Electronic Journal of Qualitative Theory of Differential Equations

A general continuation principle for the n-th order vector asymptotic boundary value problems with multivalued right-hand sides is newly developed. This continuation principle is then applied to guarantee the existence and localization of solutions to the given asymptotic problems. The obtained results are finally supplied by two illustrative examples.


Introduction
Asymptotic boundary value problems (b.v.p.) for higher-order differential equations and inclusions are important for many applications. For instance, they occur in the problems dealing with radially symmetric solutions of elliptic equations, semiconductor circuits and soil mechanics, fluid dynamics or in the boundary layer theory (see, e.g., [1,2,15], and the references therein). Furthermore, it is well known (see, e.g., [5]) that the n-th order asymptotic control problems x (n) (t) = f t, x(t), . . . , where S is a suitable constraint (e.g. asymptotic boundary conditions) and u ∈ U are control parameters such that u(t) ∈ R k , for all t ≥ t 0 , can be converted into the equivalent multivalued problems x (n) (t) ∈ F t, x(t), . . . , x (n−1) (t) , where the multivalued mapping F, representing the right-hand side (r.h.s.), is defined by F(t, x, . . . , x (n−1) ) := { f (t, x, . . . , x (n−1) , u)} u∈U .
Although boundary value problems for higher-order (mainly, the second-order) vector systems have been already intensively studied since the 70's (see e.g. [19,21,22,30,33]), there are only several papers devoted to noncompact (possibly infinite) intervals (see e.g. [4, 7, 10, 11, 14, 15, 18, 23, 24, 26-29, 31, 32], and the references therein). In these papers, various fixed point theorems, topological degree theory, shooting methods, upper and lower solution technique, etc., have been applied for the solvability of given problems. In the majority of mentioned papers, the second-order problems were considered and/or the right-hand sides of systems under consideration were single-valued, often even continuous. The aim of this paper is to investigate the n-th order problem (1.1) on non-compact intervals with the right-hand sides governed by upper-Carathéodory multivalued mappings. Besides the existence of solutions, also their localization in a given set will be studied. Following the ideas in [3][4][5][16][17][18], our approach is not sequential as traditionally, but direct. This means to consider the solutions as fixed points of the associated operators in Fréchet spaces. In this way, however, the bound sets technique like e.g. in [6] cannot be applied jointly with the degree arguments, because bounded subsets of nonnormable Fréchet spaces are, according to the Kolmogorov theorem, equal to their boundaries. On the other hand, a bigger variety of asymptotic boundary value problems can be so taken into account.
The paper is organized as follows. Firstly, the basic properties of multivalued mappings which are employed in the sequel are recalled. On this basis, we formulate the general continuation principle for the n-th order asymptotic boundary value problems with multivalued right-hand sides in a rather abstract way. Then this principle is applied in order to obtain the existence and localization of solutions. Finally, two illustrative examples are supplied.

Preliminaries
We start this section with some standard definitions and notations. At first, we recall some geometric notions of particular subsets of metric spaces and the notions of retracts. If (X, d) is an arbitrary metric space and A ⊂ X its subset, we shall mean by Int A, A and ∂A the interior, the closure and the boundary of A, respectively. For a subset A ⊂ X and ε > 0, we define Let X, Y be two metric spaces. We say that X is an absolute retract (AR-space) if, for each Y and every closed A ⊂ Y, each continuous mapping f : A → X is extendable over Y. If f is extendable over some neighborhood of A, for each closed A ⊂ Y and each continuous mapping f : A → X, then X is an absolute neighborhood retract (ANR-space). Let us note that X is an ANR-space if and only if it is a retract of an open subset of a normed space and that X is an AR-space if and only if it is a retract of some normed space.
We say that a nonempty subset A ⊂ X is contractible, provided there exist x 0 ∈ A and a homotopy h : A × [0, 1] → A such that h(x, 0) = x and h(x, 1) = x 0 , for every x ∈ A. A nonempty subset A ⊂ X is called an R δ -set if there exists a decreasing sequence {A n } ∞ n=1 of compact AR-spaces such that A = ∩{A n ; n = 1, 2, . . .}.
Note that any R δ -set is nonempty, compact and connected.
The following hierarchies hold for nonempty subsets of a metric space: and all the above inclusions are proper. A nonempty, compact subset A of a metric space X is called ∞-proximally connected if, for every ε > 0, there exists δ = δ(ε) > 0 such that, for every n ∈ N and for any map g : ∂ n → N δ (A), there exits a mapg : n → N ε (A) such that g(x) =g(x), for every x ∈ ∂ n , where ∂ n := {x ∈ R n+1 | |x| = 1} and n := {x ∈ R n+1 | |x| ≤ 1}. On ANR-spaces, the notions of ∞-proximally connected sets and R δ -sets coincide. For more details about the above subsets of metric spaces, see, e.g., [5,12,20].
Our problems under consideration naturally lead to the notion of a Fréchet space. Let us recall that by a Fréchet space, we understand a complete (metrizable) locally convex vector space. Its topology can be generated by a countable family of seminorms or by a metric (see e.g. [5,Chapter I.1]). If a Fréchet space is normable, then it becomes a Banach space. Fréchet spaces considered below will be as follows: • the space C(J, R k ) of continuous functions x : J → R k with the family of seminorms • the space C n−1 (J, R k ) of functions x : J → R k having continuous (n − 1)-st derivatives endowed with the system of seminorms p n−1 i (q) : C n−1 (J, R k ) → R defined by where {K i } is a sequence of compact subintervals of J satisfying (2.1) and (2.2), • the space AC n−1 loc (J, R n ) of functions x : J → R n with locally absolutely continuous (n − 1)-st derivatives endowed with the family of seminorms p n−1 where {K i } is a sequence of compact subintervals of J satisfying (2.1) and (2.2).
The topologies in Fréchet spaces mentioned above can be generated by the metrics , respectively.
Let J ⊂ R be compact. By H n,1 (J, R k ), we will denote the Banach space of all C n−1 functions x : J → R k with absolutely continuous (n − 1)-st derivative.
In the sequel, we also need the following definitions and notions from the multivalued theory.
Let X, Y be two metric spaces. We say that F is a multivalued mapping from X to Y (written F : X Y) if, for every x ∈ X, a nonempty subset F(x) of Y is given. We associate with F its graph Γ F , i.e. the subset of X × Y defined by The connections between upper semicontinuous mappings and mappings with closed graphs are summarized in the following propositions (see, e.g., [5,20]). Proposition 2.1. Let X,Y be metric spaces and F : X → Y be a multivalued mapping with the closed graph Γ F such that F(X) ⊂ K, where K is a compact set. Then F is u.s.c. Proposition 2.2. Let X,Y be metric spaces and F : X → Y be an u.s.c. multivalued mapping with closed values, then Γ F is a closed subset of X × Y.
A multivalued mapping F : X Y is called compact if the set F(X) = x∈X F(x) is contained in a compact subset of Y and it is called closed if the set F(B) is closed in Y, for every closed subset B of X.
We say that a multivalued mapping F : X Y is an R δ -mapping if it is a u.s.c. mapping with R δ -values.
We say that a multivalued map ϕ : X Y is a J-mapping (written, ϕ ∈ J(X, Y)) if it is a u.s.c. mapping and ϕ(x) is ∞-proximally connected, for every x ∈ X. If the space Y is a neighbourhood retract of a Fréchet space (i.e. an ANR-space), then ϕ ∈ J(X, Y), provided ϕ is an R δ -mapping, as already pointed out (cf. [5,20]).
Let Y be a separable metric space and (Ω, U , ν) be a measurable space, i.e. a nonempty set Ω equipped with a σ-algebra U of its subsets and a countably additive measure ν on U .
We say that mapping F : J × R m R n , where J ⊂ R, is an upper-Carathéodory mapping if the map F(·, x) : J R n is measurable on every compact subinterval of J, for all x ∈ R m , the map F(t, ·) : R m R n is u.s.c., for almost all (a.a.) t ∈ J, and the set F(t, x) is compact and convex, for all (t, x) ∈ J × R m .
We recall now some results which are employed in the sequel. Proposition 2.3 (cf., e.g., [7]). Let F : [a, b] × R m R n be 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 : 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.
. Let J ⊂ R be a compact interval. Assume that the sequence of absolutely continuous functions x n : J → R k satisfies the following conditions: (i) the set {x n (t) | n ∈ N} is bounded, for every t ∈ J, (ii) there exists a function α : J → R, integrable in the sense of Lebesque, such that |ẋ n (t)| ≤ α(t), for a.a. t ∈ J and for all n ∈ N.
Then there exists a subsequence of {x n } (for the sake of simplicity denoted as the sequence) convergent to an absolutely continuous function x : J → R k in the following sense: E 2 be a multivalued mapping satisfying the following conditions: (i) F(·, x) has a strongly measurable selection, for every x ∈ E 1 , i.e. there exists a sequence of step multivalued maps F n (·, Assume in addition that, for every nonempty, bounded set Let us define the Nemytskii operator ], E 2 ) in the following way: For more details concerning multivalued theory see e.g. [8,9,20,25]. In order to develop the continuation principle for the n-th order asymptotic problems, the following important arguments will be also needed (cf. [4,5,[16][17][18]). Let us assume that X is a retract of a Fréchet space E (by which X is an AR-space; cf. [12]) and D is an open subset of X (by which D is an ANR-space; cf. [12]). Let G ∈ J(D, E) be locally compact, let Fix(G) be compact and let the following condition hold: 3) The class of locally compact J-mappings from D to E with a compact fixed point set and satisfying (2.3) will be denoted by J A (D, E). We say that G 1 , 3. for every x ∈ D and every t ∈ [0, 1], the following condition holds: Remark 2.6. Note that condition (2.4) is equivalent to the following one: Remark 2.7 (see e.g. [3]). If E = X is a Banach space, then condition (2.4) can be replaced by The following proposition, which will be applied below for obtaining the existence of a solution of the studied b.v.p., follows immediately from the results in [4,5].

Proposition 2.8. Let X be a retract of a Fréchet space E, D be an open subset of X and H be a homotopy in J
Then there exists x ∈ D such that x ∈ H(x, 1).
As a direct consequence of Proposition 2.8, it is possible to obtain the following result. Corollary 2.9. Let X be a retract of a Fréchet space E, H be a homotopy in J A (X, E) such that H(x, 0) ⊂ X, for every x ∈ X, and let H(·, 0) be compact. Then H(·, 1) has a fixed point.

Continuation principle
In this section, we consider the n-th order boundary value problem in the following form where J is a given real (possibly noncompact) interval, F : J × R kn R k is a multivalued upper-Carathéodory mapping and S ⊂ AC n−1 loc (J, R k ). By a solution of problem (3.1), we mean a function x : J → R k belonging to AC n−1 loc (J, R k ) and satisfying (3.1), for almost all t ∈ J.
For our main result, the following proposition is pivotal.
where l : C n−1 (J, R k ) × C n−2 (J, R k ) . . . C(J, R k ) → R kn is a linear bounded operator. Assume that (i) there exists a subset Q of C n−1 (J, R k ) such that, for any q ∈ Q, the set T(q) of all solutions of the boundary value problem q(t), . . . , q (n−1) (t)), for a.a. t ∈ J, x ∈ S is nonempty, (ii) there exist t * ∈ J and a constant M > 0 such that for all x ∈ T(Q), (iii) there exists a nonnegative, locally integrable function α : a.e. in J, for any (q, x) ∈ Γ T .
Then T(Q) is a relatively compact subset of C n−1 (J, R k ). Moreover, the multivalued operator T : Q S is u.s.c. with compact values if the following condition is satisfied: . Since . .
Condition (iv) implies that x ∈ S, and therefore Γ T is closed. Moreover, it follows immediately from Proposition 2.1 that the operator T is u.s.c.
Since T is a compact mapping, T(q) is, for each q ∈ Q, a relatively compact set. Moreover, the operator T has a closed graph which implies that T(q) is, for each q ∈ Q, closed, and therefore T has compact values.
Then problem (3.1) has a solution.
Proof. At first, we show that all the assumptions of Proposition 3.1 are satisfied. Conditions (i), (ii) and (iv) in Theorem 3.1 guarantee conditions (i), (ii) and (iii) in Proposition 3.1. 1] be arbitrary. Then, since x m ∈ S 1 , x m → x and S 1 is closed, it holds that x ∈ S 1 . Therefore, condition (iv) from Proposition 3.1 holds as well. Thus, T : Q × [0, 1] S 1 is, according to Proposition 3.1, a compact u.s.c. mapping with compact values. According to assumption (i), T has R δ -values, and so it belongs to the class J(Q × [0, 1], C n−1 (I, R k )). Assumption (v) implies that T is a homotopy in J A (Q, C n−1 (I, R k )). From Corollary 2.9, it follows that there exists a fixed point of T(·, 1) in Q. Moreover, by the inclusion (3.6) and since S 1 ⊂ S, the fixed point of T(·, 1) is a solution of the original b.v.p. (3.1).

Existence and localization results
Let us consider the b.v.p.
for a.a. t ∈ J, a suitable locally integrable function a : J → [0, ∞), and all q ∈ Q, where Q ⊂ C n−1 (J, R k ), (iii) F : J × R kn R k is an upper-Carathéodory mapping, (iv) S is a subset of AC n−1 loc (J, R k ). If the problems associated to (4.1) are fully linearized, we obtain the following result when applying the continuation principle from the previous section.
Then the b.v.p. (4.1) has a solution in S 1 ∩ Q.
Proof. Since the associated problems are, for all (q, λ) ∈ Q × [0, 1], fully linearized, the mapping F has convex values and S 1 is convex, the set T(q, λ) is also convex, for all (q, λ) ∈ Q × [0, 1]. Therefore, all assumptions of Theorem 3.2 are satisfied, and so the problem (4.1) has a solution in S 1 ∩ Q.
Making use of the result in [13], dealing with the equivalency of norms in the Banach space H n,1 (J, R k ), we are able to improve condition (ii) from Theorem 4.1 as follows. Let us show that, for any compact interval I ⊂ J, there exists a constant M > 0 such that According to Lemma 2.36 in [13] and the remarks below that lemma, the following two norms in H n,1 (I, R k ) It is obvious that p I (x) ≤ x and, by the above mentioned equivalency of norms, there exists a constant c > 0 such that Therefore, T(Q × [0, 1]) is also bounded in C n−1 (J, R k ) which, in particular, ensures the validity of condition (ii) from Theorem 4.1.

Remark 4.3.
Let us note that Corollary 4.2 cannot be deduced by a simple transformation of a studied problem to the first-order problem. The obtained result is a vector generalization of Corollary 2.37 in [4] and it also generalizes the vector result for the second-order b.v.p. in [7], where A i did not depend on x. Moreover, a more restrictive condition (ii) was used there.
Observe that condition (v) in Theorem 4.1 hold when S 1 ⊂ Q, by which Theorem 4.1 can be simplified in the following way, suitable for practical applications.  (4.1), where J is a given real interval, F : J × R kn R k is an upper-Carathéodory mapping and S is a subset of AC n−1 loc (J, R k ). Assume that (i) there exists a retract Q of C n−1 (J, R k ) such that S ∩ Q is closed and convex and that the associated problem (ii) there exists a nonnegative, locally integrable function α : J → R such that F t, q(t),q(t), . . . , q (n−1) (t) ≤ α(t), a.e. in J, for any q ∈ Q, (iii) there exist t * ∈ J and a constant M > 0 such that for any x ∈ T(Q) (or T(Q) is bounded in C(J, R k )).
Then the b.v.p. (4.1) has a solution in S ∩ Q.

Remark 4.5.
Let us note that Corollary 4.4 generalizes the results in [4] and [7] as well as Proposition 2.1 in [18] which was (unlike our result) obtained only as a vector modification of the scalar result in [4].

Illustrative examples
Let us finally illustrate the application of Corollary 4.4 by two examples. The first one concerns the n-th order vector target (terminal) problem.
Example 5.1. Let us consider the n-th order target problem where, for all i = 1, . . . , k, F i : [0, ∞) × R k R are upper-Carathéodory mappings and l i ∈ R. Moreover, let there exist K > 0 such that, for all i = 1, . . . , k, Then it is possible to apply Corollary 4.4 and obtain that the target problem (5.1) has a solution x = (x 1 , . . . , x k ) satisfying |x i (t)| ≤ K, for all i = 1, . . . , k and all t ∈ [0, ∞). More concretely, let us define the set Q of candidate solutions as Q := (q 1 , . . . , q k ) ∈ C n−1 ([0, ∞), R k ) |q i (t)| ≤ K, for all t ∈ [0, ∞) and all i = 1, . . . , k , and let us consider the family of fully linearized associated problems At first, let us verify condition (i) from Corollary 4.4. If q = (q 1 , . . . , q k ) ∈ Q is arbitrary, then F i (t, q(t)) admits, for all i = 1, . . . , k, according to Proposition 2.3, a single-valued selection f q,i (t), measurable on every compact subinterval of [0, ∞). The corresponding problem has a solution x = (x 1 , . . . , x k ) such that This solution belongs to Q, according to (5.2), and so the assumption (i) from Corollary 4.4 is satisfied.
As the second illustrative example, let us study the n-th order multivalued Sturm-Liouville b.v.p.
The validity of assumption (ii) from Corollary 4.4 follows immediately from the properties of mapping F and the definition of the set Q. Moreover, all solutions of (5.8) belong, for arbitrary q ∈ Q, to the closed, bounded subset of X, namely x ∈ X x i ≤ L i + K i ∞ 0 α(t) dt, i = 0, 1, . . . , n − 1 , which implies that T(Q) is bounded in X. Therefore, assumption (iii) from Corollary 4.4 is satisfied as well.
Summing up, all assumptions of Corollary 4.4 are satisfied, by which the Sturm-Liouville problem (5.5) admits a solution in Q.