Continuity in a parameter of solutions to generic boundary-value problems

We introduce the most general class of linear boundary-value problems for systems of first-order ordinary differential equations whose solutions belong to the complex H\"older space $C^{n+1,\alpha}$, with $0\leq n\in\mathbb{Z}$ and $0\leq\alpha\leq1$. The boundary conditions can contain derivatives $y^{(r)}$, with $1\leq r\leq n+1$, of the solution $y$ to the system. For parameter-dependent problems from this class, we obtain constructive criterion under which their solutions are continuous in the normed space $C^{n+1,\alpha}$ with respect to the parameter.


Introduction
Questions concerning the validity of passage to the limit in parameter-dependent differential equations arise in various problems. These questions are best clear up for the Cauchy problem for systems of first-order ordinary differential equations. Gikhman [3], Krasnosel'skii and S. Krein [15], Kurzweil and Vorel [16] obtained fundamental results relating to continuous dependence in the parameter of solutions to the Cauchy problem for nonlinear systems. For linear systems, these results were refined and supplemented by Levin [17], Opial [23], Reid [24], and Nguyen The Hoan [22].
Parameter-dependent boundary-value problems are far less studied than the Cauchy problem. Kiguradze [9][10][11] and Ashordia [1] introduced and investigated the class of general linear boundary-value problems for systems of first-order ordinary differential equations. The solutions y to these problems are supposed to be absolutely continuous on a compact interval [a, b], and the boundary condition is of the form By = q where B is an arbitrary linear continuous operator from C([a, b], R m ) to R m (m is the number of differential equations in the system). Kiguradze and Ashordia obtained conditions under which the solutions to a parameter-dependent problem from this class are continuous in C([a, b], R m ) with respect to the parameter. Recently these results were refined and generalized to complex-valued functions and systems of higher-order differential equations [14,19,21].
In this paper we introduce a new class of boundary-value problems for systems of firstorder linear differential equations. In contrast to the usual boundary-value problems, this class relates to a given function space. We consider systems whose coefficients and right-hand sides belong to the Hölder space (C n,α ) m := C n,α ([a, b], C m ), with 0 ≤ n ∈ Z and 0 ≤ α ≤ 1.
Since the solutions y to each of these systems run through the whole space (C n+1,α ) m , we consider the most general boundary condition of the form By = q where B is an arbitrary linear continuous operator from (C n+1,α ) m to C m . This condition can contain the derivatives y (r) , with 1 ≤ r ≤ n, of the solution. We say that these boundary-value problems are generic with respect to the Hölder space C n+1,α .
We investigate parameter-dependent boundary-value problems from the class introduced. We will find sufficient and necessary constructive conditions under which the solutions to these problems are continuous in (C n+1,α ) m with respect to the parameter. We will also prove a two-sided estimate for the degree of convergence of the solutions.
Note that the generic boundary-value problems with respect to the Sobolev spaces and to the spaces C (n+1) were introduced in [13,20] and [18,26], where sufficient conditions for continuous dependence in parameter of solutions were obtained. These results were applied to the investigation of multipoint boundary-value problems [12], Green's matrices of boundaryvalue problems [14,19], in the spectral theory of differential operators with distributional coefficients [5][6][7].
The approach developed in this paper can be applied to generic boundary-value problems with respect to other function spaces.

Main results
Throughout the paper we arbitrarily fix a compact interval [a, b] ⊂ R, integers n ≥ 0 and m ≥ 1, and a real number α such that 0 ≤ α ≤ 1. We use the Hölder spaces with 0 ≤ l ∈ Z. They consist respectively of all vector-valued functions and (m × m)-matrixvalued functions whose components belong to C l,α := C l,α ([a, b], C), with the norm of the functions being equal to the sum of the norms in C l,α of all their components. We denote all these norms by · l,α . Certainly, if α = 0, then C l,α stands for the space C (l) := C (l) ([a, b], C) of all l times continuously differentiable functions on [a, b]. We will recall the definition of the Hölder space C l,α at the end of this section. Let the parameter ε runs through the set [0, ǫ 0 ), with the number ǫ 0 > 0 being fixed. We consider the family of the following linear boundary value-problems: Here, for every ǫ ∈ [0, ǫ 0 ), we suppose that y(·, ǫ) ∈ (C n+1,α ) m is an unknown vector-valued function, whereas the matrix-valued function A(·, ǫ) ∈ (C n,α ) m×m , vector-valued function f (·, ǫ) ∈ (C n,α ) m , continuous linear operator and vector q(ǫ) ∈ C m are arbitrarily given. Note that we interpret vectors as columns. The boundary condition (2.3) with the continuous operator (2.4) is the most general for equation (2.2) in view of Lemma 3.2, which will be given in the next section. This condition covers all the classical types of boundary conditions such as initial conditions in the Cauchy problem, various multipoint conditions, integral conditions, conditions used in mixed boundary-value problems, and also nonclassical conditions containing the derivatives y (r) (·, ε) with 1 ≤ r ≤ n + 1.
By analogy with papers [13] and [26] we say that the boundary-value problem (2.2), (2.3) is generic with respect to the space C n+1,α . (In these papers, the notion of a generic (or, in other words, total) boundary-value problem is introduced with respect to the Sobolev spaces and spaces of continuously differentiable functions respectively).
According to (2.6), the error and discrepancy of the solution y(·, ǫ) to the boundary-value problem (2.2), (2.3) are of the same degree. Here, we consider y(·, 0) as an approximate solution to this problem.
We will prove Main Theorem and Theorem 2.1 in Section 4.
is an m × m-matrix-valued function formed by scalar functions that are of bounded variation on [a, b], right-continuous on (a, b), and equal to zero at t = a. Here, the integral is understood in the Riemann-Stieltjes sense. This representation follows from the known description of the dual of C (n+1) ; see, e.g., [2, p. 344]. Applying (2.7), we can reformulate Limit Condition (II) in an explicit form. Namely, Limit Condition (II) is equivalent to that the following four conditions are fulfilled as ε → 0+: (Here, of course, the convergence is considered in C m×m .) This equivalence follows from the Riesz theorem on the weak convergence of linear continuous functionals on [25, Ch. III, Sect. 55]). It is useful to compare these conditions with the criterion of the norm convergence of the operators B(ε) → B(0) as ε → 0+. According to this criterion, the latter convergence is equivalent to that condition (2a) and are fulfilled as ε → 0+. Condition (2.8) is much stronger than the system of conditions (2b)-(2d). This becomes especially clear if we observe that condition (2.8) implies the uniform convergence of the functions Φ(t, ε) to Φ(t, 0) on [a, b] as ε → 0+ whereas conditions (2b)-(2d) do not imply the pointwise convergence of these functions at least in one point of (a, b).
At the end of this section, we recall the definition of the Hölder spaces and discuss some notions and designations relating to these spaces. Let an integer l ≥ 0. We endow the Banach By definition, the Hölder space C l,α := C l,α ([a, b], C), with 0 < α ≤ 1, consists of all functions x ∈ C (l) such that This space is Banach with respect to the norm For the sake of uniformity, we use the designations C l,0 := C (l) and · l,0 := · l .
Note that each C l,α , with 0 ≤ α ≤ 1, is a Banach algebra with respect to a certain norm which is equivalent to · l,α . Of course, this is true for the spaces (2.1) as well. The norms in them are also denoted by · l,α . It will be always clear from the context to which space (scalar or vector-valued or matrix-valued functions) these norms relate.

Preliminaries
In this section, we investigate the boundary-value problem (2.2), (2.3) provided that the parameter ε is fixed. We establish basic properties of this problem; they will be used in our proof of Main Theorem. Fixing and then omitting ε in the problem (2.2), (2.3), we write it in the form So, we consider an arbitrary boundary-value problem (3.1), (3.2) which is generic with respect to the function space C n+1,α . The latter means that y ∈ In connection with this theorem, we recall that a linear continuous operator T : E 1 → E 2 between Banach spaces E 1 and E 2 is called Fredholm if its kernel ker T and co-kernel E 2 /T (E 1 ) are both finite-dimensional. If this operator is Fredholm, then its range T (E 1 ) is closed in E 2 (see, e.g., [8,Lemma 19.1.1]). The finite index of the Fredholm operator T is defined by the formula ind T := dim ker T − dim(E 2 /T (E 1 )). Before we prove Theorem 3.1, let us establish Moreover, if f runs through the whole space (C n,α ) m , then the solutions to (3.1) run through the whole space (C n+1,α ) m .
Proof. We suppose that a differentiable function y is a solution to equation (3.1) for certain f ∈ (C n,α ) m . Let us prove that y ∈ (C n+1,α ) m . Since A and f are at least continuous on [a, b], we get y ′ = f − Ay ∈ (C (0) ) m . Hence, y ∈ (C (1) ) m ⊂ (C 0,α ) m . Moreover, Indeed, if y ∈ (C l,α ) m for some integer l ∈ [0, n], then y ′ = f − Ay ∈ (C l,α ) m and therefore y ∈ (C l+1,α ) m . Now the inclusion y ∈ (C 0,α ) m and property (3.4) imply the required inclusion y ∈ (C n+1,α ) m . Let us now prove the last assertion of this Lemma. For arbitrary f ∈ (C n,α ) m there exists a solution y to equation (3.1). We have just proved that y ∈ (C n+1,α ) m . This in view of the evident implication y ∈ (C n+1,α ) m ⇒ Ly ∈ (C n,α ) m means the last assertion of Lemma 3.2.
Proof of Theorem 3.1. Let Cy := y(a) for arbitrary y ∈ (C n+1,α ) m . The linear mapping y → (Ly, Cy), with y ∈ (C n+1,α ) m , sets an isomorphism Indeed, for arbitrary f ∈ (C n,α ) m and q ∈ C m , the Cauchy problem (3.1) and y(a) = q has a unique solution y. Owing to Lemma 3.2 we have the inclusion y ∈ (C n+1,α ) m . Therefore the above-mentioned mapping sets the one-to-one linear operator (3.5). Since this operator is continuous, it is an isomorphism by the Banach theorem on inverse operator. We now note that the operator ( Here and below, I m is the identity (m × m)-matrix. Note that Y ∈ (C n+1,α ) m×m due to Lemma 3.2.
. This set is endowed with the metric from the Banach space (C n+1,α ) m×m .
Theorem 3.4. Consider the nonlinear mapping Υ : A → Y that associates with any A ∈ (C n,α ) m×m the unique solution Y ∈ (C n+1,α ) m×m to the boundary-value problem (3.6), (3.7). This mapping is a homeomorphism of the Banach space (C n,α ) m×m onto the metric space Y n+1,α t 0 .
Our proof of this theorem uses some properties of an integral operator V A associated with the boundary-value problem (3.6), (3.7) with A ∈ (C n,α ) m×m . This operator transforms any function Y ∈ (C n+1,α ) m×m into the function (3.9) Note that ΥA = Y ⇔ (I + V A )Y = I m (3.10) for arbitrarily given Y ∈ (C n+1,α ) m×m . Henceforth I denotes the identity operator in the relevant space.
Lemma 3.5. Let A ∈ (C n,α ) m×m . Then the linear operator V A is compact on (C n+1,α ) m×m , and its norm satisfies the inequality

11)
where κ is a certain positive number that does not depend on A. Besides, this operator is quasinilpotent; i.e., for every λ ∈ C we have an isomorphism

12)
Proof. Given Y ∈ (C n,α ) m×m we can write Here c 0 and c 1 are certain positive numbers that do not depend on A and Y . Hence, . Thus, the mapping Y → V A Y is a bounded operator from (C n,α ) m×m to (C n+1,α ) m×m , and its norm does not exceed c 2 A n,α . Hence, owing to the compact embedding (C n+1,α ) m×m ֒→ (C n,α ) m×m , we conclude that the restriction of this mapping on (C n+1,α ) m×m is a compact operator on (C n+1,α ) m×m and that the norm of this operator satisfies (3.11) with κ := c 2 c 3 . Here, c 3 is the norm of this compact embedding.
Let λ ∈ C and deduce the isomorphism (3.12). Since A is at least continuous on [a, b], the mapping Y → Y − λV A Y , with Y ∈ (C (0) ) m×m , is an isomorphism of the space (C (0) ) m×m onto itself. This fact is well known if m = 1; its proof for m ≥ 2 is analogous to those for m = 1. The restriction of this isomorphism to (C 0,α ) m×m is a bounded injective operator on (C 0,α ) m×m according to what we have proved in the previous paragraph. This operator is also surjective. Indeed, if F ∈ (C 0,α ) m×m , then there exists a function Y ∈ (C (0) ) m×m such that Thus, we have the isomorphism in the case of k = 0. We now choose an integer l ∈ [0, n] arbitrarily and assume that the isomorphism (3.13) holds true for k = l. Reasoning analogously, we can deduce from our assumption that this isomorphism holds for k = l + 1. Namely, the restriction of the isomorphism (3.13) for k = l to (C l+1,α ) m×m is a bounded injective operator on (C l+1,α ) m×m . Besides, if F ∈ (C l+1,α ) m×m , then there exists a function Y ∈ (C l,α ) m×m such that Y − λV A Y = F ; hence, Therefore we obtain the isomorphism (3.13) for k = l + 1.
Thus, we have proved the required isomorphism (3.12) by the induction with respect to the integer k ∈ [0, n] in (3.13).
Proof of Theorem 3.4. If A ∈ (C n,α ) m×m , then Y := ΥA ∈ Y n+1,α t 0 according to Lemma 3.2 and the Liouville-Jacobi formula and then A(t) = −Y ′ (t)(Y (t)) −1 for every t ∈ [a, b]. Therefore we have the injective mapping (3.14) Let us show that the operator (3.14) is continuous. Assume that A k → A in (C n,α ) m×m as k → ∞. According to Lemma 3.5 we get the norm convergence I + V A k → I + V A , as k → ∞, of the bounded operators on (C n+1,α ) m×m . Hence, owing to the same lemma and equivalence (3.10), we conclude that in (C n+1,α ) m×m as k → ∞. Thus, the operator (3.14) is continuous.
Its inverse is also continuous.

Proof of the main results
We will divide Main Theorem into three lemmas and prove them. We associate the continuous linear operator with the boundary-value problem (2.2), (2.3), where ε ∈ [0, ε 1 ). According to Theorem 3.1, this operator is Fredholm with index zero. Therefore Condition (0) is equivalent to that the operator (4.1) for ǫ = 0 is an isomorphism for each ε ∈ [0, ε 0 ). Proof. We divide it into two steps.
The existence and uniqueness of these three solutions is due to Lemma 4.1.