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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.
The aim of this paper is to study of monotonic and nonnegative solutions of the nonlinear quadratic Volterra-Stieltjes integral equation having the form where and , 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.
At the beginning, we provide some basic facts concerning functions of bounded variation and the Riemann-Stieltjes integral. We refer to  or  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 , where , 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.
(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 and are functions being Riemann-Stieltjes integrable on the interval with respect to nondecreasing function and , for . Then,
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  (see also ) for a more detailed discussion. Assume that is a real Banach space. Denote by the closed ball centered at and with radius . Instead of , 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 the family of all nonempty and bounded subsets of . The symbol stands for the subfamily of consisting of all relatively compact sets. We will accept the following definition of a measure of noncompactness.
Definition 2. A mapping will be called a measure of noncompactness in the space if it satisfies the following conditions:(1)the family is nonempty and ;(2);(3);(4), for ;(5)if is a sequence of closed sets belonging to such that , for , and if , then the intersection is nonempty.
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 be a nonempty, bounded, closed, and convex subset of the space and let be a continuous transformation. Assume that there exists a constant such that for any nonempty subset of . Then, has at least one fixed point in the set . Moreover, the set of all fixed points of belonging to is a member of the family .
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 is a given function. For any function , we can define the function by putting The operator defined in such a way is called the superposition operator generated by the function .
3. Main Result
In this section, we will investigate the nonlinear quadratic Volterra-Stieltjes integral equation which has the form where is fixed number. Obviously, in our further considerations the interval can be replaced by any interval . We look for monotonic and nonnegative solutions of this equation in the space . In our study, we will need some results obtained in [1, 2].
At the beginning, let us consider the following conditions.(i)The functions are continuous and there exist nondecreasing functions such that for any and for all , where is an arbitrary fixed number.Observe that, on the basis of the above condition, we may define the finite constants , by putting Let denote the following triangle: (ii)The function is continuous. Moreover, there exists a continuous function such that for all and .(iii)The function is continuous with respect to the variable on the interval , where is fixed.(iv)For any , the function is of bounded variation on the interval .(v)For each , there exists 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.(vi)The operator is continuous and there exists a nondecreasing function such that , for any .(vii)There exists a positive real number which satisfies the inequalities
Remark 5. Observe that if is a positive solution of the first inequality from condition (vii) and if one of the terms and does not vanish, then the second inequality from (vii) is automatically satisfied.
Now, let us consider the operators , , and defined on the space by the following formulas:
Theorem 6. Let conditions (i)–(vii) hold. Then, the operator is well defined and continuous and has at least one fixed point, which gives that (7) has at least one solution in the ball , where is a number appearing in condition (vii).
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 functions are such that(1);(2)the function is nondecreasing on , for any fixed ;(3)the function is nondecreasing on , for any fixed .()(a) The function is such that(1);(2)the function is nondecreasing on , for any fixed and ;(3)for each such that , the function is nondecreasing on ;(4)for any function which is nonnegative and nondecreasing on, the function is nonnegative on , where is a number appearing in condition (vii). Or (b) The function is such that(1);(2)the function is nondecreasing on , for any fixed and ;(3)the function is nondecreasing on , for any fixed and ;(4)the function is nondecreasing on for any fixed ;(5)for each such that , the function is nondecreasing on ;(6);(7)for any function which is nonnegative and nondecreasing on , the function is nonnegative and nondecreasing on , where is a number appearing in condition (vii).()For each the function is nondecreasing on .
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 which is nonnegative and nondecreasing, where is a number appearing in condition (vii).
Proof. Let denote set of all nonnegative and nondecreasing functions from the ball . It is clear that is nonempty, bounded, closed, and convex. From Theorem 6, we conclude that the operator is continuous. We show that . To this end, fix and take such that . Since and
we obtain , for .
It is easily seen that so it suffices to check monotonicity of the operator . We get Further proving process depends on which of conditions ((a)) or ((b)) is satisfied.
Assume that condition ((a)) holds. Then based on Theorem 1, the two last integrals in estimation (17) are nonnegative and indeed .
Now, assume that condition ((b)) is satisfied. Coming back to estimation (17), we obtain and, consequently, . Finally, we have . Using Theorems 3 and 6, we obtain the existence of a fixed point of the operator in . This means that (7) has at least one nonnegative and nondecreasing solution in , and the proof is complete.
Remark 9. It can be shown (see for instance ) that if the function is continuous on the triangle and for arbitrarily fixed such that , the function is monotonic (nondecreasing or nonincreasing) on the interval ; then 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 where denotes the Euler gamma function and . It is the well-known integral equation of fractional order. If we take on the set 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), (), and (). Additionally, we have , where is the constant appearing in Remark 4. Making use of the fact that for , condition (vii) in this situation takes the following form:()there exists a positive real number which satisfies the inequalities where and is a function chosen for based on condition (i).
Obviously, when , (19) reduces to the classical nonlinear quadratic Volterra integral equation.
Now, let us consider the equation It is the Volterra counterpart of the quadratic integral equation of Chandrasekhar type. This equation is also a special case of (7), in which Using, as before, Remark 9 and the standard methods of differential calculus, we can show that this function satisfies conditions (iii)–(v), ((a)), and (). Additionally, we have , where is the constant appearing in Remark 4.
Let us observe that if we put in (7), we obtain the classical functional equation of the first order on the interval .
We finish by providing an example illustrating Theorem 8.
Example 1. Let us consider the following integral equation: Obviously, this equation is a special case of (19) if we put and In is easy to check that conditions (i)–(vi), (), ((b)), and () of Theorem 8 are satisfied and , , , , , and . Using standard estimation for and taking , we verify that condition () 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 of the space .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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