Krasnosel'skii type formula and translation along trajectories method for evolution equations

The Krasnosel'skii type degree formula for the equation $\dot u = - Au + F(u)$ where $A:D(A)\to E$ is a linear operator on a separable Banach space $E$ such that $-A$ is a generator of a $C_0$ semigroup of bounsed linear operators of $E$ and $F:E\to E$ is a locally Lipschitz $k$-set contraction, is provided. Precisely, it is shown that if $V$ is an open bounded subset of $E$ such that $0\not\in (-A+F)(\partial V \cap D(A))$, then the topological degree of $-A+F$ with respect to $V$ is equal to the fixed point index of the operator of translation along trajectories for sufficiently small positive time. The obtained degree formula is crucial for the method of translation along trajctories. It is applied to the nonautonomous periodic problem and an average principle is derived. As an application a first order system of partial differential equations is considered.


Introduction
We are concerned with the periodic problem associated with a differential equation of the form where A : D(A) → E is a linear operator on a separable Banach space such that −A generates a C 0 semigroup {S A (t) : E → E} t≥0 of bounded linear operators such that S A (t) ≤ e −ωt , for some fixed ω > 0 and any t ≥ 0, and F : [0, T ]×E → E is a locally Lipschitz map with respect to the second variable, having sublinear growth and with k ∈ [0, ω) such that, for any bounded Ω ⊂ E, β(F (Ω)) ≤ kβ(Ω) where β stands for the Hausdorff measure of noncompactness.
In general, one may speak of two topological approaches to the T -periodic problem associated with (P A,F ) (T > 0 is a fixed period). One is to formulate it as a fixed point problem for a proper solution operator in a suitable space of functions (see e.g. [10]) and the second way for finding T -periodic solutions, which we follow, is to seek fixed points of the translation along trajectory operator Φ T : E → E associated with the equation (P A,F ) (see e.g. [4] and [3]). The crucial point in our approach is the proper version of a degree formula stating that, if F is time-independent, then the fixed point index of the operator Φ t : E → E, for small t > 0, is equal to the topological degree of −A + F (with respect to proper open bounded subsets of E). We shall prove such a degree formula, which is an infinite dimensional extension of the Krasnosel'skii theorem for ordinary differential equations in finite dimensional spaces (see [12]). By use of the obtained formula, we prove a criterion for finding periodic solutions of (P A,F ) in the spirit of Mawhin ([14]), Schiaffino-Schmitt ( [17]) and Kamenskii-Obukhovskii-Zecca ( [10]). The obtained abstract method is applied to a system of partial differential equations. The paper is organized as follows. In Section 2 we provide basic definitions and facts concerning the topological degrees for k-set contractions and perturbations of operators generating C 0 semigroups of contractions. In Section 3, we prove the degree formula, which is to be used in one of the next sections, for the the equationu = −u + F (u), where F is a k-set contraction. Section 4 is devoted to general compactness properties of solution operators foru = −Au + F (t, u) with initial conditions. In Section 5 we prove the main result of the paper -the degree formula. We reduce it, by a proper homotopy, to the result of Section 3. Finally, Section 6 provides an abstract continuation principle for periodic solutions and its application.

Topological degree
In the first part of this section we provide basic information on the degree theory for k-set contraction, mainly due to [15]. In the second part of the section we carry out a standard construction of the degree for perturbations of generators of C 0 semigroups of contractions. For a survey on the topological degree theory for perturbations of accretive operators we refer to [11].
Recall that a continuous map F : X → E, where X is a closed bounded subset of a Banach space E, is called a k-set contraction if there exists a constant k ∈ [0, 1) such that, for any Ω ⊂ X, β(F (Ω)) ≤ kβ (Ω) where β stands for the Hausdorff measure of noncompactness given by β(Ω) := inf{r > 0 | Ω can be covered with a finite number of balls (in E) of radius r} (see e.g. [1], [7] or [10]). A compact convex set C ⊂ E is said to be a fundamental set (for F ) provided F (X ∩ C) ⊂ C and, for any x ∈ X, the condition x ∈ conv ({F (x)} ∪ C) implies x ∈ C.
Remark 2.1 Let F : X → E be as above and define (C n ) n≥0 a sequence by C 0 := conv ({x 0 } ∪ F (X)), C n := conv ({x 0 } ∪ F (X ∩ C n−1 )), for n ≥ 1 where x 0 is an arbitrary point from X. Then, one can verify that, C n ⊂ C n−1 , for n ≥ 1, and C := ∞ n=1 C n is a fundamental set for F .
A vector field I − F : U → E, where U ⊂ E is an open bounded set, is called admissible if F is a k-set contraction (with some k ∈ [0, 1)) and F has no fixed points in the boundary ∂U. Let C be any fundamental set for F and F C : U → E be a continuous extension of F |U∩C such that F C (U) ⊂ conv F (U ∩ C), existing due to the Dugundji extension theorem (see [6]). Then, by the invariance of C, F C (U ) ⊂ C, which implies that F C is compact. Moreover, if, for some x ∈ U , F C (x) = x, then x ∈ C, which gives F (x) = F C (x) = x and x ∈ U. Therefore the following definition of the topological degree makes sense where deg LS stands for the Leray-Schauder degree (with respect to 0 ∈ E). One may prove that the degree is independent of the choice of a fundamental set and an extension F C . It has the following standard properties of a topological degree: (D3) Let H : U × [0, 1] → E be an admissible homotopy, i.e. a continuous map such that (i) there exists k ∈ [0, 1) such that, for any Ω ⊂ U , Remark 2.2 It will be useful in the sequel to know that if I − F : U → E is an admissible k-set contraction vector field, then there exists a locally Lipschitz compact vector field I − F L : U → E homotopic to I − F via an admissible homotopy. Indeed, find F C as in (1). Since F is a k-set contraction, its fixed point set is compact and one has ρ := inf{ x − F (x) | x ∈ ∂U} > 0. By the Lasota-Yorke theorem there exists a locally Lipschitz map F L : Clearly, for any Ω ⊂ U , one gets The topological degree can be extended onto the class of condensing vector fields. For any admissible condensing vector field I − F : Indeed, if we suppose to the contrary, then there are λ n → 0 + and (x n ) ⊂ ∂U such that x n = (1 − λ n )F (x n ), for each n ≥ 1. Then, either β({x n } n≥1 ) = 0 or we note that But the latter case gives a contradiction and, therefore, (x n ) is relatively compact. Assume that x n → x 0 ∈ ∂U. By continuity, it is clear that x 0 = F (x 0 ), a contradiction with the admissibility of I − F , and the proof of (2) is finished.
We shall now pass to the version of topological degree for perturbations of generators of C 0 semigroups of contractions. We say that a mapping −A + F By an admissible homotopy we shall understand a mapping • for any λ ∈ [0, 1], 0 ∈ (−A(λ) + F (·, λ))(∂U ∩ D(A(λ))).
Therefore, if −A + F is admissible with respect to an open bounded U ⊂ E, then, by Lemma 2.4, we may put properly where ν ≥ 0 is arbitrary. By the continuity of the resolvent and the homotopy invariance of deg it is clear that the definition (4) is independent of ν. It has also all the expected properties, i.e. the existence, additivity, homotopy invariance and normalization. The latter property states that: if v 0 ∈ A(U ∩ D(A)), then Deg(−A + v 0 , U) = 1. It is a straightforward conclusion from (4) and (D4).
To define the topological degree of −A+F in the situation k = ω it is sufficient to apply in (4) the topological degree for condensing vector fields.
3 Degree formula for k-set contracting and condensing fields Before we shall deal with the degree formula, we verify basic properties of the translation along trajectories operator associated with the parameterized problem with an open bounded subset U of a separable Banach space E, satisfy the following conditions (G1) G is locally Lipschitz in the first variable uniformly with respect to the second one, i.e. for any x ∈ U there exist δ x > 0 and L x > 0 such that, for any x 1 , x 2 ∈ B(x, δ x ) ∩ U and λ ∈ [0, 1], (G3) G is a k-set contraction, i.e. there is k ∈ [0, 1) such that, for any bounded Ω ⊂ U, where u(·, 0, x, λ) stands for the (local) solution of (P G,λ ) with the initial condition u(0) = x.
Proposition 3.1 Under the above assumptions (i) there exists t > 0 such that, for any x ∈ V and λ ∈ [0, 1], there exists a unique solution u(·; 0, t, x, λ) of (P G,λ ) on [0, t] starting at x and the map In the proof we shall use the following property of the measure of noncompactness.
Proposition 3.2 (see [7] or [10]) Let E be a separable Banach space, Proof of Proposition 3.1. (i) By the standard local existence and uniqueness theorem for ordinary differential equations in a Banach space, we infer that, for any x ∈ V and λ ∈ [0, 1], there exists a unique solution u of (P G,λ ) on some This means that if t ∈ [0, ρ/(4K)] and t < ω x , then u(t) ∈ V + D(0, ρ/2) ⊂ U, which means that ω x > t := ρ/(4K). The continuity of (x, λ) → u(·; 0, t, x, λ) is a classical property of parameterized ordinary differential equations in Banach spaces. The proof of the part (i) is completed.
We pass now to the first degree formula for the equation Then there exists t > 0 such that, for any t ∈ (0, t], (i) the translation along trajectories operator for Proof. We reduce the proof to the case when F is a compact map (see e.g. [5]). Due to Remark 2.2, there is a locally Lipschitz compact map F L : V → E homotopic to F via an admissible homotopy H : Consider the parameterized problem given by Obviously, H(·, λ) is locally Lipschitz and, by Proposition 3.1 (i), there exists t 1 > 0 such that, for any x ∈ V and λ ∈ [0, 1], (P H,λ ) admits a solution on [0, where u(·; 0, t 1 , x, λ) stands for the solution of (P H,λ ) with the initial condition u(0) = x. By Proposition 3.1 (ii), for any Ω ⊂ V , Now we claim that there exists t 2 ∈ (0, t 1 ] such that, for any t ∈ (0, t 2 ], Indeed, suppose to the contrary. Then there exist sequences t n → 0 + , (λ n ) ⊂ [0, 1] and (x n ) ⊂ ∂V such that Ψ tn (x n , λ n ) = x n for each n ≥ 1. Note that this implies Ψ jtn (λ n , x n ) = x n for integer j > 0 such that jt n ≤ t 1 and, in particular, which means that β({x n } n≥1 ) = 0. Hence, without loss of generality, one may assume that x n → x 0 ∈ ∂V and λ n → λ 0 ∈ [0, 1]. Denote u n := u(·; 0, t 1 , x n , λ n ), for n ≥ 0, and observe that using the t n -periodicity of u n , for any t ∈ [0, Furthermore, by Proposition 3.1 (i), u n → u 0 in C([0, t 1 , E]) and, this together with (8) provides u 0 (t) = x 0 for each t ∈ [0, t 1 ], i.e. 0 = −x 0 + H(x 0 , λ 0 ), a contradiction, since I − H is an admissible homotopy. Thus, the proof of (7) is completed. As an immediate conclusion of (7), one obtains By the compact version of the theorem -see [5,Prop. 4.3], there exists t ∈ (0, t 2 ] such that, for any t ∈ (0, t], Finally, combining (6), (10) and (9), we end the proof.  Then there exists t > 0 such that, for any t ∈ (0, t], (i) the translation along trajectories operator for (P F ), Φ t : V → E is a welldefined condensing map and Φ t (x) = x for any x ∈ ∂V ; is the unique solution of (P F,λ ) at time 0 from x, is well-defined, continuous and, for any Ω ⊂ V , We shall show that there exists t ∈ [0, t 1 ], such that, for t ∈ [0, t], I − Θ t is an admissible homotopy in the sense of the topological degree for condensing maps. First, we need to prove that for any t ∈ (0, Clearly, either β(Θ τ (Ω 0 × Λ 0 )) = 0, for a.e. τ ∈ [0, t] and (11) follows clearly or there exists J ⊂ [0, t] of positive Lebesgue measure such that β(Θ τ (Ω 0 × Λ 0 )) > 0 for any τ ∈ J. Hence which implies (11).
Next, one has to show that there exist λ 1 ∈ (0, 1] and t 2 ∈ (0, t 1 ] such that, for any t ∈ (0, t 2 ], If we suppose to the contrary that, then there exist (x n ) ⊂ ∂V , λ n → 0 + and t n → 0 + such that Θ tn (x n , λ n ) = x n for each n ≥ 1. This clearly implies that Θ jtn (x n , λ n ) = x n for any integer j > 0 such that jt n ≤ t 1 , which gives, for large n, Θ t 1 /2+rn (x n , λ n ) = x n where r n := ([(t 1 /2)/t n ] + 1)t n − (t 1 /2). Clearly 0 < r n < t n → 0 + and The latter inequality gives a contradiction and we infer that (x n ) is relatively compact. Without loss of generality we may assume that x n → x 0 ∈ ∂V . If one puts u n := u(·; 0, t 1 , x n , λ n ), then, by use of Proposition 3.1 (i), u n → u 0 in C([0, t 1 ], E) and, by the same argument as in (8) Finally, according to (3)  (A2) if λ n → λ 0 (as n → ∞) in [0, 1], then A(λ n ) res → A(λ 0 ) (with respect to graphs or resolvent), i.e. for any ν > −ω and u ∈ E, R(ν : −A(λ n ))u → R(ν; −A(λ 0 ))u, and that F : [0, T ] × E × [0, 1] → E, where T > 0, is a continuous mapping satisfying the following conditions (F1) F is locally Lipschitz in the second variable uniformly with respect to the others, i.e. for any x ∈ E there exist δ x > 0 and L x > 0 such that, for any Below we collect some basic facts concerning mild solutions (see [16]) for the family of differential equations then, for any bounded Ω ⊂ E, Proof. The proof of (i) follows from the arguments of [16, Ch. 6, Th. 2.2], the local existence theorem and Remark 4.2.

Remark 4.3
In the above theorem, it is possible to relax or get rid of the separability assumption on E at cost of stronger assumptions on k or geometry of E -see e.g. [3].

Krasnosel'skii type degree formula
Now we are concerned with the differential equation where A : D(A) → E is a densely defined linear operator on a separable Banach space such that −A is an infinitesimal generator of a C 0 semigroup {S A (t)} t≥0 of bounded linear operators and F : E → E is a locally Lipschitz map and suppose the following properties hold (H1) there exists ω > 0 such that S A (t) L(E,E) ≤ e −ωt for any t ≥ 0; (H2) F has sublinear growth, i.e. there exists c > 0 such that F (x) ≤ c(1 + x ) for any x ∈ E; (H3) there is k ∈ [0, ω) such that, for any bounded Ω ⊂ E, β(F (Ω)) ≤ kβ(Ω).
By Φ t : E → E, for t > 0, denote the translation along trajectories operator given by Φ t (x) := u(t; 0, x) where u(·; 0, x) is a mild solution of (P A,F ) with the initial value condition u(0) = x. It is well-defined due to Proposition 4.1.
Let us now state one of the main results of the paper, i.e. the Krasnosel'skii type formula for (P A,F ).
Step 2. Now suppose that F satisfies (H3) with k ∈ [0, ω). Obviously, A −1 F : E → E is a k-set contraction with the constant k/ω < 1. Let C ⊂ E be a fundamental set for A −1 F | V (see Remark 2.1) and F 0 : E → E be a continuous extension of F | C∩V such that F 0 (E) ⊂ conv F (C ∩ V ) -existing by the Dugundji extension theorem (see e.g. [6]). Furthermore let F L : E → E be a locally Lipschitz map such that, for any x ∈ E, , and, since C is fundamental for A −1 F | V , we infer that x ∈ C; hence, by use of (19), which contradicts the definition of ρ and proves (20). Using the homotopy invariance of the degree for −A + F we obtain 0, 1, x, λ) is a solution ofu = −Au+F (u, λ) on [0, 1] with the initial condition u(0) = x. In view of Proposition 4.1, Υ t is well-defined and In view of (20) and Lemma 5.2, there exists t ∈ (0, 1] such that, for any t ∈ (0, t], Using the homotopy invariance of the degree, one has On the other hand, by Step 1, there exists t ∈ (0, t], such that, for each t ∈ (0, t], which together with (23) and (21) ends the proof. Remark 5.3 Theorem 5.1 is a version of Th. 5.1 from [5], where −A was assumed to generate a compact C 0 semigroup and F to be locally Lipschitz with sublinear growth (and without any compactness properties).
In the rest of the section we extend Theorem 5.1 to the case k = ω, i.e. we assume that (H1) and (H2) hold and, instead of (H3), we suppose that Lemma 5.5 Let T n : E → E, n ≥ 1 be a sequence of bounded linear operators such that, for any x ∈ E, (T n x) is a Cauchy sequence. Then, for any bounded set {x n } n≥1 ⊂ E, Proof. Observe first that, by the Banach-Steinhaus uniform boundedness theorem, the sequence ( T n ) is bounded. Next take any ε > 0 and choose y 1 , y 2 , . . . , y k ∈ E making a β({x n } n≥1 ) + ε net for {x n } n≥1 . There exists an integer N ≥ 1 such that, for any l, n ≥ N and i = 1, . . . , k, T n y i − T l y i ≤ ε and T n ≤ g + ε with g := lim sup m→∞ T m . Hence, for any n ≥ N, Since ε > 0 was arbitrary, we are done.

Periodic problem
In this section we shall apply the degree formula from Theorem 5.1 to the periodic problem of the form Our approach is based on the idea of branching periodic solutions from the equilibrium point of the the right-hand side (see [8] and [5]) in the parameterized differential equation. To this end consider the following parameterized family of equations By u(·; x, λ) denote the unique mild solution of (P T,λ ) with the initial condition The Krasnosel'skii type degree formula allows us to derive a continuation principle.
The proof of the theorem is based on the following averaging formula. Proof. Let F : [0, +∞) × E → E be a continuous extension of F given by and, in consequence, and A is closed, we get −Ax 0 + F (x 0 ) = 0, which is a contradiction ending the proof of (31). Thus, using the homotopy invariance of the topological degree, we get, for any λ ∈ (0, λ 1 ], where Φ λ t is the translation along trajectory operator for the equationu = −λAu+ λ F (u). In view of Theorem 5.1, there exists λ 0 ∈ (0, λ 1 ] such that, for any λ ∈ (0, Either Φ 1 T has a fixed point in ∂U or, by the assumption and the homotopy invariance of the degree, and there exists x ∈ U such that (x, 1) is a periodic point. In both cases one gets the exists of a T -periodic solution starting (and ending) at a point of U .
Remark 6.5 (a) If U := B(x 0 , r) for some x 0 ∈ D(A) and r > 0 such that −Ax 0 + F (x 0 ) = 0, and Deg(−A + F , B(x 0 , r)) = 0 and (P T,λ ) has no periodic points in ∂B(x 0 , r) × (0, 1), then the thesis of Theorem 6.3 follows also from Theorem 6.2.1 in [10]. It is worth mentioning that in [10] the authors work with set-valued F and a periodic solution comes up as a fixed point of a certain operator in the space of periodic functions from C([0, T ], E).
(b) For the set-valued upper semicontinuous F with compact convex values, the translation along trajectories approach also applies. Obviously, in general, the inclusionu ∈ −Au + F (u) has not a unique solution, i.e. the translation along trajectories operator Φ t : E ⊸ E has the form where L(x, −A+F ) is the set of solution starting from x and e t : C([0, T ], E) → E is the evaluation at t. However, due to results of [3], the solution set L(x, −A+F ), for any x ∈ E, is cell-like (or of R δ type). This enables to apply the fixed point index theory for a general class of maps given by a superposition e • L : E → E where L : E → E 1 is a set-valued upper semicontinuous map with R δ values and e : E 1 → E is a continuous map -see [13] and [2]. Using this degree and the related approximation method, we reduce the problem to the single-valued one. Roughly speaking, taking sufficiently close graph approximation f of F , we know that the topological degree of −A + f is equal to that of −A + F and the degree of Applying Theorem 6.3, we obtain the following simple existence criterion. Theorem 6.6 Suppose (H1) and (H3b) hold, F (0, x) = F (T, x) for any x ∈ E, and there exists K > 0 such that then the periodic problem (P T ) admits a mild solution u with u(0) = u(T ) ≤ K/ω.
To compute the degree Deg(−A + F , B(0, R)) consider the homotopy A + µ F . It is admissible, since β( µ∈[0,1] µ F (Ω)) ≤ kβ(Ω) for any bounded Ω ⊂ E. If Since the translation along trajectory operator Φ T is a k-set contraction and R was arbitrary, we can conclude that (P T ) has a solution in B(0, K/ω).
We end the paper with an example of applications to a system of partial differential equations.
We finish with an example showing that the results of the paper may be also applied to parabolic partial differential equations where the semigroup of the proper differential operator may not be compact and the results of [5] do not apply. (Ω) | ∆u ∈ L 2 (Ω)} and Au := −∆u. Then −A generates a C 0 semigroup S A on L 2 (Ω) such that S A (t) ≤ e −ωt for some ω > 0 and all t ≥ 0. Indeed, by the Poincaré inequality, which guarantees that −A generates a C 0 semigroup such that S A (t) ≤ e −ωt with ω := 4d −2 .