Piotr Kokocki. Periodic solutions for nonlinear evolution equations at resonance. Uniwersytet M. Kopernika w Toruniu

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1 Piotr Kokocki Uniwersytet M. Kopernika w Toruniu Periodic solutions for nonlinear evolution equations at resonance Praca semestralna nr 1 (semestr zimowy 21/11) Opiekun pracy: Wojciech Kryszewski

2 Periodic solutions for nonlinear evolution equations at resonance Piotr Kokocki * Faculty of Mathematics and Computer Science Nicolaus Copernicus University ul. Chopina 12/18, 87-1 Toruń, Poland Abstract We are concerned with periodic problems for nonlinear evolution equations at resonance of the form u(t) = Au(t) + F (t, u(t)), where a densely defined linear operator A: D(A) X on a Banach space X is such that A generates a compact C semigroup and F : [, + ) X X is a nonlinear perturbation. Imposing an appropriate Landesman Lazer type conditions on the nonlinear term F, we prove a formula expressing the fixed point index of the associated translation along trajectories operator in terms of a time averaging of F restricted to Ker A. Then we show that the translation operator has a nonzero fixed point index and, in consequence, we prove that the equation admits a periodic solution. 1 Introduction Consider a periodic problem (1.1) { u(t) = Au(t) + F (t, u(t)), t > u(t) = u(t + T ) t, where T > is a fixed period, A: D(A) X is a linear operator such that A generates a C semigroup of bounded linear operators on a Banach space X and F : [, + ) X X is a continuous mapping. Periodic problems of this form are the abstract formulations of many differential equations including the parabolic partial differential equations on an open set Ω R n, n 1 u t = Au + f(t, x, u) in (, + ) Ω (1.2) Bu = on [, + ) Ω u(t, x) = u(t + T, x) in [, + ) Ω, where Au = D i (a ij D j u) + a k D k u + a u * Corresponding author. address: p.kokocki@mat.umk.pl. The researches supported by the MNISzW Grant no. N N Mathematical Subject Classification: 47J35, 35B1, 37L5 Key words: semigroup, evolution equation, topological degree, periodic solution, resonance 1

3 is such that a ij = a ji C 1 (Ω), a k, a C(Ω), a ij (x)ξ i ξ j θ ξ 2 for ξ = (ξ 1, ξ 2,..., ξ n ) R n, x Ω, f : [, + ) Ω R R is a continuous mapping and B stands for the Dirichlet or Neumann boundary conditions. Given x X, let u(t; x) be a (mild) solution of u(t) = Au(t) + F (t, u(t)), t > such that u(; x) = x. We look for the T -periodic solutions of (1.1) as the fixed points of the translation along trajectory operator Φ T : X X given by Φ T (x) := u(t ; x). One of the effective methods used to prove the existence of the fixed points of Φ T is the averaging principle involving the equations (1.3) u(t) = λau(t) + λf (t, u(t)), t > where λ > is a parameter. Let Θ λ T : X X be the translation operator for (1.3). It is clear that Φ T = Θ 1 T. Define the mapping F : X X by F (x) := 1 T F (s, x) ds for x X. The averaging principle says that for every open bounded set U X such that / ( A + F )(D(A) U), one has that Θ λ T (x) x for x U and deg(i Θ λ T, U) = deg( A + F, U) provided λ > is sufficiently small. In the above formula deg stands for the appropriate topological degrees. Therefore, if deg( A + F, U), then using suitable a priori estimates and the continuation argument, we infer that Θ 1 T has a fixed point and, in consequence, (1.1) admits a periodic solution starting from U. The averaging principle for periodic problems on finite dimensional manifolds was studied in [16]. The principle for the equations on any Banach spaces has been recently considered in [8] in the case when A generates a compact C semigroup and in [9] for A being an m-accretive operator. In [1], a similar results were obtained when A generates a semigroup of contractions and F is condensing. For the results when the operator A is replaced by a time-dependent family {A(t)} t see [11]. However there are examples of equations where the averaging principle in the above form is not applicable. Therefore, in this paper, motivated by [5], [17] and [22], we use the method of translation along trajectories operator to derive its counterpart in the particular situation when the equation (1.1) is at resonance i.e., Ker A and F is bounded. Let N := Ker A and assume that the C semigroup {S A (t)} t generated by A is compact. Then it is well known that (real) eigenvalues of S A (T ) make a sequence which is either finite or converges to and the algebraic multiplicity of each of them is finite. Denote by µ the sum of the algebraic multiplicities of eigenvalues of S A (T ): X X lying in (1, + ). Furthermore it follows that the operator A has compact resolvents and, in consequence, dim N < +. Let M be a subspace of X such that N M = X with S A (t)m M for t. Define a mapping g : N N by (1.4) g(x) := P F (s, x) ds for x N where P : X X is a topological projection onto N with Ker P = M. First, we are concerned with an equation u(t) = Au(t) + λf (t, u(t)), t > 2

4 where λ [, 1] is a parameter. Denoting by Φ λ T the translation along trajectory operator associated with this equation, we shall show that, if V M is an open bounded set, with V and U N is an open bounded set in N such that g(x) for x from the boundary N U of U in N, then for small λ (, 1), Φ λ T (x) x for x (U V ) and (1.5) deg LS (I Φ λ T, U V ) = ( 1) µ+dim N deg B (g, U). Here deg LS and deg B stand for the Leray Schauder and Brouwer degree, respectively. The equation (1.5) will be called the resonant averaging formula. Further, for an open and bounded set Ω R n, we shall use this formula to study the periodic problem { u(t) = Au(t) + λu(t) + F (t, u(t)), t > (1.6) u(t) = u(t + T ) t, where A: D(A) X is a linear operator on the Hilbert space X := L 2 (Ω) with a real eigenvalue λ and F : [, + ) X X is a continuous mapping. As before we assume that A generates a compact C semigroup {S A (t)} t on X. The mapping F is associated with a bounded and continuous f : [, + ) Ω R R as follows (1.7) F (t, u)(x) := f(t, x, u(x)) for t [, + ), x Ω. Additionally we suppose that the following kernel coincidence holds true (which is more general than to assume that A is self-adjoint) N λ := Ker (A λi) = Ker (A λi) = Ker (I e λt S A (T )). Let Ψ t : X X be the translation along trajectories operator associated with the equation u(t) = Au(t) + λu(t) + F (t, u(t)), t >. The resonant averaging formula, under a suitable Landesman Lazer type conditions, gives an effective criterion for the existence of T -periodic solutions of (1.6). Namely, we prove that there is an open bounded set W X such that g(x) for x N λ \ (W N λ ), Ψ T (x) x for x X \ W and (1.8) deg LS (I Ψ T, W ) = ( 1) µ(λ)+dim N λ deg B (g, W N λ ) where µ(λ) is the sum of the algebraic multiplicities of the eigenvalues of e λt S A (T ) lying in (1, + ) and g : N λ N λ is given by (1.4) with P being the orthogonal projection on N λ. Additionally, we compute deg B (g, W N λ ), which may be important in the study of problems concerning to the multiplicity of periodic solutions. Obtained applications correspond to those from [5], [17], where a different approach were used to prove the existence of periodic solutions for equations at resonance. Notation and terminology. Throughout the paper we use the following notational conveniences. If (X, ) is a normed linear space, Y X is a subspace and U Y is a subset, then by cl Y U and Y U we denote the closure and boundary of U in Y, respectively, while by cl U (U) and U we denote the closure and boundary of U in X, respectively. If Z is a subspace of X such that X = Y Z, then for subsets U Y and V Z we write U V := {x + y x U, y V } for their algebraic sum. We recall also that a C semigroup {S(t): X X} t is compact if S(t)V is relatively compact for every bounded V X and t >. 3

5 2 Translation along trajectories operator Consider the following differential problem (2.9) { u(t) = Au(t) + F (λ, t, u(t)), t > u() = x where λ is a parameter from a metric space Λ, A: D(A) X is a linear operator on a Banach space (X, ) and F : Λ [, + ) X X is a continuous mapping. In this section X is assumed to be real, unless otherwise stated. Suppose that A generates a compact C semigroup {S A (t)} t and the mapping F is such that (F1) for any λ Λ and x X there is a neighborhood V X of x and a constant L > such that for any x, y V F (λ, t, x) F (λ, t, y) L x y for t [, + ); (F2) there is a continuous function c: [, + ) [, + ) such that F (λ, t, x) c(t)(1 + x ) for λ Λ, t [, + ), x X. A mild solution of the problem (2.9) is, by definition, a continuous mapping u: [, + ) X such that u(t) = S A (t)x + t S A (t s)f (λ, s, u(s)) ds for t. It is well known (see e.g. [21]) that for any λ Λ and x X, there is unique mild solution u( ; λ, x): [, + ) X of (2.9) such that u(; λ, x) = x and therefore, for any t, one can define the translation along trajectories operator Φ t : Λ X X by Φ t (λ, x) := u(t ; λ, x) for λ Λ, x X. As we need the continuity and compactness properties of Φ t, we recall the following Theorem 2.1. Let A: D(A) X be a linear operator such that A generates a compact C semigroup and let F : Λ [, + ) X X be a continuous mapping such that conditions (F1) and (F2) hold. (a) If sequences (λ n ) in Λ and (x n ) in X are such that λ n λ and x n x, as n +, then u(t; λ n, x n ) u(t; λ, x ) as n +, uniformly for t from bounded intervals in [, + ). (b) For any t >, the operator Φ t : Λ X X is completely continuous, i.e. Φ t (Λ V ) is relatively compact, for any bounded V X. Remark 2.2. The above theorem is slightly different from Theorem 2.14 in [8], where it is proved in the case when linear operator is dependent on parameter as well, and moreover the parameter space Λ is compact. However, if A is free of parameters, then compactness of Λ may be omitted. 4

6 Before we start the proof we prove the following technical lemma Lemma 2.3. Let Ω X be a bounded set. Then (a) for every t > the set {u(t ; λ, x) t [, t ], λ Λ, x Ω} is bounded; (b) for every t > and ε > there is δ > such that if t, t [, t ], t < t and t t < δ, then t S A (t s)f (λ, s, u(s; λ, x)) ds ε for λ Λ, x Ω; (c) for every t > the set is bounded. t S(t ) := { t S A (t s)f (λ, s, u(s; λ, x)) ds } λ Λ, x Ω Proof. Throughout the proof we assume that the constants M 1 and ω R are such that S A (t) Me ωt for t. (a) Let R > be such that Ω B(, R). Then by condition (F2), for every t [, t ] u(t ; λ, x) S A (t)x + Me ω t x + t t RMe ω t + t KMe ω t + S A (t s)f (λ, s, u(s; λ, x)) ds Me ω (t s) c(s)(1 + u(s; λ, x) ) ds t where K := sup s [,t ] c(s). By the Gronwall inequality KMe ω t u(s; λ, x) ds, (2.1) u(t ; λ, x) C e t C 1 for t [, t ], λ Λ x Ω, where C := RMe ω t + t KMe ω t and C 1 := KMe ω t. (b) From (a) it follows that there is C > such that u(t ; λ, x) C for t [, t ], λ Λ and x Ω. Therefore, if t, t [, t ] are such that t < t, then t t S A (t s)f (λ, s, u(s; λ, x)) ds t t t t Me ω(t s) F (λ, s, u(s; λ, x)) ds Me ω (t s) c(s)(1 + u(s; λ, x) ) ds = (t t)mke ω t (1 + C). Taking δ := ε(mke ω t (1 + C)) 1 we obtain the assertion. (c) For any λ Λ and x Ω t t S A (t s)f (λ, s, u(s ; λ, x)) ds Me ω(t s) c(s)(1 + u(s ; λ, x) ) ds t MKe ω t (1 + u(s ; λ, x)) ) ds t MKe ω t (1 + C) := R(t ). 5

7 Hence S(t ) is contained in a ball of radius R(t ) and is bounded as claimed. Proof of Theorem 2.1. Let Ω X be a bonded set and let t (, + ). We shall prove first that the set Φ t (Λ Ω) is relatively compact. Let ε >. For < t < t, λ Λ and x Ω ( t u(t; λ, x) = S A (t)x + S A (t t ) and, in consequence, t + S A (t s)f (λ, s, u(s; λ, x)) ds, t ) S A (t s)f (λ, s, u(s; λ, x)) ds (2.11) {u(t; λ, x) λ Λ, x Ω} S A (t) S A (t t )D t { t } + S A (t s)f (λ, s, u(s; λ, x)) ds λ Λ, x Ω, t where D t := { t S A (t s)f (λ, s, u(s; λ, x)) ds } λ Λ, x Ω. Applying Lemma 2.3 (b), we infer that there is t (, t) such that t (2.12) S A (t s)f (λ, s, u(s; λ, x)) ds ε for λ Λ, x Ω. t From the point (c) of this lemma it follows that D t (2.12) yields is bounded. Combining (2.11) with Φ t (Λ Ω) = {u(t; λ, x) λ Λ, x Ω} V ε + B(, ε) where V ε := S A (t) S A (t t )D t. This implies that V ε is relatively compact, since {S A (t)} t is a compact semigroup and the sets Ω, D t are bounded. On the other hand ε > may be chosen arbitrary small and therefore the set Φ t (Λ Ω) is also relatively compact. Let (λ n ) in Λ and (x n ) in X be sequences such that λ n λ Λ and x n x X. We prove that u(t; λ n, x n ) u(t; λ, x ) as n + uniformly on [, t ] where t >. For every n 1 write u n := u( ; λ n, x n ). We claim that (u n ) is an equicontinuous sequence of functions. Indeed, take t [, + ) and let ε >. If h > then, by the integral formula, (2.13) u n (t + h) u n (t) = S A (h)u n (t) u n (t) + t+h t S A (t + h s)f (λ n, s, u n (s)) ds. Note that for every t [, + ) set {u n (t) n 1} is relatively compact. For t = it follows from the convergence of (x n ), while for t (, + ) it is a consequence of the fact that the set Φ t (Λ {x n n 1}) is relatively compact. From the continuity of semigroup there is δ > such that (2.14) S A (t + h)u n (t) S A (t)u n (t) ε/2 for h (, δ ), n 1. 6

8 By Lemma 2.3 (b) there is δ (, δ ) such that for h (, δ) and n 1 t+h (2.15) S A (t + h s)f (λ n, s, u n (s)) ds ε/2. t Combining (2.13), (2.14) and (2.15), for h (, δ) we infer that, u n (t + h) u n (t) S A (h)u n (t) u n (t) t+h + S A (t + h s)f (λ n, s, u n (s)) ds ε/2 + ε/2 = ε t for every n 1. We have thus proved that (u n ) is right side equicontinuous on [, + ). It remains to show that (u n ) is equicontinuous on the left. To this end take t (, + ) and ε >. If < h δ < t then (2.16) u n (t) u n (t h) u n (t) S A (δ)u n (t δ) + S A (δ)u n (t δ) S A (δ h)u n (t δ) + S A (δ h)u n (t δ) u n (t h), and consequently, for any n 1, (2.17) u n (t) u n (t h) t t δ S A (t s)f (λ n, s, u n (s)) ds + S A (δ)u n (t δ) S A (δ h)u n (t δ) t h + S A (t h s)f (λ n, s, u n (s)) ds. t δ By Lemma 2.3 (b) there is δ (, t) such that for every t 1, t 2 [, t] with t 2 > t 1 and t 1 t 2 < δ, we have t2 (2.18) S A (t 2 s)f (λ n, s, u n (s)) ds ε/3 for n 1. t 1 Using again the relative compactness of {u n (t) n 1} where t [, + ) we can choose δ 1 (, δ) such that for every h (, δ 1 ) and n 1 (2.19) S A (δ)u n (t δ) S A (δ h)u n (t δ) ε/3. Taking into account (2.17), (2.18), (2.19), for h (, δ 1 ) t u n (t) u n (t h) S A (t s)f (λ n, s, u n (s)) ds t δ + S A (δ)u n (t δ) S A (δ h)u n (t δ) t h + S A (t h s)f (λ n, s, u n (s)) ds ε, t δ and finally the sequence (u n ) is left side equicontinuous on (, + ). Hence (u n ) is equicontinuous at every t [, + ) as claimed. For every n 1 write w n := u n [,t ]. We shall prove that w n w in C([, t ], X) where w = u( ; λ, x ) [,t ]. It is enough to show that every subsequence of (w n ) contains 7

9 a subsequence which is convergent to w. Let (w nk ) be a subsequence of (w n ). Since (w nk ) is equicontinuous on [, t ] and the set {w nk (s) n 1} = {u nk (s) n 1} is relatively compact for any s [, t ], by the Ascoli-Arzela Theorem, we infer that (w nk ) has a subsequence (w nkl ) such that w nkl w in C([, t ], X) as l +. For every l 1 define a mapping φ l : [, t ] X by φ l (s) := S A (t s)f (λ nkl, s, w nkl (s)). From the continuity of {S A (t)} t and F, we deduce that φ l φ in C([, t ], X), where φ : [, t ] X is given by φ (s) = S A (t s)f (λ, s, w (s)). It is clear that t w nkl (t ) = S A (t )x + φ l (s) ds for t [, t ], and therefore, passing to the limit with l, we infer that for t [, t ] t t w (t ) = S A (t )x + φ (s) ds = S A (t )x + S A (t s)f (λ, s, w (s)) ds. By the uniqueness of mild solutions, w (t) = u(t ; λ, x ) for t [, t ] and we conclude that w nkl w = u( ; λ, x ) as l and finally that w n w in C([, t], X). This completes the proof of point (a). If A : D(A) X is defined on a complex space X, then the point spectrum of A is the set σ p (A) := {λ C there exists z X \ {} such that λz Az = }. For a linear operator A defined on a real space X, it is possible to consider its complex point spectrum (see [2] or [12]). By the complexification of X we mean a complex linear space (X C, +, ), where X C := X X, with the operations of addition +: X C X C C and multiplication by complex scalars : C X C C given by (x 1, y 1 ) + (x 2, y 2 ) := (x 1 + x 2, y 1 + y 2 ) for (x 1, y 1 ), (x 2, y 2 ) X C, and (α + βi) (x, y) := (αx βy, αy + βx) for α + βi C, (x, y) X C, respectively. For convenience, denote the elements (x, y) of X C by x + yi. If X is a space with a norm, then the mapping C : X C C given by x + yi C := sup sin θx + cos θy θ [ π,π] is a norm on X C, and (X C, C ) is a Banach space, provided X is it. The complexification of A is a linear operator A C : D(A C ) X C given by D(A C ) := D(A) D(A) and A C (x + yi) := Ax + Ayi for x + yi D(A C ). Now, one can define the complex point spectrum of A by σ p (A) := σ p (A C ). Remark 2.4. If A is a generator of a C semigroup {S A (t)} t, then it is easy to check that the family {S A (t) C } t of the complexified operators is a C semigroup of bounded linear operators on X C with the generator A C. In the following proposition we mention some spectral properties of C semigroups 8

10 Proposition 2.5. (see [18, Theorem ]) If A is the generator of a C semigroup {S A (t)} t of bounded linear operators on a complex Banach space X, then σ p (S A (t)) = e tσp(a) \ {} for t >. Furthermore, if λ σ p (A) then for every t > ( ) (2.2) Ker (e λt I S A (t)) = span Ker (λ k,t I A) where λ k,t := λ + (2kπ/t)i for k Z. k Z 3 Averaging principle at the resonance In this section we are interested in the periodic problems of the form (3.21) { u(t) = Au(t) + εf (t, u(t)), t > u(t) = u(t + T ) t where T > is a fixed period, ε [, 1] is a parameter, A: D(A) X is a linear operator on a real Banach space X and F : [, + ) X X is a continuous mapping. Suppose that F satisfies (F1) and (F2) and A generates a compact C semigroup {S A (t)} t such that (A1) Ker A = Ker (I S A (T )) {}; (A2) there is a closed subspace M X, M {} such that X = Ker A M and S A (t)m M for t. Remark 3.1. (a) If A is any linear operator such that A generates a C semigroup {S A (t)} t, then it is immediate that Ker A Ker (I S A (t)) for t. (b) Condition (A1) can be characterized in terms of the point spectrum. Namely, (A1) is satisfied if and only if (3.22) {(2kπ/T )i k Z, k } σ p (A) =. To see this suppose first that (A1) holds. If (2kπ/T )i σ p (A) for some k, then there is z = x + yi X C \ {} such that (3.23) A C z = (2kπ/T )zi. We actually know that A C is a generator of the C semigroup {S AC (t)} t with S AC (t) = S A (t) C for t. Therefore, by Proposition 2.5, we find that z Ker (I S AC (T )) and, in consequence, S A (T )x + S A (T )yi = x + yi. By (A1), we get Ax = Ay = and finally A C z =, contrary to (3.23). Conversely, suppose that (3.22) is satisfied. Operator A C as a generator of a C semigroup is closed, and hence Ker A C is a closed subspace of X C. On the other hand, by (2.2) and (3.22), Ker (I S A (T ) C ) = Ker (I S AC (T )) = cl Ker A C = Ker A C, which implies that Ker (I S A (T )) = Ker A, i.e. (A1) is satisfied. 9

11 Since X is a Banach space and M, N are closed subspaces, there are projections P : X X and Q: X X such that P 2 = P, Q 2 = Q, P + Q = I and Im P = N, Im Q = M. By Φ ε T we denote the translation along trajectories operator associated with (3.21). Remark 3.2. The compactness of the semigroup {S A (t)} t, implies that the non-zero real eigenvalues of S A (T ) form a sequence which is either finite or converges to and the algebraic multiplicity of each of them is finite. In both cases, only a finite number of eigenvalues is greater than 1 and let µ denote the sum of their algebraic multiplicities. We are ready to formulate the main result of this section Theorem 3.3. Let g : N N be a mapping given by g(x) := P F (s, x) ds for x N and let U N and V M with V, be open bounded sets. If g(x) for x N U, then there is ε (, 1) such that for any ε (, ε ] and x (U V ), Φ ε T (x) x and deg LS (I Φ ε T, U V ) = ( 1) µ+dim N deg B (g, U) where deg LS and deg B stand for the Leray Schauder and the Brouwer topological degree, respectively. Proof. Throughout the proof, we write W := U V and Λ := [, 1] [, 1] W. For any (ε, s, y) Λ consider the differential equation (3.24) u(t) = Au(t) + G(ε, s, y, t, u(t)), t > where G: Λ [, + ) X X is defined by G(ε, s, y, t, x) := εp F (t, sx + (1 s)p y) + εsqf (t, x). We check that G satisfies condition (F1). Indeed, fix (ε, s, y) Λ and take x X. If s = then G(ε, s, y, t, ) is constant, hence we may suppose that s. There are constants L, L 1 > and neighborhoods V, V 1 X of points sx + (1 s)p y and x, respectively, such that and F (t, x 1 ) F (t, x 2 ) L x 1 x 2 for x 1, x 2 V, t [, + ) F (t, x 1 ) F (t, x 2 ) L 1 x 1 x 2 for x 1, x 2 V 1, t [, + ). Then V := 1 s (V (1 s)p y) V 1 is open, x V and, for any x 1, x 2 V, G(ε, s, y, t, x 1 ) G(ε, s, y, t, x 2 ) ε P F (t, sx 1 + (1 s)p y) F (t, sx 2 + (1 s)p y) + sε Q F (t, x 1 ) F (t, x 2 ) εl P x 1 x 2 + sεl 1 Q x 1 x 2 = (L P + L 1 Q ) x 1 x 2, 1

12 i.e. (F1) is clearly satisfied. An easy computation shows that condition (F2) also holds true. If (ε, s, y) Λ and x X, then by u( ; ε, s, y, x): [, + ) X we denote unique mild solution of (3.24) starting at x. For t, let Θ t : Λ X X be the translation along trajectories operator given by Θ t (ε, s, y, x) := u(t ; ε, s, y, x) for (ε, s, y) Λ, x X, t [, + ). For every ε (, 1) we define the mapping M ε : [, 1] W X by M ε (s, x) := Θ T (ε, s, x, x). Clearly M ε is completely continuous for every ε (, 1). Indeed, by Theorem 2.1 the operator Θ T is completely continuous and, consequently, the set Θ T (Λ W ) X is relatively compact. Since M ε ([, 1] W ) = Θ T ({ε} [, 1] W W ) Θ T (Λ W ), the set M ε ([, 1] W ) is relatively compact as well. Now we claim that there is ε (, 1) such that (3.25) M ε (s, x) x for x W, s [, 1], ε (, ε ]. Suppose to the contrary that there are sequences (ε n ) in (, 1), (s n ) in [, 1] and (x n ) in W such that ε n and (3.26) Θ T (ε n, s n, x n, x n ) = M εn (s n, x n ) = x n for n 1. We may assume that s n s with s [, 1]. By (3.26) and the boundedness of (x n ) W, the complete continuity of Θ T implies that (x n ) has convergent subsequence. Without lost of generality we may assume that x n x as n +, for some x W. After passing to the limit in (3.26), by Theorem 2.1 (a), it follows that (3.27) Θ T (, s, x, x ) = x. On the other hand (3.28) Θ t (, s, x, x ) = S A (t)x for t, which together with (3.27) implies that x = S A (T )x. Condition (A1) yields x Ker A = N and hence Qx =. Since V, and the equality (U V ) = N U cl M V cl N U M V holds true, we infer that x N U. By the use of Remark 3.1 (a) and (3.28) we also find that (3.29) Θ t (, s, x, x ) = S A (t)x = x for t. For every n 1, write u n := u( ; ε n, s n, x n, x n ) for brevity. As a consequence of (3.26) (3.3) x n = S A (T )x n + ε n S A (T τ)p F (τ, s n u n (τ) + (1 s n )P x n ) dτ + ε n s n S A (T τ)qf (τ, u n (τ)) dτ for n 1. 11

13 The fact that the spaces M, N X are closed and S A (t)n N, S A (t)m M, for t, leads to (3.31) ε n S A (T τ)p F (τ, s n u n (τ) + (1 s n )P x n ) dτ N and ε n s n S A (T τ)qf (τ, u n (τ)) dτ M for n 1. Combining (3.3) with (3.31) gives P x n = S A (T )P x n + ε n S A (T τ)p F (τ, s n u n (τ) + (1 s n )P x n ) dτ for n 1, and therefore (3.32) P F (τ, s n u n (τ) + (1 s n )P x n ) dτ = for n 1, since P x n Ker A = Ker (I S A (T )) for n 1. By Theorem 2.1 (a) and (3.29) the sequence (u n ) converges uniformly on [, T ] to the constant mapping equal to x, hence, passing to the limit in (3.32), we infer that g(x ) = P F (τ, x ) dτ =. This contradicts the assumption, since x N U, and proves (3.25). By the homotopy invariance of topological degree we have (3.33) deg LS (I Φ ε T, W ) = deg LS (I M ε (1, ), W ) = deg LS (I M ε (, ), W ) for ε (, ε ]. Let the mappings M 1 ε : U N and M 2 ε : V M be given by M 1 ε (x 1 ) := x 1 + ε P F (s, x 1 ) ds for x 1 U, M 2 ε (x 2 ) := S A (T ) M x 2 for x 2 V and let M ε : U V N M be their product For ε (, 1) and x X M ε (x 1, x 2 ) := ( M ε 1 (x 1 ), M ε 2 (x 2 )) for (x 1, x 2 ) U V. M ε (, x) = S A (T )x + ε S A (T τ)p F (τ, P x) dτ = S A (T )x + ε P F (τ, P x) dτ. and therefore it is easily seen that the mappings M ε (, ) and M ε are topologically conjugate. By the compactness of the C semigroup {S A (t) : M M} t and the fact that Ker (I S A (T ) M ) =, we infer that the mapping I M ε 2 : M M 12

14 is a linear isomorphism. By use of the multiplication property of topological degree, for any ε (, 1), deg LS (I M ε (, ), W ) = deg LS (I M ε, U V ) Combining this with (3.33), we conclude that = deg B (I M ε 1, U) deg LS (I M ε 2, V ). deg LS (I Φ ε T, W ) = deg B ( ε g, U) deg LS (I S A (T ) M, V ) = ( 1) dim N deg B (g, U) deg LS (I S A (T ) M, V ), for ε (, ε ]. If λ 1 and k 1 is an integer then, by (A1) and (A2), Ker (λi S A (T )) k M = Ker (λi S A(T )) k. Hence σ p (S A (T ) M ) = σ p (S A (T )) \ {1} and the algebraic multiplicities of the corresponding eigenvalues are the same. Therefore, by the standard spectral properties of compact operators (see e.g. [14, Theorem ]), deg LS (I S A (T ) M, V ) = ( 1) µ, and finally deg LS (I Φ ε T, W ) = ( 1) µ+dim N deg B (g, U), for every ε (, ε ], which completes the proof. An immediate consequence of Theorem 3.3 is the following Corollary 3.4. Let U N and V M with V, be open bounded sets such that g(x) for x N U. If deg B (g, U), then there is ε (, 1) such that for any ε (, ε ] problem (3.21) admits a mild solution. 4 Periodic problems with the Landesman Lazer type conditions Let Ω R n, n 1, be an open bounded set and let X := L 2 (Ω). By and, we denote the standard norm and the scalar product on X, respectively. Assume that f : [, + ) Ω R R satisfies conditions (a) there is a constant m > such that f(t, x, y) m for t [, + ), x Ω, y R; (b) there is a constant L > such that for any t [, + ), x Ω and y 1, y 2 R f(t, x, y 1 ) f(t, x, y 2 ) L y 1 y 2 ; (c) f(t, x, y) = f(t + T, x, y) for t [, + ), x Ω and y R; (d) there are continuous functions f +, f : [, + ) Ω R such that f + (t, x) = for t [, + ) and x Ω. lim f(t, x, y) and f (t, x) = lim f(t, x, y) y + y 13

15 Consider the following periodic differential problem (4.34) { u(t) = Au(t) + λu(t) + F (t, u(t)), t > u(t) = u(t + T ) t where A: D(A) X is a linear operator such that A generates a compact C semigroup {S A (t)} t of bounded linear operators on X, λ is its real eigenvalue and F : [, + ) X X is a continuous mapping given by the formula Additionally, we suppose that F (t, u)(x) := f(t, x, u(x)) for t [, + ), x Ω. (A3) Ker (A λi) = Ker (A λi) = Ker (I e λt S A (T )). Recall that by assumptions (a) and (b), the mapping F is well defined, bounded, continuous and Lipschitz uniformly with respect to time. Therefore, the translations along trajectories operator Ψ t : X X associated with the equation (4.34) is well-defined and completely continuous for t >, as a consequence of Theorem 2.1. Let N λ := Ker (λi A) and define g : N λ N λ by g(u) := P F (t, u) dt for u N λ, where P : X X is the orthogonal projection onto N λ. Since {S A (t)} t is compact, A has compact resolvents and dim N λ <. Furthermore note that, for any u, z N λ, (4.35) g(u), z = = P F (t, u), z dt = Ω f(t, x, u(x))z(x) dxdt. We are ready to state the main result of this section F (t, u), z dt Theorem 4.1. Suppose that f : [, + ) Ω R R satisfies one of the following Landesman Lazer type conditions: (4.36) {u>} for any u N λ with u = 1, or (4.37) {u>} f + (t, x)u(x) dxdt + f + (t, x)u(x) dxdt + {u<} {u<} f (t, x)u(x) dxdt >, f (t, x)u(x) dxdt <, for any u N λ with u = 1. Then the problem (4.34) admits a T -periodic mild solution. In the proof of preceding theorem, we use the following Theorem 4.2. Let f : [, + ) Ω R R satisfy the following condition: (4.38) {u>} f + (t, x)u(x) dxdt + 14 {u<} f (t, x)u(x) dxdt

16 for every u N λ with u = 1. Then there is a bounded open set W X such that Ψ T (u) u for u X \ W, g(u) for u N λ \ (W N λ ) and (4.39) deg LS (I Ψ T, W ) = ( 1) µ(λ)+dim N λ deg B (g, W N λ ) where µ(λ) is the sum of the algebraic multiplicities of the eigenvalues of e λt S A (T ): X X lying in (1, + ). We shall use the following lemma Lemma 4.3. If f : [, + ) Ω R R satisfies (4.38), then there is R > such that g(u) for u N λ with u R. Proof. Suppose the assertion is false. Then there is a sequence (u n ) N λ such that g(u n ) = for n 1 and u n + as n +. Define z n := u n / u n for n 1. Since (z n ) N λ and N λ is a finite dimensional space, (z n ) is relatively compact. We can assume that there is z N λ with z = 1 such that z n z as n +. Additionally, we can suppose that z n (x) z (x) as n + for almost every x Ω. Let (4.4) := {x Ω z (x) > } and Ω := {x Ω z (x) < }. Then, by (4.35), we have = g(u n ), z = and therefore (4.41) f(t, x, z n (x) u n )z (x) dxdt + Ω f(t, x, u n (x))z (x) dxdt, for n 1 Ω f(t, x, z n (x) u n )z (x) dxdt =, for n 1. Note that, for fixed t [, T ], the convergence f(t, x, z n (x) u n ) f + (t, x) by n + occurs for almost every x. Since the domain Ω has finite measure, z L 2 (Ω) L 1 (Ω). From the boundedness of f and the dominated convergence theorem, we infer that, for any t [, T ], (4.42) f(t, x, z n (x) u n )z (x) dx f + (t, x)z (x) dx as n +. The function ϕ + n : [, T ] R given by ϕ + n (t) := f(t, x, z n (x) u n )z (x) dx = F (t, u n ), max(z, ) for t [, T ] is continuous and furthermore ϕ + n (t) m z L 1 (Ω) < + for t [, T ]. By use of (4.42) and the dominated convergence theorem, we deduce that f(t, x, z n (x) u n )z (x) dxdt f + (t, x)z (x) dxdt as n +. Proceeding in the same way, we also find that f(t, x, z n (x) u n )z (x) dxdt f (t, x)z (x) dxdt Ω Ω 15

17 as n +. In consequence, after passing to the limit in (4.41) f + (t, x)z (x) dxdt + Ω f (t, x)z (x) dxdt = for z N λ with z = 1, contrary to (4.38), which completes the proof. Proof of Theorem 4.2. Consider the following differential problem u(t) = Au(t) + λu(t) + εf (t, u(t)), t > where ε is a parameter from [, 1] and let Υ t : [, 1] X X be the translations along trajectories operator for this equation. The previous lemma shows that there is R > such that g(u) for u N λ with u R. We claim that there is R 1 R such that (4.43) Υ T (ε, u) u for ε (, 1], u X, u R 1. Conversely, suppose that there are sequences (ε n ) in (, 1] and (u n ) in X such that (4.44) Υ T (ε n, u n ) = u n for n 1 and u n + as n +. For every n 1, set w n := w( ; ε n, u n ) where w( ; ε, u) is a mild solution of ẇ(t) = Aw(t) + λw(t) + εf (t, w(t)) starting at u. Then t (4.45) w n (t) = e λt S A (t)u n + ε n e λ(t s) S A (t s)f (s, w n (s)) ds for n 1 and t [, + ). Putting t := T in the above equation, by (4.44), we infer that (4.46) z n = e λt S A (T )z n + v n (T ), with z n := u n / u n and v n (t) := ε n u n t e λ(t s) S A (t s)f (s, w n (s)) ds for n 1, t [, + ). Observe that, for any t [, T ] and n 1, we have (4.47) v n (t) 1 u n t Me (ω+λ)(t s) F (s, w n (s)) ds mν(ω) 1/2 Me T ( ω + λ ) / u n where the constants M 1 and ω R are such that S A (t) Me ωt for t and ν stands for the Lebesgue measure. Hence (4.48) v n (t) for t [, T ] as n +, and, in particular, set {v n (T )} n 1 is relatively compact. In view of (4.46) (4.49) {z n } n 1 e λt S A (T ) ({z n } n 1 ) + {v n (T )} n 1, 16

18 and therefore, by the compactness of {S A (t)} t we see that {z n } n 1 has convergent subsequence. Without loss of generality we may assume that z n z as n + and z n (x) z (x) for almost every x Ω, where z X is such that z = 1. Passing to the limit in (4.46), as n +, and using (4.48), we find that z = e λt S A (T )z, hence that z Ker (I e λt S A (T )) and finally, by condition (A3), that (4.5) z Ker (λi A) = Ker (λi A ). Thus Remark 3.1 (a) leads to (4.51) z Ker (I e λt S A (t)) for t. From (4.45) we deduce that 1 u n (w n(t) u n ) = e λt S A (t)z n z n + v n (t) for t [, T ], which by (4.48) and (4.51) gives (4.52) 1 u n (w n(t) u n ) for t [, T ] as n +. If we again take t := T in (4.45) and act with the scalar product operation, z, we obtain u n, z = e λt S A (T )u n, z + ε n e λ(t s) S A (T s)f (s, w n (s)), z ds. Since X is Hilbert space, by [21, Corollary 1.1.6], the family {S A (t) } t of the adjoint operators is a C semigroup on X with the generator A, i.e. (4.53) S A (t) = S A (t) for t. Remark 3.1 (a) and (4.5) imply that z Ker (I e λt S A (t)) for t and consequently, by (4.53), z Ker (I e λt S A (t) ) for t. Thus and therefore u n, z = u n, e λt S A (T ) z + ε n e λ(t s) F (s, w n (s)), S A (T s) z ds We have further (4.54) = = = u n, z + ε n F (s, w n (s)), z ds, Ω F (s, w n (s)), z ds = for n 1. f(s, x, w n (s)(x))z (x) dxds f(s, x, w n (s)(x))z (x) dxds + Ω f(s, x, w n (s)(x))z (x) dxds, 17

19 where the sets and Ω are given by (4.4). Given s [, T ], we claim that (4.55) ϕ + n (s) := f(s, x, w n (s)(x))z (x) dx f + (s, x)z (x) dx and (4.56) ϕ n (s) := f(s, x, w n (s)(x))z (x) dx Ω f (s, x)z (x) dx Ω as n. Since the proofs of (4.55) and (4.56) are analogous, we consider only the former limit. We show that every sequence (n k ) has a subsequence (n kl ) such that (4.57) f(s, x, (h nkl (s, x) + z nkl (x)) u nkl )z (x) dx f + (s, x)z (x) dx as n + with h n (s, x) := (w n (s)(x) u n (x))/ u n for x Ω, n 1. Due to (4.52), one can choose a subsequence (h nkl (s, )) of (h nk (s, )) such that h nkl (s, x) for almost every x Ω. Hence (4.58) h nkl (s, x) + z nkl (x) z (x) > as n + for almost every x and consequently (4.59) f(s, x, (h nkl (s, x) + z nkl (x)) u nkl ) f + (s, x) as n + for almost every x. Since z L 2 (Ω) L 1 (Ω) and f is bounded, from the Lebesgue dominated convergence theorem, we have the convergence (4.57) and hence (4.55). Further, for any s [, T ] and n 1, one has (4.6) ϕ + n (s) f(s, x, w n (s)(x))z (x) dx m z (x) dx m z L 1 (Ω). and similarly (4.61) ϕ n (s) m z L 1 (Ω) for t [, T ] and n 1. Since ϕ + n (s) = F (s, w n (s)), max(z, ) and ϕ n (s) = F (s, w n (s)), min(z, ) for s [, T ] and n 1, functions ϕ + n and ϕ n are continuous on [, T ]. Using (4.55), (4.56), (4.6), (4.61) and the dominated convergence theorem, after passing to the limit in (4.54), we infer that (4.62) f + (s, x)z (x) dxds + f + (s, x)z (x) dxds =, which contradicts (4.38), since z N λ and z = 1 and, in consequence, proves (4.43). By the homotopy invariance of topological degree, for any ε (, 1], we have (4.63) deg LS (I Ψ T, W ) = deg LS (I Υ T (1, ), B(, R)) = deg LS (I Υ T (ε, ), B(, R)), 18

20 for all R R 1. Since A has compact resolvents Ker (A λi) = Im (A λi) and therefore, by (A3), X admits the direct sum decomposition X = N λ Im (A λi). Clearly the range and kernel of A are invariant under S A (t) for t, hence putting M := Im (λi A), condition (A2) is satisfied for A λi. Moreover R 1 R and therefore, we also have that g(u) for u N λ with u R 1. Let W := B(, R 1 ), U := W N λ and V := W M. Then g(u) for u Nλ U and clearly (4.64) W U V. Therefore, by Theorem 3.3, there is ε (, 1) such that, for any ε (, ε ] and u (U V ), Υ T (ε, u) u and (4.65) deg LS (I Υ T (ε, ), U V ) = ( 1) µ(λ)+dim N λ deg B (g, U), where µ(λ) is the sum of algebraic multiplicities of eigenvalues of S A λi (T ) in (1, + ). In view of (4.64) and the choice of the number R 1 >, we infer that and, by the excision property, {u U V Υ T (ε, u) = u} W (4.66) deg LS (I Υ T (ε, ), U V ) = deg LS (I Υ T (ε, ), W ). Combining (4.65) with (4.66) yields (4.67) deg LS (I Υ T (ε, ), W ) = ( 1) µ(λ)+dim N λ deg B (g, U), which together with (4.63) implies (4.68) deg LS (I Ψ T, W ) = ( 1) µ(λ)+dim N λ deg B (g, U) and the proof is complete. The following proposition allow us to determine the Brouwer degree of the mapping g. Proposition 4.4. (i) If condition (4.36) holds then there is R > such that g(u) for u N λ with u R and deg B (g, B(, R)) = 1 for R R. (ii) If condition (4.37) holds then there is R > such that g(u) for u N λ with u R and deg B (g, B(, R)) = ( 1) dim N λ for R R. Proof. (i) We begin by proving that there exists R > such that (4.69) g(u), u > for u N λ, u R. 19

21 Arguing by contradiction, suppose that there is a sequence (u n ) N λ such that u n + as n + and g(u n ), u n, for n 1. For every n 1, write z n := u n / u n. Since (z n ) is bounded and contained in the finite dimensional space N λ, it contains a convergent subsequence. Without loss of generality we may assume that there is z N λ with z = 1 such that z n z as n + and z n (x) z (x) as n + for almost every x Ω. Recalling the notational convention (4.4), we have (4.7) g(u n ), z n = g(u n ), z n z + g(u n ), z = f(t, x, u n (x))z (x) dxdt + g(u n ), z n z = Ω + f(t, x, z n (x) u n )z (x) dxdt Ω f(t, x, z n (x) u n )z (x) dxdt + g(u n ), z n z. On the other hand, if we fix t [, T ], then, by the condition (d), we have (4.71) f(t, x, z n (x) u n ) f + (t, x) as n + for almost every x. Since f is assumed to be bounded, by the dominated convergence theorem, (4.71) shows that (4.72) f(t, x, z n (x) u n )z (x) dx f + (t, x)z (x) dx as n. Let ϕ + n : [, T ] R be given by ϕ + n (t) := f(t, x, z n (x) u n )z (x) dx = F (t, u n ), max(z, ) for t [, T ]. The function ϕ + n is evidently continuous and ϕ + n (t) m z L 1 (Ω) for t [, T ]. Applying (4.72) and the dominated convergence theorem, we find that (4.73) f(t, x, z n (x) u n )z (x) dxdt as n +. Proceeding in the same way, we infer that (4.74) Ω f(t, x, z n (x) u n )z (x) dxdt as n +. Since the sequence (g(u n )) is bounded, we see that f + (t, x) dxdt, Ω f (t, x) dxdt, (4.75) g(u n ), z n z g(u n ) z n z as n +. By (4.73), (4.74), (4.75), letting n + in (4.7), we assert that f + (t, x)z (x) dxdt + Ω f (t, x)z (x) dxdt, 2

22 contrary to (4.36). Now, for any R > R, the mapping H : [, 1] N λ N λ given by H(s, u) := sg(u) + (1 s)u for u N λ, has no zeros for s [, 1] and u N λ with u = R. If it were not true, then there would be H(s, u) =, for some s [, 1] and u N λ with u = R, and in consequence, = H(s, u), u = s g(u), u + (1 s) u, u. If s = then = u 2 = R 2, which is impossible. If s (, 1], then g(u), u, which contradicts (4.69). By the homotopy invariance of the topological degree deg B (g, B(, R)) = deg B (H(1, ), B(, R)) = deg B (H(, ), B(, R)) = deg B (I, B(, R)) = 1, and the proof of (i) is complete. (ii) Proceeding by analogy to (i), we obtain the existence of R > such that (4.76) g(u), u < for u R. This implies, that for every R > R, the homotopy H : [, 1] N λ N λ given by H(s, u) := sg(u) (1 s)u for u N λ is such that H(s, u) for s [, 1] and u N λ with u = R. Indeed, if H(s, u) = for some s [, 1] and u N λ with u = R, then = H(s, u), u = s g(u), u (1 s) u, u. Hence, if s (, 1], then g(u), u, contrary to (4.76). If s =, then R 2 = u 2 =, and again a contradiction. In consequence, by the homotopy invariance, deg B (g, B(, R)) = deg B ( I, B(, R)) = ( 1) dim N λ, as desired. Proof Theorem 4.1. Theorem 4.2 asserts that there is an open bounded set W X such that Ψ T (u) u for u X \ W, g(u) for u N λ \ (W N λ ) and (4.77) deg LS (I Ψ T, W ) = ( 1) µ(λ)+dim N λ deg B (g, W N λ ). In view of Proposition 4.4, we obtain, the existence of R > such that W B(, R) and either deg(g, B(, R) N λ ) = 1, when (4.36) is satisfied or deg(g, B(, R) N λ ) = ( 1) dim N λ, in the case of condition (4.37). By the inclusion {u B(, R) N λ g(u) = } W N λ and (4.77) we infer that deg LS (I Ψ T, W ) = ( 1) µ(λ)+dim N λ deg B (g, W N λ ) = ( 1) µ(λ)+dim N λ deg(g, B(, R) N λ ) = ±1. Thus, by the existence property of the topological degree, we find that there is a fixed point of Ψ T and in consequence a T -periodic mild solution of (4.34). 21

23 In the particular case when the linear operator A is self-adjoint and A is a generator of a compact C semigroup {S A (t)} t of bounded linear operators on X, the spectrum σ(a) is real and consists of eigenvalues λ 1 < λ 2 < λ 3 <... < λ k <... (not counting the multiplicities) which form a sequence convergent to infinity. By Proposition 2.5, for every t >, {e λ kt } k 1 is the sequence of nonzero eigenvalues of S A (t) and (4.78) Ker (λ k I A) = Ker (e λ kt I S A (t)) for k 1. In consequence, we see that (A3) holds. Corollary 4.5. Let A be a self-adjoint operator such that A is a generator of a compact C semigroup {S A (t)} t and let f : [, + ) Ω R R satisfy the Landesman Lazer type condition (4.38). If λ = λ k for some k 1, then there is a bounded open set W X such that Ψ T (u) u for u X \ W, g(u) for u N λk \ (W N λk ) and (4.79) deg LS (I Ψ T, W ) = ( 1) d k deg B (g, W N λk ), where d k := k 1 i=1 dim Ker (λ ii A) for k 1. In particular, if either condition (4.36) or (4.37) is satisfied then (4.34) has mild solution. Proof. To see (4.79), it is enough to check that d k = µ(λ k ) + dim N λk for k 1. Since e (λ k λ 1 )T > e (λ k λ 2 )T >... > e (λ k λ k 1 )T are eigenvalues of e λkt S A (T ) which are greater than 1, for k = 1 it is evident that µ(λ k ) = and d 1 = µ(λ k ) + dim N λk. The operator S A (T ) is also self-adjoint and therefore the geometric and the algebraic multiplicity of each eigenvalue coincide. Hence k 1 (4.8) µ(λ k ) = dim Ker (e λit I S A (T )) for k 2. i=1 From (4.78) and (4.8), we deduce that k 1 µ(λ k ) = dim Ker (λ i I A) = d k dim N λk i=1 and finally that d k = µ(λ k ) + dim N λk for every k 1, as desired. The formula (4.79) together with Proposition 4.4 leads to existence of mild solution of (4.34) in the case when condition (4.36) or (4.37) is satisfied. 5 Applications Let Ω R n, n 1, be an open bounded connected set with C 1 boundary. We recall that and, denote, similarly as before, the norm and the scalar product on X = L 2 (Ω), respectively. For u H 1 (Ω), we will denote by D k u, the k-th weak derivative of u. 22

24 Laplacian with the Neumann boundary conditions We begin with the T -periodic parabolic problem u = u + εf(t, x, u) in (, + ) Ω t (5.81) u (t, x) = on [, + ) Ω n u(t, x) = u(t + T, x) in [, + ) Ω, where ε [, 1] is a parameter and f : [, + ) Ω R R is a continuous mapping which is required to satisfy conditions (a), (b) and (c) from the previous section. We put (5.81) into an abstract setting. To this end let A: D(A) X be a linear operator such that A is the Laplacian with the Neumann boundary conditions, i.e. { D(A) := u H 1 (Ω) there is g L 2 (Ω) such that } u h dx = gh dx for h H 1 (Ω), Au := g, where g is as above, and define F : [, + ) X X to be a mapping given by the formula (5.82) F (t, u)(x) := f(t, x, u(x)) for t [, + ), x Ω. Ω Then by the assumptions (a) and (b), it is well defined, continuous, bounded and Lipschitz uniformly with respect to time. Problem (5.81) may be considered in the abstract form Ω (5.83) { u(t) = Au(t) + εf (t, u(t)), t > u(t) = u(t + T ) t where ε [, 1] is a parameter. Solutions of (5.81) will be understand as mild solutions of (5.83). Theorem 5.1. Let g : R R be given by g (y) := Ω f(t, x, y) dxdt for y R. If real numbers a and b are such that a < b and g (a) g (b) <, then there is ε > such that for ε (, ε ], the problem (5.81) admits a solution. Proof. Since the spectrum of A is real, condition (A1) is satisfied as a consequence of Remark 3.1. It is known that A generates a compact C semigroup on X, N := Ker A is a one dimensional space. If we take M := Im A, then M = N and hence A satisfies also condition (A2). Let P : X X be the orthogonal projection onto N given by P (u) := 1 (u, e) e for u X ν(ω) 23

25 where e L 2 (Ω) represents the constant equal to 1 function and ν stands for the Lebesgue measure. Set U := {s e s (a, b)}, V := {u N u < 1} and let g : N N be defined by Then g(u) := P F (t, u) dt for u N. g (y) = ν(ω) K 1 (g(k(y))) for y R, where K : R N is the linear homeomorphism given by K(y) := y e. Since g (a) g (b) <, we have deg B (g, U) = deg B (g, (a, b)) and hence, by Corollary 3.4, there is ε (, 1) such that, for ε (, ε ], problem (5.81) admits a solution as desired. Uniformly elliptic differential operator with the Dirichlet boundary conditions Suppose that a ij = a ji C 1 (Ω) for 1 i, j n and let θ > be such that a ij (x)ξ i ξ j θ ξ 2 for ξ = (ξ 1, ξ 2,..., ξ n ) R n, x Ω. We assume that A: D(A) X is a linear operator given by the formula { D(A) := u H 1 (Ω) there is g L 2 (Ω) such that } a ij (x)d i ud j h dx = gh dx for h H 1 (Ω), Au := g, where g is as above. Ω It is well known that A is self-adjoint and generates a compact C semigroup on X = L 2 (Ω). Let λ 1 < λ 2 <... < λ k <... be the sequence of eigenvalues of A (not counting the multiplicities). We are concerned with a periodic parabolic problem of the form (5.84) u t = D i (a ij D j u) + λ k u + f(t, x, u) in (, + ) Ω u(t, x) = on [, + ) Ω u(t, x) = u(t + T, x) in [, + ) Ω, where λ k is k-th eigenvalue of A and f : [, + ) Ω R R is as above. We write problem (5.84) in the abstract form { u(t) = Au(t) + λk u(t) + F (t, u(t)), t > u(t) = u(t + T ) t where F : [, + ) X X is given by the formula (5.82). An immediate consequence of Corollary 4.5 is the following Theorem 5.2. Suppose that f : [, + ) Ω R R is such that: f + (t, x)u(x) dxdt + f (t, x)u(x) dxdt >, {u>} for any u ker A with u = 1, or f + (t, x)u(x) dxdt + {u>} {u<} {u<} Ω f (t, x)u(x) dxdt <, for any u ker A with u = 1. Then the problem (5.84) admits a T -periodic mild solution. 24

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