On Some Classes of Linear Volterra Integral Equations

Abstract

The sufficient conditions are obtained for the existence and uniqueness of continuous solution to the linear nonclassical Volterra equation that appears in the integral models of developing systems. The Volterra integral equations of the first kind with piecewise smooth kernels are considered. Illustrative examples are presented.

1. 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 [3–5].

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). In particular, a one-sector version of the Glushkov’s model applied to the power engineering problems was considered in [10–12]. In the recent years the researchers have got attracted by the equation (see [13] and references therein) that in a general case has the following form: ^𝑛_𝑖=1^𝑎𝑖−1_𝑎^(𝑡)_𝑖(𝑡)𝐾_𝑖(𝑡,𝑠)𝑥(𝑠)𝑑𝑠=𝑦(𝑡),𝑡∈[0,𝑇], (1) where 0≤𝑎_𝑛(𝑡)<𝑎_𝑛−1(𝑡)<⋯<𝑎₀(𝑡)𝑎≡𝑡,_𝑖(0)=0,𝑖=0,𝑛; (2) kernels 𝐾_𝑖 and right-hand side 𝑦(𝑡) are given, and 𝑥(𝑡) is an unknown desired solution.

At 𝑛=1 the problems of the existence and uniqueness of solution to (1) in the space 𝐶_[0,𝑇], 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 𝑛>1. Further, for simplicity, we will consider only the case 𝑛=2, since many results are easily generalized for the case 𝑛>2.

2. Sufficient Conditions for the Correctness of (1) at 𝑛=2 in Pair (𝐶_[0,𝑇], 𝐶^∘(1)_[0,𝑇])

For convenience, present (1) with 𝑛=2 in operator form 𝑉₁𝑥+𝑉₂𝑥≜^𝑡_𝑎1(𝑡)𝐾₁(𝑡,𝑠)𝑥(𝑠)𝑑𝑠+^𝑎1₀^(𝑡)𝐾₂(𝑡,𝑠)𝑥(𝑠)𝑑𝑠=𝑦(𝑡),𝑡∈[0,𝑇] (3) (in (3) 𝑎₂(𝑡)=0 is assumed with no loss of generality).

Let kernels 𝐾₁ and 𝐾₂ be continuous in arguments and continuously differentiable with respect to 𝑡 in regions Δ₁={(𝑡,𝑠)∶0≤𝑎₁(𝑡)≤𝑠≤𝑡≤𝑇} and Δ₂={(𝑡,𝑠)∶0≤𝑠≤𝑎₁(𝑡)}, respectively, so that Δ₁∪Δ₂=Δ, Δ={(𝑡,𝑠)∶0≤𝑠≤𝑡≤𝑇}, Δ₁∩Δ₂=𝑙, 𝑙={(𝑡,𝑠)∶𝑠=𝑎₁(𝑡)}. We will assume that 𝑎^′₁(𝑡)∈𝐶_+[0,𝑇],𝑎^′₁(0)<1. (4)

In particular, (4) holds true for 𝑎₁(𝑡)=𝛼𝑡, 𝛼∈(0,1). 𝐶^∘(1)_[0,𝑇] is further taken to mean the space of continuously differentiable functions 𝑦(𝑡) on [0,𝑇] with the norm ‖𝑦(𝑡)‖_∘𝐶⁽¹⁾_[0,𝑇]=max_{0≤𝑡≤𝑇}{|𝑦(𝑡)|+|𝑦^′(𝑡)|} and additional condition 𝑦(0)=0. If min_{𝑡∈[0,𝑇]}𝐾₁(𝑡,𝑡)=𝑘>0, (5) then, as established in [5, page 106], the following estimate is true: 𝑉₁⁻¹_∘𝐶⁽¹⁾_{[0,𝑇]→𝐶[0,𝑇]}≤𝑆𝑘⁻¹𝑒^{𝑘−1𝐿1𝑇}, (6) where 𝐿₁=max_{(𝑡,𝑠)∈Δ1}𝐾^′_1𝑡(𝑡,𝑠),𝑆=^∞𝑗_𝑗=0𝑖=1𝛾_𝑖𝛾≥1,_𝑖=𝛽_𝑖+𝑧_𝑖−𝑧_𝑖+1𝐿₁𝑘⁻¹,𝑧_𝑖=𝑎^𝑖₁(𝑇)=𝑎₁𝑎₁⋯𝑎₁(𝑇),𝑎⁰₁(𝑇)𝛽=𝑇,_𝑖=max_{𝑡∈[𝑧𝑖,𝑧𝑖−1]}𝑎^′₁(𝑡)𝐾₁(𝑡,𝑎(𝑡))𝐾₁(.𝑡,𝑡) (7)

Estimating (6) makes it possible to obtain the sufficient condition for the existence, uniqueness, and stability of the solution to (3) in pair (𝐶_[0,𝑇],𝐶^∘(1)_[0,𝑇]).

Theorem 1. Let the following inequality hold true: 𝑎₁(𝑇)𝑀₂+𝐿₂+𝐴₁𝑀₂<𝑘𝑆⁻¹𝑒^{−𝑘−1𝐿1𝑇}, (8) where 𝐴₁=max_{[]𝑡∈0,𝑇}𝑎^′₁(𝑡);𝑀₂=max_{(𝑡,𝑠)∈Δ2}𝐾₂(;𝐿𝑡,𝑠)₂=max_{(𝑡,𝑠)∈Δ2}𝐾^′_2𝑡(𝑡,𝑠), (9) Then (3) is correct in the sense of Hadamard in pair (𝐶_[0,𝑇],𝐶^∘(1)_[0,𝑇]).

Proof. By virtue of a well-known theorem of functional analysis (see, e.g., [14, page 212]), if 𝑉_{2𝐶[0,𝑇]→∘𝐶}⁽¹⁾_[0,𝑇]<1𝑉₁⁻¹_∘𝐶⁽¹⁾_{[0,𝑇]→𝐶[0,𝑇]}, (10) then the operator 𝑉=𝑉₁+𝑉₂ has a bounded inverse, and, consequently, (3) is correct in the sense of Hadamard in pair (𝐶_[0,𝑇],𝐶^∘(1)_[0,𝑇]). We show that under (8)-(9) inequality (10) holds true.
As𝑉₂𝑥_∘𝐶⁽¹⁾_[0,𝑇]=max_{0≤𝑡≤𝑇}^𝑎1₀^(𝑡)𝐾₂()𝑥(𝑠)+𝑎𝑡,𝑠𝑑𝑠_′1(𝑡)𝐾₂𝑡,𝑎₁(𝑡)+^𝑎1₀^(𝑡)𝐾^′_2𝑡()𝑥(𝑠)≤𝑎𝑡,𝑠𝑑𝑠₁(𝑇)𝑀₂+𝐿₂+𝐴₁𝑀₂‖𝑥(𝑡)‖_𝐶[0,𝑇], (11) then 𝑉_{2𝐶[0,𝑇]→∘𝐶}⁽¹⁾_[0,𝑇]≤𝑎₁(𝑇)𝑀₂+𝐿₂+𝐴₁𝑀₂ (12) and (10) follows from (6) and (12).

Condition (8) was obtained in the assumption that kernel 𝐾₁ is defined on Δ₁. If it is possible to expand the domain of definition 𝐾₁ to Δ, so that Δ₁∩Δ₂=Δ∩Δ₂=Δ₂, then the sufficient condition for the correctness of (3) is modified in the following way. Represent the first term in (3) in the form ^𝑡_𝑎1(𝑡)𝐾₁(𝑡,𝑠)𝑥(𝑠)𝑑𝑠=^𝑡₀𝐾₁(𝑡,𝑠)𝑥(𝑠)𝑑𝑠−^𝑎1₀^(𝑡)𝐾₁(𝑡,𝑠)𝑥(𝑠)𝑑𝑠. (13) Then (3) can be represented as 𝑉₁𝑥+𝑉₂𝑥≜^𝑡₀𝐾₁(𝑡,𝑠)𝑥(𝑠)𝑑𝑠+^𝑎1₀^(𝑡)𝐾₂(𝑡,𝑠)−𝐾₁()𝑥(𝑠)(𝑡),[].𝑡,𝑠𝑑𝑠=𝑦𝑡∈0,𝑇 (14)

Since (see [5, page 12]) 𝑉₁⁻¹_∘𝐶⁽¹⁾_{[0,𝑇]→𝐶[0,𝑇]}≤𝑘⁻¹𝑒^{𝑘−1𝐿1𝑇}, (15) where 𝐿₁=max_{(𝑡,𝑠)∈Δ}𝐾^′_1𝑡(𝑡,𝑠), (16) then sufficient conditions for the correctness of (14) give the following theorem.

Theorem 2. Let inequality 𝑎₁(𝑇)𝑀₂+𝐿₂+𝐴₁𝑀₂<𝑘𝑒⁻¹𝑒^{−𝑘−1𝐿1𝑇}, (17) where 𝑀₂=max_{(𝑡,𝑠)∈Δ2}𝐾₂(𝑡,𝑠)−𝐾₁(𝑡,𝑠),𝐿₂=max_{(𝑡,𝑠)∈Δ2}𝐾^′_2𝑡(𝑡,𝑠)−𝐾^′_1𝑡(𝑡,𝑠), (18)(19) hold true. Then (14) is correct in the sense of Hadamard in pair (𝐶_[0,𝑇],𝐶^∘(1)_[0,𝑇]).

Proof. With obvious changes, repeat the proof of Theorem 1.

Let us illustrate the obtained results with the following example.

Consider the equation ^𝑡_𝛼𝑡𝑥(𝑠)𝑑𝑠+𝜖₀^𝛼𝑡𝑥(𝑠)𝑑𝑠=𝑦(𝑡),𝑡∈[0,𝑇]. (20) Here by (5)–(7) 𝑘=1, 𝑀₂=|𝜖|, 𝑀₂=|1−𝜖|, 𝐿₁=𝐿₁=𝐿₂=0, 𝑎₁(𝑇)=𝛼𝑇, 𝐴₁=𝛼, 𝛾_𝑖=𝛽_𝑖=𝛼, and 𝑆=1/(1−𝛼); therefore based on (8) inequality 𝛼𝑇|𝜖|+𝛼|𝜖|<1−𝛼 (21) and based on (17) inequality 𝛼𝑇|1−𝜖|+𝛼|1−𝜖|<1 (22) give the following estimates 𝜖, which guarantee the existence, uniqueness, and stability of solution to (20) in the space 𝐶_[0,𝑇]: |𝜖|<1−𝛼𝛼();||<11+𝑇1−𝜖𝛼().1+𝑇 (23) It is useful to compare (23) with the estimate obtained by shifting from (20) to the equivalent functional equation. Differentiation of (20) gives 𝑥(𝑡)=𝛼(1−𝜖)𝑥(𝛼𝑡)+𝑦^′(𝑡), (24) whence 𝑥(𝑡)=lim_𝑛→∞𝛼^𝑛(1−𝜖)^𝑛𝑥𝛼^𝑛𝑡+^𝑛−1_𝑗=0𝛼^𝑗(1−𝜖)^𝑗𝑦^′𝛼^𝑗𝑡 (25) and condition ||<11−𝜖𝛼 (26) provides convergence of series (25) to continuous function 𝑥(𝑡) on [0,𝑇].

If in (20) 1𝜖=1−𝛼, (27) then condition (26) is violated. Then it is easy to see that the homogeneous equation ^𝑡_𝛼𝑡𝑥(𝑠)1𝑑𝑠+1−𝛼₀^𝛼𝑡𝑥(𝑠)𝑑𝑠=0 (28) has a nontrivial solution 𝑥(𝑡)=const, and if, for example, 𝑦(𝑡)=𝑡, the solution to the nonhomogeneous equation ^𝑡_𝛼𝑡𝑥(𝑠)1𝑑𝑠+1−𝛼₀^𝛼𝑡𝑥(𝑠)𝑑𝑠=𝑡,𝑡∈[0,𝑇] (29) is a one-parameter family: 𝑥(𝑡)=−ln𝑡(1).ln𝛼+𝑥 (30) Let now 1𝜖=1+𝛼. (31) Then, according to (24), 𝑥(𝑡)=−𝑥(𝛼𝑡)+𝑦^′(𝑡), (32) whence 𝑥(𝑡)=lim_𝑛→∞(−1)^𝑛𝑥𝛼^𝑛𝑡+^𝑛−1_𝑗=0(−1)^𝑗𝑦^′𝛼^𝑗𝑡 (33) so that for the right-hand side of (20) 𝑦(𝑡)=𝑦(𝑡)=𝑡^𝑘/𝑘, 𝑘=1,2,3,…, from (33) we obtain 𝑥(𝑡)=𝑡^𝑘−11+𝛼^𝑘−1,𝑘=1,2,…. (34)

In conclusion of this section it should be noted that inequalities (8) and (17) can be interpreted as constraints on the value 𝑇,which guarantee at given 𝐾₁(𝑡,𝑠),𝐾₂(𝑡,𝑠), and 𝑎₁(𝑡) the correct solvability of (3) in 𝐶_[0,𝑇]. Since all parameters in the left-hand side of (8) and (17) are nondecreasing functions of 𝑇 and the right-hand side of (8) and (17) at 𝐿₁≠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 𝑇 exists and is unique if 𝑎^′(0) is sufficiently small. In some special cases this root can be found analytically in terms of the Lambert function 𝑊 [15, 16].

In [17–22] 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.

3. The Volterra Integral Equations of the First Kind with Discontinuous Kernels

Equation (2) can be written in the form of Volterra integral equation of the first kind: ^𝑡₀𝐾(𝑡,𝑠)𝑥(𝑠)𝑑𝑠=𝑦(𝑡),𝑡∈[0,𝑇], (35) with discontinuous kernel 𝐾()=𝐾𝑡,𝑠₁(𝑡,𝑠),𝑎₁(𝑡)𝐾<𝑠≤𝑡;_𝑖(𝑡,𝑠),𝑎_𝑖(𝑡)<𝑠<𝑎_𝑖−1(𝑡),𝑖=𝐾2,𝑛−1;_𝑖(𝑡,𝑠)+𝐾_𝑖+1(𝑡,𝑠)2,𝑠=𝑎_𝑖(𝑡),𝑖=𝐾1,𝑛−1;_𝑛(𝑡,𝑠),0≤𝑠<𝑎_𝑛−1(𝑡). (36)

To illustrate the fundamental difference between (35), (36), and classical Volterra equation of the first kind with smooth kernel, we confine ourselves to (20) that has the form of (35) at 𝐾(𝑡,𝑠)=1,𝛼𝑡<𝑠≤𝑡;1+𝜖2,𝑠=𝛼𝑡;𝜖,0≤𝑠<𝛼𝑡, (37) where 𝜖≠0,1, and 𝛼∈(0,1). In particular, at 𝛼=1/2, 𝜖=−1, 𝐾()𝑡𝑡,𝑠=sign𝑠−2=𝑡1,𝑠>2;𝑡0,𝑠=2;𝑡−1,𝑠<2. (38)

For this case the solution to (35) with 𝑦(𝑡)=𝑡 given in [23] is 𝑥(𝑡)=ln𝑡(𝑠).ln2+𝑥 (39)

For kernel (38) 𝐾(0,0)=0,𝐾(𝑡,𝑡)≠0,𝑡>0. (40) If 𝐾(𝑡,𝑠) is continuous in arguments and continuously differentiable with respect to 𝑡 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 [25–28].

In particular, the author of [25–28] studies the structure of one- or many-parameter family of solutions to (35).

If 𝐾(𝑡,𝑠) is discontinuous, then the solution to (35) may be nonunique, even if 𝐾(𝑡,𝑡)≠0∀𝑡≥0.

For example, if 𝛼≠1/2 and 𝜖=1−(1/𝛼)≠−1, the solution to equation ^𝑡_𝛼𝑡𝑥(𝑠)𝑑𝑠+𝜖₀^𝛼𝑡𝑥(𝑠)𝑑𝑠=𝑡,𝑡∈[0,𝑇], (41) is a one-parameter family: 𝑥(𝑡)=−ln𝑡(1),ln𝛼+𝑥 (42) but, by (37) 𝐾(0,0)=(1+𝜖)/2≠0, 𝐾(𝑡,𝑡)=1,  𝑡>0.

Now we show that there can be a nonunique solution to (35) and (36) even in the case 𝐾(𝑡,𝑡)≡1. Let 𝐾(𝑡,𝑠)=1,𝑠≥𝛼𝑡,𝜖,𝑠<𝛼𝑡, (43) so that condition 𝐾(𝑡,𝑡)≡1 is true.

We prove that solutions to (35), (37) and (35), (43) coincide. It suffices to show that the equivalent functional equations for (35), (37) and (35), (43) coincide. Recall that for (35), (37) the equivalent functional equation is (24).

Theorem 3. The equivalent functional equations for (35), (37) and (35), (43) coincide.

Proof. Let us represent (43) by 𝐾(𝑡,𝑠)≡1+(𝜖−1)𝑒(𝛼𝑡−𝑠), (44) where 𝑒(⋅)– is a Heaviside function: 𝑒(𝜈)=1,𝜈≥0,0,𝜈<0. (45)
Substitution of (44) in (35) gives ^𝑡₀𝑥(𝑠)𝑑𝑠+(𝜖−1)^𝑡₀𝑒()𝑥(𝑠)(𝑡),[].𝛼𝑡−𝑠𝑑𝑠=𝑦𝑡∈0,𝑇 (46)
Transform the second integral. Let 𝜈=𝛼𝑡−𝑠. Then ^𝑡₀𝑒(𝛼𝑡−𝑠)𝑥(𝑠)𝑑𝑠=^𝛼𝑡_{(𝛼−1)𝑡}𝑒(𝜈)𝑥()=𝛼𝑡−𝜈𝑑𝜈_0𝛼𝑡𝑥(𝛼𝑡−𝜈)𝑑𝜈. (47) By virtue of (47), differentiation of (46) results in 𝑥(𝑡)+(𝜖−1)𝛼𝑥(0)+(𝜖−1)₀^𝛼𝑡𝑥^′_𝑡(𝛼𝑡−𝜈)𝑑𝜈=𝑦^′(𝑡). (48) But 𝑥^′_𝑡(𝛼𝑡−𝜈)=−𝛼𝑥^′_𝜈(𝛼𝑡−𝜈). (49) By virtue of (49) we have 𝑥(𝑡)+(𝜖−1)𝛼𝑥(0)−(𝜖−1)𝛼𝑥(𝛼𝑡−𝜈)|₀^𝛼𝑡=𝑦^′(𝑡), (50) from (48), whence finally 𝑥(𝑡)+𝛼(𝜖−1)𝑥(𝛼𝑡)=𝑦^′(𝑡), (51) and (51) coincides with (24).

The solution to (35), (43) in the class of piecewise continuous functions with a jump on line 𝑠=𝛼𝑡 is interesting from the application perspective.

It is easy to see that this solution is 𝑥()=𝑦𝑡,𝑠^′(𝑠)1,𝑠≥𝛼𝑡,𝜖𝑦^′(𝑠),𝑠<𝛼𝑡. (52)

At last consider the concept of 𝛼-convolution. Volterra integral equations of convolution type 𝐾(𝑡)∗𝑥(𝑡)=^Δ𝑡₀𝐾(𝑡−𝑠)𝑥(𝑠)𝑑𝑠=𝑦(𝑡),𝑡∈[0,𝑇] (53) are important for application.

Examples (38) and (44) show the usefulness of the 𝛼-convolution concept: 𝐾(𝑡)∗^𝛼𝑥(𝑡)=^Δ𝑡₀𝐾()𝑥(𝑠)(𝑡)(][].𝛼𝑡−𝑠𝑑𝑠=𝑦,𝛼∈0,1,𝑡∈0,𝑇 (54)

Give some inversion formulas of the integral equation 𝐾(𝑡)∗^𝛼𝑥(𝑡)=𝑦(𝑡),𝑡∈[0,𝑇]. (55)(1)If𝐾(𝑡)=𝛿(𝑡), 𝑦(𝑡)∈𝐶_[0,𝑇], and 𝛼∈(0,1], then 𝑥(𝛼𝑡)=𝑦(𝑡). (56)(2)If𝐾(𝑡)=𝑒(𝑡),  𝑦(𝑡)∈𝐶^∘(1)_[0,𝑇], and  𝛼∈(0,1], then 𝑥()=1𝛼𝑡𝛼𝑦^′(𝑡). (57) (3)If𝐾(𝑡)=sign𝑡, 𝑦(𝑡)=𝑡, and 𝛼∈(0,1), then 𝑥(𝑡)=ln𝑡(1).ln𝛼+𝑥 (58) At 𝐾(𝑡)=𝑡^𝑛,𝑛≥1, (55) is Volterra integral equation of the third kind.(4)If𝐾(𝑡)=𝑡,  𝑦(𝑡)=𝑡²/2, and  𝛼=1/2, then 𝑥(𝑡)=−2ln𝑡+𝑥(1). (59)(5)If𝐾(𝑡)=𝑡,𝑦(𝑡)=𝑡²/2, and𝛼∈(0,1), 𝛼≠1/2, then 𝑥(𝑡)=𝑥(1)𝑡^{(2𝛼−1)/(𝛼−1)}. (60)

4. Conclusion

As is mentioned in the introduction, the main results of this study can be easily applied to the case 𝑛>2 in (1). The equations of type (1) not only are of theoretical interest, but also play an important role in the mathematical modeling of developing dynamic systems. Moreover, by 𝑦(𝑡), we can mean some criterion that characterizes the level of development of the system as a whole, and the 𝑖th term in (1) represents a contribution of the system components 𝑥(𝑠) of the 𝑖th age group, whose operation is reflected by the efficiency coefficient 𝐾_𝑖(𝑡−𝑠). As a rule, 𝐾₁≥⋯≥𝐾_𝑛≥0. Such an approach is implemented, for instance, in [29, 30], in the problem of the analysis of strategies for the long-term expansion of the Russian electric power system, with the consideration of aging of the power plants equipment.

Conflict of Interests

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

Acknowledgments

The author wishes to thank the reviewers for their helpful notes. The study is supported by the Russian Foundation for Basic Research, Grant no. 12-01-00722a.