Topological structure of solution sets to asymptotic n-th order vector boundary value problems

Topological structure of solution sets to asymptotic n-th order vector boundary value problems

Jan Andres

and Martina Pavlaˇcková

Deptartment of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic

Received 14 June 2018, appeared 14 August 2018 Communicated by Alberto Cabada

Abstract. The R_δ-structure of solutions is investigated for asymptotic, higher-order, vector boundary value problems. Using the inverse limit technique, the topological structure is also studied, as the first step, on compact intervals. The main theorems are supplied by illustrative examples. One of them is finally applied, on the basis of our recently developed principle, to nontrivial existence problems.

Keywords:asymptoticn-th order vector problems, topological structure,R_δ-set, inverse limit technique, Hukuhara–Kneser–Aronszajn type results.

2010 Mathematics Subject Classification: 34A60, 34B15, 34B40, 47H04.

1 Introduction

The investigation of a topological structure of sets of solutions to asymptotic boundary value problems is a delicate problem. The majority of the Hukuhara–Kneser–Aronszajn type results about (possibly special) continua of solutions is related to Cauchy initial value problems on compact intervals (see e.g. [5, Chapter III.2], [18–20], [22, Chapters 3 and 4], [26] and the references therein). Less results for Cauchy problems on non-compact and, in particular, infinite intervals were obtained by various techniques, e.g., in [1–4], [5, Chapter III.2], [12, 17,20,21,24,26,33,37,40]. Rather rare results for boundary vale problems concern again almost exclusively those on compact intervals (see e.g. [11,15,38] and [5, Chapter III.3], [22, Chapter 6], where the whole chapters are devoted to this problem). The quite unique related results on non-compact intervals can be found, as far as we know, in [7,28], [29, Chapter III.13].

The knowledge of a topological structure of solutions for further boundary value problems on non-compact intervals would be therefore highly appreciated.

The reason why such results are so rare consists in counter-examples presented in [3], [5, Example II.2.12], [23], demonstrating the impossibility of asymptotic analogies to the situation on compact intervals. These troubles are due to an “unpleasant” related topology of non- normable Fréchet spaces. For instance, a contractivity of a given operator with respect to a metric need not follow from a contractivity with respect to all seminorms. That is why the

BCorresponding author. Email: jan.andres@upol.cz

main theorem in [28] might be empty. On the other hand, although this difficulty can be overcame by means of the inverse limit method, sometimes also called the projective limit technique (see e.g. [3,4,23,27,34]), the class of appropriate boundary value problems seems to be rather narrow. In fact, besides for the Cauchy initial value problems, simple nontrivial examples for asymptotic boundary value problems via the inverse limit method, were given only by ourselves in [7].

Let us note that, unlike in [29], where (the Hukuhara–Kneser type) continua of solutions were received in a purely analytic way, (the Aronszajn type) R_δ-structure was detected in [7].

The main advantage of these special continua consists in its further possible application to the existence results for nonlinear asymptotic problems (see Section 5 below). The same is true for all special continua (except compact, connected sets themselves) from the proper inclusion scheme (2.1) below.

Hence, our paper is organized as follows. After the auxiliary definitions, lemmas and propositions in Section 2, the topological structure is firstly studied on compact intervals in Section 3. The first obtained result can be regarded in a certain sense as a final theorem.

Nevertheless, it will only take there the form of proposition, because it is further employed, via an inverse limit method in Section 4, as a preliminary step for the investigation of structure on non-compact intervals. This knowledge is finally applied, on the basis of our principle developed recently in [6], to the solvability of nontrivial existence asymptotic problems. All the main theorems are supplied by illustrative examples.

2 Preliminaries

At first, we recall some geometric notions of subsets of metric spaces; in particular, of compact absolute retracts, compact contractible sets andR_δ-sets. For more details, see, e.g., [5,14,25].

For a subset A ⊂ X of a metric space X = (X,d)and ε > 0, we define the set Nε(A) := {x∈ X| ∃a∈ A:d(x,a)<ε}, i.e. N_ε(A)is an open neighborhood of the setAin X. A subset A⊂ Xis called aretractof Xif there exists a retractionr :X → A, i.e. a continuous function satisfyingr(x) = x, for every x∈ A.

We say that a metric space X is an absolute retract (AR-space) if, for each metric space Y and every closed A ⊂ Y, each continuous mapping f : A → X is extendable over Y. Let us note thatXis an AR-space if and only if it is a retract of some normed space. Moreover, if X is a retract of a convex set in a Fréchet space, then it is anAR-space, too. So, in particular, for an arbitrary interval J ⊂ _Randk,n ∈N, the spacesC(J,R^k)_,Cⁿ(J,R^k)_, ACⁿ_loc(J,R^k)_are AR- spaces as well as their convex subsets. The foregoing symbols denote, as usually, the spaces of functions f : J → _R^k which are continuous, have continuous n-th derivatives and locally absolutely continuousn-th derivatives, respectively, endowed with the respective topologies.

We say that a nonempty subset A of a metric spaceX is contractibleif there exist a point x₀ ∈ Aand a homotopy h : A×[0, 1] → A such thath(x, 0) = x and h(x, 1) = x₀, for every x∈ A. A nonempty setA⊂ Xis called anR_δ-setif there exists a decreasing sequence{A_n}^∞_n=1 of compactAR-spaces (or, despite of the hierarchy (2.1) below, compact, contractible sets) such that

A=

\∞ n=₁

A_n.

Note that anyR_δ-set is nonempty, compact and connected. The following hierarchy holds for

nonempty compact subsets of a metric space:

compact + convex⊂compact AR-space⊂compact + contractible⊂ R_δ-set

⊂compact + acyclic⊂compact + connected, (2.1) and all the above inclusions are proper.

We also employ the following definitions and statements from the multivalued analysis in the sequel. LetXandYbe arbitrary metric spaces. We say thatFis amultivalued mappingfrom X toY (written F : X ( ^Y) if, for every x ∈ X, a nonempty subset F(x)of Y is prescribed.

We associate with FitsgraphΓF, the subset of X×Y, defined by ΓF:={(x,y)∈X×Y|y∈ F(x)}.

A multivalued mapping F : X ( ^{Y is called} upper semicontinuous (shortly, u.s.c.) if, for each openU ⊂Y, the set{x∈ X|F(x)⊂U}is open inX. Every upper semicontinuous map with closed values has a closed graph.

A multivalued mapping F : X ( ^X with bounded values is called Lipschitzian if there exists a constant L>0 such that

d_H(F(x),F(y))≤ Ld(x,y), for every x,y∈X, where

d_H(A,B):=inf{r >0| A⊂ N_r(B)andB⊂ N_r(A)}

stands for the Hausdorff distance; for its properties, see, e.g., [5,25].

We say that a multivalued mappingF:X(^Xwith bounded values is acontractionif it is Lipschitzian with a Lipschitz constant L∈[0, 1).

LetY be a separable metric space and(_Ω,U,ν)_{be a} measurable space, i.e. a nonempty set Ωequipped with a suitableσ-algebra U of its subsets and a countably additive measureνon U. A multivalued mapping F : Ω( ^{Y is calledmeasurable if} {ω ∈ _Ω | F(ω) ⊂ V} ∈ U, for each open setV⊂Y.

We say that the mappingF : J×_R^m (Rⁿ, where J ⊂ _{R, is}upper-Carathéodoryif the map F(·,x) : J ( Rⁿ is measurable on every compact subinterval of J, for all x ∈ _R^m, the map F(t,·):R^m (Rⁿis u.s.c., for almost all (a.a.) t∈ J, and the setF(t,x)is compact and convex, for all (t,x)∈ J×_R^m.

We will employ the following selection statement.

Lemma 2.1 (cf., e.g., [9]). Let F : [a,b]×_R^m (Rⁿbe an upper-Carathéodory mapping satisfying

|y| ≤ r(t)(₁+|x|)_, for every(t,x)∈ [a,b]×_R^m_,and every y ∈ F(t,x)_,where r : [a,b] → [_0,∞) is an integrable function. Then the composition F(t,q(t)) admits, for every q ∈ C([a,b],R^m), a single-valued measurable selection.

IfX∩Y 6=_∅and F:X(Y, then a pointx ∈X∩Yis called afixed pointof Fif x∈ F(x). The set of all fixed points of Fwill be denoted by Fix(F), i.e.

Fix(F):={x∈X |x∈ F(x)}. It will be also convenient to recall the following results.

Proposition 2.2 (cf. [35]). Let X be a closed, convex subset of a Banach space E and letφ : X (^X be a contraction with compact, convex values. ThenFix(φ)is a nonempty, compact AR-space.

Lemma 2.3(cf. [10, Theorem 0.3.4]). Let[a,b]⊂_Rbe a compact interval. Assume that the sequence of absolutely continuous functions x_k :[a,b]→_Rⁿsatisfies the following conditions:

(i) the set{x_k(t)|k ∈_N}is bounded, for every t∈ [a,b],

(ii) there exists a functionα:[a,b]→_R,integrable in the sense of Lebesgue, such that

|x˙_k(t)| ≤α(t), for a.a. t∈[a,b]and for all k ∈_N.

Then there exists a subsequence of {x_k} (for the sake of simplicity, denoted in the same way as the sequence)converging to an absolutely continuous function x:[a,b]→_Rⁿin the following way:

1. {x_k}converges uniformly to x,

2. {x˙_k}converges weakly in L¹([a,b]_,Rⁿ)tox.˙

The following lemma is a slight modification of the well known result.

Lemma 2.4(cf. [39, p. 88]). Let[a,b] ⊂ _R be a compact interval, E₁, E2 be Euclidean spaces and F:[a,b]×E₁(^E2be an upper-Carathéodory mapping.

Assume in addition that, for every nonempty, bounded set B ⊂ E₁, there exists ν = ν(B) ∈ L¹([a,b],[0,∞))such that

|F(t,x)| ≤ν(t), for a.a. t∈ [a,b]and every x∈ B.

Let us define the Nemytskiˇı operator N_F :C([a,b],E₁)( ^L1([a,b],E₂)in the following way:

N_F(x):={f ∈ L¹([a,b],E₂)| f(t)∈F(t,x(t)), a.e. on[a,b]},

for every x ∈ C([a,b],E₁). Then, if sequences {x_i} ⊂ C([a,b],E₁)and {f_i} ⊂ L¹([a,b],E₂), f_i ∈ N_F(x_i), i ∈ _N, are such that x_i → x in C([a,b],E₁) and f_i → f weakly in L¹([a,b],E₂), then

f ∈ NF(x).

3 Topological structure on compact intervals

At first, let us consider the constraint problems for linear systems

x⁽ⁿ⁾(_t) +_A1(_t)_x⁽ⁿ⁻¹⁾(_t) +· · ·+_An(_t)_x(_t)∈ C(_t)_{, for a.a.t}∈ [_0,m]_, x∈ Sm,

(3.1) x⁽ⁿ⁾(t) +A₁(t)x⁽ⁿ⁻¹⁾(t) +· · ·+An(t)x(t)∈ C(t), for a.a.t∈ [0,m],

(x, ˙x, . . . ,x⁽ⁿ⁻¹⁾)∈ S⁰_m,

(3.2) where

(i) A_i : [_0,m] → _R^k×k are integrable matrix functions such that |A_i(t)| ≤ a_i(t)_{, for a.a.}

t∈[0,m]and suitable nonnegative functions a_i ∈L¹([0,m],R), for alli=1, . . . ,n, (ii) S_m is a closed, convex subset of ACⁿ⁻¹([0,m],R^k) (S⁰_m is a closed, convex subset of

ACⁿ⁻¹([0,m],R^k)×ACⁿ⁻²([0,m],R^k)× · · · ×AC([0,m],R^k)),

(iii) C :[0,m](R^k is an integrable mapping with convex closed values such that|C(t)| ≤ c(t)_{, for a.a.} t∈[_0,m]and a suitable nonnegative function c∈ L¹([_0,m]_,R)_,

(iv) there existt₀∈[0,m]and a constantMsuch that|x(t₀)| ≤M,|x˙(t₀)| ≤M, . . . ,|x⁽ⁿ⁻¹⁾(t₀)|

≤ M, for all solutions of problem (3.1)(all solutions of problem (3.2)).

Proposition 3.1. Under the above assumptions(i)–(iv), the solution set of problem(3.1) (the set of solutions and their derivatives up to the(n−1)-st order of problem(3.2))is convex and compact).

Proof. Let us prove that the set of solutions and their derivatives of the b.v.p. (3.2) is convex and compact. By the similar reasoning, it is possible to obtain that the solution set of problem (3.1) is convex and compact as well.

Let us denote by P(t,x(t), ˙x(t), . . . ,x⁽ⁿ⁻¹⁾(t)) := C(t)−A₁(t)x⁽ⁿ⁻¹⁾(t)− · · · −A_n(t)x(t). If x₁,x₂ are solutions of problem (3.2), then it follows from the integral representation of a solution that, for a.a.t ∈[0,m], we have

x₁(t)∈ x₁(t0) +x˙₁(t0)(t−t0) + ¹

2x¨₁(t0)(t−t0)²+· · ·+ ¹

(n−1)!x₁⁽ⁿ⁻¹⁾(t0)(t−t0)ⁿ⁻¹

+ ¹

(n−1)! Z _t

t0

(t−s)ⁿ⁻¹P(s,x₁(s), ˙x₁(s), . . . ,x⁽₁ⁿ⁻¹⁾(s))ds, and

x₂(t)∈ x₂(t₀) +x˙₂(t₀)(t−t₀) + ¹

2x¨₂(t₀)(t−t₀)²+· · ·+ ¹

(n−1)!x₂⁽ⁿ⁻¹⁾(t₀)(t−t₀)ⁿ⁻¹

+ ¹

(n−1)! Z _t

t0

(t−s)ⁿ⁻¹P(s,x₂(s)_{, ˙}x₂(s)_{, . . . ,}x⁽₂ⁿ⁻¹⁾(s))ds.

Letθ ∈[0, 1]be arbitrary. Then

θx₁(t) + (1−θ)x₂(t)∈θx₁(t₀) + (1−θ)·x₂(t₀) + [θx˙₁(t₀) + (1−θ)x˙₂(t₀)](t−t₀) +. . .

+ ¹

(n−1)! Z _t

t0

(t−s)ⁿ⁻¹θ·P(s,x₁(s), ˙x₁(s), . . . ,x₁⁽ⁿ⁻¹⁾(s))ds

+ ¹

(n−1)! Z _t

t0

(t−s)ⁿ⁻¹(1−θ)P(s,x₂(s), ˙x₂(s), . . . ,x⁽₂ⁿ⁻¹⁾(s))ds

=θx₁(t₀) + (1−θ)x₂(t₀) + [θx˙₁(t₀) + (1−θ)x˙₂(t₀)](t−t₀) +. . .

+ ¹

(n−1)! Z _t

t₀

(t−s)ⁿ⁻¹P(s,θx₁(s) + (1−θ)x₂(s), . . . ,θx⁽₁ⁿ⁻¹⁾(s) + (1−θ)x⁽₂ⁿ⁻¹⁾(s))ds.

Moreover, for allk=1, . . . ,n−1,

x⁽₁^k)(t)∈ x⁽₁^k)(t₀) +x₁^(k+1)(t₀)(t−t₀) +· · ·+ ¹ (n−1−k)_!^x

(n−1−k)

1 (t₀)(t−t₀)^n−1−k

+ ¹

(n−1−k)! Z _t

t₀

(t−s)^n−1−kP(s,x₁(s), ˙x₁(s), . . . ,x₁⁽ⁿ⁻¹⁾(s))ds, and

x⁽₂^k)(t)∈ x⁽₂^k)(t₀) +x₂^(k+1)(t₀)(t−t₀) +· · ·+ ¹

(n−1−k)!x⁽₂^n−1−k)(t₀)(t−t₀)^n−1−k

+ ¹

(n−₁−k)_!

Z _t

t0

(t−s)^n−1−kP(s,x₂(s), ˙x₂(s), . . . ,x₂⁽ⁿ⁻¹⁾(s))ds.

By similar arguments as before, we can obtain, for an arbitrary θ ∈ [0, 1] and all k = 1, . . . ,n−1, that

θx₁^(k)(t) + (1−θ)x₂^(k)(t)∈θx⁽₁^k)(t₀) + (1−θ)x⁽₂^k)(t₀) + [θx₁^(k+1)(t₀) + (1−θ)x⁽₂^k)(t₀)](t−t₀) +. . .

+ ¹

(n−1−k)! Z _t

t0

(t−s)^n−1−kP(s,θx₁(s) + (1−θ)x₂(s), . . . ,θx⁽₁ⁿ⁻¹⁾(s) + (1−θ)x⁽₂ⁿ⁻¹⁾(s))ds.

Finally, because of convexity ofS⁰_m, we obtain that

θx₁+ (1−θ)x₂, θx˙₁+ (1−θ)x˙₂, . . . ,θx⁽₁ⁿ⁻¹⁾+ (1−θ)x⁽₂ⁿ⁻¹⁾

∈S⁰_m and, therefore, the set of solutions of (3.2) and their derivatives is convex.

Let us also prove that the set of solutions of (3.2) and their derivatives is relatively compact.

It follows from the well known Arzelà–Ascoli lemma that the set of solutions is relatively compact inCⁿ⁻¹([0,m],R^k)if and only if it is bounded and all solutions and their derivatives (up to the(n−1)-st order) are equi-continuous.

At first, let us show that the set of solutions of (3.2) is bounded inCⁿ⁻¹([0,m],R^k). Letx be a solution of (3.2) and lett∈[0,m]be arbitrary.

Since

x⁽ⁿ⁻¹⁾(t) =x⁽ⁿ⁻¹⁾(t₀) +

Z _t

t0

x⁽ⁿ⁾(s)ds, for a.a. t∈ [0,m], ...

˙

x(t) =x˙(t₀) +

Z _t

t0

¨

x(s)ds, for a.a. t ∈[_0,m]_, x(t) =x(t₀) +

Z _t

t0

˙

x(s)ds, for a.a. t ∈[0,m], it holds, according to conditions(i),(iii)and(iv), that

|x(t)|+|x˙(t)|+· · ·+|x⁽ⁿ⁻¹⁾(t)|

≤ |x(t0)|+|x˙(t0)|+· · ·+|x⁽ⁿ⁻¹⁾(t0)|+

Z _t

t0

|x˙(s)|+|x¨(s)|+· · ·+|x⁽ⁿ⁾(s)|ds

≤nM+

Z _m

0

|x˙(s)|+|x¨(s)|+· · ·+|x⁽ⁿ⁻¹⁾(s)|+c(s) +a₁(s)|x⁽ⁿ⁻¹⁾(s)|+· · ·+a_n(s)|x(s)|ds

≤nM+

Z _m

0 c(s)ds+

Z _m

0 an(s)|x(s)|+ (1+a_n−1(s))|x˙(s)|+· · ·+ (1+a₁(s))|x⁽ⁿ⁻¹⁾(s)|ds

≤nM+

Z _m

0

c(s)ds+

Z _m

0

k(s)(|x(s)|+|x˙(s)|+· · ·+|x⁽ⁿ⁻¹⁾(s)|)ds

where, for all s ∈ [0,m], k(s) := max{1+a₁(s), . . . , 1+a_n−1(s),an(s)}. Therefore, by Gron- wall’s lemma (cf. [30]),

|x(t)|+|x˙(t)|+· · ·+|x⁽ⁿ⁻¹⁾(t)| ≤

nM+

Z _m

0 c(s)ds

e^R0mk(s)ds, for a.a.t∈ [0,m]. (3.3) Therefore, the set of solutions of (3.2) and their derivatives (up to the (n−1)-st order) is bounded inCⁿ⁻¹([0,m],R^k).

Let us now show that all solutions x of (3.2) and their derivatives ˙x, . . . ,x⁽ⁿ⁻¹⁾ are also equi-continuous. So, letxbe a solution of (3.2) andt₁,t₂ ∈[0,m]be arbitrary. Then, we have

|x(t₁)−x(t2)| ≤

Z _t2

t1

|x˙(τ)|dτ

≤

Z _t2

t1

nM+

Z _m

0 c(s)ds

e

R_m

0 k(s)ds dτ

. (3.4) Analogously, we can get, for eachk ∈ {1, . . . ,n−2}, that

|x^(k)(t₁)−x^(k)(t₂)| ≤

Z _t2

t₁

|x^(k+1)(τ)|dτ

≤

Z _t2

t₁

nM+

Z _m

0 c(s)ds

e^R0mk(s)ds dτ

. (3.5)

Moreover,

|x⁽ⁿ⁻¹⁾(t₁)−x⁽ⁿ⁻¹⁾(t₂)| ≤

Z _t2

t1

c(τ) +a₁(τ)|x⁽ⁿ⁻¹⁾(τ)|+· · ·+a_n(τ)|x(τ)|dτ

≤

Z _t2

t₁ c(τ) +l(τ)

nM+

Z _m

0 c(s)ds

e^R0mk(s)ds dτ

, (3.6) where, for all τ∈[0,m],l(τ):=max{a₁(τ), . . . ,a_n(τ)}.

Taking into account estimates (3.4)–(3.6),x, ˙x, . . . ,x⁽ⁿ⁻¹⁾are equi-continuous, becausec(·), k(·), l(·) ∈ L¹([0,m],R). Thus, the set of solutions of (3.2) and their derivatives is relatively compact.

We will still show that the set of solutions of (3.2) and their derivatives (up to the(n−1)-st order) is closed. Let {x_i}be a sequence of solutions of (3.2) such that{(x_i, ˙x_i, . . . ,x⁽_iⁿ⁻¹⁾)} → (x, ˙x, . . . ,x⁽ⁿ⁻¹⁾). By conditions (i), (iii) and estimate (3.3), the sequences {x_i}, {x˙_i}, . . . , {x⁽_iⁿ⁻¹⁾} satisfy the assumptions of Lemma 2.3. Thus, there exists a subsequence of {x_i}, for the sake of simplicity denoted as the sequence, uniformly convergent to x on[0,m], such that{x˙_i}, . . . ,{x_i⁽ⁿ⁻¹⁾}converges uniformly to ˙x, . . . ,x⁽ⁿ⁻¹⁾on[0,m]and that{x⁽_iⁿ⁾}converges weakly to x⁽ⁿ⁾ in L¹ [_0,m]_{, R}^k_.

If we set z_i := x_i, ˙x_i, . . . ,x_i⁽ⁿ⁻¹⁾

, then ˙z_i → x, ¨˙ x, . . . ,x⁽ⁿ⁾

weakly in L¹ [0,m], R^k . Let us now consider the following system

z˙_i(t)∈G(t,z_i(t)), for a.a.t∈ [0,m], (3.7) where

G(t,z_i(t)) =x˙_i, . . . ,x⁽_iⁿ⁾,P(t,z_i(t)).

Using Lemma2.4, for f_i :=z˙_i, f := (x, ¨˙ x, . . .x⁽ⁿ⁾), x_i := (z_i), it follows that (x˙(t)_{, ¨}x(t)_{, . . . ,}x⁽ⁿ⁾(t))∈ G

t,x(t)_{, ˙}x(t)_{, . . . ,}x⁽ⁿ⁻¹⁾(t)_, for a.a. t∈[0,m], i.e.

x⁽ⁿ⁾(t)∈P

t,x(t), ˙x(t), . . . ,x⁽ⁿ⁻¹⁾(t), for a.a.t∈ [0,m].

Moreover, since the set S⁰_m is closed,(x_i, . . .x⁽_iⁿ⁻¹⁾)∈ S⁰_m, for alli∈ _{N, and}(x_i, . . . ,x⁽_iⁿ⁻¹⁾)→ (x, . . .x⁽ⁿ⁻¹⁾), it also holds that (x, ˙x, . . . ,x⁽ⁿ⁻¹⁾) ∈ S⁰_m. After all, the set of solutions of (3.2) and their derivatives is convex and compact, as claimed.

Remark 3.2. The statement of Proposition 3.1 also holds if we consider multivalued matrix mappings A_i, i.e. if we replace problems (3.1), (3.2) by

x⁽ⁿ⁾(t)∈ C(t)−A₁(t)x⁽ⁿ⁻¹⁾(t)− · · · −An(t)x(t), for a.a. t ∈[0,m], x∈S_m,

(3.8) x⁽ⁿ⁾(t)∈ C(t)−A₁(t)x⁽ⁿ⁻¹⁾(t)− · · · −An(t)x(t), for a.a. t ∈[0,m],

(x, ˙x, . . . ,x⁽ⁿ⁻¹⁾)∈S⁰_m,

(3.9) where A_i : [0,m] ( R^k×k are integrable multivalued matrix mappings such that |A_i(t)| ≤ a_i(t)_{, for a.a.} t ∈[_0,m]and suitable nonnegative functionsa_i ∈L¹([_0,m]_,R)_{, for all}i=_{1, . . . ,}n.

Furthermore, let us study the structure of a solution set, on a compact interval, to a semi- linear problem.

Hence, letm∈_Nand consider the b.v.p.

x⁽ⁿ⁾(t) +A₁(t)x⁽ⁿ⁻¹⁾(t) +· · ·+An(t)x(t)∈C(t,x(t), . . . ,x⁽ⁿ⁻¹⁾(t)), for a.a. t∈ [0,m],

l(x, ˙x, . . . ,x⁽ⁿ⁻¹⁾) =0,





 (Pm)

where

(i) A_i ∈L¹([0,m],R^k×k)are such that|A_i(t)| ≤a_i(t), for allt ∈[0,m]and suitable integrable functionsa_i :[0,m]→[0,∞), for alli=1, . . . ,n,

(ii) l:Cⁿ⁻¹([0,m],R^k)× · · · ×C([0,m],R^k)→_R^kn is a linear bounded operator, (iii) the associated homogeneous problem

x⁽ⁿ⁾(t) +A₁(t)x⁽ⁿ⁻¹⁾(t) +· · ·+An(t)x(t) =0, for a.a. t ∈[0,m], l(x, ˙x, . . . ,x⁽ⁿ⁻¹⁾) =₀

(H_m) has only the trivial solution,

(iv) C:[0,m]×_R^kn (R^k is an upper-Carathéodory mapping, (v) there exists an integrable functionα :[0,m] →[0,∞), with Rm

0 α(t)dtsufficiently small, such that

dH(C(t,x₁,x2, . . . ,xn),C(t,y₁,y2, . . . ,yn)≤α(t)·(|x₁−y₁|+· · ·+|xn−yn|), for a.a. t∈[0,m]and all x₁, . . . ,x_n,y₁, . . . ,y_n∈_R^k,

(vi) there exist a point(x₁, . . . ,x_n)∈_R^kn and a constant C₀ ≥0 such that

|C(t,x₁, . . . ,x_n)| ≤C₀·α(t) holds, for a.a.t∈[0,m]

(vi)

=⇒ |C(t,x₁, . . . ,x_n)|:= sup{|z| | z ∈ C(t,x₁, . . . ,x_n)} ≤ α(t) C₀+|x₁|+· · ·+|x_n|+

|x₁|+· · ·+|x_n| holds, for a.a. t∈ [0,m]and allx_i ∈_R^k, i=1, . . . ,n .

Theorem 3.3. Under the above assumptions (i)–(vi), the set of solutions of the b.v.p. (P_m) is a nonempty, compact AR-space.

Proof. Problem(Pm)is equivalent to the first-order problem

ξ˙(t) +D(t)ξ(t)∈K(t,ξ(t)), for a.a. t∈[0,m], l(ξ) =0,

(P^˜_m) where

ξ(t)_kn×1= (x(t), ˙x(t), . . . ,x⁽ⁿ⁻¹⁾(t))^T, D(t)_kn×kn =

0_(kn−k)×k −I_{(kn−k)×(kn−k)}

An(t) A_n−1(t) . . . A₁(t)

and

K(t,ξ)_kn×1= (_{0(kn−k)×1,}C(t,x, ˙x, . . . ,x⁽ⁿ⁻¹⁾(t)))^T_.

Similarly, the associated homogeneous problem (H_m)is equivalent to the first-order problem ξ˙(t) +D(t)ξ(t) =0, for a.a. t∈ [0,m],

l(ξ) =0.

(H^˜m)

The Fredholm alternative implies (see, e.g., [30]) that there exists the Green function ˜Gfor the homogeneous problem(H^˜_m)such that each solution ξ(·)of (P^˜_m)can be expressed by the formula ξ(t) = Rm

0 G˜(t,s)k(s) ds, where k(·) is a suitable measurable selection of K(·,ξ(·)) (cf. Lemma2.1). If we denote by ˜Gthe block matrix

G˜_kn×kn =







G˜¹¹_k×k G˜¹²_k×k . . . G˜_k¹ⁿ_×k ... ... . .. ... G˜ⁿ¹_k×k G˜ⁿ²_k×k . . . G˜_kⁿⁿ_×k





, (3.10)

then each solutionx(·)of(Pm)and its derivatives can be expressed as x(t) =

Z _m

0

G˜¹ⁿ(t,s)c(s)ds,

˙ x(t) =

Z _m

0

G˜²ⁿ(t,s)c(s)ds, ...

x⁽ⁿ⁻¹⁾(t) =

Z _m

0

G˜ⁿⁿ(t,s)c(s)ds,

wherec(·)is a suitable measurable selection ofC(·,x(·), ˙x(·), . . . ,x⁽ⁿ⁻¹⁾(·)). Moreover, in view of (v)and(vi),

|x(t)|+· · ·+|x⁽ⁿ⁻¹⁾(t))| ≤

Z _m

0 Gα(s)^hC0+|x₁|+· · ·+|xn|+|x(s)|+· · ·+|x⁽ⁿ⁻¹⁾(s)|ⁱ ds, for a.a. t∈[0,m], where G:=sup_{(t,s)∈[0,m]×[0,m]}

G˜¹ⁿ(t,s)+G˜²ⁿ(t,s)+· · ·+G˜ⁿⁿ(t,s) . Therefore,

tmax∈[0,m]{|x(t)|+|x˙(t)|+· · ·+|x⁽ⁿ⁻¹⁾(t))|} ≤ ^G·(C₀+|x₁|+. . .+|x_n|)·Rm

0 α(s)ds 1−GRm

0 α(s)ds =: M,

provided

Z _m

0 α(s)ds< ¹

G. (3.11)

Therefore, ifRm

0 α(s)dsis small enough, namely if the inequality (3.11) holds, then the set of solutions of (P_m)is equal to the set of solutions of the problem

x⁽ⁿ⁾(t) +A₁(t)x⁽ⁿ⁻¹⁾(t) +· · ·+An(t)x(t)∈C^∗(t,x(t), . . .x⁽ⁿ⁻¹⁾(t)), for a.a. t ∈[0,m],

l(x, ˙x, . . .x⁽ⁿ⁻¹⁾) =0,





 (R_m)

whereC^∗ satisfies conditions(iv)–(v)in Theorem3.3withCreplaced byC^∗, but this time C^∗(t,x₁, . . . ,x_n):=

(C(t,x₁, . . . ,x_n)_{, for}|x_i| ≤ M, i=_{1, . . . ,}n, C(t,M₁, . . . ,M_n), otherwise,

where M₁, . . . ,M_n are suitable vectors such that |M₁|= · · · = |M_n|= M. It follows immedi- ately from its definition thatC^∗ satisfies

|C^∗(t,x₁, . . .x_n)|:= sup{|z| |z ∈C^∗(t,x₁, . . . ,x_n)}

= sup{|z| |z ∈C(t,x₁, . . . ,x_n), where|x_i| ≤ M, i=1, . . . ,n}

≤α(t)(C₀^∗+|x^∗₁|+· · ·+|x^∗_n|+nM) =:β(t), (3.12) where(x^∗₁, . . . ,x^∗_n)∈_R^kn is such that|C^∗(t,x^∗₁, . . . ,x^∗_n)| ≤C₀^∗α(t), for a.a. t∈[0,m].

Let us denote by G(·,··) := G^˜12(·,··)the Green function associated to the homogeneous problem(H_m)and define the Nemytskiˇi operator

N:Cⁿ⁻¹([0,m],R^k)(^Cn−1([0,m],R^k) by the formula

Nx :=ⁿh∈C¹([0,m],R^k)|h(·) =

Z _m

0 G(·,s)f(s)ds, where f ∈L¹([0,m],R^k), f(t)∈ C^∗(t,x(t), ˙x(t), . . . ,x⁽ⁿ⁻¹⁾(t)), for a.a. t∈[0,m]^o. Let us note that Nx 6= _{∅, for all} x ∈ Cⁿ⁻¹([0,m],R^k), because, for all x ∈ Cⁿ⁻¹([0,m],R^k), C^∗(t,x(t), ˙x(t), . . . ,x⁽ⁿ⁻¹⁾(t)) possesses a measurable selection (again, according to Lemma 2.1).

It is evident that the set of solutions of problem (R_m) is equal to the set of fixed points of the operator N. In order to show that Fix(N)is, by means of Proposition2.2, a nonempty, compact AR-space, we will proceed in three steps.

(a) At first, let us show that the operator N has convex values. If h₁,h₂ ∈ Nx, then there exist integrable selections f₁(·),f₂(·) of C^∗(·,x(·), ˙x(·), . . . ,x⁽ⁿ⁻¹⁾(·)) such that, for a.a.

t∈[0,m],

h₁(t) =

Z _m

0

G(t,s)f₁(s)ds and

h2(t) =

Z _m

0 G(t,s)f2(s)ds.

Letλ∈[_{0, 1}]be arbitrary. Then, for a.a.t∈ [_0,m]_, λh₁(t) + (₁−λ)h₂(t) =

Z _m

0

G(t,s) [λf₁(s) + (₁−λ)f₂(s)] ds.

Since the mappingC^∗ has convex values,

λf₁(s) + (1−λ)f2(s)∈C^∗(s,x(s), ˙x(s), . . . ,x⁽ⁿ⁻¹⁾(s)),

for a.a.s ∈[0,m]. Therefore,λh₁+ (1−λ)h₂ ∈ Nx, i.e. the operatorNhas convex values, as claimed.

(b) Secondly, let us show that the operatorN has compact values. Let x ∈ Cⁿ⁻¹([_0,m]_,R^k) be arbitrary and letvbe an arbitrary integrable function such that

v(t)∈C^∗(t,x(t)_{, ˙}x(t)_{, . . . ,}x⁽ⁿ⁻¹⁾(t))_,

for a.a.t ∈[0,m].

Let us consider the elementhof Nx defined, for a.a. t∈ [0,m], by h(_t)_:=

Z _m

0 G(_t,s)_v(_s)_ds.

Ift,τ∈[0,m]are arbitrary, then

|h(t)−h(τ)|=

Z _m

0 G(t,s)v(s)ds−

Z _m

0 G(τ,s)v(s)ds

≤

Z _m

0

|G(t,s)−G(τ,s)| · |v(s)|ds≤

Z _m

0

|G(t,s)−G(τ,s)| ·β(s)ds. (3.13) Since β(·)is, by the definition, an integrable function, estimate (3.13) implies the equi- continuity of h. Moreover, it immediately follows from condition (3.12) and the proper- ties of the Green function that h is also bounded. Therefore, the well known Arzelà–

Ascoli lemma implies that the set Nxis relatively compact.

The relative compactness of values follows also alternatively from the contractivity of N which will be proved in the next step(3). It is namely well known that contractivity implies condensity.

The closedness of values follows from the fact that, according to [31],Ncan be expressed as the closed graph composition of operators φ ◦ S_C^∗, where S_C^∗ : Cⁿ⁻¹([0,m],R^k) ( L¹([0,m],R^k)andφ:L¹([0,m],R^k)→Cⁿ⁻¹([0,m],R^k)are defined by

S_C^∗(x):= ⁿf ∈ L¹([0,m],R^k)| f(t)∈C^∗(t,x(t), . . . ,x⁽ⁿ⁻¹⁾(t)), for a.a.t∈[0,m]^o and

φ(f):=

h∈ Cⁿ⁻¹([0,m],R^k)|h(t) =

Z _m

0 G(t,s)f(s)ds, for a.a.t∈[0,m]

. (c) In order to show that the operator Nis a contraction, let us consider the Banach space

Cⁿ⁻¹([_0,m]_,R^k)endowed with the norm

|x|_Cn−1 := sup

t∈[0,m]

n|x(t)|+|x˙(t)|+. . .+|x⁽ⁿ⁻¹⁾(t)|^o,

where | · | stands for the Euclidean norm in R^k. If x,y ∈ Cⁿ⁻¹([0,m],R^k) are arbitrary, then there exist h_x ∈ Nx, h_y ∈ Ny and integrable selections (cf. Lemma 2.1) f_x(·) of C^∗(·,x(·), ˙x(·), . . . ,x⁽ⁿ⁻¹⁾(·))and fy(·)of C^∗(·,y(·), ˙y(·), . . . ,y⁽ⁿ⁻¹⁾(·))such that

d_H(Nx,Ny) =|h_x−h_y|_Cn−1 =

Z _m

0 G(t,s)f_x(s)ds−

Z _m

0 G(t,s)f_y(s)ds Cⁿ⁻¹

= sup

t∈[0,m]

Z _m

0 G(t,s)[f_x(s)− f_y(s)]ds

+· · ·+

Z _m

0

∂ⁿ⁻¹

∂tⁿ⁻¹G(t,s)[f_x(s)− f_y(s)]ds

≤ sup

t∈[0,m] Z _m

0

|G(t,s)|+

∂

∂tG(t,s)

+· · ·+

∂ⁿ⁻¹

∂tⁿ⁻¹G(t,s)

·fx(s)− fy(s) ds

≤ sup

t∈[0,m]

{|x(t)−y(t)|+· · ·+|xⁿ⁻¹(t)−yⁿ⁻¹(t)|}

× sup

(t,s)∈[0,m]×[0,m]

|G(t,s)|+· · ·+

∂ⁿ⁻¹

∂tⁿ⁻¹G(t,s)

·

Z _m

0 α(t)dt

= sup

(t,s)∈[0,m]×[0,m]

|G(t,s)|+

∂

∂tG(t,s)

+· · ·+

∂ⁿ⁻¹

∂tⁿ⁻¹G(t,s)

×

Z _m

0 α(t)dt· |x−y|_Cn−1. If the integralRm

0 α(t)dtis small enough, namely if L:= sup

(t,s)∈[0,m]×[0,m]

|G(t,s)|+· · ·+

∂ⁿ⁻¹

∂tⁿ⁻¹G(t,s)

·

Z _m

0 α(t)dt<1, (3.14) then the operatorN is a desired contraction with a Lipschitz constantL∈[0, 1).

Finally, sinceNis a contraction with compact and convex values, the set Fix(N)is, accord- ing to Proposition2.2, a nonempty, compact AR-space which completes the proof.

Remark 3.4. It follows from the proof of Theorem 3.3 that the smallness of the integral Rm

0 α(t) dt in conditions (v) and (vi) is given by the identical inequalities (3.11) and (3.14), namely

Z _m

0 α(t)dt< ¹

sup

(t,s)∈[0,m]×[0,m]

n|G(t,s)|+

∂

∂tG(t,s)

+· · ·+

∂ⁿ⁻¹

∂tⁿ⁻¹G(t,s)

o. (3.15)

Remark 3.5. If the mapping C(t,·)is Lipschitzian with a sufficiently small constant L, i.e. if condition(v)takes the form

(v⁰) there exists a sufficiently small constantL≥0 such that

d_H(C(t,x₁, . . . ,x_n−1),C(t,y₁, . . . ,y_n−1))≤ L·(|x₁−y₁|+· · ·+|x_n−1−y_n−1|), for a.a. t ∈ [0,m]and all x_i,y_i ∈ _R^k, i = 1, . . . ,n−1, then the same conclusion holds, provided

L< ¹

sup

t∈[0,m]

Rm

0 |G(t,s)|+

∂

∂tG(t,s)

+· · ·+

∂ⁿ⁻¹

∂tⁿ⁻¹G(t,s) ds

. (3.16)

Remark 3.6. Theorem3.3reduces, forn =1, to [11, Theorem 4] and, forn =2, to [6, Lemma 3.2]. Moreover, unlike in [11], the smallness ofRm

0 α(t)dt, resp. L, is expressed here explicitly in (3.15), (3.16) (see also (3.20), (3.21), (3.23) and (3.25) below).

Remark 3.7. For scalar (k = 1) problem(P_m), the topological structure of the set of solutions was studied in [38]. Our related conditions(i)_–(vi)are, in this particular case, more explicit and our conclusion is more precise, because in [38] anR_δ-set was obtained.

Example 3.8. Consider then-point vector interpolation b.v.p. inR^k:

x⁽ⁿ⁾(t) +A₁(t)x⁽ⁿ⁻¹⁾(t) +· · ·+An(t)x(t)∈ C(t,x(t), . . . ,x⁽ⁿ⁻¹⁾(t)), for a.a.t∈ [0,m],

x(t₁) =· · ·=x(t_n) =0,







(3.17)