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

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.


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  () = ( 1 ) () + ( 2 ) () ∫  0  (, , () ())    (, ) , (1) where  ∈ [, ] and  1 ,  2 are superposition operators defined on the function space [, ].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.

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  is a real function defined on the interval [, ].The symbol ⋁    stands for the variation of the function  on the interval [, ].In case of a function (, ) =  :  → R , where  ⊂ R 2 , the symbol ⋁  = (, ) denotes the variation of the function  → (, ) on the interval [, ] which is contained in the domain of this function, where the variable  is fixed.Further, assume that ,  are given real functions defined on the interval [, ].Then, under some additional conditions imposed on  and , we can define the Riemann-Stieltjes integral of the function  with respect to the function .In such a case, we say that  is integrable in the Riemann-Stieltjes sense on the interval [, ] with respect to .Now, we recall two useful properties of the Riemann-Stieltjes integral, which will be employed in the sequel.
In what follows we will use the Riemann-Stieltjes integral of the form where the symbol   indicates the integration with respect to the variable  and  is fixed.Let us mention that, in some situations, lower and upper limit of the integration can also depend upon the variable .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  is a real Banach space.Denote by (, ) the closed ball centered at  and with radius .Instead of (0, ), we will write   .If  is a subset of , then the symbols  and Conv denote the closure and convex closed hull of the set , respectively.Further, denote by M  the family of all nonempty and bounded subsets of .The symbol N  stands for the subfamily of M  consisting of all relatively compact sets.We will accept the following definition of a measure of noncompactness.Definition 2. A mapping  : M  → R + = [0, +∞) will be called a measure of noncompactness in the space  if it satisfies the following conditions: (1) the family ker  = { ∈ M  : () = 0} is nonempty and ker  ⊂ N  ; (2)  ⊂  ⇒ () ≤ (); An important example of a measure of noncompactness is the Hausdorff measure of noncompactness defined by the formula 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  and let  : Ω → Ω be a continuous transformation.Assume that there exists a constant  ∈ [0, 1) such that () ≤ () for any nonempty subset  of Ω.Then,  has at least one fixed point in the set Ω.Moreover, the set Fix  of all fixed points of  belonging to Ω is a member of the family ker .
The considerations in this paper will be placed in the Banach space [, ] consisting of all real functions defined and continuous on the bounded interval [, ] 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  : [, ]×R → R is a given function.For any function () =  : [, ] → R, we can define the function  by putting The operator  defined in such a way is called the superposition operator generated by the function .

Main Result
In this section, we will investigate the nonlinear quadratic Volterra-Stieltjes integral equation which has the form where  > 0 is fixed number.Obviously, in our further considerations the interval  = [0, ] can be replaced by any interval [, ].We look for monotonic and nonnegative solutions of this equation in the space [0, ].In our study, we will need some results obtained in [1,2].At the beginning, let us consider the following conditions.
(i) The functions   :  × R → R ( = 1, 2) are continuous and there exist nondecreasing functions for any  ∈  and for all ,  ∈ [−, ], where  ≥ 0 is an arbitrary fixed number.
Observe that, on the basis of the above condition, we may define the finite constants  1 ,  2 by putting Let Δ  denote the following triangle: (ii) The function  : Δ  × R → R is continuous.Moreover, there exists a continuous function Φ : for all (, ) ∈ Δ  and  ∈ R.
(iii) The function  : Δ  → R is continuous with respect to the variable  on the interval [0, ], where  ∈  is fixed.
(v) For each  > 0, there exists  > 0 such that, for all ,  ∈  and | − | ≤ , the following inequality holds Remark 4. It can be shown (see [1,2]) that the constant is well defined and finite.
(vii) There exists a positive real number  0 which satisfies the inequalities Remark 5. Observe that if  0 is a positive solution of the first inequality from condition (vii) and if one of the terms  1 and  2 Φ(Ψ( 0 )) does not vanish, then the second inequality from (vii) is automatically satisfied.Now, let us consider the operators   ( = 1, 2), , and  defined on the space () by the following formulas: (15) Theorem 6.Let conditions (i)-(vii) hold.Then, the operator  |  0 :   0 →   0 is well defined and continuous and has at least one fixed point, which gives that (7) has at least one solution in the ball   0 , where  0 is a number appearing in condition (vii).
The basic idea of the proof of Theorem 6 is to study behaviour of the operator  with respect to the Hausdorff measure of noncompactness in connection with Theorem 3.

Remark 7.
Additionally, all solutions of (7) from the ball   0 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.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   0 which is nonnegative and nondecreasing, where  0 is a number appearing in condition (vii).
Remark 9.It can be shown (see for instance [1]) that if the function  : Δ  → R is continuous on the triangle Δ  and for arbitrarily fixed ,  ∈  such that  < , the function  → (, ) − (, ) is monotonic (nondecreasing or nonincreasing) on the interval [0, ]; then  satisfies condition (v).

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 where Γ denotes the Euler gamma function and  > 0. It is the well-known integral equation of fractional order.If we take on the set Δ  the function  defined by 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  satisfies conditions (iii)-(v), (ii  ), and (iii  ).Additionally, we have  = (1/)  , where  is the constant appearing in Remark 4. Making use of the fact that Γ( + 1) = Γ() for  > 0, condition (vii) in this situation takes the following form: (vii * ) there exists a positive real number  0 which satisfies the inequalities where F2 = max{| f2 (, 0)| :  ∈ } and k2 is a function chosen for f2 based on condition (i).
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  =  ln 2, where  is the constant appearing in Remark 4.
Let us observe that if we put  2 (, ) ≡ 0 in (7), we obtain the classical functional equation of the first order on the interval .
We finish by providing an example illustrating Theorem 8.

Theorem 1 .
(a) If  is  continuous function and  is a function of bounded variation on the interval [, ], then  is Riemann-Stieltjes integrable on [, ] with respect to .(b) Suppose that  1 and  2 are functions being Riemann-Stieltjes integrable on the interval [, ] with respect to  nondecreasing function  and  1 () ≤  2 (), for  ∈ [, ].