THE KRASNOSEL’SKII FORMULA FOR PARABOLIC DIFFERENTIAL INCLUSIONS WITH STATE CONSTRAINTS

. We consider a constrained semilinear evolution inclusion of parabolic type involving an m -dissipative linear operator and a source term of multivalued type in a Banach space and topological properties of the solution map. We establish the R δ -description of the set of solutions surviving in the constraining area and show a relation between the ﬁxed point index of the Krasnosel’skii–Poincar´e operator of translation along trajectories associ- ated with the problem and the appropriately deﬁned constrained degree of the right-hand side in the equation. This provides topological tools appropriate to obtain results on the existence of periodic solutions to studied diﬀerential problems.

1. Introduction. In the paper we present a topological approach to study dynamics and periodic behaviour of a reaction-diffusion system of the form Our interest is concerned with a chemical system of N reacting components, u i (t, x) is the concentration at time t and location x in the unstirred bounded reactor Ω of the i-th reactant (i = 1, ..., N ), subject to diffusion and the source term f i depending on (t, x, u). A natural requirement is that the initial u i (·, 0) as well as the intermediate concentrations u i (·, t), i = 1, ..., N , are nonnegative. There are natural upper bounds, too, e.g. some threshold values R i > 0 beyond which the component u i is saturated. Sometimes implicit bounds follow e.g. from the mass constraint: the total mass cannot exceed the threshold value R. It, therefore, makes sense to look for solutions u = u(x, t) taking values in a rectangle P := {u ∈ R N | 0 ≤ u i ≤ R i , i = 1, ..., N } or in a simplex S := {u ∈ R N | N i=1 u i ≤ R, u i ≥ 0, i = 1, ..., N }. Of course the imposed state constraints must be compatible with initial boundary conditions.
The presence of state constraints puts also certain limitations to the source term f i responsible for the increase (production) or decrease (vanishing) of u i . Firstly it may make sense to define f for u in the constraining set only; secondly if, for instance, we postulate that solutions are to take values in the rectangle P , then it is clear that we need to require f i (t, x, u) ≥ 0 if u i = 0 since the concentration u i may only increase in this case, and f i (t, x, u) ≤ 0 if u i = R i since the concentration may only decrease then. The same argument implies that if we impose state constraints in the simplex S, then the total mass can only decrease in this situation and f i (t, x, u) ≥ 0 if u i = 0. In both cases the natural requirements on f may be subsumed by saying, for instance, that f (t, x, u) ∈ T P (u) for all ∈ [0, T ], x ∈ Ω and u ∈ P , where T P (u) stands for the tangent cone to P at u (see (4)). Similar assumptions should be made if the state of the system is constrained to the simplex S.
In general we will look for solutions to (1) in a prescribed closed convex set C ⊂ R N of state constraints requiring additionally that f is tangent to C, i.e., f (t, x, u) ∈ T C (u) for all t ∈ [0, T ], x ∈ Ω and u ∈ C.
We will also admit discontinuous or even multivalued nonlinear term f . This is motivated by numerous applications when f is e.g. accretive, or when system data are determined by measurements, or when systems are subject to the presence of Coulomb friction, or phase transitions, or the studying phenomena display the hysteresis effect. In this case it is usual to replace f by an appropriate set-valued regularization ϕ : [0, T ] × Ω × C R N (see e.g. [29]) and to deal with problems of the form u t ∈ Lu + ϕ(t, x, u), u(0, ·) = u 0 , where u 0 (x) ∈ C for x ∈ Ω, subject to Dirichlet of Neumann boundary conditions. We are going to study the existence and the structure of (strong) solutions to (2) and the existence of periodic solutions. To this end, we shall put the problem into an appropriate abstract setting of a constrained semilinear differential inclusion in a Banach space (see (14)) and consider an approach based on the t-Poincaré map (or the translation along trajectories operator) assigning to the initial state x 0 the set Σ t (x 0 ) of reachable states of the system after time t > 0. A standard observation is that fixed points of Σ T correspond to 'periodic' solutions u, i.e. such that u(0) = u(T ). This approach to periodic orbits is well-known an used by numerous authors (see the book [39] and references therein). The important obstacle here is that we deal with generic nonuniqueness of solutions and the presence of constraints. In order to apply appropriate fixed point theoretical results relying on the fixed point index approach and get conditions guaranteeing the existence of fixed points of the Poincaré operator associated to (2) we shall characterize the set of its solutions and study its dependence on initial conditions. Finally we shall establish a counterpart of the famous Krasnosel'skii formula for such problems. In this way we will obtain a convenient topological tool to study the dynamics of (2) as well as its abstract versions and the existence of periodic solutions as well as the branching (or bifurcation) of periodic solutions in the case of the parameterized problems.
Recall the classical Krasnosel'skii formula that concerns an ODĖ with locally Lipschitz f : [0, T ] × R N → R N , admitting global solutions. If U ⊂ R N is open bounded and f (0, u) = 0 for u in the boundary ∂U of U , then P t (x) = x for x ∈ ∂U and sufficiently small t > 0, where P t (x) = u(t) with u(·) being the unique solution to ( * ) starting at x ∈ U (P t is the t-Poincaré operator associated to ( * )), and the Brouwer degrees are equal with I denoting the identity operator on R N (cf. [41,Lem. 13.1,13.2]). The Krasnosel'skii formula shows that the right-hand side data allows to compute the fixed point index of the Poincaré operator associated to ( * ). An infinite dimensional variant of the Krasnosel'skii formula was obtained in [21,Thm. 4.5] in the case of (14) with K = E, single-valued, time-independent and locally Lipschitz nonlinearity and in [22,Thm. 4.1] for K being a closed convex cone (see also [23]). After this introduction the paper is organized as follows: below we recall some relevant concepts and auxiliary results, present standing assumptions concerning (2) and introduce the convenient functional setting. In the the second section we present the abstract version of (2), namely problem (14), and present a result on a topological characterization of the set of its solutions. In the third section we introduce topological tools necessary to establish the announced version of the Krasnosel'skii formula. The setting we propose is convenient and general. In the fourth section the Kranosel'skii formula is proven, while the fifth section is devoted to the study of periodic orbits. We present several results for a bounded, as well unbounded set of constraints. In the unbounded case we introduce a guiding function approach. This attitude is well-known in the finite-dimensional situation (see the extensive and up to date references in [45]). To the best of our knowledge however, this approach was used here in the infinite-dimensional case for the first time.
Remark 1. In the paper we deal mostly with the set-valued setting. It is to be noted however that the results we propose are new in the single-valued case, too.
1.1. Preliminaries. The notation used in this paper is rather standard and so is the use of functional spaces (L p , Sobolev etc.), the theory of operators, etc. In what follows (E, · ) denotes a real Banach space, while E is the normed topological dual of E; we write x, p instead of p(x) for x ∈ E, p ∈ E ; L(E) denotes the space of bounded linear operators on E. By L p (0, T ; E), p ≥ 1 (resp. C([0, T ], E)) we denote the space of Bochner p-integrable (resp. continuous) functions u : in particular B X (x, r) (resp. D X (x, r)) stands for the open (resp. closed) ball at x ∈ X of radius r > 0. If X ⊂ E, Y is a topological space, then a continuous f : X → Y is compact or completely continuous if f (B) is relatively compact for each bounded B ⊂ X. Set-valued analysis terminology and detailed discussion of many 'set-valued' objects is taken from [7]; in particular a set-valued map ϕ : X Y assigns to each x ∈ X a nonempty subset ϕ (x) ⊂ Y . If X, Y are topological spaces, then ϕ is upper semicontinuous or usc (resp. lower semicontinuous or lsc)  [31] for details and examples concerning set-valued maps).
If K is a closed convex subset of a Banach space E and x ∈ K, then We present now two auxiliary results in a form adopted for our needs.
We will look for strong solutions to (2), i.e., functions u : [0, T ]×Ω → C such that u(0, ·) = u 0 a.e. on Ω, u(t, ·) ∈ H 2 (Ω, R N ) with u(t, ·) = 0 (or ∂u(t,·) ∂n = 0) on ∂Ω in the sense of trace for all t ∈ (0, T ], u ∈ W 1,1 loc ((0, T ], L 2 (Ω, R N )) and for a.a. x ∈ Ω and all t ∈ (0, T ], u t (t, It is convenient and natural to put (2) into the functional setting of a semilinear differential inclusion in E := L 2 (Ω, R N ) where Let us collect some properties of K, A and F introduced above. This is important since it justifies assumptions undertaken in the next abstract sections.
is the (modified) resolvent of A (in the Dirichlet case this holds provided 0 ∈ C). For the sake of completeness we give an argument in the Dirichlet case. Assume that 0 ∈ C. In view of [3,Cor. 7.49], C = n≥1 C n , where C n := x ∈ R N | p n · x ≤ a n for some p n ∈ R N and a n ≥ 0. We thus have K = ∞ n=1 K n , where K n = {u ∈ E | u(x) ∈ C n for a.a x ∈ Ω} and the countable collection plays a role. We shall show that J h (K n ) ⊂ K n for each n ≥ 1. Then To this end we assume that C = {x ∈ R N | p · x ≤ a}, where p ∈ R N , a ≥ 0, and K = {u ∈ E | p · u(x) ≤ a for a.a x ∈ Ω}. Let f ∈ K and put u = J h (f ). By definition u ∈ D(A) and u − hAu = f . Letf (x) := p · f (x),ū(x) := p · u(x) for x ∈ Ω. Thenf ≤ a a.e.,ū ∈ H 2 ∩ H 1 0 (Ω) and for every ξ ∈ H 1 0 (Ω) Henceū ≤ a a.e. and J h (f ) ∈ K. By the Post-Widder formula (see [28,Corollary 5.5,5.6]) K is also semigroup invariant, i.e. S(t)(K) ⊂ K for all t ≥ 0. The physical meaning is that if the reaction term F vanishes, then the pure diffusion process u(t) := S(t)x, x ∈ K, survives in K.
(b) In view of (ϕ 1 ), for each t ∈ [0, T ] and u ∈ K, the set F (t, u) is nonempty since the map Ω x ϕ(t, x, u(x)) ⊂ R N is measurable and, thus, has a selector v which belongs to E in view of (ϕ 2 ). Moreover, the set F (t, u) is is closed and convex; F (t, u) is weakly compact since, by (ϕ 2 ), it is bounded and E is reflexive. Observe, however, that F (t, u) is never compact. Arguing as in [42,Lemma 4.1], one proves that F is H-upper semicontinuous. Therefore (see e.g. [14,Prop. 2.3]), F is upper semicontinuous when the target space is endowed with the weak topology: . This implies that given a weakly closed C ⊂ E, the preimage F −1 (C) is closed. As a consequence, we see that if U ⊂ E is open then F −1 (U ) is an F σ -set since U may be represented as the union of countable many closed balls. Hence F is product-measurable.
(c) It is well-known that C y T C (y) is lower semicontinuous. Let u ∈ K and t ∈ [0, T ]. Then maps T C (u(·)) and ϕ(t, ·, u(·)) are measurable and so is the intersection T C (u(·)) ∩ ϕ(t, ·, u(·)). In view of (ϕ 3 ) and the Kuratowski-Ryll-Nardzewski theorem, there is v ∈ E such that v(x) ∈ ϕ(t, u(x)) ∩ T C (u(x)) for almost all x ∈ Ω. Hence v ∈ F (t, u) ∩ T K (u) since, by [7,Cor. 8.5.2], T K (u) = {v ∈ E | v(x) ∈ T C (u(x)) a.e. in Ω}. In other words, for all t ∈ [0, T ] and u ∈ K, (d) Recall that a continuous u : [0, T ] → K is a mild solution to (7) if where w ∈ L 1 (0, T ; E) and w(s) ∈ F (s, u(s)) a.e. on [0, T ] (it is easy to see that such w exists; observe that w belongs actually to L 2 (0, T ; E) in view of (9)). In particular u is a unique mild solution tȯ Observe that (in both cases of the Dirichlet or Neumann boundary conditions) −A is the subdifferential of the lower semicontinuous quadratic functional γ : where , E) and u (t) = Au(t) + w(t) for a.a. t ∈ (0, T ) (u (t) denotes the strong derivative of u which exists a.a. since u ∈ W 1,1 loc ). This implies that actually any mild solution u of (7) (treated as a function of variables t ∈ [0, T ] and x ∈ Ω) is actually a required strong solution to problem (2).

2.
Existence and structure of solutions. In this Section we are going to consider an abstract version of (7), i.e. a semilinear differential inclusioṅ subject to the initial value problem where: E has convex weakly compact values; (F 2 ) F is product measurable and for every t ∈ [0, T ], the map K u F (t, u) ⊂ E is H-upper semicontinuous; (F 3 ) there are a ∈ L 1 (0, T ), b ≥ 0 such that sup y∈F (t,u) y ≤ a(t) + b u for t ∈ [0, T ] and u ∈ K; (F 4 ) F is weakly tangent to K, i.e. F (t, u) ∩ T K (u) = ∅ for all t ∈ [0, T ] and u ∈ K (see (4)).
Let us briefly comment on these assumptions. (c) It is standard to see (comp. e.g. [42,Rem. 3.5.]) that (K) holds if and only if K is resolvent invariant, i.e. J h (K) ⊂ K for h > 0 with hω < 1.
Suppose that E is reflexive. Then for any x ∈ E, the metric projection is nonempty. The following characterization is well-known: y ∈ π K (x) if and only if for any p ∈ J(x − y) and v ∈ K, v − y, p ≤ 0, where J stands for the duality map in E (see e.g. [51]). Arguing similarly as in [4,Th. (6.2)] we show that: if for any x ∈ D(A) there is y ∈ π K (x) and p ∈ J(x − y) such that then K is resolvent invariant. If M = 1 (see (a) above) and K is semigroup invariant, then for each x ∈ D(A), y ∈ π K (x) and p ∈ J(x − y) condition (16) is satisfied.
(d) The relevance of the tangency assumption (F 4 ) and the resolvent invariance has already been discussed. These assumptions are not only justified by the model application, as it was explained in the Introduction, but they are responsible for the so-called viability properties. It is easy to see that, as a consequence of (5), (K) implies that for all u ∈ K Hence (K) together with (F 4 ) imply that The sets T A K (x), u ∈ K, were introduced by Pavel [46] and condition (17) was shown to be necessary and sufficient for the viablity, i.e. the existence of (mild) solutions to (14) with single-valued continuous F surviving in K. This condition is also sufficient for the existence in case of a H-upper semicontinuous set-valued perturbation F (see [14, §4.5] and [47]); see also [17,Chap. 9] and [5] for a detailed discussion of different tangency issues and relations to viability.
We are going to show that the set of all (mild) solutions to (14) surviving in K is an R δ -subset of C([0, T ], K), the space of continuous functions on [0, T ] taking values in K, i.e. can be represented as the intersection of a decreasing sequence of compact absolute retracts (see also e.g. [31, p. 14] and [37] for a detailed discussion of the class of R δ -sets). In particular R δ -sets are nonempty.
The question of topological characterization of solution to (14) with K = E was discussed by numerous authors (see e.g. monographies [35,36,39,27] and Górniewicz [32] and references therein) and a diversity of results have been obtained. Most of these results relied on a restrictive assumption about the compactness of values of F . As noted in Remark 2 (b) the abstract substitution operator F related to compact convex valued perturbation ϕ has only convex weakly compact values.
In Bothe [14] (see also some of his earlier papers) and Chen et al. [19] the case of weakly upper semicontinuous and weakly compact convex case was studied.
Here we deal additionally with the presence of constraints. Suppose for a while that F is defined on [0, T ]×E. As mentioned above assumptions (K) and (F 4 ) imply the existence of solutions in K, but certainly do not prevent that some solutions escape from K.
Hence results concerning the structure of the set of 'unconstrained' solutions gives no information about the structure of solutions surviving in K. Such problems have been addressed by Bader [9], Bader, Kryszewski [10] and Bothe [14,Thm. 5.2]. The important drawback is that in [9] there is an assumption int K = ∅, in both [9,10] F has compact values, while in [14, Thm. 5.2] F is H-upper semicontinuous and there is an extra assumption that E is uniformly convex.
Proof. In the proof the following characterization will be used (see [14,Lem. 5

.1.]):
If a decreasing family {X n } ∞ n=1 of closed contractible subset of a metric space is such that the Hausdorff measure of noncompactness β(X n ) → 0, then X 0 := ∞ n=1 X n is an R δ -set.
Step 1. Take a sequence (ε n ) n≥1 in (0, 1) such that ε n 0. Since K x T K (x) ⊂ E is lower semicontinuous we can apply Theorem 1.2: for every n ≥ 1, there is a Caratheódory f n : For each k ≥ 1, by a version of the Scorza-Dragoni theorem (cf. [8,43]), there is a closed subset I k ⊂ [0, T ] such that the Lebesgue measure µ([0, T ] \ I k ) ≤ min{ε m /2 k−m+1 | m = 1, ..., k} and the restriction f k I k ×K is continuous (with respect to both variables). Let I n := k≥n I k , n ≥ 1. The family {I n } increases, consists of compact sets and f n In×K is continuous.
Obviously, f n is continuous and for t ∈ [0, T ] and x ∈ K (below conv stands for the convex hull) For any n ≥ 1 we easily find a continuous α n : where and (20) holds, i.e., In view of (21), (A) and [12, Thm 1, Lemma 2 (b)], the probleṁ For any n ≥ 1, let X n be the set of mild solutions (in K) of the problem By (22) and (18), (19), g n (t, x) ∈ F n (t, x) on [0, T ] × K. Henceū n ∈ X n and thus X n = ∅. Clearly, Step 2. We shall see that given a sequence (u n ), where u n ∈ X n for n ≥ 1, then (up to a subsequence) u n → u 0 ∈ X 0 . To this end, observe that for each n ≥ 1 there is w n ∈ L 1 (0, T ; E) such that w n (t) ∈ F n (t, u n (t)) for a.e. t ∈ [0, T ] and Observe now that the set {χ n w n } n≥1 , where χ n is the indicator of I n , is integrably bounded. If t ∈ n≥1 I n , i.e., t ∈ I n for large n, say n ≥ N , then Hence up to a subsequence, χ n w n w 0 ∈ L 1 (0, T ; E) (weakly) and, hence, On the other hand In view of (24) and the 'convergence theorem' [6,Th. 3 The assertion we have just proved and (23) implies that sup v∈Xn d X0 (v) → 0. Arguing as above we show that X 0 is compact and, hence, the measure of noncompactness β(cl X n ) → 0 and X 0 = ∞ n=1 cl X n .
Step 3. Now we shall show that the closure cl X n is contractible. To see this, fix n ≥ 1 and recall the above constructed locally Lipschitz g n : [0, T ] × K → E being tangent to K and having the sublinear growth. Take z ∈ [0, T ] and y ∈ K. The problem Let us consider the homotopy h : cl where u ∈ cl X n , λ ∈ [0, 1] and s ∈ [0, T ]. It is easy to see that h is well-defined, continuous (comp. [14,Th. 5.1]) and h(X n × [0, 1]) ⊂ X n since g n is the selection of F n ; thus h(cl X n × [0, 1]) ⊂ cl X n . Furthermore, h(·, 0) = v(·; 0, u(0)) = v(·; 0, x 0 ) and h(·, 1) = id cl Xn proving the contractibility of cl X n .
Example 2. If we are in the setting of Subsection 1.2 and problem (2) then it follows that the set of all strong solution to (2) 3. Some topological tools. Let the solution map Σ : K C([0, T ], K) assign to x ∈ K the set of all solutions in K to (14) starting at x. For a fixed t ∈ [0, T ], the evaluation e t : C([0, T ], K) → K, e t (u) := u(t) for u ∈ C([0, T ], K), is well-defined and continuous. With (14) we associate the Poincaré t-operator Σ t : K K, By Theorem 2.1, Σ t is the superposition of a set-valued map Σ with R δ -values with a continuous map e t . It makes sense to discuss the class of such maps carefully and introduce appropriate tools to study their fixed points. Here, after [30], we recall some relevant concepts and results (comp. also [31]).
3.1. c -admissible maps. A compact metric space space S is cell-like if it can be represented as the intersection of a decreasing sequence of compact contractible spaces. The following conditions are equivalent (see e.g. [37,44]): S is cell-like; S has the shape of a point; S is an R δ -set; S has the U V ∞ -property, i.e. if S is embedded into an ANR, then it is contractible in any of its neighbourhoods. It is clear that cell-like sets are acyclic with respect to any continuous homology or cohomology theory, e.g. theČech homology, theČech or the Alexander-Spanier cohomology. Theorem 2.1 states that the set of all (mild) solutions to (14) starting at x 0 ∈ K and surviving in K is cell-like Let X, Y be metric spaces; an upper semicontinuous map ϕ : Properties of a c-admissible ϕ strongly depend on a decomposition ϕ = f • ψ or a pair (p, q) representing it. When studying c-admissible maps one has to take into account representing pairs (for a detailed discussion of c-admissible maps, related topics and some references -see [30]). In particular: Y is compact if and only if it is represented by a compact c-admissible pair.
Lemma 3.1. The solution map Σ is cell-like and maps bounded sets onto bounded ones.
Proof. The second assertion follows from the Gronwall inequality and (F 3 ). In view of Theorem 2.1, we only need to show that Σ is upper semicontinuous. Let x n → x ∈ K and u n ∈ Σ(x n ) for n ≥ 1. Then u n = S(·)x n +J 0 (w n ) for some w n ∈ L 1 (0, T ; E) such that w n (t) ∈ F (t, u n (t)) for a.e. t ∈ [0, T ]. (F 3 ) and the Gronwall inequality imply that {u n } n≥1 is bounded, so {w n } n≥1 is integrably bounded. As in Step 2 of the proof of Theorem 2.1, (up to a subsequence) u n − S(·)x n → u − S(·)x in C([0, T ], E). Thus, again up to a subsequence w n w ∈ L 1 (0, T ; E) and w(t) ∈ F (t, u(t)) for a.a. t ∈ [0, T ]. As a result, u = S(·)x + J 0 (w) ∈ Σ(x).
In what follows Σ will be identified with its canonical pair this means that Γ : (2) For any numbers 0 Observe, in particular, that Σ t and the pair (p t , q t ) are compact.
Then all above results remain true: for any x ∈ K, the set Σ(λ, x) of all mild solutions to this problem starting at x is an R δ -set; arguing as in Lemma 3.1 one shows that the solution map Σ : Λ × K C([0, T ], K) is cell-like. Hence, for all t ∈ [0, T ], the Poincaré map Σ t : Λ × K K given by Σ t (λ, x) = {u(t) | u ∈ Σ(λ, x)} is compact and c-admissible: it is represented by the pair where here Γ is the graph of Σ, p is the projection onto Λ × K and q t (λ, x, u) = u(t) for any (λ, x, u) ∈ Γ (i.e. u ∈ Σ(λ, x), λ ∈ Λ, x ∈ K).

Fixed point index for c-admissible maps. Given a compact c-admissible
) for x ∈ ∂V , the fixed point index Ind((p, q), V ) is well-defined (cf. [30,Th. 4.5]). This index has the usual properties such as: the existence (of fixed points), the localization, the additivity and the homotopy invariance (see [30]).
It is easy to get a generalization to a constrained case in a standard way. Let K ⊂ E be convex closed and let U ⊂ K be (relatively) open and bounded. Let r : E → K be an arbitrary retraction and j : K → E be the inclusion. Given a c-admissible compact pair K p ← − Γ q − → K such that x / ∈ q(p −1 (x)) for x ∈ ∂ K U (cl K U and ∂ K U denote the closure and the boundary of U in K, respectively), we let U r := r −1 (U ) ∩ B, where B is open bounded and B ⊃ U , Γ r := {(x, γ) ∈ cl U r × Γ | r(x) = p(γ)}, p r : Γ r → cl U r and q r : Γ r → E by p r (x, γ) := x and q r (x, γ) := j •q(γ) for (x, γ) ∈ Γ r . Note that q r •p −1 r = j •q •p −1 •r cl Ur : cl U r E, the pair (p r , q r ) is c-admissible and x ∈ q r (p −1 r (x)) for x ∈ ∂U r . Thus, we are in a position to define the constrained fixed point index by Ind K ((p, q), U ) := Ind((p r , q r ), U r ).
It is easy to see that this definition is correct, i.e. it does not depend on the choice of r and B; furthermore Ind K has the same properties as Ind does.
(ii) If a c-admissible pair (p, q) represents a compact single-valued f : K → K (that is f (x) = q(p −1 (x)) on K) and x = f (x) for x ∈ ∂U , then it can be proved (see [30]) that Ind K ((p, q), U ) = Ind K (f, U ), where Ind K (f, U ) stands for the fixed point index as defined in [33, §12]. Observe that, in particular, f is represented by (iii) If Ind K ((p, q), U ) = 0 then the map ϕ represented by (p, q) has a fixed point, i.e. there is x ∈ U such that x ∈ q(p −1 (x)). If U = K is bounded then any cadmissible pair (p, q) is c-homotopic to the constant map f (·) ≡ u 0 ∈ K. Hence Ind K ((p, q), U ) = 1 the existence of fixed points follows.
3.3. The degree of the right hand side. We will construct a homotopy invariant (the so-called constrained topological degree) responsible for the existence of zeros of maps of the form A + G, where: E is H-upper semicontinuous, has convex weakly compact values, maps bounded sets onto bounded ones and G(x) ∩ T K (x) = ∅ for every x ∈ K, i.e. G is tangent to K; (G 2 ) K ⊂ E is convex closed; A : D(A) → E satisfies (A) and (K).
Let U ⊂ K be bounded and relatively open in K. We assume that here ∂U = ∂ K U stands for the boundary of U relative to K.
Proof. Fix h > 0 with hω < 1 and suppose to the contrary that for n ≥ 1 there is x n ∈ D(A) ∩ ∂U , y n ∈ G(x n ), where x n −x n < 1/n and ξ n ∈ E with ξ n < 1/n such that 0 = Ax n + y n + ξ n ⇐⇒ x n = J h (x n + h(y n + ξ n )).

.4]).
Lemma 3.4. Assume that 0 < α ≤ α 0 and g : K → E is a continuous and tangent Proof. If not then for each n ≥ 1 there is x n ∈ ∂U such that x n = r(u n ), where u n := J hn (x n + h n g(x n )) ∈ D(A) and 0 < h n < 1/n with h n ω < 1. For each n ≥ 1 we have u n − r(u n ) = u n − x n = h n (Au n + g(x n )) and, recalling that J h (x n ) ∈ K, Hence the sequence (Au n ) is bounded since so is (g(x n )). Clearly, (u n ) is bounded, too. Therefore, by Remark 3 (a), (up to a subsequence) u n → u 0 . Thus, x n = r(u n ) → r(u 0 ) and, by (29), u 0 = r(u 0 ) ∈ K. In view of Lemma 3.3, we see that This implies that Au n → −g(u 0 ) and since A is closed we have u 0 ∈ D(A) ∩ ∂U and Au 0 = −g(u 0 ). This contradicts Lemma 3.2.
Let α ∈ (0, α 0 ] and h ∈ (0, h 0 ] as given in above Lemmas. By Lemma 1.1, there is a locally Lipschitz g : K → E being an α-approximation of G tangent to K. Then f is compact and, by Lemma 3.4, x = f (x) for x ∈ ∂U . Thus, the fixed point index Ind K (f, U ) is well-defined (see again [33, §12]).
Lemma 3.5. The number Ind K (f, U ) does not depend on the choice of a sufficiently close approximation g, a retraction r and sufficiently small h > 0.
Thus, we are in a position to define the degree deg K by where g : K → E is a tangent and sufficiently close locally Lipschitz approximation of G.
The remaining assertions are standard and left to the reader. 4. The Krasnosel'skii type formula. In this section we will prove the following counterpart of the Krasnosel'skii formula (3) by showing a relation between the constrained degree of the operator A + F (0, ·) in the right-hand side of (14) and the fixed point index of the Poincaré operator Σ t (with sufficiently small t > 0) associated to (14); see (25), (27).
Proof. The long proof of Theorem 4.1 will be presented in a series of steps and auxiliary lemmas.  Proof. We shall show that [0, T ] × K (t, x) F ([0, t], x) ⊂ E is H-upper semicontinuous; then the assertion will follow easily. Take t 0 ∈ [0, T ], x 0 ∈ K and ε > 0. There is δ 0 > 0 such that if t ∈ [0, T ] and |t − t 0 | < δ 0 then For every 0 ≤ t ≤ t 0 + δ 0 /2 there is δ(t) > 0 such that provided s ∈ [0, T ], |s − t| < δ(t) and x ∈ B K (x 0 , δ(t)). There are t i , i = 1, ..., k such that [0, 4 Since F is H-upper semicontinuous, then, arguing as in Remark 2 (b), F is product measurable, i.e. (33) implies (F 2 ) and (F 3 ). Note also that if A is defined by the Dirichlet (or Neumann) laplacian then assumption (32) is fulfilled. 5 Here while if t i > t 0 , then t i − t 0 ≤ δ 0 /2 and, by (34), Using the same methods as in Lemma 3.2 we get: Step 2. By Lemma 1.1, there is a locally Lipschitz map f : K → E being a tangent to K α-approximation of F (0, ·). Obviously, there is a constant c > 0 such that for all u ∈ K f (u) ≤ c(1 + u ).
Hence arguing as in Lemma 3.4, we find h 0 > 0 with h 0 ω < 1 such that where r : E → K is a retraction. Since E is the Hilbert space, without loss of generality we may assume that r is a metric projection, i.e., Observe that, by definition (see (31)), Define the auxiliary set-valued map G : [0, 1] × K E by the formula Obviously, G is H-upper semicontinuous, tangent to K, has sublinear growth and convex weakly compact values. By Theorem 2.1 (see also Remark 6), for any z ∈ [0, 1] the set of all (mild) solutions tȯ starting at x ∈ K is an R δ -set.
To prove that the set {u n } n≥1 is relatively compact it is enough to show that so is {x n } n≥1 (cf. Remark 3 (b)). Take τ 0 ∈ (0, τ ) and put k n := [τ 0 /t n ] + 1. Then r n := k n t n − τ 0 → 0 and u n (k n t n ) = u n (τ 0 + r n ) = x n for all n. For sufficiently large n ≥ 1: τ 0 + r n < τ and In other words, x n ∈ Σ τ0+rn (x n ) for all n ≥ 1 and, by Remark 5 (2) {x n } is indeed relatively compact and so, up to a subsequence, x n → x 0 ∈ ∂U . Thus, again for a subsequence, u n → u 0 ∈ C([0, τ ], K) and, by the uniform equicontinuity of {u n } and w 0 (s) ∈ G(z 0 , x 0 ) for a.e. s ∈ [0, τ ]. We will show that this implies that t 0 w 0 (s) ds = w 0 (ξ). By (41) for small η > 0 we have We shall show that To this end take p ∈ E * and ε > 0. Then The dual semigroup {S * (t)} t≥0 is strongly continuous since E is reflexive. Thus, there is δ > 0 such that S * (t)p − p < εC −1 , if 0 ≤ t < δ, where C := sup y∈G(z0,x0) y . If 0 < η < δ, then for a.a. ξ ≤ s ≤ ξ + η, what proves (42). As a result, In view of [48,Th. 2.1.3], x 0 ∈ D(A) ∩ ∂U and Ax 0 = −w 0 (ξ). Hence 0 = Ax 0 + w 0 (ξ) ∈ Ax 0 + G(z 0 , x 0 ). This is a contradiction with Lemma 4.3, since Step 3. Recall the solution operator Σ : K C([0, T ], K) (see (26)) and the t-Poincaré operator Σ t : K K associated with (14) (see (26) and (27)) and consider their restrictions to cl U . By a slight abuse of notation, we will still denote these restrictions by the same symbols, i.e. Σ : Taking into account (38) and Remark 7, we are to show that for sufficiently small t > 0, h > 0, the c-admissible pairs (p t , q t ) and (id, r • J h (I + hf )), where id = id cl U stands for the identity on cl U , are c-homotopic via a compact c-homotopy without fixed points on the boundary ∂U . This will be done in two stages.
Stage 1. For any x ∈ cl U , the problem It is clear that Φ is a cell-like map (cf. Remark 6). Fix t ∈ (0, T ] and consider the Poincaré t-operator Φ t : As before Φ t is compact and c-admissible. If u ∈ Φ(z, x), for some z ∈ [0, 1] and x ∈ K, then u [0,t0] is also the solution of the problem (39) on the segment [0, t 0 ]. Hence, by Lemma 4.4, Clearly, Φ(1, ·) = Σ and Φ(0, ·) = P ; hence Φ t (1, ·) = Σ t and Φ t (0, ·) = P t on cl U . Therefore, the canonical pair (p Φ , q Φ ) representing Φ is the c-homotopy joining (p Σ , q Σ ) to the canonical pair representing P . At the same time, the pair (p Φ , e t •q Φ ) representing Φ t is a c-homotopy joining (p t , q t ) to (id cl U , P t ). We have thus shown that: , where t 0 is given by Lemma 4.4, then the pairs (p t , q t ) and (id cl U , P t ) are c-homotopic via the compact c-homotopy without fixed points on ∂U .
Stage 2. We are going now to establish the following Proposition 2. There are 0 < t 1 ≤ t 0 and h 1 ∈ (0, h 0 ], where h 0 was chosen at the beginning of Step 2 (see (37)), such that for t ∈ (0, t 1 ] and h ∈ (0, h 1 ], the maps P t and r • J h (I + hf ), see (38), are homotopic via a compact homotopy without fixed points on ∂U .
Proof. Claim 1. First we shall show that for sufficiently small h ∈ (0, h 0 ] and t > 0 the Poincaré t-operator P t associated to (43) and the Poincaré operator associated to the parameterized problemu are homotopic via a β-contracting homotopy without fixed points on ∂U ( 6 ).
To this end, fix h 1 ∈ (0, h 0 ] such that take h ∈ (0, h 1 ] and consider a parameterized semilinear probleṁ where g z : K → E is defined by For each z ∈ [0, 1], g z is locally Lipschitz, since so are f and r (recall that r, as the metric projection onto the convex K, is nonexpansive). Moreover, for any , for x ∈ K; and, hence, g z (x) ∈ T K (x), for x ∈ K. It is easy to see that, for each z ∈ [0, 1], g z has sublinear growth and the semigroup {S(zt)} t≥0 generated by zA leaves the set K invariant. Thus, in view of Remark 3 (d) and Theorem 2.1, for any z ∈ [0, 1], x ∈ K, the problem (47) (and, in particular, (45)) along with the initial condition u(0) = x has a unique mild solution Θ(z, x) : [0, T ] → K. Obviously, Θ(0, x) (resp. Θ (1, x)) is the solution to (45) (resp. (43)) starting at x. Note that Θ implicitly depends on h, too. To see that the map with sufficiently small t > 0 and h, is the required homotopy joining the Poincaré t-operators associated to problems of (45) and (43) we need to consider a different form of (47). Namely, following [21], we shall consider the following families {A z : D(A) → E} z∈[0,1] and {f z : K → E} z∈[0,1] of linear operators and maps defined for z ∈ [0, 1] by where It is easy to see that In order to establish the necessary properties of Θ t with small t > 0 we need to collect some facts about the family {A z } z∈[0,1] .
We are now ready to collect properties of Θ t .
The resolvent continuity together with compactness of f 2 implies that the set z∈[0,1] (1 − z)B 2 (z) is relatively compact. The sets B 1 (z) are uniformly bounded, i.e. there is R > 0 such that for any z ∈ [0, 1], y ≤ R for y ∈ B 1 (z).
We have shown that for any t ∈ [0, T ], Θ t is a β-contraction with constant e −t , i.e., for a bounded B ⊂ K, Observe that the contraction constant depends on t but it does not depend on h.

(No fixed points on the boundary)
We claim that there is h 1 ∈ (0, h 1 ] and t 1 > 0 such that for h ∈ (0, h 1 ] and t ∈ (0, t 1 ] Suppose to the contrary and fix a small h. There are sequences (x n ) in ∂U , (z n ) in [0, 1] and t n → 0 + such that x n = Θ tn (z n , x n ). Take t ∈ [0, T ), then x n = Θ t+rn (z n , x n ), where r n := ([t/t n ]+1)t n −t (note that t+r n < T for large n). In view of (54) and the continuity of Θ, {x n } is relatively compact. Taking subsequences if necessary, x n → x h ∈ ∂U , z n → z h ∈ [0, 1], we see that u n := Θ(z n , x n ) → u h := Θ(z h , x h ) (remember that the sequences (x n ), (z n ) as well as their existing limiting points depend on h). Having this and proceeding as in the proof of Lemma 4.4, we show that u h is constantly equal to Let Hence, by (48), (49), the equality (56) reads where If z h = 0 then h = 0 and we have x h = r(J h (y h )). Therefore, putting u h := J h (y h ), u h ∈ D(A) and If z h > 0, then h > 0. Inverting I − h A in (57), we get by (30) that Therefore, Now let h → 0 + . Arguing as in the proof of Lemma 3.4, we see that quantities (59) are bounded; hence also Au h is bounded and so is u h . By Remark 3 (a), we gather that (up to a subsequence) u h → x 0 and x h → x 0 ∈ D(A) ∩ ∂U . Hence, by Lemma 3.3, we see (58) and (59), (36).
Indeed, for a fixed t ∈ [0, T ] and for x ∈ cl U , z ∈ [0, 1] let As in [21,Prop. 4.3], one shows that Ψ t is continuous and there is t 1 ∈ (0, T ), t 1 < t 1 , such that Ψ t is β-contracting and Ψ t (z, x) = x for z ∈ [0, 1], x ∈ ∂U and t ∈ (0, t 1 ]. Take 0 < t ≤ t 1 and let Then Ψ t is continuous and β-contracting as the superposition of Ψ t with the nonexpansive retraction r. To see that for all z ∈ [0, 1], Ψ(z, ·) has no fixed points on ∂U , we shall make use of the following general observation. Proof. There are h 0 > 0 and k 0 ∈ K such that y = x + h 0 (k 0 − x). The so-called variational characterization of r (see e.g. [15,Th. 5.2]) yields that for all k ∈ K, k − r(y), y − r(y) Thus, k 0 = x and y = x.
Observe that Ψ t (0, x) = r • g 0 (x) = r • J h (x + hf (x)) and, in view of (37), For such z and x, by Lemma 4.6, Connecting β-contracting homotopies provided by Claim 1 and Claim 2 we get the β-contracting homotopy H joining r • J h (I + hf ) to P t when t > 0 and h > 0 are sufficiently small. Claim 3. There is a compact homotopy joining r • J h (I + hf ) to P t without fixed points on ∂U .
To this end we will rely on the following general result. This establishes Proposition 2, since Lemma 4.7 produces a compact homotopy out of H (recall that r • J h (I + hf ) and P t are compact).
To sum up, during the proof we have shown that for sufficiently small t > 0 and h > 0: 1. the c-admissible pair (p t , q t ) is c-homotopic to the pair (id cl U , P t ) via the compact c-homotopy without fixed point on ∂U (cf. Lemma 4.5); 2. the Poincaré t-operator P t : cl U → K is homotopic to r•J h (I+hf ) : cl U → K via the compact homotopy without fixed points on ∂U (cf. Proposition 2). Thus, in view of Remark 7 (ii) and (38), we have This concludes the proof of Theorem 4.1.
Let us finally formulate a direct single-valued counterpart of this result being a direct generalization of [21,Thm 4.5].
Corollary 1. Assume that A and U are the same as in Theorem 4.1. Additionally, let f : [0, 1] × K → E be tangent to K locally Lipschitz function with sublinear growth. If 0 = Ax + f (0, x), x ∈ ∂U , then there is t 0 ∈ (0, 1] such that for every t ∈ (0, t 0 ] where P t : cl U → K is the Poincaré t-operator associated with the problemu = Au + f (t, u).

Periodic solutions.
It is common to establish the existence of periodic solutions to different type of ODE's by applying the fixed point theory to the associated Poincaré time map. Such approach, in the case of a semilinear constrained evolution equation has been presented by e.g. Prüss [49] and, for the evolution inclusion of the form (14) by Bothe [13] (see also [14]), Bader [9], [10]. Fully nonlinear evolution equations/inclusions (also including m-dissipative 'diffusion' A), in general 'unconstrained', have been studied by numerous authors and many interesting results were obtained, see for instance Vrabie [52], Hirano et al. [34], Bothe [14],Ćwiszewski [21] and monographs [36], [39]. The quite recent paper of Aizicovici et al. [1] gives a nice historical overview of the topic. We would like to show how the theory developed in the present paper may be used in the study of periodic orbits. We shall show some results that follow immediately from was done above. They correspond directly to results of [9,10] and [52]; in these papers, however, either assumptions concerning the constraining set or the nonlinearity are stronger then ours. Proof. It is clear that if x ∈ K and x ∈ Σ T (x) then there is a solution u : [0, T ] → K of (14) such that x = u(0) = u(T ). Extending u periodically onto R one gets a periodic solution. Since K is closed convex and bounded, Σ T : K K is a compact c-admissible map (i.e., cl Σ T (K) is compact). In view of the Schauder fixed point theorem for admissible maps (see [31,Theorem (41.13)]), Σ T has a fixed point. Equivalently, one may use Remark 7 (iii).
In what follows we let K be not necessarily bounded. We begin with the following generalization of [9, Cor. 11].
Theorem 5.1. Assume that K = E and, in addition to hypotheses of Proposition 3, A satisfies condition (32). If lim sup uniformly for t ∈ [0, T ], where J : E E is the duality map (see e.g. [51]), then (14) has a periodic solution.
Proof. In view of (60), there is R > 0 such that if x ∈ E with x ≥ R then for any t ∈ [0, T ] there is y = y(t, x) ∈ F (t, x) such that for all p ∈ J(x), y, p ≤ −ω x 2 .
Consider an auxiliary problemu It is clear that G has the same properties as F does. Moreover, for any x ∈ E, the set of all solutions to (61) coincides with Σ(x). Let t ∈ [0, T ], x ∈ E and x = R. If z = y(t, x) + ωx then ∀ p ∈ J(x) z, p = y, p + ω x 2 ≤ 0.
Let ξ(x) := 1 2 x 2 , x ∈ E. It is well-known that J(x) = ∂ξ(x), the subdifferential of ξ at x. In view of [6,Prop. 7.3.16], and, thus, (62) implies that z ∈ T D (x), where D := D(0, R). Hence G(t, x) ∩ T D (x) = ∅. The semigroup generated by B is of the form {e −ωt S(t)} t≥0 . In view of (32), this semigroup consists of contractions; hence D is invariant with respect to this semigroup. The assertion follows from Proposition 3 after replacing K by D.
Remark 8. (i) In [52], the author studies (14) with a general accretive A and single-valued F . Specifying for a closed densely defined linear A, the main results asserts the existence of a periodic solution provided there is ω > 0 such that A − ωI is accretive and It is easy to see that this condition implies (60). Hence Theorem 5.1 may be considered as a (partial) generalization of [52, Thm. 1].
(ii) The result stated in Theorem 5.1 stays true if K E, ω ≤ 0 in (32) and for sufficiently large R > 0 In this case, one does not introduce B and G but shows that F (t, x) ∩ T D∩K (x) for any x ∈ D ∩ K, where D := D(0, R) with sufficiently large R > 0. For large R, K ∩ B(0, R) = ∅ and T K (u) ∩ T D (u) = T K∩D (u), in view of [6,Cor. 4.1.19]. Therefore, Moreover, D is semigroup invariant since it consists of contractions. The assertion follows from Corollary 3 after replacing K by K ∩ D.
(iii) Finally, observe that if E * is uniformly convex, then J is single-valued and the result stated in (ii) is a straightforward generalization of the (semilinear) version of Theorem 5 in [1].
In order to further deal with unbounded K we need to collect some additional facts concerning compact C 0 semigroups.
Then, for any t ≥ 0, the spectral radius of r(U (t)) of U (t) is given by where ω 0 (B) := lim t→∞ Assume (A), (K) and suppose now that 0 ∈ K and K 0 := T K (0) is the tangent cone to K at 0. Then E 0 := cl (K 0 − K 0 ) is the minimal closed subspace containing K and the so-called blade of K 0 , i.e. b(K 0 ) := K 0 ∩ (−K 0 ) is the maximal closed subspace contained in K 0 . The semigroup invariance of K implies that that E 0 and b(K 0 ) are invariant, too. Therefore, the restrictions S 0 (t) := S(t) E0 , t ≥ 0, form the compact C 0 semigroup generated by the restriction A 0 : [28,Sections I.5.12,II.2.3]. Observe that since 0 ∈ K, we have K ⊂ K 0 and S(t)x = S 0 (t)x for t ≥ 0 and x ∈ K. In view of [28, Sections I.5.13, II.2.3], the quotient operators S 0 (t) : Since, for every t ≥ 0, S 0 (t) ≤ S 0 (t) ≤ S(t) , the growth bounds of A 0 , A 0 and A are related as follows Theorem 5.2. Assume (A), (K), 0 ∈ K, the growth bound ω 0 ( A 0 ) < 0 and that no number of the form 2πikT −1 , where k ∈ N, belongs to the spectrum of A 0 . If F satisfies (F 1 ), (F 2 ), (F 4 ) and, instead of (F 3 ), F is bounded, i.e. there is c > 0 such that sup y∈F (t,x) y ≤ c, for all t ∈ [0, T ] and x ∈ K, then (14) has a periodic solution.
Proof. There is R > 0 such that, for all t ∈ [0, T ] and x ∈ K and y ∈ F (t, x), if x ≥ R then y ≤ γ x . As before, we are going to establish a priori bounds for fixed point of Σ λ T (see (66)). Assume to the contrary that there are no such bounds, i.e. for each n ∈ N there is x n ∈ K, x n ∈ Σ T (x n ) with x n ≥ n; there also exists u n ∈ Σ(x n ) such that x n = u n (0) = u n (T ).
We claim that for sufficiently large n, n ≥ n 0 say, inf t∈[0,T ] u n (t) ≥ R. If not then (up to a subsequence) for any n there is t n ∈ (0, T ) such that u n (t n ) = R and u n (t) ≥ R for t ∈ [t n , T ]. For such t where w n (s) ∈ F (s, u n (s)) for a.a. s ∈ [t n , t]. Since w n (s) ≤ γ u n (s) , the Gronwall inequality implies that u n (t) ≤ Re γT for t ∈ [t n , T ]. In particular, n ≤ u n (T ) ≤ Re γT for all n: a contradiction.
For n ≥ n 0 let y n := T 0 S(T − s)w n (s) ds. As before, we show that w n (s) ≤ γ u n (s) ≤ γ x n e γT .
Hence y n ≤ γ x n e γT 1 − e ωT −ω . Therefore, a contradiction. Now, we proceed as in the last part of the proof of Theorem 5.2.
Example 3. If, for example, we are within the setting of Subsection 1.2 for (2) subject to the Dirichlet boundary condition then ω 0 (A) < 0. Assume that 0 ∈ C. Then problem (7) has a periodic mild solution, i.e.
(2) has a T -periodic strong solution provided C is bounded. The same holds if C is not bounded, but ϕ is. If in (ϕ 2 ) we have that α ∈ L ∞ ([0, T ] × Ω) and e β β < λ 1 , where λ 1 is the first eigenvalue of the Dirichlet Laplacian −∆, then the existence of periodic solutions to (2) follows from Theorem 5.3 since, by Remark 9 (2), the growth bound of ∆ equals −λ 1 .

5.1.
The guiding function approach. One of the most powerful tools to establish the existence of periodic solution to systems of ODE's or differential inclusions in R n relies on the use of the so-called guiding potential. This method, started by Krasnosel'skii, is described in a recent monograph [45]. In this section we shall try to show how the guiding function approach may be applied in the context of partial differential equations and inclusions. We assume again that E is a separable Hilbert space, conditions (A), (K) hold true with K unbounded and let F : [0, T ] × K E satisfy conditions (F 1 ), (F 4 ) and (33). In order to establish the existence of (strong) periodic solutions, we shall assume additionally that: (A 1 ) −A is determined by the so-called elliptic quadratic form a(·, ·) (see [4,Thm. 4.3 and eq. (4.1)]); (V 1 ) there is a C 1 -functional V : E → R such that V is bounded on bounded sets, V (u) → ∞ as u → ∞; (V 2 ) there is R 0 > 0 and > ε 0 such that for u ∈ K ∩ D(A), u ≥ R 0 and t ∈ [0, T ], sup v∈F (t,u) ∇V (u), Au + v ≤ −. ε Remark 10. (a) Assumption (A 1 ) is fairly general. For instance, as discussed in Remark 2 (d), the Dirichlet (or Neumann) Laplacian satisfy this condition. Observe that in this case A − ωI, for some ω is m-dissipative. Hence, in view of the Lumer theorem (see [28,Thm. II.3.15]), assumption (32) is satisfied. Therefore, in the setting of (2) given in section 1.2 all assumptions stated above are satisfied. (b) Suppose (A 1 ). If u is a mild solution to (14) then it is a mild solution tou = (A − ωI)u + w, where w(s) ∈ F (s, u(s)) + ωu(s) on [0, T ]. In view of (33), w ∈ L 2 (0, T ; E). Since A − ωI is the subdifferential of the quadratic function determined by the form a(·, ·)+ω ·, · (see [4,Rem. 4.4]), by [16,Thm. 3.6] (see also [51,Thm.1.9.3]), we gather that u is a strong solution to this problem, in particular u(t) ∈ D(A) for 0 < t ≤ T , u ∈ W 1,1 loc ((0, T ], E) and u (t) ∈ Au(t) + F (t, u(t)) a.e. on (0, T ]. (b) If E is of finite dimension, K = E and A ≡ 0, then (V 2 ) means the −V is a strict guiding function for (14) (see [45,Definition 2.3]).
Theorem 5.4. Under the above assumptions, (14) has a periodic solution, i.e. there is a solution u to (14) such that u(0) = u(T ) and u(t) ∈ K for t ∈ [0, T ].
This, of course, as in the proof of Theorem 5.1, will conclude the argument and show that (14) has a periodic solution. If x ∈ D(A) ∩ K and x = R then 0 ∈ Ax + F (0, x) in view of (V 2 ). Hence, by Theorem 4.1, there is a small t 0 ∈ (0, T ] such that Ind K ((p t0 , q t0 ), B K (0, R)) is well-defined and equal to deg K (A + F (0, ·), B K (0, R)).
Observe that Φ t is well-defined for all t ≥ 0 since for x ∈ K solutions u to (70) starting at x and living in K are defined on the whole half-line [0, +∞)), i.e. for x ∈ K, Φ t (x) = {u(t) | u solves (70), u(s) ∈ K for all s ≥ 0, u(0) = x}.
and ends the proof of (68).