The Existence and Attractivity of Solutions of an Urysohn Integral Equation on an Unbounded Interval

Abstract

We prove a result on the existence and uniform attractivity of solutions of an Urysohn integral equation. Our considerations are conducted in the Banach space consisting of real functions which are bounded and continuous on the nonnegative real half axis. The main tool used in investigations is the technique associated with the measures of noncompactness and a fixed point theorem of Darbo type. An example showing the utility of the obtained results is also included.

1. Introduction

The theory of nonlinear functional integral equations creates an important branch of the modern nonlinear analysis. The large part of that theory describes a lot of classical nonlinear integral equations such as nonlinear Volterra integral equations, Hammerstein integral equations, and Urysohn integral equations with solutions defined on a bounded interval (cf. [1–4]).

Nevertheless, more important and simultaneously, a more difficult part of that theory is connected with the study of solutions of the mentioned integral equations defined on an unbounded domain. Obviously, there are some known results concerning the existence of solutions of those integral equations in such a setting but, in general, they are mostly obtained under rather restrictive assumptions [2, 4–8].

On the other hand, the use of some tools of nonlinear analysis enables us to obtain several valuable results under less restrictive assumptions (cf. [1, 9–15]). It turns out that the technique of measures of noncompactness creates a very convenient tool for the study of the solvability of nonlinear functional integral equations of various types. It is caused by the fact that the approach to the study of solutions of those equations with the use of the technique of measures of noncompactness gives not only the possibility to obtain existence results, but also allows us to look for solutions of mentioned equations having some desired properties such as monotonicity, attractivity, and asymptotic stability (cf. [16–19], for instance).

In the paper, we will use the above described approach associated with the technique of measures of noncompactness in order to obtain a result on the existence of solutions of a quadratic Urysohn integral equation. Applying the mentioned technique in conjunction with a fixed point theorem of Darbo type, we show that the equation in question has solutions defined, continuous, and bounded on the nonnegative real half axis ℝ₊ which are uniformly attractive (asymptotic stable) on ℝ₊.

The results obtained in this paper generalize several results obtained earlier in numerous papers treating nonlinear functional integral equations, which were quoted above. Particularly, we generalize the results concerning the Urysohn or Hammerstein integral equations obtained in the papers [9, 10, 20].

2. Notation, Definitions, and Auxiliary Facts

In this section, we establish some notations, and we collect auxiliary facts which will be used in the sequel.

By the symbol ℝ we denote the set of real numbers, while ℝ₊ stands for the half axis [0,∞). Further, assume that (𝐸,‖⋅‖) is a given real Banach space with the zero element 𝜃. For 𝑥∈𝐸 and for a fixed 𝑟>0, denote by 𝐵(𝑥,𝑟) the closed ball centered at 𝑥 and with radius 𝑟. We write 𝐵_𝑟 in order to denote the ball 𝐵(𝜃,𝑟).

Moreover, if 𝑋 and 𝑌 are nonempty subsets of 𝐸 and 𝜆∈ℝ, then we denote by 𝑋+𝑌, 𝜆𝑋, the usual algebraic operations on sets. If 𝑋 is a subset of 𝐸 then the symbols 𝑋 and Conv𝑋 denote the closure and closed convex hull of 𝑋, respectively. Apart from this, we denote by 𝔐_𝐸 the family of all nonempty and bounded subsets of 𝐸 and by 𝔑_𝐸 its subfamily consisting of all relatively compact sets.

In what follows, we will accept the following definition of the concept of a measure of noncompactness [21].

Definition 1. A mapping 𝜇∶𝔐_𝐸→ℝ₊ 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) is a sequence of closed sets from 𝔐_𝐸 such that 𝑋_𝑛+1⊂𝑋_𝑛 for 𝑛=1,2,… and if lim_𝑛→∞𝜇(𝑋_𝑛)=0, then the intersection set 𝑋_∞=⋂^∞_𝑛=1𝑋_𝑛 is nonempty.
The family ker𝜇 appearing in 1° is called the kernel of the measure of noncompactness  𝜇.
Observe that the set 𝑋_∞ from the axiom 5^° is a member of the family ker𝜇. Indeed, since 𝜇(𝑋_∞)≤𝜇(𝑋_𝑛) for any natural number 𝑛, we infer that 𝜇(𝑋_∞)=0. Consequently, 𝑋_∞∈ker𝜇. This simple observation will be essential in our further investigations.
Now, we formulate a fixed point theorem of Darbo type which will be used further on [21].

Theorem 2. Let Ω be a nonempty, bounded, closed, and convex subset of the Banach space 𝐸, and let 𝐹∶Ω→Ω be a continuous mapping. Assume that there exists a constant 𝑘∈[0,1) such that 𝜇(𝐹𝑋)≤𝑘𝜇(𝑋) for any nonempty subset 𝑋 of Ω. Then, 𝐹 has a fixed point in the set Ω.

Remark 3. Denote by Fix𝐹 the set of all fixed points of the operator 𝐹 belonging to Ω. It can be easily seen [21] that the set Fix𝐹 belongs to the family ker𝜇.
In what follows, we will work in the Banach space 𝐵𝐶(ℝ₊) consisting of all real functions 𝑥=𝑥(𝑡) defined, continuous, and bounded on ℝ₊. This space will be endowed with the standard supremum norm ‖𝑥‖=sup|𝑥(𝑡)|∶𝑡∈ℝ₊. (1)
Now, we recall the construction of a measure of noncompactness in the space 𝐵𝐶(ℝ₊) which was introduced in [21]. To this end, fix a nonempty and bounded subset 𝑋 of the space 𝐵𝐶(ℝ₊) and positive numbers 𝜀>0, 𝑇>0. For 𝑥∈𝑋, denote by 𝜔^𝑇(𝑥,𝜀)  the modulus of continuity of the function 𝑥 on the interval [0,𝑇]; that is, 𝜔^𝑇(𝑥,𝜀)=sup{|𝑥(𝑡)−𝑥(𝑠)|∶𝑡,𝑠∈[0,𝑇],|𝑡−𝑠|≤𝜀}. (2) Next, we put 𝜔^𝑇(𝑋,𝜀)=sup𝜔^𝑇(),𝜔𝑥,𝜀∶𝑥∈𝑋_𝑇0(𝑋)=lim_𝜀→0𝜔^𝑇(),𝜔𝑋,𝜀₀(𝑋)=lim_𝑇→∞𝜔^𝑇₀(𝑋). (3) Further, for a fixed number 𝑡∈ℝ₊ let us put 𝑋(𝑡)={𝑥(𝑡)},(𝑡)𝑥(𝑡)(𝑡.∶𝑥∈𝑋diam𝑋=sup−𝑦)∶𝑥,𝑦∈𝑋 (4) Finally, let us consider the function 𝜇 defined on the family 𝔐_{𝐵𝐶(ℝ+)} by the formula 𝜇(𝑋)=𝜔₀(𝑋)+limsup_𝑡→∞diam𝑋(𝑡). (5) It can be shown [21] that the function 𝜇 is a measure of noncompactness in the space 𝐵𝐶(ℝ₊). Moreover, the kernel ker𝜇of this measure consists of all nonempty and bounded subsets 𝑋 of 𝐵𝐶(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. This property in combination with Remark 3 permits us to characterize solutions of the integral equation considered in the sequel.
For further purposes, we introduce now the concept of attractivity (stability) of solutions of operator equations in the space 𝐵𝐶(ℝ₊). To this end, assume that Ω is a nonempty subset of the space 𝐵𝐶(ℝ₊). Moreover, let 𝐹 be an operator defined on Ω with values in 𝐵𝐶(ℝ₊). Let us consider the operator equation of the form 𝑥(𝑡)=(𝐹𝑥)(𝑡),𝑡∈ℝ₊. (6)

Definition 4. We say that solutions of (6) are attractive (or locally attractive) if there exists a ball 𝐵(𝑥₀,𝑟) in the space 𝐵𝐶(ℝ₊) such that 𝐵(𝑥₀,𝑟)∩Ω≠∅, and for arbitrary solutions 𝑥=𝑥(𝑡), 𝑦=𝑦(𝑡) of (6) belonging to the set 𝐵(𝑥₀,𝑟)∩Ω we have that lim_𝑡→∞𝑥(𝑡)−𝑦(𝑡)=0. (7) In the case when limit (7) is uniform with respect to the set 𝐵(𝑥₀,𝑟)∩Ω, that is, when for each 𝜀>0 there exists 𝑇>0 such that 𝑥(𝑡)−𝑦(𝑡)≤𝜀 (8) for all 𝑥,𝑦∈𝐵(𝑥₀,𝑟)∩Ω being solutions of (6) and for any 𝑡≥𝑇, we will say that solutions of (6) are uniformly attractive (or asymptotically stable).
Notice that the previous definition comes from [16, 19].

3. Main Result

We will consider the existence and asymptotic behaviour of solutions of the quadratic Urysohn integral equation having the form 𝑥(𝑡)=𝑎(𝑡)+𝑓(𝑡,𝑥(𝑡))^∞₀𝑢(𝑡,𝑠,𝑥(𝑠))𝑑𝑠 (9) for 𝑡∈ℝ₊.

In our study, we will impose the following assumptions.(i)𝑎∈𝐵𝐶(ℝ₊).(ii)𝑓∶ℝ₊×ℝ→ℝ is a continuous function, and the function 𝑡→𝑓(𝑡,0) is a member of the space 𝐵𝐶(ℝ₊).(iii)The function 𝑓=𝑓(𝑡,𝑥) satisfies the Lipschitz condition with respect to the second variable; that is, there exists a constant 𝑘>0 such that 𝑓(𝑡,𝑥)−𝑓𝑡,𝑦≤𝑘𝑥−𝑦 (10) for 𝑥,𝑦∈ℝ₊ and 𝑡∈ℝ₊.(iv)𝑢∶ℝ₊×ℝ₊×ℝ→ℝ is a continuous function, and there exists a continuous function 𝑔∶ℝ₊×ℝ₊→ℝ₊ and a continuous, nondecreasing function ℎ∶ℝ₊→ℝ₊ with lim_𝜀→0ℎ(𝜀)=0, such that 𝑢(𝑡,𝑠,𝑥)−𝑢𝑡,𝑠,𝑦≤𝑔(𝑡,𝑠)ℎ𝑥−𝑦 (11) for 𝑡,𝑠∈ℝ₊ and 𝑥,𝑦∈ℝ.(v)For each 𝑡∈ℝ₊, the functions 𝑠→𝑔(𝑡,𝑠) and 𝑠→|𝑢(𝑡,𝑠,0)| are integrable ℝ₊ and lim_𝑡→∞^∞₀𝑔(𝑡,𝑠)𝑑𝑠=0. (12) Moreover, the function 𝑡→∫^∞₀|𝑢(𝑡,𝑠,0)|𝑑𝑠 is bounded on ℝ₊.(vi)The following equalities hold: lim_𝑇→∞sup^∞_𝑇𝑔(𝑡,𝑠)𝑑𝑠∶𝑡∈[0,𝑇]=0,lim_𝑇→∞sup^∞_𝑇|𝑢(𝑡,𝑠,0)|𝑑𝑠∶𝑡∈[0,𝑇]=0. (13)

Let us observe that in view of assumptions (ii) and (v) we can define the following finite constants: 𝑓=sup𝑓(𝑡,0)∶𝑡∈ℝ₊,𝑔=sup^∞₀𝑔(𝑡,𝑠)𝑑𝑠∶𝑡∈ℝ₊,𝑢=sup^∞₀|𝑢(𝑡,𝑠,0)|𝑑𝑠∶𝑡∈ℝ₊. (14)

Remark 5. It is worthwhile mentioning that in the theory of improper Riemann integral with a parameter there has been considered the concept of the uniform convergence of the improper integral with respect to that parameter (cf. [22]). In order to recall this concept, suppose that there is a given function 𝑧(𝑡,𝑠)=𝑧∶ℝ₊×ℝ₊→ℝsuch that the improper integral ^∞₀𝑧(𝑡,𝑠)𝑑𝑠 (15) does exist for every fixed 𝑡∈ℝ₊.
We say that the integral (15) is uniformly convergent with respect to 𝑡∈ℝ₊ if lim_𝑇→∞^𝑇₀𝑧(𝑡,𝑠)𝑑𝑠=^∞₀𝑧(𝑡,𝑠)𝑑𝑠 (16) uniformly with respect to 𝑡∈ℝ₊.
Equivalently (cf. [10]), the integral (15) is uniformly convergent with respect to 𝑡∈ℝ₊ if lim_𝑇→∞sup_𝑡∈ℝ+^∞_𝑇𝑧(𝑡,𝑠)𝑑𝑠=0. (17)
Let us observe that if integrals appearing in assumption (v), that is, the integrals ^∞₀𝑔(𝑡,𝑠)𝑑𝑠,^∞₀|𝑢(𝑡,𝑠,0)|𝑑𝑠 (18) are uniformly convergent with respect to 𝑡∈ℝ₊, then the inequalities from assumption (vi) are satisfied (cf. [10]). Indeed, this conclusion follows easily from the inequalities sup_{[]𝑡∈0,𝑇}^∞_𝑇𝑔(𝑡,𝑠)𝑑𝑠≤sup_𝑡∈ℝ+^∞_𝑇𝑔(𝑡,𝑠)𝑑𝑠,sup_{𝑡∈[0,𝑇]}^∞_𝑇|𝑢(𝑡,𝑠,0)|𝑑𝑠≤sup_𝑡∈ℝ+^∞_𝑇|𝑢(𝑡,𝑠,0)|𝑑𝑠, (19) which are valid for any 𝑇>0.
It may be also shown that the converse implications are, in general, not valid [10].

Remark 6. It can be also shown [10] that the requirements concerning the function 𝑔(𝑡,𝑠) imposed in assumption (v) are independent, that is, there exist functions 𝑔_𝑖∶ℝ₊×ℝ₊→ℝ₊(𝑖=1,2) such that for each 𝑡∈ℝ₊ there exist the integrals ∫^∞₀𝑔_𝑖(𝑡,𝑠)𝑑𝑠(𝑖=1,2) and such that the integral ∫^∞₀𝑔₁(𝑡,𝑠)𝑑𝑠 is uniformly convergent but lim_𝑡→∞∫^∞₀𝑔₁(𝑡,𝑠)𝑑𝑠≠0 while lim_𝑡→∞∫^∞₀𝑔₂(𝑡,𝑠)𝑑𝑠=0, but the integral ∫^∞₀𝑔₂(𝑡,𝑠)𝑑𝑠 is not uniformly convergent.
Now, we formulate our last assumption.(vii)The inequality ‖𝑎‖+𝑘𝑔𝑟ℎ(𝑟)+𝑘𝑢𝑟+𝑓𝑔ℎ(𝑟)+𝑓𝑢≤𝑟 (20)
has a positive solution 𝑟₀ such that 𝑘𝑔ℎ𝑟₀+𝑢<1. (21)

Remark 7. Assume that 𝑟₀>0 satisfies the first inequality from assumption (vii); that is, ‖𝑎‖+𝑘𝑔𝑟₀ℎ𝑟₀+𝑘𝑢𝑟₀+𝑓𝑔ℎ𝑟₀+𝑓𝑢≤𝑟₀. (22)
Then, we obtain 𝑘𝑔ℎ𝑟₀+𝑢‖𝑎‖≤1−𝑟₀−𝑓𝑔ℎ𝑟₀𝑟₀−𝑓𝑢𝑟₀. (23) Thus, the second inequality from assumption (vii) is satisfied provided at least one of the quantities ‖𝑎‖, 𝑓𝑔ℎ(𝑟₀), and 𝑓𝑢 does not vanish.
Now, we are prepared to formulate our main result.

Theorem 8. Under assumptions (i)-(vii), (9) has at least one solution 𝑥=𝑥(𝑡) in the space 𝐵𝐶(ℝ₊). Moreover, all solutions of (9) are uniformly attractive.

Proof. Consider the operator 𝑈 defined on the space 𝐵𝐶(ℝ₊) by the formula (𝑈𝑥)(𝑡)=𝑎(𝑡)+𝑓(𝑡,𝑥(𝑡))^∞₀𝑢(𝑡,𝑠,𝑥(𝑠))𝑑𝑠,𝑡∈ℝ₊. (24) Notice that in view of assumptions (i), (ii), (iv), and (v), the function 𝑡→(𝑈𝑥)(𝑡) is well defined on the interval ℝ₊. We show that this function is continuous on ℝ₊. To this end, fix arbitrarily 𝑇>0 and 𝜀>0. Next, take arbitrary numbers 𝑡,𝑠∈[0,𝑇] with |𝑡−𝑠|≤𝜀. Then, in view of imposed assumptions we obtain 𝑡)−(𝑠≤|𝑎(𝑡)(𝑠+𝑓((𝑡|(𝑈𝑥)(𝑈𝑥)()|−𝑎)|𝑡,𝑥))^∞₀𝑢((𝜏((𝑠𝑡,𝜏,𝑥))𝑑𝜏−𝑓𝑠,𝑥))^∞₀𝑢((𝜏+𝑓((𝑠𝑡,𝜏,𝑥))𝑑𝜏𝑠,𝑥))^∞₀𝑢((𝜏((𝑠𝑡,𝜏,𝑥))𝑑𝜏−𝑓𝑠,𝑥))^∞₀𝑢(𝑠,𝜏,𝑥(𝜏))𝑑𝜏≤𝜔^𝑇()+𝑓((𝑡((𝑠+𝑓((𝑠((𝑠×𝑎,𝜀𝑡,𝑥))−𝑓𝑡,𝑥))𝑡,𝑥))−𝑓𝑠,𝑥))_∞0|𝑢((𝜏+𝑓((𝑠(+𝑓(𝑡,𝜏,𝑥))|𝑑𝜏𝑠,𝑥))−𝑓𝑠,0)𝑠,0)×^∞₀|𝑢(𝑡,𝜏,𝑥(𝜏))−𝑢(𝑠,𝜏,𝑥(𝜏))|𝑑𝜏≤𝜔^𝑇(𝑎,𝜀)+𝑘|𝑥(𝑡)−𝑥(𝑠)|+𝜔^𝑇_‖𝑥‖×𝑓,𝜀_∞0𝑢((𝜏(+|𝑢(+𝑘|𝑥(𝑠+𝑓(×[|𝑡,𝜏,𝑥))−𝑢𝑡,𝜏,0)|𝑡,𝜏,0)|]𝑑𝜏)|𝑠,0)_𝑇0|𝑢((𝜏((𝜏+𝑡,𝜏,𝑥))−𝑢𝑠,𝜏,𝑥))|𝑑𝜏_∞𝑇|𝑢(𝑡,𝜏,𝑥(𝜏))−𝑢(𝑠,𝜏,𝑥(𝜏))|𝑑𝜏≤𝜔^𝑇(𝑎,𝜀)+𝑘𝜔^𝑇(𝑥,𝜀)+𝜔^𝑇_‖𝑥‖×𝑓,𝜀_∞0𝑔()ℎ𝑥(𝜏+|𝑢(+𝑘‖𝑥‖+𝑡,𝜏(|)|)𝑡,𝜏,0)|𝑑𝜏𝑓^𝑇₀𝜔^𝑇_‖𝑥‖()+𝑢,𝜀𝑑𝜏_∞𝑇|𝑢((𝜏()()()()((𝜏𝑡,𝜏,𝑥))−𝑢𝑡,𝜏,0+𝑢𝑡,𝜏,0−𝑢𝑠,𝜏,0+𝑢𝑠,𝜏,0−𝑢𝑠,𝜏,𝑥))|𝑑𝜏≤𝜔^𝑇(𝑎,𝜀)+𝑘𝜔^𝑇(𝑥,𝜀)+𝜔^𝑇_‖𝑥‖×𝑓,𝜀_∞0𝑔()ℎ𝑥+|𝑢(+𝑘‖𝑥‖+𝑡,𝜏(‖‖)𝑡,𝜏,0)|𝑑𝜏𝑓^𝑇₀𝜔^𝑇_‖𝑥‖()+𝑢,𝜀𝑑𝜏_∞𝑇|𝑢((𝜏(+𝑡,𝜏,𝑥))−𝑢𝑡,𝜏,0)|𝑑𝜏_∞𝑇|𝑢((𝜏(+𝑠,𝜏,𝑥))−𝑢𝑠,𝜏,0)|𝑑𝜏_∞𝑇|𝑢(+𝑡,𝜏,0)|𝑑𝜏_∞𝑇|𝑢(𝑠,𝜏,0)|𝑑𝜏≤𝜔^𝑇(𝑎,𝜀)+𝑘𝜔^𝑇(𝑥,𝜀)+𝜔^𝑇_‖𝑥‖×𝑓,𝜀𝑔ℎ(‖𝑥‖)+𝑢+𝑘‖𝑥‖+𝑓×𝑇𝜔^𝑇_‖𝑥‖(𝑢,𝜀)+2^∞_𝑇𝑔()ℎ𝑥+𝑡,𝑠(‖‖)𝑑𝑠_∞𝑇|𝑢(𝑡,𝜏,0)|𝑑𝜏+^∞_𝑇|𝑢(𝑠,𝜏,0)|𝑑𝜏, (25) where we denoted 𝜔^𝑇_𝑑𝑓()(∶[],|||𝑥|},𝜔𝑓,𝜀=sup𝑡,𝑥−𝑓𝑠,𝑥)𝑡,𝑠∈0,𝑇𝑡−𝑠≤𝜀,≤𝑑_𝑇𝑑()𝑢()(∶[],|||𝑥|}.𝑢,𝜀=sup{|𝑡,𝜏,𝑥−𝑢𝑠,𝜏,𝑥)|𝑡,𝑠,𝜏∈0,𝑇𝑡−𝑠≤𝜀,≤𝑑 (26) Obviously, in the previous performed calculations we should put ‖𝑥‖ in place of 𝑑.
Further, let us notice that in view of assumptions (ii) and (iv), the function 𝑓 is uniformly continuous on the set [0,𝑇]×[−‖𝑥‖,‖𝑥‖], while the function 𝑢 is uniformly continuous on the set [0,𝑇]×[0,𝑇]×[−‖𝑥‖,‖𝑥‖]. Hence, we derive that 𝜔^𝑇_‖𝑥‖(𝑓,𝜀)→0 and 𝜔^𝑇_‖𝑥‖(𝑢,𝜀)→0 as 𝜀→0. Next, let us observe that based on assumption (vi) we can choose a number 𝑇 so large that the last terms in estimate (25), that is, the integrals ^∞_𝑇|𝑢(𝑡,𝜏,0)|𝑑𝜏,^∞_𝑇|𝑢(𝑠,𝜏,0)|𝑑𝜏, (27) are sufficiently small. The same is also true with regard to the integral ^∞_𝑇𝑔(𝑡,𝑠)𝑑𝑠. (28) Thus, taking into account the all facts established above and estimate (25), we infer that the function 𝑈𝑥 is continuous on the whole interval [0,𝑇] for each 𝑇>0 being big enough. This implies that 𝑈𝑥 is continuous on the whole interval ℝ₊.
In what follows, we show that the function 𝑈𝑥 is bounded on ℝ₊. Indeed, keeping in mind our assumptions, for an arbitrary fixed 𝑡∈ℝ₊, we get the following estimates: |(𝑈𝑥)(𝑡)|≤|𝑎(𝑡)|+𝑓(𝑡,𝑥(𝑡))^∞₀|𝑢((𝑠≤|𝑎(𝑡+𝑓((𝑡(+𝑓(×𝑡,𝑠,𝑥))|𝑑𝑠)|𝑡,𝑥))−𝑓𝑡,0)𝑡,0)_∞0𝑢((𝑠(+|𝑢(≤|𝑎(𝑡+𝑘‖𝑥‖+[|𝑡,𝑠,𝑥))−𝑢𝑡,𝑠,0)|𝑡,𝑠,0)|]𝑑𝑠)|𝑓×^∞₀𝑔(𝑡,𝑠)ℎ(|𝑥(𝑠)|)𝑑𝑠+^∞₀|𝑢(𝑡,𝑠,0)|𝑑𝑠. (29) Hence, we obtain the following evaluation |(𝑈𝑥)(𝑡)|≤‖𝑥‖+𝑘‖𝑥‖+𝑓𝑔ℎ(‖𝑥‖)+𝑢, (30) which implies that the function 𝑈𝑥 is bounded on ℝ₊. Combining this fact with the continuity of the function 𝑈𝑥 on ℝ₊, we conclude that the operator 𝑈 transforms the space 𝐵𝐶(ℝ₊) into itself.
Further, observe that from (30) we get ‖𝑈𝑥‖≤‖𝑎‖+𝑘𝑔‖𝑥‖ℎ(‖𝑥‖)+𝑘𝑢‖𝑥‖+𝑓𝑔ℎ(‖𝑥‖)+𝑓𝑢. (31) Linking the previous inequality with assumption (vii), we deduce that the operator 𝑈 maps the ball 𝐵_𝑟0 into itself, where 𝑟₀>0 is a number indicated in assumption (vii).
Now, let us take a nonempty subset 𝑋 of the ball 𝐵_𝑟0.
Fix numbers 𝜀>0 and 𝑇>0 and choose an arbitrary function 𝑥∈𝑋. Then, in virtue of estimate (25), for an arbitrary 𝑡∈[0,𝑇] we obtain 𝜔^𝑇(𝑈𝑥,𝜀)≤𝜔^𝑇(𝑎,𝜀)+𝑔ℎ𝑟₀+𝑢𝑘𝜔^𝑇()+𝑥,𝜀𝑔ℎ𝑟₀+𝑢𝜔^𝑇_𝑟0𝑓,𝜀+𝑘𝑟₀+𝑓×𝑇𝜔^𝑇_𝑟0(𝑢,𝜀)+2ℎ𝑟₀×sup^∞_𝑇𝑔(𝑡,𝑠)𝑑𝑠∶𝑡∈[0,𝑇]+2sup^∞_𝑇|𝑢(𝑡,𝜏,0)|𝑑𝜏∶𝑡∈[0,𝑇]. (32) Hence, we derive the following estimate 𝜔^𝑇(𝑈𝑋,𝜀)≤𝜔^𝑇(𝑎,𝜀)+𝑔ℎ𝑟₀+𝑢𝑘𝜔^𝑇()+𝑋,𝜀𝑔ℎ𝑟₀+𝑢𝜔^𝑇_𝑟0𝑓,𝜀+𝑘𝑟₀+𝑓×𝑇𝜔^𝑇_𝑟0(𝑢,𝜀)+2ℎ𝑟₀×sup^∞_𝑇𝑔(𝑡,𝑠)𝑑𝑠∶𝑡∈[0,𝑇]+2sup^∞_𝑇|𝑢(𝑡,𝜏,0)|𝑑𝜏∶𝑡∈[0,𝑇]. (33) Further, keeping in mind the properties of the terms of the right hand side of the perviously obtained inequality, which were mentioned earlier (cf. assumptions (ii) and (iv)), we deduce that the following estimate holds 𝜔^𝑇₀(𝑈𝑋)≤𝑘𝑔ℎ𝑟₀+𝑢𝜔^𝑇₀(𝑋)+𝑘𝑟₀+𝑓^∞_𝑇|𝑢(𝑡,𝜏,0)|𝑑𝜏∶𝑡∈[0,𝑇]2ℎ𝑟₀×sup^∞_𝑇𝑔(𝑡,𝑠)𝑑𝑠∶𝑡∈[0,𝑇]+2sup^∞_𝑇|𝑢(𝑡,𝜏,0)|𝑑𝜏∶𝑡∈[0,𝑇]. (34) Consequently, in view of assumption (vi) we have 𝜔₀(𝑈𝑋)≤𝑘𝑔ℎ𝑟₀+𝑢𝜔₀(𝑋). (35)
In what follows assume, as previously, that 𝑋 is a fixed nonempty subset of the ball 𝐵_𝑟0. Next, take arbitrary elements 𝑥,𝑦∈𝑋. Then, for an arbitrarily fixed 𝑡∈ℝ₊, in virtue of imposed assumptions, we obtain: 𝑡)−(𝑡≤𝑓((𝑡(𝑈𝑥)(𝑈𝑦)𝑡,𝑥))^∞₀𝑢((𝑠(𝑡)𝑡,𝑠,𝑥))𝑑𝑠−𝑓𝑡,𝑦^∞₀𝑢((𝑠+𝑓(𝑡𝑡,𝑠,𝑥))𝑑𝑠𝑡,𝑦)^∞₀𝑢((𝑠(𝑡)𝑡,𝑠,𝑥))𝑑𝑠−𝑓𝑡,𝑦^∞₀𝑢(𝑠)≤𝑓((𝑡(𝑡)𝑡,𝑠,𝑦𝑑𝑠𝑡,𝑥))−𝑓𝑡,𝑦^∞₀|𝑢((𝑠+𝑓(𝑡𝑡,𝑠,𝑥))|𝑑𝑠𝑡,𝑦)^∞₀𝑢((𝑠(𝑠𝑥(𝑡)(𝑡𝑡,𝑠,𝑥))−𝑢𝑡,𝑠,𝑦)𝑑𝑠≤𝑘−𝑦)^∞₀𝑢((𝑠(+|𝑢(+𝑓(𝑡)(+𝑓(×[|𝑡,𝑠,𝑥))−𝑢𝑡,𝑠,0)|𝑡,𝑠,0)|]𝑑𝑠𝑡,𝑦−𝑓𝑡,0)𝑡,0)_∞0𝑔()ℎ𝑥(𝑠)(𝑠𝑥(𝑡)(𝑡𝑡,𝑠−𝑦)𝑑𝑠≤𝑘−𝑦)^∞₀𝑔()ℎ𝑥(𝑠+𝑡,𝑠(|)|)𝑑𝑠_∞0|𝑢(+𝑘𝑦(𝑡+𝑓(×𝑡,𝑠,0)|𝑑𝑠)𝑡,0)_∞0𝑔()ℎ𝑥(𝑠)(𝑠ℎ𝑥𝑡,𝑠−𝑦)𝑑𝑠≤𝑘𝑥−𝑦(‖‖)^∞₀𝑔(𝑡,𝑠)𝑑𝑠+𝑘𝑢𝑥(𝑡)(𝑡+𝑘𝑦+−𝑦)𝑓ℎ𝑥−𝑦^∞₀𝑔(𝑡,𝑠)𝑑𝑠. (36) Hence, we get diam(𝑈𝑋)(𝑡)≤𝑘(𝑡)+𝑢diam𝑋2𝑘𝑟₀ℎ𝑟₀+𝑘𝑟₀+𝑓ℎ2𝑟₀×^∞₀𝑔(𝑡,𝑠)𝑑𝑠. (37) Taking into account assumption (v), from the previous inequality we derive the following one: limsup_𝑡→∞diam(𝑈𝑋)(𝑡)≤𝑘𝑢limsup_𝑡→∞diam𝑋(𝑡). (38) Obviously, the previous inequality implies the following estimate: limsup_𝑡→∞diam(𝑈𝑋)(𝑡)≤𝑘𝑔ℎ𝑟₀+𝑢limsup_𝑡→∞diam𝑋(𝑡). (39) Finally, let us observe that by combining (35) and (39) we have 𝜇(𝑈𝑋)≤𝑘𝑔ℎ𝑟₀+𝑢𝜇(𝑋), (40) where 𝜇 is the measure of noncompactness defined by formula (5).
In the last step of our proof, we show that the operator 𝑈 is continuous on the ball 𝐵_𝑟0. To this end, fix an arbitrary number 𝜀>0 and take 𝑥,𝑦∈𝐵_𝑟0 such that ‖𝑥−𝑦‖≤𝜀. Then, from estimate (36) we obtain 𝑈𝑥−𝑈𝑦≤𝑘𝑔ℎ𝑟₀𝜀+𝑘𝑢𝜀+𝑘𝑟₀+𝑓𝑔ℎ(𝜀). (41) The previous inequality in conjunction with assumption (iv) implies that the operator 𝑈 is a continuous self-mapping of the ball 𝐵_𝑟0.
Finally, using the above established facts and (40) and taking into account assumption (vii) and Theorem 2, we infer that the operator 𝑈 has at least one fixed point 𝑥 in the ball 𝐵_𝑟0. Obviously, every function 𝑥=𝑥(𝑡) being a fixed point of the operator 𝑈, is a solution of (9). Moreover, keeping in mind Remark 3, we conclude that the set Fix𝑈 of all fixed points of the operator 𝑈 belonging to the ball 𝐵_𝑟0 (equivalently: the set Fix𝑈 of all solutions of (9) belonging to the ball 𝐵_𝑟0) is a member of ker𝜇. Hence, in view of the description of the kernel ker𝜇 given in Section 2 we infer that all solutions of (9) belonging to the ball 𝐵_𝑟0 are uniformly attractive (asymptotically stable).
The proof is complete.

It is worthwhile mentioning that the above result generalizes those obtained in [3, 5, 8, 10, 17, 20], among others.

Now, we are going to illustrate the result contained in Theorem 8 by an example.

Example 9. Let us consider the following quadratic Urysohn integral equation: 𝑥(𝑡)=𝛼𝑒^𝑡1+𝑒^𝑡𝑡+𝛽²+sin𝑥(𝑡)1+𝑡^2∞₀³𝑡^3/2+𝑥²(𝑠)1+𝑡+𝑠²𝑑𝑠, (42) where 𝑡∈ℝ₊ and 𝛼,𝛽 are positive constants.
Observe that this equation is a special case of (9) if we put 𝑎(𝑡)=𝛼𝑒^𝑡1+𝑒^𝑡,𝑓()𝑡𝑡,𝑥=𝛽²+sin𝑥1+𝑡²,𝑢(𝑡,𝑠,𝑥)=³𝑡^3/2+𝑥²1+𝑡+𝑠². (43) It is easily to check that for the previous functions there are satisfied assumptions of Theorem 8. Indeed, the function 𝑎=𝑎(𝑡) is an element of the space 𝐵𝐶(ℝ₊) and ||𝑎||=𝛼. This means that there is verified assumption (i). Next, notice that 𝑓 is continuous on ℝ₊×ℝ and 𝑓(𝑡,0)=𝛽𝑡²/(1+𝑡²). Thus, the function 𝑡→𝑓(𝑡,0) belongs to 𝐵𝐶(ℝ₊), and there is satisfied assumption (ii). Moreover, we have that 𝑓=𝛽. Apart from this, for 𝑥,𝑦∈ℝ and for 𝑡∈ℝ₊ we obtain 𝑓(𝑡,𝑥)−𝑓𝑡,𝑦≤𝛽sin𝑥−sin𝑦1+𝑡²≤𝛽𝑥−𝑦1+𝑡²≤𝛽𝑥−𝑦. (44) This implies that the function 𝑓(𝑡,𝑥) satisfies assumption (iii) with 𝑘=𝛽.
Further, let us note that the function 𝑢(𝑡,𝑠,𝑥) is continuous on the set ℝ₊×ℝ₊×ℝ. For arbitrarily fixed 𝑡,𝑠∈ℝ₊, and 𝑥,𝑦∈ℝ we obtain 𝑢(𝑡,𝑠,𝑥)−𝑢𝑡,𝑠,𝑦≤³𝑡^3/2+𝑥²−³𝑡^3/2+𝑦²1+𝑡+𝑠². (45) Hence, using the inequality (cf. [9]) √³𝑎+𝑥²−³𝑎+𝑦²≤³𝑥−𝑦², (46) which is satisfied for all 𝑥,𝑦∈ℝ and for any fixed 𝑎, 𝑎≥0, we obtain 𝑢(𝑡,𝑠,𝑥)−𝑢𝑡,𝑠,𝑦≤𝑥−𝑦^2/31+𝑡+𝑠². (47) This implies that the function 𝑢(𝑡,𝑠,𝑥) satisfies assumption (iv) with ℎ(𝑟)=𝑟^2/3 and 𝑔(𝑡,𝑠)=1/(1+𝑡+𝑠²). Obviously, the function ℎ∶ℝ₊→ℝ₊ and is continuous and increasing on ℝ₊, while the function 𝑔(𝑡,𝑠) transforms ℝ₊×ℝ₊ into ℝ₊ and is continuous on ℝ₊×ℝ₊.
Moreover, we get ^∞₀𝑔(𝑡,𝑠)𝑑𝑠=lim_𝐴→∞^𝐴₀𝑔(𝑡,𝑠)𝑑𝑠=lim_𝐴→∞1𝑡+1^𝐴₀𝑑𝑠1+𝑠/√1+𝑡²=lim_𝐴→∞√1+𝑡𝐴1+𝑡arctan√=√1+𝑡1+𝑡⋅𝜋1+𝑡2. (48) Thus, the function 𝑔(𝑡,𝑠) is integrable over ℝ₊. Next, we obtain lim_𝑡→∞^∞₀𝑔(𝑡,𝑠)𝑑𝑠=lim_𝑡→∞√1+𝑡𝜋1+𝑡2=0. (49) Apart from this, in view of (48) we can easily to obtain that 𝑔=sup^∞₀𝑔(𝑡,𝑠)𝑑𝑠∶𝑡∈ℝ₊=𝜋2. (50) Further, let us fix arbitrarily a number 𝑇>0. Then, similarly as above, we get sup_{[]𝑡∈0,𝑇}^∞_t𝑔(𝑡,𝑠)𝑑𝑠=sup_{𝑡∈[0,𝑇]}√1+𝑡𝜋1+𝑡2𝑇−arctan√≤𝜋𝑡+12𝑇−arctan√.𝑇+1 (51) This allows us to deduce the following equality: lim_𝑇→∞sup_{[]𝑡∈0,𝑇}^∞_𝑇𝑔(𝑡,𝑠)𝑑𝑠=0. (52) Finally, let us take into account the function 𝑢()=|𝑢(=√𝑡,𝑠,0𝑡,𝑠,0)|𝑡1+𝑡+𝑠². (53) Calculating the indefinite integral of the function 𝑢(𝑡,𝑠,0), we obtain 𝑢()√𝑡,𝑠,0𝑑𝑠=𝑡²+𝑡𝑠𝑡+1arctan√.𝑡+1 (54) Hence, we get ^∞₀|𝑢(𝑡,𝑠,0)|𝑑𝑠=^∞₀𝑢(𝑡,𝑠,0)𝑑𝑠=lim_𝐴→∞^𝐴₀𝑢(𝑡,𝑠,0)𝑑𝑠=lim_𝐴→∞√𝑡²+𝑡𝐴𝑡+1arctan√=√𝑡+1𝑡²+𝑡𝜋𝑡+12. (55) Hence, taking into account that sup_𝑡∈ℝ+√𝑡²+𝑡𝑡+1=1, (56) we derive the following equality: 𝑢=sup_𝑡∈ℝ+^∞₀|𝑢(𝜋𝑡,𝑠,0)|𝑑𝑠=2. (57)
Now, fix arbitrarily a number 𝑇>0. Then, from (54) we get ^∞_𝑇|𝑢(√𝑡,𝑠,0)|𝑑𝑠=𝑡²+𝑡𝜋𝑡+12𝑇−arctan√.𝑡+1 (58) This allows us to derive the following estimate: sup_{[]𝑡∈0,𝑇}^∞_𝑇|𝑢(𝜋𝑡,𝑠,0)|𝑑𝑠≤2𝑇−arctan√.𝑇+1 (59) Consequently, this yields that the following equality holds: lim_𝑇→∞sup_{[]𝑡∈0,𝑇}^∞_𝑇|𝑢(𝑡,𝑠,0)|𝑑𝑠=0. (60)
Next, observe that taking into account (48), (52), (55), and (60), we conclude that the functions 𝑔(𝑡,𝑠) and 𝑢(𝑡,𝑠,0) satisfy assumptions (v) and (vi).
Now, we are coming to the last assumption of Theorem 8, that is, assumption (vii). Notice, that in the case of our Equation (42), in view of estimates (44), (50), and (55), the first inequality from assumption (vii) has the form 𝜋𝛼+𝛽⋅2𝑟⋅𝑟^2/3𝜋+𝛽2𝜋𝑟+𝛽⋅2𝑟^2/3𝜋+𝛽2≤𝑟. (61) Hence, after some simplification, we obtain 𝜋𝛼+2𝛽𝑟^5/3+𝑟+𝑟^2/3+1≤𝑟. (62) Thus, if for fixed positive 𝛼 and 𝛽 there exists a number 𝑟₀>0 satisfying inequality (62), then the second inequality from assumption (vii) is automatically satisfied. Indeed, it is an immediate consequence of Remark 7. In such a case (42) has solutions belonging to the ball 𝐵_𝑟0 which are uniformly attractive.
For example, it is easy to check that if we take 𝛼=𝛽=1/8 then the number 𝑟₀=4/5 satisfies inequality (62). This means that (42) with 𝛼=𝛽=1/8 has solutions belonging to the ball 𝐵_4/5 being uniformly attractive.

Acknowledgment

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (470/363/1433). The authors, therefore, acknowledge with thanks DSR technical and financial support.

Conflict of Interests

The authors declare that there is no conflict of interests in the submitted paper.