On Monotonic and Nonnegative Solutions of a Nonlinear Volterra-Stieltjes Integral Equation

Abstract

We study the existence of monotonic and nonnegative solutions of a nonlinear quadratic Volterra-Stieltjes integral equation in the space of real functions being continuous on a bounded interval. The main tools used in our considerations are the technique of measures of noncompactness in connection with the theory of functions of bounded variation and the theory of Riemann-Stieltjes integral. The obtained results can be easily applied to the class of fractional integral equations and Volterra-Chandrasekhar integral equations, among others.

1. Introduction

The aim of this paper is to study of monotonic and nonnegative solutions of the nonlinear quadratic Volterra-Stieltjes integral equation having the form 𝑥(𝑡)=𝐹₁𝑥(𝑡)+𝐹₂𝑥(𝑡)^𝑡₀𝑢(𝑡,𝜏,(𝑇𝑥)(𝜏))𝑑_𝜏𝑔(𝑡,𝜏), (1) where 𝑡∈[𝑎,𝑏] and 𝐹₁,𝐹₂ are superposition operators defined on the function space 𝐶[𝑎,𝑏]. The precise definitions will be given later. We show the existence of such solutions of the previous equation under some reasonable and handy assumptions. In our considerations, we use the technique associated with measures of noncompactness and the Riemann-Stieltjes integral with a kernel depending on two variables. Moreover, the theory of functions of bounded variation is also employed.

The main result of the paper is contained in Theorem 8. That theorem covers, as particular cases, the classical Volterra integral equation, the integral equation of fractional order, and the Volterra counterpart of the famous integral equation of Chandrasekhar type. It is worth pointing out that differential and integral equations of fractional order create an important branch of nonlinear analysis and the theory of integral equations. Moreover, these equations have found a lot of applications connected with real world problems. Integral equations of Chandrasekhar type can be often encountered in several applications as well.

This paper can be considered as a continuation of [1, 2] (cf. also [3–5]).

2. Preliminaries

At the beginning, we provide some basic facts concerning functions of bounded variation and the Riemann-Stieltjes integral. We refer to [6] or [7] for more information about this subject. Assume that 𝑥 is a real function defined on the interval [𝑎,𝑏]. The symbol ⋁^𝑏_𝑎𝑥 stands for the variation of the function 𝑥 on the interval [𝑎,𝑏]. In case of a function 𝑢(𝑡,𝜏)=𝑢∶𝐴→ℝ, where 𝐴⊂ℝ², the symbol ⋁^𝑞_𝜏=𝑝𝑢(𝑡,𝜏) denotes the variation of the function 𝜏→𝑢(𝑡,𝜏) on the interval [𝑝,𝑞] which is contained in the domain of this function, where the variable 𝑡 is fixed. Further, assume that 𝑥,𝜑 are given real functions defined on the interval [𝑎,𝑏]. Then, under some additional conditions imposed on 𝑥 and 𝜑, we can define the Riemann-Stieltjes integral ^𝑏_𝑎𝑥(𝑡)𝑑𝜑(𝑡) (2) of the function 𝑥 with respect to the function 𝜑. In such a case, we say that 𝑥 is integrable in the Riemann-Stieltjes sense on the interval [𝑎,𝑏] with respect to 𝜑.

Now, we recall two useful properties of the Riemann-Stieltjes integral, which will be employed in the sequel.

Theorem 1. (a) If 𝑥 is 𝑎 continuous function and 𝜑 is a function of bounded variation on the interval [𝑎,𝑏], then 𝑥 is Riemann-Stieltjes integrable on [𝑎,𝑏] with respect to 𝜑.
(b) Suppose that 𝑥₁ and 𝑥₂ are functions being Riemann-Stieltjes integrable on the interval [𝑎,𝑏] with respect to 𝑎 nondecreasing function 𝜑 and 𝑥₁(𝑡)≤𝑥₂(𝑡), for 𝑡∈[𝑎,𝑏]. Then, ^𝑏_𝑎𝑥₁(𝑡)𝑑𝜑(𝑡)≤^𝑏_𝑎𝑥₂(𝑡)𝑑𝜑(𝑡). (3)

In what follows we will use the Riemann-Stieltjes integral of the form ^𝑏_𝑎𝑥(𝜏)𝑑_𝜏𝑔(𝑡,𝜏), (4) where the symbol 𝑑_𝜏 indicates the integration with respect to the variable 𝜏 and 𝑡 is fixed. Let us mention that, in some situations, lower and upper limit of the integration can also depend upon the variable 𝑡.

Now, we deal with the discussion of basic facts connected with measures of noncompactness. We refer to [8] (see also [9]) for a more detailed discussion. Assume that 𝐸 is a real Banach space. Denote by 𝐵(𝑥,𝑟) the closed ball centered at 𝑥 and with radius 𝑟. Instead of 𝐵(0,𝑟), we will write 𝐵_𝑟. If 𝑋 is a subset of 𝐸, then the symbols 𝑋 and Conv𝑋 denote the closure and convex closed hull of the set 𝑋, respectively. Further, denote by 𝔐_𝐸 the family of all nonempty and bounded subsets of 𝐸. The symbol 𝔑_𝐸 stands for the subfamily of 𝔐_𝐸 consisting of all relatively compact sets. We will accept the following definition of a measure of noncompactness.

Definition 2. A mapping 𝜇∶𝔐_𝐸→ℝ₊=[0,+∞) will be called a measure of noncompactness in the space 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(𝑋)=𝜇(Conv𝑋);(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 belonging to 𝔐_𝐸 such that 𝑋_𝑛+1⊂𝑋_𝑛, for 𝑛=1,2,…, and if lim_𝑛→∞𝜇(𝑋_𝑛)=0, then the intersection 𝑋_∞=⋂^∞_𝑛=1𝑋_𝑛 is nonempty.

An important example of a measure of noncompactness is the Hausdorff measure of noncompactness defined by the formula 𝜒(𝑋)=inf{𝜀>0∶𝑋hasafinite𝜀−netin𝐸},𝑋∈𝔐_𝐸. (5)

The key role in our studies will be played by Darbo’s fixed point theorem.

Theorem 3. Let Ω be a nonempty, bounded, closed, and convex subset of the space 𝐸 and let 𝑄∶Ω→Ω be a continuous transformation. Assume that there exists a constant 𝑘∈[0,1) such that 𝜇(𝑄𝑋)≤𝑘𝜇(𝑋) for any nonempty subset 𝑋 of Ω. Then, 𝑄 has at least one fixed point in the set Ω. Moreover, the set Fix𝑄 of all fixed points of 𝑄 belonging to Ω is a member of the family ker𝜇.

The considerations in this paper will be placed in the Banach space 𝐶[𝑎,𝑏] consisting of all real functions defined and continuous on the bounded interval [𝑎,𝑏] with the standard maximum norm.

Finally, we turn our attention to the superposition (or Nemytskii) operator which appears very frequently in nonlinear analysis. We refer to monographs [6, 10] for detailed information covering the properties of this operator. To define the operator in question, suppose that 𝑓∶[𝑎,𝑏]×ℝ→ℝ is a given function. For any function 𝑥(𝑡)=𝑥∶[𝑎,𝑏]→ℝ, we can define the function 𝐹𝑥 by putting (𝐹𝑥)(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝑡∈[𝑎,𝑏]. (6) The operator 𝐹 defined in such a way is called the superposition operator generated by the function 𝑓.

3. Main Result

In this section, we will investigate the nonlinear quadratic Volterra-Stieltjes integral equation which has the form 𝑥(𝑡)=𝑓₁(𝑡,𝑥(𝑡))+𝑓₂(𝑡,𝑥(𝑡))×^𝑡₀𝑢(𝑡,𝜏,(𝑇𝑥)(𝜏))𝑑_𝜏𝑔(𝑡,𝜏),𝑡∈𝐼=[0,𝑀], (7) where 𝑀>0 is fixed number. Obviously, in our further considerations the interval 𝐼=[0,𝑀] can be replaced by any interval [𝑎,𝑏]. We look for monotonic and nonnegative solutions of this equation in the space 𝐶[0,𝑀]. In our study, we will need some results obtained in [1, 2].

At the beginning, let us consider the following conditions.(i)The functions 𝑓_𝑖∶𝐼×ℝ→ℝ  (𝑖=1,2) are continuous and there exist nondecreasing functions 𝑘_𝑖∶ℝ₊→ℝ₊ such that 𝑓_𝑖(𝑡,𝑥)−𝑓_𝑖𝑡,𝑦≤𝑘_𝑖(𝑟)𝑥−𝑦(𝑖=1,2), (8) for any 𝑡∈𝐼 and for all 𝑥,𝑦∈[−𝑟,𝑟], where 𝑟≥0 is an arbitrary fixed number.Observe that, on the basis of the above condition, we may define the finite constants 𝐹₁,𝐹₂ by putting 𝐹_𝑖=max𝑓_𝑖(𝑡,0)∶𝑡∈𝐼(𝑖=1,2). (9) Let Δ_𝑀 denote the following triangle: Δ_𝑀=(𝑡,𝜏)∈ℝ²∶0≤𝜏≤𝑡≤𝑀. (10)(ii)The function 𝑢∶Δ_𝑀×ℝ→ℝ is continuous. Moreover, there exists a continuous function Φ∶ℝ₊→ℝ₊ such that |𝑢(𝑡,𝜏,𝑥)|≤Φ(|𝑥|), (11) for all (𝑡,𝜏)∈Δ_𝑀 and 𝑥∈ℝ.(iii)The function 𝑔∶Δ_𝑀→ℝ is continuous with respect to the variable 𝜏 on the interval [0,𝑡], where 𝑡∈𝐼 is fixed.(iv)For any 𝑡∈𝐼, the function 𝜏→𝑔(𝑡,𝜏) is of bounded variation on the interval [0,𝑡].(v)For each 𝜀>0, there exists 𝛿>0 such that, for all 𝑡,𝑠∈𝐼 and |𝑠−𝑡|≤𝛿, the following inequality holds ^{min{𝑡,𝑠}}_𝜏=0𝑔(𝑠,𝜏)−𝑔(𝑡,𝜏)≤𝜀. (12)

Remark 4. It can be shown (see [1, 2]) that the constant 𝐾=max^𝑡_𝜏=0𝑔(𝑡,𝜏)∶𝑡∈𝐼 (13) is well defined and finite.(vi)The operator 𝑇∶𝐶(𝐼)→𝐶(𝐼) is continuous and there exists a nondecreasing function Ψ∶ℝ₊→ℝ₊ such that ‖𝑇𝑥‖≤Ψ(‖𝑥‖), for any 𝑥∈𝐶(𝐼).(vii)There exists a positive real number 𝑟₀ which satisfies the inequalities 𝑟𝑘₁(𝑟)+𝐹₁+𝐾𝑟𝑘₂(𝑟)+𝐹₂Φ(Ψ(𝑟𝑘))≤𝑟,₁(𝑟)+𝐾𝑘₂(𝑟)Φ(Ψ(𝑟))<1. (14)

Remark 5. Observe that if 𝑟₀ is a positive solution of the first inequality from condition (vii) and if one of the terms 𝐹₁ and 𝐾𝐹₂Φ(Ψ(𝑟₀)) does not vanish, then the second inequality from (vii) is automatically satisfied.

Now, let us consider the operators 𝐹_𝑖(𝑖=1,2), 𝑈, and 𝑉 defined on the space 𝐶(𝐼) by the following formulas: 𝐹_𝑖𝑥(𝑡)=𝑓_𝑖((𝑡),(𝑡)=𝑡,𝑥))(𝑖=1,2𝑈𝑥)(^𝑡₀𝑢(𝑡,𝜏,(𝑇𝑥)(𝜏))𝑑_𝜏𝑔(),(𝑡)=𝐹𝑡,𝜏𝑉𝑥)(₁𝑥(𝑡)+𝐹₂𝑥(𝑡)(𝑈𝑥)(𝑡). (15)

Theorem 6. Let conditions (i)–(vii) hold. Then, the operator 𝑉_|𝐵𝑟0∶𝐵_𝑟0→𝐵_𝑟0 is well defined and continuous and has at least one fixed point, which gives that (7) has at least one solution in the ball 𝐵_𝑟0, where 𝑟₀ is a number appearing in condition (vii).

The basic idea of the proof of Theorem 6 is to study behaviour of the operator 𝑉 with respect to the Hausdorff measure of noncompactness in connection with Theorem 3.

Remark 7. Additionally, all solutions of (7) from the ball 𝐵_𝑟0 are equicontinuous. This observation results directly from the Arzela-Ascoli theorem and Theorem 3.

We can now formulate our main result about monotonicity and nonnegativity of the solutions of (7). In our study, we will consider the following conditions.(i^′)The functions 𝑓_𝑖∶𝐼×ℝ→ℝ(𝑖=1,2) are such that(1)𝑓_𝑖(𝐼×ℝ₊)⊂ℝ₊;(2)the function 𝑡→𝑓_𝑖(𝑡,𝑥) is nondecreasing on 𝐼, for any fixed 𝑥∈ℝ₊;(3)the function 𝑥→𝑓_𝑖(𝑡,𝑥) is nondecreasing on ℝ₊, for any fixed 𝑡∈𝐼.(ii^′)(a) The function 𝑢∶Δ_𝑀×ℝ→ℝ is such that(1)𝑢(Δ_𝑀×ℝ₊)⊂ℝ₊;(2)the function 𝑡→𝑢(𝑡,𝜏,𝑥) is nondecreasing on [𝜏,𝑀], for any fixed 𝜏∈𝐼 and 𝑥∈ℝ₊;(3)for each 𝑡,𝑠∈𝐼 such that 𝑡<𝑠, the function 𝜏→𝑔(𝑠,𝜏)−𝑔(𝑡,𝜏) is nondecreasing on [0,𝑡];(4)for any function 𝑥∈𝐵_𝑟0 which is nonnegative and nondecreasing on𝐼, the function 𝑇𝑥 is nonnegative on 𝐼, where 𝑟₀ is a number appearing in condition (vii). Or (b) The function 𝑢∶Δ_𝑀×ℝ→ℝ is such that(1)𝑢(Δ_𝑀×ℝ₊)⊂ℝ₊;(2)the function 𝑡→𝑢(𝑡,𝜏,𝑥) is nondecreasing on [𝜏,𝑀], for any fixed 𝜏∈𝐼 and 𝑥∈ℝ₊;(3)the function 𝜏→𝑢(𝑡,𝜏,𝑥) is nondecreasing on [0,𝑡], for any fixed 𝑡∈𝐼 and 𝑥∈ℝ₊;(4)the function 𝑥→𝑢(𝑡,𝜏,𝑥) is nondecreasing on ℝ₊ for any fixed (𝑡,𝜏)∈Δ_𝑀;(5)for each 𝑡,𝑠∈𝐼 such that 𝑡<𝑠, the function 𝜏→𝑔(𝑠,𝜏)−𝑔(𝑡,𝜏) is nondecreasing on [0,𝑡];(6)𝑔(𝑠,𝑠)−𝑔(𝑡,𝑡)+𝑔(𝑡,0)−𝑔(𝑠,0)≥0;(7)for any function 𝑥∈𝐵_𝑟0 which is nonnegative and nondecreasing on 𝐼, the function 𝑇𝑥 is nonnegative and nondecreasing on 𝐼, where 𝑟₀ is a number appearing in condition (vii).(iii^′)For each 𝑡∈𝐼 the function 𝜏→𝑔(𝑡,𝜏) is nondecreasing on [0,𝑡].

The following theorem is a completion of Theorem 6.

Theorem 8. Suppose that conditions (i)–(vii) and (i^')–(iii^') are fulfilled. Then, (7) has at least one solution in 𝐵_𝑟0 which is nonnegative and nondecreasing, where 𝑟₀ is a number appearing in condition (vii).

Proof. Let 𝐵⁺_𝑟0 denote set of all nonnegative and nondecreasing functions from the ball 𝐵_𝑟0. It is clear that 𝐵⁺_𝑟0 is nonempty, bounded, closed, and convex. From Theorem 6, we conclude that the operator 𝑉_|𝐵⁺_𝑟0 is continuous. We show that 𝑉(𝐵⁺_𝑟0)⊂𝐵⁺_𝑟0. To this end, fix 𝑥∈𝐵⁺_𝑟0 and take 𝑡,𝑠∈𝐼 such that 𝑠>𝑡. Since (𝐹_𝑖𝑥)(𝑡)=𝑓_𝑖(𝑡,𝑥(𝑡))≥0 and 𝐹_𝑖𝑥(𝑠)−𝐹_𝑖𝑥(𝑡)=𝑓_𝑖(𝑠,𝑥(𝑠))−𝑓_𝑖(𝑡,𝑥(𝑡))≥𝑓_𝑖(𝑡,𝑥(𝑠))−𝑓_𝑖(𝑡,𝑥(𝑡))≥0, (16) we obtain 𝐹_𝑖(𝐵⁺_𝑟0)⊂𝐵⁺_𝑟0, for 𝑖=1,2.
It is easily seen that (𝑈𝑥)(𝑡)≥0 so it suffices to check monotonicity of the operator 𝑈. We get (𝑠)−(𝑡)=𝑈𝑥)(𝑈𝑥)(_𝑠0𝑢(𝑠,𝜏,(𝑇𝑥)(𝜏))𝑑_𝜏𝑔()−𝑠,𝜏_𝑡0𝑢(𝑡,𝜏,(𝑇𝑥)(𝜏))𝑑_𝜏𝑔()=𝑡,𝜏_𝑡0𝑢(𝑠,𝜏,(𝑇𝑥)(𝜏))𝑑_𝜏𝑔()+𝑠,𝜏_𝑠𝑡𝑢(𝑠,𝜏,(𝑇𝑥)(𝜏))𝑑_𝜏𝑔()−𝑠,𝜏_𝑡0𝑢(𝑡,𝜏,(𝑇𝑥)(𝜏))𝑑_𝜏𝑔()≥𝑡,𝜏_𝑡0𝑢(𝑠,𝜏,(𝑇𝑥)(𝜏))𝑑_𝜏𝑔()−𝑠,𝜏_𝑡0𝑢(𝑠,𝜏,(𝑇𝑥)(𝜏))𝑑_𝜏𝑔()+𝑡,𝜏_𝑠𝑡𝑢(𝑠,𝜏,(𝑇𝑥)(𝜏))𝑑_𝜏𝑔()=𝑠,𝜏_𝑡0𝑢(𝑠,𝜏,(𝑇𝑥)(𝜏))𝑑_𝜏𝑔()(+𝑠,𝜏−𝑔𝑡,𝜏)_𝑠𝑡𝑢(𝑠,𝜏,(𝑇𝑥)(𝜏))𝑑_𝜏𝑔(𝑠,𝜏). (17) Further proving process depends on which of conditions (ii^′(a)) or (ii^′(b)) is satisfied.
Assume that condition (ii^′(a)) holds. Then based on Theorem 1, the two last integrals in estimation (17) are nonnegative and indeed (𝑈𝑥)(𝑠)−(𝑈𝑥)(𝑡)≥0.
Now, assume that condition (ii^′(b)) is satisfied. Coming back to estimation (17), we obtain (𝑠)−(𝑡)≥𝑈𝑥)(𝑈𝑥)(_𝑡0𝑢(𝑠,𝑡,(𝑇𝑥)(𝑡))𝑑_𝜏𝑔()(+𝑠,𝜏−𝑔𝑡,𝜏)_𝑠𝑡𝑢(𝑠,𝑡,(𝑇𝑥)(𝑡))𝑑_𝜏𝑔()((𝑡𝑔()()−𝑔()()()(((𝑡𝑔()()()(𝑠,𝜏=𝑢𝑠,𝑡,𝑇𝑥)())𝑠,𝑡−𝑔𝑡,𝑡𝑠,0−𝑔𝑡,0+𝑔𝑠,𝑠−𝑔𝑠,𝑡)=𝑢𝑠,𝑡,𝑇𝑥)())𝑠,𝑠−𝑔𝑡,𝑡+𝑔𝑡,0−𝑔𝑠,0)≥0 (18) and, consequently, 𝑈(𝐵⁺_𝑟0)⊂𝐵⁺_𝑟0. Finally, we have 𝑉(𝐵⁺_𝑟0)⊂𝐵⁺_𝑟0. Using Theorems 3 and 6, we obtain the existence of a fixed point of the operator 𝑉 in 𝐵⁺_𝑟0. This means that (7) has at least one nonnegative and nondecreasing solution in 𝐵_𝑟0, and the proof is complete.

Remark 9. It can be shown (see for instance [1]) that if the function 𝑔∶Δ_𝑀→ℝ is continuous on the triangle Δ_𝑀 and for arbitrarily fixed 𝑡,𝑠∈𝐼 such that 𝑡<𝑠, the function 𝜏→𝑔(𝑠,𝜏)−𝑔(𝑡,𝜏) is monotonic (nondecreasing or nonincreasing) on the interval [0,𝑡]; then 𝑔 satisfies condition (v).

4. Applications and an Example

The topic of this section is to present some applications of Theorem 8 in the situation of the classical integral equations.

Let us consider the equation 𝑥(𝑡)=𝑓₁((𝑡+𝑓𝑡,𝑥))²(𝑡,𝑥(𝑡))Γ(𝛼)×^𝑡₀𝑢(𝑡,𝜏,(𝑇𝑥)(𝜏))(𝑡−𝜏)^1−𝛼𝑑𝜏,𝑡∈𝐼, (19) where Γ denotes the Euler gamma function and 𝛼>0. It is the well-known integral equation of fractional order. If we take on the set Δ_𝑀 the function 𝑔 defined by 𝑔()=1𝑡,𝜏𝛼𝑡^𝛼−(𝑡−𝜏)^𝛼, (20) then it is easy to check that (19) is a special case of (7). Using Remark 9 and the standard methods of differential calculus, we can show that the function 𝑔 satisfies conditions (iii)–(v), (ii^′), and (iii^′). Additionally, we have 𝐾=(1/𝛼)𝑀^𝛼, where 𝐾 is the constant appearing in Remark 4. Making use of the fact that Γ(𝛼+1)=𝛼Γ(𝛼) for 𝛼>0, condition (vii) in this situation takes the following form:(vii^∗)there exists a positive real number 𝑟₀ which satisfies the inequalities 𝑟𝑘₁(𝑟)+𝐹₁+𝑀^𝛼Γ()𝑟𝑘𝛼+1²(𝑟)+𝐹₂Φ(Ψ(𝑟𝑘))≤𝑟,₁(𝑟)+𝑀^𝛼Γ()𝑘𝛼+1²(𝑟)Φ(Ψ(𝑟))<1, (21)where 𝐹₂=max{|𝑓₂(𝑡,0)|∶𝑡∈𝐼} and 𝑘₂ is a function chosen for 𝑓₂ based on condition (i).

Obviously, when 𝛼=1, (19) reduces to the classical nonlinear quadratic Volterra integral equation.

Now, let us consider the equation 𝑥(𝑡)=𝑓₁(𝑡,𝑥(𝑡))+𝑓₂(𝑡,𝑥(𝑡))×^𝑡₀𝑡𝑢((𝜏𝑡+𝜏𝑡,𝜏,𝑇𝑥)())𝑑𝜏,𝑡∈𝐼. (22) It is the Volterra counterpart of the quadratic integral equation of Chandrasekhar type. This equation is also a special case of (7), in which 𝑔()=𝜏𝑡,𝜏𝑡ln1+𝑡,(𝑡,𝜏)∈Δ_𝑀⧵{(0,0)}0,𝑡=𝜏=0. (23) Using, as before, Remark 9 and the standard methods of differential calculus, we can show that this function satisfies conditions (iii)–(v), (ii^′(a)), and (iii^′). Additionally, we have 𝐾=𝑀ln2, where 𝐾 is the constant appearing in Remark 4.

Let us observe that if we put 𝑓₂(𝑡,𝑥)≡0 in (7), we obtain the classical functional equation of the first order on the interval 𝐼.

We finish by providing an example illustrating Theorem 8.

Example 1. Let us consider the following integral equation: 𝑥(𝑡)=𝑡𝑒^−𝑡+𝑡²+𝑥(𝑡)Γ()×2/3_𝑡012𝜋|𝑥(𝜏)|arctg4+𝑡²+𝜏²₃√[].𝑡−𝜏𝑑𝜏,𝑡∈0,1 (24) Obviously, this equation is a special case of (19) if we put 𝛼=2/3 and 𝑓₁(𝑡,𝑥)=𝑡𝑒^−𝑡,𝑓₂𝑓,𝑥=𝑡²𝑢()=1+𝑥,𝑡,𝜏,𝑥2𝜋|𝑥|arctg4+𝑡²+𝜏²,𝑇𝑥=𝑥. (25) In is easy to check that conditions (i)–(vi), (i^′), (ii^′(b)), and (iii^′) of Theorem 8 are satisfied and 𝑘₁(𝑟)=0, 𝐹₁=1/𝑒, 𝑘₂(𝑟)=1, 𝐹₂=1, Φ(𝑟)=(1/4)√𝑟, and Ψ(𝑟)=𝑟. Using standard estimation Γ(𝛼)>0.8856 for 𝛼>0 and taking 𝑟₀=1, we verify that condition (vii^∗) is also satisfied. Therefore, in case of (24), we can apply Theorem 8. This means that (24) has at least one nonnegative and nondecreasing solution belonging to the ball 𝐵₁ of the space 𝐶[0,1].

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.