On Attractivity and Positivity of Solutions for Functional Integral Equations of Fractional Order

Abstract

We investigate a class of functional integral equations of fractional order given by 𝑥(𝑡)=𝑞(𝑡)+𝑓₁(𝑡,𝑥(𝛼₁(𝑡)),𝑥(𝛼₂(𝑡)))+(𝑓₂(𝑡,𝑥(𝛽₁(𝑡)),𝑥(𝛽₂(𝑡)))/Γ(𝛼))×∫^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₃(𝑡,𝑠,𝑥(𝛾₁(𝑠)), 𝑥(𝛾₂(𝑠)))d𝑠: sufficient conditions for the existence, global attractivity, and ultimate positivity of solutions of the equations are derived. The main tools include the techniques of measures of noncompactness and a recent measure theoretic fixed point theorem of Dhage. Our investigations are placed in the Banach space of continuous and bounded real-valued functions defined on unbounded intervals. Moreover, two examples are given to illustrate our results.

1. Introduction

Nonlinear functional integral equations with bounded intervals have been studied extensively in the literature as regards various qualitative properties. This includes existence, uniqueness, stability, boundedness, monotonicity and extremality of solutions. But the study of nonlinear functional integral equations with unbounded intervals is relatively rare and exploited for the characteristics of attractivity and asymptotic attractivity of solutions. There are two methods for dealing with these characteristics of solutions, namely, classical fixed point theorems involving the hypotheses from analysis and topology and the fixed point theorems involving the use of measure of noncompactness. Each one of these methods has some advantages and disadvantages over the others [1–11].

The theory of integral equations of fractional order plays a very important role in describing some real world problems; it has recently received a lot of attention and now constitutes a significant branch of nonlinear analysis. In recent years, differential and integral equations of fractional order have found wide applications in physics, mechanics, engineering, electro chemistry, economics, and other fields [12–19]; meanwhile, numerous research papers and monographs have appeared devoted to differential and integral equations of fractional order [20–38]. These papers contain various qualitative properties such as existence, uniqueness, stability, and asymptotic behavior for equations of fractional order.

The aim of this paper is to study the existence, global attractivity, and positivity of solutions for a functional integral equation of fractional order. The mentioned equation has rather general form and contains, as particular cases, a lot of fractional functional equations and nonlinear fractional integral equations of Volterra type. The main technique used in our considerations is the measures of noncompactness and a fixed point theorem of Dhage [3]. Our investigations will be situated in the Banach space of real functions which are defined, continuous, and bounded on the right-hand real half axis ℝ₊.

The measures of noncompactness used in this paper allow us not only to obtain the existence of solutions of the mentioned fractional functional integral equations but also to characterize those solutions in terms of global attractivity and positivity on unbounded intervals. This assertion means that all possible solutions of the equations in question are globally uniformly attractive and positive in the sense which will be defined further on.

It is worthwhile mentioning that the novelty of our approach consists mainly in the possibility of obtaining the global attractivity, asymptotic attractivity, and positivity of solutions for the considered fractional functional integral equations. We hope that the concept of measure of noncompactness considered here may be a stimulant for further investigations concerning solutions of nonlinear fractional differential and integral equations of other types.

2. Notations, Definitions, and Auxiliary Facts

This section is devoted to collect some notations, definitions, and auxiliary facts which will be used in the further considerations of this paper.

Let 𝐸 be a Banach space, 𝒫(𝐸), a class of subsets of 𝐸 and let 𝒫_𝑝(𝐸) denote the class of all nonempty subsets of 𝐸 with property 𝑝. Here 𝑝 may be 𝑝=closed (in short cl), 𝑝=bounded (in short bd), 𝑝=relativelycompact (in short rcp), and so forth. Thus, 𝒫_cl(𝐸), 𝒫_bd(𝐸), 𝒫_cl,bd(𝐸), and 𝒫_rcp(𝐸) denote, respectively, the classes of closed, bounded, closed and bounded, and relatively compact subsets of 𝐸. A function 𝑑_𝐻∶𝒫(𝐸)×𝒫(𝐸)→ℝ₊ defined by 𝑑_𝐻(𝐴,𝐵)=maxsup_𝑎∈𝐴𝐷(𝑎,𝐵),sup_𝑏∈𝐵𝐷(𝑏,𝐴) (1) satisfies all the conditions of a metric on 𝒫(𝐸) and is called a Hausdorff-Pompeiu metric on 𝐸, where 𝐷(𝑎,𝐵)=inf{‖𝑎−𝑏‖∶𝑏∈𝐵}. It is known that the hyperspace (𝒫_cl(𝐸),𝑑_𝐻) is a complete metric space.

The auxiliary way of defining the measures of noncompactness has been adopted in several papers in the literature; see Akhmerov et al. [39], Appell [40], Banaśand Goebel [41], in the works Väth [42] and the references therein. In this paper, we adopt the following axiomatic definition of the measure of noncompactness in a Banach space given by Dhage [3]. The other useful forms appear in the work of Banaś and Goebel [41] and the references therein.

Before giving definition of measure of noncompactness, we need the following definitions.

Definition 1 (see [43]). A sequence {𝐴_𝑛} of nonempty sets in 𝒫_𝑝(𝐸) is said to converge to a set 𝐴, called the limiting set if 𝑑_𝐻(𝐴_𝑛,𝐴)→0 as 𝑛→∞. A mapping 𝜇∶𝒫_𝑝(𝐸)→ℝ₊ is called continuous if for any sequence {𝐴_𝑛} in 𝒫_𝑝(𝐸) one has that 𝑑_𝐻𝐴_𝑛,𝐴⟶0⟹𝜇𝐴_𝑛−𝜇(𝐴)⟶0as𝑛⟶∞. (2)

Definition 2 (see [43]). A mapping 𝜇∶𝒫_𝑝(𝐸)→ℝ₊ is called nondecreasing, if for 𝐴,𝐵∈𝒫_𝑝(𝐸) with 𝐴⊆𝐵, then 𝜇(𝐴)≤𝜇(𝐵), where ⊆ is an order relation by inclusion 𝒫_𝑝(𝐸).

Now we are equipped with the necessary details to define the measures of noncompactness for a bounded subset of the Banach space 𝐸.

Definition 3 (see [43]). A function 𝜇∶𝒫_𝑝(𝐸)→ℝ₊ is called a measure of noncompactness if it satisfies 1^𝑜∅≠𝜇⁻¹(0)⊂𝒫_rcp(𝐸), 2^𝑜𝜇(𝐴)=𝜇(𝐴), where 𝐴 is the closure of 𝐴, 3^𝑜𝜇(𝐴)=𝜇(Conv(𝐴)), where Conv(𝐴) is the convex hull of 𝐴 and 4^𝑜𝜇 is nondecreasing, and 5^𝑜 if {𝐴_𝑛} is a decreasing sequence of sets in 𝒫_bd(𝐸) such that lim_𝑛→∞(𝐴_𝑛)=0, then the limiting set 𝐴_∞=lim_𝑛→∞𝐴_𝑛=∩^∞_𝑛=0𝐴_𝑛 is nonempty.
The family ker𝜇 described in 1^𝑜 is said to be the kernel of 𝜇 and ker𝜇=𝐴∈𝒫_bd(𝐸)∣𝜇(𝐴)=0⊂𝒫_rcp(𝐸). (3)
A measure 𝜇 is called complete or full if the kernel ker𝜇 of 𝜇 consists of all possible relatively compact subsets of 𝐸. Next, a measure 𝜇 is called sublinear if it satisfies 6^𝑜𝜇()=|𝜆|𝜇(𝐴)7𝜆𝐴for𝜆∈ℝ,_𝑜𝜇(𝐴+𝐵)≤𝜇(𝐴)+𝜇(𝐵). (4)
There do exist the sublinear measures of noncompactness on Banach spaces 𝐸. Indeed, the Kuratowskii and Hausdorff measures of noncompactness are sublinear in 𝐸. A good collection of different types of measures of noncompactness appears in Appell [40].
Observe that the limiting set 𝐴_∞ from 5^𝑜 is a member of the family ker𝜇. In fact, since 𝜇𝐴_∞≤𝜇𝐴_𝑛=𝜇𝐴_𝑛,forany𝑛, (5) one infers that 𝜇(𝐴_∞)=0. This yields that 𝐴_∞∈ker𝜇. This simple observation will be essential in our further investigations.
Now we state a key fixed point theorem of Dhage [3] which will be used in the sequel. Before stating this fixed point result, we give a useful definition.

Definition 4 (see [43]). A mapping 𝑄∶𝐸→𝐸 is called 𝐷-set-Lipschitz if there exists a continuous nondecreasing function 𝜑∶ℝ₊→ℝ₊ such that 𝜇(𝑄(𝐴))≤𝜑(𝜇(𝐴)) for all 𝐴∈𝒫_bd(𝐸) with 𝑄(𝐴)∈𝒫_bd(𝐸), where 𝜑(0)=0. Sometimes we call the function 𝜑 to be a 𝐷-function of 𝑄 on 𝐸. In the special case, when 𝜑(𝑟)=𝑘𝑟, 𝑘>0, 𝑄 is called a 𝑘-set-Lipschitz mapping, and if 𝑘<1, then 𝑄 is called a 𝑘-set-contraction on 𝐸. Further, if 𝜑(𝑟)<𝑟 for 𝑟>0, then 𝑄 is called a nonlinear 𝐷-set-contraction on 𝐸.

Theorem 5 (see, Dhage [43]). Let 𝐶 be a nonempty, closed, convex, and bounded subset of a Banach space 𝐸 and let 𝑄∶𝐶→𝐶 be a continuous and nonlinear 𝐷-set-contraction. Then 𝑄 has a fixed point.

Remark 6. Denote by Fix(𝑄) the set of all fixed points of the operator 𝑄 which belong to 𝐶. It can be shown that the set Fix(𝑄) existing in Theorem 5 belongs to the family ker𝜇. In fact if Fix(𝑄)∉ker𝜇, then 𝜇(Fix(𝑄))>0 and 𝑄(Fix(𝑄))=Fix(𝑄). Now from nonlinear 𝐷-set-contraction it follows that 𝜇(𝑄(Fix(𝑄)))≤𝜑(𝜇(Fix(𝑄))) which is a contradiction, since 𝜑(𝑟)<𝑟 for 𝑟>0. Hence, Fix(𝑄)∈ker𝜇.
Our further considerations will be placed in the Banach space 𝐵𝐶(ℝ₊,ℝ) consisting of all real functions 𝑥=𝑥(𝑡) defined, continuous, and bounded on ℝ₊. This space is equipped with the standard supremum norm ‖𝑥‖=sup{|𝑥(𝑡)|∶𝑡∈ℝ₊}.
For our purposes we will use the Hausdorff or ball measure of noncompactness in 𝐵𝐶(ℝ₊,ℝ). A handy formula for Hausdorff measure of noncompactness useful in application is defined as follows. Fix a nonempty and bounded subset 𝑋 of the space 𝐵𝐶(ℝ₊,ℝ) and a positive number 𝑇. For 𝑥∈𝑋 and 𝜀>0, denote by 𝜔^𝑇(𝑥,𝜀) the modulus of continuity of the function 𝑥 on the closed and bounded interval [0,𝑇] defined by 𝜔^𝑇(𝑥,𝜀)=sup{|𝑥(𝑡)−𝑥(𝑠)|∶𝑡,𝑠∈[0,𝑇],|𝑡−𝑠|<𝜀}. (6) Next, put 𝜔^𝑇(𝑋,𝜀)=sup𝜔^𝑇(),𝜔𝑥,𝜀∶𝑥∈𝑋_𝑇0(𝑋)=lim_𝜀→0𝜔^𝑇(𝑋,𝜀). (7)
It is known that 𝜔^𝑇₀ is a measure of noncompactness in the Banach space 𝐶([0,𝑇],ℝ) of continuous and real-valued functions defined on a closed and bounded interval [0,𝑇] in ℝ which is equivalent to Hausdorff or ball measure 𝜒 of noncompactness in it. In fact, one has 𝜒(𝑋)=(1/2)𝜔^𝑇₀(𝑋) for any bounded subset 𝑋 of 𝐶([0,𝑇],ℝ) (see Banaś and Goebel [41] and the references therein). Finally, define 𝜔₀(𝑋)=lim_𝑇→∞𝜔^𝑇₀(𝑋).
Now, for a fixed number 𝑡∈ℝ₊, denote 𝑋(𝑡)={𝑥(𝑡)},‖𝑋(𝑡𝑥(𝑡},‖𝑋(𝑡)‖𝑥(𝑡)|}.∶𝑥∈𝑋)‖=sup{|)|∶𝑥∈𝑋−𝑐=sup{|−𝑐∶𝑥∈𝑋 (8) Finally, consider the functions 𝜇’s defined on the family 𝒫_cl,bd(𝑋) by the formulas 𝜇_𝑎(𝑋)=max𝜔₀(𝑋),limsup_𝑡→∞(𝑡,𝜇diam𝑋)_𝑏(𝑋)=max𝜔₀(𝑋),limsup_𝑡→∞‖𝑋(𝑡,𝜇)‖_𝑐(𝑋)=max𝜔₀(𝑋),limsup_𝑡→∞‖𝑋(𝑡)−𝑐‖. (9)(10)(11)
Let 𝑇>0 be fixed. Then for any 𝑥∈𝐵𝐶(ℝ₊,ℝ), define 𝛿_𝑇(𝑥)=sup{||𝑥(𝑡)|−𝑥(𝑡)|∶𝑡≥𝑇}. Similarly, for any bounded subset 𝑋 of 𝐵𝐶(ℝ₊,ℝ), define 𝛿_𝑇(𝑋)=sup𝛿_𝑇(𝑥),𝛿(𝑋)∶𝑥∈𝑋=lim_𝑇→∞𝛿_𝑇(𝑋). (12) Define the functions 𝜇_ad,𝜇_bd,𝜇_cd∶𝒫_bd(𝐸)→ℝ₊ by the formulas 𝜇_ad(𝑋)=max𝜇_𝑎(𝑋)(𝑋,𝜇,𝛿)_bd(𝑋)=max𝜇_𝑏(𝑋)(𝑋,𝜇,𝛿)_cd(𝑋)=max𝜇_𝑐(𝑋),𝛿(𝑋), (13) for all 𝑋∈𝒫_cl,bd(𝐸).

Remark 7. It can be shown as in Banaś and Goebel [41] that the functions 𝜇_𝑎, 𝜇_𝑏, 𝜇_𝑐, 𝜇_ad, 𝜇_bd, and 𝜇_cd are measures of noncompactness in the space 𝐵𝐶(ℝ₊,ℝ). The kernels ker𝜇_𝑎, ker𝜇_𝑏, and ker𝜇_𝑐 of the measures 𝜇_𝑎, 𝜇_𝑏, and 𝜇_𝑐 consist of nonempty and bounded subsets 𝑋 of 𝐵𝐶(ℝ₊,ℝ) such that functions from 𝑋 are locally equicontinuous on ℝ₊ and the thickness of the bundle formed by functions from 𝑋 tends to zero at infinity. Moreover, the functions from ker𝜇_𝑐 come closer along a line 𝑦(𝑡)=𝑐 and the functions from ker𝜇_𝑏 come closer along the line 𝑦(𝑡)=0 as 𝑡 increases to ∞ through ℝ₊. A similar situation is also true for the kernels ker𝜇_ad, ker𝜇_bd, and ker𝜇_cd of the measures of noncompactness 𝜇_ad, 𝜇_bd, and 𝜇_cd. Moreover, these measures 𝜇_ad, 𝜇_bd, and 𝜇_cd characterize the ultimate positivity of the functions belonging to the kernels of ker𝜇_ad, ker𝜇_bd, and ker𝜇_cd. The above expressed property of ker𝜇_𝑎, ker𝜇_𝑏, ker𝜇_𝑐, and ker𝜇_ad, ker𝜇_bd, ker𝜇_cd permits us to characterize solutions of the fractional functional integral equations considered in the sequel.
In order to introduce further concepts used in this paper, let us assume that 𝐸=𝐵𝐶(ℝ₊,ℝ) and let Ω be a subset of 𝑋. Let 𝑄∶𝐸→𝐸 be an operator and consider the following operator equation in 𝐸: 𝑄𝑥(𝑡)=𝑥(𝑡),∀𝑡∈ℝ₊. (14)
Below we give different characterizations of the solutions for (14) on ℝ₊.

Definition 8. We say that solutions of (14) are locally attractive if there exists a closed ball 𝐵_𝑟(𝑥₀) in the space 𝐵𝐶(ℝ₊,ℝ) for some 𝑥₀∈𝐵𝐶(ℝ₊,ℝ) such that for arbitrary solutions 𝑥=𝑥(𝑡) and 𝑦=𝑦(𝑡) of (14) belonging to 𝐵_𝑟(𝑥₀)∩Ω one has that lim_𝑡→∞𝑥(𝑡)−𝑦(𝑡)=0. (15) In the case when the limit (15) is uniform with respect to the set 𝐵_𝑟(𝑥₀)∩Ω, that is, when for each 𝜀>0 there exists 𝑇>0 such that 𝑥(𝑡)−𝑦(𝑡)<𝜀, (16) for all 𝑥,𝑦∈𝐵_𝑟(𝑥₀)∩Ω being solutions of (14) and for all 𝑡≥𝑇, we will say that solutions of (14) are uniformly locally attractive on ℝ₊.

Definition 9. The solution 𝑥=𝑥(𝑡) of (14) is said to be globally attractive if (15) holds for each solution 𝑦=𝑦(𝑡) of (14) on Ω. In other words, we may say that solutions of (14) are globally attractive, if for arbitrary solutions 𝑥(𝑡) and 𝑦(𝑡) of (14) on Ω, the condition (15) is satisfied. In the case when the condition (15) is satisfied uniformly with respect to the set Ω, that is, if for every 𝜀>0 there exists 𝑇>0 such that the inequality (16) is satisfied for all 𝑥,𝑦∈Ω being the solutions of (14) and for all 𝑡≥𝑇, we will say that solutions of (14) are uniformly globally attractive on ℝ₊.

The following definitions appear in the work of Dhage [7].

Definition 10. A line 𝑦(𝑡)=𝑐, where 𝑐 is a real number, is called an attractor for a solution 𝑥∈𝐵𝐶(ℝ₊,ℝ) to (14) if lim_𝑡→∞[𝑥(𝑡)−𝑐]=0. In this case the solution 𝑥 to (14) is also called to be asymptotic to the line 𝑦(𝑡)=𝑐 and the line is an asymptote for the solution 𝑥 on ℝ₊.

Now we introduce the following definitions which are useful in the sequel.

Definition 11. The solutions of (14) are said to be globally asymptotically attractive if for any two solutions 𝑥=𝑥(𝑡) and 𝑦=𝑦(𝑡) of (14), the condition (15) is satisfied, and there is a line which is a common attractor to them on ℝ₊. In the case when condition (15) is satisfied uniformly, that is, if for every 𝜀>0 there exists 𝑇>0 such that the inequality (16) is satisfied for all 𝑡≥𝑇 and for all 𝑥,𝑦 being the solutions of (14) and having a line as a common attractor, we will say that solutions of (14) are uniformly globally asymptotically attractive on ℝ₊.

Definition 12. A solution 𝑥 of (14) is called locally ultimately positive if there exists a closed ball 𝐵_𝑟(𝑥₀) in 𝐵𝐶(ℝ₊,ℝ) for some 𝑥₀∈𝐵𝐶(ℝ₊,ℝ) such that 𝑥∈𝐵_𝑟(𝑥₀) and lim_𝑡→∞||𝑥(𝑡)|−𝑥(𝑡)|=0. (17) In the case when the limit (17) is uniform with respect to the solution set of the operator equation (14), that is, when for each 𝜀>0 there exists 𝑇>0 such that ||𝑥(𝑡)|−𝑥(𝑡)|<𝜀, (18) for all 𝑥 being solutions of (14) and for all 𝑡≥𝑇, we will say that solutions of (14) are uniformly locally ultimately positive on ℝ₊.

Definition 13. A solution 𝑥∈𝐶(ℝ₊,ℝ) of (14) is called globally ultimately positive if (17) is satisfied. In the case when the limit (17) is uniform with respect to the solution set of the operator equation (14) in 𝐶(ℝ₊,ℝ), that is, when for each 𝜀>0 there exists 𝑇>0 such that (18) is satisfied for all 𝑥 being solutions of (14) and for all 𝑡≥𝑇, we will say that solutions of (14) are uniformly globally ultimately positive on ℝ₊.

Remark 14. Note that the global attractivity and global asymptotic attractivity imply, respectively, the local attractivity and local asymptotic attractivity of the solutions for the operator equation (14) on ℝ₊. Similarly, global ultimate positivity implies local ultimate positivity of the solutions for the operator equation (14) on unbounded intervals. However, the converse of the above two statements may not be true. A few details of ultimate positivity are given in the work of Dhage [44].

Finally, we introduce the concept of the fraction integral and the Riemann-Liouville fractional derivative.

Definition 15 (see [45, 46]). The fractional integral of order 𝛼>0 with the lower limit 𝑡₀ for a function 𝑓 is defined as 𝐼^𝛼𝑓(𝑡)=1Γ(𝛼)^𝑡_𝑡0(𝑡−𝑠)^𝛼−1𝑓(𝑠)d𝑠,𝑡>𝑡₀, (19) provided the right-hand side is pointwise on [𝑡₀,∞), where Γ(𝛼) is the Gamma function.

Definition 16 (see [45, 46]). The Riemann-Liouville derivative of order 𝛼>0 with the lower limit 𝑡₀ for a function 𝑓∶[𝑡₀,∞)→ℝ can be written as 𝐷^𝛼_𝑡𝑓(𝑡)=1Γ()d𝑛−𝛼d𝑡^𝑛𝑛𝑡_𝑡0(𝑡−𝑠)^{𝛼+1−𝑛}𝑓(𝑠)d𝑠,𝑡>𝑡₀,𝑛−1<𝛼<𝑛. (20)
The first and maybe the most important property of the Riemann-Liouville fractional derivative is that for 𝑡>𝑡₀ and 𝛼>0, one has 𝐷^𝛼(𝐼^𝛼𝑓(𝑡))=𝑓(𝑡), which means that the Riemann-Liouville fractional differentiation operator is a left inverse to the Riemann-Liouville fractional integration operator of the same order 𝛼.

In the following section we prove the main results of this paper.

3. Attractivity and Positivity Results

In this section we will investigate the following functional integral equation of fractional order with deviating arguments: 𝑥(𝑡)=𝑞(𝑡)+𝑓₁𝑡,𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡)+𝑓₂𝑡,𝑥𝛽₁(𝑡),𝑥𝛽₂(𝑡)Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)d𝑠, (21) where 𝑡∈ℝ₊, 𝑞∶ℝ₊→ℝ, 𝑓₁,𝑓₂∶ℝ₊×ℝ×ℝ→ℝ, 𝑓₃∶ℝ₊×ℝ₊×ℝ×ℝ→ℝ, 𝛼₁,𝛼₂,𝛽₁,𝛽₂,𝛾₁,𝛾₂∶ℝ₊→ℝ₊, 𝛼∈(0,1), and Γ(𝛼) denotes the Gamma function.

Equation (21) has rather general form, when 𝑓₁𝑡,𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡)𝑓=0,₂𝑡,𝑥𝛽₁(𝑡),𝑥𝛽₂(𝑡)((𝑡,𝑓=𝑓𝑡,𝑥))₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)=𝑢(𝑡,𝑠,𝑥(𝑡)). (22) Equation (21) reduces to the following quadratic Volterra integral equation of fractional order 𝑥(𝑡)(𝑡)+𝑓((𝑡=𝑞𝑡,𝑥))Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑢(𝑡,𝑠,𝑥(𝑠))d𝑠. (23) Equation (23) has been studied in the work of Banaś and O’Regan [47] for the existence and local attractivity of solutions via classical hybrid fixed point theory, when 𝑞(𝑡)𝑓=0,₁𝑡,𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡)𝜂(𝑡),𝑓=𝑔𝑡,𝑥₂𝑡,𝑥𝛽₁(𝑡),𝑥𝛽₂(𝑡𝛽(𝑡,𝑓)=𝑓𝑡,𝑥)₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)=ℎ(𝑡,𝑠)𝑢𝑠,𝑥𝛾(𝑠). (24) Equation (21) reduces to the following functional integral equation of fractional order considered in Balachandran et al. [48] for the local attractivity of solutions 𝑥(𝑡)𝜂(𝑡)+𝑓𝛽(𝑡)=𝑔𝑡,𝑥𝑡,𝑥Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1ℎ(𝑡,𝑠)𝑢𝑠,𝑥𝛾(𝑠)d𝑠. (25) Therefore, (21) is more general and contains as particular cases a lot of fractional functional equations and nonlinear fractional integral equations of Volterra type.

By a solution of (21) we mean a function 𝑥∈𝐶(ℝ₊,ℝ) that satisfies (21), where 𝐶(ℝ₊,ℝ) is the space of continuous real-valued functions defined on ℝ₊.

Equation (21) will be considered under the following assumptions.

(𝐻₀) the functions 𝛼₁,𝛼₂,𝛽₁,𝛽₂∶ℝ₊→ℝ₊ are continuous and satisfy 𝛼₁(𝑡),𝛼₂(𝑡),𝛽₁(𝑡),𝛽₂(𝑡)𝛼≥𝑡,₁(𝑡),𝛼₂(𝑡),𝛽₁(𝑡),𝛽₂(𝑡)⟶∞as𝑡⟶∞. (26)

(𝐻₁) the function 𝑞∶ℝ₊→ℝ is continuous and bounded.

(𝐻₂) the function 𝑓₁∶ℝ₊×ℝ×ℝ→ℝ is continuous and there exists a bounded function ℓ∶ℝ₊→ℝ with bound 𝐿 and a positive constant 𝑀 such that 𝑓₁𝑡,𝑥₁,𝑦₁−𝑓₁𝑡,𝑥₂,𝑦₂≤ℓ(𝑡)max𝑥₁−𝑥₂,𝑦₁−𝑦₂𝑀+max𝑥₁−𝑥₂,𝑦₁−𝑦₂, (27)

for all 𝑡∈ℝ₊ and 𝑥₁,𝑥₂,𝑦₁,𝑦₂∈ℝ. Moreover, we assume that 𝐿≤𝑀.

(𝐻₃) the function 𝑓₂∶ℝ₊×ℝ×ℝ→ℝ is continuous and there exists a function 𝑚∶ℝ₊→ℝ₊ being continuous on ℝ₊ and such that 𝑓₂𝑡,𝑥₁,𝑦₁−𝑓₂𝑡,𝑥₂,𝑦₂≤𝑚(𝑡)𝑥₁−𝑥₂+𝑦₁−𝑦₂, (28)

for all 𝑡∈ℝ₊ and 𝑥₁,𝑥₂,𝑦₁,𝑦₂∈ℝ.

(𝐻₄) the function 𝐹₁(𝑡)=𝑓₁(𝑡,0,0) is bounded with 𝐹₁=sup{|𝑓₁(𝑡,0,0)|∶𝑡∈ℝ₊}.

(𝐻₅) the function 𝑓₃∶ℝ₊×ℝ₊×ℝ×ℝ→ℝ is continuous and there exists a function 𝑛∶ℝ₊→ℝ₊ being continuous on ℝ₊ and a function 𝜑∶ℝ₊→ℝ₊ being continuous and nondecreasing on ℝ₊ with 𝜑(0)=0 such that 𝑓₃𝑡,𝑠,𝑥₁,𝑦₁−𝑓₃𝑡,𝑠,𝑥₂,𝑦₂≤𝑛(𝑡)𝜑𝑥₁−𝑥₂+𝑦₁−𝑦₂, (29)

for all 𝑡∈ℝ₊ and 𝑥₁,𝑥₂,𝑦₁,𝑦₂∈ℝ.

For further purposes, define the functions 𝐹₂∶ℝ₊→ℝ₊ by putting 𝐹₂(𝑡)=𝑓₂(𝑡,0,0) and 𝐹₃∶ℝ₊→ℝ₊ by putting 𝐹₃(𝑡)=max{|𝑓₃(𝑡,𝑠,0,0)|∶0≤𝑠≤𝑡}. Obviously the functions 𝐹₂ and 𝐹₃ are continuous on ℝ₊.

(𝐻₆) the functions 𝑎,𝑏,𝑐,𝑑∶ℝ₊→ℝ₊ defined by the formulas 𝑎(𝑡)=2𝑚(𝑡)𝑛(𝑡)𝑡^𝛼,𝑏(𝑡)=2𝑚(𝑡)𝐹₃(𝑡)𝑡^𝛼,𝑐(𝑡)=𝑛(𝑡)𝐹₂(𝑡)𝑡^𝛼,𝑑(𝑡)=𝐹₂(𝑡)𝐹₃(𝑡)𝑡^𝛼 (30)

are bounded on ℝ₊ and the functions 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡), 𝑑(𝑡) vanish at infinity, that is, lim_𝑡→∞𝑎(𝑡)=lim_𝑡→∞𝑏(𝑡)=lim_𝑡→∞𝑐(𝑡)=lim_𝑡→∞𝑑(𝑡)=0. (31) Keeping in mind assumption (𝐻₆), define the following finite constants: 𝐴=sup𝑎(𝑡)∶𝑡∈ℝ₊,𝐵=sup𝑏(𝑡)∶𝑡∈ℝ₊,𝐶=sup𝑐(𝑡)∶𝑡∈ℝ₊,𝐷=sup𝑑(𝑡)∶𝑡∈ℝ₊. (32)

Now we formulate the last assumption.

(𝐻₇) There exists a positive solution 𝑟₀ of the inequality 𝑞+𝐿𝑟𝑀+𝑟+𝐹₁+𝐴𝑟𝜑(2𝑟)+𝐵𝑟+𝐶𝜑(2𝑟)+𝐷Γ(𝛼+1)<𝑟. (33)

Remark 17. Hypothesis (𝐻₂) is satisfied if in particular 𝑓₁ satisfies the condition, 𝑓₁𝑡,𝑥₁,𝑦₁−𝑓₁𝑡,𝑥₂,𝑦₂≤ℓ(𝑡)𝑥₁−𝑥₂+𝑦₁−𝑦₂2𝑀+𝑥₁−𝑥₂+𝑦₁−𝑦₂, (34) for all 𝑡∈ℝ₊ and 𝑥₁,𝑥₂,𝑦₁,𝑦₂∈ℝ, where 𝐿≤𝑀, and the function ℓ is defined as in hypothesis (𝐻₂) which further yields the usual Lipschitz condition on the function 𝑓₁, 𝑓₁𝑡,𝑥₁,𝑦₁−𝑓₁𝑡,𝑥₂,𝑦₂≤ℓ(𝑡)𝑥2𝑀¹−𝑥₂+𝑦₁−𝑦₂, (35) for all 𝑡∈ℝ₊ and 𝑥₁,𝑥₂,𝑦₁,𝑦₂∈ℝ provided 𝐿<𝑀. As mentioned in the work of Dhage [11], our hypothesis (𝐻₂) is more general than that existing in the literature.

Now, consider the operators 𝐹, 𝐺, 𝑈, and 𝑄 defined on the space 𝐵𝐶(ℝ₊,ℝ): (𝐺𝑥)(𝑡)=𝑞(𝑡)+𝑓₁𝑡,𝑥𝛼₁(𝑡),𝛼₂(𝑡),(𝐹𝑥)(𝑡)=𝑓₂𝑡,𝑥𝛽₁(𝑡),𝛽₂(𝑡),(𝑡)=1𝑈𝑥)(Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)(𝑡)=(𝑡)+(𝑡𝑡).d𝑠,𝑄𝑥)(𝐺𝑥)(𝐹𝑥)()(𝑈𝑥)( (36)

Then one has the following lemma.

Lemma 18. Under the above assumptions the operator 𝑄 transforms the ball 𝐵_𝑟0 in the space 𝐵𝐶(ℝ₊,ℝ) into itself. Moreover, all solutions of (21) belonging to the space 𝐵𝐶(ℝ₊,ℝ) are fixed points of the operator 𝑄.

Proof. Observe that for any function 𝑥∈𝐵𝐶(ℝ₊,ℝ), 𝐺𝑥 and 𝐹𝑥 are continuous on ℝ₊. We show that the same holds also for 𝑈𝑥. Take an arbitrary function 𝑥∈𝐵𝐶(ℝ₊,ℝ) and fix 𝑇>0, 𝜀>0. Next assume that 𝑡₁,𝑡₂∈[0,𝑇] such that |𝑡₂−𝑡₁|<𝜀. Without loss of generality one can assume that 𝑡₁<𝑡₂. Then, in view of imposed assumptions, one has (𝑈𝑥)𝑡₂−(𝑈𝑥)𝑡₁=1Γ(𝛼)^𝑡1₀𝑡₂−𝑠^𝛼−1𝑓₃𝑡₂,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)+d𝑠_𝑡2𝑡1𝑡₂−𝑠^𝛼−1𝑓₃𝑡₂,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)−d𝑠_𝑡10𝑡₁−𝑠^𝛼−1𝑓₃𝑡₁,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)≤1d𝑠Γ(𝛼)^𝑡1₀𝑡₂−𝑠^𝛼−1𝑓₃𝑡₂,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)−𝑡₂−𝑠^𝛼−1𝑓₃𝑡₁,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)+1d𝑠Γ(𝛼)^𝑡1₀𝑡₂−𝑠^𝛼−1𝑓₃𝑡₁,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)−𝑡₁−𝑠^𝛼−1𝑓₃𝑡₁,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)+1d𝑠Γ(𝛼)^𝑡2_𝑡1𝑡₂−𝑠^𝛼−1𝑓₃𝑡₂,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠≤1)d𝑠Γ(𝛼)^𝑡1₀𝑡₂−𝑠^𝛼−1𝑓₃𝑡₂,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)−𝑓₃𝑡₁,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)+1d𝑠Γ(𝛼)^𝑡1₀𝑓₃𝑡₁,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)×𝑡₁−𝑠^𝛼−1−𝑡₂−𝑠^𝛼−1+1d𝑠Γ(𝛼)^𝑡2_𝑡1𝑡₂−𝑠^𝛼−1𝑓₃𝑡₂,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)≤1d𝑠Γ(𝛼)^𝑡1₀𝜔^𝑇₁𝑓₃,𝜀,‖𝑥‖𝑡₂−𝑠^𝛼−1+1d𝑠Γ(𝛼)^𝑡1₀𝑓₃𝑡₁,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)−𝑓₃𝑡₁,𝑠,0,0+𝐹₃𝑡₁×𝑡₁−𝑠^𝛼−1−𝑡₂−𝑠^𝛼−1+1d𝑠Γ(𝛼)^𝑡2_𝑡1𝑡₂−𝑠^𝛼−1𝑓₃𝑡₂,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)−𝑓₃𝑡₂,𝑠,0,0+𝐹₃𝑡₂≤𝜔d𝑠_𝑇1𝑓₃,𝜀,‖𝑥‖Γ(𝛼)𝑡^𝛼₂−𝑡₂−𝑡₁^𝛼𝛼+1Γ(𝛼)×^𝑡1₀𝑛𝑡₁𝜑(2‖𝑥‖)+𝐹₃𝑡₁×𝑡₁−𝑠^𝛼−1−𝑡₂−𝑠^𝛼−1+1d𝑠Γ(𝛼)^𝑡2_𝑡1𝑡₂−𝑠^𝛼−1𝑛𝑡₂𝜑(2‖𝑥‖)+𝐹₃𝑡₂≤𝜔d𝑠_𝑇1𝑓₃,𝜀,‖𝑥‖Γ()𝑡𝛼+1^𝛼1+𝑛𝑡₁𝜑(2‖𝑥‖)+𝐹₃𝑡₁Γ()×𝑡𝛼+1_𝛼1−𝑡^𝛼₂+𝑡₂−𝑡₁^𝛼+𝑛𝑡₂𝜑(2‖𝑥‖)+𝐹₃𝑡₂Γ()𝑡𝛼+1²−𝑡₁^𝛼≤1Γ()𝑡𝛼+1^𝛼1𝜔^𝑇₁𝑓₃,𝜀,‖𝑥‖+𝑡₂−𝑡₁^𝛼×𝑛𝑡₁𝜑(2‖𝑥‖)+𝐹₃𝑡₁+𝑡₂−𝑡₁^𝛼𝑛𝑡₂𝜑(2‖𝑥‖)+𝐹₃𝑡₂, (37) where 𝜔^𝑇₁𝑓₃‖𝑥𝑓,𝜀,‖=sup₃𝑡₂,𝑠,𝑦₁,𝑦₂−𝑓₃𝑡₁,𝑠,𝑦₁,𝑦₂∶𝑠,𝑡₁,𝑡₂∈[0,𝑇],𝑠≤𝑡₁,𝑠≤𝑡₂,𝑡₂−𝑡₁≤𝜀,𝑦₁≤‖𝑥‖,𝑦₂≤‖𝑥‖. (38) Obviously, in view of the uniform continuity of 𝑓₃(𝑡,𝑠,𝑦₁,𝑦₂) on the set [0,𝑇]×[0,𝑇]×[−‖𝑥‖,‖𝑥‖]×[−‖𝑥‖,‖𝑥‖], one has that 𝜔^𝑇₁(𝑓₃,𝜀,‖𝑥‖)→0 as 𝜀→0. In what follows, denote 𝑛(𝑇)=max{𝑛(𝑡)∶𝑡∈[0,𝑇]},𝐹₃(𝑇)=max𝐹₃(𝑡)∶𝑡∈[0,𝑇]. (39) Then, keeping in mind the estimate (37) one obtains (𝑈𝑥)𝑡₂−(𝑈𝑥)𝑡₁≤1Γ()𝑇𝛼+1^𝛼𝜔^𝑇₁𝑓₃,𝜀,‖𝑥‖+2𝜀^𝛼𝑛(𝑇)𝜑(2‖𝑥‖)+𝐹₃(𝑇). (40) From the above inequality one can infer that the function 𝑈𝑥 is continuous on the interval [0,𝑇] for any 𝑇>0. This yields the continuity of 𝑈𝑥 on ℝ₊.
Finally, combining the continuity of the functions 𝐺𝑥, 𝐹𝑥, and 𝑈𝑥, one deduces that the function 𝑄𝑥 is continuous on on ℝ₊.
Now, taking an arbitrary function 𝑥∈𝐵𝐶(ℝ₊,ℝ), then, using our assumptions, for a fixed 𝑡∈ℝ₊, one has |(𝑡≤𝑞(𝑡+𝑓𝑄𝑥)|)₁𝑡,𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡)+𝑓₂𝑡,𝑥𝛽₁(𝑡),𝑥𝛽₂(𝑡)Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)≤𝑞(𝑡+𝑓d𝑠)₁𝑡,𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡)−𝐹₁(𝑡)+𝐹₁+1Γ(𝛼)𝑓₂𝑡,𝑥𝛽₁(𝑡),𝛽₂(𝑡)−𝐹₂(𝑡)+𝐹₂(𝑡×_𝑡0)(𝑡−𝑠)^𝛼−1𝑓₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)−𝑓₃(𝑡,𝑠,0,0)+𝐹₃(𝑡≤𝑞+ℓ(𝑡)𝑥𝛼)d𝑠max₁(𝑡),𝑥𝛼₂(𝑡)𝑀+max𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡)+𝐹₁+2‖𝑥‖𝑚(𝑡)+𝐹₂(𝑡)Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1𝑛(𝑡)𝜑(2‖𝑥‖)+𝐹₃(𝑡≤𝑞+𝐿‖𝑥‖)d𝑠𝑀+‖𝑥‖+𝐹₁+2‖𝑥‖𝑚(𝑡)+𝐹₂(𝑡)Γ(𝛼)×𝑛(𝑡)𝜑(2‖𝑥‖)+𝐹₃(𝑡)^𝑡₀(𝑡−𝑠)^𝛼−1≤𝑞+𝐿‖𝑥‖d𝑠𝑀+‖𝑥‖+𝐹₁+𝑡^𝛼(𝑡)𝑛(𝑡𝑥‖𝜑(2‖𝑥‖𝑥‖𝑚(𝑡)𝐹2𝑚)‖‖)+2₃(𝑡)+𝐹₂(𝑡𝑛(𝑡)𝜑(2‖𝑥+𝐹)‖)₂(𝑡)𝐹₃(𝑡)×(Γ(𝛼+1))⁻¹≤𝑞+𝐿‖𝑥‖𝑀+‖𝑥‖+𝐹₁+𝑎(𝑡)‖𝑥‖𝜑(2‖𝑥‖)+𝑏(𝑡)‖𝑥‖+𝑐(𝑡)𝜑(2‖𝑥‖)+𝑑(𝑡)Γ().𝛼+1 (41) Hence, in view of assumption (𝐻₆) one can infer that the function 𝑄𝑥 is bounded on ℝ₊. This assertion in conjunction with the continuity of 𝑄𝑥 on ℝ₊ allows us to conclude that 𝑄𝑥∈𝐵𝐶(ℝ₊,ℝ). Moreover, from the estimate (41) one obtains |(𝑡≤𝑞+𝐿‖𝑥‖𝑄𝑥)|𝑀+‖𝑥‖+𝐹₁+𝐴‖𝑥‖𝜑(2‖𝑥‖)+𝐵‖𝑥‖+𝐶𝜑(2‖𝑥‖)+𝐷Γ().𝛼+1 (42) Linking this estimate with assumption (𝐻₇), one deduces that there exists 𝑟₀>0 such that the operator 𝑄 transforms the ball 𝐵_𝑟0 into itself.
Finally, let us notice that the second assertion of our lemma is obvious in the light of the fact that the operator 𝑄 transforms the space 𝐵𝐶(ℝ₊,ℝ) into itself. The proof is complete.

Now, we are prepared to state and prove our main theorem of this section.

Theorem 19. Under the above assumptions (H₀)–(H₇), (21) has at least one solution in the space 𝐵𝐶(ℝ₊,ℝ). Moreover, these solutions are globally uniformly attractive on ℝ₊.

Proof. In what follows we will consider the operator 𝑄 as a mapping from 𝐵_𝑟0 into itself. Now we show that the operator 𝑄 is continuous on the ball 𝐵_𝑟0. To do this, fix arbitrarily 𝜀>0 and take 𝑥,𝑦∈𝐵_𝑟0 such that ‖𝑥−𝑦‖<𝜀. Then one gets (𝑡)(𝑡=𝑓𝑄𝑥−𝑄𝑦)₁𝑡,𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡)−𝑓₁𝑡,𝑦𝛼₁(𝑡),𝑦𝛼₂(𝑡)+𝑓₂𝑡,𝑥𝛽₁(𝑡),𝑥𝛽₂(𝑡)Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)−𝑓d𝑠₂𝑡,𝑦𝛽₁(𝑡),𝑦𝛽₂(𝑡)Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₃𝑡,𝑠,𝑦𝛾₁(𝑠),𝑦𝛾₂(𝑠)≤𝑓d𝑠₁𝑡,𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡)−𝑓₁𝑡,𝑦𝛼₁(𝑡),𝑦𝛼₂(𝑡)+𝑓₂𝑡,𝑥𝛽₁(𝑡),𝑥𝛽₂(𝑡)−𝑓₂𝑡,𝑦𝛽₁(𝑡),𝑦𝛽₂(𝑡)×(Γ(𝛼))⁻¹×^𝑡₀𝑓₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)(𝑡−𝑠)^1−𝛼+𝑓d𝑠₂𝑡,𝑦𝛽₁(𝑡),𝑦𝛽₂(𝑡)Γ(𝛼)×^𝑡₀𝑓₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)−𝑓₃𝑡,𝑠,𝑦𝛾₁(𝑠),𝑦𝛾₂(𝑠)×(𝑡−𝑠)^1−𝛼−1≤ℓ(𝑡)𝑥𝛼d𝑠max₁(𝑡)−𝑦𝛼₁(𝑡),𝑥𝛼₂(𝑡)−𝑦𝛼₂(𝑡)×𝑀+max𝑥𝛼₁(𝑡)−𝑦𝛼₁(𝑡),𝑥𝛼₂(𝑡)−𝑦𝛼₂(𝑡)⁻¹+𝑚(𝑡)𝑥𝛽₁(𝑡)−𝑦𝛽₁(𝑡)+𝑥𝛽₂(𝑡)−𝑦𝛽₂(𝑡)×(Γ(𝛼))⁻¹×^𝑡₀𝑓₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)−𝑓₃(𝑡,𝑠,0,0)+𝐹₃(𝑡)×(𝑡−𝑠)^1−𝛼−1+𝑓d𝑠₂𝑡,𝑦𝛽₁(𝑡),𝑦𝛽₂(𝑡)−𝐹₂(𝑡)+𝐹₂(𝑡)Γ(𝛼)×^𝑡₀𝑛(𝑡)𝜑𝑥𝛾₁(𝑠)−𝑦𝛾₁(𝑠)+𝑥𝛾₂(𝑠)−𝑦𝛾₂(𝑠)×(𝑡−𝑠)^1−𝛼−1≤𝐿d𝑠𝑥−𝑦+(𝑡𝑀+𝑥−𝑦2𝑚)𝑥−𝑦Γ(𝛼)×𝑛(𝑡)𝜑(2‖𝑥‖)+𝐹₃(𝑡)^𝑡₀(𝑡−𝑠)^𝛼−1+(𝑡𝑦+𝐹d𝑠2𝑚)₂(𝑡)Γ(𝛼)𝑛(𝑡)𝜑2×𝑥−𝑦_𝑡0(𝑡−𝑠)^𝛼−1d𝑠. (43) Now, linking the above established facts one concludes that the operator 𝑄 maps continuously the closed ball 𝐵_𝑟0 into itself.
Further, taking a nonempty subset 𝑋 of the ball 𝐵_𝑟0, fixing arbitrarily 𝑇>0 and 𝜀>0 and choosing 𝑥∈𝑋 and 𝑡₁,𝑡₂∈[0,𝑇] with |𝑡₂−𝑡₂|<𝜀, without loss of generality we may assume that 𝑡₁<𝑡₂. Then, taking into account our assumptions (𝐻₂) and (𝐻₄), one gets (𝑄𝑥)𝑡₂−(𝑄𝑥)𝑡₁≤𝑞𝑡₁−𝑞𝑡₂+𝑓₁𝑡₂,𝑥𝛼₁𝑡₂,𝛼₂𝑡₂−𝑓₁𝑡₂,𝑥𝛼₁𝑡₁,𝛼₂𝑡₁+𝑓₁𝑡₂,𝑥𝛼₁𝑡₁,𝛼₂𝑡₁−𝑓₁𝑡₁,𝑥𝛼₁𝑡₁,𝛼₂𝑡₁+(𝐹𝑥)𝑡₂(𝑈𝑥)𝑡₂−(𝐹𝑥)𝑡₁(𝑈𝑥)𝑡₂+(𝐹𝑥)𝑡₁(𝑈𝑥)𝑡₂−(𝐹𝑥)𝑡₁(𝑈𝑥)𝑡₁≤𝜔^𝑇𝑞,𝜀+ℓ(𝑡)max𝑥𝛼₁𝑡₂−𝑥𝛼₁𝑡₁,𝑥𝛼₂𝑡₂−𝑥𝛼₂𝑡₁×𝑀+max𝑥𝛼₁𝑡₂−𝑥𝛼₁𝑡₁,𝑥𝛼₂𝑡₂−𝑥𝛼₂𝑡₁⁻¹+𝜔^𝑇₁𝑓₁+𝑓,𝜀₂𝑡₂,𝑥𝛽₁𝑡₂,𝑥𝛽₂𝑡₂−𝑓₂𝑡₁,𝑥𝛽₁𝑡₁,𝑥𝛽₂𝑡₁×(Γ(𝛼))⁻¹×^𝑡2₀𝑓₃𝑡₂,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)𝑡₂−𝑠^1−𝛼+𝑓d𝑠₂𝑡₁,𝑥𝛽₁𝑡₁,𝑥𝛽₂𝑡₁Γ()×𝑇𝛼+1_𝛼𝜔^𝑇₁𝑓₃,𝜀,𝑟₀+2𝜀^𝛼𝑛(𝑇)𝜑2𝑟₀+𝐹₃(𝑇)≤𝜔^𝑇+ℓ(𝑡)𝑥𝛼𝑞,𝜀max₁𝑡₂−𝑥𝛼₁𝑡₁,𝑥𝛼₂𝑡₂−𝑥𝛼₂𝑡₁×𝑀+max𝑥𝛼₁𝑡₂−𝑥𝛼₁𝑡₁,𝑥𝛼₂𝑡₂−𝑥𝛼₂𝑡₁⁻¹+𝜔^𝑇₁𝑓₁+1,𝜀Γ(𝛼)𝑓₂𝑡₂,𝑥𝛽₁𝑡₂,𝑥𝛽₂𝑡₂−𝑓₂𝑡₂,𝑥𝛽₁𝑡₁,𝑥𝛽₂𝑡₁+𝑓₂𝑡₂,𝑥𝛽₁𝑡₁,𝑥𝛽₂𝑡₁−𝑓₂𝑡₁,𝑥𝛽₁𝑡₁,𝑥𝛽₂𝑡₁×^𝑡2₀𝑓₃𝑡₂,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)−𝑓₃𝑡₂,𝑠,0,0+𝐹₃𝑡₂×𝑡₂−𝑠^1−𝛼−1+𝑓d𝑠₂𝑡₁,𝑥𝛽₁𝑡₁,𝑥𝛽₂𝑡₁−𝐹₂𝑡₁+𝐹₂𝑡₁Γ()×𝑇𝛼+1_𝛼𝜔^𝑇₁𝑓₃,𝜀,𝑟₀+2𝜀^𝛼𝑛(𝑇)𝜑2𝑟₀+𝐹₃(𝑇)≤𝜔^𝑇+𝜔𝑞,𝜀𝐿max^𝑇𝑥,𝜔^𝑇𝛼₁,𝜀,𝜔^𝑇𝑥,𝜔^𝑇𝛼₂,𝜀𝑀+max𝜔^𝑇𝑥,𝜔^𝑇𝛼₁,𝜀,𝜔^𝑇𝑥,𝜔^𝑇𝛼₂,𝜀+𝜔^𝑇₁𝑓₁+𝑚𝑡,𝜀₂𝑥𝛽₁𝑡₂−𝑥𝛽₁𝑡₁+𝑥𝛽₂𝑡₂−𝑥𝛽₂𝑡₂×(Γ(𝛼))⁻¹+𝜔^𝑇₁𝑓₂,𝜀^𝑡2₀𝑛𝑡₂𝜑(2‖𝑥‖)+𝐹₃𝑡₂𝑡₂−𝑠^1−𝛼+𝑚𝑡d𝑠₁𝑥𝛽₁𝑡₁+𝑥𝛽₂𝑡₁+𝐹₂𝑡₁Γ()×𝑇𝛼+1_𝛼𝜔^𝑇₁𝑓₃,𝜀,𝑟₀+2𝜀^𝛼𝑛(𝑇)𝜑2𝑟₀+𝐹₃(𝑇)≤𝜔^𝑇+𝜔𝑞,𝜀𝐿max^𝑇𝑥,𝜔^𝑇𝛼₁,𝜀,𝜔^𝑇𝑥,𝜔^𝑇𝛼₂,𝜀𝑀+max𝜔^𝑇𝑥,𝜔^𝑇𝛼₁,𝜀,𝜔^𝑇𝑥,𝜔^𝑇𝛼₂,𝜀+𝜔^𝑇₁𝑓₁+1,𝜀Γ()𝑚𝑡𝛼+1²𝜔^𝑇𝑥,𝜔^𝑇𝛽₁,𝜀+𝜔^𝑇𝑥,𝜔^𝑇𝛽₂,𝜀+𝜔^𝑇₁𝑓₂,𝜀𝑡^𝛼₂𝑛𝑡₂𝜑2𝑟₀+𝐹₃𝑡₂+2𝑚𝑡₁𝑟₀+𝐹₂𝑡₁Γ()×𝑇𝛼+1_𝛼𝜔^𝑇₁𝑓₃,𝜀,𝑟₀+2𝜀^𝛼𝑛(𝑇)𝜑2𝑟₀+𝐹₃(𝑇)≤𝜔^𝑇+𝜔𝑞,𝜀𝐿max^𝑇𝑥,𝜔^𝑇𝛼₁,𝜀,𝜔^𝑇𝑥,𝜔^𝑇𝛼₂,𝜀𝑀+max𝜔^𝑇𝑥,𝜔^𝑇𝛼₁,𝜀,𝜔^𝑇𝑥,𝜔^𝑇𝛼₂,𝜀+𝜔^𝑇₁𝑓₁+𝑡,𝜀_𝛼2𝑚𝑡₂𝑛𝑡₂𝜑2𝑟₀+𝑚𝑡₂𝐹₃𝑡₂×𝜔^𝑇𝑥,𝜔^𝑇𝛽₁,𝜀+𝜔^𝑇𝑥,𝜔^𝑇𝛽₂×(Γ(,𝜀𝛼+1))⁻¹+𝜔^𝑇₁𝑓₂,𝜀𝑡^𝛼₂𝑛𝑡₂𝜑2𝑟₀+𝐹₃𝑡₂Γ()+𝑡𝛼+12𝑚₁𝑟₀+𝐹₂𝑡₁Γ()×𝑇𝛼+1_𝛼𝜔^𝑇₁𝑓₃,𝜀,𝑟₀+2𝜀^𝛼𝑛(𝑇)𝜑2𝑟₀+𝐹₃(𝑇)≤𝜔^𝑇+𝜔𝑞,𝜀𝐿max^𝑇𝑥,𝜔^𝑇𝛼₁,𝜀,𝜔^𝑇𝑥,𝜔^𝑇𝛼₂,𝜀𝑀+max𝜔^𝑇𝑥,𝜔^𝑇𝛼₁,𝜀,𝜔^𝑇𝑥,𝜔^𝑇𝛼₂,𝜀+𝜔^𝑇₁𝑓₁+,𝜀𝐴𝜑2𝑟₀+𝐵𝜔^𝑇𝑥,𝜔^𝑇𝛽₁,𝜀+𝜔^𝑇𝑥,𝜔^𝑇𝛽₂,𝜀×(2Γ(𝛼+1))⁻¹+𝜔^𝑇₁𝑓₂,𝜀𝑡^𝛼₂𝑛𝑡₂𝜑2𝑟₀+𝐹₃𝑡₂Γ()+𝑡𝛼+12𝑚₁𝑟₀+𝐹₂𝑡₁Γ()×𝑇𝛼+1_𝛼𝜔^𝑇₁𝑓₃,𝜀,𝑟₀+2𝜀^𝛼𝑛(𝑇)𝜑2𝑟₀+𝐹₃(𝑇), (44) where 𝜔^𝑇𝑞𝑡𝑞,𝜀=sup₂−𝑞𝑡₁∶𝑡₁,𝑡₂∈[0,𝑇],𝑡₂−𝑡₁,𝜔<𝜀_𝑇1𝑓₁𝑓,𝜀=sup₁𝑡₂,𝑥,𝑦−𝑓₁𝑡₁,𝑥,𝑦∶𝑡₁,𝑡₂∈[],𝑡0,𝑇₂−𝑡₁<𝜀,𝑥,𝑦∈−𝑟₀,𝑟₀,𝜔^𝑇₁𝑓₂𝑓,𝜀=sup₂𝑡₂,𝑥,𝑦−𝑓₂𝑡₁,𝑥,𝑦∶𝑡₁,𝑡₂∈[],𝑡0,𝑇₂−𝑡₁<𝜀,𝑥,𝑦∈−𝑟₀,𝑟₀. (45) Moreover, mention that other notations used in the above estimate were introduced earlier.
From the above estimate one can derive the following inequality: 𝜔^𝑇(𝑄𝑥,𝜀)≤𝜔^𝑇+𝜔𝑞,𝜀𝐿max^𝑇𝑥,𝜔^𝑇𝛼₁,𝜀,𝜔^𝑇𝑥,𝜔^𝑇𝛼₂,𝜀𝑀+max𝜔^𝑇𝑥,𝜔^𝑇𝛼₁,𝜀,𝜔^𝑇𝑥,𝜔^𝑇𝛼₂,𝜀+𝜔^𝑇_𝑟𝑓₁+,𝜀𝐴𝜑2𝑟₀+𝐵𝜔^𝑇𝑥,𝜔^𝑇𝛽₁,𝜀+𝜔^𝑇𝑥,𝜔^𝑇𝛽₂,𝜀()+𝜔2Γ𝛼+1_𝑇1𝑓₂,𝜀𝑡^𝛼₂𝑛𝑡₂𝜑2𝑟₀+𝐹₃𝑡₂Γ()+𝑡𝛼+12𝑚₁𝑟₀+𝐹₂𝑡₁Γ()×𝑇𝛼+1_𝛼𝜔^𝑇₁𝑓₃,𝜀,𝑟₀+2𝜀^𝛼𝑛(𝑇)𝜑2𝑟₀+𝐹₃(𝑇). (46) Observe that 𝜔^𝑇(𝑞,𝜀)→0, 𝜔^𝑇₁(𝑓₁,𝜀)→0, 𝜔^𝑇₁(𝑓₂,𝜀)→0 and 𝜔^𝑇₁(𝑓₃,𝜀)→0 as 𝜀→0, which is a simple consequence of the uniform continuity of the functions 𝑞, 𝑓₁, 𝑓₂, and 𝑓₃ on the sets [0,𝑇], [0,𝑇]×[−𝑟₀,𝑟₀]×[−𝑟₀,𝑟₀], [0,𝑇]×[−𝑟₀,𝑟₀]×[−𝑟₀,𝑟₀] and [0,𝑇]×[0,𝑇]×[−𝑟₀,𝑟₀]×[−𝑟₀,𝑟₀], respectively. Moreover, it is obvious that the constant𝜔^𝑇𝛼₁,𝜀⟶0,𝜔^𝑇𝛼₂𝜔,𝜀⟶0,_𝑇𝛽₁,𝜀⟶0,𝜔^𝑇𝛽₂,𝜀⟶0,as𝜀⟶0. (47) Thus, linking the established facts with the estimate (46) one gets 𝜔^𝑇₀(𝑄𝑋)≤𝐿𝜔^𝑇₀(𝑋)𝑀+𝜔^𝑇₀(𝑋). (48) Now, taking into account our assumptions, for arbitrarily fixed 𝑇∈ℝ₊ as well as for 𝑥₁,𝑥₂,𝑦₁,𝑦₂∈𝑋 one can deduce the following estimate (cf. the estimate (41)–(44)): (𝑡)(𝑡≤𝑥𝛼𝑄𝑥−𝑄𝑦)𝐿max₁(𝑡)−𝑦𝛼₁(𝑡),𝑥𝛼₂(𝑡)−𝑦𝛼₂(𝑡)×𝑀+max𝑥𝛼₁(𝑡)−𝑦𝛼₁(𝑡,𝑥𝛼₂)(𝑡)−𝑦𝛼₂(𝑡)⁻¹+𝑚(𝑡)𝑥𝛽₁(𝑡)−𝑦𝛽₁(𝑡+𝑥𝛽₂)(𝑡)−𝑦𝛽₂(𝑡)×(Γ(𝛼))⁻¹×^𝑡₀𝑓₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)−𝑓₃(𝑡,𝑠,0,0)+𝐹₃(𝑡)×(𝑡−𝑠)^1−𝛼−1+𝑓d𝑠₂𝑡,𝑦𝛽₁(𝑡),𝑦𝛽₂(𝑡)−𝐹₂(𝑡)+𝐹₂(𝑡)Γ(𝛼)×^𝑡₀𝑛(𝑡)𝜑𝑥𝛾₁(𝑠)−𝑦𝛾₁(𝑠)+𝑥𝛾₂(𝑠)−𝑦𝛾₂(𝑠)×(𝑡−𝑠)^1−𝛼−1≤𝛼d𝑠𝐿maxdiam𝑋₁(𝑡),diam𝑋𝛼₂(𝑡)𝑀+maxdiam𝑋𝛼₁(𝑡),diam𝑋𝛼₂(𝑡)+𝑚(𝑡)𝑥𝛽₁(𝑡)−𝑦𝛽₁(𝑡)+𝑥𝛽₂(𝑡)−𝑦𝛽₂(𝑡)Γ(𝛼)×^𝑡₀𝑛(𝑡)𝜑𝑥𝛾₁(𝑠)+𝑥𝛾₂(𝑠)+𝐹₃(𝑡)(𝑡−𝑠)^1−𝛼+𝑚(𝑡)𝑦𝛽d𝑠₁(𝑡)+𝑦𝛽₂(𝑡)+𝐹₂(𝑡)Γ(𝛼)×^𝑡₀𝑛(𝑡)𝜑𝑥𝛾₁(𝑠)−𝑦𝛾₁(𝑠)+𝑥𝛾₂(𝑠)−𝑦𝛾₂(𝑠)×(𝑡−𝑠)^1−𝛼−1≤𝛼d𝑠𝐿maxdiam𝑋₁(𝑡),diam𝑋𝛼₂(𝑡)𝑀+maxdiam𝑋𝛼₁(𝑡),diam𝑋𝛼₂(𝑡+𝑚(𝑡)𝑛(𝑡)𝑥𝛽₁)(𝑡)+𝑦𝛽₁(𝑡)+𝑥𝛽₂(𝑡)+𝑦𝛽₂(𝑡)×(Γ(𝛼))⁻¹×^𝑡₀𝜑𝑥𝛾₁(𝑠)+𝑥𝛾₂(𝑠)(𝑡−𝑠)^1−𝛼+𝑚(𝑡)𝐹d𝑠₃(𝑡)𝑥𝛽₁(𝑡)+𝑦𝛽₁(𝑡)+𝑥𝛽₂(𝑡)+𝑦𝛽₂(𝑡)×(Γ(𝛼))⁻¹×^𝑡₀𝜑𝑥𝛾₁(𝑠)+𝑥𝛾₂(𝑠)(𝑡−𝑠)^1−𝛼+𝑚(𝑡)𝑛(𝑡)𝑦𝛽d𝑠₁(𝑡)+𝑦𝛽₂(𝑡)Γ(𝛼)×^𝑡₀𝜑𝑥𝛾₁(𝑠)+𝑦𝛾₁(𝑠)+𝑥𝛾₂(𝑠)+𝑦𝛾₂(𝑠)×(𝑡−𝑠)^1−𝛼−1+𝑛(𝑡𝐹d𝑠)₂(𝑡)𝑦𝛽₁(𝑡)+𝑦𝛽₂(𝑡)Γ(𝛼)×^𝑡₀𝜑𝑥𝛾₁(𝑠)+𝑦𝛾₁(𝑠)+𝑥𝛾₂(𝑠)+𝑦𝛾₂(𝑠)×(𝑡−𝑠)^1−𝛼−1≤𝛼d𝑠𝐿maxdiam𝑋₁(𝑡),diam𝑋𝛼₂(𝑡)𝑀+maxdiam𝑋𝛼₁(𝑡),diam𝑋𝛼₂(𝑡)+4𝑚(𝑡)𝑛(𝑡)𝑟₀𝜑2𝑟₀Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1+(𝑡)𝐹d𝑠4𝑚₃(𝑡)𝑟₀𝜑2𝑟₀Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1+(𝑡)𝑛(𝑡)𝑟d𝑠2𝑚₀𝜑4𝑟₀Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1+𝑛(𝑡𝐹d𝑠)₂(𝑡)𝜑4𝑟₀Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1=𝛼d𝑠𝐿maxdiam𝑋₁(𝑡),diam𝑋𝛼₂(𝑡)𝑀+maxdiam𝑋𝛼₁(𝑡),diam𝑋𝛼₂(𝑡)+2𝑎(𝑡)𝑟₀𝜑2𝑟₀Γ()+𝑎(𝑡)𝑟𝛼+1⁰𝜑4𝑟₀Γ()+𝑐(𝑡)𝜑𝛼+14𝑟₀Γ()+(𝑡)𝑟𝛼+12𝑏₀𝜑2𝑟₀Γ().𝛼+1 (49) In view of assumptions (𝐻₀) and (𝐻₆) this yields limsup_𝑡→∞(𝑡)≤diam𝑄𝑋𝐿limsup_𝑡→∞maxdiam𝑋𝛼₁(𝑡),diam𝑋𝛼₂(𝑡)𝑀+limsup_𝑡→∞maxdiam𝑋𝛼₁(𝑡),diam𝑋𝛼₂(𝑡)≤𝐿limsup_𝑡→∞diam𝑋(𝑡)𝑀+limsup_𝑡→∞(𝑡).diam𝑋 (50) Further, using the measure of noncompactness 𝜇_𝑎 defined by the formula (9) and keeping in mind the estimates (48) and (50), one obtains 𝜇_𝑎()𝜔𝑄𝑋=max₀(𝑄𝑋),limsup_𝑡→∞maxdiam(𝑄𝑋)≤max𝐿𝜔₀(𝑋)𝑀+𝜔₀(𝑋),𝐿limsup_𝑡→∞diam𝑋(𝑡)𝑀+limsup_𝑡→∞(𝑡)≤𝜔diam𝑋𝐿max₀(𝑋),limsup_𝑡→∞diam𝑋(𝑡)𝑀+max𝜔₀(𝑋),limsup_𝑡→∞(𝑡=diam𝑋)𝐿𝜇_𝑎(𝑋)𝑀+𝜇_𝑎(𝑋). (51) Since 𝐿≤𝑀 in view of assumption (𝐻₂), from the above estimate one infers that 𝜇_𝑎(𝑄𝑋)≤𝜑(𝜇_𝑎(𝑋)), where 𝜑(𝑟)=𝐿𝑟/(𝑀+𝑟)<𝑟 for 𝑟>0. Hence, apply Theorem 5 to deduce that the operator 𝑄 has a fixed point 𝑥 in the ball 𝐵_𝑟0. On the other hand, from Remark 6 one concludes that the set Fix(𝑄) belongs to the family ker𝜇_𝑎. Now, taking into account the description of sets belonging to ker𝜇_𝑎 (given in Section 2) one deduces that all solutions for (21) are globally uniformly attractive on ℝ₊. This completes the proof.

Remark 20. When 𝑞≡0, 𝑓₁(𝑡,𝑥,𝑦)=𝑓₁(𝑡,𝑥), 𝑓₂(𝑡,𝑥,𝑦)=𝑓₂(𝑡,𝑥) and 𝑓₃(𝑡,𝑠,𝑥,𝑦)=𝑓₃(𝑡,𝑠,𝑥), according to our Theorem 19, one can obtain the global attractivity result for (23) which has been studied by Banaś and O’Regan in [47]. Meanwhile when 𝑞(𝑡)𝑓=0,₁𝑡,𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡)𝜂(𝑡),𝑓=𝑔𝑡,𝑥₂𝑡,𝑥𝛽₁(𝑡),𝑥𝛽₂(𝑡𝛽(𝑡,𝑓)=𝑓𝑡,𝑥)₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)=ℎ(𝑡,𝑠)𝑢𝑠,𝑥𝛾(𝑠), (52) from our Theorem 19, one can obtain the global attractivity result for (23) which has been studied by Balachandran et al. in [48].
To prove next result concerning the asymptotic positivity of the attractive solutions, we need the following hypothesis in the sequel.

(𝐻₈) The functions 𝑞 and 𝑓₁ satisfy lim_𝑡→∞[|𝑞(𝑡)|−𝑞(𝑡)]=0, lim_𝑡→∞[|𝑓₁(𝑡,𝑥,𝑦)|−𝑓₁(𝑡,𝑥,𝑦)]=0 for all 𝑥,𝑦∈ℝ.

Theorem 21. Under the hypotheses of Theorem 19 and (H₈), (21) has at least one solution on ℝ₊. Moreover, these solutions are uniformly globally attractive and ultimately positive on ℝ₊.

Proof. Consider the closed ball 𝐵_𝑟0 in the Banach space 𝐵𝐶(ℝ₊,ℝ), where the real number 𝑟₀ is given as in the proof of Theorem 19, and define a mapping 𝑄∶𝐵𝐶(ℝ₊,ℝ)→𝐵𝐶(ℝ₊,ℝ) by (36). Then it is shown as in the proof of Theorem 19 that 𝑄 defines a continuous mapping from the space 𝐵𝐶(ℝ₊,ℝ) into 𝐵_𝑟0. In particular, 𝑄 maps 𝐵_𝑟0 into itself. Next we show that 𝑄 is a nonlinear 𝐷-set-contraction with respect to the measure 𝜇_ad of noncompactness in 𝐵𝐶(ℝ₊,ℝ). We know that, for any 𝑥,𝑦∈ℝ, one has the inequality, |𝑥|+|𝑦|≥|𝑥+𝑦|≥𝑥+𝑦, and therefore, −≤𝑥|+𝑦−≤𝑥||+𝑦𝑥+𝑦𝑥+𝑦|𝑥+𝑦||−𝑥−𝑦 (53) for all 𝑥,𝑦∈ℝ. Now for any 𝑥∈𝐵_𝑟0, one has (𝑡(𝑡≤𝑞(𝑡(𝑡+𝑓||𝑄𝑥)|−𝑄𝑥)|)−𝑞)₁𝑡,𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡)−𝑓₁𝑡,𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡)+𝑓₂𝑡,𝑥𝛽₁(𝑡),𝑥𝛽₂(𝑡)Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)−𝑓d𝑠₂𝑡,𝑥𝛽₁(𝑡),𝑥𝛽₂(𝑡)Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)d𝑠≤𝛿_𝑇𝑞+𝛿_𝑇𝑓₁𝑓+2²𝑡,𝑥𝛽₁(𝑡),𝑥𝛽₂(𝑡)Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)d𝑠≤𝛿_𝑇𝑞+𝛿_𝑇𝑓₁+2𝑎(𝑡𝑥‖𝜑(2‖𝑥(𝑡𝑥‖(𝑡)𝜑(2‖𝑥(𝑡×(Γ()‖‖)+𝑏)‖+𝑐‖)+𝑑)𝛼+1))⁻¹. (54) From the above inequality, it follows that 𝛿_𝑇(𝑋)≤𝛿_𝑇𝑞+𝛿_𝑇𝑓₁+2𝑎(𝑡𝑥‖𝜑(2‖𝑥(𝑡𝑥‖(𝑡)𝜑(2‖𝑥(𝑡×(Γ()‖‖)+𝑏)‖+𝑐‖)+𝑑)𝛼+1))⁻¹. (55) For all closed 𝑋⊂𝐵_𝑟0. Taking the limit superior as 𝑇→∞, one obtains limsup_𝑇→∞𝛿_𝑇(𝑋)≤limsup_𝑇→∞𝛿_𝑇𝑞+limsup_𝑇→∞𝛿_𝑇𝑓₁ (56) for all closed 𝑋⊂𝐵_𝑟0. Hence 𝛿(𝑄𝑋)=lim_𝑇→∞𝛿_𝑇(𝑋)=0 for all closed subsets 𝑋 of 𝐵_𝑟0. Further, using the measure of noncompactness 𝜇_𝑎 defined by the formula (9) and keeping in mind the estimates (50) and (51), one obtains 𝜇_ad(𝑄𝑋)=max𝜇_𝑎(𝑄𝑋),𝛿(𝑄𝑋)≤max𝐿𝜇_𝑎(𝑋)𝑀+𝜇_𝑎(𝑋)≤𝜇,0𝐿max_𝑎(𝑋),0𝑀+max𝜇_𝑎(𝑋)≤,0𝐿𝜇_ad(𝑋)𝑀+𝜇_ad(𝑋). (57) Since 𝐿≤𝑀 in view of assumption (𝐻₂), from the above estimate one infers that 𝜇_ad(𝑄𝑋)≤𝜑(𝜇_ad(𝑄𝑋)), where 𝜑(𝑟)=𝐿𝑟/(𝑀+𝑟)<𝑟 for 𝑟>0. Hence, applying Theorem 5 to deduce that the operator 𝑄 has a fixed point 𝑥 in the ball 𝐵_𝑟0. Obviously 𝑥 is a solution of (21). Now, taking into account the description of sets belonging to ker𝜇_ad (given in Section 2) one deduces that all solutions of (21) are uniformly globally attractive and ultimately positive on ℝ₊. This completes the proof.

Next we prove the global asymptotic attractivity results for (21). We need the following hypotheses in the sequel.

(𝐻₉) The function 𝑞∶ℝ₊→ℝ is continuous and lim_𝑡→∞𝑞(𝑡)=𝑐.

(𝐻₁₀)𝑓₁(𝑡,0,0)=0 for all 𝑡∈ℝ₊, and

(𝐻₁₁)lim_𝑡→∞ℓ(𝑡)=0, where the function ℓ is defined as in hypothesis (𝐻₂).

Theorem 22. Assume that the hypotheses (H₀), (H₂)–(H₇) and (H₉)–(H₁₁) hold. Then (21) has at least one solution in the space 𝐵𝐶(ℝ₊,ℝ). Moreover, these solutions are uniformly globally asymptotically attractive on ℝ₊.

Proof. Consider the closed ball 𝐵_𝑟0 in the Banach space 𝐵𝐶(ℝ₊,ℝ), where the real number 𝑟₀ is given as in the proof of Theorem 19 and define a mapping 𝑄∶𝐵_𝑟0→𝐵_𝑟0 by (36). Then 𝑄 is continuous and maps the space 𝐵𝐶(ℝ⁺,ℝ) and, in particular, 𝐵_𝑟0 into 𝐵_𝑟0. We show that 𝑄 is a nonlinear 𝐷-set-contraction with respect to the measure 𝜇_𝑐 of noncompactness in 𝐵𝐶(ℝ₊,ℝ). Let 𝑥∈𝐵_𝑟0 be arbitrary. Then one has |(𝑡)|≤𝑞(𝑡)+𝑓𝑄𝑥−𝑐−𝑐₁𝑡,𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡)+𝑓₂𝑡,𝑥𝛽₁(𝑡),𝑥𝛽₂(𝑡)Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)≤𝑞(𝑡)+𝑓d𝑠−𝑐₁𝑡,𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡+𝑓₂)𝑡,𝑥𝛽₁(𝑡),𝑥𝛽₂(𝑡)Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)≤𝑞(𝑡)+𝑓d𝑠−𝑐₁𝑡,𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡)−𝐹₁(𝑡+1)Γ(𝛼)𝑓₂𝑡,𝑥𝛽₁(𝑡),𝛽₂(𝑡)−𝐹₂(𝑡)+𝐹₂(𝑡×_𝑡0)(𝑡−𝑠)^𝛼−1𝑓₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)−𝑓₃(𝑡,𝑠,0,0)+𝐹₃(𝑡≤𝑞(𝑡)+ℓ(𝑡)𝑥𝛼)d𝑠−𝑐max₁(𝑡),𝑥𝛼₂(𝑡)𝑀+max𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡)+2‖𝑥‖𝑚(𝑡)+𝐹₂(𝑡)Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1𝑛(𝑡)𝜑(2‖𝑥‖)+𝐹₃(𝑡≤𝑞(𝑡)+ℓ(𝑡)𝑥𝛼)d𝑠−𝑐max₁(𝑡),𝑥𝛼₂(𝑡)𝑀+max𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡)+2‖𝑥‖𝑚(𝑡)+𝐹₂(𝑡)Γ(𝛼)𝑛(𝑡)𝜑(2‖𝑥‖)+𝐹₃(𝑡×_𝑡0)(𝑡−𝑠)^𝛼−1≤𝑞(𝑡)+ℓ(𝑡)𝑥𝛼d𝑠−𝑐max₁(𝑡),𝑥𝛼₂(𝑡)𝑀+max𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡)+𝑡^𝛼2𝑚(𝑡)𝑛(𝑡)‖𝑥‖𝜑(2‖𝑥‖)+2‖𝑥‖𝑚(𝑡)𝐹₃(𝑡)+𝐹₂(𝑡𝑛(𝑡)𝜑(2‖𝑥+𝐹)‖)₂(𝑡)𝐹₃(𝑡)×(Γ(𝛼+1))⁻¹≤𝑞(𝑡)+ℓ(𝑡)𝑥𝛼−𝑐max₁(𝑡),𝑥𝛼₂(𝑡)𝑀+max𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡)+𝑎(𝑡)‖𝑥‖𝜑(2‖𝑥‖)+𝑏(𝑡)‖𝑥‖+𝑐(𝑡)𝜑(2‖𝑥‖)+𝑑(𝑡)Γ()=𝑞(𝑡)+ℓ(𝑡)𝑥𝛼𝛼+1−𝑐max₁(𝑡),𝑥𝛼₂(𝑡)𝑀+max𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡)+𝑣(𝑡), (58) for all 𝑡∈ℝ₊. This further implies that ‖𝑄𝑋(𝑡)−𝑐‖≤|𝑞(𝑡)−𝑐|+ℓ(𝑡)+𝑣(𝑡). Taking the limit superior in the above inequality, one obtains limsup_𝑡→∞‖𝑄𝑋(𝑡)−𝑐‖≤limsup_𝑡→∞𝑞(𝑡)−𝑐+limsup_𝑡→∞ℓ(𝑡)+limsup_𝑡→∞𝑣(𝑡). (59) Further, using the measure of noncompactness 𝜇_𝑐 defined by the formula (11) and keeping in mind the estimates (48) and (59), one obtains 𝜇_𝑐(𝑄𝑋)=max𝜔₀(𝑄𝑋),limsup_𝑡→∞‖𝑄𝑋(𝑡)−𝑐‖≤max𝐿𝜔₀(𝑋)𝑀+𝜔₀(𝑋)≤𝜔,0𝐿max₀(𝑋),0𝑀+max𝜔₀(𝑋)≤,0𝐿𝜇_𝑐(𝑋)𝑀+𝜇_𝑐(𝑋). (60) Since 𝐿≤𝑀 in view of assumption (𝐻₂), from the above estimate one infers that 𝜇_𝑐(𝑄𝑋)≤𝜑(𝜇_𝑐(𝑄𝑋)), where 𝜑(𝑟)=𝐿𝑟/(𝑀+𝑟)<𝑟 for 𝑟>0. Hence, apply Theorem 5 to deduce that the operator 𝑄 has a fixed point 𝑥 in the ball 𝐵_𝑟0. Obviously 𝑥 is a solution of the fractional functional integral equation (21). Now, taking into account the description of sets belonging to ker𝜇_𝑐 (given in Section 2) one deduces that all solutions of (21) are uniformly globally asymptotically attractive on ℝ₊. This completes the proof.

Theorem 23. Under the hypotheses of Theorem 22 and (H₈), (21) has at least one solution on ℝ₊. Moreover, solutions of (21) are uniformly globally asymptotically attractive and ultimately positive on ℝ₊.

Proof. The proof is similar to Theorem 21 with appropriate modifications. Now the desired conclusion follows an application of the measure of noncompactness 𝜇_cd in 𝐵𝐶(ℝ₊,ℝ). This completes the proof.

4. Applications

In what follows, we show that the assumptions imposed in Theorems 19 and 21 admit some natural realizations. First, we indicate some possible forms for expressing the function 𝑓₁ that satisfies the hypothesis (𝐻₂). Define a class Θ of functions 𝜃∶ℝ₊→ℝ₊ satisfying the following properties: (i) 𝜃 is continuous; (ii) 𝜃 is nondecreasing (iii) 𝜃 is subadditive, that is, 𝜃(𝑥+𝑦)≤𝜃(𝑥)+𝜃(𝑦) for all 𝑥,𝑦∈ℝ₊.

Notice that if 𝜃∈Θ, then after simple computation it can be shown that |𝜃(𝑥)−𝜃(𝑦)|≤𝜃(|𝑥−𝑦|) for all 𝑥,𝑦∈ℝ₊. Now consider the function 𝑓₁∶ℝ₊×ℝ×ℝ→ℝ defined by 𝑓₁(𝑡)𝜃𝑡,𝑥,𝑦=ℓ¹(|𝑥|)+𝜃₂𝑦2𝑀+𝜃₁(|𝑥|)+𝜃₂𝑦+𝑚(𝑡), (61) where the functions ℓ,𝑚∶ℝ₊→ℝ are continuous and bounded on ℝ₊, that is, ℓ,𝑚∈𝐵𝐶(ℝ₊,ℝ) with sup_𝑡≥0ℓ(𝑡)=𝐿, 𝜃₁,𝜃₂∈Θ satisfying 𝜃₁(𝑟)≤𝑟, 𝜃₂(𝑟)≤𝑟, and 𝑀 is a positive constant such that 𝐿≤𝑀. It is shown as in the work of Dhage [11] that the function 𝑓₁ satisfies the condition (34) and consequently the hypothesis (𝐻₂). There do exist functions 𝜃 given in the expression (61). Indeed, the following functions 𝜃(𝑟)(𝑟)(),𝜃(𝑟)(𝑟)(𝑟)√=𝑟,𝜃=ln1+𝑟=arctan,𝜃=21+𝑟−1 (62) satisfy all the requirements of 𝜃₁ and 𝜃₂ given in (61) (cf. in the work of Banaś and Dhage [49]).

Finally, we provide two examples of the nonlinear fractional functional integral equations of the form (21) for which there are global attractive and ultimate positive solutions.

Example 24. Consider the following nonlinear functional integral equation: 𝑥(𝑡)=12𝑡𝑒^−𝑡2/2+𝑡²+1𝑡²×𝑥(𝑡𝑥(+4arctan(|)|)+arctan(|2𝑡)|)𝑥(𝑡𝑥(+√9+arctan(|)|)+arctan(|2𝑡)|)𝑡+𝑡[𝑥(𝑡)+𝑥(2𝑡)]Γ()×2/3_𝑡0𝑒^{−3𝑡−𝑠3𝑥2(𝑠)+𝑥2(2𝑠)/4}+1/10𝑡^8/3+1(𝑡−𝑠)^1/3d𝑠. (63)
Observe that the above equation is a special case of the fractional functional integral equation (21). Indeed, if we put 𝛼=2/3 and 𝛼₁(𝑡)=𝛽₁(𝑡)=𝛾₁(𝑡)𝛼=𝑡,₂(𝑡)=𝛽₂(𝑡)=𝛾₂(𝑡)(𝑡)=1=2𝑡,𝑞2𝑡𝑒^−𝑡2/2,𝑓₁𝑡,𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡=𝑡₂)+1𝑡²×𝑥(𝑡𝑥(+4arctan(|)|)+arctan(|2𝑡)|)𝑥(𝑡𝑥(,𝑓9+arctan(|)|)+arctan(|2𝑡)|)₂𝑡,𝑥𝛽₁(𝑡),𝑥𝛽₂(𝑡)=𝑡+√𝑡[𝑥(𝑡)(,𝑓+𝑥2𝑡)]₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)=𝑒^{𝑒3−3𝑡−𝑠𝑥22(𝑠)+𝑥(2𝑠)/4}+110𝑡^8/3.+1 (64) Obviously the functions 𝛼₁, 𝛼₂, 𝛽₁, 𝛽₂ and 𝛾₁, 𝛾₂ satisfy hypothesis (𝐻₀). Further notice that the function 𝑞(𝑡)=(1/2)𝑡𝑒^−𝑡2/2 is continuous and bounded on ℝ₊ with ‖𝑞‖=𝑞(1)=(1/2)𝑒^−1/2=0.30327…. Thus assumption (𝐻₁) is satisfied. On the other hand, the function 𝑓₁(𝑡,𝑥(𝛼₁(𝑡)),𝑥(𝛼₂(𝑡))) has the form (61) with ℓ(𝑡)=(𝑡²+1)/(𝑡²+4). Moreover, 𝜃(𝑟)=arctan(𝑟) and 𝑀=9/2. Since ‖ℓ‖=𝐿=1 one has that 𝐿≤𝑀. Additionally one has that the function 𝜃 satisfies above discussed requirements of the class of functions Θ, so the function 𝑓₁(𝑡,𝑥(𝛼₁(𝑡)),𝑥(𝛼₂(𝑡))) satisfies assumption (𝐻₂) and (𝐻₄).
Further observe that the function 𝑓₂(𝑡,𝑥(𝛽₁(𝑡)),𝑥(𝛽₂(𝑡))) satisfies assumption (𝐻₃) with 𝑚(𝑡)=√𝑡 and |𝐹₂(𝑡)|=𝑓₂(𝑡,0,0)=𝑡. Next, the function 𝑓₃(𝑡,𝑠,𝑥(𝛾₁(𝑠)),𝑥(𝛾₂(𝑠))) satisfies assumption (𝐻₅), where 𝑛(𝑡)=𝑒^−3𝑡, 𝜑(𝑟)=√³𝑟²/4 and 𝑓₃(𝑡,𝑠,0,0)=1/(10𝑡^8/3+1). Thus 𝐹₃(𝑡)=1/(10𝑡^8/3+1). To check that assumption (𝐻₆) is satisfied let us observe that the functions 𝑎, 𝑏, 𝑐, 𝑑 appearing in that assumption take the form 𝑎(𝑡)=2𝑡^7/6𝑒^−3𝑡,𝑏(𝑡)=2𝑡^7/610𝑡^8/3,𝑐(𝑡)+1=𝑡^5/3𝑒^−3𝑡(𝑡)=𝑡,𝑑^5/310𝑡^8/3.+1 (65) Thus, it is easily seen that 𝑎(𝑡)→0 as 𝑡→∞ and 𝐴=𝑎(7/18)=2(7/18)^7/6𝑒^−7/6=0.2069266…. Further one has that 𝑏(𝑡)→0 as 𝑡→∞ and 𝐵=𝑏((7/90)^3/8)=0.0594821…. It is also easy to check that 𝑐(𝑡)→0 as 𝑡→∞. Moreover, one has that 𝐶=𝑐(5/9)=(5/9)^5/3𝑒^−5/3=0.0709235…. Also one sees that 𝑑(𝑡)→0 as 𝑡→∞ and 𝐷=𝑑((1/6)^3/8)=0.1223733…. Finally, let us note that the inequality from assumption (𝐻₇) has the form 12𝑒^−1/2+𝑟+()()9+𝑟𝐴𝑟𝜑2𝑟+𝐵𝑟+𝐶𝜑2𝑟+𝐷Γ(5/3)<𝑟. (66) Let us write this inequality in the form 12Γ53𝑒^−1/2+Γ(5/3)𝑟()()59+𝑟+𝐴𝑟𝜑2𝑟+𝐵𝑟+𝐶𝜑2𝑟+𝐷<Γ3𝑟. (67) Denoting by 𝐿(𝑟) the left-hand side of this inequality, that is, 𝐿(𝑟)=12Γ53𝑒^−1/2+Γ(5/3)𝑟()()9+𝑟+𝐴𝑟𝜑2𝑟+𝐵𝑟+𝐶𝜑2𝑟+𝐷 (68) and keeping in mind the above established values of 𝐴, 𝐵, 𝐶, 𝐷, for 𝑟=1 one obtains 𝐿(1)=12Γ53𝑒^−1/2+Γ(5/3)(2)(2)=110+𝐴𝜑+𝐵+𝐶𝜑+𝐷2Γ53Γ()0.60653⋯+5/310+0.963727…. (69) Hence, taking into account that Γ(5/3)>0.8856 (cf. [50]), one obtains that the number 𝑟₀=1 is a solution of the inequality (63).
Now, based on Theorem 19 one can conclude that the functional integral equation (63) has solutions in the space 𝐵𝐶(ℝ₊,ℝ) and all solutions of this equation are uniformly globally attractive on ℝ₊. Furthermore, 𝑓₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)=𝑒^{−3𝑡−𝑠3𝑥2(𝑠)+𝑥2(2𝑠)/4}+110𝑡^8/3+1=𝑓₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠) (70) for all 𝑡∈ℝ₊ and 𝑥,𝑦∈ℝ. Hence the functions 𝑞 and 𝑓₃(𝑡,𝑠,𝑥(𝛾₁(𝑠)),𝑥(𝛾₂(𝑠))) satisfy the hypothesis (𝐻₈). Hence by Theorem 21, solutions of (63) are uniformly globally attractive and ultimately positive on ℝ₊.

Example 25. Consider the following nonlinear functional integral equation: 𝑥(𝑡)=12𝑡𝑒^−𝑡2/2+𝑒^−𝑡arctan(|𝑥(𝑡)|)+arctan(|𝑥(2𝑡)|)𝑥(𝑡𝑥(+√9+arctan(|)|)+arctan(|2𝑡)|)𝑡+𝑡[𝑥(𝑡)+𝑥(2𝑡)]Γ()×2/3_𝑡0𝑒^{−3𝑡−𝑠3𝑥2(𝑠)+𝑥2(2𝑠)/4}+1/10𝑡^8/3+1(𝑡−𝑠)^1/3d𝑠. (71) Observe that (71) is a special case of (21), where one has 𝛼₁(𝑡)=𝛽₁(𝑡)=𝛾₁(𝑡)𝛼=𝑡,₂(𝑡)=𝛽₂(𝑡)=𝛾₂(𝑡)(𝑡)=1=2𝑡,𝑞2𝑡𝑒^−𝑡2/2,𝑓₁𝑡,𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡)=𝑒^−𝑡arctan(|𝑥(𝑡)|)+arctan(|𝑥(2𝑡)|)𝑥(𝑡𝑥(,𝑓9+arctan(|)|)+arctan(|2𝑡)|)₂𝑡,𝑥𝛽₁(𝑡),𝑥𝛽₂(𝑡)=𝑡+√𝑡[𝑥(𝑡)(,𝑓+𝑥2𝑡)]₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)=𝑒^{−3𝑡−𝑠3𝑥2(𝑠)+𝑥2(2𝑠)/4}+110𝑡^8/3.+1 (72) Obviously, the functions 𝛼₁, 𝛼₂, 𝛽₁, 𝛽₂ and 𝛾₁, 𝛾₂ satisfy hypothesis (𝐻₀) and it is shown as in Example 24 that assumption (𝐻₁) is satisfied. Further, notice that the function 𝑓₁(𝑡,𝑥(𝛼₁(𝑡)),𝑥(𝛼₂(𝑡))) has the form (61) with ℓ(𝑡)=𝑒^−𝑡 and lim_𝑡→∞ℓ(𝑡)=lim_𝑡→∞𝑒^−𝑡=0. Moreover, 𝜃(𝑟)=arctan(𝑟), 𝑀=9/2. Since ‖ℓ‖=1 one has that 𝐿≤𝑀. Additionally one has that 𝜃∈Θ, so the function 𝑓₁(𝑡,𝑥(𝛼₁(𝑡)),𝑥(𝛼₂(𝑡))) satisfies assumption (𝐻₂) and (𝐻₄).
Finally, it is shown as in Example 24 that the functions 𝑓₂(𝑡,𝑥(𝛽₁(𝑡)),𝑥(𝛽₂(𝑡))) and 𝑓₃(𝑡,𝑠,𝑥(𝛾₁(𝑠)),𝑥(𝛾₂(𝑠))) are continuous on ℝ₊×ℝ×ℝ and ℝ₊×ℝ₊×ℝ×ℝ, respectively; moreover, they satisfy hypotheses (𝐻₃) and (𝐻₅)--(𝐻₇). Now, based on Theorem 22 one concludes that the fractional functional integral equation (61) has solutions in the space 𝐵𝐶(ℝ₊,ℝ) and all solutions of this equation are uniformly globally asymptotically attractive on ℝ₊. Furthermore, 𝑓₁𝑡,𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡)=𝑒^−𝑡arctan(|𝑥(𝑡)|)+arctan(|𝑥(2𝑡)|)9+arctan(|𝑥(𝑡)|)+arctan(|𝑥(2𝑡)|)=𝑓₁𝑡,𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡) (73) for all 𝑡∈ℝ₊ and 𝑥,𝑦∈ℝ. Hence the functions 𝑞 and 𝑓₁(𝑡,𝑥(𝛼₁(𝑡)),𝑥(𝛼₂(𝑡))) satisfy the hypotheses (𝐻₈)--(𝐻₁₁). Hence by Theorem 23, solutions of (71) are uniformly globally asymptotically attractive and ultimately positive on ℝ₊.

Remark 26. Note that the global existence as well as attractivity and positivity results of (21) can be extended to the following fractional functional integral equation: 𝑥(𝑡)=𝑞(𝑡)+𝑓₁𝑡,𝑥𝛼₁(𝑡),𝑥𝛼₂(𝑡),…,𝑥𝛼_𝑛(𝑡)+𝑓₂𝑡,𝑥𝛽₁(𝑡),𝑥𝛽₂(𝑡),…,𝑥𝛽_𝑛(𝑡)Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₃𝑡,𝑠,𝑥𝛾₁(𝑠),𝑥𝛾₂(𝑠)𝑥𝛾,…,_𝑛(𝑠)d𝑠, (74) with similar method under appropriate modifications. Then so obtained results are useful in determining the global attractivity and positivity and global asymptotic attractivity and positivity of solutions for the fractional functional integral equations defined, respectively, by 𝑥(𝑡)=12𝑡𝑒^−𝑡2/2+𝑡²+1𝑡²×∑+4^𝑛𝑖=1arctan(|𝑥(𝑖𝑡)|)9+arctan∑^𝑛_𝑖=1|𝑥(+√𝑖𝑡)|𝑡+𝑡∑^𝑛_𝑖=1𝑥(𝑖𝑡)Γ()×2/3_𝑡0𝑒^{−3𝑡−𝑠3∑𝑛𝑖=1𝑥2(𝑖𝑠)/4}+1/10𝑡^8/3+1(𝑡−𝑠)^1/3𝑥(𝑡)=1d𝑠,2𝑡𝑒^−𝑡2/2+𝑒^−𝑡∑^𝑛_𝑖=1arctan(|𝑥(𝑖𝑡)|)9+arctan∑^𝑛_𝑖=1𝑥(+√(|𝑖𝑡)|)𝑡+𝑡∑^𝑛_𝑖=1𝑥(𝑖𝑡)Γ()×2/3_𝑡0𝑒^{−3𝑡−𝑠3∑𝑛𝑖=1𝑥2(𝑖𝑠)/4}+1/10𝑡^8/3+1(𝑡−𝑠)^1/3d𝑠. (75)

Acknowledgment

This work is supported by the Guangdong Natural Science Foundation (no. S2011010004447).