Solvability of Nonlinear Integral Equations of Volterra Type

Abstract

This paper deals with the existence of continuous bounded solutions for a rather general nonlinear integral equation of Volterra type and discusses also the existence and asymptotic stability of continuous bounded solutions for another nonlinear integral equation of Volterra type. The main tools used in the proofs are some techniques in analysis and the Darbo fixed point theorem via measures of noncompactness. The results obtained in this paper extend and improve essentially some known results in the recent literature. Two nontrivial examples that explain the generalizations and applications of our results are also included.

1. Introduction

It is well known that the theory of nonlinear integral equations and inclusions has become important in some mathematical models of real processes and phenomena studied in mathematical physics, elasticity, engineering, biology, queuing theory economics, and so on (see, [1–3]). In the last decade, the existence, asymptotical stability, and global asymptotical stability of solutions for various Volterra integral equations have received much attention, see, for instance, [1, 4–22] and the references therein.

In this paper, we are interested in the following nonlinear integral equations of Volterra type: 𝑥(𝑡)=𝑓𝑡,𝑥(𝑎(𝑡)),(𝐻𝑥)(𝑏(𝑡)),₀^𝛼(𝑡)𝑢(𝑡,𝑠,𝑥(𝑐(𝑠)))𝑑𝑠,∀𝑡∈ℝ₊,𝑥(𝑡)=ℎ𝑡,𝑥(𝑡),₀^𝛼(𝑡)𝑢(𝑡,𝑠,𝑥(𝑐(𝑠)))𝑑𝑠,∀𝑡∈ℝ₊, (1.1)(1.2) where the functions 𝑓,ℎ,𝑢,𝑎,𝑏,𝑐,𝛼 and the operator 𝐻 appearing in (1.1) are given while 𝑥=𝑥(𝑡) is an unknown function.

To the best of our knowledge, the papers dealing with (1.1) and (1.2) are few. But some special cases of (1.1) and (1.2) have been investigated by a lot of authors. For example, Arias et al. [4] studied the existence, uniqueness, and attractive behaviour of solutions for the nonlinear Volterra integral equation with nonconvolution kernels 𝑥(𝑡)=^𝑡₀𝑘(𝑡,𝑠)𝑔(𝑥(𝑠))𝑑𝑠,∀𝑡∈ℝ₊. (1.3)

Using the monotone iterative technique, Constantin [13] got a sufficient condition which ensures the existence of positive solutions of the nonlinear integral equation 𝑥(𝑡)=𝐿(𝑡)+^𝑡₀𝑀(𝑡,𝑠)𝑥(𝑠)+𝐾(𝑡,𝑠)𝑔(𝑥(𝑠))𝑑𝑠,∀𝑡∈ℝ₊. (1.4)

Roberts [21] examined the below nonlinear Volterra integral equation 𝑥(𝑡)=^𝑡₀𝑘(𝑡−𝑠)𝐺((𝑥(𝑠),𝑠))𝑑𝑠,∀𝑡∈ℝ₊, (1.5) which arose from certain models of a diffusive medium that can experience explosive behavior; utilizing the Darbo fixed point theorem and the measure of noncompactness in [7], Banaś and Dhage [6], Banaś et al. [8], Banaś and Rzepka [9, 10], Hu and Yan [16] and Liu and Kang [19] investigated the existence and/or asymptotic stability and/or global asymptotic stability of solutions for the below class of integral equations of Volterra type: 𝑥(𝑡)=(𝑇𝑥)(𝑡)^𝑡₀𝑢(𝑡,𝑠,𝑥(𝑠))𝑑𝑠,∀𝑡∈ℝ₊,𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡))+^𝑡₀𝑢(𝑡,𝑠,𝑥(𝑠))𝑑𝑠,∀𝑡∈ℝ₊,𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡))^𝑡₀𝑢(𝑡,𝑠,𝑥(𝑠))𝑑𝑠,∀𝑡∈ℝ₊,𝑥(𝑡)=𝑔(𝑡,𝑥(𝑡))+𝑥(𝑡)^𝑡₀𝑢(𝑡,𝑠,𝑥(𝑠))𝑑𝑠,∀𝑡∈ℝ₊,𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡))+𝑔(𝑡,𝑥(𝑡))^𝑡₀𝑢(𝑡,𝑠,𝑥(𝑠))𝑑𝑠,∀𝑡∈ℝ₊,𝑥(𝑡)=𝑓(𝑡,𝑥(𝛼(𝑡)))+₀^𝛽(𝑡)𝑔𝑡,𝑠,𝑥𝛾(𝑠)𝑑𝑠,∀𝑡∈ℝ₊, (1.6) respectively. By means of the Schauder fixed point theorem and the measure of noncompactness in [7], Banaś and Rzepka [11] studied the existence of solutions for the below nonlinear quadratic Volterra integral equation: 𝑥(𝑡)=𝑝(𝑡)+𝑓(𝑡,𝑥(𝑡))^𝑡₀𝑣(𝑡,𝑠,𝑥(𝑠))𝑑𝑠,∀𝑡∈ℝ₊. (1.7)

Banaś and Chlebowicz [5] got the solvability of the following functional integral equation 𝑥(𝑡)=𝑓₁𝑡,^𝑡₀𝑘(𝑡,𝑠)𝑓₂(𝑠,𝑥(𝑠))𝑑𝑠,∀𝑡∈ℝ₊ (1.8) in the space of Lebesgue integrable functions on ℝ₊. El-Sayed [15] studied a differential equation of neutral type with deviated argument, which is equivalent to the functional-integral equation 𝑥(𝑡)=𝑓𝑡,₀^𝐻(𝑡)𝑥(𝑠)𝑑𝑠,𝑥(ℎ(𝑡)),∀𝑡∈ℝ₊ (1.9) by the technique linking measures of noncompactness with the classical Schauder fixed point principle. Using an improvement of the Krasnosel’skii type fixed point theorem, Taoudi [22] discussed the existence of integrable solutions of a generalized functional-integral equation 𝑥(𝑡)=𝑔(𝑡,𝑥(𝑡))+𝑓₁𝑡,^𝑡₀𝑘(𝑡,𝑠)𝑓₂(𝑠,𝑥(𝑠))𝑑𝑠,∀𝑡∈ℝ₊. (1.10)

Dhage [14] used the classical hybrid fixed point theorem to establish the uniform local asymptotic stability of solutions for the nonlinear quadratic functional integral equation of mixed type 𝑥(𝑡)=𝑓(𝑡,𝑥(𝛼(𝑡)))𝑞(𝑡)+₀^𝛽(𝑡)𝑢𝑡,𝑠,𝑥𝛾(𝑠)𝑑𝑠,∀𝑡∈ℝ₊. (1.11)

The purpose of this paper is to prove the existence of continuous bounded solutions for (1.1) and to discuss the existence and asymptotic stability of continuous bounded solutions for (1.2). The main tool used in our considerations is the technique of measures of noncompactness [7] and the famous fixed point theorem of Darbo [23]. The results presented in this paper extend proper the corresponding results in [6, 9, 10, 15, 16, 19]. Two nontrivial examples which show the importance and the applicability of our results are also included.

This paper is organized as follows. In the second section, we recall some definitions and preliminary results and prove a few lemmas, which will be used in our investigations. In the third section, we state and prove our main results involving the existence and asymptotic stability of solutions for (1.1) and (1.2). In the final section, we construct two nontrivial examples for explaining our results, from which one can see that the results obtained in this paper extend proper several ones obtained earlier in a lot of papers.

2. Preliminaries

In this section, we give a collection of auxiliary facts which will be needed further on. Let ℝ=(−∞,∞) and ℝ₊=[0,∞). Assume that (𝐸,‖⋅‖) is an infinite dimensional Banach space with zero element 𝜃 and 𝐵_𝑟 stands for the closed ball centered at 𝜃 and with radius 𝑟. Let 𝐵(𝐸) denote the family of all nonempty bounded subsets of 𝐸.

Definition 2.1. Let 𝐷 be a nonempty bounded closed convex subset of the space 𝐸. A operator 𝑓∶𝐷→𝐸 is said to be a Darbo operator if it is continuous and satisfies that 𝜇(𝑓𝐴)≤𝑘𝜇(𝐴) for each nonempty subset 𝐴 of 𝐷, where 𝑘∈[0,1) is a constant and 𝜇 is a measure of noncompactness on 𝐵(𝐸).

The Darbo fixed point theorem is as follows.

Lemma 2.2 (see [23]). Let 𝐷 be a nonempty bounded closed convex subset of the space 𝐸 and let 𝑓∶𝐷→𝐷 be a Darbo operator. Then 𝑓 has at least one fixed point in 𝐷.

Let 𝐵𝐶(ℝ₊) denote the Banach space of all bounded and continuous functions 𝑥∶ℝ₊→ℝ equipped with the standard norm ‖𝑥‖=sup|𝑥(𝑡)|∶𝑡∈ℝ₊. (2.1)

For any nonempty bounded subset 𝑋 of 𝐵𝐶(ℝ₊),𝑥∈𝑋,𝑡∈ℝ₊,𝑇>0 and 𝜀≥0, define 𝜔^𝑇()𝑥𝑝𝑞[],𝜔𝑥,𝜀=sup−𝑥∶𝑝,𝑞∈0,𝑇with𝑝−𝑞≤𝜀_𝑇(𝑋,𝜀)=sup𝜔^𝑇(𝑥,𝜀)∶𝑥∈𝑋,𝜔^𝑇₀(𝑋)=lim_𝜀→0𝜔^𝑇(),𝜔𝑋,𝜀₀(𝑋)=lim_𝑇→+∞𝜔^𝑇₀(𝑋)(𝑡)={𝑥(𝑡)},(𝑡)𝑥(𝑡)(𝑡,𝜇(𝑋),𝑋∶𝑥∈𝑋diam𝑋=sup−𝑦)∶𝑥,𝑦∈𝑋=𝜔₀(𝑋)+limsup_𝑡→+∞diam𝑋(𝑡). (2.2)

It can be shown that the mapping 𝜇 is a measure of noncompactness in the space 𝐵𝐶(ℝ₊) [4].

Definition 2.3. Solutions of an integral equation are said to be asymptotically stable if there exists a ball 𝐵_𝑟 in the space 𝐵𝐶(ℝ₊) such that for any 𝜀>0, there exists 𝑇>0 with 𝑥(𝑡)−𝑦(𝑡)≤𝜀 (2.3) for all solutions 𝑥(𝑡),𝑦(𝑡)∈𝐵_𝑟 of the integral equation and any 𝑡≥𝑇.
It is clear that the concept of asymptotic stability of solutions is equivalent to the concept of uniform local attractivity [9].

Lemma 2.4. Let 𝜑 and 𝜓∶ℝ₊→ℝ₊ be functions with lim_𝑡→+∞𝜓(𝑡)=+∞,limsup_𝑡→+∞𝜑(𝑡)<+∞. (2.4)
Then limsup_𝑡→+∞𝜑𝜓(𝑡)=limsup_𝑡→+∞𝜑(𝑡). (2.5)

Proof. Let limsup_𝑡→+∞𝜑(𝑡)=𝐴. It follows that for each 𝜀>0, there exists 𝑇>0 such that 𝜑(𝑡)<𝐴+𝜀,∀𝑡≥𝑇. (2.6) Equation (2.4) means that there exists 𝐶>0 satisfying 𝜓(𝑡)≥𝑇,∀𝑡≥𝐶. (2.7)
Using (2.6) and (2.7), we infer that 𝜑𝜓(𝑡)<𝐴+𝜀,∀𝑡≥𝐶, (2.8) that is, (2.5) holds. This completes the proof.

Lemma 2.5. Let 𝑎∶ℝ₊→ℝ₊ be a differential function. If for each 𝑇>0, there exists a positive number 𝑎_𝑇 satisfying 0≤𝑎^′(𝑡)≤𝑎_𝑇,∀𝑡∈[0,𝑇], (2.9) then 𝜔^𝑇(𝑥∘𝑎,𝜀)≤𝜔^𝑎(𝑇)𝑥,𝑎_𝑇𝜀,∀(𝑥,𝜀)∈𝐵𝐶𝑅₊×(0,+∞). (2.10)

Proof. Let 𝑇>0. It is clear that (2.9) yields that the function 𝑎 is nondecreasing in [0,𝑇] and for any 𝑡,𝑠∈[0,𝑇], there exists 𝜉∈(0,𝑇) satisfying |𝑎(𝑡)−𝑎(𝑠)|=𝑎^′(𝜉)|𝑡−𝑠|≤𝑎_𝑇|𝑡−𝑠| (2.11) by the mean value theorem. Notice that (2.9) means that 𝑎(𝑡)∈[𝑎(0),𝑎(𝑇)]⊆[0,𝑎(𝑇)] for each 𝑡∈[0,𝑇], which together with (2.11) gives that 𝜔^𝑇()𝑥(𝑎(𝑡(𝑎(𝑠[],||}𝑥𝑝𝑞[𝑎(0)(𝑇,≤𝑥∘𝑎,𝜀=sup{|))−𝑥))|∶𝑡,𝑠∈0,𝑇𝑡−𝑠≤𝜀≤sup−𝑥∶𝑝,𝑞∈,𝑎)]𝑝−𝑞𝑎_𝑇𝜀≤𝜔^𝑎(𝑇)𝑥,𝑎_𝑇𝜀,∀(𝑥,𝜀)∈𝐵𝐶𝑅₊×(0,+∞), (2.12) which yields that (2.10) holds. This completes the proof.

Lemma 2.6. Let 𝜑∶ℝ₊→ℝ₊ be a function with lim_𝑡→+∞𝜑(𝑡)=+∞ and 𝑋 be a nonempty bounded subset of 𝐵𝐶(ℝ₊). Then 𝜔₀(𝑋)=lim_𝑇→+∞𝜔₀^𝜑(𝑇)(𝑋). (2.13)

Proof. Since 𝑋 is a nonempty bounded subset of 𝐵𝐶(ℝ₊), it follows that 𝜔₀(𝑋)=lim_𝑇→+∞𝜔^𝑇₀(𝑋). That is, for given 𝜀>0, there exists 𝑀>0 satisfying 𝜔^𝑇₀(𝑋)−𝜔₀(𝑋)<𝜀,∀𝑇>𝑀. (2.14)
It follows from lim_𝑡→+∞𝜑(𝑡)=+∞ that there exists 𝐿>0 satisfying 𝜑(𝑇)>𝑀,∀𝑇>𝐿. (2.15)
By means of (2.14) and (2.15), we get that 𝜔₀^𝜑(𝑇)(𝑋)−𝜔₀(𝑋)<𝜀,∀𝑇>𝐿, (2.16) which yields (2.13). This completes the proof.

3. Main Results

Now we formulate the assumptions under which (1.1) will be investigated.(H1)𝑓∶ℝ₊×ℝ³→ℝ is continuous with 𝑓(𝑡,0,0,0)∈𝐵𝐶(ℝ₊) and 𝑓=sup{|𝑓(𝑡,0,0,0)|∶ 𝑡∈ℝ₊};(H2)𝑎,𝑏,𝑐,𝛼∶ℝ₊→ℝ₊ satisfy that 𝑎 and 𝑏 have nonnegative and bounded derivative in the interval [0,𝑇] for each 𝑇>0,𝑐 and 𝛼 are continuous and 𝛼 is nondecreasing and lim_𝑡→+∞𝑎(𝑡)=lim_𝑡→+∞𝑏(t)=+∞; (3.1)(H3)𝑢∶ℝ²₊×ℝ→ℝ is continuous;(H4) there exist five positive constants 𝑟,𝑀,𝑀₀,𝑀₁, and 𝑀₂ and four continuous functions 𝑚₁, 𝑚₂,𝑚₃,𝑔∶ℝ₊→ℝ₊ such that 𝑔 is nondecreasing and 𝑓(𝑡,𝑣,𝑤,𝑧)−𝑓𝑡,𝑝,𝑞,𝑦≤𝑚₁(𝑡)𝑣−𝑝+𝑚₂(𝑡)𝑤−𝑞+𝑚₃(𝑡)𝑧−𝑦,∀𝑡∈ℝ₊,𝑣,𝑝∈[−𝑟,𝑟],𝑤,𝑞∈−𝑔(𝑟),𝑔(𝑟),𝑧,𝑦∈[−𝑀,𝑀],lim_𝑡→+∞sup𝑚₃(𝑡)₀^𝛼(𝑡)|𝑢(𝑡,𝑠,𝑣(𝑐(𝑠)))−𝑢(𝑡,𝑠,𝑤(𝑐(𝑠)))|𝑑𝑠∶𝑣,𝑤∈𝐵_𝑟𝑚=0,sup₃(𝑡)₀^𝛼(𝑡)|𝑢(𝑡,𝑠,𝑣(𝑐(𝑠)))|𝑑𝑠∶𝑡∈ℝ₊,𝑣∈𝐵_𝑟≤𝑀₀,sup₀^𝛼(𝑡)|𝑢(𝑡,𝑠,𝑣(𝑐(𝑠)))|𝑑𝑠∶𝑡∈ℝ₊,𝑣∈𝐵_𝑟𝑚≤𝑀,sup_𝑖(𝑡)∶𝑡∈ℝ₊≤𝑀_𝑖{},𝑀,𝑖∈1,2₁𝑟+𝑀₂𝑔(𝑟)+𝑀₀+𝑓≤𝑟,𝑀₁+𝑄𝑀₂<1; (3.2)(3.3)(3.4)(3.5)(3.6)(3.7)(H5)𝐻∶𝐵𝐶(ℝ₊)→𝐵𝐶(𝑅₊) satisfies that 𝐻∶𝐵_𝑟→𝐵𝐶(ℝ₊) is a Darbo operator with respect to the measure of noncompactness of 𝜇 with a constant 𝑄 and |(𝐻𝑥)(𝑏(𝑡))|≤𝑔(|𝑥(𝑏(𝑡))|),∀(𝑥,𝑡)∈𝐵_𝑟×ℝ₊. (3.8)

Theorem 3.1. Under Assumptions (H1)−(H5), (1.1) has at least one solution 𝑥=𝑥(𝑡)∈𝐵_𝑟.

Proof. Let 𝑥∈𝐵_𝑟 and define (𝐹𝑥)(𝑡)=𝑓𝑡,𝑥(𝑎(𝑡)),(𝐻𝑥)(𝑏(𝑡)),₀^𝛼(𝑡)𝑢(𝑡,𝑠,𝑥(𝑐(𝑠)))𝑑𝑠,∀𝑡∈ℝ₊. (3.9)
It follows from (3.9) and Assumptions (H1)−(H5) that 𝐹𝑥 is continuous on ℝ₊ and that |(𝐹𝑥)(𝑡)|≤𝑓𝑡,𝑥(𝑎(𝑡)),(𝐻𝑥)(𝑏(𝑡)),₀^𝛼(𝑡)𝑢(𝑡,𝑠,𝑥(𝑐(𝑠)))𝑑𝑠−𝑓(𝑡,0,0,0)+𝑓(𝑡,0,0,0)≤𝑚₁(𝑡)|𝑥(𝑎(𝑡))|+𝑚₂(𝑡)|(𝐻𝑥)(𝑏(𝑡))|+𝑚₃(𝑡)₀^𝛼(𝑡)𝑢(𝑡,𝑠,𝑥(𝑐(𝑠)))𝑑𝑠+𝑓≤𝑀₁𝑟+𝑀₂𝑔(𝑟)+𝑀₀+𝑓≤𝑟,∀𝑡∈ℝ₊, (3.10) which means that 𝐹𝑥 is bounded on ℝ₊ and 𝐹(𝐵_𝑟)⊆𝐵_𝑟.
We now prove that 𝜇(𝐹𝑋)≤𝑀₁+𝑀₂𝑄𝜇(𝑋),∀𝑋⊆𝐵_𝑟. (3.11)
Let 𝑋 be a nonempty subset of 𝐵_𝑟. Using (3.2), (3.6), and (3.9), we conclude that 𝑡)−(𝑡=𝑓(𝑎(𝑡,(𝑏(𝑡(𝐹𝑥)(𝐹𝑦)𝑡,𝑥))𝐻𝑥)()),₀^𝛼(𝑡)𝑢((𝑐(𝑠(𝑎(𝑡,𝑏(𝑡,𝑡,𝑠,𝑥)))𝑑𝑠−𝑓𝑡,𝑦))𝐻𝑦())₀^𝛼(𝑡)𝑢𝑡,𝑠,𝑦(𝑐(𝑠))𝑑𝑠≤𝑚₁(𝑡)𝑥(𝑎(𝑡))−𝑦(𝑎(𝑡))+𝑚₂(𝑡)(𝐻𝑥)(𝑏(𝑡))−𝐻𝑦(𝑏(𝑡))+𝑚₃(𝑡)₀^𝛼(𝑡)𝑢(𝑡,𝑠,𝑥(𝑐(𝑠)))−𝑢𝑡,𝑠,𝑦(𝑐(𝑠))𝑑𝑠≤𝑀₁𝑥(𝑎(𝑡))−𝑦(𝑎(𝑡))+𝑀₂𝑏(𝑡−(𝑏(𝑡𝑚(𝐻𝑥)())𝐻𝑦))+sup₃(𝑡)₀^𝛼(𝑡)|𝑢(𝑡,𝑠,𝑤(𝑐(𝑠)))−𝑢(𝑡,𝑠,𝑧(𝑐(𝑠)))|𝑑𝑠∶𝑤,𝑧∈𝐵_𝑟,∀𝑥,𝑦∈𝑋,𝑡∈ℝ₊, (3.12) which yields that diam(𝐹𝑋)(𝑡)≤𝑀₁diam𝑋(𝑎(𝑡))+𝑀₂(𝑏(𝑡𝑚diam𝐻𝑋)())+sup₃(𝑡)₀^𝛼(𝑡)|𝑢(𝑡,𝑠,𝑤(𝑐(𝑠)))−𝑢(𝑡,𝑠,𝑧(𝑐(𝑠)))|𝑑𝑠∶𝑤,𝑧∈𝐵_𝑟,∀𝑡∈ℝ₊, (3.13) which together with (3.3), Assumption (H2) and Lemma 2.4 ensures that limsup_𝑡→+∞diam(𝐹𝑋)(𝑡)≤𝑀₁limsup_𝑡→+∞diam𝑋(𝑎(𝑡))+𝑀₂limsup_𝑡→+∞diam(𝐻𝑋)(𝑏(𝑡))+limsup_𝑡→+∞sup𝑚₃(𝑡)₀^𝛼(𝑡)|𝑢(𝑡,𝑠,𝑤(𝑐(𝑠)))−𝑢(𝑡,𝑠,𝑧(𝑐(𝑠)))|𝑑𝑠∶𝑤,𝑧∈𝐵_𝑟=𝑀₁limsup_𝑡→+∞diam𝑋(𝑡)+𝑀₂limsup_𝑡→+∞diam(𝐻𝑋)(𝑡), (3.14) that is, limsup_𝑡→+∞diam(𝐹𝑋)(𝑡)≤𝑀₁limsup_𝑡→+∞diam𝑋(𝑡)+𝑀₂limsup_𝑡→+∞diam(𝐻𝑋)(𝑡). (3.15)
For each 𝑇>0 and 𝜀>0, put 𝑀_3𝑇=sup𝑚₃𝑝[,𝑢∶𝑝∈0,𝑇]_𝑇𝑟𝑢[][(𝑇[,𝜔=sup𝑝,𝑞,𝑣∶𝑝∈0,𝑇,𝑞∈0,𝛼)],𝑣∈−𝑟,𝑟]_𝑇𝑟()𝑢[],[(𝑇[,𝜔𝑢,𝜀=sup𝑝,𝜏,𝑣−𝑢𝑞,𝜏,𝑣∶𝑝,𝑞∈0,𝑇𝑝−𝑞≤𝜀,𝜏∈0,𝛼)],𝑣∈−𝑟,𝑟]_𝑇()𝑢[][(𝑇[],||,𝜔𝑢,𝜀,𝑟=sup𝑝,𝑞,𝑣−𝑢𝑝,𝑞,𝑤∶𝑝∈0,𝑇,𝑞∈0,𝛼)],𝑣,𝑤∈−𝑟,𝑟𝑣−𝑤≤𝜀_𝑇𝑟(𝑟)𝑓[],[](𝑟)(𝑟[.𝑓,𝜀,𝑔=sup𝑝,𝑣,𝑤,𝑧−𝑓𝑞,𝑣,𝑤,𝑧∶𝑝,𝑞∈0,𝑇𝑝−𝑞≤𝜀,𝑣∈−𝑟,𝑟,𝑤∈−𝑔,𝑔),𝑧∈−𝑀,𝑀] (3.16)
Let 𝑇>0,𝜀>0,𝑥∈𝑋 and 𝑡,𝑠∈[0,𝑇] with |𝑡−𝑠|≤𝜀. It follows from (H2) that there exist 𝑎_𝑇 and 𝑏_𝑇 satisfying 0≤𝑎^′(𝑡)≤𝑎_𝑇,0≤𝑏^′(𝑡)≤𝑏_𝑇,∀𝑡∈[0,𝑇]. (3.17)
In light of (3.2), (3.6), (3.9), (3.16), (3.17), and Lemma 2.5, we get that 𝑡)−(𝑠≤𝑓(𝑎(𝑡,(𝑏(𝑡|(𝐹𝑥)(𝐹𝑥)()|𝑡,𝑥))𝐻𝑥)()),₀^𝛼(𝑡)𝑢((𝑐(𝜏(𝑎(𝑠,(𝑏(𝑠𝑡,𝜏,𝑥)))𝑑𝜏−𝑓𝑡,𝑥))𝐻𝑥)()),₀^𝛼(𝑠)𝑢((𝑐(𝜏+𝑓(𝑎(𝑠,(𝑏(𝑠𝑠,𝜏,𝑥)))𝑑𝜏𝑡,𝑥))𝐻𝑥)()),₀^𝛼(𝑠)𝑢((𝑐(𝜏(𝑎(𝑠,(𝑏(𝑠,𝑠,𝜏,𝑥)))𝑑𝜏−𝑓𝑠,𝑥))𝐻𝑥)())₀^𝛼(𝑠)𝑢(𝑠,𝜏,𝑥(𝑐(𝜏)))𝑑𝜏≤𝑚₁(𝑡)|𝑥(𝑎(𝑡))−𝑥(𝑎(𝑠))|+𝑚₂(𝑡)|(𝐻𝑥)(𝑏(𝑡))−(𝐻𝑥)(𝑏(𝑠))|+𝑚₃(𝑡)_𝛼(𝑠)^𝛼(𝑡)|𝑢(𝑡,𝜏,𝑥(𝑐(𝜏)))|𝑑𝜏+₀^𝛼(𝑠)|𝑢((𝑐(𝜏((𝑐(𝜏𝑓[],[](𝑟)(𝑟[𝑡,𝜏,𝑥)))−𝑢𝑠,𝜏,𝑥)))|𝑑𝜏+sup𝑝,𝑣,𝑤,𝑧−𝑓𝑞,𝑣,𝑤,𝑧∶𝑝,𝑞∈0,𝑇𝑝−𝑞≤𝜀,𝑣∈−𝑟,𝑟,𝑤∈−𝑔,𝑔),𝑧∈−𝑀,𝑀]≤𝑀₁𝜔^𝑇(𝑥∘𝑎,𝜀)+𝑀₂𝜔^𝑇((𝐻𝑥)∘𝑏,𝜀)+𝑀_3𝑇|𝛼(𝑡)−𝛼(𝑠)|sup𝑢𝑝,𝜏,𝑣∶𝑝∈[0,𝑇],𝜏∈[0,𝛼(𝑇)],𝑣∈[−𝑟,𝑟]+𝑀_3𝑇𝛼(𝑇)sup𝑢𝑝,𝜏,𝑣−𝑢𝑞,𝜏,𝑣∶𝑝,𝑞∈[0,𝑇],𝑝−𝑞≤𝜀,𝜏∈[0,𝛼(𝑇)],𝑣∈[−𝑟,𝑟]+𝜔^𝑇_𝑟𝑓,𝜀,𝑔(𝑟)≤𝑀₁𝜔^𝑎(𝑇)𝑥,𝑎_𝑇𝜀+𝑀₂𝜔^𝑏(𝑇)𝐻𝑥,𝑏_𝑇𝜀+𝑀_3𝑇𝜔^𝑇(𝛼,𝜀)𝑢^𝑇_𝑟+𝑀_3𝑇𝛼(𝑇)𝜔^𝑇_𝑟(𝑢,𝜀)+𝜔^𝑇_𝑟𝑓,𝜀,𝑔(𝑟), (3.18) which implies that 𝜔^𝑇(𝐹𝑥,𝜀)≤𝑀₁𝜔^𝑎(𝑇)𝑥,𝑎_𝑇𝜀+𝑀₂𝜔^𝑏(𝑇)𝐻𝑥,𝑏_𝑇𝜀+𝑀_3𝑇𝜔^𝑇(𝛼,𝜀)𝑢^𝑇_𝑟+𝑀_3𝑇𝛼(𝑇)𝜔^𝑇_𝑟(𝑢,𝜀)+𝜔^𝑇_𝑟𝑓,𝜀,𝑔(𝑟),∀𝑇>0,𝜀>0,𝑥∈𝑋. (3.19)
Notice that Assumptions (H1)−(H3) imply that the functions 𝛼=𝛼(𝑡),𝑓=𝑓(𝑡,𝑝,𝑞,𝑣) and 𝑢=𝑢(𝑡,𝑦,𝑧) are uniformly continuous on the sets [0,𝑇],  [0,𝑇]×[−𝑟,𝑟]×[−𝑔(𝑟),𝑔(𝑟)]×[−𝑀,𝑀] and [0,𝑇]× [0,𝛼(𝑇)]×[−𝑟,𝑟], respectively. It follows that lim_𝜀→0𝜔^𝑇(𝛼,𝜀)=lim_𝜀→0𝜔^𝑇_𝑟(𝑢,𝜀)=lim_𝜀→0𝜔^𝑇_𝑟𝑓,𝜀,𝑔(𝑟)=0. (3.20)
In terms of (3.19) and (3.20), we have 𝜔^𝑇₀(𝐹𝑋)≤𝑀₁𝜔₀^𝑎(𝑇)(𝑋)+𝑀₂𝜔₀^𝑏(𝑇)(𝐻𝑋), (3.21)
letting 𝑇→+∞ in the above inequality, by Assumption (H2) and Lemma 2.6, we infer that 𝜔₀(𝐹𝑋)≤𝑀₁𝜔₀(𝑋)+𝑀₂𝜔₀(𝐻𝑋). (3.22)
By means of (3.15), (3.22), and Assumption (H5), we conclude immediately that 𝜇(𝐹𝑋)=𝜔₀(𝐹𝑋)+limsup_𝑡→+∞diam(𝐹𝑋)(𝑡)≤𝑀₁𝜔₀(𝑋)+𝑀₂𝜔₀(𝐻𝑋)+𝑀₁limsup_𝑡→+∞diam𝑋(𝑡)+𝑀₂limsup_𝑡→+∞diam(𝐻𝑋)(𝑡)=𝑀₁𝜇(𝑋)+𝑀₂𝜔₀(𝐻𝑋)+limsup_𝑡→+∞(𝑡)≤𝑀diam𝐻𝑋)(₁+𝑀₂𝑄𝜇(𝑋), (3.23) that is, (3.11) holds.
Next we prove that 𝐹 is continuous on the ball 𝐵_𝑟. Let 𝑥∈𝐵_𝑟 and {𝑥_𝑛}_𝑛≥1⊂𝐵_𝑟 with lim_𝑛→∞𝑥_𝑛=𝑥. It follows from (3.3) that for given 𝜀>0, there exists a positive constant 𝑇 such that sup𝑚₃(𝑡)₀^𝛼(𝑡)|𝑢(𝑡,𝑠,𝑣(𝑐(𝑠)))−𝑢(𝑡,𝑠,𝑤(𝑐(𝑠)))|𝑑𝑠∶𝑣,𝑤∈𝐵_𝑟<𝜀3,∀𝑡>𝑇. (3.24)
Since 𝑢=𝑢(𝑡,𝑠,𝑣) is uniformly continuous in [0,𝑇]×[0,𝛼(𝑇)]×[−𝑟,𝑟], it follows from (3.16) that there exists 𝛿₀>0 satisfying 𝜔^𝑇()<𝜀𝑢,𝛿,𝑟1+3𝑀_3𝑇𝛼(𝑇),∀𝛿∈0,𝛿₀. (3.25)
By Assumption (H5) and lim_𝑛→∞𝑥_𝑛=𝑥, we know that there exists a positive integer 𝑁 such that 1+𝑀₁𝑥_𝑛−𝑥+𝑀₂𝐻𝑥_𝑛<1−𝐻𝑥2min𝜀,𝛿₀,∀𝑛>𝑁. (3.26)
In view of (3.2), (3.6), (3.9), (3.24)–(3.26), and Assumption (H2), we gain that for any 𝑛>𝑁 and 𝑡∈ℝ₊𝐹𝑥_𝑛(𝑡)−(𝑡=𝑓𝐹𝑥)()𝑡,𝑥_𝑛(𝑎(𝑡)),𝐻𝑥_𝑛(𝑏(𝑡)),₀^𝛼(𝑡)𝑢𝑡,𝑠,𝑥_𝑛(𝑐(𝑠(𝑎(𝑡,(𝑏(𝑡,))𝑑𝑠−𝑓𝑡,𝑥))𝐻𝑥)())₀^𝛼(𝑡)𝑢(𝑡,𝑠,𝑥(𝑐(𝑠)))𝑑𝑠≤𝑚₁(𝑡)𝑥_𝑛(𝑎(𝑡))−𝑥(𝑎(𝑡))+𝑚₂(𝑡)𝐻𝑥_𝑛(𝑏(𝑡))−(𝐻𝑥)(𝑏(𝑡))+𝑚₃(𝑡)₀^𝛼(𝑡)𝑢𝑡,𝑠,𝑥_𝑛(𝑐(𝑠))−𝑢(𝑡,𝑠,𝑥(𝑐(𝑠)))𝑑𝑠≤𝑀₁𝑥_𝑛−𝑥+𝑀₂𝐻𝑥_𝑛−𝐻𝑥+maxsup_𝜏>𝑇sup𝑚₃(𝜏)₀^𝛼(𝜏)𝑢(𝜏,𝑠,𝑧(𝑐(𝑠)))−𝑢𝜏,𝑠,𝑦(𝑐(𝑠))𝑑𝑠∶𝑧,𝑦∈𝐵_𝑟,sup_{[]𝜏∈0,𝑇}sup𝑚₃(𝜏)₀^𝛼(𝜏)𝑢(𝜏,𝑠,𝑧(𝑐(𝑠)))−𝑢𝜏,𝑠,𝑦(𝑐(𝑠))𝑑𝑠∶𝑧,𝑦∈𝐵_𝑟,≤𝛿𝑧−𝑦⁰2<𝜀2𝜀+max3,sup_{[]𝜏∈0,𝑇}𝑚₃(𝜏)₀^𝛼(𝜏)𝜔^𝑇𝛿𝑢,⁰2<𝜀,𝑟𝑑𝑠2𝜀+max3,𝑀_3𝑇𝛼(𝑇)𝜀1+3𝑀_3𝑇𝛼(𝑇)=5𝜀6, (3.27) which yields that 𝐹𝑥_𝑛−𝐹𝑥<𝜀,∀𝑛>𝑁, (3.28) that is, 𝐹 is continuous at each point 𝑥∈𝐵_𝑟.
Thus Lemma 2.2 ensures that 𝐹 has at least one fixed point 𝑥=𝑥(𝑡)∈𝐵_𝑟. Hence (1.1) has at least one solution 𝑥=𝑥(𝑡)∈𝐵_𝑟. This completes the proof.

Now we discuss (1.2) under below hypotheses:(H6)ℎ∶ℝ₊×ℝ²→ℝ is continuous with ℎ(𝑡,0,0)∈𝐵𝐶(ℝ₊) and ℎ=sup{|ℎ(𝑡,0,0)|∶𝑡∈ℝ₊};(H7)𝑐,𝛼∶ℝ₊→ℝ₊ are continuous and 𝛼 is nondecreasing;(H8) there exist two continuous functions 𝑚₁,𝑚₃∶ℝ₊→ℝ₊ and four positive constants 𝑟,𝑀,𝑀₀ and 𝑀₁ satisfying (3.3)–(3.5), ℎ(𝑡,𝑣,𝑤)−ℎ𝑡,𝑝,𝑞≤𝑚₁(𝑡)𝑣−𝑝+𝑚₃(𝑡)𝑤−𝑞,∀𝑡∈ℝ₊[][],𝑚,𝑣,𝑝∈−𝑟,𝑟,𝑤,𝑞∈−𝑀,𝑀sup₁(𝑡)∶𝑡∈ℝ₊≤𝑀₁,𝑀₀+ℎ≤𝑟1−𝑀₁. (3.29)(3.30)(3.31)

Theorem 3.2. Under Assumptions (H3) and (H6)−(H8), (1.2) has at least one solution 𝑥=𝑥(𝑡)∈𝐵_𝑟. Moreover, solutions of (1.2) are asymptotically stable.

Proof. As in the proof of Theorem 3.1, we conclude that (1.2) possesses at least one solution in 𝐵_𝑟.
Now we claim that solutions of (1.2) are asymptotically stable. Note that 𝑟,𝑀₀, and 𝑀₁ are positive numbers and ℎ≥0, it follows from (3.31) that 𝑀₁<1. In terms of (3.3), we infer that for given 𝜀>0, there exists 𝑇>0 such that sup𝑚₃(𝑡)₀^𝛼(𝑡)|𝑢(𝑡,𝑠,𝑣(𝑐(𝑠)))−𝑢(𝑡,𝑠,𝑤(𝑐(𝑠)))|𝑑𝑠∶𝑣,𝑤∈𝐵_𝑟<𝜀1−𝑀₁,∀𝑡≥𝑇. (3.32)
Let 𝑧=𝑧(𝑡),𝑦=𝑦(𝑡) be two arbitrarily solutions of (1.2) in 𝐵_𝑟. According to (3.29)–(3.32), we deduce that 𝑧(𝑡)(𝑡=ℎ(𝑡),−𝑦)𝑡,𝑧₀^𝛼(𝑡)𝑢(𝑡,𝜏,𝑧(𝑐(𝜏)))𝑑𝜏−ℎ𝑡,𝑦(𝑡),₀^𝛼(𝑡)𝑢𝑡,𝜏,𝑦(𝑐(𝜏))𝑑𝜏≤𝑚₁(𝑡)𝑧(𝑡)−𝑦(𝑡)+𝑚₃(𝑡)₀^𝛼(𝑡)𝑢(𝑡,𝜏,𝑧(𝑐(𝜏)))−𝑢𝑡,𝜏,𝑦(𝑐(𝜏))𝑑𝜏≤𝑀₁𝑧(𝑡)−𝑦(𝑡)+sup𝑚₃(𝑡)₀^𝛼(𝑡)|𝑢(𝑡,𝜏,𝑣(𝑐(𝜏)))−𝑢(𝑡,𝜏,𝑤(𝑐(𝜏)))|𝑑𝜏∶𝑣,𝑤∈𝐵_𝑟<𝑀₁𝑧(𝑡)−𝑦(𝑡)+𝜀1−𝑀₁,∀𝑡≥𝑇, (3.33) which means that 𝑧(𝑡)−𝑦(𝑡)<𝜀 (3.34) whenever 𝑧,𝑦 are solutions of (1.2) in 𝐵_𝑟 and 𝑡≥𝑇. Hence solutions of (1.2) are asymptotically stable. This completes the proof.

Remark 3.3. Theorems 3.1 and 3.2 generalize Theorem  3.1 in [6], Theorem  2 in [9], Theorem  3 in [10], Theorem  1 in [15], Theorem  2 in [16], and Theorem  3.1 in [19]. Examples 4.1 and 4.2 in the fourth section show that Theorems 3.1 and 3.2 substantially extend the corresponding results in [6, 9, 10, 15, 16, 19].

4. Examples

In this section, we construct two nontrivial examples to support our results.

Example 4.1. Consider the following nonlinear integral equation of Volterra type: 𝑥(𝑡)=110+ln³1+𝑡³+𝑡𝑥²1+3𝑡²()+12001+𝑡10√3+20𝑡+3𝑥𝑡³+19+√1+𝑡^𝑡20𝑠²𝑥(4𝑠)sin√1+𝑡³𝑠⁵−(𝑡−𝑠)⁴𝑥³(4𝑠)1+𝑡¹²+𝑠𝑡𝑥³(4𝑠)−3𝑠³+7(𝑡−2𝑠)²𝑥²(4𝑠)𝑑𝑠³,∀𝑡∈ℝ₊. (4.1) Put 𝑓()=1𝑡,𝑣,𝑤,𝑧10+ln³1+𝑡³+𝑡𝑣²()+12001+𝑡10√|𝑤|+𝑧3+20𝑡+9+√³,𝑢=𝑠1+𝑡𝑡,𝑠,𝑝²𝑝sin√1+𝑡³𝑠⁵−(𝑡−𝑠)⁴𝑝³1+𝑡¹²+𝑠𝑡𝑝³−3𝑠³+7(𝑡−2𝑠)²𝑝²,𝑎(𝑡)=1+3𝑡²,𝑏(𝑡)=𝑡³,𝑐(𝑡)=4𝑡,𝛼(𝑡)=𝑡²,∀𝑡,𝑠∈ℝ₊,𝑣,𝑤,𝑧,𝑝∈ℝ. (4.2)
Let √𝑟∈9−512,9+√512,𝑀=3,𝑀₀=9𝑟10,𝑀₁=𝑟100,𝑀₂=1,3001𝑓=,10𝑄=3,𝑚₁(𝑡)=𝑟𝑡100(1+𝑡),𝑚₂(𝑡)=110√3+20𝑡²,𝑚₃(𝑡)=3𝑀²9+√,(𝑡)(𝑡)(𝑡)1+𝑡𝐻𝑦=3𝑦,𝑔=3𝑡,∀𝑡∈ℝ₊,𝑦∈𝐵𝐶ℝ₊. (4.3)
It is easy to verify that (3.6) and Assumptions (H1)−(H3) and (H5) are satisfied. Notice that 𝑀₁𝑟+𝑀₂𝑔(𝑟)+𝑀₀+√𝑓≤𝑟⟺𝑟∈9−512,9+√512,𝑓()≤𝑡𝑡,𝑣,𝑤,𝑧−𝑓𝑡,𝑝,𝑞,𝑦()𝑣2001+𝑡²−𝑝²+110√|𝑤|−13+20𝑡+10√𝑞+13+20𝑡+9+√𝑧1+𝑡³−𝑦³≤𝑚₁(𝑡)𝑣−𝑝+𝑚₂(𝑡)𝑤−𝑞+𝑚₃(𝑡)𝑧−𝑦,∀𝑡∈ℝ₊[](𝑟)(𝑟[],𝑚,𝑣,𝑝∈−𝑟,𝑟,𝑤,𝑞∈−𝑔,𝑔),𝑧,𝑦∈−𝑀,𝑀sup₃(𝑡)₀^𝛼(𝑡)|𝑢(𝑡,𝑠,𝑣(𝑐(𝑠)))−𝑢(𝑡,𝑠,𝑤(𝑐(𝑠)))|𝑑𝑠∶𝑣,𝑤∈𝐵_𝑟=sup3𝑀²9+√1+𝑡^𝑡20𝑠²𝑣(4𝑠)sin√1+𝑡³𝑠⁵−(𝑡−𝑠)⁴𝑣³(4𝑠)1+𝑡¹²+𝑠𝑡𝑣³(4𝑠)−3𝑠³+7(𝑡−2𝑠)²𝑣²()−𝑠4𝑠₂𝑤(4𝑠)sin√1+𝑡³𝑠⁵−(𝑡−𝑠)⁴𝑤³(4𝑠)1+𝑡¹²+𝑠𝑡𝑤³(4𝑠)−3𝑠³+7(𝑡−2𝑠)²𝑤²(4𝑠)𝑑𝑠∶𝑣,𝑤∈𝐵_𝑟≤3𝑀²9+√1+𝑡^𝑡202𝑠²1+𝑡¹²²𝑟1+𝑡¹²+𝑟³𝑟³𝑡³+3𝑟⁶+7𝑡+2𝑡²²≤𝑑𝑠2𝑀²𝑡⁶9+√1+𝑡1+𝑡¹²²𝑟1+𝑡¹²+𝑟³𝑟³𝑡³+3𝑟⁶+7𝑡+2𝑡²²𝑚→0as𝑡→+∞,sup₃(𝑡)₀^𝛼(𝑡)|𝑢(𝑡,𝑠,𝑣(𝑐(𝑠)))|𝑑𝑠∶𝑡∈ℝ₊,𝑣∈𝐵_𝑟=sup3𝑀²9+√1+𝑡^𝑡20𝑠²𝑣(4𝑠)sin√1+𝑡³𝑠⁵−(𝑡−𝑠)⁴𝑣³(4𝑠)1+𝑡¹²+𝑠𝑡𝑣³(4𝑠)−3𝑠³+7(𝑡−2𝑠)²𝑣²(4𝑠)𝑑𝑠∶𝑡∈ℝ₊,𝑣∈𝐵_𝑟≤sup3𝑀²10^𝑡₀𝑠²𝑑𝑠1+𝑡¹²∶𝑡∈ℝ⁺=𝑟𝑀²20<𝑀₀,sup₀^𝛼(𝑡)|𝑢(𝑡,𝑠,𝑣(𝑐(𝑠)))|𝑑𝑠∶𝑡∈ℝ₊,𝑣∈𝐵_𝑟=sup3𝑀²9+√1+𝑡^𝑡20𝑠²𝑣(4𝑠)sin√1+𝑡³𝑠⁵−(𝑡−𝑠)⁴𝑣³(4𝑠)1+𝑡¹²+𝑠𝑡𝑣³(4𝑠)−3𝑠³+7(𝑡−2𝑠)²𝑣²(4𝑠)𝑑𝑠∶𝑡∈ℝ₊,𝑣∈𝐵_𝑟≤sup3𝑀²10^𝑡₀𝑠²𝑑𝑠1+𝑡¹²∶𝑡∈ℝ⁺=𝑟6≤𝑀, (4.4) that is, (3.2)–(3.5) and (3.7) hold. Hence all Assumptions of Theorem 3.1 are fulfilled. Consequently, Theorem 3.1 ensures that (4.1) has at least one solution 𝑥=𝑥(𝑡)∈𝐵_𝑟. However Theorem  3.1 in [6], Theorem  2 in [9],Theorem  3 in [10], Theorem  1 in [15], Theorem  2 in [16], and Theorem  3.1 in [19] are unapplicable for (4.1).

Example 4.2. Consider the following nonlinear integral equation of Volterra type: 𝑥(𝑡)=3+sin⁴√1+𝑡²+𝑥16²(𝑡)8+𝑡²+11+𝑡^√𝑡₀𝑠𝑥³1+𝑠²1+𝑡²+𝑠cos²1+𝑡³𝑠⁷𝑥²1+𝑠²𝑑𝑠²,∀𝑡∈ℝ₊. (4.5)
Put ℎ(𝑡,𝑣,𝑤)=3+sin⁴√1+𝑡²+𝑣168+²𝑡²+𝑤²,𝑢=1+𝑡𝑡,𝑠,𝑝𝑠𝑝³1+𝑡²+𝑠cos²1+𝑡³𝑠⁷𝑝²,𝛼(𝑡)=√𝑡,𝑐(𝑡)=1+𝑡²,𝑚₁(𝑡)=18+𝑡²,𝑚₃(𝑡)=41+𝑡,∀𝑡,𝑠∈ℝ₊1,𝑣,𝑤,𝑝∈ℝ,𝑟=2,𝑀=2,𝑀₀=𝑀₁=18,1ℎ=4. (4.6)
It is easy to verify that (3.30), (3.31) and Assumptions (H6) and (H7) are satisfied. Notice that ℎ()≤1𝑡,𝑣,𝑤−ℎ𝑡,𝑝,𝑞8+𝑡²𝑣²−𝑝²+1𝑤1+𝑡²−𝑞²≤𝑚₁(𝑡)𝑣−𝑝+𝑚₃(𝑡)𝑧−𝑦,∀𝑡∈ℝ₊[][],𝑚,𝑣,𝑝∈−𝑟,𝑟,𝑤,𝑞∈−𝑀,𝑀sup₃(𝑡)₀^𝛼(𝑡)|𝑢(𝑡,𝑠,𝑣(𝑐(𝑠)))−𝑢(𝑡,𝑠,𝑤(𝑐(𝑠)))|𝑑𝑠∶𝑥,𝑦∈𝐵_𝑟4=sup1+𝑡^√𝑡₀𝑠𝑣³1+𝑠²1+𝑡²+𝑠cos²1+𝑡³𝑠⁷𝑣²1+𝑠²−𝑠𝑤³1+𝑠²1+𝑡²+𝑠cos²1+𝑡³𝑠⁷𝑤²1+𝑠²𝑑𝑠𝑑𝑠∶𝑣,𝑤∈𝐵_𝑟≤4𝑟³𝑡1+√𝑡+𝑡²(1+𝑡)1+𝑡²²𝑚→0as𝑡→+∞,sup₃(𝑡)₀^𝛼(𝑡)|𝑢(𝑡,𝑠,𝑣(𝑐(𝑠)))|𝑑𝑠∶𝑡∈ℝ₊,𝑣∈𝐵_𝑟4=sup1+𝑡^√𝑡₀𝑠𝑣³1+𝑠²1+𝑡²+𝑠cos²1+𝑡³𝑠⁷𝑣²1+𝑠²𝑑𝑠∶𝑡∈ℝ₊,𝑣∈𝐵_𝑟≤sup4𝑟³1+𝑡^√𝑡₀𝑠1+𝑡²𝑑𝑠∶𝑡∈ℝ⁺𝑡≤sup4(1+𝑡)1+𝑡²∶𝑡∈ℝ⁺≤18=𝑀₁,sup₀^𝛼(𝑡)|𝑢(𝑡,𝑠,𝑣(𝑐(𝑠)))|𝑑𝑠∶𝑡∈ℝ₊,𝑣∈𝐵_𝑟=sup^√𝑡₀𝑠𝑣³1+𝑠²1+𝑡²+𝑠cos²1+𝑡³𝑠⁷𝑣²1+𝑠²𝑑𝑠∶𝑡∈ℝ₊,𝑣∈𝐵_𝑟≤sup𝑟^3√𝑡₀𝑠1+𝑡²𝑑𝑠∶𝑡∈ℝ⁺=132<𝑀, (4.7) which yield (3.3)–(3.5). That is, all Assumptions of Theorem 3.2 are fulfilled. Therefore, Theorem 3.2 guarantees that (4.5) has at least one solution 𝑥=𝑥(𝑡)∈𝐵_𝑟. Moreover, solutions of (4.5) are asymptotically stable. But Theorem  2 in [9], Theorem  3 in [10], Theorem  1 in [15], Theorem  2 in [16], and Theorem  3.1 in [19] are invalid for (4.5).

Acknowledgments

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