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∈ℕ