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

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

Our aim is to examine the solvability of the equation : BV([, ], R  ) → BV([, ], R  ) is a, generally speaking, nonlinear operator and  : BV([, ], R  ) → R  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  : [, ] → R. We refer to [1][2][3][4][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  ∈ BV([, ], R  ), 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  : [, ] → R  which has bounded variation and satisfies (1) on the interval [, ].
Equation ( 1) is an extension of a measure differential equation studied systematically, for example, in [2,[8][9][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 () = ,  ∈ [, ], (1) takes the form and, thus, in the absolutely continuous case, reduces to the nonlocal boundary value problem for a functional differential equation whose various particular types are the object of investigation of many authors (see, e.g., [15][16][17][18][19]).A more general choice of  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] where Δ() := (+) − (−) for any function  from BV([, ], R  ) (in fact, Δ() = (+) − () if, as is customary [11] in that context, a solution is assumed to be left continuous).Here,  = (  )  =1 : BV([, ], R  ) → BV([, ], R  ), the jumps may occur at the preassigned times  1 ,  2 , . . .,   , and their action is described by the operators   : R  → R  ,  = 1, . . ., .By the usual integration argument [11], one can represent (4) alternatively in the form It follows, in particular, from [14,Lemma 2.4] that ( 5) is equivalent to the measure functional differential equation with () =  + ∑  =1  (  ,] (),  ∈ [,], and f : 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  suitably [13], one arrives at the equation In (8), ℎ : ([−, 0], R  ) × [, ] → R  is a functional in the first variable,  is from the space ([−, 0], R  ) of continuous functions on [−, 0], and the Krasovsky notation Finally, eliminating the initial function  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([, ], R  ).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][25][26].The key assumption is that certain linear operators associated with the nonlinear operator  appearing in (1) possess the following property.Definition 1.Let ℎ : BV([, ], R  ) → R  be a linear mapping.One says that a linear operator  : has a unique solution  for any  from BV([, ], R  ), and, moreover, the solution  is nonnegative for any nonnegative .
The property described by Definition 1, in fact, means that the linear operator associated with ( 9) is positively invertible on BV([, ], R  ), and thus it corresponds to the existence and positivity of Green's operator for a boundary value problem [15].
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][18][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][28][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 neutraltype functional differential equations [23] can be formulated in form (1); see also [4,30]).

Unique Solvability Conditions
We are going to show that the knowledge of the property  ∈ S ℎ ([, ], R  ) for certain linear operators  and ℎ associated with (1) allows one to guarantee its unique solvability.

Nonlinear Equations.
The following statement is true.Theorem 3. Assume that there exist certain linear operators for arbitrary functions  : Furthermore, let the inclusions be fulfilled with some linear functionals Then (1) has a unique solution for an arbitrary  such that 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
and such that (13) and the inequality is true for arbitrary functions  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([, ], R  ).Note that no monotonicity assumptions are imposed on  1 in Theorem 4. In the cases where the positivity of certain linear majorants is known, the following statement may be of use.
Corollary 6 allows one to obtain, in particular, the following statements.
Corollary 8.The assertion of Corollary 7 is true with (18) replaced by the condition The statements formulated above express fairly general properties of (1) and extend, in particular, the corresponding results of [25,27,29,31].

Linear Equations.
Let us now assume that  : 1) is an affine mapping, and, therefore, hold, and the estimate is satisfied for any nonnegative  ∈ BV ([, ], R  ).Then (20) has a unique solution.
We also have the following.
Corollary 10.Let there exist positive linear operators   : BV([, ], R  ) → BV([, ], R  ),  = 0,1, satisfying the inclusions and such that the inequalities We conclude this note by considering the case where  in (20) is a linear mapping admitting a decomposition into the sum of its positive and negative parts; that is, where   : BV([, ], R  ) → BV([, ], R  ),  = 0, 1, are linear and positive.In that case, for the equation of the form where ℎ : BV([, ], R  ) → R  is linear and  ∈ BV([, ], R  ), the following result is obtained.
It is interesting to observe the second condition in (27); it thus turns out that property S ℎ ([, ], R  ) for one half of the operator under the integral sign in (26) ensures the unique solvability of the original equation ( 26).

Proofs
Let ⟨, ‖ ⋅ ‖⟩ be real Banach space, let  ∈  be a given vector, and let  :  →  be a mapping.Let   ⊂ ,  = 1, 2, be closed cones inducing the corresponding partial orderings ≦   , so that  ≦    if and only if  −  ∈   .The following statement [32,33] on the abstract equation will be used below.
Recall that  2 is normal if all the sets order bounded with respect to ≦  2 are also norm bounded and that  1 is generating if and only if { − V | {, V} ⊂  1 } =  (see, e.g., [33,34]).
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 ++ ([, ], R  ) that it is a cone in BV([, ], R  ), 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] The last estimate shows that the norms of all such  are uniformly bounded.
then  ,ℎ is invertible and, moreover, its inverse  −1 ,ℎ satisfies the inclusion We will also use the obvious identity which is valid for any linear for any  from BV([, ], R  ).Then (1) takes the form of ( 28) with  = 0. Since  and  are both from BV([, ], R  ), it follows (see, e.g., [30]) that the function also belongs to BV([, ], R  ).Therefore,  given by ( 38) is an operator acting in .

Proof of Theorem 4.
Rewriting relations (15) in the form and putting we find that  admits estimate (10) with  1 and  2 defined by (52).Therefore, it remains only to note that assumption (14) ensures the validity of inclusions (12), and to apply Theorem 3.

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

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 nor by the condition no matter how small  ∈ (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 with a positive .The same holds for the other inequalities and constants.

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].
has a unique solution  for any  from BV([, ], R  ) and, moreover, the solution  is nonnegative for nonnegative .
In that case, an analogue of the assertion of Theorem 3 is obtained if assumption (12) is replaced by the pair of conditions