Measure Functional Differential Equations in the Space of Functions of Bounded Variation

Abstract

We establish general conditions for the unique solvability of nonlinear measure functional differential equations in terms of properties of suitable linear majorants.

1. Introduction, Motivation, and Problem Setting

Let ℝ=(−∞,∞), ℝ^𝑛∋𝑥=(𝑥_𝑘)^𝑛_𝑘=1↦‖𝑥‖∶=max_{1≤𝑘≤𝑛}|𝑥_𝑘| be the norm in ℝ^𝑛, and let BV([𝑎,𝑏],ℝ^𝑛) be the Banach space of functions of bounded variation with the standard norm BV([𝑎,𝑏],ℝ^𝑛)∋𝑢↦‖𝑢‖_BV∶=|𝑢(𝑎)|+Var_[𝑎,𝑏]𝑢, where −∞<𝑎<𝑏<∞.

Our aim is to examine the solvability of the equation 𝑢(𝑡)=𝜑(𝑢)+^𝑡_𝑎𝑓𝑢(𝑠)𝑑𝑔(𝑠),𝑡∈[𝑎,𝑏]; (1)𝑓∶BV([𝑎,𝑏],ℝ^𝑛)→BV([𝑎,𝑏],ℝ^𝑛) is a, generally speaking, nonlinear operator and 𝜑∶BV([𝑎,𝑏],ℝ^𝑛)→ℝ^𝑛 is a nonlinear vector functional. The integral on the right-hand side of (1) is the Kurzweil-Stieltjes integral with respect to a nondecreasing function 𝑔∶[𝑎,𝑏]→ℝ. We refer to [1–5] for the definition and properties of this kind of an integral, recalling only that (1) is a particular case of a generalised differential equation [2, 6]. It is important to note that, for any 𝑢∈BV([𝑎,𝑏],ℝ^𝑛), the Kurzweil-Stieltjes integral in (1) exists (see, e.g., [4, 7]) and, therefore, the equation itself makes sense.

By a solution of (1), we mean a vector function 𝑢∶[𝑎,𝑏]→ℝ^𝑛 which has bounded variation and satisfies (1) on the interval [𝑎,𝑏].

Equation (1) is an extension of a measure differential equation studied systematically, for example, in [2, 8–10]. It is a fairly general object which includes many other types of equations such as differential equations with impulses [11] or functional dynamic equations on time scales [12] (see, e.g., [13, 14]). In particular, if 𝑔(𝑠)=𝑠, 𝑠∈[𝑎,𝑏], (1) takes the form 𝑢(𝑡)=𝜑(𝑢)+^𝑡_𝑎𝑓𝑢(𝑠)𝑑𝑠,𝑡∈[𝑎,𝑏], (2) and, thus, in the absolutely continuous case, reduces to the nonlocal boundary value problem for a functional differential equation 𝑢^′(𝑡)=𝑓𝑢(𝑡),𝑡∈[𝑎,𝑏],𝑢(𝑎)=𝜑(𝑢), (3) whose various particular types are the object of investigation of many authors (see, e.g., [15–19]). A more general choice of 𝑔 in (1) allows one to cover further cases where solutions lose their absolute continuity at some points. For example, consider the impulsive functional differential equation [16, page 191] 𝑢^′(𝑡)=𝑓𝑢(𝑡),𝑡∈[𝑎,𝑏]⧵𝑡₁,𝑡₂,…,𝑡_𝑚,Δ𝑢(𝑡)=𝐼_𝑖(𝑢(𝑡))for𝑡=𝑡_𝑖,𝑖∈{1,2,…,𝑚}, (4) where Δ𝑢(𝑡)∶=𝑢(𝑡+)−𝑢(𝑡−) for any function 𝑢 from BV([𝑎,𝑏],ℝ^𝑛) (in fact, Δ𝑢(𝑡)=𝑢(𝑡+)−𝑢(𝑡) if, as is customary [11] in that context, a solution is assumed to be left continuous). Here, 𝑓=(𝑓_𝑘)^𝑛_𝑘=1∶BV([𝑎,𝑏],ℝ^𝑛)→BV([𝑎,𝑏],ℝ^𝑛), the jumps may occur at the preassigned times 𝑡₁,𝑡₂,…,𝑡_𝑚, and their action is described by the operators 𝐼_𝑖∶ℝ^𝑛→ℝ^𝑛, 𝑖=1,…,𝑚. By the usual integration argument [11], one can represent (4) alternatively in the form 𝑢(𝑡)=𝑢(𝑎)+^𝑡_𝑎𝑠)+𝑓𝑢(𝑑𝑠_{𝑘∶𝑎<𝑡𝑘<𝑡,}𝐼_𝑘𝑢𝑡_𝑘,𝑡∈[𝑎,𝑏]. (5) It follows, in particular, from [14, Lemma  2.4] that (5) is equivalent to the measure functional differential equation 𝑢(𝑡)=𝑢(𝑎)+^𝑡_𝑎𝑓𝑢(𝑠)𝑑𝑔(𝑠),𝑡∈[𝑎,𝑏], (6) with 𝑔(𝑠)=𝑠+∑^𝑚_𝑖=1𝜒_{(𝑡𝑖,𝑏]}(𝑠), 𝑠∈[𝑎,𝑏], and 𝑓∶BV([𝑎,𝑏],ℝ^𝑛)→BV([𝑎,𝑏],ℝ^𝑛) defined by the relation 𝑠)=(𝑠)[]⧵𝑡𝑓𝑢(𝑓𝑢if𝑠∈𝑎,𝑏₁,𝑡₂,…,𝑡_𝑚,𝐼_𝑖(𝑢(𝑠))if𝑠∈𝑡₁,𝑡₂,…,𝑡_𝑚. (7) Thus, system (4) can be considered as a particular case of (1). Likewise, an appropriate construction [13, 20] allows one to regard differential equations on time scales [12] as measure differential equations. The same is true for equations involving functional components; in the case of a differential equation on a time scale with retarded argument, by choosing 𝑔 suitably [13], one arrives at the equation 𝑢(𝑡)=𝑢(𝑎)+^𝑡_𝑎ℎ𝑢_𝑠(𝑠)[],𝑢,𝑠𝑑𝑔,𝑡∈𝑎,𝑏_𝑎=𝜙. (8) In (8), ℎ∶𝐶([−𝑟,0],ℝ^𝑛)×[𝑎,𝑏]→ℝ^𝑛 is a functional in the first variable, 𝜙 is from the space 𝐶([−𝑟,0],ℝ^𝑛) of continuous functions on [−𝑟,0], and the Krasovsky notation 𝑢_𝑡∶[−𝑟,0]∋𝑠↦𝑢(𝑡+𝑠), 𝑟>0, is used [21, Chapter VI]. Finally, eliminating the initial function 𝜙 from (8) in a standard way by transforming it to a forcing term (see [15]), we conclude that the resulting equation falls into the class of equations of form (1).

Note that, by measure functional differential equations, the Volterra type equations of form (8) are usually meant in the existing bibliography on the subject (see, e.g., [8, 13, 22]), whereas equations with more general types of argument deviation are rather scarce (we can cite, perhaps, only [4, page 217]). Comparing (8) with (1), we find that the latter includes non-Volterra cases as well.

This list of examples can be continued. It is interesting to observe that solutions of problems of type (3) studied in the literature up to now are always assumed, at least locally, to be absolutely continuous [16], or even continuously differentiable [23]. In contrast to this, the gauge integral involved in (1) allows one to deal with a considerably wider class of solutions, which are, in fact, assumed to be of bounded variation only. A possible noteworthy consequence for systems with impulses may be that the unpleasant effect of the so-called pulsation phenomenon [11, page 5] might be more natural to be dealt with in the framework of the space BV([𝑎,𝑏],ℝ^𝑛). Our interest in (1), originally motivated by a relation to problems of type (3), has strengthened still further due to the last observation.

The general character of the object represented by (1) suggests a natural idea to examine its solvability by comparing it to simpler linear equations with suitable properties. Here, we show that such statements can indeed be obtained rather easily by analogy to [24–26]. The key assumption is that certain linear operators associated with the nonlinear operator 𝑓 appearing in (1) possess the following property.

Definition 1. Let ℎ∶BV([𝑎,𝑏],ℝ^𝑛)→ℝ^𝑛 be a linear mapping. One says that a linear operator 𝑝∶BV([𝑎,𝑏],ℝ^𝑛)→BV([𝑎,𝑏],ℝ^𝑛) belongs to the set 𝒮_ℎ([𝑎,𝑏],ℝ^𝑛) if the equation 𝑢(𝑡)=ℎ(𝑢)+^𝑡_𝑎𝑠)(𝑠)(𝑡),[]𝑝𝑢(𝑑𝑔+𝑟𝑡∈𝑎,𝑏 (9) has a unique solution 𝑢 for any 𝑟 from BV([𝑎,𝑏],ℝ^𝑛), and, moreover, the solution 𝑢 is nonnegative for any nonnegative 𝑟.

The property described by Definition 1, in fact, means that the linear operator associated with (9) is positively invertible on BV([𝑎,𝑏],ℝ^𝑛), and thus it corresponds to the existence and positivity of Green’s operator for a boundary value problem [15].

Remark 2. The inclusion 𝑝∈𝒮_ℎ([𝑎,𝑏],ℝ^𝑛), generally speaking, does not imply that 𝜆𝑝∈𝒮_ℎ([𝑎,𝑏],ℝ^𝑛) for 𝜆≠1!

The question on the unique solvability of (1) is thus reduced to estimating the nonlinearities suitably, so that the appropriate majorants generate linear equations with a unique solution depending monotonously on forcing terms. The problem of finding such majorants is a separate topic not discussed here. We only note that, in a number of cases, the existing results on differential inequalities can be applied (see, e.g., [17–19]).

Note that, due to the nature of the techniques used, statements of this kind available in the literature on problems of type (3), as a rule, are established separately in every concrete case (see, e.g., [27–29]). Here, we provide a simple unified proof, which is, in a sense, independent on the character of the equation and also allows one to gain a considerable degree of generality. The results may be useful in studies of the solvability of various measure functional differential equations and, in particular, of problem (3) and its generalisations (note that, e.g., rather complicated neutral-type functional differential equations [23] can be formulated in form (1); see also [4, 30]).

2. Unique Solvability Conditions

We are going to show that the knowledge of the property 𝑝∈𝒮_ℎ([𝑎,𝑏],ℝ^𝑛) for certain linear operators 𝑝 and ℎ associated with (1) allows one to guarantee its unique solvability.

2.1. Nonlinear Equations

The following statement is true.

Theorem 3. Assume that there exist certain linear operators 𝑝_𝑖∶BV([𝑎,𝑏],ℝ^𝑛)→BV([𝑎,𝑏],ℝ^𝑛), 𝑖=1,2, such that 𝑝₂(𝑢−𝑣)(𝑡)≤𝑓𝑢(𝑡)−𝑓𝑣(𝑡)≤𝑝₁(𝑢−𝑣)(𝑡),𝑡∈[𝑎,𝑏], (10) for arbitrary functions 𝑢∶[𝑎,𝑏]→ℝ^𝑛, 𝑣∶[𝑎,𝑏]→ℝ^𝑛 with the property 𝑢(𝑡)≥𝑣(𝑡)∀𝑡∈[𝑎,𝑏]. (11) Furthermore, let the inclusions 𝑝₁∈𝒮_ℎ1[𝑎,𝑏],ℝ^𝑛,12𝑝₁+𝑝₂∈𝒮_{(1/2)(ℎ1+ℎ2)}[𝑎,𝑏],ℝ^𝑛 (12) be fulfilled with some linear functionals ℎ_𝑖∶BV([𝑎,𝑏],ℝ^𝑛)→ℝ, 𝑖=1,2. Then (1) has a unique solution for an arbitrary 𝜑 such that ℎ₂(𝑢−𝑣)≤𝜑(𝑢)−𝜑(𝑣)≤ℎ₁(𝑢−𝑣) (13) whenever (11) holds.

The inequality sign and modulus for vectors in (10), (11), (13), and similar relations below are understood componentwise. The theorem as well as the other statements formulated below will be proved later.

Theorem 4. Let there exist certain linear operators 𝑙_𝑖∶BV([𝑎,𝑏],ℝ^𝑛)→BV([𝑎,𝑏],ℝ^𝑛), 𝑖=1,2, and linear functionals ℎ_𝑖∶BV([𝑎,𝑏],ℝ^𝑛)→ℝ, 𝑖=1,2 satisfying the inclusions 𝑙₁+𝑙₂∈𝒮_ℎ1[𝑎,𝑏],ℝ^𝑛,𝑙₁∈𝒮_{(1/2)(ℎ1+ℎ2)}[𝑎,𝑏],ℝ^𝑛, (14) and such that (13) and the inequality 𝑓𝑢(𝑡)−𝑓𝑣(𝑡)−𝑙₁(𝑢−𝑣)(𝑡)≤𝑙₂(𝑢−𝑣)(𝑡),𝑡∈[𝑎,𝑏], (15) is true for arbitrary functions 𝑢 and 𝑣 of bounded variation with property (11). Then (1) is uniquely solvable.

Theorem 4 is, in fact, an alternative form of Theorem 3, where the estimate of a “linear part” is more visible.

In other statements, we need the following natural notion of positivity of a linear operator in the space BV([𝑎,𝑏],ℝ^𝑛).

Definition 5. A linear operator 𝑞∶BV([𝑎,𝑏],ℝ^𝑛)→BV([𝑎,𝑏],ℝ^𝑛) will be called positive if 𝑞𝑢 is a nonnegative function for an arbitrary nonnegative 𝑢 from BV([𝑎,𝑏],ℝ^𝑛).

Note that no monotonicity assumptions are imposed on 𝑙₁ in Theorem 4. In the cases where the positivity of certain linear majorants is known, the following statement may be of use.

Corollary 6. Assume that there exist some positive linear operators 𝑞_𝑖∶BV([𝑎,𝑏],ℝ^𝑛)→BV([𝑎,𝑏],ℝ^𝑛), 𝑖=1,2, such that the inequalities 𝑓𝑢(𝑡)−𝑓𝑣(𝑡)+𝑞₂(𝑢−𝑣)(𝑡)≤𝑞₁(𝑢−𝑣)(𝑡) (16) hold on [𝑎,𝑏] for any 𝑢 and 𝑣 from BV([𝑎,𝑏],ℝ^𝑛) with property (11). Moreover, let one can specify linear functionals ℎ_𝑖∶BV([𝑎,𝑏],ℝ^𝑛)→ℝ, 𝑖=1,2, satisfying (13), and such that the inclusions 𝑞₁+(1−2𝜃)𝑞₂∈𝒮_ℎ1[𝑎,𝑏],ℝ^𝑛,−𝜃𝑞₂∈𝒮_{(1/2)(ℎ1+ℎ2)}[𝑎,𝑏],ℝ^𝑛 (17) hold for a certain 𝜃∈(0,1). Then (1) has a unique solution.

Corollary 6 allows one to obtain, in particular, the following statements.

Corollary 7. Assume that, for arbitrary 𝑢 and 𝑣 from BV([𝑎,𝑏],ℝ^𝑛) with property (11), 𝑓 and 𝜑 satisfy estimates (13) and (16) with some linear functionals ℎ_𝑖∶BV([𝑎,𝑏],ℝ^𝑛)→ℝ, 𝑖=1,2 and positive linear operators 𝑞_𝑖∶BV([𝑎,𝑏],ℝ^𝑛)→BV([𝑎,𝑏],ℝ^𝑛), 𝑖=1,2. Then the inclusions 𝑞₁∈𝒮_ℎ1[𝑎,𝑏],ℝ^𝑛1,−2𝑞₂∈𝒮_{(1/2)(ℎ1+ℎ2)}[𝑎,𝑏],ℝ^𝑛 (18) guarantee that (1) is uniquely solvable.

Corollary 8. The assertion of Corollary 7 is true with (18) replaced by the condition 𝑞₁+12𝑞₂∈𝒮_ℎ1[𝑎,𝑏],ℝ^𝑛,−14𝑞₂∈𝒮_{(1/2)(ℎ1+ℎ2)}[𝑎,𝑏],ℝ^𝑛. (19)

The statements formulated above express fairly general properties of (1) and extend, in particular, the corresponding results of [25, 27, 29, 31].

2.2. Linear Equations

Let us now assume that 𝑓∶BV([𝑎,𝑏],ℝ^𝑛)→BV([𝑎,𝑏],ℝ^𝑛) in (1) is an affine mapping, and, therefore, (1) has the form 𝑢(𝑡)=ℎ(𝑢)+^𝑡_𝑎(𝑙𝑢)(𝑠)𝑑𝑔(𝑠)+𝑦(𝑡),𝑡∈[𝑎,𝑏], (20) where 𝑙∶BV([𝑎,𝑏],ℝ^𝑛)→BV([𝑎,𝑏],ℝ^𝑛) and ℎ∶BV([𝑎,𝑏],ℝ^𝑛)→ℝ^𝑛 are linear, and 𝑦∈BV([𝑎,𝑏],ℝ^𝑛) is a given function.

Corollary 9. Assume that there exist certain linear operators 𝑝_𝑖∶BV([𝑎,𝑏],ℝ^𝑛)→BV([𝑎,𝑏],ℝ^𝑛), 𝑖=0,1, and a linear mapping ℎ∶BV([𝑎,𝑏],ℝ^𝑛)→ℝ^𝑛 such that the inclusions 𝑝₁∈𝒮_ℎ[𝑎,𝑏],ℝ^𝑛,𝑝₀+𝑝₁∈𝒮_ℎ[𝑎,𝑏],ℝ^𝑛 (21) hold, and the estimate (𝑙𝑢)(𝑡)−𝑝₁𝑢(𝑡)≤𝑝₀𝑢(𝑡),𝑡∈[𝑎,𝑏] (22) is satisfied for any nonnegative 𝑢∈BV([𝑎,𝑏],ℝ^𝑛). Then (20) has a unique solution.

We also have the following.

Corollary 10. Let there exist positive linear operators 𝑞_𝑖∶BV([𝑎,𝑏],ℝ^𝑛)→BV([𝑎,𝑏],ℝ^𝑛), 𝑖=0,1, satisfying the inclusions 𝑞₁∈𝒮_ℎ[𝑎,𝑏],ℝ^𝑛1,−2𝑞₂∈𝒮_ℎ[𝑎,𝑏],ℝ^𝑛, (23) and such that the inequalities (𝑙𝑢)(𝑡)+𝑞₂𝑢(𝑡)≤𝑞₁𝑢(𝑡),𝑡∈[𝑎,𝑏] (24) are true for an arbitrary nonnegative function 𝑢∶[𝑎,𝑏]→ℝ^𝑛 of bounded variation. Then (20) has a unique solution for any 𝑦∈BV([𝑎,𝑏],ℝ^𝑛).

We conclude this note by considering the case where 𝑙 in (20) is a linear mapping admitting a decomposition into the sum of its positive and negative parts; that is, 𝑙=𝑙₀−𝑙₁, (25) where 𝑙_𝑘∶BV([𝑎,𝑏],ℝ^𝑛)→BV([𝑎,𝑏],ℝ^𝑛), 𝑘=0,1, are linear and positive. In that case, for the equation of the form 𝑢(𝑡)=ℎ(𝑢)+^𝑡_𝑎𝑙₀𝑢(𝑠)−𝑙₁𝑢𝑠(𝑠)(𝑡),[],()𝑑𝑔+𝑦𝑡∈𝑎,𝑏 (26) where ℎ∶BV([𝑎,𝑏],ℝ^𝑛)→ℝ^𝑛 is linear and 𝑦∈BV([𝑎,𝑏],ℝ^𝑛), the following result is obtained.

Corollary 11. Let the linear vector functional ℎ∶BV([𝑎,𝑏],ℝ^𝑛)→ℝ^𝑛 and the linear positive operators 𝑙_𝑖∶BV([𝑎,𝑏],ℝ^𝑛)→BV([𝑎,𝑏],ℝ^𝑛),   𝑖=1,2, be such that the inclusions 𝑙₀∈𝒮_ℎ[𝑎,𝑏],ℝ^𝑛,12𝑙₀−𝑙₁∈𝒮_ℎ[𝑎,𝑏],ℝ^𝑛 (27) are satisfied. Then (26) has a unique solution for any 𝑦∈BV([𝑎,𝑏],ℝ^𝑛).

It is interesting to observe the second condition in (27); it thus turns out that property 𝒮_ℎ([𝑎,𝑏],ℝ^𝑛) for one half of the operator under the integral sign in (26) ensures the unique solvability of the original equation (26).

3. Proofs

Let ⟨𝐸,‖⋅‖⟩ be real Banach space, let 𝑧∈𝐸 be a given vector, and let 𝐹∶𝐸→𝐸 be a mapping. Let 𝐾_𝑖⊂𝐸, 𝑖=1,2, be closed cones inducing the corresponding partial orderings ≦_𝐾𝑖, so that 𝑥≦_𝐾𝑖𝑦 if and only if 𝑦−𝑥∈𝐾_𝑖. The following statement [32, 33] on the abstract equation 𝐹𝑢=𝑧 (28) will be used below.

Theorem 12 (see [33], Theorem  49.4). Let the cone 𝐾₂ be normal and generating. Furthermore, let 𝐵_𝑘∶𝐸→𝐸, 𝑘=1,2, be linear operators such that 𝐵₁⁻¹ and (𝐵₁+𝐵₂)⁻¹ exist and possess the properties 𝐵₁⁻¹𝐾₂⊂𝐾₁,𝐵₁+𝐵₂⁻¹𝐾₂⊂𝐾₁, (29) and, furthermore, let the order relation 𝐵₁𝑥−𝑦≦_𝐾2𝐹𝑥−𝐹𝑦≦_𝐾2𝐵₂𝑥−𝑦 (30) be satisfied for any pair (𝑥,𝑦) such that 𝑦≦_𝐾1𝑥. Then (28) has a unique solution for an arbitrary element 𝑧∈𝐸.

Recall that 𝐾₂ is normal if all the sets order bounded with respect to ≦_𝐾2 are also norm bounded and that 𝐾₁ is generating if and only if {𝑢−𝑣∣{𝑢,𝑣}⊂𝐾₁}=𝐸 (see, e.g., [33, 34]).

Let BV⁺([𝑎,𝑏],ℝ^𝑛) (resp., BV⁺⁺([𝑎,𝑏],ℝ^𝑛)) be the set of all the nonnegative (resp., nonnegative and nondecreasing) functions from BV([𝑎,𝑏],ℝ^𝑛).

Lemma 13. (1) The set BV⁺([𝑎,𝑏],ℝ^𝑛) is a cone in the space BV([𝑎,𝑏],ℝ^𝑛).
(2) The set BV⁺⁺([𝑎,𝑏],ℝ^𝑛) is a normal and generating cone in BV([𝑎,𝑏],ℝ^𝑛).

Proof. The first assertion of the lemma being obvious, only the second one should be verified.
It follows directly from the definition of the set BV⁺⁺([𝑎,𝑏],ℝ^𝑛) that it is a cone in BV([𝑎,𝑏],ℝ^𝑛), which is also generating due to the Jordan decomposition of a function of bounded variation (see, e.g., [3]). In order to verify its normality, it will be sufficient to show [32, Theorem  4.1] that the set 𝐴𝛼,𝛽∶=𝑥∈BV[𝑎,𝑏],ℝ^𝑛∶𝑥−𝛼,𝛽−𝑥⊂BV⁺⁺[𝑎,𝑏],ℝ^𝑛 (31) is bounded for any {𝛼,𝛽}⊂BV([𝑎,𝑏],ℝ^𝑛). Indeed, if 𝑥∈𝐴(𝛼,𝛽), then the functions 𝑥−𝛼 and 𝛽−𝑥 are both nonnegative and nondecreasing. Therefore, Var_[]𝑎,𝑏(𝑥−𝛼)=𝛼(𝑎)−𝛼(𝑏)+𝑥(𝑏)−𝑥(𝑎), (32) and, hence, ‖𝑥‖_BV≤‖𝛼‖_BV+‖𝑥−𝛼‖_BV=‖𝛼‖_BV+|𝑥(𝑎)−𝛼(𝑎)|+Var_[]𝑎,𝑏()=‖𝛼‖𝑥−𝛼_BV+𝑥(𝑏)−𝛼(𝑏)≤‖𝛼‖_BV+𝛽(𝑏)−𝛼(𝑏). (33) The last estimate shows that the norms of all such 𝑥 are uniformly bounded.

Let 𝑝∶BV([𝑎,𝑏],ℝ^𝑛)→BV([𝑎,𝑏],ℝ^𝑛) be a linear operator and ℎ∶BV([𝑎,𝑏],ℝ^𝑛)→ℝ a linear functional. Let us put 𝑉_𝑝,ℎ𝑢∶=𝑢−^⋅_𝑎𝑝𝑢(𝜉)𝑑𝑔(𝜉)−ℎ(𝑢) (34) for any 𝑢 from BV([𝑎,𝑏],ℝ^𝑛). It follows immediately from Definition 1 that the linear operator 𝑉_𝑝,ℎ∶BV([𝑎,𝑏],ℝ^𝑛)→BV([𝑎,𝑏],ℝ^𝑛) defined by (34) has the following property.

Lemma 14. If 𝑝 is a linear operator such that 𝑝∈𝒮_ℎ[𝑎,𝑏],ℝ^𝑛, (35) then 𝑉_𝑝,ℎ is invertible and, moreover, its inverse 𝑉⁻¹_𝑝,ℎ satisfies the inclusion 𝑉⁻¹_𝑝,ℎBV⁺⁺[𝑎,𝑏],ℝ^𝑛⊂BV⁺[𝑎,𝑏],ℝ^𝑛. (36)

We will also use the obvious identity 𝑉_𝑝1,ℎ1+𝑉_𝑝2,ℎ2=2𝑉_{(1/2)(𝑝1+𝑝2),(1/2)(ℎ1+ℎ2)}, (37) which is valid for any linear 𝑝_𝑖∶BV([𝑎,𝑏],ℝ^𝑛)→BV([𝑎,𝑏],ℝ^𝑛), 𝑖=1,2.

3.1. Proof of Theorem 3

Let us set 𝐸=BV([𝑎,𝑏],ℝ^𝑛) and put (𝐹𝑢)(𝑡)∶=𝑢(𝑡)−^𝑡_𝑎𝑓𝑢(𝑠)𝑑𝑔(𝑠)−𝜑(𝑢),𝑡∈[𝑎,𝑏], (38) for any 𝑢 from BV([𝑎,𝑏],ℝ^𝑛). Then (1) takes the form of (28) with 𝑧=0. Since 𝑓𝑢 and 𝑔 are both from BV([𝑎,𝑏],ℝ^𝑛), it follows (see, e.g., [30]) that the function [𝑎,𝑏]∋𝑡↦^𝑡_𝑎𝑓𝑢(𝑠)𝑑𝑔(𝑠) (39) also belongs to BV([𝑎,𝑏],ℝ^𝑛). Therefore, 𝐹 given by (38) is an operator acting in 𝐸.

Note that relation (10) is equivalent to inequalities −𝑝₁(𝑢−𝑣)(𝑡)≤−𝑓𝑢(𝑡)+𝑓𝑣(𝑡)≤−𝑝₂(𝑢−𝑣)(𝑡), (40) for any 𝑡∈[𝑎,𝑏] and {𝑢,𝑣} from BV([𝑎,𝑏],ℝ^𝑛) with properties (11). Integrating (40) with respect to 𝑔, we obtain −^𝑡_𝑎𝑝₁(𝑢−𝑣)(𝑠)𝑑𝑔(𝑠)≤−^𝑡_𝑎𝑠)(𝑠)+𝑓𝑢(𝑑𝑔_𝑡𝑎𝑓𝑣(𝑠)𝑑𝑔(𝑠)≤−^𝑡_𝑎𝑝₂(𝑢−𝑣)(𝑠)𝑑𝑔(𝑠), (41) and, therefore, according to (38), 𝑢(𝑡)−𝑣(𝑡)−^𝑡_𝑎𝑝₁(𝑠)(𝑠)(𝑢)(𝑣)≤(𝑡)−(𝑡)(𝑡)(𝑡)−𝑢−𝑣)(𝑑𝑔−𝜑+𝜑𝐹𝑢)(𝐹𝑣)(≤𝑢−𝑣^𝑡_𝑎𝑝₂(𝑢−𝑣)(𝑠)𝑑𝑔(𝑠)−𝜑(𝑢)+𝜑(𝑣), (42) for all 𝑡∈[𝑎,𝑏]. Taking assumption (13) into account and using notation (34), we get 𝑉_𝑝1,ℎ1(𝑢−𝑣)(𝑡)≤(𝐹𝑢)(𝑡)−(𝐹𝑣)(𝑡)≤𝑉_𝑝2,ℎ2(𝑢−𝑣)(𝑡), (43) for all 𝑡∈[𝑎,𝑏] and 𝑢 and 𝑣 from BV([𝑎,𝑏],ℝ^𝑛) with properties (11). Furthermore, it follows immediately from (34) and (38) that, for any 𝑡∈[𝑎,𝑏], (𝐹𝑢)(𝑡)−(𝐹𝑣)(𝑡)−𝑉_𝑝1,ℎ1(𝑡)(𝑣)(𝑢)+𝑢−𝑣)(=𝜑−𝜑_𝑡𝑎𝑝₁(𝑢−𝑣)(𝑠)−𝑓𝑢(𝑠)+𝑓𝑣(𝑠)𝑑𝑔(𝑠). (44) Therefore, by virtue of inequality (43) and assumption (10), the function 𝐹𝑢−𝐹𝑣−𝑉_𝑝1,ℎ1(𝑢−𝑣) is nonnegative and nondecreasing and, hence, 𝐹𝑢−𝐹𝑣−𝑉_𝑝1,ℎ1(𝑢−𝑣)∈BV⁺⁺[𝑎,𝑏],ℝ^𝑛. (45) In the same manner, one shows that 𝑉_𝑝2,ℎ2(𝑢−𝑣)−𝐹𝑢+𝐹𝑣∈BV⁺⁺[𝑎,𝑏],ℝ^𝑛. (46) Considering (45) and (46), we conclude that 𝐹 satisfies condition (30) with 𝐵_𝑖=𝑉_{𝑝𝑖,ℎ𝑖}, (47)𝑖=1,2, and 𝐾₁=BV⁺[𝑎,𝑏],ℝ^𝑛,𝐾₂=BV⁺⁺[𝑎,𝑏],ℝ^𝑛. (48) By virtue of Lemma 13, 𝐾₂ is a normal and generating cone in BV([𝑎,𝑏],ℝ^𝑛).

Since, by assumption (12), 𝑝₁∈𝒮_ℎ1, it follows that 𝑉_𝑝1,ℎ1 is invertible and the inclusion 𝑉_𝑝⁻¹_1,ℎ1𝐾₂⊂𝐾₁ (49) holds. Furthermore, by (12) and Lemma 14, the operator (1/2)𝑉⁻¹_{(1/2)(𝑝1+𝑝2),(1/2)(ℎ1+ℎ2)} exists and coincides with the inverse operator to 𝑉_𝑝1,ℎ1+𝑉_𝑝2,ℎ2. It is moreover positive in the sense that 𝑉_𝑝1,ℎ1+𝑉_𝑝2,ℎ2⁻¹𝐾₂⊂𝐾₁. (50) Combining (49) and (50), we see that the inverse operators 𝐵⁻¹ and (𝐵₁+𝐵₂)⁻¹ exist and possess properties (29) with respect to cones (48). Applying now Theorem 12, we prove the unique solvability of (28) and, hence, that of (1).

3.2. Proof of Theorem 4

Rewriting relations (15) in the form 𝑙₁(𝑢−𝑣)(𝑡)−𝑙₂(𝑡)≤(𝑡)−(𝑡)𝑢−𝑣)(𝑓𝑢𝑓𝑣≤𝑙₂(𝑢−𝑣)(𝑡)+𝑙₁(𝑢−𝑣)(𝑡),𝑡∈[𝑎,𝑏], (51) and putting 𝑝_𝑖∶=𝑙₁−(−1)^𝑖𝑙₂,𝑖=1,2, (52) we find that 𝑓 admits estimate (10) with 𝑝₁ and 𝑝₂ defined by (52). Therefore, it remains only to note that assumption (14) ensures the validity of inclusions (12), and to apply Theorem 3.

3.3. Proof of Corollary 6

It turns out that, under assumptions (16) and (17), the operators 𝑙_𝑖∶BV([𝑎,𝑏],ℝ^𝑛)→BV([𝑎,𝑏],ℝ^𝑛), 𝑖=1,2, defined by the formulae 𝑙₁∶=−𝜃𝑞₂,𝑙₂∶=𝑞₁+(1−𝜃)𝑞₂ (53) with 𝜃∈(0,1), satisfy conditions (14) and (15) of Theorem 4. Indeed, estimate (16) and the positivity of the operator 𝑞₂ imply that, for any 𝑢 and 𝑣 with properties (11) and all 𝑡∈[𝑎,𝑏], the relations 𝑓𝑢(𝑡)−𝑓𝑣(𝑡)+𝜃𝑞₂(𝑡=𝑡)−𝑡)𝑢−𝑣)()𝑓𝑢(𝑓𝑣(+𝑞₂(𝑡)−()𝑞𝑢−𝑣)(1−𝜃₂(𝑢−𝑣)(𝑡)≤𝑞₁(𝑢−𝑣)(𝑡)+(1−𝜃)𝑞₂(𝑢−𝑣)(𝑡)=𝑞₁(𝑢−𝑣)(𝑡)+(1−𝜃)𝑞₂(𝑢−𝑣)(𝑡) (54) are true. This means that 𝑓 admits estimate (15) with the operators 𝑙₁ and 𝑙₂ of form (53). It is easy to verify that assumption (17) ensures the validity of inclusions (14) for operators (53), and, therefore, Theorem 4 can be applied.

3.4. Proof of Corollaries 7 and 8

The results follow directly from Corollary 6 if one puts 𝜃=(1/2) and 𝜃=(1/4), respectively.

3.5. Proof of Corollary 9

If 𝑦=0, one should apply Theorem 4 with 𝑓=𝑙, 𝑙₁=𝑝₁, 𝑙₂=𝑝₀, and ℎ₁=ℎ, ℎ₂=ℎ. For a nonzero 𝑦∈BV([𝑎,𝑏],ℝ^𝑛), one can modify the theorem slightly by incorporating the forcing term 𝑦 directly into (1) similarly to (20). Then we find that the argument of Section 3.1 remains almost unchanged.

3.6. Proof of Corollary 10

Corollary 7 with 𝑓=𝑙, ℎ₁=ℎ, and ℎ₂=ℎ is applied.

3.7. Proof of Corollary 11

It is sufficient to note that, under these assumptions, the linear operators 𝑝_𝑖∶BV([𝑎,𝑏],ℝ^𝑛)→BV([𝑎,𝑏],ℝ^𝑛),   𝑖=1,2, defined by the formulae 𝑝₀1∶=2𝑙₀+𝑙₁,𝑝₁1∶=2𝑙₀−𝑙₁, (55) satisfy conditions (21) and (22) of Corollary 9.

4. Comments

The following can be pointed out in relation to the above said.

4.1. Remark on Constants

The conditions presented in Sections 2.1 and 2.2 are, in a sense, optimal and cannot be improved. For example, it follows from [26] that assumption (14) of Corollary 7 can be replaced neither by the condition (1−𝜀)𝑙₁∈𝒮_ℎ[𝑎,𝑏],ℝ^𝑛,𝑙₀+𝑙₁∈𝒮_ℎ[𝑎,𝑏],ℝ^𝑛 (56) nor by the condition 𝑙₁∈𝒮_ℎ[𝑎,𝑏],ℝ^𝑛,(1−𝜀)𝑙₀+𝑙₁∈𝒮_ℎ[𝑎,𝑏],ℝ^𝑛, (57) no matter how small 𝜀∈(0,∞) may be. Likewise, counterexamples show that the assertion of Corollary 11 is not true any more if condition (27) is replaced by either of its weaker versions (1−𝜀)𝑙₀∈𝒮_ℎ[𝑎,𝑏],ℝ^𝑛,12𝑙₀−𝑙₁∈𝒮_ℎ[𝑎,𝑏],ℝ^𝑛, (58) and 𝑙₀∈𝒮_ℎ[𝑎,𝑏],ℝ^𝑛,1𝑙2+𝜀⁰−𝑙₁∈𝒮_ℎ[𝑎,𝑏],ℝ^𝑛 (59) with a positive 𝜀. The same holds for the other inequalities and constants.

4.2. Equations with Matrix-Valued Functions

It is clear from the proofs given above that similar statements can also be obtained in the case where the integrals of matrix-valued functions are considered in (1), as described, for example, in [3, 4].

4.3. The Case of a Nonmonotone Measure

Results similar to those stated above can also be formulated in the case where the function 𝑔 involved in (1) is of bounded variation only and not necessarily nondecreasing. For this purpose, one should use the representation 𝑔=𝑔₁−𝑔₂, (60) where 𝑔_𝑘, 𝑘=1,2, are nondecreasing functions, and modify the definition of the set 𝒮_ℎ([𝑎,𝑏],ℝ^𝑛) in the following way.

Definition 15. A pair of operators (𝑞₁,𝑞₂) is said to belong to 𝒮_ℎ([𝑎,𝑏],ℝ^𝑛) if the equation 𝑢(𝑡)=ℎ(𝑢)+^𝑡_𝑎𝑞₁𝑢(𝑠)𝑑𝑔₁(𝑠)−^𝑡_𝑎𝑞₂𝑢(𝑠)𝑑𝑔₂(𝑠)+𝑟(𝑡),𝑡∈[𝑎,𝑏], (61) has a unique solution 𝑢 for any 𝑟 from BV([𝑎,𝑏],ℝ^𝑛) and, moreover, the solution 𝑢 is nonnegative for nonnegative 𝑟.

In that case, an analogue of the assertion of Theorem 3 is obtained if assumption (12) is replaced by the pair of conditions 𝑝₁,𝑝₂∈𝒮_ℎ1[𝑎,𝑏],ℝ^𝑛,12𝑝₁+𝑝₂,12𝑝₁+𝑝₂∈𝒮_{(1/2)(ℎ1+ℎ2)}[𝑎,𝑏],ℝ^𝑛. (62)

The proof of this fact is pretty similar to the argument given in Section 3.1 and uses Theorem 12 with the operators 𝐵_𝑘∶BV([𝑎,𝑏],ℝ^𝑛)→BV([𝑎,𝑏],ℝ^𝑛), 𝑘=1,2, 𝐵_𝑘𝑢(𝑡)∶=𝑢(𝑡)−^𝑡_𝑎𝑝_𝑘𝑢(𝑠)𝑑𝑔₁(𝑠)+^𝑡_𝑎𝑝_3−𝑘𝑢(𝑠)𝑑𝑔₂(𝑠)−ℎ_𝑘(𝑢),𝑡∈[𝑎,𝑏], (63) instead of those defined by (47).

Acknowledgment

This research is supported in part by RVO: 67985840 (A. Rontó).