On Some Classes of Linear Volterra Integral Equations

and Applied Analysis 3 Proof. With obvious changes, repeat the proof of Theorem 1. Let us illustrate the obtained results with the following example. Consider the equation


Introduction
Volterra integral equations of the first kind with variable upper and lower limits of integration were studied by Volterra himself [1].The publications on this topic in the first half of the 20th century were reviewed in [2] and later studies were discussed in [3][4][5].
A noticeable impetus to the development of this area is related to the research [6] which suggested a macroeconomic two-sector integral model.The Glushkov's models of developing systems were further extended in [7,8] and used in many applications (see [9] and references therein).
At  = 1 the problems of the existence and uniqueness of solution to (1) in the space  [0,] , as well as the numerical methods, are studied in detail in [5].In this paper we will be interested in the same problems for (1) at  > 1.Further, for simplicity, we will consider only the case  = 2, since many results are easily generalized for the case  > 2.

Sufficient Conditions for the Correctness of
For convenience, present (1) with  = 2 in operator form (in (3)  2 () = 0 is assumed with no loss of generality).
[0,] is further taken to mean the space of continuously differentiable functions () on [0, ] with the norm then, as established in [5, page 106], the following estimate is true: where Estimating ( 6) makes it possible to obtain the sufficient condition for the existence, uniqueness, and stability of the [0,] ).

Theorem 1. Let the following inequality hold true:
where Then (3) is correct in the sense of Hadamard in pair [0,] ).
Condition (8) was obtained in the assumption that kernel  1 is defined on Δ 1 .If it is possible to expand the domain of definition  1 to Δ, so that Δ 1 ∩ Δ 2 = Δ ∩ Δ 2 = Δ 2 , then the sufficient condition for the correctness of ( 3) is modified in the following way.Represent the first term in (3) in the form Since (see [5, page 12]) where then sufficient conditions for the correctness of ( 14) give the following theorem.

Theorem 2. Let inequality
where M2 = max hold true.Then (14) is correct in the sense of Hadamard in pair [0,] ).
Proof.With obvious changes, repeat the proof of Theorem 1.
Let us illustrate the obtained results with the following example.
Consider the equation Here by ( 5)-( 7 and based on (17) inequality give the following estimates , which guarantee the existence, uniqueness, and stability of solution to (20) in the space  [0,] : It is useful to compare (23) with the estimate obtained by shifting from (20) to the equivalent functional equation.Differentiation of (20) gives whence and condition provides convergence of series (25) to continuous function If in ( 20) then condition (26) is violated.Then it is easy to see that the homogeneous equation has a nontrivial solution () = const, and if, for example, () = , the solution to the nonhomogeneous equation is a one-parameter family: Let now Then, according to (24), whence so that for the right-hand side of ( 20) () = () =   /,  = 1, 2, 3, . .., from (33) we obtain In conclusion of this section it should be noted that inequalities ( 8) and ( 17) can be interpreted as constraints on the value , which guarantee at given  1 (, ),  2 (, ), and  1 () the correct solvability of (3) in  [0,] .Since all parameters in the left-hand side of ( 8) and ( 17) are nondecreasing functions of  and the right-hand side of ( 8) and ( 17) at  1 ̸ = 0 ( L1 ̸ = 0), on the contrary, monotonously decreases, then the real positive root of corresponding nonlinear equation that gives a guaranteed lower-bound estimate of  exists and is unique if   (0) is sufficiently small.In some special cases this root can be found analytically in terms of the Lambert function  [15,16].
In [17][18][19][20][21][22] the authors studied the characteristic of continuous solution locality and the role of the Lambert function as applied to the polynomial (multilinear) Volterra equations of the first kind.The calculations of the test examples show that the locality feature of the solution to the linear equation ( 3) is not the result of the inaccuracy of estimates ( 8) and (17) and reflects the specifics of the considered class of problems.In this paper we do not dwell on the problem of numerically solving (3).It is of independent interest and deserves special consideration.
If (, ) is continuous in arguments and continuously differentiable with respect to  in Δ, then condition (40) means that (35) is Volterra integral equation of the third kind.The theory (whose foundation was laid by Volterra (see [24, pages 104-106])) of such equations is developed in the research done by Magnitsky [25][26][27][28].