠),…,𝑧₁𝑓_𝑙(𝑠)−𝑓𝑠,𝑧₂𝑓₁(𝑠),…,𝑧₂𝑓_𝑙(𝑠)𝑑𝑠𝑑𝑢−𝐻_𝑖𝑇^+∞_3𝑢^+∞𝑠^{𝑛−𝑚−𝑖−1}𝑢^𝑚−1|𝑎(𝑢×ℎ)|𝑠,𝑧₁𝑓₁(𝑠),…,𝑧₁𝑓_𝑙(𝑠)−ℎ𝑠,𝑧₂𝑓₁(𝑠),…,𝑧₂𝑓_𝑙(𝑠)≥1𝑑𝑠𝑑𝑢𝑏()×𝐿𝑡+𝜏₁−𝐿₂−𝑧₁−𝑧₂×_𝑇^+∞_3𝑢^+∞𝑢^𝑚−1|𝑎(𝑢𝐻)|⁰𝑠^{𝑛−𝑚−1}𝑃(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑅(𝑠>1)𝑑𝑠𝑑𝑢𝑏()𝐿𝑡+𝜏¹−𝐿₂−𝑧₁−𝑧₂𝐿₁−𝐿₂1+2𝑧₁−𝑧₂>𝐿₁−𝐿₂2𝑏(𝑡+𝜏)>0,(59) which yields that 𝑧₁≠𝑧₂. This completes the proof.

Theorem 6. Assume that there exist three constants 𝑀, 𝑁, and 𝑏₀ and four functions 𝑃,𝑄,𝑅,𝑊∈𝐶([𝑡₀,+∞),ℝ⁺) satisfying (18), (46), (47), and 𝑀0<𝑁<𝑀,𝑀−𝑁<𝑏₀,𝑏(𝑡)≤−𝑏₀𝑒𝑣𝑒𝑛𝑡𝑢𝑎𝑙𝑙𝑦.(60) Then
(a) for any 𝐿∈(𝑁,(1−1/𝑏₀)𝑀), there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| such that for each 𝑥₀∈Ω₃(𝑁,𝑀), the Mann iterative sequence {𝑥_𝑘}_𝑘∈ℕ0 generated by the following scheme 𝑥_𝑘+1(𝑡)=1−𝛼_𝑘𝑥_𝑘(𝑡)+𝛼_𝑘𝑏()×𝑡+𝜏−𝐿−𝑥_𝑘(𝑡+𝜏)+(−1)^𝑛𝐻₀×∫^+∞_𝑡+𝜏∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑡−𝜏)^𝑚−1𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥_𝑘𝑓₁(𝑠𝑥),…,_𝑘𝑓_𝑙(𝑠+())𝑑𝑠𝑑𝑢−1^{𝑛−𝑖−1}𝐻_𝑖×∫^+∞_𝑡+𝜏∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑡−𝜏)^𝑚−1𝑎(𝑢)×ℎ𝑠,𝑥_𝑘ℎ₁(𝑠),…,𝑥_𝑘ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢,𝑡≥𝑇,𝑘∈ℕ₀,1−𝛼_𝑘𝑥_𝑘(𝑇)+𝛼_𝑘𝑏()×𝑡+𝜏−𝐿−𝑥_𝑘(𝑇+𝜏)+(−1)^𝑛𝐻₀×∫^+∞_𝑇+𝜏∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑇−𝜏)^𝑚−1𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥_𝑘𝑓₁(𝑠𝑥),…,_𝑘𝑓_𝑙(𝑠+())𝑑𝑠𝑑𝑢−1^{𝑛−𝑖−1}𝐻_𝑖×∫^+∞_𝑇+𝜏∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑇−𝜏)^𝑚−1𝑎(𝑢)×ℎ𝑠,𝑥_𝑘ℎ₁(𝑠),…,𝑥_𝑘ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢,𝛾≤𝑡<𝑇,𝑘∈ℕ₀(61) converges to a positive solution 𝑥∈Ω₃(𝑁,𝑀) of (5) and has the error estimate (20), where {𝛼_𝑘}_𝑘∈ℕ0 is an arbitrary sequence in [0,1] satisfying (21);
(b) Equation (5) has uncountably many positive solutions in Ω₃(𝑁,𝑀).

Proof. Put 𝐿∈(𝑁,(1−1/𝑏₀)𝑀). It follows from (18) and (60) that there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| satisfying (50) and 𝑏(𝑡)≤−𝑏₀,∀𝑡≥𝑇;_{𝑇+∞𝑢+∞}𝑢^𝑚−1|𝑎(𝑢𝐻)|⁰𝑠^{𝑛−𝑚−1}𝑔(𝑠)+𝑄(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑊(𝑠𝑀1)𝑑𝑠𝑑𝑢<min1−𝑏₀−𝐿,𝐿−𝑁.(62) Define a mapping 𝑆_𝐿∶Ω₃(𝑁,𝑀)→CB([𝛾,+∞),ℝ) by 𝑆_𝐿𝑥(𝑡)=−𝐿𝑏()−𝑥()𝑡+𝜏𝑡+𝜏𝑏()+()𝑡+𝜏−1^𝑛𝐻₀𝑏(𝑡+𝜏)×∫^+∞_𝑡+𝜏∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑡−𝜏)^𝑚−1𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥𝑓₁(𝑠)𝑥𝑓,…,_𝑙(𝑠)+()𝑑𝑠𝑑𝑢−1^{𝑛−𝑖−1}𝐻_𝑖𝑏(𝑡+𝜏)×∫^+∞_𝑡+𝜏∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑡−𝜏)^𝑚−1𝑎(𝑢)×ℎ𝑠,𝑥ℎ₁(𝑠),…,𝑥ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢,𝑡≥𝑇,𝑥∈Ω₃(),𝑆𝑁,𝑀_𝐿𝑥(𝑇),𝛾≤𝑡<𝑇,𝑥∈Ω₃(𝑁,𝑀).(63) By virtue of (47), (62), and (63), we know that for any 𝑥∈Ω₃(𝑁,𝑀) and 𝑡≥𝑇𝑆_𝐿𝑥(𝑡)≤1𝑏()×()𝑡+𝜏−𝐿−𝑥𝑡+𝜏−𝐻₀×^+∞_{𝑡+𝜏𝑢}^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑡−𝜏)^𝑚−1|𝑎(𝑢×𝑔(𝑠+𝑓𝑓)|)𝑠,𝑥₁(𝑠)𝑥𝑓,…,_𝑙(𝑠)𝑑𝑠𝑑𝑢−𝐻_𝑖^+∞_{𝑡+𝜏𝑢}^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑡−𝜏)^𝑚−1|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠),…,𝑥ℎ_𝑙(𝑠)≤1𝑑𝑠𝑑𝑢𝑏()×𝑀𝑡+𝜏−𝐿+𝑏()−𝑡+𝜏_{𝑇+∞𝑢+∞}𝑢^𝑚−1|𝑎(𝑢×𝐻)|₀𝑠^{𝑛−𝑚−1}𝑔(𝑠)+𝑄(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑊(𝑠<1)𝑑𝑠𝑑𝑢𝑏()𝑀𝑡+𝜏−𝐿+𝑏()𝑀1𝑡+𝜏−min1−𝑏₀≤−𝐿,𝐿−𝑁−𝑀𝑏(),𝑆𝑡+𝜏_𝐿𝑥(𝑡)≥1𝑏()×()𝑡+𝜏−𝐿−𝑥𝑡+𝜏+𝐻₀×^+∞_{𝑡+𝜏𝑢}^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑡−𝜏)^𝑚−1|𝑎(𝑢×𝑔(𝑠+𝑓𝑓)|)𝑠,𝑥₁(𝑠)𝑥𝑓,…,_𝑙(𝑠)𝑑𝑠𝑑𝑢+𝐻_𝑖^+∞_{𝑡+𝜏𝑢}^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑡−𝜏)^𝑚−1|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠),…,𝑥ℎ_𝑙(𝑠)≥1𝑑𝑠𝑑𝑢𝑏()×𝑁𝑡+𝜏−𝐿+𝑏()+𝑡+𝜏_{𝑇+∞𝑢+∞}𝑢^𝑚−1|𝑎(𝑢×𝐻)|₀𝑠^{𝑛−𝑚−1}𝑔(𝑠)+𝑄(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑊(𝑠>1)𝑑𝑠𝑑𝑢𝑏()𝑁𝑡+𝜏−𝐿+𝑏()𝑀1𝑡+𝜏+min1−𝑏₀≥−𝐿,𝐿−𝑁−𝑁𝑏(),𝑡+𝜏(64) which imply that 𝑆_𝐿(Ω₃(𝑁,𝑀))⊆Ω₃(𝑁,𝑀). The rest of the proof is identical with the proof of Theorem 5 and hence is omitted. This completes the proof.

Theorem 7. Let 𝑚≥2. Assume that there exist two constants 𝑀, 𝑁 with 𝑀>𝑁>0 and four functions 𝑃,𝑄,𝑅,𝑊∈𝐶([𝑡₀,+∞),ℝ⁺) satisfying (16)–(18) and 𝑏(𝑡)=1𝑒𝑣𝑒𝑛𝑡𝑢𝑎𝑙𝑙𝑦.(65) Then
(a) for any 𝐿∈(𝑁,𝑀), there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| such that for each 𝑥₀∈Ω₁(𝑁,𝑀), the Mann iterative sequence {𝑥_𝑘}_𝑘∈ℕ0 generated by the following scheme 𝑥_𝑘+1(𝑡)=1−𝛼_𝑘𝑥_𝑘(𝑡)+𝛼_𝑘×𝐿+(−1)^𝑛(𝑚−1)𝐻₀×∑^∞_𝑗=1∫^{𝑡+2𝑗𝜏}_{𝑡+(2𝑗−1)𝜏}∫_𝑟^+∞∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑟)^𝑚−2×(𝑎(𝑢))⁻¹×𝑔(𝑠)−𝑓𝑠,𝑥_𝑘𝑓₁(𝑠)𝑥,…,_𝑘𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢𝑑𝑟−1^{𝑛−𝑖−1}(𝑚−1)𝐻_𝑖×∑^∞_𝑗=1∫^{𝑡+2𝑗𝜏}_{𝑡+(2𝑗−1)𝜏}∫_𝑟^+∞∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑟)^𝑚−2×(𝑎(𝑢))⁻¹×ℎ𝑠,𝑥_𝑘ℎ₁(𝑠)𝑥,…,_𝑘ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢𝑑𝑟,𝑡≥𝑇,𝑘∈ℕ₀,1−𝛼_𝑘𝑥_𝑘(𝑇)+𝛼_𝑘×𝐿+(−1)^𝑛(𝑚−1)𝐻₀×∑^∞_𝑗=1∫^{𝑇+2𝑗𝜏}_{𝑇+(2𝑗−1)𝜏}∫_𝑟^+∞∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑟)^𝑚−2×(𝑎(𝑢))⁻¹×𝑔(𝑠)−𝑓𝑠,𝑥_𝑘𝑓₁(𝑠)𝑥,…,_𝑘𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢𝑑𝑟−1^{𝑛−𝑖−1}(𝑚−1)𝐻_𝑖×∑^∞_𝑗=1∫^{𝑇+2𝑗𝜏}_{𝑇+(2𝑗−1)𝜏}∫_𝑟^+∞∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑟)^𝑚−2×(𝑎(𝑢))⁻¹×ℎ𝑠,𝑥_𝑘ℎ₁(𝑠)𝑥,…,_𝑘ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢𝑑𝑟,𝛾≤𝑡<𝑇,𝑘∈ℕ₀(66) converges to a positive solution 𝑥∈Ω₁(𝑁,𝑀) of (5) and has the error estimate (20), where {𝛼_𝑘}_𝑘∈ℕ0 is an arbitrary sequence in [0,1] with (21);
(b) Equation (5) has uncountably many positive solutions in Ω₁(𝑁,𝑀).

Proof. Let 𝐿∈(𝑁,𝑀). It follows from (18) and (65) that there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| satisfying 𝑏(𝑡)=1,∀𝑡≥𝑇;(67)𝜃=_𝑇^+∞_𝑢^+∞𝑢^𝑚−1|𝑎(𝑢𝐻)|⁰𝑠^{𝑛−𝑚−1}𝑃(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑅(𝑠)𝑑𝑠𝑑𝑢;(68)_{𝑇+∞𝑢+∞}𝑢^𝑚−1|𝑎(𝑢𝐻)|⁰𝑠^{𝑛−𝑚−1}𝑔(𝑠)+𝑄(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑊(𝑠{}.)𝑑𝑠𝑑𝑢<min𝑀−𝐿,𝐿−𝑁(69) Define a mapping 𝑆_𝐿∶Ω₁(𝑁,𝑀)→CB([𝛾,+∞),ℝ) by 𝑆_𝐿𝑥(𝑡)=𝐿+(−1)^𝑛(𝑚−1)𝐻₀×∑^∞_𝑗=1∫_𝜏^{𝑡+2𝑗𝜏}_{𝑡+2𝑗−1}∫_𝑟^+∞∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑟)^𝑚−2×(𝑎(𝑢))⁻¹×𝑔(𝑠)−𝑓𝑠,𝑥𝑓₁(𝑠),…,𝑥𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢𝑑𝑟−1^{𝑛−𝑖−1}(𝑚−1)𝐻_𝑖×∑^∞_𝑗=1∫_𝜏^{𝑡+2𝑗𝜏}_{𝑡+2𝑗−1}∫_𝑟^+∞∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑟)^𝑚−2×(𝑎(𝑢))⁻¹×ℎ𝑠,𝑥ℎ₁(𝑠),…,𝑥ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢𝑑𝑟,𝑡≥𝑇,𝑥∈Ω₁(),𝑆𝑁,𝑀_𝐿𝑥(𝑇),𝛾≤𝑡<𝑇,𝑥∈Ω₁(𝑁,𝑀).(70) With a view to (16), (68), and (70), we derive that for any 𝑥,𝑦∈Ω₁(𝑁,𝑀) and 𝑡≥𝑇𝑆_𝐿𝑥(𝑡)−𝑆_𝐿𝑦(𝑡≤()𝐻)𝑚−1₀×^∞_𝑗=1^{𝑡+2𝑗𝜏}_{𝑡+(2𝑗−1)𝜏𝑟}^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑟)^𝑚−2|𝑎(𝑢×𝑓𝑓)|𝑠,𝑥₁(𝑠),…,𝑥𝑓_𝑙(𝑠)−𝑓𝑠,𝑦𝑓₁(𝑠),…,𝑦𝑓_𝑙(𝑠)+()𝐻𝑑𝑠𝑑𝑢𝑑𝑟𝑚−1_𝑖×^∞_𝑗=1^{𝑡+2𝑗𝜏}_{𝑡+(2𝑗−1)𝜏𝑟}^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑟)^𝑚−2|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠),…,𝑥ℎ_𝑙(𝑠)−ℎ𝑠,𝑦ℎ₁(𝑠),…,𝑦ℎ_𝑙(𝑠)≤()𝐻𝑑𝑠𝑑𝑢𝑑𝑟𝑚−1₀×𝑥−𝑦_{𝑡+∞𝑟+∞𝑢+∞}(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑟)^𝑚−2|𝑎(𝑢𝑃(𝑠)+()𝐻)|𝑑𝑠𝑑𝑢𝑑𝑟𝑚−1_𝑖×𝑥−𝑦_{𝑡+∞𝑟+∞𝑢+∞}(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑟)^𝑚−2|𝑎(𝑢𝑅(𝑠))|𝑑𝑠𝑑𝑢𝑑𝑟=𝐻₀𝑥−𝑦×_𝑡^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑡)^𝑚−1|𝑎(𝑢𝑃(𝑠))|𝑑𝑠𝑑𝑢+𝐻_𝑖×𝑥−𝑦_{𝑡+∞𝑢+∞}(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑡)^𝑚−1|𝑎(𝑢𝑅(𝑠)≤×)|𝑑𝑠𝑑𝑢𝑥−𝑦_{𝑇+∞𝑢+∞}𝑢^𝑚−1|𝑎(𝑢𝐻)|⁰𝑠^{𝑛−𝑚−1}𝑃(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑅(𝑠,)𝑑𝑠𝑑𝑢=𝜃𝑥−𝑦(71) which gives (27). By virtue of (17), (69), and (70), we deduce that for any 𝑥∈Ω₁(𝑁,𝑀) and 𝑡≥𝑇𝑆_𝐿𝑥(𝑡)≤𝐿+(𝑚−1)𝐻₀×_𝑡^+∞_𝑟^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑟)^𝑚−2|𝑎(𝑢×𝑔(𝑠+𝑓𝑓)|)𝑠,𝑥₁(𝑠),…,𝑥𝑓_𝑙(𝑠)+()𝐻𝑑𝑠𝑑𝑢𝑑𝑟𝑚−1_𝑖×_𝑡^+∞_𝑟^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑟)^𝑚−2|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠),…,𝑥ℎ_𝑙(𝑠)()𝐻𝑑𝑠𝑑𝑢𝑑𝑟≤𝐿+𝑚−1₀×_𝑡^+∞_𝑟^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑟)^𝑚−2|𝑎(𝑢×𝑔(𝑠(𝑠+()𝐻)|)+𝑄)𝑑𝑠𝑑𝑢𝑑𝑟𝑚−1_𝑖×_𝑡^+∞_𝑟^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑟)^𝑚−2|𝑎(𝑢𝑊(𝑠))|𝑑𝑠𝑑𝑢𝑑𝑟=𝐿+𝐻_0𝑡^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑡)^𝑚−1|𝑎(𝑢×𝑔(𝑠(𝑠)|)+𝑄)𝑑𝑠𝑑𝑢+𝐻_𝑖𝑡^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑡)^𝑚−1|𝑎(𝑢(𝑠))|×𝑊𝑑𝑠𝑑𝑢≤𝐿+_𝑇^+∞_𝑢^+∞𝑢^𝑚−1|𝑎(𝑢×𝐻)|₀𝑠^{𝑛−𝑚−1}𝑔(𝑠)+𝑄(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑊(𝑠{}𝑆)𝑑𝑠𝑑𝑢<𝐿+min𝑀−𝐿,𝐿−𝑁≤𝑀,_𝐿𝑥(𝑡)≥𝐿−(𝑚−1)𝐻₀×_𝑡^+∞_𝑟^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑟)^𝑚−2|𝑎(𝑢×𝑔(𝑠+𝑓𝑓)|)𝑠,𝑥₁(𝑠)𝑥𝑓,…,_𝑙(𝑠)−()𝐻𝑑𝑠𝑑𝑢𝑑𝑟𝑚−1_𝑖×_𝑡^+∞_𝑟^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑟)^𝑚−2|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠),…,𝑥ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢𝑑𝑟≥𝐿−_𝑇^+∞_𝑢^+∞𝑢^𝑚−1|𝑎(𝑢×𝐻)|₀𝑠^{𝑛−𝑚−1}𝑔(𝑠)+𝑄(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑊(𝑠{})𝑑𝑠𝑑𝑢>𝐿−min𝑀−𝐿,𝐿−𝑁≥𝑁,(72) which mean that 𝑆_𝐿(Ω₁(𝑁,𝑀))⊆Ω₁(𝑁,𝑀). Coupled with (27) and (68), we get that 𝑆_𝐿 is a contraction mapping in Ω₁(𝑁,𝑀) and it possesses a unique fixed point 𝑥∈Ω₁(𝑁,𝑀). Clearly, 𝑥∈Ω₁(𝑁,𝑀) is a positive solution of (5).
From (27), (66), and (70), we gain that 𝑥_𝑘+1(𝑡)(𝑡=−𝑥)1−𝛼_𝑘𝑥_𝑘(𝑡)+𝛼_𝑘𝐿+(−1)^𝑛(𝑚−1)𝐻₀×^∞_𝑗=1^{𝑡+2𝑗𝜏}_{𝑡+(2𝑗−1)𝜏𝑟}^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑟)^𝑚−2𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥_𝑘𝑓₁(𝑠),…,𝑥_𝑘𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢𝑑𝑟−1^{𝑛−𝑖−1}(𝑚−1)𝐻_𝑖×^∞_𝑗=1^{𝑡+2𝑗𝜏}_{𝑡+(2𝑗−1)𝜏𝑟}^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑟)^𝑚−2𝑎(𝑢)×ℎ𝑠,𝑥_𝑘ℎ₁(𝑠),…,𝑥_𝑘ℎ_𝑙(𝑠)(𝑡)≤𝑑𝑠𝑑𝑢𝑑𝑟−𝑥1−𝛼_𝑘𝑥_𝑘(𝑡)−𝑥(𝑡)+𝛼_𝑘𝑆_𝐿𝑥_𝑘(𝑡)−𝑆_𝐿𝑥(𝑡≤)1−𝛼_𝑘𝑥_𝑘(𝑡)−𝑥(𝑡)+𝛼_𝑘𝜃𝑥_𝑘(𝑡)(𝑡=()𝛼−𝑥)1−1−𝜃_𝑘𝑥_𝑘(𝑡)−𝑥(𝑡)≤𝑒^{−(1−𝜃)𝛼𝑘}𝑥_𝑘−𝑥≤𝑒^{−(1−𝜃)∑𝑘𝑝=0𝛼𝑝}𝑥₀−𝑥,∀𝑘∈ℕ₀,𝑡≥𝑇,(73) which yields (20). It follows from (20) and (21) that lim_𝑘→∞𝑥_𝑘=𝑥.
Now we prove that (b) holds. Let 𝐿₁,𝐿₂∈(𝑁,𝑀) and 𝐿₁≠𝐿₂. As in the proof of (a), we conclude that for each 𝑝∈{1,2}, there exist 𝜃_𝑝∈(0,1), 𝑇_𝑝>1+|𝑡₀|+𝜏+|𝛾| and 𝑆_𝐿𝑝∶Ω₁(𝑁,𝑀)→Ω₁(𝑁,𝑀) satisfying (69)–(77), where 𝐿, 𝜃, 𝑇, and 𝑆_𝐿 are replaced by 𝐿_𝑝, 𝜃_𝑝, 𝑇_𝑝, and 𝑆_𝐿𝑝, respectively, and 𝑆_𝐿𝑝 has a unique fixed point 𝑧_𝑝∈Ω₁(𝑁,𝑀), which is a positive solution of (5) in Ω₁(𝑁,𝑀), that is, 𝑧_𝑝(𝑡)=𝐿_𝑘+(−1)^𝑛(𝑚−1)𝐻₀×^∞_𝑗=1𝜏^{𝑡+2𝑗𝜏}_{𝑡+2𝑗−1𝑟}^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑟)^𝑚−2𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑧_𝑝𝑓₁(𝑠),…,𝑧_𝑝𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢𝑑𝑟−1^{𝑛−𝑖−1}(𝑚−1)𝐻_𝑖×^∞_𝑗=1^{𝑡+2𝑗𝜏}_{𝑡+(2𝑗−1)𝜏𝑟}^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑟)^𝑚−2𝑎(𝑢)×ℎ𝑠,𝑧_𝑝ℎ₁(𝑠),…,𝑧_𝑝ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢𝑑𝑟,∀𝑡≥𝑇_𝑝,𝑝∈{1,2}.(74) For purpose of proving (b), we just need to show that 𝑧₁≠𝑧₂. It follows from (16), (27), (68), and (74) that 𝑧₁(𝑡)−𝑧₂(𝑡≥𝐿₁)−𝐿₂−(𝑚−1)𝐻₀𝑧₁−𝑧₂×_𝑡^+∞_𝑟^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑟)^𝑚−2×(|𝑎(𝑢)|)⁻¹𝑃(𝑠)−()𝐻𝑑𝑠𝑑𝑢𝑑𝑟𝑚−1_𝑖𝑧₁−𝑧₂×_𝑡^+∞_𝑟^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑟)^𝑚−2×(|𝑎(𝑢)|)⁻¹𝑅(𝑠)≥𝐿𝑑𝑠𝑑𝑢𝑑𝑟₁−𝐿₂−𝑧₁−𝑧₂×_𝑇^+∞_{max1,𝑇2𝑢}^+∞𝑢^𝑚−1|𝑎(𝑢×𝐻)|₀𝑠^{𝑛−𝑚−1}𝑃(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑅(𝑠>𝐿)𝑑𝑠𝑑𝑢₁−𝐿₂−max𝜃₁,𝜃₂𝑧₁−𝑧₂,∀𝑡≥max𝑇₁,𝑇₂,(75) which yields that 𝑧₁−𝑧₂≥𝐿₁−𝐿₂1+max𝜃₁,𝜃₂>0,(76) that is, 𝑧₁≠𝑧₂. This completes the proof.

Theorem 8. Let 𝑚=1. Assume that there exist two constants 𝑀, 𝑁 with 𝑀>𝑁>0 and four functions 𝑃,𝑄,𝑅,𝑊∈𝐶([𝑡₀,+∞),ℝ⁺) satisfying (16), (17), (65), and _𝑡^+∞_0𝑢^+∞1|𝑎(𝑢×𝑠|)||^𝑛−2𝑃(𝑠)(𝑠),𝑔(𝑠+|𝑠|max,𝑄)_{𝑛−𝑖−2}max{𝑅(𝑠),𝑊(𝑠)}𝑑𝑠𝑑𝑢<+∞.(77) Then
(a) for any 𝐿∈(𝑁,𝑀), there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| such that for each 𝑥₀∈Ω₁(𝑁,𝑀), the Mann iterative sequence {𝑥_𝑘}_𝑘∈ℕ0 generated by the following scheme 𝑥_𝑘+1(𝑡)=1−𝛼_𝑘𝑥_𝑘(𝑡)+𝛼_𝑘()𝐿+−1^𝑛()!×𝑛−2_∞𝑗∑₌₁∫_𝜏^{𝑡+2𝑗𝜏}_{𝑡+2𝑗−1}∫_𝑢^+∞(𝑠−𝑢)^𝑛−2𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥_𝑘𝑓₁(𝑠),…,𝑥_𝑘𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢−1^{𝑛−𝑖−1}()!×𝑛−𝑖−2_∞𝑗∑₌₁∫_𝜏^{𝑡+2𝑗𝜏}_{𝑡+2𝑗−1}∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑖−2}𝑎(𝑢)×ℎ𝑠,𝑥_𝑘ℎ₁(𝑠),…,𝑥_𝑘ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢,𝑡≥𝑇,𝑘∈ℕ₀,1−𝛼_𝑘𝑥_𝑘(𝑇)+𝛼_𝑘()𝐿+−1^𝑛()!×𝑛−2_∞𝑗∑₌₁∫_𝜏^{𝑇+2𝑗𝜏}_{𝑇+2𝑗−1}∫_𝑢^+∞(𝑠−𝑢)^𝑛−2𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥_𝑘𝑓₁(𝑠),…,𝑥_𝑘𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢−1^{𝑛−𝑖−1}()!×𝑛−𝑖−2_∞𝑗∑₌₁∫_𝜏^{𝑇+2𝑗𝜏}_{𝑇+2𝑗−1}∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑖−2}𝑎(𝑢)×ℎ𝑠,𝑥_𝑘ℎ₁(𝑠),…,𝑥_𝑘ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢,𝛾≤𝑡<𝑇,𝑘∈ℕ₀(78) converges to a positive solution 𝑥∈Ω₁(𝑁,𝑀) of (5) and has the error estimate (20), where {𝛼_𝑘}_𝑘∈ℕ0 is an arbitrary sequence in [0,1] with (21);
(b) Equation (5) has uncountably many positive solutions in Ω₁(𝑁,𝑀).

Proof. Let 𝐿∈(𝑁,𝑀). It follows from (65) and (77) that there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| satisfying (67), 𝜃=_𝑇^+∞_𝑢^+∞1|𝑎(𝑢×𝑠)|_𝑛−2()!𝑃(𝑠)+𝑠𝑛−2(^{𝑛−𝑖−2})!𝑅(𝑠𝑛−𝑖−2)𝑑𝑠𝑑𝑢,(79)_{𝑇+∞𝑢+∞}1|𝑎(𝑢𝑠)|(^𝑛−2)!𝑔(𝑠(𝑠)+𝑠𝑛−2)+𝑄_{𝑛−𝑖−2}()!𝑊(𝑠{}.𝑛−𝑖−2)𝑑𝑠𝑑𝑢<min𝑀−𝐿,𝐿−𝑁(80) Define a mapping 𝑆_𝐿∶Ω₁(𝑁,𝑀)→CB([𝛾,+∞),ℝ) by 𝑆_𝐿𝑥(𝑡)=()𝐿+−1^𝑛(𝑛−2)!^∞_𝑗∑₌₁∫^{𝑡+2𝑗𝜏}_{𝑡+(2𝑗−1)𝜏}∫_𝑢^+∞(𝑠−𝑢)^𝑛−2𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥𝑓₁(𝑠),…,𝑥𝑓_𝑙(𝑠+())𝑑𝑠𝑑𝑢−1^{𝑛−𝑖−1}(𝑛−𝑖−2)!^∞_𝑗∑₌₁∫^{𝑡+2𝑗𝜏}_{𝑡+(2𝑗−1)𝜏}∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑖−2}𝑎(𝑢)×ℎ𝑠,𝑥ℎ₁(𝑠),…,𝑥ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢,𝑡≥𝑇,𝑥∈Ω₁(),𝑆𝑁,𝑀_𝐿𝑥(𝑇),𝛾≤𝑡<𝑇,𝑥∈Ω₁(𝑁,𝑀).(81) By virtue of (16), (79), and (81), we derive that for any 𝑥,𝑦∈Ω₁(𝑁,𝑀) and 𝑡≥𝑇𝑆_𝐿𝑥(𝑡)−𝑆_𝐿𝑦(𝑡≤1)(𝑛−2)!^∞_𝑗=1^{𝑡+2𝑗𝜏}_{𝑡+(2𝑗−1)𝜏𝑢}^+∞(𝑠−𝑢)^𝑛−2|𝑎(𝑢×𝑓𝑓)|𝑠,𝑥₁(𝑠),…,𝑥𝑓_𝑙(𝑠)−𝑓𝑠,𝑦𝑓₁(𝑠),…,𝑦𝑓_𝑙(𝑠+1)𝑑𝑠𝑑𝑢(𝑛−𝑖−2)!^∞_𝑗=1^{𝑡+2𝑗𝜏}_{𝑡+(2𝑗−1)𝜏𝑢}^+∞(𝑠−𝑢)^{𝑛−𝑖−2}|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠),…,𝑥ℎ_𝑙(𝑠)−ℎ𝑠,𝑦ℎ₁(𝑠),…,𝑦ℎ_𝑙(𝑠)≤1𝑑𝑠𝑑𝑢(𝑛−2)!𝑥−𝑦_𝑇^+∞_𝑢^+∞(𝑠−𝑢)^𝑛−2|𝑎(𝑢𝑃(𝑠)+1)|𝑑𝑠𝑑𝑢()!×𝑛−𝑖−2𝑥−𝑦_{𝑇+∞𝑢+∞}(𝑠−𝑢)^{𝑛−𝑖−2}|𝑎(𝑢𝑅(𝑠)≤)|𝑑𝑠𝑑𝑢𝑥−𝑦_𝑇^+∞_𝑢^+∞1|𝑎(𝑢𝑠)|(^𝑛−2)!𝑃(𝑠)+𝑠𝑛−2_{𝑛−𝑖−2}()!𝑅(𝑠,𝑛−𝑖−2)𝑑𝑠𝑑𝑢=𝜃𝑥−𝑦(82) which gives (27). It follows from (17), (80), and (81) that for any 𝑥∈Ω₁(𝑁,𝑀) and 𝑡≥𝑇𝑆_𝐿𝑥(𝑡)1≤𝐿+(𝑛−2)!^∞_𝑗=1𝜏^{𝑡+2𝑗𝜏}_{𝑡+2𝑗−1𝑢}^+∞(𝑠−𝑢)^𝑛−2|𝑎(𝑢×𝑔(𝑠+𝑓𝑓)|)𝑠,𝑥₁(𝑠),…,𝑥𝑓_𝑙(𝑠)+1𝑑𝑠𝑑𝑢(𝑛−𝑖−2)!^∞_𝑗=1^{𝑡+2𝑗𝜏}_{𝑡+(2𝑗−1)𝜏𝑢}^+∞(𝑠−𝑢)^{𝑛−𝑖−2}|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠),…,𝑥ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢≤𝐿+_𝑇^+∞_𝑢^+∞1|𝑎(𝑢×𝑠)|_𝑛−2()!𝑔(𝑠(𝑠)+𝑠𝑛−2)+𝑄^{𝑛−𝑖−2}()!𝑊(𝑠{}𝑆𝑛−𝑖−2)𝑑𝑠𝑑𝑢<𝐿+min𝑀−𝐿,𝐿−𝑁≤𝑀,_𝐿𝑥(𝑡)1≥𝐿−(𝑛−2)!^∞_𝑗=1𝜏^{𝑡+2𝑗𝜏}_{𝑡+2𝑗−1𝑢}^+∞(𝑠−𝑢)^𝑛−2|𝑎(𝑢×𝑔(𝑠+𝑓𝑓)|)𝑠,𝑥₁(𝑠),…,𝑥𝑓_𝑙(𝑠)−1𝑑𝑠𝑑𝑢(𝑛−𝑖−2)!^∞_𝑗=1^{𝑡+2𝑗𝜏}_{𝑡+(2𝑗−1)𝜏𝑢}^+∞(𝑠−𝑢)^{𝑛−𝑖−2}|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠),…,𝑥ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢≥𝐿−_𝑇^+∞_𝑢^+∞1|𝑎(𝑢×𝑠)|_𝑛−2()!𝑔(𝑠(𝑠)+𝑠𝑛−2)+𝑄^{𝑛−𝑖−2}()!𝑊(𝑠{}𝑛−𝑖−2)𝑑𝑠𝑑𝑢>𝐿−min𝑀−𝐿,𝐿−𝑁≥𝑁,(83) which mean that 𝑆_𝐿(Ω₁(𝑁,𝑀))⊆Ω₁(𝑁,𝑀). Combined with (27) and (79), we know that 𝑆_𝐿 is a contraction mapping in Ω₁(𝑁,𝑀) and it possesses a unique fixed point 𝑥∈Ω₁(𝑁,𝑀). Obviously, 𝑥∈Ω₁(𝑁,𝑀) is a positive solution of (5).
In light of (27), (78), and (81), we gain that 𝑥_𝑘+1(𝑡)(𝑡=−𝑥)1−𝛼_𝑘𝑥_𝑘(𝑡)+𝛼_𝑘()𝐿+−1^𝑛(𝑛−2)!^∞_𝑗=1^{𝑡+2𝑗𝜏}_{𝑡+(2𝑗−1)𝜏𝑢}^+∞(𝑠−𝑢)^𝑛−2𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥_𝑘𝑓₁(𝑠),…,𝑥_𝑘𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢−1^{𝑛−𝑖−1}(𝑛−𝑖−2)!^∞_𝑗=1𝜏^{𝑡+2𝑗𝜏}_{𝑡+2𝑗−1𝑢}^+∞(𝑠−𝑢)^{𝑛−𝑖−2}𝑎(𝑢)×ℎ𝑠,𝑥_𝑘ℎ₁(𝑠),…,𝑥_𝑘ℎ_𝑙(𝑠)(𝑡)≤𝑑𝑠𝑑𝑢−𝑥1−𝛼_𝑘𝑥_𝑘(𝑡)−𝑥(𝑡)+𝛼_𝑘𝑆_𝐿𝑥_𝑘(𝑡)−𝑆_𝐿𝑥(𝑡≤)1−𝛼_𝑘𝑥_𝑘(𝑡)−𝑥(𝑡)+𝛼_𝑘𝜃𝑥_𝑘(𝑡)(𝑡=()𝛼−𝑥)1−1−𝜃_𝑘𝑥_𝑘(𝑡)−𝑥(𝑡)≤𝑒^{−(1−𝜃)𝛼𝑘}𝑥_𝑘−𝑥≤𝑒^{−(1−𝜃)∑𝑘𝑝=0𝛼𝑝}𝑥₀−𝑥,∀𝑘∈ℕ₀,𝑡≥𝑇,(84) which yields (20). It follows from (20) and (21) that lim_𝑘→∞𝑥_𝑘=𝑥.
Now we prove that (b) holds. Let 𝐿₁,𝐿₂∈(𝑁,𝑀) and 𝐿₁≠𝐿₂. As in the proof of (a), we conclude that for each 𝑝∈{1,2}, there exist 𝜃_𝑝∈(0,1), 𝑇_𝑝>1+|𝑡₀|+𝜏+|𝛾| and 𝑆_𝐿𝑝∶Ω₁(𝑁,𝑀)→Ω₁(𝑁,𝑀) satisfying (67) and (79)–(81), where 𝐿, 𝜃, 𝑇, and 𝑆_𝐿 are replaced by 𝐿_𝑝, 𝜃_𝑝, 𝑇_𝑝, and 𝑆_𝐿𝑝, respectively, and 𝑆_𝐿𝑝 has a unique fixed point 𝑧_𝑝∈Ω₁(𝑁,𝑀), which is a positive solution of (5) in Ω₁(𝑁,𝑀), that is, 𝑧_𝑝(𝑡)=𝐿_𝑝+(−1)^𝑛()!×𝑛−2_{∞𝑗=1𝜏𝑡+2𝑗𝜏𝑡+2𝑗−1𝑢+∞}(𝑠−𝑢)^𝑛−2𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑧_𝑝𝑓₁(𝑠),…,𝑧_𝑝𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢−1^{𝑛−𝑖−1}(𝑛−𝑖−2)!^∞_𝑗=1^{𝑡+2𝑗𝜏}_{𝑡+(2𝑗−1)𝜏𝑢}^+∞(𝑠−𝑢)^{𝑛−𝑖−2}𝑎(𝑢)×ℎ𝑠,𝑧_𝑝ℎ₁(𝑠),…,𝑧_𝑝ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢,∀𝑡≥𝑇_𝑝,𝑝∈{1,2}.(85) In order to prove (b), we just need to show that 𝑧₁≠𝑧₂. In view of (16), (27), (79), and (85), we get that 𝑧₁(𝑡)−𝑧₂(𝑡≥𝐿₁)−𝐿₂−1(𝑛−2)!^∞_𝑗=1^{𝑡+2𝑗𝜏}_{𝑡+(2𝑗−1)𝜏𝑢}^+∞(𝑠−𝑢)^𝑛−2|𝑎(𝑢×𝑓)|𝑠,𝑧₂𝑓₁(𝑠),…,𝑧₂𝑓_𝑙(𝑠)−𝑓𝑠,𝑧₁𝑓₁(𝑠),…,𝑧₁𝑓_𝑙(𝑠−1)𝑑𝑠𝑑𝑢(𝑛−𝑖−2)!^∞_𝑗=1^{𝑡+2𝑗𝜏}_{𝑡+(2𝑗−1)𝜏𝑢}^+∞(𝑠−𝑢)^{𝑛−𝑖−2}|𝑎(𝑢×ℎ)|𝑠,𝑧₁ℎ₁(𝑠),…,𝑧₁ℎ_𝑙(𝑠)−ℎ𝑠,𝑧₂ℎ₁(𝑠),…,𝑧₂ℎ_𝑙(𝑠)≥𝐿𝑑𝑠𝑑𝑢₁−𝐿₂−𝑧₁−𝑧₂(𝑛−2)!_𝑡^+∞_𝑢^+∞𝑠^𝑛−2|𝑎(𝑢𝑃(𝑠)−𝑧)|𝑑𝑠𝑑𝑢₁−𝑧₂(𝑛−𝑖−2)!_𝑡^+∞_𝑢^+∞𝑠^{𝑛−𝑖−2}|𝑎(𝑢𝑅(𝑠)≥𝐿)|𝑑𝑠𝑑𝑢₁−𝐿₂−𝑧₁−𝑧₂×_𝑇^+∞_{max1,𝑇2𝑢}^+∞1|𝑎(𝑢×𝑠)|_𝑛−2()!𝑃(𝑠)+𝑠𝑛−2(^{𝑛−𝑖−2})!𝑅(𝑠>𝐿𝑛−𝑖−2)𝑑𝑠𝑑𝑢₁−𝐿₂−max𝜃₁,𝜃₂𝑧₁−𝑧₂,∀𝑡≥max𝑇₁,𝑇₂,(86) which implies that 𝑧₁−𝑧₂≥𝐿₁−𝐿₂1+max𝜃₁,𝜃₂>0,(87) that is, 𝑧₁≠𝑧₂. This completes the proof.

Theorem 9. Let 𝑚≥2. Assume that there exist two constants 𝑀, 𝑁 with 𝑀>𝑁>0 and four functions 𝑃,𝑄,𝑅,𝑊∈𝐶([𝑡₀,+∞),ℝ⁺) satisfying (16), (17), _𝑡^+∞_0𝑟^+∞_𝑢^+∞|𝑟||𝑢|^𝑚|𝑎(𝑢×𝑠|)||^{𝑛−𝑚−1}𝑃(𝑠)(𝑠),𝑔(𝑠+|𝑠|max,𝑄)_{𝑛−𝑚−𝑖−1}{𝑅(𝑠)(𝑠𝑏(𝑡)max,𝑊)}𝑑𝑠𝑑𝑢𝑑𝑟<+∞,(88)=−1𝑒𝑣𝑒𝑛𝑡𝑢𝑎𝑙𝑙𝑦.(89) Then
(a) for any 𝐿∈(𝑁,𝑀), there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| such that for each 𝑥₀∈Ω₁(𝑁,𝑀), the Mann iterative sequence {𝑥_𝑘}_𝑘∈ℕ0 generated by the following scheme 𝑥_𝑘+1(𝑡)=1−𝛼_𝑘𝑥_𝑘(𝑡)+𝛼_𝑘𝐿+(−1)^𝑛−1(𝑚−1)𝐻₀×∑^∞_𝑗=1∫^+∞_{𝑡+𝑗𝜏}∫_𝑟^+∞∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑟)^𝑚−2𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥_𝑘𝑓₁(𝑠),…,𝑥_𝑘𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢𝑑𝑟−1^𝑛−𝑖(𝑚−1)𝐻_𝑖×∑^∞_𝑗=1∫^+∞_{𝑡+𝑗𝜏}∫_𝑟^+∞∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑟)^𝑚−2𝑎(𝑢)×ℎ𝑠,𝑥_𝑘ℎ₁(𝑠),…,𝑥_𝑘ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢𝑑𝑟,𝑡≥𝑇,𝑘∈ℕ₀,1−𝛼_𝑘𝑥_𝑘(𝑇)+𝛼_𝑘𝐿+(−1)^𝑛−1(𝑚−1)𝐻₀×∑^∞_𝑗=1∫^+∞_{𝑇+𝑗𝜏}∫_𝑟^+∞∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑟)^𝑚−2𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥_𝑘𝑓₁(𝑠),…,𝑥_𝑘𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢𝑑𝑟−1^𝑛−𝑖(𝑚−1)𝐻_𝑖×∑^∞_𝑗=1∫^+∞_{𝑇+𝑗𝜏}∫_𝑟^+∞∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑟)^𝑚−2𝑎(𝑢)×ℎ𝑠,𝑥_𝑘ℎ₁(𝑠),…,𝑥_𝑘ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢𝑑𝑟,𝛾≤𝑡<𝑇,𝑘∈ℕ₀(90) converges to a positive solution 𝑥∈Ω₁(𝑁,𝑀) of (5) and has the error estimate (20), where {𝛼_𝑘}_𝑘∈ℕ0 is an arbitrary sequence in [0,1] with (21);
(b) Equation (5) has uncountably many positive solutions in Ω₁(𝑁,𝑀).

Proof. Set 𝐿∈(𝑁,𝑀). In view of (88) and (89), there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| such that 𝑏(𝑡)=−1,∀𝑡≥𝑇;(91)𝜃=𝑚−1𝜏_𝑇^+∞_𝑟^+∞_𝑢^+∞𝑟𝑢^𝑚−2|𝑎(𝑢×𝐻)|₀𝑠^{𝑛−𝑚−1}𝑃(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑅(𝑠)𝑑𝑠𝑑𝑢𝑑𝑟;(92)𝑚−1𝜏_𝑇^+∞_𝑟^+∞_𝑢^+∞𝑟𝑢^𝑚−2|𝑎(𝑢×𝐻)|₀𝑠^{𝑛−𝑚−1}𝑔(𝑠)+𝑄(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑊(𝑠{}.)𝑑𝑠𝑑𝑢𝑑𝑟<min𝑀−𝐿,𝐿−𝑁(93) Define a mapping 𝑆_𝐿∶Ω₁(𝑁,𝑀)→CB([𝛾,+∞),ℝ) by 𝑆_𝐿𝑥(𝑡)=𝐿+(−1)^𝑛−1(𝑚−1)𝐻₀×∑^∞_𝑗=1∫^+∞_{𝑡+𝑗𝜏}∫_𝑟^+∞∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑟)^𝑚−2𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥𝑓₁(𝑠),…,𝑥𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢𝑑𝑟−1^𝑛−𝑖(𝑚−1)𝐻_𝑖×∑^∞_𝑗=1∫^+∞_{𝑡+𝑗𝜏}∫_𝑟^+∞∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑟)^𝑚−2𝑎(𝑢)×ℎ𝑠,𝑥ℎ₁(𝑠),…,𝑥ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢𝑑𝑟,𝑡≥𝑇,𝑥∈Ω₁(),𝑆𝑁,𝑀_𝐿𝑥(𝑇),𝛾≤𝑡<𝑇,𝑥∈Ω₁(𝑁,𝑀).(94) By virtue of (16), (92), (94), and Lemma 1, we acquire that for any 𝑥,𝑦∈Ω₁(𝑁,𝑀) and 𝑡≥𝑇𝑆_𝐿𝑥(𝑡)−𝑆_𝐿𝑦(𝑡≤()𝐻)𝑚−1₀^∞_𝑗=1^+∞_{𝑡+𝑗𝜏𝑟}^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑟)^𝑚−2|𝑎(𝑢×𝑓𝑓)|𝑠,𝑥₁(𝑠),…,𝑥𝑓_𝑙(𝑠)−𝑓𝑠,𝑦𝑓₁(𝑠),…,𝑦𝑓_𝑙(𝑠+()𝐻)𝑑𝑠𝑑𝑢𝑑𝑟𝑚−1_𝑖^∞_𝑗=1^+∞_{𝑡+𝑗𝜏𝑟}^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑟)^𝑚−2|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠),…,𝑥ℎ_𝑙(𝑠)−ℎ𝑠,𝑦ℎ₁(𝑠),…,𝑦ℎ_𝑙(𝑠)≤()𝐻𝑑𝑠𝑑𝑢𝑑𝑟𝑚−1⁰𝜏𝑥−𝑦×_𝑇^+∞_𝑟^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑟)^𝑚−2𝑟|𝑎(𝑢𝑃(𝑠)+()𝐻)|𝑑𝑠𝑑𝑢𝑑𝑟𝑚−1^𝑖𝜏×𝑥−𝑦_{𝑇+∞𝑟+∞𝑢+∞}(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑟)^𝑚−2𝑟|𝑎(𝑢𝑅(𝑠)≤)|𝑑𝑠𝑑𝑢𝑑𝑟𝑚−1𝜏_𝑇^+∞_𝑟^+∞_𝑢^+∞𝑟𝑢^𝑚−2|𝑎(𝑢×𝐻)|₀𝑠^{𝑛−𝑚−1}𝑃(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑅(𝑠,)𝑑𝑠𝑑𝑢𝑑𝑟=𝜃𝑥−𝑦(95) which yields that (27) holds. From (17), (94), (98), and Lemma 1, we obtain that for any 𝑥∈Ω₁(𝑁,𝑀) and 𝑡≥𝑇𝑆_𝐿𝑥(𝑡)≤()𝐻−𝐿𝑚−1₀^∞_𝑗=1^+∞_{𝑡+𝑗𝜏𝑟}^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑟)^𝑚−2|𝑎(𝑢×𝑔(𝑠+𝑓𝑓)|)𝑠,𝑥₁(𝑠),…,𝑥𝑓_𝑙(𝑠)+()𝐻𝑑𝑠𝑑𝑢𝑑𝑟𝑚−1_𝑖^∞_𝑗=1^+∞_{𝑡+𝑗𝜏𝑟}^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑟)^𝑚−2|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠),…,𝑥ℎ_𝑙(𝑠)≤()𝐻𝑑𝑠𝑑𝑢𝑑𝑟𝑚−1⁰𝜏_𝑇^+∞_𝑟^+∞_𝑢^+∞𝑟𝑠^{𝑛−𝑚−1}𝑢^𝑚−2|𝑎(𝑢×𝑔(𝑠(𝑠+()𝐻)|)+𝑄)𝑑𝑠𝑑𝑢𝑑𝑟𝑚−1^𝑖𝜏_𝑇^+∞_𝑟^+∞_𝑢^+∞𝑟𝑠^{𝑛−𝑚−𝑖−1}𝑢^𝑚−2|𝑎(𝑢(𝑠)=)|×𝑊𝑑𝑠𝑑𝑢𝑑𝑟𝑚−1𝜏_𝑇^+∞_𝑟^+∞_𝑢^+∞𝑟𝑢^𝑚−2|𝑎(𝑢×𝐻)|₀𝑠^{𝑛−𝑚−1}𝑔(𝑠)+𝑄(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑊(𝑠{},)𝑑𝑠𝑑𝑢𝑑𝑟<min𝑀−𝐿,𝐿−𝑁(96) which means that 𝑆_𝐿(Ω₁(𝑁,𝑀))⊆Ω₁(𝑁,𝑀). It follows from (27) and (92) that 𝑆_𝐿 is a contraction mapping and it has a unique fixed point 𝑥∈Ω₁(𝑁,𝑀). It is clear that 𝑥∈Ω₁(𝑁,𝑀) is a positive solution of (5).
On the basis of (27), (90), and (94), we deduce that 𝑥_𝑘+1(𝑡)(𝑡=−𝑥)1−𝛼_𝑘𝑥_𝑘(𝑡)+𝛼_𝑘𝐿+(−1)^𝑛−1(𝑚−1)𝐻₀×^∞_𝑗=1^+∞_{𝑡+𝑗𝜏𝑟}^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑟)^𝑚−2𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥_𝑘𝑓₁(𝑠)𝑥,…,_𝑘𝑓_𝑙(𝑠+())𝑑𝑠𝑑𝑢𝑑𝑟−1^𝑛−𝑖(𝑚−1)𝐻_𝑖×^∞_𝑗=1^+∞_{𝑡+𝑗𝜏𝑟}^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑟)^𝑚−2𝑎(𝑢)×ℎ𝑠,𝑥_𝑘ℎ₁(𝑠),…,𝑥_𝑘ℎ_𝑙(𝑠(𝑡)≤)𝑑𝑠𝑑𝑢𝑑𝑟−𝑥1−𝛼_𝑘𝑥_𝑘(𝑡)−𝑥(𝑡)+𝛼_𝑘𝑆_𝐿𝑥_𝑘(𝑡)−𝑆_𝐿𝑥(𝑡≤)1−𝛼_𝑘𝑥_𝑘(𝑡)−𝑥(𝑡)+𝛼_𝑘𝜃𝑥_𝑘(𝑡)(𝑡=()𝛼−𝑥)1−1−𝜃_𝑘𝑥_𝑘(𝑡)−𝑥(𝑡)≤𝑒^{−(1−𝜃)𝛼𝑘}𝑥_𝑘−𝑥≤𝑒^{−(1−𝜃)∑𝑘𝑝=0𝛼𝑝}𝑥₀−𝑥,∀𝑘∈ℕ₀,𝑡≥𝑇,(97) which signifies that (20) holds. It follows from (20) and (21) and that lim_𝑘→∞𝑥_𝑘=𝑥.
Now we show that (b) holds. Let 𝐿₁,𝐿₂∈(𝑁,𝑀) and 𝐿₁≠𝐿₂. As in the proof of (a), we conclude that for each 𝑝∈{1,2}, there exist 𝜃_𝑝∈(0,1), 𝑇_𝑝>1+|𝑡₀|+𝜏+|𝛾| and 𝑆_𝐿𝑝∶Ω₁(𝑁,𝑀)→Ω₁(𝑁,𝑀) satisfying (91)–(94), where 𝐿, 𝜃, 𝑇, and 𝑆_𝐿 are replaced by 𝐿_𝑝, 𝜃_𝑝, 𝑇_𝑝, and 𝑆_𝐿𝑝, respectively, and 𝑆_𝐿𝑝 has a unique fixed point 𝑧_𝑝∈Ω₁(𝑁,𝑀), which is a positive solution of (5) in Ω₁(𝑁,𝑀), that is, 𝑧_𝑝(𝑡)=𝐿_𝑝+(−1)^𝑛−1(𝑚−1)𝐻₀×^∞_𝑗=1^+∞_{𝑡+𝑗𝜏𝑟}^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−1}(𝑢−𝑟)^𝑚−2𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑧_𝑝𝑓₁(𝑠),…,𝑧_𝑝𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢𝑑𝑟−1^𝑛−𝑖(𝑚−1)𝐻_𝑖×^∞_𝑗=1^+∞_{𝑡+𝑗𝜏𝑟}^+∞_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑚−𝑖−1}(𝑢−𝑟)^𝑚−2𝑎(𝑢)×ℎ𝑠,𝑧_𝑝ℎ₁(𝑠),…,𝑧_𝑝ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢𝑑𝑟,∀𝑡≥𝑇_𝑝,𝑝∈{1,2}.(98) In order to prove (b), it is sufficient to show that 𝑧₁≠𝑧₂. Note that (16), (92), (98), and Lemma 1 lead to 𝑧₁(𝑡)−𝑧₂(𝑡≥𝐿₁)−𝐿₂−()𝐻𝑚−1⁰𝜏𝑧₁−𝑧₂×^+∞_{𝑡+𝜏𝑟}^+∞_𝑢^+∞𝑟𝑠^{𝑛−𝑚−1}𝑢^𝑚−2|𝑎(𝑢𝑃(𝑠)−()𝐻)|𝑑𝑠𝑑𝑢𝑑𝑟𝑚−1^𝑖𝜏𝑧₁−𝑧₂×^+∞_{𝑡+𝜏𝑟}^+∞_𝑢^+∞𝑟𝑠^{𝑛−𝑚−𝑖−1}𝑢^𝑚−2|𝑎(𝑢𝑅(𝑠)≥𝐿)|𝑑𝑠𝑑𝑢𝑑𝑟₁−𝐿₂−(𝑚−1)𝑧₁−𝑧₂𝜏×^+∞_{max{𝑇1,𝑇2}𝑟}^+∞_𝑢^+∞𝑟𝑢^𝑚−2|𝑎(𝑢×𝐻)|₀𝑠^{𝑛−𝑚−1}𝑃(𝑠)+𝐻_𝑖𝑠^{𝑛−𝑚−𝑖−1}𝑅(𝑠>𝐿)𝑑𝑠𝑑𝑢𝑑𝑟₁−𝐿₂−max𝜃₁,𝜃₂𝑧₁−𝑧₂,∀𝑡≥max𝑇₁,𝑇₂,(99) which means that 𝑧₁−𝑧₂≥𝐿₁−𝐿₂1+max𝜃₁,𝜃₂>0,(100) that is, 𝑧₁≠𝑧₂. This completes the proof.

Theorem 10. Let 𝑚=1. Assume that there exist two constants 𝑀, 𝑁 with 𝑀>𝑁>0 and four functions 𝑃,𝑄,𝑅,𝑊∈𝐶([𝑡₀,+∞),ℝ⁺) satisfying (16), (17), (89), and _𝑡^+∞_0𝑢^+∞|𝑢||𝑎(𝑢𝑠|)||^𝑛−2𝑃(𝑠)(𝑠),𝑔(𝑠+|𝑠|max,𝑄)_{𝑛−𝑖−2}max{𝑅(𝑠),𝑊(𝑠)}𝑑𝑠𝑑𝑢<+∞.(101) Then
(a) for any 𝐿∈(𝑁,𝑀), there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| such that for each 𝑥₀∈Ω₁(𝑁,𝑀), the Mann iterative sequence {𝑥_𝑘}_𝑘∈ℕ0 generated by the following scheme 𝑥_𝑘+1(𝑡)=1−𝛼_𝑘𝑥_𝑘(𝑡)+𝛼_𝑘()𝐿+−1^𝑛−1(𝑛−2)!^∞_𝑗∑₌₁∫^+∞_{𝑡+𝑗𝜏}∫_𝑢^+∞(𝑠−𝑢)^𝑛−2𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥_𝑘𝑓₁(𝑠),…,𝑥_𝑘𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢−1^𝑛−𝑖(𝑛−𝑖−2)!^∞_𝑗∑₌₁∫^+∞_{𝑡+𝑗𝜏}∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑖−2}𝑎(𝑢)×ℎ𝑠,𝑥_𝑘ℎ₁(𝑠),…,𝑥_𝑘ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢,𝑡≥𝑇,𝑘∈ℕ₀,1−𝛼_𝑘𝑥_𝑘(𝑇)+𝛼_𝑘()𝐿+−1^𝑛−1(𝑛−2)!^∞_𝑗∑₌₁∫^+∞_{𝑇+𝑗𝜏}∫_𝑢^+∞(𝑠−𝑢)^𝑛−2𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥_𝑘𝑓₁(𝑠),…,𝑥_𝑘𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢−1^𝑛−𝑖(𝑛−𝑖−2)!^∞_𝑗∑₌₁∫^+∞_{𝑇+𝑗𝜏}∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑖−2}𝑎(𝑢)×ℎ𝑠,𝑥_𝑘ℎ₁(𝑠),…,𝑥_𝑘ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢,𝛾≤𝑡<𝑇,𝑘∈ℕ₀,(102) converges to a positive solution 𝑥∈Ω₁(𝑁,𝑀) of (5) and has the error estimate (20), where {𝛼_𝑘}_𝑘∈ℕ0 is an arbitrary sequence in [0,1] with (21);
(b) Equation (5) has uncountably many positive solutions in Ω₁(𝑁,𝑀).

Proof. Set 𝐿∈(𝑁,𝑀). Due to (101), there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| satisfying (91), 1𝜃=𝜏_𝑇^+∞_𝑢^+∞𝑢|𝑎(𝑢×𝑠)|_𝑛−2()!𝑃(𝑠)+𝑠𝑛−2(^{𝑛−𝑖−2})!𝑅(𝑠1𝑛−𝑖−2)𝑑𝑠𝑑𝑢,(103)𝜏_𝑇^+∞_𝑢^+∞𝑢|𝑎(𝑢×𝑠)|_𝑛−2()!𝑔(𝑠(𝑠)+𝑠𝑛−2)+𝑄^{𝑛−𝑖−2}()!𝑊(𝑠{}.𝑛−𝑖−2)𝑑𝑠𝑑𝑢<min𝑀−𝐿,𝐿−𝑁(104) Define a mapping 𝑆_𝐿∶Ω₁(𝑁,𝑀)→CB([𝛾,+∞),ℝ) by 𝑆_𝐿𝑥(𝑡)=()𝐿+−1^𝑛−1(𝑛−2)!^∞_𝑗∑₌₁∫^+∞_{𝑡+𝑗𝜏}∫_𝑢^+∞(𝑠−𝑢)^𝑛−2𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥𝑓₁(𝑠),…,𝑥𝑓_𝑙(𝑠+())𝑑𝑠𝑑𝑢−1^𝑛−𝑖(𝑛−𝑖−2)!^∞_𝑗∑₌₁∫^+∞_{𝑡+𝑗𝜏}∫_𝑢^+∞(𝑠−𝑢)^{𝑛−𝑖−2}𝑎(𝑢)×ℎ𝑠,𝑥ℎ₁(𝑠),…,𝑥ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢,𝑡≥𝑇,𝑥∈Ω₁(),𝑆𝑁,𝑀_𝐿𝑥(𝑇),𝛾≤𝑡<𝑇,𝑥∈Ω₁(𝑁,𝑀).(105) In view of (16), (103), (105), and Lemma 1, we achieve that for any 𝑥,𝑦∈Ω₁(𝑁,𝑀) and 𝑡≥𝑇𝑆_𝐿𝑥(𝑡)−𝑆_𝐿𝑦(𝑡≤1)(𝑛−2)!^∞_𝑗=1^+∞_{𝑡+𝑗𝜏𝑢}^+∞(𝑠−𝑢)^𝑛−2|𝑎(𝑢×𝑓𝑓)|𝑠,𝑥₁(𝑠),…,𝑥𝑓_𝑙(𝑠)−𝑓𝑠,𝑦𝑓₁(𝑠),…,𝑦𝑓_𝑙(𝑠+1)𝑑𝑠𝑑𝑢(𝑛−𝑖−2)!^∞_𝑗=1^+∞_{𝑡+𝑗𝜏𝑢}^+∞(𝑠−𝑢)^{𝑛−𝑖−2}|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠),…,𝑥ℎ_𝑙(𝑠)−ℎ𝑠,𝑦ℎ₁(𝑠),…,𝑦ℎ_𝑙(𝑠)≤𝑑𝑠𝑑𝑢𝑥−𝑦𝜏(𝑛−2)!^+∞_{𝑡+𝜏𝑢}^+∞𝑢𝑠^𝑛−2|𝑎(𝑢𝑃(𝑠)+)|𝑑𝑠𝑑𝑢𝑥−𝑦𝜏(𝑛−𝑖−2)!^+∞_{𝑡+𝜏𝑢}^+∞𝑢𝑠^{𝑛−𝑖−2}|𝑎(𝑢𝑅(𝑠)≤)|𝑑𝑠𝑑𝑢𝑥−𝑦𝜏_𝑇^+∞_𝑢^+∞𝑢|𝑎(𝑢×𝑠)|_𝑛−2()!𝑃(𝑠)+𝑠𝑛−2(^{𝑛−𝑖−2})!𝑅(𝑠,𝑛−𝑖−2)𝑑𝑠𝑑𝑢=𝜃𝑥−𝑦(106) which means that (27) holds. It follows from (17), (104), (105), and Lemma 1 that for any 𝑥∈Ω₁(𝑁,𝑀) and 𝑡≥𝑇𝑆_𝐿𝑥(𝑡)≤1−𝐿(𝑛−2)!^∞_𝑗=1^+∞_{𝑡+𝑗𝜏𝑢}^+∞(𝑠−𝑢)^𝑛−2|𝑎(𝑢×𝑔(𝑠+𝑓𝑓)|)𝑠,𝑥₁(𝑠),…,𝑥𝑓_𝑙(𝑠)+1𝑑𝑠𝑑𝑢(𝑛−𝑖−2)!^∞_𝑗=1^+∞_{𝑡+𝑗𝜏𝑢}^+∞(𝑠−𝑢)^{𝑛−𝑖−2}|𝑎(𝑢×ℎℎ)|𝑠,𝑥₁(𝑠),…,𝑥ℎ_𝑙(𝑠)≤1𝑑𝑠𝑑𝑢𝜏(𝑛−2)!^+∞_{𝑡+𝜏𝑢}^+∞𝑢𝑠^𝑛−2|𝑎(𝑢𝑔(𝑠(𝑠)+1)|)+𝑄𝑑𝑠𝑑𝑢𝜏(𝑛−𝑖−2)!^+∞_{𝑡+𝜏𝑢}^+∞𝑢𝑠^{𝑛−𝑖−2}|𝑎(𝑢𝑊(𝑠)≤1)|𝑑𝑠𝑑𝑢𝜏_𝑇^+∞_𝑢^+∞𝑢|𝑎(𝑢×𝑠)|_𝑛−2()!𝑔(𝑠(𝑠)+𝑠𝑛−2)+𝑄^{𝑛−𝑖−2}()!𝑊(𝑠{},𝑛−𝑖−2)𝑑𝑠𝑑𝑢<min𝑀−𝐿,𝐿−𝑁(107) which means that 𝑆_𝐿(Ω₁(𝑁,𝑀))⊆Ω₁(𝑁,𝑀). Coupled with (27), we know that 𝑆_𝐿 is a contraction mapping and it has a unique fixed point 𝑥∈Ω₁(𝑁,𝑀). It follows that 𝑥∈Ω₁(𝑁,𝑀) is a positive solution of (5).
In view of (27), (102), and (105), we deduce that 𝑥_𝑘+1(𝑡)(𝑡=−𝑥)1−𝛼_𝑘𝑥_𝑘(𝑡)+𝛼_𝑘()𝐿+−1^𝑛−1(𝑛−2)!^∞_𝑗=1^+∞_{𝑡+𝑗𝜏𝑢}^+∞(𝑠−𝑢)^𝑛−2𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑥_𝑘𝑓₁(𝑠),…,𝑥_𝑘𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢−1^𝑛−𝑖(𝑛−𝑖−2)!^∞_𝑗=1^+∞_{𝑡+𝑗𝜏𝑢}^+∞(𝑠−𝑢)^{𝑛−𝑖−2}𝑎(𝑢)×ℎ𝑠,𝑥_𝑘ℎ₁(𝑠),…,𝑥_𝑘ℎ_𝑙(𝑠)(𝑡)≤𝑑𝑠𝑑𝑢−𝑥1−𝛼_𝑘𝑥_𝑘(𝑡)−𝑥(𝑡)+𝛼_𝑘𝑆_𝐿𝑥_𝑘(𝑡)−𝑆_𝐿𝑥(𝑡≤)1−𝛼_𝑘𝑥_𝑘(𝑡)−𝑥(𝑡)+𝛼_𝑘𝜃𝑥_𝑘(𝑡)(𝑡=()𝛼−𝑥)1−1−𝜃_𝑘𝑥_𝑘(𝑡)−𝑥(𝑡)≤𝑒^{−(1−𝜃)𝛼𝑘}𝑥_𝑘−𝑥≤𝑒^{−(1−𝜃)∑𝑘𝑝=0𝛼𝑝}𝑥₀−𝑥,∀𝑘∈ℕ₀,𝑡≥𝑇,(108) which signifies that (20) holds. It follows from (20) and (21) that lim_𝑘→∞𝑥_𝑘=𝑥.
Now we show that (b) holds. Let 𝐿₁,𝐿₂∈(𝑁,𝑀) and 𝐿₁≠𝐿₂. As in the proof of (a), we conclude that for each 𝑝∈{1,2}, there exist 𝜃_𝑝∈(0,1), 𝑇_𝑝>1+|𝑡₀|+𝜏+|𝛾| and 𝑆_𝐿𝑝∶Ω₁(𝑁,𝑀)→Ω₁(𝑁,𝑀) satisfying (91) and (103)–(105), where 𝐿, 𝜃, 𝑇, and 𝑆_𝐿 are replaced by 𝐿_𝑝, 𝜃_𝑝, 𝑇_𝑝 and 𝑆_𝐿𝑝, respectively, and 𝑆_𝐿𝑝 has a unique fixed point 𝑧_𝑝∈Ω₁(𝑁,𝑀), which is a positive solution of (5) in Ω₁(𝑁,𝑀). It follows that for any 𝑡≥𝑇_𝑝 and 𝑝∈{1,2}𝑧_𝑝(𝑡)=𝐿_𝑝+(−1)^𝑛−1(𝑛−2)!^∞_𝑗=1^+∞_{𝑡+𝑗𝜏𝑢}^+∞(𝑠−𝑢)^𝑛−2𝑎(𝑢)×𝑔(𝑠)−𝑓𝑠,𝑧_𝑝𝑓₁(𝑠),…,𝑧_𝑝𝑓_𝑙(𝑠)+()𝑑𝑠𝑑𝑢−1^𝑛−𝑖(𝑛−𝑖−2)!^∞_𝑗=1^+∞_{𝑡+𝑗𝜏𝑢}^+∞(𝑠−𝑢)^{𝑛−𝑖−2}𝑎(𝑢)×ℎ𝑠,𝑧_𝑝ℎ₁(𝑠),…,𝑧_𝑝ℎ_𝑙(𝑠)𝑑𝑠𝑑𝑢.(109) In order to prove (b), we just need to show that 𝑧₁≠𝑧₂. Notice that (16), (103), (109), and Lemma 1 ensure that 𝑧₁(𝑡)−𝑧₂(𝑡≥𝐿₁)−𝐿₂−1(𝑛−2)!^∞_𝑗=1^+∞_{𝑡+𝑗𝜏𝑢}^+∞(𝑠−𝑢)^𝑛−2|𝑎(𝑢×𝑓)|𝑠,𝑧₂𝑓₁(𝑠),…,𝑧₂𝑓_𝑙(𝑠)−𝑓𝑠,𝑧₁𝑓₁(𝑠),…,𝑧₁𝑓_𝑙(𝑠−1)𝑑𝑠𝑑𝑢(𝑛−𝑖−2)!^∞_𝑗=1^+∞_{𝑡+𝑗𝜏𝑢}^+∞(𝑠−𝑢)^{𝑛−𝑖−2}|𝑎(𝑢×ℎ)|𝑠,𝑧₁ℎ₁(𝑠),…,𝑧₁ℎ_𝑙(𝑠)−ℎ𝑠,𝑧₂ℎ₁(𝑠),…,𝑧₂ℎ_𝑙(𝑠)≥𝐿𝑑𝑠𝑑𝑢₁−𝐿₂−𝑧₁−𝑧₂𝜏(𝑛−2)!^+∞_{𝑡+𝜏𝑢}^+∞𝑢𝑠^𝑛−2|𝑎(𝑢𝑃(𝑠)−𝑧)|𝑑𝑠𝑑𝑢₁−𝑧₂𝜏(𝑛−𝑖−2)!^+∞_{𝑡+𝜏𝑢}^+∞𝑢𝑠^{𝑛−𝑖−2}|𝑎(𝑢𝑅(𝑠)≥𝐿)|𝑑𝑠𝑑𝑢₁−𝐿₂−𝑧₁−𝑧₂𝜏^+∞_{max{𝑇1,𝑇2}𝑢}^+∞𝑢|𝑎(𝑢×𝑠)|_𝑛−2()!𝑃(𝑠)+𝑠𝑛−2(^{𝑛−𝑖−2})!𝑅(𝑠>𝐿𝑛−𝑖−2)𝑑𝑠𝑑𝑢₁−𝐿₂−max𝜃₁,𝜃₂𝑧₁−𝑧₂,(110) which yields that 𝑧₁−𝑧₂≥𝐿₁−𝐿₂1+max𝜃₁,𝜃₂>0,(111) that is, 𝑧₁≠𝑧₂. This completes the proof.

3. Remark and Examples

Remark 11. Theorems 2–10 extend, improve, and unifies Theorems 1–4 in [1], the theorem in [2], Theorems 2.1–2.4 in [4], Theorems 2.1–2.5 in [5, 8], Theorems 1–3 in [9], and Theorems 1–4 in [11], respectively. The examples below prove that Theorems 2–10 extend substantially the corresponding results in [1, 2, 4, 5, 8, 9, 11]. Note that none of the known results can be applied to these examples.

Example 12. Consider the higher order nonlinear neutral delay differential equation 𝑡^𝑚+1+1𝑥(𝑡)+sin2𝑡²−cos𝑡⁵−17+2sin8𝑡³𝑥()+2𝑡−1𝑡−𝜏^{(𝑚)(𝑛−𝑚)}+𝑡²𝑥(𝑡−3)𝑥²(𝑡−4)𝑡^{𝑛−𝑚−𝑖+3}+𝑡²+1^(𝑖)+𝑡³𝑥³𝑡²−𝑡−𝑥⁴(𝑡−1)𝑡^{𝑛−𝑚+4}=+𝑡+2𝑡ln1+𝑡²−cos²𝑡²−𝑡+1𝑡^{2𝑛−𝑚+3}+1,∀𝑡≥2,(112) where 𝜏>0 and 𝑖≤𝑛−𝑚−1. Let 𝑙=2, 𝑡₀=2, 𝛾=min{2−𝜏,−2}, 𝑀=10, 𝑁=1, 𝑏₀=2/5 and ℎ₁(𝑡)=𝑡−3,ℎ₂(𝑡)𝑓=𝑡−4,₁(𝑡)=𝑡²−𝑡,𝑓₂(𝑡)𝑎(𝑡)=𝑡−1,=𝑡^𝑚+1+1,𝑏(𝑡)=sin2𝑡²−cos𝑡⁵−17+2sin8𝑡³,ℎ()=𝑡+2𝑡−1𝑡,𝑢,𝑣²𝑢𝑣²𝑡^{𝑛−𝑚−𝑖+3}+𝑡²,𝑓()=𝑡+1𝑡,𝑢,𝑣³𝑢³−𝑣⁴𝑡^{𝑛−𝑚+4},𝑔(𝑡)=+𝑡+2𝑡ln1+𝑡²−cos²𝑡²−𝑡+1𝑡^{2𝑛−𝑚+3},𝑃(𝑡)=𝑀+1₂3𝑡³+4𝑀𝑡^{𝑛−𝑚+4}(𝑡)=𝑀+𝑡+2,𝑄³𝑡³+𝑀𝑡^{𝑛−𝑚+4},𝑅(𝑡)=+𝑡+23𝑀²𝑡²𝑡^{𝑛−𝑚−𝑖+3}+𝑡²,𝑊(𝑡)=𝑀+1₃𝑡²𝑡^{𝑛−𝑚−𝑖+3}+𝑡²,∀()∈𝑡+1𝑡,𝑢,𝑣₀,+∞×[𝑁,𝑀]².(113) It is easy to verify that the conditions of Theorem 2 are satisfied. Thus Theorem 2 ensures that (112) has uncountably many positive solutions in Ω₁(1,10), and for any 𝐿∈(5,6), there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| such that the Mann iterative sequence {𝑥_𝑘}_𝑘∈ℕ0 generated by (19) and (21) converges to a positive solution 𝑥∈Ω₁(1,10) of (112) and has the error estimate (20).

Example 13. Consider the higher order nonlinear neutral delay differential equation 𝑡^𝑛+1𝑥(𝑡)+3𝑡²4𝑡²𝑥()+3𝑡−𝜏^{(𝑚)(𝑛−𝑚)}+𝑡²𝑥(𝑡ln𝑡)−(𝑡+1)𝑥(𝑡ln𝑡)𝑥√2𝑡𝑡^{𝑛+𝑚−𝑖+3(𝑖)}+2−𝑡²arctan𝑡+𝑡𝑥³𝑡³+𝑡²𝑥²𝑡²𝑡^𝑛+𝑚+3+𝑥²𝑡²=1−8𝑡³+13𝑡⁵+5𝑡⁶cos𝑡³−1𝑡^2𝑛+𝑚+4,∀𝑡≥1,(114) where 𝜏>0 and 𝑖≤𝑛−𝑚−1. Let 𝑙=2, 𝑡₀=1, 𝛾=min{1−𝜏,0}, 𝑀=6, 𝑁=1, 𝑏₀=3/4 and ℎ₁(𝑡)=𝑡ln𝑡,ℎ₂(𝑡)=√𝑓2𝑡,₁(𝑡)=𝑡³+𝑡²,𝑓₂(𝑡)=𝑡²,𝑎(𝑡)=𝑡^𝑛+1,𝑏(𝑡)=3𝑡²4𝑡²,ℎ()=𝑡+3𝑡,𝑢,𝑣²𝑢−(𝑡+1)𝑢𝑣𝑡^{𝑛+𝑚−𝑖+3},𝑓(𝑡,𝑢,𝑣)=2−𝑡²arctan𝑡+𝑡𝑢³𝑣²𝑡^𝑛+𝑚+3+𝑣²,𝑔(𝑡)=1−8𝑡³+13𝑡⁵+5𝑡⁶cos𝑡³−1𝑡^2𝑛+𝑚+4,𝑃(𝑡)=5𝑀⁴𝑡^𝑛+𝑚+4+2𝑀2+𝑡²arctan𝑡+5𝑀⁶𝑡𝑡^𝑛+𝑚+3+𝑁²²,𝑄(𝑡)=2+𝑡²arctan𝑡+𝑀⁵𝑡𝑡^𝑛+𝑚+3+𝑁²,𝑅(𝑡)=𝑡²+2𝑀(𝑡+1)𝑡^{𝑛+𝑚−𝑖+3},𝑊(𝑡)=𝑀𝑡²+𝑀²(𝑡+1)𝑡^{𝑛+𝑚−𝑖+3},∀(𝑡,𝑢,𝑣)∈𝑡₀,+∞×[𝑁,𝑀]².(115) It is easy to check that the conditions of Theorem 3 are satisfied. Therefore (114) has uncountably many positive solutions in Ω₁(1,6), and for any 𝐿∈(11/2,6), there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| such that the Mann iterative sequence {𝑥_𝑘}_𝑘∈ℕ0 generated by (19) and (21) converges to a positive solution 𝑥∈Ω₁(1,6) of (114) and has the error estimate (20).

Example 14. Consider the higher order nonlinear neutral delay differential equation (𝑡+1)𝑥(𝑡)−arctan𝑡2𝑥(𝑡−𝜏)^{(𝑚)(𝑛−𝑚)}+𝑡²𝑥𝑡³+𝑡−𝑥²𝑡³−1𝑡^{2𝑛+𝑚−𝑖+3}+𝑥²𝑡³+𝑡^(𝑖)+𝑡𝑥(𝑡−1)sin(𝑡𝑥(𝑡−sin𝑡))𝑡^2𝑛+𝑚+3+𝑡³=𝑡√+2𝑡+1sin²𝑡²+2𝑡+1𝑡^2𝑛+𝑚+3+1,∀𝑡≥0,(116) where 𝜏>0 and 𝑖≤𝑛−𝑚−1. Let 𝑙=2, 𝑡₀=0, 𝛾=min{−𝜏,−1}, 𝑀=8, 𝑁=1/2, 𝑏₀=7/8 and ℎ₁(𝑡)=𝑡³+𝑡,ℎ₂(𝑡)=𝑡³𝑓−1,₁(𝑡)=𝑡−1,𝑓₂(𝑡)𝑎(𝑡)(𝑡)1=𝑡−sin𝑡,=𝑡+1,𝑏=−2ℎ()=𝑡arctan𝑡,𝑡,𝑢,𝑣²𝑢−𝑣²𝑡^{2𝑛+𝑚−𝑖+3}+𝑢²,𝑓()=()𝑡,𝑢,𝑣𝑡𝑢sin𝑡𝑣𝑡^2𝑛+𝑚+3+𝑡³,𝑔(𝑡)=𝑡√+2𝑡+1sin²𝑡²+2𝑡+1𝑡^2𝑛+𝑚+3,𝑃(𝑡)=+1𝑀𝑡²+𝑡𝑡^2𝑛+𝑚+3+𝑡³(𝑡)=+2,𝑄𝑀𝑡𝑡^2𝑛+𝑚+3+𝑡³,𝑅(𝑡)=𝑡+2_{2𝑛+𝑚−𝑖+5}+𝑀²𝑡²+2𝑀𝑡^{2𝑛+𝑚−𝑖+3}+4𝑀³𝑡^{2𝑛+𝑚−𝑖+3}+𝑁²²,𝑊(𝑡)=𝑀𝑡²+𝑀²𝑡^{2𝑛+𝑚−𝑖+3}+𝑁²,∀(𝑡,𝑢,𝑣)∈𝑡₀,+∞×[𝑁,𝑀]².(117) It is easy to prove that the conditions of Theorem 4 are satisfied. Hence (116) has uncountably many positive solutions in Ω₁(1/2,8), and for any 𝐿∈(1/2,1), there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| such that the Mann iterative sequence {𝑥_𝑘}_𝑘∈ℕ0 generated by (19) and (21) converges to a positive solution 𝑥∈Ω₁(1/2,8) of (116) and has the error estimate (20).

Example 15. Consider the higher order nonlinear neutral delay differential equation 𝑡^2𝑖+1𝑥(𝑡)+2^𝑡2+1𝑥(𝑡−𝜏)^{(𝑚)(𝑛−𝑚)}+√𝑡+1𝑥²(𝑡−1)𝑥𝑡²𝑡^𝑛+𝑖+1+𝑡ln(1+|𝑥(𝑡−2)|)+4^(𝑖)+𝑡²𝑥(𝑡−12)𝑥𝑡²−9𝑡^2𝑛+3|𝑥(=𝑡+𝑡𝑡−3)|+3₂cos(2𝑡)+arctan𝑡³𝑡^𝑛+4+sin²1−𝑡³+𝑡⁴+1,∀𝑡≥3,(118) where 𝜏>0 and 𝑖≤𝑛−𝑚−1. Let 𝑙=3, 𝑡₀=0, 𝛾=min{3−𝜏,−9}, 𝑀=12, 𝑁=5, 𝑏₀=2 and ℎ₁(𝑡)=𝑡−1,ℎ₂(𝑡)=𝑡²,ℎ₃(𝑡)𝑓=𝑡−2,₁(𝑡)=𝑡−12,𝑓₂(𝑡)=𝑡²−9,𝑓₃(𝑡)𝑎(𝑡)=𝑡−3,=𝑡^2𝑖+1,𝑏(𝑡)=2^𝑡2+1,ℎ()=√𝑡,𝑢,𝑣,𝑤𝑡+1𝑢²𝑣𝑡^𝑛+𝑖+1(|𝑤,𝑔(𝑡)=𝑡+𝑡ln1+|)+4₂cos(2𝑡)+arctan𝑡³𝑡^𝑛+4+sin²1−𝑡³+𝑡⁴,𝑓()=𝑡+1𝑡,𝑢,𝑣,𝑤²𝑢𝑣𝑡^2𝑛+3|𝑤|,𝑃(𝑡)=+𝑡+3𝑀𝑡²6𝑏₀+3𝑀²𝑡+2𝑏₀𝑡^2𝑛+3𝑏²₀𝑡^2𝑛+3+3²,𝑄(𝑡)=𝑀²𝑡²𝑏²₀𝑡^2𝑛+3,𝑅(𝑡)=𝑀+3₂√𝑡+1𝑏²₀𝑡^𝑛+𝑖+1+4²×3𝑡^𝑛+𝑖+1𝑀+12+𝑏₀𝑀𝑡+3𝑡ln1+𝑏₀,𝑊(𝑡)=𝑀³√𝑡+1𝑏³₀𝑡^𝑛+𝑖+1,∀()∈𝑡+4𝑡,𝑢,𝑣,𝑤₀×𝑀,+∞0,𝑏₀³.(119) It is easy to verify that the conditions of Theorem 5 are satisfied. Hence Theorem 5 ensures that (118) has uncountably many positive solutions in Ω₂(5,12), and, for any 𝐿∈(11,12), there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| such that the Mann iterative sequence {𝑥_𝑘}_𝑘∈ℕ0 generated by (48) and (21) converges to a positive solution 𝑥∈Ω₂(5,12) of (118) and has the error estimate (20).

Example 16. Consider the higher order nonlinear neutral delay differential equation 𝑡^3𝑛+2𝑡^𝑛+1+1𝑥(𝑡)−𝑡²+2𝑡+4𝑥(𝑡−𝜏)^{(𝑚)(𝑛−𝑚)}+𝑡𝑥(𝑡−3)𝑥²(𝑡−4)𝑡^𝑛+10+𝑥²(𝑡−3)^(𝑖)+√𝑡+1𝑥³(𝑡ln𝑡)−𝑡𝑥⁴𝑡²−𝑡(𝑡+1)^𝑛+3+𝑡𝑥²()=𝑡ln𝑡𝑡ln1+𝑡²−√𝑡+3cos³𝑡³+1𝑡^𝑛+5+4𝑡³−1−𝑡sin⁵𝑡²−3,∀𝑡≥1,(120) where 𝜏>0, and 𝑖≤𝑛−𝑚−1. Let 𝑙=2, 𝑡₀=1, 𝛾=min{1−𝜏,−3}, 𝑀=6, 𝑁=2, 𝑏₀=3 and ℎ₁(𝑡)=𝑡−3,ℎ₂(𝑡)𝑓=𝑡−4,₁(𝑡)=𝑡ln𝑡,𝑓₂(𝑡)=𝑡²𝑎(𝑡)−𝑡,=𝑡^3𝑛+2𝑡^𝑛+1+1,𝑏(𝑡)=−𝑡²ℎ()=−2𝑡−4,𝑡,𝑢,𝑣𝑡𝑢𝑣²𝑡^𝑛+10+𝑢²,𝑓()=√𝑡,𝑢,𝑣𝑡+1𝑢³−𝑡𝑣⁴(𝑡+1)^𝑛+3+𝑡𝑢²,𝑔(𝑡)=𝑡ln1+𝑡²−√𝑡+3cos³𝑡³+1𝑡^𝑛+5+4𝑡³−1−𝑡sin⁵𝑡²,𝑃(𝑡)=𝑀−3₂𝑏²₀(𝑡+1)^2𝑛+6×3(𝑡+1)^𝑛+7/2+𝑀²𝑏²₀𝑡√𝑀𝑡+1+4𝑏₀𝑡(𝑡+1)^𝑛+3𝑀+6³𝑏³₀𝑡²,𝑄(𝑡)=𝑀³𝑏³₀(𝑡+1)^𝑛+8√𝑡+1+𝑀𝑏₀𝑡,𝑅(𝑡)=3𝑀²𝑏²₀𝑡^2𝑛+19𝑡^𝑛+10+𝑀²𝑏²₀,𝑊(𝑡)=𝑀³𝑏³₀𝑡^𝑛+9,∀(𝑡,𝑢,𝑣)∈𝑡₀×𝑀,+∞0,𝑏₀².(121) It is easy to check that the conditions of Theorem 6 are satisfied. Thus Theorem 6 ensures that (120) has uncountably many positive solutions in Ω₃(2,6), and, for any 𝐿∈(2,4), there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| such that the Mann iterative sequence {𝑥_𝑘}_𝑘∈ℕ0 generated by (61) and (21) converges to a positive solution 𝑥∈Ω₃(2,6) of (120) and has the error estimate (20).

Example 17. Consider the higher order nonlinear neutral delay differential equation 𝑡²ln𝑡(𝑥(𝑡)+𝑥(𝑡−𝜏))^{(𝑚)(𝑛−𝑚)}+𝑥√𝑡−2+𝑥(2𝑡−1)𝑡^{3𝑛−𝑚−𝑖+2}+𝑥²(𝑡−cos𝑡)^(𝑖)+𝑥²(𝑡−4)+𝑥√𝑡−1𝑥²𝑡−sin𝑡⁹+1𝑡^{2𝑛−𝑚+3}=+1sin¹³𝑡⁵−√𝑡+1𝑡^𝑛+7/2+1,∀𝑡≥4,(122) where 𝜏>0, 𝑚≥2 and 𝑖≤𝑛−𝑚−1. Let 𝑙=3, 𝑡₀=4, 𝛾=min{4−𝜏,0}, 𝑀=100, 𝑁=1 and ℎ₁(𝑡)=√𝑡−2,ℎ₂(𝑡)=2𝑡−1,ℎ₃(𝑡)𝑓=𝑡−cos𝑡,₁(𝑡)=𝑡−4,𝑓₂(𝑡)=√𝑡−1,𝑓₃(𝑡)=𝑡−sin𝑡⁹,𝑎(𝑡)+1=𝑡²(𝑡)ℎ()=ln𝑡,𝑏=1,𝑡,𝑢,𝑣,𝑤𝑢+𝑣𝑡^{3𝑛−𝑚−𝑖+2}+𝑤²,𝑓()=𝑢𝑡,𝑢,𝑣,𝑤²+𝑣𝑤²𝑡^{2𝑛−𝑚+3},𝑔(𝑡)=+1sin¹³𝑡⁵−√𝑡+1𝑡^𝑛+7/2(𝑡)=+1,𝑃2𝑀+3𝑀²𝑡^{2𝑛−𝑚+3},𝑄(𝑡)=𝑀+1₂+𝑀³𝑡^{2𝑛−𝑚+3}(𝑡)=+1,𝑅2𝑡^{3𝑛−𝑚−𝑖+2}+6𝑀²𝑡^{3𝑛−𝑚−𝑖+2}+𝑁²²,𝑊(𝑡)=2𝑀𝑡^{3𝑛−𝑚−𝑖+2}+𝑁²,∀(𝑡,𝑢,𝑣,𝑤)∈𝑡₀,+∞×[𝑁,𝑀]³.(123) It is easy to check that the conditions of Theorem 7 are satisfied. Thus (122) has uncountably many positive solutions in Ω₁(1,100), and, for any 𝐿∈(1,100), there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| such that the Mann iterative sequence {𝑥_𝑘}_𝑘∈ℕ0 generated by (66) and (21) converges to a positive solution 𝑥∈Ω₁(1,100) of (122) and has the error estimate (20).

Example 18. Consider the higher order nonlinear neutral delay differential equation 2+sin𝑡+√𝑡𝑡²(𝑥(𝑡)+𝑥(𝑡−𝜏))^{′(𝑛−1)}+𝑥³(𝑡−2)−𝑡²𝑥⁴√𝑡+1−1𝑡^5𝑛+3+𝑡+1^(𝑖)+𝑥³(𝑡−3)−𝑡𝑡^3𝑛+4+𝑥²𝑡−(−1)^𝑛=sin𝑡⁴−√𝑡²+1𝑡^𝑛+3+ln𝑡,∀𝑡≥3,(124) where 𝜏>0, 𝑚=1 and 𝑖≤𝑛−2. Let 𝑙=2, 𝑡₀=3, 𝛾=min{3−𝜏,0}, 𝑀=10, 𝑁=9 and ℎ₁(𝑡)=𝑡−2,ℎ₂(𝑡)=√𝑓𝑡+1−1,₁(𝑡)=𝑡−3,𝑓₂(𝑡)=𝑡−(−1)^𝑛,𝑎(𝑡)=2+sin𝑡+√𝑡𝑡²(𝑡)ℎ()=𝑢,𝑏=1,𝑡,𝑢,𝑣³−𝑡²𝑣⁴𝑡^5𝑛+3,𝑓()=𝑢+𝑡+1𝑡,𝑢,𝑣³−𝑡𝑡^3𝑛+4+𝑣²,𝑔(𝑡)=sin𝑡⁴−√𝑡²+1𝑡^𝑛+3,𝑃(𝑡)=𝑀+ln𝑡5𝑀³+2𝑡+2𝑀𝑡^3𝑛+4𝑡^3𝑛+4+𝑁²²,𝑄(𝑡)=𝑀³+𝑡𝑡^3𝑛+4+𝑁²,𝑅(𝑡)=𝑀²3+4𝑀𝑡²𝑡^5𝑛+3(𝑡)=𝑀+𝑡+1,𝑊³1+𝑀𝑡²𝑡^5𝑛+3,∀()∈𝑡+𝑡+1𝑡,𝑢,𝑣₀,+∞×[𝑁,𝑀]².(125) It is easy to check that the conditions of Theorem 8 are satisfied. Thus Theorem 8 ensures that (124) has uncountably many positive solutions in Ω₁(9,10), and, for any 𝐿∈(9,10), there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| such that the Mann iterative sequence {𝑥_𝑘}_𝑘∈ℕ0 generated by (78) and (21) converges to a positive solution 𝑥∈Ω₁(9,10) of (124) and has the error estimate (20).

Example 19. Consider the higher order nonlinear neutral delay differential equation 𝑡^𝑚+1ln4+sin𝑡²−√𝑡(𝑥(𝑡)−𝑥(𝑡−𝜏))^{(𝑚)(𝑛−𝑚)}+𝑡𝑡^{𝑛−𝑚+4}+𝑥⁴𝑡−√𝑡^(𝑖)+sin𝑡³−2𝑡+√𝑡³+1𝑡^𝑛+𝑚+|(=𝑡−𝑥𝑡−3)|𝑡cos⁵𝑡⁷−𝑡⁴+1𝑡^𝑛+2𝑡−cos³𝑡²−3,∀𝑡≥4,(126) where 𝜏>0, 𝑚≥2 and 𝑖≤𝑛−𝑚−1. Let 𝑙=1, 𝑡₀=4, 𝛾=min{4−𝜏,1}, 𝑀=7, 𝑁=5 and ℎ₁(𝑡)=𝑡−√𝑡,𝑓₁(𝑡)𝑎(𝑡)=𝑡−3,=𝑡^𝑚+1ln4+sin𝑡²−√𝑡,𝑏(𝑡)()=𝑡=−1,ℎ𝑡,𝑢𝑡^{𝑛−𝑚+4}+𝑢⁴,𝑓(𝑡,𝑢)=sin𝑡³−2𝑡+√𝑡³+1𝑡^𝑛+𝑚+||,𝑔(𝑡)=𝑡−𝑢𝑡cos⁵𝑡⁷−𝑡⁴+1𝑡^𝑛+2𝑡−cos³𝑡²,𝑃(𝑡)=1−3𝑡^2𝑛+2𝑚(𝑡)=1,𝑄𝑡^𝑛+𝑚,𝑅(𝑡)=4𝑀³𝑡^{2𝑛−2𝑚+7}(𝑡)=1,𝑊𝑡^{𝑛−𝑚+3},∀(𝑡,𝑢)∈𝑡₀,+∞×[𝑁,𝑀]².(127) It is easy to check that the conditions of Theorem 9 are satisfied. Thus Theorem 9 ensures that (126) has uncountably many positive solutions in Ω₁(5,7), and, for any 𝐿∈(5,7), there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| such that the Mann iterative sequence {𝑥_𝑘}_𝑘∈ℕ0 generated by (90) and (21) converges to a positive solution 𝑥∈Ω₁(5,7) of (126) and has the error estimate (20).

Example 20. Consider the higher order nonlinear neutral delay differential equation 1𝑡³(𝑥(𝑡)−𝑥(𝑡−𝜏))^{′(𝑛−1)}+𝑡−sin𝑡⁸−4𝑡⁵−1𝑡^𝑛+7+𝑥(𝑡−1)−𝑥³(𝑡−2)^(𝑖)+ln1+𝑥²(𝑡−arctan𝑡)𝑡^2𝑛+6+𝑥²()=𝑡𝑡−4𝑡ln𝑡+cos³−1𝑡^𝑛+8+1,∀𝑡≥5,(128) where 𝜏>0, 𝑚=1 and 𝑖≤𝑛−2. Let 𝑙=2, 𝑡₀=5, 𝛾=min{5−𝜏,1}, 𝑀=4, 𝑁=2 and ℎ₁(𝑡)=𝑡−1,ℎ₂(𝑡)=𝑡−2,𝑓₁(𝑡)𝑓=𝑡−arctan𝑡,₂(𝑡)(𝑡)=1=𝑡−4,𝑎𝑡³(𝑡)ℎ()=𝑡,𝑏=−1,𝑡,𝑢,𝑣𝑡−sin⁸−4𝑡⁵−1𝑡^𝑛+7+𝑢−𝑣³,𝑓(𝑡,𝑢,𝑣)=ln1+𝑢²𝑡^2𝑛+6+𝑣²(𝑡)=𝑡,𝑔𝑡ln𝑡+cos³−1𝑡^𝑛+8,𝑃(𝑡)=+12𝑀2𝑀²+𝑡^2𝑛+6𝑡^4𝑛+12(𝑡)=𝑀,𝑄²𝑡^2𝑛+6,𝑅(𝑡)=2+6𝑀²𝑡^2𝑛+13(𝑡)=2,𝑊𝑡^𝑛+6,∀(𝑡,𝑢,𝑣)∈𝑡₀,+∞×[𝑁,𝑀]².(129) It is easy to check that the conditions of Theorem 10 are satisfied. Thus Theorem 10 ensures that (128) has uncountably many positive solutions in Ω₁(2,4), and, for any 𝐿∈(2,4), there exist 𝜃∈(0,1) and 𝑇>1+|𝑡₀|+𝜏+|𝛾| such that the Mann iterative sequence {𝑥_𝑘}_𝑘∈ℕ0 generated by (102) and (21) converges to a positive solution 𝑥∈Ω₁(2,4) of (128) and has the error estimate (20).

Acknowledgments

The authors would like to thank the referees for useful comments and suggestions. This paper was supported by the Science Research Foundation of Educational Department of Liaoning province (L2012380).