Global Asymptotic Stability for Nonlinear Functional Integral Equation of Mixed Type

Abstract

The existence results of global asymptotic stability of the solution are proved for functional integral equation of mixed type. The measure of noncompactness and the fixed-point theorem of Darbo are the main tools in carrying out our proof. Furthermore, some examples are given to show the efficiency and usefulness of the main findings.

1. Introduction

It is well known that functional integral equation of various types creates an important subject of numerous mathematical investigations and constitutes a significant branch of nonlinear analysis. It has great applications in physics, engineering, economics, and biologyed modelling problems connect with real world. With the help of several tools of functional analysis, topology, and fixed-point theory, many authors have made important contributions to this theory [1–5].

Let ℝ denote the real line and ℝ⁺=[0,+∞). In this paper, we investigate the nonlinear functional integral equation of mixed type, namely, 𝑥(𝑡)=𝐹𝑡,𝑓(𝑡,𝑥(𝛼(𝑡)))_0𝛽(𝑡),𝑢𝑡,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔(𝑡,𝑠_∞0),𝑣(𝑡,𝑠,𝑥(𝛿(𝑠)))𝑑𝑠,𝑡∈ℝ⁺. (1) It is worthwhile mentioning that (1) contains numerous functions and integral and functional integral equations encountered in nonlinear analysis. For example, the classical Volterra integral equation, the famous Chandrasekhar integral equation, the Urysohn integral equation, and the Volterra-Stieltjes integral equation are the special cases of (1).

The goal of this paper is to investigate the existence and asymptotic behavior of solutions for (1). Using the technique associated with a suitable measure of noncompactness and the fixed-point theorem of Darbo, we show that (1) has at least one solution under rather general and convenient assumptions. We also obtain some asymptotic characterization of the solutions of (1). The results of the present paper generalize several ones obtained previously in the papers [1, 3, 6–8] and references therein.

2. Preliminaries

In this section, we introduce some notations, definitions, and preliminary facts which are used in this paper. Let 𝑥 be a real function defined on [𝑎,𝑏], and let ⋁^𝑏_𝑎𝑥 denotes the variation of 𝑥 on [𝑎,𝑏]; if ⋁^𝑏_𝑎𝑥 is finite, we say that 𝑥 is of bounded variation. Let 𝑔(𝑡,𝑠)∶[𝑎,𝑏]×[𝑐,𝑑]→ℝ be a function, and let ⋁^𝑞_𝑡=𝑝𝑔(𝑡,𝑠) indicate the variation of the function 𝑡→𝑔(𝑡,𝑠) on [𝑝,𝑞]⊂[𝑎,𝑏], where 𝑠 is arbitrarily fixed in [𝑐,𝑑]. Similarly, define ⋁^𝑞_𝑠=𝑝𝑔(𝑡,𝑠). If 𝑥 and 𝜑 are two real functions defined on [𝑎,𝑏], define the Stieltjes integral ∫^𝑏_𝑎𝑥(𝑡)𝑑𝜑(𝑡) and say that 𝑥 is Stieltjes integrable on [𝑎,𝑏] with respect to 𝜑.

Lemma 1 (see [9]). If 𝑥 is Stieltjes integrable on [𝑎,𝑏] with respect to 𝜑 of bounded variation; then ^𝑏_𝑎𝑥(𝑡)𝑑𝜑(𝑡)≤^𝑏_𝑎|𝑥(𝑡)|𝑑^𝑡_𝑎𝜑. (2)

Lemma 2 (see [9]). Let 𝑥₁, 𝑥₂ be Stieltjes integrable functions on [𝑎,𝑏] with respect to a nondecreasing function 𝜑 such that 𝑥₁(𝑡)≤𝑥₂(𝑡) for 𝑡∈[𝑎,𝑏], then ^𝑏_𝑎𝑥₁(𝑡)𝑑𝜑(𝑡)≤^𝑏_𝑎𝑥₂(𝑡)𝑑𝜑(𝑡). (3)

We also consider Stieltjes integrals ∫^𝑏_𝑎𝑥(𝑠)𝑑_𝑠𝑔(𝑡,𝑠), where 𝑔∶[𝑎,𝑏]×[𝑎,𝑏]→ℝ and 𝑑_𝑠 indicates the integration with respect to 𝑠. The details concerning the integral of this type will be given later.

Let (𝐸,‖⋅‖) be a real Banach space with zero element 𝜃. Denote by 𝐵(𝑥,𝑟) the closed ball centered at 𝑥 and with radius 𝑟, 𝐵_𝑟 stands for the ball 𝐵(𝜃,𝑟). If 𝑋 is a nonempty subset of 𝐸, we denote by 𝑋 and Conv𝑋 the closure and the closed convex hull of 𝑋, respectively. The family of all nonempty and bounded subsets of 𝐸 is denoted by 𝔐_𝐸 and its subfamily consisting of all relatively compact sets is denoted by 𝔑_𝐸.

Definition 3 (see [10]). A function 𝜇∶𝔐_𝐸→ℝ⁺∶=[0,∞) is said to be a measure of noncompactness in 𝐸 if it satisfies the following conditions. (1)The family ker𝜇={𝑋∈𝔐_𝐸∶𝜇(𝑋)=0} is nonempty and ker𝜇⊂𝔑_𝐸. (2)𝑋⊂𝑌⇒𝜇(𝑋)≤𝜇(𝑌). (3)𝜇(𝑋)=𝜇(Conv𝑋)=𝜇(𝑋). (4)𝜇(𝜆𝑋+(1−𝜆)𝑌)≤𝜆𝜇(𝑋)+(1−𝜆)𝜇(𝑌) for 𝜆∈[0,1]. (5)If {𝑋_𝑛}_𝑛 is a sequence of closed sets from 𝔐_𝐸 such that 𝑋_𝑛+1⊂𝑋_𝑛(𝑛=1,2,…) and if lim_𝑛→∞𝜇(𝑋_𝑛)=0, then the set 𝑋_∞=⋂^∞_𝑛=1𝑋_𝑛 is nonempty.
The family ker𝜇 in (1) is called the kernel of 𝜇. Note that 𝑋_∞ from (5) is a member of ker𝜇.
Consider the Banach space BC(ℝ⁺) of all real functions defined, bounded and continuous on ℝ⁺, and equipped with the maximum norm ‖𝑥‖=sup{|𝑥(𝑡)|∶𝑡≥0}. Fix a nonempty bounded subset 𝑋 of BC(ℝ⁺) and 𝑇>0, and for 𝑥∈𝑋, 𝜀>0, define 𝜔^𝑇(𝑥,𝜀)∶=sup{|𝑥(𝑡)−𝑥(𝑠)|∶𝑡,𝑠∈[0,𝑇],|𝑡−𝑠|≤𝜀}. (4) Let 𝜔^𝑇(𝑋,𝜀)=sup𝜔^𝑇(),𝜔𝑥,𝜀∶𝑥∈𝑋_𝑇0(𝑋)=lim_𝜀→0𝜔^𝑇(),𝜔𝑋,𝜀₀(𝑋)=lim_𝑇→∞𝜔^𝑇₀(𝑋). (5) Moreover, for a fixed 𝑡∈ℝ⁺, define 𝑋(𝑡)={𝑥(𝑡)},(𝑡)𝑥(𝑡)(𝑡.∶𝑥∈𝑋diam𝑋=sup−𝑦)∶𝑥,𝑦∈𝑋 (6) Finally, define the function 𝜇 on 𝔐_BC(ℝ⁺₎ by 𝜇(𝑋)=𝜔₀(𝑋)+limsup_𝑡→∞diam𝑋(𝑡). (7) It can be shown that 𝜇 is a measure of noncompactness in BC(ℝ⁺) [10].

Remark 4. ker𝜇 contains nonempty and bounded sets 𝑋 such that functions from 𝑋 are locally equicontinuous on ℝ⁺ and tend to zero at infinity uniformly with respect to 𝑋.

Assume Ω is a nonempty subset of BC(ℝ⁺) and 𝑄 is an operator defined on Ω with values in BC(ℝ⁺). Consider the operator equation 𝑥(𝑡)=(𝑄𝑥)(𝑡),𝑡∈ℝ⁺. (8)

Definition 5 (see [8]). The solution 𝑥(𝑡) of (8) is said to be locally attractive if there exists a ball 𝐵(𝑥₀,𝑟) in BC(ℝ⁺) such that for arbitrary solutions 𝑥=𝑥(𝑡) and 𝑦=𝑦(𝑡) of (8) belonging to 𝐵(𝑥₀,𝑟)∩Ω such that lim_𝑡→∞𝑥(𝑡)−𝑦(𝑡)=0. (9)

Definition 6 (see [3]). The solution 𝑥=𝑥(𝑡) of (8) is said to be globally attractive, if (9) holds for each solutions 𝑦=𝑦(𝑡) of (8). If (9) is satisfied uniformly with respect to Ω, we say that solutions of (8) are globally asymptotically stable (or uniformly globally attractive).

Theorem 7 (Darbo fixed-point theorem [10]). Let Ω be a nonempty bounded closed and convex subset of Banach space 𝐸 and let 𝑄∶Ω→Ω be a continuous mapping such that 𝜇(𝑄𝑋)≤𝑘𝜇(𝑋) for any nonempty subset of 𝑋 of Ω, where 𝑘∈[0,1) is a constant. Then 𝑄 has a fixed-point in Ω.

From Theorem 7, it can be shown that the set fix 𝑄 of fixed-points of 𝑄 is a member of ker𝜇.

3. Main Results

In this section, we will investigate the existence and asymptotic behavior of solutions for (1) in BC(ℝ⁺). Assume that the following conditions are satisfied. (𝐻₁)𝛼,𝛽,𝛾,𝛿∶ℝ⁺→ℝ⁺ are continuous functions.(𝐻₂)𝑔∶ℝ⁺×ℝ⁺→ℝ satisfies the following conditions. (𝐻_2.𝑎) For all 𝑡₁,𝑡₂∈ℝ⁺ such that 𝑡₁<𝑡₂, the function 𝑠→𝑔(𝑡₂,𝑠)−𝑔(𝑡₁,𝑠) is nondecreasing on ℝ⁺.(𝐻_2.𝑏) The function 𝑠→𝑔(0,𝑠) is nondecreasing on ℝ⁺.(𝐻_2.𝑐) The functions 𝑠→𝑔(𝑡,𝑠) and 𝑡→𝑔(𝑡,𝑠) are continuous on ℝ⁺ for each fixed 𝑡∈ℝ⁺ or 𝑠∈ℝ⁺, respectively. (𝐻₃)𝑢∶ℝ⁺×ℝ⁺×ℝ→ℝ is a continuous function and there exist a continuous function 𝑘(𝑡,𝑠)∶ℝ⁺×ℝ⁺→ℝ⁺ and a continuous, nondecreasing function 𝜑∶ℝ⁺→ℝ⁺ such that |𝑢(𝑡,𝑠,𝑥)|≤𝑘(𝑡,𝑠)𝜑(|𝑥|),∀𝑡,𝑠∈ℝ⁺,𝑥∈ℝ. (10)(𝐻₄)𝑣∶ℝ⁺×ℝ⁺×ℝ→ℝ is a continuous function, 𝑣(𝑡,𝑠,0)=0 and there exist a continuous function 𝑝(𝑡,𝑠)∶ℝ⁺×ℝ⁺→ℝ⁺ and a continuous, nondecreasing functions 𝜓∶ℝ⁺→ℝ⁺ such that 𝜓(0)=0 and 𝑣()()𝜓𝑡,𝑠,𝑥−𝑣𝑡,𝑠,𝑦≤𝑝𝑡,𝑠𝑥−𝑦,∀𝑡,𝑠∈ℝ⁺,𝑥,𝑦∈ℝ. (11)(𝐻₅)𝐹∶ℝ⁺×ℝ×ℝ×ℝ→ℝ is a continuous function and there exist a constant 𝑀>0 and continuous functions 𝑚₁(𝑡),𝑚₂(𝑡)∶ℝ⁺→ℝ⁺ such that 𝐹𝑡,𝑥₁,𝑥₂,𝑥₃−𝐹𝑡,𝑦₁,𝑦₂,𝑦₃≤𝑀𝑥₁−𝑦₁+𝑚₁(𝑡)𝑥₂−𝑦₂+𝑚₂(𝑡)𝑥₃−𝑦₃, (12) for all 𝑡∈ℝ⁺, 𝑥₁,𝑥₂,𝑥₃,𝑦₁,𝑦₂,𝑦₃∈ℝ. Moreover, the function 𝑡→𝐹(𝑡,0,0,0) is bounded with 𝐹₀=sup_𝑡≥0|𝐹(𝑡,0,0,0)|.(𝐻₆)𝑓∶ℝ⁺×ℝ→ℝ is a continuous function and there exists a constant 𝜆>0 such that 𝑓(𝑡,𝑥)−𝑓𝑡,𝑦≤𝜆𝑥−𝑦,∀𝑥,𝑦∈ℝ,𝑡∈ℝ⁺. (13) Moreover, 𝑓(𝑡,0)∈BC(ℝ⁺) and 𝑓₀=sup_𝑡≥0|𝑓(𝑡,0)|.(𝐻₇)lim_𝑡→∞𝑚₁(𝑡)∫₀^𝛽(𝑡)𝑘(𝑡,𝑠)𝑑_𝑠𝑔(𝑡,𝑠)=0, lim_𝑡→∞𝑚₂(𝑡)∫^∞₀𝑝(𝑡,𝑠)𝑑𝑠=0.(𝐻₈) Consider lim_𝑇→∞sup^∞_𝑇𝑝(𝑡,𝑠)𝑑𝑠∶𝑡∈ℝ⁺=0. (14)(𝐻₉) There exists a positive solution 𝑟₀ of the inequality 𝑀𝜆𝑟+𝑀𝑓₀+𝑀₁𝜑(𝑟)+𝑀₂𝜓(𝑟)+𝐹₀≤𝑟 (15) such that 𝑀𝜆<1, where 𝑀₁=sup𝑚₁(𝑡)₀^𝛽(𝑡)𝑘(𝑡,𝑠)𝑑_𝑠𝑔(),𝑀𝑡,𝑠∶𝑡≥0₂=sup𝑚₂(𝑡)^∞₀𝑝(𝑡,𝑠)𝑑𝑠∶𝑡≥0. (16)

Remark 8. There are many examples of function 𝑔(𝑡,𝑠) which verify (𝐻₂), such as(i)𝑔(𝑡,𝑠)=𝑠, (ii)𝑔(𝑡,𝑠)=𝑠ln(𝑡+1), (iii)𝑔(𝑡,𝑠)=^{𝑡ln((𝑡+𝑠)/𝑡),for𝑡>0,𝑠≥0}_{0,for𝑡=0,𝑠≥0,}(iv)𝑔(𝑡,𝑠)=∫^𝑡₀(∫^𝑠₀𝑝(𝑧,𝑦)𝑑𝑦)𝑑𝑧, where 𝑝∶ℝ⁺×ℝ⁺→ℝ⁺ is a bounded and integrable function. One can see [2] for more details.

In what follows, we provide some properties of the function 𝑔(𝑡,𝑠).

Lemma 9 (see [9]). Under assumptions (𝐻_2.𝑎) and (𝐻_2.𝑏), the function 𝑠→𝑔(𝑡,𝑠) is nondecreasing on ℝ⁺ for any fixed 𝑡∈ℝ⁺.

Lemma 10 (see [9]). Assume that the function 𝑔 satisfies the assumption (𝐻_2.𝑎); then for arbitrarily fixed 𝑠₁,𝑠₂∈ℝ⁺ such that 𝑠₁<𝑠₂, the function 𝑡→𝑔(𝑡,𝑠₂)−𝑔(𝑡,𝑠₁) is nondecreasing on ℝ⁺.

Remark 11. Observe that if 𝑟₀>0 satisfies (15), then 𝑀𝜆≤1−𝑀𝑓₀𝑟₀−𝐹₀𝑟₀−𝑀₁𝜑𝑟₀𝑟₀−𝑀₂𝜓𝑟₀𝑟₀. (17) Thus, if one of the terms of 𝑀𝑓₀, 𝐹₀, 𝑀₁, and 𝑀₂ dose not vanish, then 𝑀𝜆<1 is automatically satisfied.

For the sake of convenience and simplicity, we will use the following notations in the paper. For fixed 𝑇>0, let 𝛼^𝑇{𝛼(𝑡)[,𝛽=sup∶𝑡∈0,𝑇]}_𝑇𝛽(𝑡)[,𝑚=sup∶𝑡∈0,𝑇]_1𝑇=inf𝑚₁(𝑡)[,𝑚∶𝑡∈0,𝑇]_2𝑇=inf𝑚₂(𝑡)[,𝑚∶𝑡∈0,𝑇]_𝑇1=sup𝑚₁(𝑡)[,𝑘∶𝑡∈0,𝑇]_𝑇=sup𝑘(𝑡,𝑠)∶𝑡∈[0,𝑇],𝑠∈0,𝛽^𝑇. (18)

The main result of this section is the following theorem.

Theorem 12. Under assumptions (𝐻₁)--(𝐻₉), (1) has at least one solution 𝑥∈𝐵𝐶(ℝ⁺). Moreover, solutions of (1) are globally asymptotically stable.

Proof. Consider the operator ℱ defined on BC(ℝ⁺) by (ℱ𝑥)(𝑡)=𝐹𝑡,𝑓(𝑡,𝑥(𝛼(𝑡)))_0𝛽(𝑡),𝑢𝑡,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔(𝑡,𝑠_∞0),𝑣(𝑡,𝑠,𝑥(𝛿(𝑠)))𝑑𝑠,𝑡∈ℝ⁺. (19) Taking into account (𝐻₂), (𝐻₃), and Lemma 9, we deduce that ℱ is well defined on BC(ℝ⁺).
First, we show that ℱ maps BC(ℝ⁺) into itself. It is clear that ℱ𝑥 is continuous on ℝ⁺ for each 𝑥∈BC(ℝ⁺). By (𝐻₃)--(𝐻₆) and Lemma 1, for 𝑡∈ℝ⁺, one has 𝑡≤𝐹((𝛼(𝑡,|(ℱ𝑥)()|𝑡,𝑓𝑡,𝑥)))_0𝛽(𝑡)𝑢𝑡,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔(𝑡,𝑠_∞0),𝑣((𝛿(𝑠≤𝐹((𝛼(𝑡,𝑡,𝑠,𝑥)))𝑑𝑠𝑡,𝑓𝑡,𝑥)))_0𝛽(𝑡)𝑢𝑡,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔(𝑡,𝑠_∞0),𝑣((𝛿(𝑠(()𝑓((𝛼(𝑡𝑡,𝑠,𝑥)))𝑑𝑠−𝐹𝑡,0,0,0)+𝐹𝑡,0,0,0≤𝑀𝑡,𝑥)))+𝑚₁(𝑡)×₀^𝛽(𝑡)𝑢𝑡,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔(𝑡,𝑠)+𝑚₂(𝑡)^∞₀𝑣((𝛿(𝑠+|𝐹(|𝑥(𝛼(𝑡𝑓(𝑡,𝑠,𝑥)))𝑑𝑠𝑡,0,0,0)|≤𝑀𝜆))|+𝑀𝑡,0)+𝑚₁(𝑡)×₀^𝛽(𝑡)𝑢𝑡,𝑠,𝑥𝛾(𝑠)𝑑_𝑠^𝑠_𝑝=0𝑔𝑡,𝑝+𝑚₂(𝑡)^∞₀|𝑣((𝛿(𝑠|𝐹(‖𝑥‖𝑓(𝑡,𝑠,𝑥)))|𝑑𝑠+𝑡,0,0,0)|≤𝑀𝜆+𝑀𝑡,0)+𝑚₁(𝑡)×₀^𝛽(𝑡)𝑘(𝑡,𝑠)𝜑𝑥𝛾(𝑠)𝑑_𝑠𝑔(𝑡,𝑠)+𝑚₂(𝑡)^∞₀𝑝()𝜓𝑥(𝛿(𝑠|𝐹(‖𝑥‖𝑡,𝑠(|))|)𝑑𝑠+𝑡,0,0,0)|≤𝑀𝜆+𝑀𝑓₀+𝜑(‖𝑥‖)𝑚₁(𝑡)×₀^𝛽(𝑡)𝑘(𝑡,𝑠)𝑑_𝑠𝑔()𝑥𝑚𝑡,𝑠+𝜓(‖‖)₂(𝑡)^∞₀𝑝(𝑡,𝑠)𝑑𝑠+𝐹₀≤𝑀𝜆‖𝑥‖+𝑀𝑓₀+𝑀₁𝜑(‖𝑥‖)+𝑀₂𝜓(‖𝑥‖)+𝐹₀, (20) By (𝐻₉), it is clear that ℱ(𝐵_𝑟0)⊂𝐵_𝑟0.
Next, we show that ℱ is continuous on 𝐵_𝑟0. Let 𝜀>0 be given, and since (𝐻₇) holds, there exists 𝑇>0, such that 𝑚₁(𝑡)∫₀^𝛽(𝑡)𝑘(𝑡,𝑠)𝑑_𝑠𝑔(𝑡,𝑠)<𝜀 and 𝑚₂(𝑡)∫^∞₀𝑝(𝑡,𝑠)𝑑𝑠<𝜀 for 𝑡≥𝑇. Consider the following two cases.
Case 1. For 𝑡≥𝑇, 𝑥,𝑦∈𝐵_𝑟0 with ‖𝑥−𝑦‖≤𝜀, 𝑡)−(𝑡≤𝐹((𝛼(𝑡,(ℱ𝑥)(ℱ𝑦)𝑡,𝑓𝑡,𝑥)))_0𝛽(𝑡)𝑢𝑡,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔(𝑡,𝑠_∞0),𝑣((𝛿(𝑠(𝛼(𝑡,𝑡,𝑠,𝑥)))𝑑𝑠−𝐹𝑡,𝑓𝑡,𝑦))_0𝛽(𝑡)𝑢𝑡,𝑠,𝑦𝛾(𝑠)𝑑_𝑠𝑔(𝑡,𝑠_∞0),𝑣(𝛿(𝑠𝑓((𝛼(𝑡(𝛼(𝑡𝑡,𝑠,𝑦))𝑑𝑠≤𝑀𝑡,𝑥)))−𝑓𝑡,𝑦))+𝑚₁(𝑡)×₀^𝛽(𝑡)𝑢𝛾(𝑠𝛾(𝑠𝑑𝑡,𝑠,𝑥)−𝑢𝑡,𝑠,𝑦)_𝑠𝑔(𝑡,𝑠)+𝑚₂(𝑡)^∞₀𝑣((𝛿(𝑠(𝛿(𝑠𝑥(𝛼(𝑡(𝛼(𝑡𝑡,𝑠,𝑥)))−𝑣𝑡,𝑠,𝑦))𝑑𝑠≤𝑀𝜆))−𝑦))+𝑚₁(𝑡)×₀^𝛽(𝑡)𝑢𝑡,𝑠,𝑥𝛾(𝑠)−𝑢𝑡,𝑠,𝑦𝛾(𝑠)𝑑_𝑠×^𝑠_𝑝=0𝑔𝑡,𝑝+𝑚₂(𝑡)×^∞₀𝑣(𝑡,𝑠,𝑥(𝛿(𝑠)))−𝑣𝑡,𝑠,𝑦(𝛿(𝑠))𝑑𝑠≤𝑀𝜆𝑥−𝑦+𝑚₁(𝑡)×₀^𝛽(𝑡)𝑢𝑡,𝑠,𝑥𝛾(𝑠)−𝑢𝑡,𝑠,𝑦𝛾(𝑠)𝑑_𝑠𝑔(𝑡,𝑠)+𝑚₂(𝑡)^∞₀|𝑣(𝑡,𝑠,𝑥(𝛿(𝑠)))|𝑑𝑠+𝑚₂(𝑡)^∞₀𝑣𝑡,𝑠,𝑦(𝛿(𝑠))𝑑𝑠≤𝑀𝜆𝑥−𝑦+𝑚₁(𝑡)×₀^𝛽(𝑡)𝑘(𝑡,𝑠)𝜑𝑥𝛾(𝑠)𝑑_𝑠𝑔(𝑡,𝑠)+𝑚₁(𝑡)₀^𝛽(𝑡)𝑘(𝑡,𝑠)𝜑𝑦𝛾(𝑠)𝑑_𝑠𝑔(𝑡,𝑠)+𝑚₂(𝑡)^∞₀𝑝(𝑡,𝑠)𝜓(|𝑥(𝛿(𝑠))|)𝑑𝑠+𝑚₂(𝑡)^∞₀𝑝()𝜓𝑦(𝛿(𝑠𝑥𝑚𝑡,𝑠))𝑑𝑠≤𝑀𝜆𝑥−𝑦+𝜑(‖‖)₁(𝑡)×₀^𝛽(𝑡)𝑘(𝑡,𝑠)𝑑_𝑠𝑔(𝑡,𝑠)+𝜑𝑦𝑚₁(𝑡)×₀^𝛽(𝑡)𝑘(𝑡,𝑠)𝑑_𝑠𝑔()𝑥𝑚𝑡,𝑠+𝜓(‖‖)₂(𝑡)^∞₀𝑝()𝑦𝑚𝑡,𝑠𝑑𝑠+𝜓₂(𝑡)^∞₀𝑝()𝑟𝑡,𝑠𝑑𝑠≤𝑀𝜆𝑥−𝑦+2𝜑₀𝑚₁(𝑡)×₀^𝛽(𝑡)𝑘(𝑡,𝑠)𝑑_𝑠𝑔()𝑟𝑡,𝑠+2𝜓₀𝑚₂(𝑡)^∞₀𝑝()𝑟𝑡,𝑠𝑑𝑠≤𝑀𝜆𝜀+2𝜑₀𝜀+2𝜓𝑟₀𝜀. (21)
Case 2. For 𝑡∈[0,𝑇], 𝑥,𝑦∈𝐵_𝑟0 with ‖𝑥−𝑦‖≤𝜀, by Lemmas 9 and 10, one has𝑡)−(𝑡𝑓((𝛼(𝑡(𝛼(𝑡(ℱ𝑥)(ℱ𝑦)≤𝑀𝑡,𝑥)))−𝑓𝑡,𝑦))+𝑚₁(𝑡)×₀^𝛽(𝑡)𝑢𝛾(𝑠𝛾(𝑠)𝑑𝑡,𝑠,𝑥)−𝑢𝑡,𝑠,𝑦_𝑠𝑔(𝑡,𝑠)+𝑚₂(𝑡)×^∞₀𝑣((𝛿(𝑠(𝛿(𝑠𝑥(𝛼(𝑡(𝛼(𝑡𝑡,𝑠,𝑥)))−𝑣𝑡,𝑠,𝑦))𝑑𝑠≤𝑀𝜆))−𝑦))+𝑚₁(𝑡)×₀^𝛽(𝑡)𝑢𝑡,𝑠,𝑥𝛾(𝑠)−𝑢𝑡,𝑠,𝑦𝛾(𝑠)𝑑_𝑠×^𝑠_𝑝=0𝑔𝑡,𝑝+𝑚₂(𝑡)×^∞₀𝑣(𝑡,𝑠,𝑥(𝛿(𝑠)))−𝑣𝑡,𝑠,𝑦(𝛿(𝑠))𝑑𝑠≤𝑀𝜆𝑥−𝑦+𝑚₁(𝑡)×₀^𝛽(𝑡)𝑢𝑡,𝑠,𝑥𝛾(𝑠)−𝑢𝑡,𝑠,𝑦𝛾(𝑠)𝑑_𝑠𝑔(𝑡,𝑠)+𝑚₂(𝑡)^∞₀𝑝(𝑡,𝑠)𝜓𝑥(𝛿(𝑠))−𝑦(𝛿(𝑠))𝑑𝑠≤𝑀𝜆𝜀+𝑚^𝑇₁𝜔(𝑢,𝜀)₀^𝛽(𝑡)𝑑_𝑠𝑔()𝑚𝑡,𝑠+𝜓𝑥−𝑦₂(𝑡)^∞₀𝑝(𝑡,𝑠)𝑑𝑠≤𝑀𝜆𝜀+𝑚^𝑇₁𝜔()×𝑔(𝑡(𝑢,𝜀𝑡,𝛽)−𝑔𝑡,0)+𝑀₂𝜓(𝜀)≤𝑀𝜆𝜀+𝑚^𝑇₁𝜔()×𝑔(𝑡)()𝑢,𝜀𝑇,𝛽−𝑔𝑇,0+𝑀₂𝜓(𝜀)≤𝑀𝜆𝜀+𝑚^𝑇₁𝜔()×𝑔𝑢,𝜀𝑇,𝛽^𝑇−𝑔(𝑇,0)+𝑀₂𝜓(𝜀), (22) where 𝜔(𝑢,𝜀)=sup𝑢(𝑡,𝑠,𝑥)−𝑢𝑡,𝑠,𝑦∶𝑡∈[0,𝑇],𝑠∈0,𝛽^𝑇,𝑥,𝑦∈−𝑟₀,𝑟₀,𝑥−𝑦≤𝜀. (23) By (21) and (22), 𝑡)−(𝑡𝑟(ℱ𝑥)(ℱ𝑦)≤max𝑀𝜆𝜀+2𝜑₀𝜀+2𝜓𝑟₀𝜀,𝑀𝜆𝜀+𝑚^𝑇₁𝜔()×𝑔𝑢,𝜀𝑇,𝛽^𝑇−𝑔(𝑇,0)+𝑀₂𝜓(𝜀). (24) Note that 𝜔(𝑢,𝜀)→0, 𝜓(𝜀)→0 as 𝜀→0, and ℱ is continuous on 𝐵_𝑟0 by (24).
Now, and let 𝑋 be a nonempty set of 𝐵_𝑟0 and 𝑥∈𝑋, fix arbitrarily 𝑇>0 and 𝜀>0, let 𝑡₁,𝑡₂∈[0,𝑇] such that 𝑡₁<𝑡₂ and 𝑡₂−𝑡₁≤𝜀, (ℱ𝑥)𝑡₂−(ℱ𝑥)𝑡₁≤𝐹𝑡₂,𝑓𝑡₂,𝑥𝛼𝑡₂,_𝛽(𝑡2)0𝑢𝑡₂,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₂,𝑠_∞0,𝑣𝑡₂(𝛿(𝑠𝑡,𝑠,𝑥))𝑑𝑠−𝐹₁,𝑓𝑡₁,𝑥𝛼𝑡₁,_𝛽(𝑡1)0𝑢𝑡₁,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₁,𝑠_∞0,𝑣𝑡₁(𝛿(𝑠≤𝐹𝑡,𝑠,𝑥))𝑑𝑠₂,𝑓𝑡₂,𝑥𝛼𝑡₂,_𝛽(𝑡2)0𝑢𝑡₂,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₂,𝑠_∞0,𝑣𝑡₂(𝛿(𝑠𝑡,𝑠,𝑥))𝑑𝑠−𝐹₂,𝑓𝑡₁,𝑥𝛼𝑡₁,_𝛽(𝑡2)0𝑢𝑡₂,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₂,𝑠_∞0,𝑣𝑡₂(𝛿(𝑠+𝐹𝑡,𝑠,𝑥))𝑑𝑠₂,𝑓𝑡₁,𝑥𝛼𝑡₁,_𝛽(𝑡2)0𝑢𝑡₂,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₂,𝑠_∞0,𝑣𝑡₂(𝛿(𝑠𝑡,𝑠,𝑥))𝑑𝑠−𝐹₂,𝑓𝑡₁,𝑥𝛼𝑡₁,_𝛽(𝑡1)0𝑢𝑡₁,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₁,𝑠_∞0,𝑣𝑡₂(𝛿(𝑠+𝐹𝑡,𝑠,𝑥))𝑑𝑠₂,𝑓𝑡₁,𝑥𝛼𝑡₁,_𝛽(𝑡1)0𝑢𝑡₁,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₁,𝑠_∞0,𝑣𝑡₂(𝛿(𝑠𝑡,𝑠,𝑥))𝑑𝑠−𝐹₂,𝑓𝑡₁,𝑥𝛼𝑡₁,_𝛽(𝑡1)0𝑢𝑡₁,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₁,𝑠_∞0,𝑣𝑡₁(𝛿(𝑠+𝐹𝑡,𝑠,𝑥))𝑑𝑠₂,𝑓𝑡₁,𝑥𝛼𝑡₁,_𝛽(𝑡1)0𝑢𝑡₁,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₁,𝑠_∞0,𝑣𝑡₁(𝛿(𝑠𝑡,𝑠,𝑥))𝑑𝑠−𝐹₁,𝑓𝑡₁,𝑥𝛼𝑡₁,_𝛽(𝑡1)0𝑢𝑡₁,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₁,𝑠_∞0,𝑣𝑡₁(𝛿(𝑠𝑓𝑡,𝑠,𝑥))𝑑𝑠≤𝑀₂,𝑥𝛼𝑡₂−𝑓𝑡₁,𝑥𝛼𝑡₁+𝑚^𝑇₁^𝛽(𝑡2)₀𝑢𝑡₂,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₂−,𝑠_𝛽(𝑡1)0𝑢𝑡₁,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₁,𝑠+𝑚^𝑇₂^∞₀𝑣𝑡₂(𝛿(𝑠−,𝑠,𝑥))𝑑𝑠_∞0𝑣𝑡₁,𝑠,𝑥(𝛿(𝑠))𝑑𝑠+𝜔^𝑇_𝑟0(𝐹,𝜀)∶=𝑀𝜗₀+𝑚^𝑇₁𝜗₁+𝑚^𝑇₂𝜗₂+𝜔^𝑇_𝑟0(𝐹,𝜀), (25) where 𝜗₀=𝑓𝑡₂,𝑥𝛼𝑡₂−𝑓𝑡₁,𝑥𝛼𝑡₁,𝜗₁=^𝛽(𝑡2)₀𝑢𝑡₂,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₂,𝑠−^𝛽(𝑡1)₀𝑢𝑡₁,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₁,𝜗,𝑠₂=^∞₀𝑣𝑡₂,𝑠,𝑥(𝛿(𝑠))𝑑𝑠−^∞₀𝑣𝑡₁(𝛿(𝑠,𝜔,𝑠,𝑥))𝑑𝑠_𝑇𝑟0()𝐹𝑡𝐹,𝜀=sup₂,𝑥,𝑦,𝑧−𝐹𝑡₁,𝑥,𝑦,𝑧∶𝑡₁,𝑡₂∈[0,𝑇],𝑡₂−𝑡₁≤𝜀,𝑥∈−𝜆𝑟₀−𝑓₀,𝜆𝑟₀+𝑓₀,−𝜑𝑟𝑦∈⁰𝑀₁𝑚_1𝑇,𝜑𝑟₀𝑀₁𝑚_1𝑇,−𝑀𝑧∈²𝜓𝑟₀𝑚_2𝑇,𝑀₂𝜓𝑟₀𝑚_2𝑇. (26) Moreover, 𝜗₀=𝑓𝑡₂,𝑥𝛼𝑡₂−𝑓𝑡₁,𝑥𝛼𝑡₁≤𝑓𝑡₂,𝑥𝛼𝑡₂−𝑓𝑡₂,𝑥𝛼𝑡₁+𝑓𝑡₂,𝑥𝛼𝑡₁−𝑓𝑡₁,𝑥𝛼𝑡₁≤𝜆𝑥𝛼𝑡₂−𝑥𝛼𝑡₁+𝑓𝑡₂,𝑥𝛼𝑡₁−𝑓𝑡₁,𝑥𝛼𝑡₁≤𝜆𝜔^𝑇𝑥,𝜔^𝑇(𝛼,𝜀)+𝜔^𝑇_𝑟0𝑓,𝜀, (27) where 𝜔^𝑇𝑥,𝜔^𝑇(𝑥𝜃𝛼,𝜀)=sup₂−𝑥𝜃₁∶𝜃₁,𝜃₂∈0,𝛼^𝑇,𝜃₂−𝜃₁<𝜔^𝑇(,𝜔𝛼,𝜀)_𝑇𝑟0𝑓𝑡𝑓,𝜀=sup₂,𝑥−𝑓𝑡₁,𝑥∶𝑡₁,𝑡₂∈[],𝑡0,𝑇₂−𝑡₁≤𝜀,𝑥∈−𝑟₀,𝑟₀. (28) By Lemma 1, one has 𝜗₁=^𝛽(𝑡2)₀𝑢𝑡₂,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₂−,𝑠_𝛽(𝑡1)0𝑢𝑡₁,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₁≤,𝑠_𝛽(𝑡2)0𝑢𝑡₂,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₂−,𝑠_𝛽(𝑡1)0𝑢𝑡₂,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₂+,𝑠_𝛽(𝑡1)0𝑢𝑡₂,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₂−,𝑠_𝛽(𝑡1)0𝑢𝑡₁,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₂+,𝑠_𝛽(𝑡1)0𝑢𝑡₁,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₂−,𝑠_𝛽(𝑡1)0𝑢𝑡₁,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₁,𝑠≤^𝛽(𝑡2)_𝛽𝑡1𝑢𝑡₂,𝑠,𝑥𝛾(𝑠)𝑑_𝑠×^𝑠_{𝑡𝑝=𝛽1}𝑔𝑡₂+,𝑝_𝛽(𝑡1)0𝑢𝑡₂,𝑠,𝑥𝛾(𝑠)−𝑢𝑡₁,𝑠,𝑥𝛾(𝑠)𝑑_𝑠×^𝑠_𝑝=0𝑔𝑡₂+,𝑝_𝛽(𝑡1)0𝑢𝑡₁,𝑠,𝑥𝛾(𝑠)𝑑_𝑠×^𝑠_𝑝=0𝑔𝑡₂,𝑝−𝑔𝑡₁≤,𝑝_{𝛽(𝑡2)𝛽𝑡1}𝑢𝑡₂,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₂+,𝑠_𝛽(𝑡1)0𝑢𝑡₂𝛾(𝑠)𝑡,𝑠,𝑥−𝑢₁,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₂,𝑠+^𝛽(𝑡1)₀𝑢𝑡₁,𝑠,𝑥𝛾(𝑠)𝑑_𝑠×𝑔𝑡₂,𝑠−𝑔𝑡₁≤,𝑠_{𝛽(𝑡2)𝛽𝑡1}𝑘𝑡₂,𝑠𝜑𝑥𝛾(𝑠)𝑑_𝑠𝑔𝑡₂+,𝑠_𝛽(𝑡1)0𝜔^𝑇_𝑟0(𝑢,𝜀)𝑑_𝑠𝑔𝑡₂+,𝑠_𝛽(𝑡1)0𝑘𝑡₁,𝑠𝜑𝑥𝛾(𝑠)𝑑_𝑠×𝑔𝑡₂,𝑠−𝑔𝑡₁,𝑠≤𝑘_𝑇𝜑𝑟₀^𝛽(𝑡2)_𝛽(𝑡1)𝑑_𝑠𝑔𝑡₂,𝑠+𝜔^𝑇_𝑟0(𝑢,𝜀)^𝛽(𝑡1)₀𝑑_𝑠𝑔𝑡₂,𝑠+𝑘_𝑇𝜑𝑟₀^𝛽(𝑡1)₀𝑑_𝑠𝑔𝑡₂,𝑠−𝑔𝑡₁,𝑠=𝑘_𝑇𝜑𝑟₀𝑔𝑡₂,𝛽𝑡₂−𝑔𝑡₂,𝛽𝑡₁+𝜔^𝑇_𝑟0(𝑢,𝜀)𝑔𝑡₂,𝛽𝑡₁−𝑔𝑡₂,0+𝑘_𝑇𝜑𝑟₀𝑔𝑡₂,𝛽𝑡₁−𝑔𝑡₁,𝛽𝑡₁−𝑔𝑡₂,0−𝑔𝑡₁,0≤𝑘_𝑇𝜑𝑟₀𝑔𝑇,𝛽𝑡₂−𝑔𝑇,𝛽𝑡₁+𝜔^𝑇_𝑟0(𝑢,𝜀)𝑔𝑇,𝛽𝑡₁−𝑔(𝑇,0)+𝑘_𝑇𝜑𝑟₀𝑔𝑡₂,𝛽^𝑇−𝑔𝑡₁,𝛽^𝑇+𝑔𝑡₂,0−𝑔𝑡₁,0≤𝑘_𝑇𝜑𝑟₀𝜔^𝑇𝑔(𝑇,⋅),𝜔^𝑇𝛽,𝜀+𝜔^𝑇_𝑟0(𝑢,𝜀)𝑔𝑇,𝛽^𝑇−𝑔(𝑇,0)+𝑘_𝑇𝜑𝑟₀𝜔^𝑇𝑔⋅,𝛽^𝑇,𝜀+𝜔^𝑇𝑔(⋅,0),𝜀, (29) where 𝜔^𝑇𝑔(𝑇,⋅),𝜔^𝑇𝑔𝛽,𝜀=sup𝑇,𝜃₂−𝑔𝑇,𝜃₁∶𝜃₁,𝜃₂∈0,𝛽^𝑇,𝜃₂−𝜃₁<𝜔^𝑇,𝜔𝛽,𝜀_𝑇𝑟0()𝑢𝑡𝑢,𝜀=sup₂,𝑠,𝑥−𝑢𝑡₁,𝑠,𝑥∶𝑡₁,𝑡₂∈[0,𝑇],𝑠∈0,𝛽^𝑇,𝑡₂−𝑡₁≤𝜀,𝑥∈−𝑟₀,𝑟₀. (30) By (𝐻₄), one has 𝜗₂=^∞₀𝑣𝑡₂(𝛿(𝑠−,𝑠,𝑥))𝑑𝑠_∞0𝑣𝑡₁(𝛿(𝑠≤,𝑠,𝑥))𝑑𝑠_∞0𝑣𝑡₂(𝛿(𝑠𝑡,𝑠,𝑥))𝑑𝑠−𝑣₁,𝑠,𝑥(𝛿(𝑠))𝑑𝑠≤^𝑇₀𝑣𝑡₂(𝛿(𝑠𝑡,𝑠,𝑥))𝑑𝑠−𝑣₁(𝛿(𝑠+,𝑠,𝑥))𝑑𝑠_∞𝑇𝑣𝑡₂,𝑠,𝑥(𝛿(𝑠))𝑑𝑠−𝑣𝑡₁,𝑠,𝑥(𝛿(𝑠))𝑑𝑠≤𝑇𝜔^𝑇_𝑟0(𝑣,𝜀)+^∞_𝑇𝑣𝑡₂(𝛿(𝑠+,𝑠,𝑥))𝑑𝑠_∞𝑇𝑣𝑡₁,𝑠,𝑥(𝛿(𝑠))𝑑𝑠≤𝑇𝜔^𝑇_𝑟0(𝑣,𝜀)+^∞_𝑇𝑝𝑡₂𝜓𝑥(𝛿(𝑠+,𝑠(|))|)𝑑𝑠_∞𝑇𝑝𝑡₁,𝑠𝜓(|𝑥(𝛿(𝑠))|)𝑑𝑠≤𝑇𝜔^𝑇_𝑟0(𝑣,𝜀)+2sup^∞_𝑇𝑝(𝑡,𝑠)𝑑𝑠∶𝑡∈ℝ⁺⋅𝜓𝑟₀, (31) where 𝜔^𝑇_𝑟0()𝑣𝑡𝑣,𝜀=sup₂,𝑠,𝑥−𝑣𝑡₁,𝑠,𝑥∶𝑡₁,𝑡₂[],[],𝑡,∈0,𝑇𝑠∈0,𝑇₂−𝑡₁≤𝜀,𝑥∈−𝑟₀,𝑟₀. (32) By (25), (27), (29), and (31), one has 𝜔^𝑇(ℱ𝑥,𝜀)≤𝑀𝜆𝜔^𝑇𝑥,𝜔^𝑇(𝛼,𝜀)+𝑀𝜔^𝑇_𝑟0𝑓,𝜀+𝑚^𝑇₁𝑘_𝑇𝜑𝑟₀𝜔^𝑇𝑔(𝑇,⋅),𝜔^𝑇𝛽,𝜀+𝑚^𝑇₁𝜔^𝑇_𝑟0(𝑢,𝜀)𝑔𝑇,𝛽^𝑇−𝑔(𝑇,0)+𝑚^𝑇₁𝑘_𝑇𝜑𝑟₀𝜔^𝑇𝑔⋅,𝛽^𝑇,𝜀+𝜔^𝑇𝑔(⋅,0),𝜀+𝑚^𝑇₂𝑇𝜔^𝑇_𝑟0(𝑣,𝜀)+2𝑚^𝑇₂𝜓𝑟₀×sup^∞_𝑇𝑝(𝑡,𝑠)𝑑𝑠∶𝑡∈ℝ⁺+𝜔^𝑇_𝑟0(𝐹,𝜀), (33) and then 𝜔^𝑇(ℱ𝑋,𝜀)≤𝑀𝜆𝜔^𝑇𝑋,𝜔^𝑇(𝛼,𝜀)+𝑀𝜔^𝑇_𝑟0𝑓,𝜀+𝑚^𝑇₁𝑘_𝑇𝜑𝑟₀𝜔^𝑇×𝑔(𝑇,⋅),𝜔^𝑇𝛽,𝜀+𝑚^𝑇₁𝜔^𝑇_𝑟0(𝑢,𝜀)𝑔𝑇,𝛽^𝑇−𝑔(𝑇,0)+𝑚^𝑇₁𝑘_𝑇𝜑𝑟₀×𝜔^𝑇𝑔⋅,𝛽^𝑇,𝜀+𝜔^𝑇𝑔(⋅,0),𝜀+𝑚^𝑇₂𝑇𝜔^𝑇_𝑟0(𝑣,𝜀)+2𝑚^𝑇₂𝜓𝑟₀×sup^∞_𝑇𝑝(𝑡,𝑠)𝑑𝑠∶𝑡∈ℝ⁺+𝜔^𝑇_𝑟0(𝐹,𝜀). (34) By the uniform continuity of the functions 𝑓, 𝑔, 𝛽, 𝑢, 𝑣, and 𝐹 on compact set, one has 𝜔^𝑇_𝑟0𝑓,𝜀⟶0,𝜔^𝑇_𝑟0()𝜔𝑢,𝜀⟶0,_𝑇𝑟0(𝑣,𝜀)⟶0,𝜔^𝑇_𝑟0()𝜔𝐹,𝜀⟶0,as𝜀⟶0,_𝑇𝑔(𝑇,⋅),𝜔^𝑇𝜔𝛽,𝜀⟶0,_𝑇𝑔⋅,𝛽^𝑇,𝜀⟶0,𝜔^𝑇𝑔(⋅,0),𝜀⟶0,as𝜀⟶0, (35) so 𝜔^𝑇₀(ℱ𝑋)≤𝑀𝜆𝜔^𝑇₀(𝑋)+2𝑚^𝑇₂𝜓𝑟₀×sup^∞_𝑇𝑝(𝑡,𝑠)𝑑𝑠∶𝑡∈ℝ⁺, (36) and then, by (𝐻₈), 𝜔₀(ℱ𝑋)≤𝑀𝜆𝜔₀(𝑋). (37) On the other hand, take arbitrary 𝑥,𝑦∈𝑋 and fix 𝑡∈ℝ⁺; one has 𝑡)−(𝑡𝑓((𝛼(𝑡(𝛼(𝑡(ℱ𝑥)(ℱ𝑦)≤𝑀𝑡,𝑥)))−𝑓𝑡,𝑦))+𝑚₁(𝑡)×₀^𝛽(𝑡)𝑢𝑡,𝑠,𝑥𝛾(𝑠)−𝑢𝑡,𝑠,𝑦𝛾(𝑠)𝑑_𝑠𝑔(𝑡,𝑠)+𝑚₂(𝑡)^∞₀𝑣((𝛿(𝑠(𝛿(𝑠𝑥(𝛼(𝑡(𝛼(𝑡𝑡,𝑠,𝑥)))−𝑣𝑡,𝑠,𝑦))𝑑𝑠≤𝑀𝜆))−𝑦))+𝑚₁(𝑡)×₀^𝛽(𝑡)𝑢𝑡,𝑠,𝑥𝛾(𝑠)−𝑢𝑡,𝑠,𝑦𝛾(𝑠)𝑑_𝑠×^𝑠_𝑝=0𝑔𝑡,𝑝+𝑚₂(𝑡)^∞₀𝑣((𝛿(𝑠(𝛿(𝑠(𝑡)𝑡,𝑠,𝑥)))−𝑣𝑡,𝑠,𝑦))𝑑𝑠≤𝑀𝜆diam𝑋+𝑚₁(𝑡)×₀^𝛽(𝑡)𝑘(𝑡,𝑠)𝜑𝑥𝛾(𝑠)𝑑_𝑠𝑔(𝑡,𝑠)+𝑚₁(𝑡)₀^𝛽(𝑡)𝑘(𝑡,𝑠)𝜑𝑦𝛾(𝑠)𝑑_𝑠𝑔(𝑡,𝑠)+𝑚₂(𝑡)^∞₀𝑝(𝑡,𝑠)𝜓(|𝑥(𝛿(𝑠))|)𝑑𝑠+𝑚₂(𝑡)^∞₀𝑝()𝜓𝑦(𝛿(𝑠(𝑡)𝑟𝑡,𝑠))𝑑𝑠≤𝑀𝜆diam𝑋+2𝜑₀𝑚₁(𝑡)×₀^𝛽(𝑡)𝑘(𝑡,𝑠)𝑑_𝑠𝑔()𝑟𝑡,𝑠+2𝜓₀𝑚₂(𝑡)^∞₀𝑝(𝑡,𝑠)𝑑𝑠, (38) and hence (𝑡)(𝑡)𝑟diamℱ𝑋)(≤𝑀𝜆diam𝑋+2𝜑₀𝑚₁(𝑡)×₀^𝛽(𝑡)𝑘(𝑡,𝑠)𝑑_𝑠𝑔()𝑟𝑡,𝑠+2𝜓₀𝑚₂(𝑡)^∞₀𝑝(𝑡,𝑠)𝑑𝑠. (39) By (𝐻₇), one has limsup_𝑡→∞diam(ℱ𝑋)(𝑡)≤𝑀𝜆limsup_𝑡→∞diam𝑋(𝑡). (40) By (37) and (40), one has 𝜇(ℱ𝑋)≤𝑀𝜆𝜇(𝑋), (41) where 𝜇 is the measure of noncompactness defined in (7). Hence, in view of Theorem 7, we conclude that ℱ has at least one fixed-point 𝑥∈𝐵_𝑟0, which is a solution of (1). Moreover, taking into account the description of sets belonging to ker𝜇 (see Remark 4), we deduce that the solutions of (1) are globally asymptotically stable. This completes the proof.

Remark 13. Equation (1) is quite general and includes some nonlinear integral equations studied earlier by various authors as special cases. For example, we have the following. (i) If 𝑓(𝑡,𝑥)=𝑥, 𝑔(𝑡,𝑠)=𝑠, 𝛼(𝑡)=𝛽(𝑡)=𝛾(𝑡)=𝛿(𝑡)=𝑡, (1) reduces to the nonlinear functional integral equation which studied by [7], that is, 𝑥(𝑡)=𝐹𝑡,𝑥(𝑡),^𝑡₀𝑢(𝑡,𝑠,𝑥(𝑠))𝑑𝑠,_∞0𝑣(𝑡,𝑠,𝑥(𝑠))𝑑𝑠,𝑡∈ℝ⁺. (42)(ii) If 𝐹(𝑡,𝑥,𝑦,𝑧)=𝑞(𝑡)+𝑥𝑦, 𝛼(𝑡)=𝛽(𝑡)=𝛾(𝑡)=𝛿(𝑡)=𝑡, 𝑣(𝑡,𝑠,𝑥)=0, (1) reduces to the nonlinear quadratic integral equation of Volterra-Stieltjes type which is studied by the author of [2], that is, 𝑥(𝑡)=𝑞(𝑡)+𝑓(𝑡,𝑥(𝑡))×^𝑡₀𝑢(𝑡,𝑠,𝑥(𝑠))𝑑_𝑠𝑔(𝑡,𝑠),𝑡∈ℝ⁺. (43)(iii) If 𝐹(𝑡,𝑥,𝑦,𝑧)=𝑞(𝑡)+𝑥𝑧, 𝑔(𝑡,𝑠)=𝑠, 𝛼(𝑡)=𝛽(𝑡)=𝛾(𝑡)=𝛿(𝑡)=𝑡, 𝑢(𝑡,𝑠,𝑥)=0, (1) reduces to the quadratic Urysohn integral equation which is studied by the author of [22], that is, 𝑥(𝑡)(𝑡)((𝑡×=𝑞+𝑓𝑡,𝑥))_∞0𝑣(𝑡,𝑠,𝑥(𝑠))𝑑𝑠,𝑡∈ℝ⁺. (44)For more detail, one can see Table 1.

Table 1 

Some of papers studied earlier as special cases, where 𝑞, 𝜓, 𝜑, 𝑘, ℎ are suitable continuous functions.

4. Examples

In this section, we present some examples to illustrate our main result.

Example 14. Consider the nonlinear functional integral equation of Volterra-Stieltjes type 𝑥(𝑡)((𝛼(𝑡+=𝑓𝑡,𝑥)))_0𝛽(𝑡)𝑢𝑡,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔(𝑡,𝑠),𝑡∈ℝ⁺. (45)
Equation (45) is a special case of (1), where 𝐹(𝑡,𝑥,𝑦,𝑧)=𝑥+𝑦, 𝑣(𝑡,𝑠,𝑥)=0, 𝛿(𝑡)=0. Since 𝐹(𝑡,𝑥,𝑦,𝑧)=𝑥+𝑦, (𝐻₅) is satisfied with 𝑀=𝑚₁(𝑡)=1, 𝑚₂(𝑡)=0, and 𝐹₀=0. By Theorem 12, one has the following.

Theorem 15. Assume that (𝐻₁)--(𝐻₃), (𝐻₆), (𝐻₇), and (𝐻₉) hold; then (45) has at least one solution in 𝐵𝐶(ℝ⁺) which is globally asymptotically stable.

Remark 16. When 𝑔(𝑡,𝑠)=𝑠, (45) is investigated by [3], and if 𝑔(𝑡,𝑠)=𝑠, 𝛼(𝑡)=𝛽(𝑡)=𝛾(𝑡)=𝑡, it is investigated by the author of [6], see Table 1 for more details. Theorem 15 improves, and generalizes the results of [3, 6].

Remark 17. If 𝐹(𝑡,𝑥,𝑦,𝑧)=𝑞(𝑡)+𝑥𝑦, the following quadratic nonlinear functional integral equation of Volterra-Stieltjes type: 𝑥(𝑡)(𝑡)((𝛼(𝑡×=𝑞+𝑓𝑡,𝑥)))_0𝛽(𝑡)𝑢𝑡,𝑠,𝑥𝛾(𝑠)𝑑_𝑠𝑔(𝑡,𝑠),𝑡∈ℝ⁺. (46) When 𝛼(𝑡)=𝛽(𝑡)=𝛾(𝑡)=𝑡 in (46), it is investigated by the authors of [2]. For (46), by Theorem 12, one has a similar result. Here we omit it.

Example 18. Consider the quadratic Urysohn functional integral equation 𝑥(𝑡)(𝑡)((𝛼(𝑡×=𝑞+𝑓𝑡,𝑥)))_∞0𝑣(𝑡,𝑠,𝑥(𝛿(𝑠)))𝑑𝑠𝑡∈ℝ⁺. (47)
Equation (47) is a special case of (1), where 𝐹(𝑡,𝑥,𝑦,𝑧)=𝑞(𝑡)+𝑥𝑧, 𝑢(𝑡,𝑠,𝑥)=0, and 𝛽(𝑡)=𝛾(𝑡)=0. By Theorem 12, one has the following theorem.

Theorem 19. Assume that (𝐻₁) and (𝐻₄)--(𝐻₉) hold, and (47) has at least one solution in 𝐵𝐶(ℝ⁺) which is globally asymptotically stable.

Finally, we provide an example of concrete nonlinear functional integral equation of (1) which satisfies all the assumptions of Theorem 12.

Example 20. Consider the functional integral equation 𝑥(𝑡)=1𝑒32^−𝑡2+𝑡²+232𝑡²𝑥()+1+12𝑡1+𝑡^2𝑡2₀𝑥²(𝑠)1+𝑡⁴+𝑠²+1𝑑𝑠4sin²𝑡^∞₀𝑥(𝑠)𝑒^{−𝑠𝑒𝑡}𝑑𝑠,𝑡∈ℝ⁺. (48)
Observe that (48) is a special case of (1), where 𝐹(𝑡,𝑥,𝑦,𝑧)=𝑥+(1/(1+𝑡²))𝑦+𝑧sin²𝑡, 𝑓(𝑡,𝑥)=(1/32)𝑒^−𝑡2+((𝑡²+2)/(32(𝑡²+1)))𝑥, 𝑢(𝑡,𝑠,𝑥)=𝑥²/(1+𝑡⁴+𝑠²), 𝑣(𝑡,𝑠,𝑥)=(1/4)𝑥𝑒^{−𝑠𝑒𝑡}, 𝑔(𝑡,𝑠)=𝑠, 𝛼(𝑡)=2𝑡, 𝛽(𝑡)=𝑡², and 𝛾(𝑡)=𝛿(𝑡)=𝑡. Then |𝑢(≤1𝑡,𝑠,𝑥)|1+𝑡⁴+𝑠²𝑥²,𝑣()≤1𝑡,𝑠,𝑥−𝑣𝑡,𝑠,𝑦4𝑒^{−𝑠𝑒𝑡}𝑥−𝑦, (49) so 𝑘(𝑡,𝑠)=1/(1+𝑡⁴+𝑠²), 𝑝(𝑡,𝑠)=(1/4)𝑒^{−𝑠𝑒𝑡}, 𝜑(𝑥)=𝑥², and 𝜓(𝑥)=𝑥, and then (𝐻₁)--(𝐻₄) hold.
By 𝐹(𝑡,𝑥,𝑦,𝑧)=𝑥+(1/(1+𝑡²))𝑦+𝑧sin²𝑡 and 𝑓(𝑡,𝑥)=(1/32)𝑒^−𝑡2+((𝑡²+2)/(32(𝑡²+1)))𝑥, it is easy to see that (𝐻₅) and (𝐻₆) are satisfied with 𝑀=1, 𝑚₁(𝑡)=1/(1+𝑡²), 𝑚₂(𝑡)=sin²𝑡, 𝜆=1/16, and 𝐹₀=0, 𝑓₀=1/32. Since 11+𝑡⁴+𝑠²≤11+𝑠²_∞0,𝑝()1𝑡,𝑠𝑑𝑠=4𝑒^−𝑡, (50) which implies that lim_𝑡→∞0^𝛽(𝑡)𝑘(𝑡,𝑠)𝑑_𝑠𝑔(𝑡,𝑠)≤lim_𝑡→∞^𝑡2₀11+𝑠²𝜋𝑑𝑠=2,lim_𝑡→∞^∞₀𝑝(𝑡,𝑠)𝑑𝑠=0, (51) so (𝐻₇) holds. It can be easily seen that 𝑝(𝑡,𝑠) satisfies (𝐻₈). Moreover, 𝑀₁1=sup1+𝑡^2𝑡2₀11+𝑡⁴+𝑠²≤𝜋𝑑𝑠∶𝑡≥02,𝑀₂1=sup4sin²𝑡^∞₀𝑒^{−𝑠𝑒𝑡}1𝑑𝑠∶𝑡≥0=sup4𝑒^−𝑡sin²≤1𝑡∶𝑡≥04, (52) then the inequality from (𝐻₉) has the following form: 1116𝑟++𝜋322𝑟²+14𝑟≤𝑟. (53) Obviously, 𝑟₀=1/8 is a positive solution of this inequality for which 𝑀𝜆<1, and thus (𝐻₉) holds. By Theorem 12, (48) has at least one solution in BC(ℝ⁺) which is globally asymptotically stable.

Acknowledgment

This paper is supported by Zhejiang Provincial Natural Science Foundation of China under Grant no. LQ13A010015.