ON BOUNDARY VALUE PROBLEMS FOR DEGENERATE DIFFERENTIAL INCLUSIONS IN BANACH SPACES

We consider the applications of the theory of condensing set-valued maps, the theory of set-valued linear operators, and the topological degree theory of the existence of mild solutions for a class of degenerate di ﬀ erential inclusions in a reﬂexive Banach space. Further, these techniques are used to obtain the solvability of general boundary value problems for a given class of inclusions. Some particular cases including periodic problems are considered.


Introduction
In the last decades the theory of degenerate differential equations in Banach spaces attracted the attention of a large number of researchers (see, e.g., Favini and Yagi [5], Showalter [13], and [6] and the references therein).One of the main reasons is that many partial differential equations arising in mathematical physics and in applied sciences may be naturally presented in this form.
In this paper, we introduce a class of degenerate differential inclusions in a reflexive Banach space and define the notion of mild solution for such inclusion.Applying the techniques of the theory of condensing multimaps (see Kamenskii et al. [8]) and the methods of the theory of multivalued linear operators [3,5], we first prove existence results for the Cauchy problem for various types of multivalued nonlinearities.Then we consider the solvability of general boundary value problems and discuss some particular cases including periodic problem.It should be noted that, starting from the paper of Zecca and Zezza [15], nonlinear boundary value problems for nondegenerate differential inclusions in Banach spaces were studied in a number of papers (see, e.g., [4,9,10,11,12]) under compactness conditions on the evolution operator generated by the linear part of the problem.In our paper we do not suppose compactness for the linear part nor for the multivalued nonlinearity.We only assume some regularity conditions expressed in terms of the measures of noncompactness.

Multivalued linear operators.
We present some necessary definitions and results from the theory of multivalued linear operators.Details can be found in [3,5].
Let E be a complex Banach space.
Definition 2.1.A multivalued map (multimap) A : E → 2 E is said to be a multivalued linear operator (MLO) in E if (1) (2) the following linearity relations hold: It is an easy consequence of the definition to note that Ax + Ay = A(x + y) for all x, y ∈ D(A) and λAx = A(λx) for all x ∈ D(A), λ = 0.

It is easy to verify that
Definition 2.3.Let A and B be two MLOs in E. The sum and the product of A and B are defined, respectively, by the relations One can observe that A + B and AB are MLOs in E and that for some positive constant C.

V. Obukhovskii and P. Zecca 771
The following statement gives an example of MLO satisfying (H-Y).Let E * denote a space dual to E and J : E → 2 E * be the duality map.
Proposition 2.6.Given an MLO A, suppose that for every x ∈ D(A) there exists x * ∈ J(x) such that Re y,x * ≤ β x 2  (2.5) for all y ∈ Ax, where β is a real number.Let also for some λ 0 > β.Then (β,+∞) ⊂ ρ(A) and the resolvent (λI The proof follows the lines of [5,Theorem 2.7]. Let A be an MLO satisfying (H-Y).
Definition 2.7.For an integer n > β, the bounded linear operator where The definition is motivated by the following facts.First of all, following [5, Theorem 2.6], it is easy to verify that given a function f ∈ L 1 ((0,T);E), every Carathéodory solution to the problem is necessarily of the form Furthermore, the function takes its values in the subspace At last, in the nondegenerate case M = I, the given definition agrees with the notion of mild solution for a semilinear differential inclusion (see, e.g., [8]).
To present sufficient conditions under which the MLO A = LM −1 satisfies (H-Y), we recall that, in the Banach space E, a semiscalar product can be defined as [u,v] = u,v * with v * ∈ J(v) (see [14]).

Multimaps and measures of noncompactness.
Let X be a metric space and Y a normed space.Let P(Y ) denote the collection of all nonempty subsets of Y .We denote We recall some notions (see, e.g., [8] for further details).
A function β : (2.16) for every a ∈ Ᏹ and Ω ∈ P(Ᏹ); (iii) invariant with respect to reflection through the origin if β(−Ω) = β(Ω) for every As an example of MNC satisfying all the above properties, we can consider the Hausdorff MNC χ(Ω) = inf{ε > 0 : Ω has a finite ε-net}. (2.17) Another example can be presented by the following MNC φ defined on the space of continuous functions C([0,T];E) with the values in a Banach space E: where χ E is the Hausdorff MNC in E and Ω(t) = {y(t) : y ∈ Ω}.The MNC φ satisfies all the above-mentioned properties except regularity.It is known (see [8]) that for every set where χ C is the Hausdorff MNC in C([0,T];E).Moreover, φ coincides with χ C on equicontinuous sets Ω (see [8]).
Let Ᏹ and Ᏹ be Banach spaces with MNCs β and β , respectively, and let N : Ᏹ → Ᏹ be a bounded linear operator.Definition 2.14 (see [1]).The value In particular, if β and β are the Hausdorff MNCs χ and χ , then the value N (χ,χ ) is denoted by N (χ) and is called the χ-norm of N. The χ-norm may be evaluated by the formula (see [1]) where S is a unit sphere and B is a unit ball in Ᏹ.The above formula easily implies Let W ⊂ Ᏹ be an open set, ⊆ Ᏹ a closed convex subset, β a monotone MNC in Ᏹ, and F : W → Kv() an u.s.c.multimap such that x / ∈ F(x) for all x ∈ ∂W , where W and ∂W denote, respectively, the closure and the boundary of the set W = W ∩ in the relative topology of the space .
In such a setting, the relative topological degree deg (F, W) satisfying the standard properties is defined (see [8,Chapter 3]).

Cauchy problem.
We consider the Cauchy problem (2.9) under the assumptions that the operators M and L satisfy condition (ML) and A = ML −1 satisfies condition (H-Y).
On the multivalued nonlinearity F : [0,T] × E E, we first suppose the following assumptions: (F0) F has nonempty, compact, and convex values; (F1) the multifunction F(•,x) : [0,T] → Kv(E) has a strongly measurable selection for every for every bounded set D ⊂ E.

C([0,T],E) C([0,T],E) as
To prove the theorem, it is sufficient to verify that the fixed-point set Fix Γ is nonempty and compact.
Consider the closed, convex set = {y : y(0) = y 0 } ⊂ C([0,T];E) and the parametrized family of multioperators Γ : ×[0,1] such that Following [8] it is easy to verify that the multimap Γ has compact, convex values; it is u.s.c. and ν-condensing on bounded sets of , where ν is the MNC in C([0,T];E) with values in the naturally partially ordered R 2 + , defined as where ∆(Ω) is the collection of all denumerable subsets of Ω, given by b > 0 large enough and mod C (Ᏸ) is the modulus of equicontinuity of Ᏸ defined as Furthermore, using condition (F3) and the standard technique based on the Gronwall-type inequality, it is possible to prove that the set of solutions of the family of inclusions is a priori bounded in norm by the constant where Hence, if we take W as an open ball in C([0,T];E) centered at the origin with radius r > r 0 , from the basic properties of the topological degree, we have and the result follows.
We now consider the case when the nonlinearity F : [0,T] × E → K(E) has nonconvex values, but instead of upper Carathéodory conditions (F1) and (F2), it satisfies the almost lower semicontinuity assumption that (F L ) there exists a sequence of disjoint compact sets {I n }, I n ⊆ [0,T], such that (i) meas([0,T]\I) = 0, where I = ∪ n I n ; (ii) the restriction of F on each set I n × E is l.s.c.We also assume that the space E is separable.
It is easy to see that the fixed points of Q coincide with the functions y(•) defined by the equality y(t) = Mx(t), where x(•) is a mild solution of problem (3.14) and (3.15).
To study further properties of the multioperator Q, we recall some known facts from [8].

([0,T];E) be an abstract operator satisfying the following conditions:
(S1 ) there exists K ≥ 0 such that

If ᏼ F is a superposition multioperator generated by a multimap F satisfying (F0), (F1), (F2), (F3), and (F4), then the composition Sᏼ F is an u.s.c. multimap with compact values.
Following the lines of [8, Lemma 4.2.1], it is possible to verify the following proposition.

Proposition 3.4. The generalized Cauchy operator G satisfies properties (S1 ) and (S2).
Since (I − ΛᏮ) is a bounded linear operator, we obtain as an immediate consequence the following proposition.
Proposition 3.5.The multioperator Q is u.s.c. and has compact convex values.
Our goal is to give conditions under which the multioperator Q is condensing on bounded subsets of C([0,T];E).
We assume that the linear operator Ꮾ satisfies the following condition: (Ꮾ) the operator Ꮾ is (b,φ,χ E )-bounded, that is, there exists a constant b ≥ 0 such that for every bounded set Ω ⊂ C([0,T];E) we have that where φ is the MNC in C([0,T];E) defined by (2.18).

V. Obukhovskii and P. Zecca 779
We further suppose that the linear operators U(t) satisfy the following estimate for the χ-norm: where h : [0,T] → R + is a summable function.At last, let the following condition holds: where k(•) is the function from condition (F4).
Note that condition (C) is satisfied when the generalized semigroup U(t) is compact (h(•) = 0) or when F is completely u.s.c. in the second argument (in the sense that k(•) = 0).
Consider the MNC on the space C([0,T];E) with values in the cone R + 2 , where mod C (Ω) is the modulus of equicontinuity (see Section 2).The MNC ϕ is monotone, regular, and invariant with respect to union with a compact set.Proposition 3.6.Under conditions (Ꮾ) and (C), the multioperator Q is ϕ-condensing on bounded subsets of C([0,T];E).
in the sense of the order generated by the cone R 2 + .We estimate the value of φ(Q(Ω)).Since the operator Ꮿ is completely continuous, we need to estimate only We have To estimate φ(Gᏼ F (Ω)), note that (3.27) Hence, by the theorem on χ-estimates for a multivalued integral (see [8, Theorem 4.2.3]),we have for every t ∈ [0,T] It follows that φ(Ω) = 0 and hence Now, we show that the set Ω is equicontinuous.Note that from mod C (Ω) ≤ mod C Q(Ω) , (3.32) it follows that it is sufficient to show that Q(Ω) is equicontinuous.It is also sufficient to verify that the set (I − ΛᏮ)Gᏼ F (Ω) is equicontinuous.This is equivalent to show that every sequence {g n } ⊂ (I − ΛᏮ)Gᏼ F (Ω) satisfies this property.Given a sequence {g n }, there exist a sequence {y n } ⊂ Ω and a sequence of We can now observe that it is possible to apply the topological degree theory introduced in Section 2.3 to the multioperator Q.We can then formulate the following general existence principle.Proof.We prove that there exists a closed ball B r ⊂ C([0,T];E) such that Q(B r ) ⊆ B r .Supposing the contrary, we have the sequences  (see [7,8]).Condition (C) can be written as (C T ) (1 + 2 [U(T) − I] −1 (χ) • sup 0≤t≤T h(t)) • sup 0≤t≤T h(t − s)k(s)ds < 1.As a corollary of Theorem 3.8, we obtain the following statement on the existence of a periodic solution.
Corollary 3.3.1])to the restriction Q : B r → Kv(B r ).