HARTMAN-TYPE CONDITIONS FOR MULTIVALUED DIRICHLET PROBLEM IN ABSTRACT SPACES

The classical Hartman’s Theorem in [18] for the solvability of the vector Dirichlet problem will be generalized and extended in several directions. We will consider its multivalued versions for Marchaud and upper-Carathéodory right-hand sides with only certain amount of compactness in Banach spaces. Advanced topological methods are combined with a bound sets technique. Besides the existence, the localization of solutions can be obtained in this way.

On the other hand, condition (2) can be assumed on the boundary of a convex subset of R n or even on the boundary of its homeomorphic image or retract (cf.also [1]), which we regard rather as a generalization than an improvement of Theorem 1.1.
In the present paper, we will show that such extensions and generalizations are also possible for multivalued problems in abstract spaces.We will consider separately Dirichlet problems with not necessarily completely continuous Marchaud (i.e.globally upper semicontinuous) and upper-Carathéodory r.h.s. in Banach and, in particular, Hilbert spaces.
We would like to improve especially the statements in Remarks 1.3 and 1.4 and generalize them into the multivalued setting in abstract spaces.
The paper is organized as follows.After this introduction, we recall in the next section some elements of functional and multivalued analyses, jointly with the definition of a measure of non-compactness and condensing maps.Since our main theorems rely essentially on our earlier results in [3]- [6], [8], we formulate them in the form of propositions.The Nagumotype auxiliary results are given separately in Section 3. The main results are formulated in the form of three theorems in Section 4. Section 5 contains an application to a control problem.Finally, in Section 6, the special finite-dimensional single-valued case of obtained results is compared in discussion with their classical analogous versions of the other authors.
Thus, the differentiability (derivatives) will be entirely understood in a strong sense of Fréchet.Similarly, solutions of differential inclusions will be considered in a (strong) Carathéodory sense, i.e. belonging to the AC 1 -class.Of course, in the single-valued case, solutions of ordinary differential equations with continuous r.h.s. will automatically become C 2 -solutions, like in Theorem 1.1.
We will denote by E the Banach space dual to E and by •, • the pairing (the duality relation) between E and E , i.e., for all Φ ∈ E and x ∈ E, we put Φ(x) := Φ, x .
We shall also need the following definitions and notions from the multivalued analysis.Let X, Y be two metric spaces.We say that F is a multivalued mapping from X to Y (written We say that a multivalued mapping F : [0, T ] Y with closed values is a step multivalued mapping if there exists a finite family of disjoint measurable subsets Y with closed values is called strongly measurable if there exists a sequence of step multivalued mappings {F n } such that d H (F n (t), F (t)) → 0 as n → ∞, for a.a.t ∈ [0, T ], where d H stands for the Hausdorff distance.
A multivalued mapping Y is strongly measurable, for all x ∈ X, the map F (t, •) : X Y is u.s.c., for almost all t ∈ [0, T ], and the set F (t, x) is compact and convex, for all (t, x) ∈ [0, T ] × X.
In the sequel, the measure of non-compactness will also be employed.
Let N be a partially ordered set, E be a Banach space and let P (E) denote the family of all nonempty subsets of E. A function β : P (E) → N is called a measure of non-compactness (m.n.c.) in E if β(co Ω) = β(Ω), for all Ω ∈ P (E), where co Ω denotes the closed convex hull of Ω.
The typical example of an m.n.c. is the Hausdorff measure of noncompactness γ defined, for all Ω ⊂ E by Let E be a Banach space and X ⊂ E. A multivalued mapping F : X E with compact values is called condensing with respect to an m.n.c.β (shortly, β-condensing) if, for every Ω ⊂ X such that β(F (Ω)) ≥ β(Ω), it holds that Ω is relatively compact.
We will also need the following slight modification of the result presented in [8].
is well defined.Moreover, it holds that Vx , ψ(x) = 1, for all x in its domain.
We will also apply the following continuation principle developed in [4].
(v) For each λ ∈ (0, 1), the solution mapping T(•, λ) has no fixed points on the boundary ∂Q of Q.Then the b.v.p. (11) has a solution in Q.
In [3] and [5], we proved in detail the following three results, stated here in the form of propositions, which deal with the existence and the localization of a solution of a multivalued Dirichlet problem.The first result concerns the case of globally upper semicontinuous r.h.s. and strictly localized (so called bounding) function V (cf.conditions ( 15) and ( 16) below).The second result concerns the case of upper-Carathéodory r.h.s. and non-strictly localized bounding function V (cf.conditions ( 20) and ( 21) below).The third result, obtained in [5] via the Scorza-Dragoni approximation technique, deals with the case of upper-Carathéodory r.h.s. and strictly localized bounding function V (cf.condition (23) below).
where E is a Banach space having the Radon-Nikodym property and F : [0, T ] × E × E E is an upper semicontinuous mapping with compact, convex values.Assume that K ⊂ E is an open, bounded, convex set containing 0. Moreover, let the following conditions be satisfied: for a.a.t ∈ [0, T ] and all (x, y) ∈ Ω × E.
Remark 2.2.The typical case occurs when E = H is a Hilbert space, •, • denotes the scalar product and for some R > 0. In this case, V ∈ C 2 (H, R) and it is not difficult to see that condition ( 16) becomes y, y + λ x, w > 0 (18) with t, x, y as in Proposition 2.3 (where K := {x ∈ H| ||x|| < R }) and w ∈ F (t, x, y).Moreover, since the condition ( 18) is satisfied, when holds, for t, x, y and w as above, we can assume, instead of (18), condition (19) which is more convenient for applications and especially more suitable for comparison with analogous results.
We will present the conditions under which the strict inequality in the condition ( 16) can be replaced by a non-strict one.In the last section, we will discuss the case when Finally, let there exist a function V ∈ C 1 (E, R) with a locally Lipschitzian Fréchet derivative V satisfying conditions (H1) and (H2) of Proposition 2.1 and at least one of the conditions lim sup for a suitable ε > 0, all x ∈ K ∩ B(∂K, ε), t ∈ (0, T ), y ∈ E, λ ∈ (0, 1) and w ∈ λF (t, x, y).
Remark 2.3.Let us note that also in this case the analogy of previous remark holds, i.e. if E = H is a Hilbert space, •, • denotes the scalar product and V is defined by (17), then conditions ( 20) and ( 21) take the form (18).They can be satisfied e.g. by the condition (19) which is more convenient for comparisons with other results.
We will also present the conditions under which the strict inequality in the condition (18) can be replaced by a non-strict one.E is an upper-Carathéodory mapping which is either globally measurable or quasi-compact.Let K ⊂ E be an open, bounded, convex set containing 0 and let the conditions (1 i ), (1 ii ) and the following condition (2 iii ) be satisfied: Furthermore, let there exist ε > 0 and a function V ∈ C 2 (E, R) with Fréchet derivative V Lipschitzian in B(∂K, ε) satisfying (H1), (H2) and (H3) of Proposition 2.1.Let there still exist h > 0 such that where Vx (v) denotes the second Fréchet derivative of V at x in the direction for a.a.t ∈ (0, T ) and all x ∈ ∂K, v ∈ E, and w ∈ F (t, x, v).
3. Nagumo-type auxiliary results.Following the ideas in [34], for x ∈ C 2 ([0, 1], E) and F single-valued and completely continuous, we can state the following lemma which this time concerns x ∈ AC 1 ([0, T ], E) and a multivalued upper-Carathéodory r.h.s.F .For the sake of completeness, we shall prove it in detail.
Proof.Let x ∈ AC 1 ([0, T ], E) be a solution of the inclusion (25) satisfying ] is arbitrary such that t 0 + a ∈ [0, T ], it holds that Using the per-partes formula and the integration in the sense of Lebesque, we then get that Therefore, because the function φ is nondecreasing and satisfies condition (29).Moreover, since We will now show that q ≤ M , where For this purpose, let us assume that q > Q(R).Then q 2 4R > φ(q), and so If 4R q ≥ 1 2 , then q ≤ 8R, and if 4R q < 1 2 , then we obtain the contradiction to (27), when setting |a| = 4R q , because in such a case Therefore, if q > Q(R), then q ≤ 8R, and so q = || ẋ|| ≤ M , where M is defined by (26).
It was shown in [19] that if (24) can be improved as follows.
be a continuous function satisfying condition (4), where φ : [0, ∞) → (0, ∞) is a continuous function satisfying condition (5).Moreover, let there exist α, β ≥ 0 such that, for a.a.t ∈ [0, T ] and all x ∈ R n with ||x|| ≤ R and y ∈ R n , condition (3) holds. If satisfying ||x(t)|| ≤ R, then there exists M > 0 (depending only on R, α, β and φ) such that Let us note that the standard usage of the Nagumo technique, i.e.Lemma 3.1 and Lemma 3.2 here, consists in its application to the inclusion ẍ ∈ F (t, x, r( ẋ)), where r : E → B M is a retraction to the ball B M := {y ∈ E : y ≤ M } such that r| B M = id| B M .Since x ≤ R implies ẋ ≤ M , the obtained results can be then adopted to the original inclusion ẍ ∈ F (t, x, ẋ).
4. Main results.In this section, we will show conditions under which the strict inequalities ( 16), ( 20)-( 21), (23) in Propositions 2.3-2.5 can be replaced by non-strict ones and we will also formulate the versions of the obtained results, when E is a Hilbert space.In the following, we always assume that F satisfies the Nagumo-type condition, i.e.
Let K ⊂ E be an open, bounded, convex set containing 0 and let there exist a function V ∈ C 1 (E, R) with a locally Lipschitzian Fréchet derivative V satisfying (H1), (H2) and (H3) of Proposition 2.1.Furthermore, let, for all x ∈ ∂K, t ∈ (0, T ), λ ∈ (0, 1) and y ∈ E such that Vx , y = 0, (31) the following condition holds, for all w ∈ λF (t, x, y).Finally, let (29) be valid with φ as in (24) and Then Since ψ is continuous, the mapping F n is u.s.c., for all n ∈ N. We will show that F n satisfies condition (1 i ) with g n = g + L ψ n .In such a case, for a sufficiently large n, we have that and so (1 iii ) holds.In order to prove condition (1 i ), for F n , let Ω 1 ⊂ K and Ω 2 ⊂ E be bounded.Firstly, notice that , for every bounded Ω ⊂ E (see e.g.[6]).Therefore, according to the properties of the Hausdorff m.n.c., Moreover, according to (1 i ) and the additivity of the Hausdorff m.n.c., we obtain that for all n ∈ N. Hence, F n satisfies (1 i ), for all n ∈ N, with g n as announced.From now on, we assume that condition (1 iii ) is satisfied, for all n ∈ N.
It will be shown in the following theorem that if F is an upper-Carathéodory multivalued mapping and the transversality condition is not strictly localized, i.e. not required exclusively for x ∈ ∂K, but assumed for all x ∈ K ∩ B(∂K, ε), it is possible to replace the strict inequalities in ( 20) and ( 21) by non-strict ones.
Proof.Consider the function ψ introduced in the proof of Theorem 4.1 and put ψ := jψ with j > 0. Let us note that ψ is continuous and bounded with sup x∈E ψ(x) = jm.
We will now apply the continuation principle of Proposition 2.2 in the case when ϕ(t, x, y) : = F (t, x, y), S := {x ∈ AC 1 ([0, T ], E) : x(0) = x(T ) = 0} and At first, let us observe that both ϕ and H are upper-Carathéodory mappings and that condition ( 12) is obviously satisfied.In what follows, we will check, step by step, that all the assumptions of Proposition 2.2 are satisfied.ad (i) Let S 1 = S and put Q = C1 ([0, T ], K).It is clear that S 1 is closed and, according to the properties of K, the set Q is closed, convex and has a nonempty interior.Moreover, given q ∈ Q, λ ∈ [0, 1] and f ∈ L 1 ([0, T ], E) with f (t) ∈ F (t, q(t), q(t)), for a.a.t ∈ [0, T ], consider the Dirichlet b.v.p.
This problem is uniquely solvable (see e.g. the proof of [3, Theorem 5.1]) which implies that the set T(q, λ) of its solutions is nonempty.Since F is convex-valued, T(q, λ) is also convex.ad (ii) Condition (ii) is a direct consequence of (2 ii ), when applied Lemma 3.1, and the boundedness of ψ.More concretely, if Ω ⊂ E × E is bounded, then there exists with G(t, s) as in the proof of [3, Theorem 5.1] and f n (t) ∈ F (t, q n (t), qn (t)), for a.a.
Therefore, a very similar reasoning as in the proof of [3,Theorem 5.1] can be used in order to show that the solution mapping T is quasicompact and µ-condensing, where µ is the monotone non-singular m.n.c.defined in [3, p. 308].ad (iv) For every q ∈ Q, T(q, 0) is the unique solution of the problem So, if j is small enough, then the ball jT 2 m 4 B ⊂ K, and so condition (iv) is also satisfied.ad (v) Assume that condition (38) is satisfied. 1If x is a fixed point of T(•, λ), for some λ ∈ (0, 1), then x is a solution of We can then apply the bound set theory (cf.[3, Propositon 4.1]) to the problem (40) in order to show that x is necessarily in the interior of Q.
All the assumptions of the continuation principle from Proposition 2.2 are so satisfied, and the proof is complete.
As a straightforward consequence of Theorem 4.2, we can obtain the following result, when E is a Hilbert space with the scalar product •, • , K := B(0, R) and V (x) := 1 2 x 2 − R 2 .In this case, both conditions (38) and (39) become x, w + y, y ≥ 0. By the similar arguments as in the proof of Theorem 4.1, the following result dealing with an upper-Carathéodory mapping F and a strictly localized bounding function V can be proven.In this case, it is possible to construct a sequence of approximating problems which satisfy assumptions of Proposition 2.5.Moreover, since the Nagumo-type condition ( 29) is used, it is guaranteed that the desired bounded solutions have bounded derivatives, which enables us to replace condition (1 ii ) in Proposition 2.5 by the less restrictive condition (2 ii ).Proof.Let us define the function ψ and the sequence of problems (P n ), where F n : [0, T ] × E × E E, in the same way as in the proof of Theorem 4.1.Since ψ is continuous, the mapping F n is upper-Carathéodory, for all n ∈ N. By the analogous reasonings as in the proof of Theorem 4.1, we can obtain that F n satisfies condition (1 i ) with g n = g + L ψ n , and so for n sufficiently large, and that also condition ( 33) is valid, with φ defined as in the proof of Theorem 4.1.Therefore, we can consider, instead of condition (1 ii ), the less restrictive condition (2 ii ), which is ensured directly by the definition of mapping F n .
For x ∈ ∂K, y ∈ E and w n ∈ F n (t, x, y), there exists w ∈ F (t, x, y) satisfying w n = w + ψ(x) n , and so for a.a.t ∈ (0, T ), which ensures the validity of condition (23).Hence, thanks to condition (29), for a sufficiently large n the Dirichlet problem (P n ) satisfies all the assumptions of Proposition 2.5, and subsequently it admits a solution x n (t) such that x n (t) ∈ K, for all t ∈ [0, T ].Moreover, according to (33)   E is an upper-Carathéodory multivalued mapping which is either globally measurable or quasi-compact.Let the conditions (1 i ) from Proposition 2.3, (2 ii ) from Theorem 4.1 and (2 iii ) from Proposition 2.5 be satisfied.Moreover, let there exist R > 0 such that x, w ≥ 0 (43) holds, for a.a.t ∈ (0, T ) and all x ∈ E satisfying ||x|| = R, v ∈ E, and w ∈ F (t, x, v).Finally, let (29) be valid, with φ as in (24)., we proposed some applications of our techniques to integro-differential equations respectively with discontinuous r.h.s. and non-homogeneous conditions.In the following, we briefly show their usage in the study of control problems.Consider the integro-differential equation where We prove now that D is also compact.Indeed, let {z n } ⊂ D(L As a consequence of (47), we have that D is bounded and let r > 0 be such that z L 2 (R) ≤ r for all z ∈ D(L 2 (R)).Let y, v ∈ L 2 (R) with y ≤ R for some R > 0; notice that F (y, v) ≤ (b + L + 1)R + r + v and so (24) and ( 29 Moreover, in the case when E = R n and f = F : [0, T ] × R n × R n → R n is continuous, the function g appearing in (1 i ) and (1 iii ) is equal to 0, and since f is continuous, it is guaranteed by the second Weierstrass theorem that there exists, for every nonempty, bounded set Ω ⊂ R n × R n , a constant M Ω such that for all (t, x, y) ∈ [0, T ] × Ω.Thus, condition (2 ii ) is in the finite-dimensional, continuous case automatically satisfied.Therefore, we are able to obtain as a direct consequence of Theorem 4.1 and Lemma 3.2 exactly Hartman's Theorem 1.1.
Moreover, we could also immediately obtain (applying Proposition 2.3, Theorem 4.1 or Theorem 4.2) the following improvements of results from Remark 1.3: • The strict inequality in (8) can be replaced by a non-strict one (2).
The result from Remark 1.4 can be also improved using Proposition 2.3, Theorem 4.1 or Theorem 4.2 as follows: • The inequality (10) can be replaced by the more general (2).
As a very particular case of Theorem 4.1, we are able to obtain the following generalization of results from Remarks 1.3 and 1.4.As already mentioned before, as a direct consequence of Theorem 4.1 and Lemma 3.2, exactly Hartman's Theorem 1.1 can be obtained.On the other hand, a variant of Theorem 4.1 in R n , with (29), (24) replaced (in view of Lemma 3.2) by ( 3), (5), is a significant generalization of Hartman's Theorem 1.1 (see also Remark 4.1).This, besides other things, demonstrates, jointly with Corollary 6.1, the power of our main theorems formulated in abstract spaces.

Theorem 4 . 1 .
Let us consider the Dirichlet b.v.p.(13), where F : [0, T ] × E × E E is an upper semicontinuous mapping with compact, convex values satisfying conditions (1 i ), (1 iii ) from Proposition 2.3 and condition (2 ii ), where: the Dirichlet b.v.p. (13) admits a solution whose values are located in K. If, moreover, 0 / ∈ F (t, 0, 0), for a.a.t ∈ [0, T ], then the obtained solution is nontrivial.Proof.Consider a continuous function τ : E → [0, 1] such that τ (x) = 0, for x ∈ ∂K + kB, while τ (x) = 1, for x ∈ ∂K + k 2 B, where the constant k was introduced in Proposition 2.1.Let ψ be as in Proposition 2.1 and let us denote by L ψ its Lipschitz constant.It is easy to see that the function ψ : E → E given by ψ(x) = τ (x)ψ(x), for x ∈ ∂K + kB, 0, otherwise, is well-defined, continuous and bounded.Let us put m := sup x∈E ψ(x) and consider the sequence of Dirichlet problems for a.a.t ∈ [0, T ].Finally, by means of a classical closure result (see e.g.[21, Lemma 5.1.1.]),we can show that x is a solution of (13) which completes the proof.

Remark 4 . 1 .
Theorem 4.1 is a significant generalization in several directions of[34,.Since in finite-dimensional spaces K in Theorem 4.1 need not be convex, but only an absolute retract space (i.e. in particular contractible), Theorem 4.1 can be also regarded as a generalization of [1, Theorem 4.5], where only strict sign conditions were applied.

2 x 2 −
(cf.Lemma 3.1), there exists M > 0 such that ẋn (t) ≤ M , for a.a.t ∈ [0, T ].Since E is separable, less restrictive estimates than (35) and (36) are valid implying S ≤ T +4 4 S g n L 1 ([0,T ],[0,∞]) with S defined as in the proof of Theorem 4.1.The convergence of obtained sequence of solutions to the solution of the original Dirichlet problem (13) can then be shown by the same arguments as in the proof of Theorem 4.1.In the case when E is a Hilbert space with the scalar product •, • , K := B(0, R) and V (x) := 1 R 2 , condition (42) becomesx, w ≥ 0, and we can immediately obtain the following consequence of Theorem 4.3.

6 .
) are satisfied, with φ(s) = (b + L + 1)R + r + s.Since F is autonomous, also condition (2 ii ) in Theorem 4.1 is true.Now we prove condition (37).So, let y, v ∈ L 2 (R) with y, v = 0 and take w ∈ F (x, y), implying w = −v +by +K(y)+ f (y)+ω 1 with ω 1 ∈ D(y).Condition (a) implies that y, K(y) ≥ − y while (b) yields that y, f (y) ≥ 0. Consequently, since b > 1 (cfr.(d)), y, w + v, v ≥ bR 2 − R 2 − Rr > 0 when R is sufficiently large.All the assumptions of Corollary 4.1 are then satisfied and so the problem (44)-(45) is solvable.Comparison with classical results.In the last section, we will discuss the relations between classical theorems (for E = R n and a continuous function f ) on one side and the main results introduced in the previous parts of the paper (formulated for a Banach space E and multivalued u.s.c. or upper-Carathéodory r.h.s.F ) on the other side.At first, let us note that if E = R n and f = F : [0, T ] × R n × R n → R n is continuous, (24) can be replaced by (3) and (5) (cf.Lemma 3.2).