with

The paper deals with a semilinear evolution equation in a reflexive and separable Banach space. The non-linear term is multivalued, upper Caratheodory and it depends on a retarded argument. The existence of global almost exact, i.e. classical, solutions is investigated. The results are based on a continuation principle for condensing multifields and the required transversalities derive from the introduction of suitable generalized guiding functions. As a consequence, the equation has a bounded globally viable set. The results are new also in the lack of retard and in the single valued case. A brief discussion of a non-local diffusion model completes this investigation.


Introduction
The paper deals with the multivalued evolution equation depending on a retarded argument and satisfying the initial condition We assume that the state space E is a reflexive Banach space; when it is needed we take E also separable. We recall that in a Banach space with the Radon-Nikodym property, in particular in a reflexive Banach space, any absolutely continuous function x(·) is almost everywhere differentiable and it satisfies the classical integral formula. Definition 1.1. We say that a multivalued function F : I × A ⊸ E 2 , where I is a closed real interval, A ⊆ E 1 and E 1 , E 2 are Banach spaces, is integrably bounded on every bounded set if, for every bounded subset Ω ⊂ A there exists µ F Ω ∈ L 1 + (I) such that y E 2 ≤ µ F Ω (t), for a.a. t ∈ I, all x ∈ Ω and y ∈ F(t, x).
We assume the following conditions  (F3) F is integrably bounded on every bounded set.
The existence of local solutions, i.e. defined on some [a, h] ⊆ [a, b], for problem (1.1)-(1.2) without delay was recently investigated both in [11], under conditions (A)-(F1-4), and in [7] where the u.s.c. in (F2) is intended in the sense of the usual weak topologies. The results depend on suitable fixed point theorems. When the nonlinearity F is sublinear, i.e. when condition (F3) is replaced by Nonlinearities F satisfying (F3) may also be superlinear in x and the use of a fixed point approach does not seem appropriate for the investigation of (1.1)- (1.2). In this paper we show that a continuation principle for condensing multifields (see Theorem 3.1 in Section 3) can be used, in alternative, and we introduce suitable generalized guiding functions in order to prove its transversality condition, i.e. condition (d) in Theorem 3.1.
We further assume that (V) V (x) ≤ 0 for all x ∈ K and V (x) ≡ 0 on ∂K.
If there exists a set of functions parametrized by x ∈ ∂K and with similar properties as V (see e.g. [25]) we say that (1.1)-(1.2) has a family of bounding functions. The notions of guiding and bounding function were first introduced by Gaines and Mawhin in [18] for the investigation of single-valued equations in Euclidean spaces and we refer to [22] for recent results and several references in this context. Important contributions to the theory of bounding functions were given by Zanolin (see e.g. [25] and references therein). The theory of guiding and bounding functions was then generalized in [2,3,4,5] for multivalued equations of first and second order also in infinite dimensional Banach spaces.
Given an arbitrary Banach space X and a positive value r, we denote with B(x, r) the open ball of X with ray r centered in the point x ∈ X and we simply write B in the case of open unit ball with center 0. The symbol τ stands for the Lebesgue measure on the real line.
When the state space is separable and we assume the existence of a guiding function, we obtain the following existence result for problem (1.1)-(1.2) which is the main result of this paper. As far as we know, it is new also when the nonlinear part is single-valued and in the non-retarded case.
The detailed proof is in Section 7 and it is divided into some parts. Thanks to a Scorza-Dragoni type result discussed in Section 3, we introduce a sequence of related initial value problems (P m ). Their solvability depends on some preliminary existence results in Section 6, the continuation principle in Section 3 combined with the generalized guiding function V . The compactness and regularity properties that we need are respectively discussed in Sections 4 and 5. The conclusion then follows from a limiting process. We remark that Theorem 1.2 asserts that K is a viable set (see e.g. Section 8) for the semilinear multivalued equation (1.1).
An alternative approach for the study of viability problems can be found in [14] where the tangency conditions depend on some new tangent sets. Section 8 contains some concrete applications of Theorem 1.2 to viability theory in the case when the norm in E is sufficiently regular.
Remark 1.1. Let L p be the usual Lebesgue space L p (Ω, Σ, µ), for some measure space (Ω, Σ, µ), with the canonical norm · p . Very frequently, when (1.1) is the standard abstract formulation of some partial differential equation, E = L p (see e.g. [8] for some concrete examples). It is known (see e.g. [15, Theorem V.1.1]) that, when p ∈ (1, +∞) and it is not an integer, the function Υ : is Hölder continuous. The symbol [p] denotes the integer part of p. If p is an integer, then Υ(x) is at least C p−1 -smooth with lipschitzian derivative of order p − 1. Therefore, for each R > 0, the function Υ(x)−R p is Fréchet differentiable, with a Lipschitzian Fréchet derivative, provided that p ≥ 2. Notice moreover that Consequently, when K is the ball centered in the origin and radius R, then Υ(x) − R p satisfies (V) and conditions (i) in Definition 1.3; hence it is a good candidate for the construction of a generalized guiding function. Of course p = 2 is a special case, since L 2 is an Hilbert space, and all previous computations can be obtained directly.
It was recently showed that, in some cases, nonlocal conditions of the type x(a) + p k=1 c k x(t k ) = x 0 with c k ∈ R \ {0} and x 0 ∈ E can better describe the evolution of some non-retarded processes (see [10] and references therein). Impulses can be very naturally included in semilinear evolution dynamics such as (1.1) (see e.g. [9,12]). We think that a generalized guiding function approach can be fruitful in both cases as well as in the investigation of half-linear equations such as those in [16].
; E , t → x t and put x n,(·) when indices are present. [a, b]. If Φ has measurable selections we say that F is superpositionally measurable and denote with S 1 F (·,q (·) ) the collection of all measurable selections. It is known that, whenever F satisfies (F1-2), then it is superpositionally measurable (see e.g. [19,Theorem 1.3.5]).

Measures of noncompactness and condensing multimaps
A m.n.c. is a function β : P(E) → N defined on the collection of all nonempty subsets of a Banach space E and taking values in a partially ordered set N such that for all Ω ⊂ E, β(coΩ) = β(Ω). A m.n.c. may enjoy the following properties: If, in addition, N is a cone in a Banach space a m.n.c. may also satisfy the property of (iii) algrebraic subadditivity: β(Ω 1 + Ω 2 ) ≤ β(Ω 1 ) + β(Ω 2 ), for all Ω 1 , Ω 2 ⊂ E.
A well-known example of m.n.c. satisfying all the above properties is the Hausdorff m.n.c. χ, defined by In the following we also need to consider m.n.c. defined on spaces of continuous functions. is a m.n.c. on C 0 ([α, β]; E). However, if Ω is not equicontinuous, the m.n.c. γ is not regular. For this reason we always pair it with the following m.n.c.
and define the m.n.c.

Continuation principle and Scorza-Dragoni type result
We propose now the continuation principle which is the basis for our investigation. We sketch here its proof as a consequence of the theory of relative topological degree for convex-valued multifields, developed in [19] but we remark that it can be also derived from the topological index introduced in [1]. (b) H is quasi-compact and µ-condensing with respect to a monotone, nonsingular m.n.c. defined on Y ; (c) H(Q, 0) ⊂ Q; (d) H(·, λ) is fixed points free on the boundary of Q for all λ ∈ [0, 1).
Then there exists y ∈ Q such that y ∈ H(y, 1).
Proof. If H(·, 1) has a fixed point in the boundary ∂Q of Q the proof is finished. Hence, we can assume that H(·, λ) is fixed points free on ∂Q for all λ ∈ [0, 1]. Since H has closed graph and it is quasi-compact, then H has compact values and hence it is u.s.c. (see e.g. [19,Theorem 1.1.12]). Under these assumptions, all the multimaps H(·, 0), H(·, 1) and H are completely, fundamentally restrictible. Therefore the relative topological degree can be defined (see e.g. [19]), for the corresponding vector-fields Φ 0 := Id − H(·, 0) and Φ 1 := Id − H(·, 1) with Φ i : Q ⊸ Y for i = 0, 1. In addition, H(·, 0) has a fixed point in Q, implying that deg Y (Φ 0 , Q) = 1. Finally, since H is a homotopy from Φ 0 to Φ 1 and H is fixed points free on ∂Q, we obtain that deg Y (Φ 0 , Q) = deg Y (Φ 1 , Q) = 1; hence H(·, 1) has a fixed point in Q and the proof is complete. 2 In Frechét spaces transversality condition (d) in previous theorem needs to be replaced by the stronger so called pushing condition. A guiding function approach was proposed, in [8], for getting pushing condition. It is useful for treating problems with regularities in (F2) given by the weak topologies and also for the investigation of boundary value problems on unbounded intervals such as those in [6,20,21].
We propose now a Scorza-Dragoni type result for the multimap F when the space E is separable. Its proof is a direct consequence of [23,Theorem 1].
possibly empty values satisfying the following conditions: possibly empty values satisfying conditions valid for F 0 in (i) and (ii) and such that According to (F1), and the property (ii) applied to F , it is possible to find N ⊂ [a, b] with τ (N) = 0 and F (t, x n ) = ∅ for all t ∈ N and all n. Now fix ε > 0 and put it is possible to find y ∈ E and a subsequence, denoted as the sequence, such that Hence F 0 has convex, compact values and it satisfies properties (i), (ii) and (iii). 2

Compactness properties of evolution operators
When assuming condition (A) the linear operators {A(t) : t ∈ [a, b]} give rise to a strongly continuous evolution operator (see e.g. [17]) a ≤ s ≤ t ≤ b} and it is always true that Consequently In the study of (1.1)-(1.2) we show that it is useful to introduce a parameterized family of linear operators {λA(t) : t ∈ [a, b]} depending on the real value λ; their corresponding evolution operators will be denoted by U λ : ∆ → L(E) . The following two lemmas show some compactness properties of {U λ : λ ∈ R}.
Proof. Take λ ∈ R and (t, s) ∈ ∆. Notice that the operators U λ (t, s) are strongly differentiable with respect to t and s and it follows that Consequently, for any λ, µ ∈ R, Integrating the previous relation we have that According to (4.1) and (4.2) we obtain that The stated result follows easily, when observing that any convergent sequence is bounded. 2 As a consequence of the previous lemma the set {U λn (t, s) : n ∈ N} is relatively compact in L(E) implying the relative compact- Proof. Fix (t, s) ∈ ∆ and α > 0. If r := χ ({f n (s) : n ∈ N}), there exist p ≥ 1 and e 1 , ..., e p ∈ E satisfying According to Remark 4.1, the set {U λn (t, s) : n ∈ N} is relatively compact in L(E). Therefore, given H := max{ e j E , j = 1, ..., p}, and Therefore, according to (4.2), we obtain that The arbitrariness of α implies the estimate. 2 Definition 4.2. We say that a sequence {f n : ]; E be a semicompact sequence and U : ∆ → L(E) a strongly continuous evolution operator.
Then the sequence

The solution multi-operator
We embed now the linearized version of (1.1) into a family of equations depending on a real parameter and we study the main features of the solution multi-operator T of the parameterized family. Precisely, given q ∈ C 0 [a − h, b]; E and λ ∈ [0, 1], we consider the multivalued equation , nonempty values and satisfying: (G1) G(·, x, λ) has a strongly measurable selection for every (x, λ) (G3) G is integrably bounded on every bounded set; ]; E and the following two propositions show its main features. We always assume conditions (A) and (F1-4).
, then it is easy to prove that (G1-5) are satisfied.
is quasi-compact, with nonempty, convex values and it has closed graph.
Proof. According to (G1-2), the multimap G(·, ·, λ) is superpositionally measurable for all λ and hence S 1 G(·,q (·) ,λ) is nonempty, implying that T (q, λ) = ∅ (see e.g. [17]); thus T is welldefined. It is easy to see that T is also convex-valued, since G is such. LetΩ ⊂ C 0 [a − h, b]; E be compact and consider {y n : n ∈ N} ⊂ T (Ω × [0, 1]). Let {q n : n ∈ N} ⊆Ω, {λ n : n ∈ N} ⊂ [0, 1] and {g n : n ∈ N}, g n ∈ S 1 G(·,q n,(·) ,λn) for all n be such that with the evolution operators U λn introduced in Section 4. The compactness ofΩ and [0, 1] guarantee the existence of a subsequence, that we continue to denote as the related sequence, such that (q n , λ n ) → (q, λ) ∈Ω × [0, 1] and from now on we restrict our attention to it. Let us re-write (5.2) as follows For t ∈ [a, b] put Θ(t) := {q n,t : n ∈ N} and let Θ : The sequence {g n : n ∈ N} is then semicompact (see Definition 4.2) and so it is weakly compact in L 1 [a, b]; E . Hence there exists a subsequence, again denoted as the sequence, and a function g ∈ L 1 [a, b]; E satisfying g n ⇀ g in L 1 [a, b]; E . According to Theorem 4.1 we also obtain that Therefore, since λ n → λ and U λn (t, s) → U λ (t, s) in L(E) uniformly in ∆ as proved in Lemma 4.1 we obtain that the first addendum in (5.3) converges to U λ (t, a)ϕ(0) and the third one converges to zero both of them in C 0 [a, b]; E . We have then proved that T is quasi-compact.
Let now {q n : n ∈ N} ⊂ C 0 [a − h, b]; E , {λ n : n ∈ N} ⊂ [0, 1] and {y n : n ∈ N} ⊂ AC 0 [a, b]; E , with y n ∈ T (q n , λ n ) for all n, be such that q n → q, λ n → λ and y n → y, each one with respect to the corresponding topology. Consequently there exists g n ∈ S 1 G(·,q n,(·) ,λn) such that y n satisfies (5.3) and y n (t) = ϕ(t − a) on [a − h, a] for all n. As showed in the first part of this proof, the sequence {g n : n ∈ N} is semicompact and there exist g ∈ L 1 [a, b]; E and a subsequence of {y n : n ∈ N}, again denoted as the sequence, which converges in C 0 [a, b]; E to the function t → p(t) := U λ (t, a)ϕ(0) + t a U λ (t, s)g(s)ds. The uniqueness of the limit implies y = p. Applying Mazur's Lemma, we may assume the existence of a sequence {g n : n ∈ N} such thatg n is a finite convex combination of {g i : i ≥ n} and g n → g in L 1 [a, b]; E . Passing to a subsequence, denoted as the sequence, we obtain that g n (t) → g(t) for a.a. t ∈ [a, b].
Let M ⊆ [a, b] be such that G(t, ·, ·) is u.s.c., g n (t) ∈ G(t, q n,t , λ n ) andg n (t) → g(t) for all t ∈ M and n ∈ N. According to (G2) and the definition of S 1 G , the set [a, b] \ M has null Lebesgue measure. Let t ∈ M be fixed. According to (G2), for each ε > 0 there exists δ > 0 such that exists n such that q n,t − q t C 0 [−h,0];E ≤ δ and |λ n − λ| ≤ δ implying g n (t) ∈ G(t, q t , λ) + εB for all n > n.
Since G(t, q t , λ) + εB is convex, we also have thatg n ∈ G(t, q t , λ) + εB for all n > n. The arbitrariness of ε and the closure of G(t, q t , λ) imply g(t) ∈ G(t, q t , λ). Therefore T has closed graph.  As a consequence of (4.2) and Lemma 4.2, for all (t, s) ∈ ∆ we have that So, according to (5.4) and the definition of the sequence {y n : n ∈ N}, if γ {y n : n ∈ N} > 0 we obtain the contradictory conclusion γ {q n : n ∈ N} ≤ γ {y n : n ∈ N} < γ {q n : n ∈ N} .
Hence γ {q n : n ∈ N} = 0 implying χ {q n (t) : n ∈ N} = 0 for all t ∈ [a − h, b]. According to (G4) we obtain that {g n (t) : n ∈ N} is relatively compact for a.a. t ∈ [a, b]. Therefore, according to (5.7), {g n : n ∈ N} is semicompact. Let {λ np : p ∈ N} be a convergent subsequence. With a similar reasoning as in the proof of Proposition 5.1 we can show the existence of a convergent subsequence {y np : p ∈ N}; this proves that the set {y n : n ∈ N} is relatively compact in C 0 [a − h, b]; E . As stated in Section 2, this implies that (0, 0) = ν {y n : n ∈ N} = ν (T (Ω × [0, 1]) ≥ ν(Ω); since ν is a regular m.n.c. we obtain that Ω is relatively compact and thus T turns out to be ν−condensing. 2

Preliminary existence results
We state now an existence result for problem (1.1)-(1.2) which is valid in an arbitrary reflexive Banach space E and it is based on the parameterizations given by (5.1). Combining it with the Scorza-Dragoni type argument given in Theorem 3.2 and assuming that E is also separable, in next Section we will prove Theorem 1.2. Let K ⊂ E be nonempty, open and bounded and V : E → R a locally Lipschitzian function on ∂K such that for a.a. t 0 ∈ (a, b] and all (ϑ 0 , λ) ∈ C 0 ([−h, 0]; K) × (0, 1) with ϑ 0 (0) ∈ ∂K, there exists δ = δ(t 0 , ϑ 0 , λ) satisfying We remark that, when V is Gateaux differentiable, the estimate (6.1) reduces itself to V G ϑ(0) (−λA(t)ϑ(0) + w) < 0.
Proof. Let T be the solution multi-operator defined in Section 5, i.e. associated to the problem (5.1)-(1.2) and put Q := C 0 [a − h, b]; K . If we are able to prove that T : Q × [0, 1] ⊸ C 0 [a − h, b]; E satisfies all the assumptions of the continuation principle given in Theorem 3.1, then T (·, 1) has a fixed point which is, according to (G5), a solution of (1.1)-(1.2) and it satisfies x(t) ∈ K for all t ∈ [a, b].
Property (a) in Theorem 3.1 derives from Propositions 5.1; since K is bounded, implying that also Q is bounded, property (b) in Theorem 3.1 comes from Propositions 5.1 and 5.2.
Since, by assumption, T (Q, 0) ⊂ int Q, then property (c) in Theorem 3.1 is true and it remains to prove the transversality condition (d) only for λ ∈ (0, 1). Let x ∈ Q be a fixed point of T (·, λ) with λ ∈ (0, 1); hence there is β λ ∈ S 1 G(·,x (·) ,λ) such that According to the properties of ϕ we have that t 0 ∈ (a, b]. Since V is locally Lipschitzian in ∂K, there exist an open set U ⊆ E with x(t 0 ) ∈ U and a constant L such that, when restricted to U, V is L−Lipschitzian. Let 0 <ĥ < min{t 0 − a, δ} be such that as in (6.1). It is easy to see that g is absolutely continuous in [t 0 −ĥ, t 0 ]. If we further prove that then we arrive to the contradictory conclusion In fact, since without loss of generality we can take x(t) ∈ K for t ∈ [a − h, t 0 ), from (V) we obtain the contradictory conclusion V (x(t 0 −ĥ)) > 0; it implies condition (d) for all λ ∈ (0, 1). So, it remains to show condition (6.3). Let t ∈ (t 0 −ĥ, t 0 ) be fixed and such that conditions (6.1) and (6.2) are valid and there is g ′ (t). Take h ∈ (t 0 − t −ĥ, 0) with h = h(t) sufficiently small so that also x(t) + hx ′ (t) ∈ U. Since h , according to the Lipschitzianity of V in U we have that ∆(h) → 0 as h → 0. According to (6.1) and (6.2) Note that for the fully linearized parametrization G(t, x, λ) = λF (t, x) condition (6.1) can be replaced by the following simpler one, independent on λ lim inf Indeed, let (ϑ, λ) as in (6.1), take t ∈ (a, b] for which (6.4) is true and w ′ ∈ λF (t, ϑ). Since So the following result is an easy consequence of previous theorem Looking at the proof of Theorem 6.1 it is easy to see that x(t 0 ) ∈ ∂K for some solution x of (5.1)-(1.2) and t 0 ∈ (a, b] leads to a contradiction. This is to say that K is a positively invariant set for (5.1) for every λ ∈ (0, 1). Consequently, the proof of Theorem 6.1 can be derived from Theorem 1.1 with no need to introduce a continuation principle. χ (Ω(s)) .
Therefore G λ (·, ·) satisfies also (F4). Consider now the bounded set Ω := {x ∈ C 0 [−h, 0]; E : x ≤ 2M}; applying condition (G3) to the multimap G we obtain a function µ G ∈ [a, b] and all x ∈ C 0 [−h, 0]; E . This shows that G λ (·, ·) is integrably bounded, hence it satisfies the sublinear growth condition (F3 ′ ). Consider a sequence {λ n : n ∈ N} with λ n → 1 − and denote with y n a corresponding sequence of solutions of the equation satisfying the initial condition (1.2). Their existence can be guaranteed by Theorem 1.1. Since, moreover, we also assume condition (6.1) and it implies that y n (t) ∈ K for all t ∈ [a, b], then each y n is indeed a solution (5.1) with corresponding parameter. Let g n ∈ S 1 G(·,y n,(·) ,λn) be such that y n satisfies (5.2) and consider the m.n.c.  a.a. t ∈ [a, b]. Notice that {y n : n ∈ N} ⊆ T ({y n : n ∈ N} × [0, 1]), implying that ν (T ({y n : n ∈ N} × [0, 1])) ≥ ν ({y n : n ∈ N}); since T is ν-condensing (see Proposition 5.2), this implies that {y n : n ∈ N} is relatively compact. Consider a subsequence, denoted as the sequence, such that y n → y ∈ C 0 [a − h, b]; E . According to (G4), it implies that {g n : n ∈ N} is semicompact and hence there is g ∈ L 1 [a, b]; E and a subsequence, again denoted as the sequence, such that g n ⇀ g in L 1 [a, b]; E . Finally, with a similar reasoning as in the proof of Proposition 5.1, we obtain that y(t) := U(t, a)ϕ(0) + t a U(t, s)g(s)ds for t ∈ [a, b] and y(t) = ϕ(t − a) on [a − h, a] with g ∈ S 1 G(·,y (·) ,1) . Therefore, according to (G5), y is a solution of (1 .1)-(1.2).
If we assume that V is a generalized guiding function on ∂K and it satisfies (V), instead of taking condition (6.1) (or (6.4) in the special case) K does no longer become a positively invariant set for any λ (see e.g. [13, Example 3.1]). For this reason our main result, i.e. Theorem 1.2, can not be derived from Theorem 1.1.

Proof of the main result
We need the following technical lemma which is a straightforward generalization of [13, Theorem 2.2] to the case of a function V only Gateaux differentiable. Hence we omit its proof, which is very technical.
Lemma 7.1. Let E be a Banach space and K ⊂ E be nonempty, bounded, open and convex. Assume that V : E → R is a generalized guiding function on ∂K (see Definition 1.3) satisfying (V). Let κ > 0 be such that V G x is Lipschitzian on ∂K +κB. Then it is possible to find ε ∈ (0, κ) and a Lipschitzian function φ : ∂K + εB → E such that V G x (φ(x)) ≡ 1. Proof of Theorem 1.2 Take ε as in the statement of Lemma 7.1. According to Urishon Lemma we can find a continuous map µ : E → [0, 1] such that µ ≡ 1 on ∂K + ε 2 B and µ ≡ 0 on E \ (∂K + εB). Considerφ : E → E defined bỹ where φ was introduced in Lemma 7.1. It is easy to see thatφ is well-defined. Since the functions: x −→ V G x and x −→ φ(x) are Lipschitzian on ∂K + εB (see Lemma 7.1) andφ ≡ 0 on E \ (∂K + εB), thenφ is continuous and bounded on its whole domain and let Mφ > 0 be where Ω ε = ϑ ∈ C 0 [−h, 0]; K : ϑ(0) ∈ (∂K + ε 2 B) ; µ F Ωε is the function given by (F3) and ∂K := sup k∈∂K k < +∞ since K is bounded. Now the proof splits into some parts. For each m ∈ N we consider the convex, compact, nonempty valued multimap and consider the multivalued equation 2. Solvability of the sequence of problems (7.1)-(1.2) Fix m ∈ N. Now we show that, for every sufficiently large m, the initial value problem (7.1)-(1.2) satisfies all the assumptions of Theorem 6.1 and hence, it is solvable.
Notice that F m satisfies (F1) and, sinceφ is continuous and bounded, also (F2) and (F3) are respectively true. Let us introduce, now, the convex, compact, nonempty valued multimap It is easy to see that G m satisfies (G1-3) and (G5). Now we prove (G4).
It proves, at the same time, that F m and G m respectively satisfy (F4) and (G4).
We investigate now the transversality condition (6.1). First of all we take (t, ϑ, λ) ∈ J m × Ω ε × (0, 1) with t = a and t ∈ J. If w m ∈ G m (t, ϕ, λ), then w m = λw 0 − (m+1)p(t) mφ (ϑ(0)) and we obtain that (0))). According to Lemma 7.1 and the definition of p we have that Since V is a generalized guiding function on ∂K (see Definition 1.3) and p(t) ≥ 1, for a.a. t we obtain that Notice that the multimap (t, ϑ, λ) ⊸ −λA(t)ϑ(0) 1] and according to the Lipschitzianity of V G (·) also the multimap . This is to say that Together with (7.3) it proves condition (6.1). It remains to investigate the case when λ = 0. So, let q ∈ C 0 [a − h, b]; K and x 0 m be a solution of the initial value problem If η > 0 is such that ϕ(0) + ηB ⊂ K, then Thus it is clear that x 0 m (t) − ϕ(0) E ≤ η, implying that x 0 m (t) ∈ K, for all t ∈ [a, b] and every sufficiently large m.
All the assumptions of Theorem 6.1 are then satisfied and hence the initial value problem (7.1)-(1.2) has a solution, denoted x m , for all m sufficiently large and it follows that x m (t) ∈ K for all t ∈ [a, b].
3. Existence and localization of a solution Consider the sequence {x m : ∈ [a, b] and according to (F4), we obtain With a similar computation as in the proof of Proposition 5.2 we have that With an argument based on Mazur's Lemma very similar to the one in the proof of Proposition 5.1 we can show that h ∈ S 1 F (·,x(·)) and the proof is complete.

An application to viability theory
As an application of Theorem 1.2 we obtain now a viability result (see Theorem 8.2) for the semilinear evolution equation (1.1). Indeed, in order to eliminate some technicalities, we restrict to the case when the nonlinearity F does not contain delays, i.e. we consider First we briefly recall the notion of viable set. Given the subset K ⊆ [a, b] × E and the multimap F : K ⊸ E, consider an almost exact, i.e. a classical, solution x : [τ, T ] → K of (8.2) satisfying x(τ ) = ξ, with (τ, ξ) ∈ K, is said to be global if Definition 8.1. The set K is said to be almost exact globally viable for the multivalued equation Necessary and even necessary and sufficient conditions in order that K is a viable set where recently obtained in [14] in an arbitrary Banach space. They involve new notions of tangent set and quasi-tangent set, which are more general than the classical Boulingand tangent vector and deal with globally u.s.c and positively sublinear terms F (see e.g. Theorem 8.2 and Theorem 9.2 in [14]). We need the state space E to be reflexive, separable and with a sufficiently regular norm; the case when E = L p with 2 ≤ p < +∞ is included. However, our analysis extends to any Bochner integrable A(t) and u-Carathéodory nonlinearity F satisfying (F4 ′ ) which is integrably bounded on bounded set.
for a.a. t ∈ (a, b], all x with x E = R and every w ∈ F (t, x), then the set RB is almost exact globally viable. Let E be a separable Hilbert space with scalar product ·, · and V (x) = 1 2 ( x, x − R 2 ) for some positive R. It is easy to see that V is Fréchet differentiable with V F x (h) = x, h and also that V F x L(E;R) = x E . Moreover V F x − V F y L(E;R) = x − y E for each x, y ∈ E, implying that V F x is Lipschitzian and we obtain the following consequence of previous result  ∈ (a, b], all x with x, x = R 2 and every w ∈ F (t, x), then the set RB is almost exact globally viable.
Since the function p contains some discontinuities, a solution of (8.5) satisfying a given initial condition u(a, x) = u 0 (x), x ∈ R (8.6) will be interpreted in the sense of Filippov. More precisely, given t ∈ [a, b], consider 1 − min{p(t, r i ), p(t, r − i ), p(t, r + i )}, max{p(t, r i ), p(t, r − i ), p(t, r + i )} if r = r i , i = 1, 2, ..., n where p(t, r ∓ i ) := lim According to (a) and (b), both A and F are well defined and it is not difficult to show that A satisfies (A) and F is nonempty, convex compact valued and satisfies (F1-3). If Ω ⊂ L 2 (R) is bounded, according to (a) we have that R ϕ(x)y(x) dx ∈ [−C, C] for some C > 0 and all y ∈ Ω. Put k F (t) = 1 + max max p(t, r), p(t, r − i ), p(t, r + i ) which exists according to (c). According to the well-known properties of then Hausdorff m.n.c.
It is easy to see that π ≤ 1 − min min p(t, r), p(t, r − i ), p(t, r + i ) . According to condition (8.8) we then derive that y, −A(t)y + πy ≤ 0 for a.a. t ∈ [a, b]. Corollary 8.1 can then be applied. It implies the global viability of the set RB ⊂ L 2 (R) for problem (8.7)-(8.6) and hence the solvability, in the sense of Filippov, of (8.5)-(8.6) provided that u 0 2 < R.