Conditions for the solvability of the Cauchy problem for linear first-order functional differential equations

Conditions for the unique solvability of the Cauchy problem for a family of scalar functional differential equations are obtained. These conditions are sufficient for the solvability of the Cauchy problem for every equation from the family and are necessary for the solvability of the Cauchy problem for all equations from the family. In contrast to many known articles, we consider equations with functional operators acting into the space of essentially bounded functions.


Introduction
In this article, the well-known integral conditions for the solvability of the Cauchy problem for linear functional differential equations (Theorem 1) are added to necessary and sufficient conditions with point-wise restrictions on functional operators (Theorem 2). Also some conditions for solvability of the Cauchy problem for a family of quasilinear equations are obtained. We use the following notation: R is the space of real numbers; C = C[a, b] is the Banach space of continuous functions x : [a, b] → R with the norm 1 (t) ≡ 1 is the unit function; an operator T : C → L is said to be positive (or isotonic in the terminology of [5]) if it maps each non-negative continuous function into an almost everywhere non-negative function.
Consider the Cauchy problem for a first-order functional differential equatioṅ x(a) = c, where T + , T − : C → L are linear positive operators, f ∈ L, c ∈ R.
A solution of (1)-(2) is a satisfying the initial conditions (2) function x ∈ AC such that equality (1) holds almost everywhere on [a, b]. Problem (1)-(2) is called uniquely solvable if it has a unique solution for every pair f ∈ L, c ∈ R.
The positiveness of operators T + and T − implies their u-boundedness (or the strong boundedness in other terminology) [6]. This property guarantees the Fredholm property of problem (1)-(2) [5,7]. From the Fredholm property it follows that problem (1)-(2) is uniquely solvable if and only if the problem has a unique solution for at least one pair f ∈ L, c ∈ R. In particular, the problem is uniquely solvable if and only if the homogeneous problem has only the trivial solution.
If linear positive operators T + , T − : C → L are Volterra type operators, then the Cauchy problem is uniquely solvable without additional conditions (see, for example, [5]). Some optimal, in a sense, conditions for the unique solvability of problem (1)-(2) are found in [1] (see also [2], [3], [4]) for, generally speaking, non-Volterra operators. We give here this result in the form of necessary and sufficient conditions for the solvability. Note, that each linear positive operator T : C → L is bounded, its norm is defined by the equality hold.
As far as we know, unimprovable conditions for the solvability of the Cauchy problem (1)-(2) in terms of norm operators T + , T − : C → L ∞ are not obtained yet.
The main result is the statement, which is similar to Theorem 1, but deals with operators acting from the space C into the space os essentially bounded functions L ∞ . The norm of linear positive operator T : C → L ∞ is defined by the equality For short, we use the notation From Theorem 1 it is easily to archive a sufficient condition for the solvability. Corollary 1. Let non-negative numbers T + , T − be given. Then, for problem (1)-(2) to be uniquely solvable for all linear positive operators T + , The forthcoming Theorem 2 shows that for all A < 1 (except A = 0) conditions (5) can be improved, and for A = 0 the necessary and sufficient condition for the solvability of the probleṁ for all linear positive operators T − : C → L ∞ with given norm T − C→L∞ = T − remains the inequality B < 3 as in Theorem 1.

The main results
Theorem 2. Let non-negative numbers T + , T − be given. Then, for problem (1)-(2) to be uniquely solvable for all linear positive operators T + , Corollary 2. Let non-negative numbers T + , T − be given. Then, for problem (1)-(2) to be uniquely solvable for all linear positive operators T + , T − : C → L ∞ with given norms T + C→L∞ = T + , T − C→L∞ = T − , it is necessary and sufficient that at least one of the following conditions be fulfilled: 3) Let us remark an obvious corollary of Theorem 2, which can be easily proved by the Schauder fixed point theorem (see, for example, [4, p. 190]). Consider the quasilinear Cauchy problem where T + , T − : C → L ∞ are linear positive operators, the operator F : C → L ∞ is continuous and bounded (maps bounded sets into bounded ones), c ∈ R.
Then the Cauchy problem (10) has a solution.

Proofs
To prove Theorem 2 and Corollary 1 we need auxiliary assertions.
Lemma 1. Let non-negative functions p + , p − ∈ L ∞ be given. To problem (1)- (2) to be uniquely solvable for all linear positive operators T + , T − : C → L ∞ satisfying the equalities T + 1 = p + and T − 1 = p − , it is necessary and sufficient that this problem be uniquely solvable for all linear positive operators T + , Proof. It is clear that only the assertion on the necessity is needed to be proven. Suppose that problem (1)- (2) is not uniquely solvable for some linear positive op- (3) has a non-trivial solution. Then for perturbated operators which satisfy the conditions the homogeneous problem x(a) = 0, has the same non-trivial solution. Hence, the corresponding non-homogeneous problems are not uniquely solvable.
Lemma 2. Let non-negative functions p + , p − ∈ L ∞ be given. For problem (1)- (2) to be uniquely solvable for all linear positive operators T + , T − : C → L ∞ such that it is necessary and sufficient that the problem to have only the trivial solutions for all points τ 1 , τ 2 ∈ [a, b] and for all functions p 1 , p 2 ∈ L ∞ satisfying the conditions Proof. To prove the sufficiency suppose that for given non-negative functions p + , p − ∈ L ∞ there exist linear positive operators T + , T − : C → L ∞ such that and problem (1)- (2) is not uniquely solvable. Then there exists a non-trivial solution y ∈ AC of the homogeneous problem (3). Let τ 1 be a point of the minimum, τ 2 be a point of the maximum of the solution y. Then Therefore, It follows that there exists a measurable function ξ : [a, b] → [0, 1] such that for the functions holds. It is clear that conditions (13) for the functions p 1 , p 2 are fulfilled and problem (12) has a non-trivial solution. Let us prove the necessity. Suppose that function p + , p − ∈ L ∞ are non-negative. Let conditions (13) be fulfilled and problem (12) have a non-trivial solution. Define linear positive solutions T + , T − : C → L ∞ by the equalities , where p + 1 and p − 1 are the positive and negative parts of the function p 1 (p + 1 = (|p 1 | + p 1 )/2, p − 1 = (|p 1 | − p 1 )/2). Then operators T + , T − satisfy equalities (11) and, moreover, problem (3) has the same non-trivial solution as well as problem (12). So, problem (1)-(2) is not uniquely solvable.
Lemma 3. Let non-negative numbers T + , T − be given. For problem (12) to have only the trivial solution for all τ 1 , τ 2 ∈ [a, b] and for all functions p 1 , p 2 ∈ L ∞ such that it is necessary and sufficient that inequalities (7), (8) to be valid.
Proof. Let us get necessary and sufficient conditions for all problems (12) provided (13) have only the trivial solution. For any solution y we have Therefore, the system of equations has a solution C 1 = y(τ 1 ), C 2 = y(τ 2 ). Conversely, the solution of the Cauchy problem (12) corresponds to every solution (C 1 , C 2 ) of system (15). Thus, problem (12) has no non-trivial solutions if and only if algebraic system (15) has no non-trivial solutions with respect to the variables C 1 , C 2 , that is, if Consider the determinant △ for a ≤ τ 1 ≤ τ 2 ≤ b and for all functions p 1 , p 2 satisfying conditions (14). We have where under conditions (14) the values can take arbitrary numbers from the following intervals Since the determinant △ is continuous with respect to α, β, τ 1 , τ 2 on the connected admissible set of these parameters and △ = 1 for admissible values α = 0, β = 0, τ 1 = τ 2 = 0, then for all problems (12) to have only the trivial solution it is necessary and sufficient that △ be positive for all admissible values of these parameters. Find the minimal value of △ for fixed T + , T − and for all rest parameters satisfying inequalities (16). All problems (12) provided conditions (14) have only the trivial solutions if and only if for the pair (T + , T − ) this minimum is positive.
Proofs of Theorem 2 and Corollary 1. Assertion 1) of Corollary 1 follows from lemmas 1, 2, and 3. If in the proof of lemma 3 not to minimize with respect to the variable τ 1 , then we direct obtain the condition of the positiveness for the minimal value M that is inequality (6) of Theorem 2. If we solve (6) with respect to the variable B, then we have condition 2) of Corollary 1, if with respect to the variable A, then we obtain condition 3).