AN APPROXIMATION SOLVABILITY METHOD FOR NONLOCAL DIFFERENTIAL PROBLEMS IN HILBERT SPACES

A new approach is developed for the solvability of nonlocal problems in Hilbert spaces associated to nonlinear differential equations. It is based on a joint combination of the degree theory with the approximation solvability method and the bounding functions technique. No compactness or condensivity condition on the nonlinearities is assumed. Some applications of the abstract result to the study of nonlocal problems for integrodifferential equations and systems of integro-differential equations are then showed. A generalization of the result by using nonsmooth bounding functions is given.


Introduction
Let H be a separable Hilbert space.In this paper we consider a differential equation of the form: x ′ (t) = f (t, x(t)) for a.e.t ∈ [0, T ], (1.1) with a nonlocal condition where f : [0, T ] × H → H and M : C([0, T ]; H) → H are nonlinear and linear maps, respectively.
It is well known (see, e.g., [?,?] and the references therein) that a great variety of partial differential and integro-differential equations can be written in the form (??).
In this paper, by combining the degree theory in abstract spaces with an improvement of the approximation solvability method and the bounding functions technique an existence theorem (see Theorem ??) for problem (??)-(??) is proved.It is then showed how the abstract result can be applied to study various problems of integro-differential equations (including periodic, anti-periodic, mean value and multi-point problems).
Compactness conditions in terms of the strong topology are usually required in order to apply the degree theory for a suitable solution operator.In this paper we obtain existence results in the lack of this compactness both on the nonlinear term f and on the nonlocal operator M .This is possible exploiting compactly embedded Gel'fand triples with a Hilbert space and Hartman-type conditions.Other techniques have previously been employed to avoid assumptions of compactness.For instance in [?] a continuation principle has been used in the weak topology in reflexive Banach spaces and in [?] the concept of weak measure of non compactness is introduced to consider also the case of non reflexive Banach spaces.
The paper is organized in the following way.In the next section we give a brief review of all the methods which will be used to study problem (??)-(??).We explain also the main idea of our technique presented in this paper.Section 3 is devoted to the Leray-Schauder topological degree and the notation.The main result is presented in Section 4. Some applications of the abstract result are given in Section 5 including the periodic problem for an integro-differential equation of the form In Section 6, equation (??) is considered for t ∈ [0, ∞) and with the initial condition x(0) = 0.The existence of a solution which is bounded on the whole interval [0, ∞) is showed.In Section 7 we deal with systems of differential equations in various Hilbert spaces with application to a system of integro-differential equations with non-local conditions ) , u(0, ξ) = ∑ m 1 α j u(t j , ξ), v(0, ξ) = ∫ 1 0 g(t)v(t, ξ) dt, for a.e.t ∈ [0, 1] and for all ξ ∈ Ω ⊂ R n (n ≥ 2), where a, b, c > 0; α j ∈ R; In Section 8 we investigate the existence of a unique solution for all the problems considered in the paper.In the last section the generalization of the main result is presented by using a nonsmooth bounding function.

A review of methods
The technique presented in this paper is based on the bounding functions method and the approximation method.So, let us give a brief review of both of them.

Bounding functions method:
The earliest version of a bounding function is a guiding function which was introduced by Krasnosel'skii and Perov (see, e.g., [?,?, ?]).They generalized the notion of the Lyapunov function to study the existence of periodic solutions of an ODE x ′ (t) = f (t, x(t)), (2.1) where f : R × R n → R n is a (globally) continuous map which is locally Lipschitzian w.r.t. the second argument.The notion of guiding function was then generalized in several directions and applied to various problems.Among a large number of papers on this subject let us recall: Mawhin [?] with studying of functional differential equations; Fonda [?] with the notion of integral guiding function; Capietto and Zanolin [?] for periodic problem in flow-invariant Euclidean Neighborhood Retracts; Górniewicz and Plaskacz [?,?] with the notion of general form of guiding functions for differential inclusions (see also [?]); Lewicka [?] for nonsmooth guiding functions; Kryszewski [?], Kryszewski and Gabor [?] and Loi [?] with the application of the method of guiding functions to bifurcation problems.The backgrounds and applications of the method of guiding functions in nonlinear analysis can be found in the recent monograph [?].However, it seems to be that the method of guiding functions can hardly be applied to study more general classes of boundary value problems than periodic problem.Also it is worth noting that usually the direct applications of the method of guiding functions were connected with finite-dimensional objects.
In [?] (see also [?]) Mawhin introduced the concept of bounding functions, this method was then developed by Gaines and Mawhin [?,?] and by Mawin and Ward [?].We recall in the following the main features of this method.Consider again equation (??) for t ∈ [a, b] with the boundary condition where f and g are continuous maps.The idea for the existence of solutions to (??)-(??) comes from the Leray-Schauder continuation principle [?], following which it is possible to consider the linearized problem with boundary condition (??), where λ ∈ [0, 1] and It is supposed that for each y(•) problem (??)-(??) has a unique solution T (y, λ) and moreover, the solution map is completely continuous.Notice that a fixed point of the map T (•, 1) is a solution of problem (??)-(??).Now, if there exists an open bounded subset then problem (??)-(??) admits a solution x ∈ Ω.
Recall (see [?]) that a C 1 −function V : R n → R is said to be a bounding function to equation (??) if The name bounding function comes from the fact that if there exists a bounding function V and if there exists a fixed point ).Therefore, the set Ω can be taken as the set of all continuous functions x : [a, b] → K.The conditions x(a) / ∈ ∂K and x(b) / ∈ ∂K usually follow from the choice of the boundary map g (see, [?] and the last section of the present paper).To obtain condition (a) in various papers (and in the present one) the authors usually consider a convex set K containing 0. Then the linearized problem is modified so that 0 should be the unique fixed point of T (•, 0), and therefore, deg From the above consideration it is clear that the bounding functions method is useful to study boundary value problems.Moreover, since by applying this method we do not need to evaluate the topological degree of bounding functions, it is possible BLMO An approximation solvability method for nonlocal differential problems in Hilbert spaces 5 to extend this method to infinite-dimensional Banach spaces.Preliminary ideas of the bounding functions methods in the framework of periodic solutions were introduced in Lefschetz [?] for second order equations in R and in Browder [?] for dynamics in Hilbert spaces with monotone nonlinearities w.r.The approximation solvability method : As already mentioned, one of the most effective tools for the investigation of the solvability of equations in Banach spaces is the topological method suggested by Leray and Schauder [?].However, from the practical point of view, it is often more important to study approximable solutions rather than usual solutions since the former ones can be localized by using the approximation methods.For simple description of approximation solvability method, we consider a separable Hilbert space H with a basis {e i } ∞ i=1 , then, denoting with H n the subspace with base {e 1 , • • • , e n }, we approximate the original problem by a family of auxiliary problems by means of the natural projections P n : H → H n (n ∈ N).Precisely, for a given n ∈ N, we prove the existence of a solution in the space W 1,1 (I, H n ) for the problem Then, by a limit argument, we obtain the existence of a solution for the original problem.
About the idea of combining these methods: A joint application of the Leray-Schauder continuation principle with the bounding functions method was recently proposed also in reflexive Banach spaces (see e.g.[?] for the study of the Floquet problem and [?] for the investigation of nonlocal conditions).A regularity assumption is needed, in this context, which is expressed in terms of the Hausdorff measure of noncompactness χ (usually denoted χ-regularity).A further restriction is required, essentially involving the rate of noncompactness of the model (see e.g.[?, condition (5.2)]).
We point out the fact that we prove the existence of a solution of problem (??) without any assumption of monotonicity or compactness nether on the nonlinearity f, nor on the nonlocal operator M.This can be done by the compact embedding of H into E and by the assumption of the continuity of f (t, •) : H → H for a.e.t ∈ I w.r.t. the topology of E, see hypothesis (f 2) below.

Preliminaries and Notation
Let X , Z be Banach spaces.A map Σ : X → Z is said to be completely continuous if it is continuous and maps every bounded subset U ⊂ X into a relatively compact subset of Z.Let us recall that if U is an open bounded subset of X and F : U → X is a completely continuous map such that x ̸ = F (x) for all x ∈ ∂U , then for the corresponding vector field i − F (where i denotes the inclusion map) the Leray-Schauder topological degree deg(i − F, U ) is well-defined (see, e.g.[?,?]).
In the sequel, by (H, ∥ • ∥ H ) we denote a separable Hilbert space which is compactly embedded into a Banach space (E, ∥ • ∥ E ) with the following relation between norms: Throughout the paper, I = [0, T ] and let C(I, H) [L 1 (I, H)] be the space of all continuous [respectively integrable] functions u : I → H with usual norms Consider the space of all absolutely continuous functions u : I → H whose generalized derivatives u ′ belong to L 1 (I, H).It is well known (see, e.g.[?]) that this space can be identified with the Sobolev space W 1,1 (I, H) endowed with the norm and the embedding W 1,1 (I, H) → C(I, H) is continuous.Definition 3.1.Let S ⊆ R be a bounded and measurable subset.A subset A ⊂ L 1 (S, H) is said uniformly integrable if for every ϵ > 0 there is δ > 0 such that Ω ⊂ S and µ(Ω) < δ implies where µ is the Lebesgue measure on R.
Theorem 3.1.Let A ⊂ L 1 (S, H) be a bounded, uniformly integrable subset.Then A is weakly relatively compact in L 1 (S, H).
For each n ∈ N define the map

Main result
To study problem (??)-(??) we assume that:  (i) M x = 0 (the general Cauchy condition x(0) = x 0 can be replaced by condition z(0) = 0 by a transformation z From conditions (f 1) and (f 3) it follows that for every x ∈ C(I, H) the superposition function f (s, x(s)) belongs to L 1 (I, H).
The main result of this paper is the following statement.Theorem 4.1.Let conditions (f 1) − (f 3) and (M ) hold.In addition, assume that Then problem (??)-(??) admits a solution with values in B H (0, R 0 ).
has only the trivial solution in the space W 1,1 (I, H n ).
We are going to show now that, for a given n ∈ N, the problem has a solution in the space W 1,1 (I, H n ).
To this aim, we choose arbitrarily r * ∈ (r 0 , R 0 ) and let has a unique solution x n ∈ W 1,1 (I, H n ): is the solution of (??).It is clear that Step 2. (a) At first, let us show that the map T n has a closed graph in the space Since H is embedded in E, from condition (f 2) we have The convergence is also dominated by (f 3).Passing to the limit m → ∞ in (??) we obtain (b) Now we will show that the set In fact, from (f 3) and the boundedness of the set ) is bounded and equicontinuous in C(I, H n ), and therefore, it is relatively compact in C(I, H n ).So, T n is a closed and compact map, and therefore, it is completely continuous.
(c) Assume that there exists (y n , λ) that is a contradiction.So, t 0 ∈ (0, T ].Therefore, we can choose a sufficiently small ε > 0 such that From the last inequalities it follows that giving the contradiction.Thus, if there is y n ∈ ∂Q (n) such that y n = T n (y n , 1), then y n is a solution to (??).If y n ̸ = T n (y n , 1) for all y n ∈ ∂Q (n) , then T n is a homotopy connecting the maps T n (•, 0) and T n (•, 1).By virtue of the homotopy invariance and normalization properties of the topological degree we have So, for every n ∈ N there exists y n ∈ Q (n) such that y n = T n (y n , 1), and hence y n satisfies (??).
Step 3. Denote f n (t) = f (t, y n (t)).From the condition ∥y n (t)∥ H ≤ r * and according to (f 3) it follows that there exists ν * ∈ L 1 (I, H) such that ∥f n (t)∥ H ≤ ν * (t) for a.e.t ∈ I and n.Therefore, the sequence {f n } is bounded and uniformly integrable in L 1 (I, H).By virtue of Theorem ?? it is relatively weakly compact in The set {y n (0) : n ∈ N} is bounded in H. So, w.l.o.g.we can assume that Define It is easy to see that  By virtue of the Lebesgue dominated convergence theorem we have for every element g ∈ L 1 (I, H).

Now we prove that
⇀ f 0 .To this aim, let Φ : L 1 (I, H) → R be a linear and bounded functional.Hence, there is φ ∈ L ∞ (I, H) such that We have and Therefore, for a.e.t ∈ I we have where ∥φ∥ ∞ is the norm of φ in L ∞ (I, H).
From the Lebesgue dominated convergence theorem it follows that Applying (??) and condition (f 2) we obtain that for a.e.t ∈ I and every ε > 0 there is an integer ) for all i ≥ i 0 , and by the convexity of the set ) f or all n ≥ i 0 .
Therefore, y ′ 0 (t) = f (t, y 0 (t)) for a.e.t ∈ I. Combining with y 0 (0) = M y 0 we obtain that y 0 is a solution to problem (??)-(??).Remark 4.2.Theorem ?? gives us not only the existence of a solution y 0 to problem (??)-(??).It also provides us an important information about this solution, that is the (weak) convergence of the sequence {y n } to y 0 .Since each y n takes its values in a finite-dimensional subspace, we can approximate the solution y 0 by finite-dimensional functions via weak approximation scheme.

Applications to semilinear differential equations in Hilbert spaces
Consider the following semilinear differential equation where A : H → H is a bounded linear operator which is E − E continuous; f and M satisfy (f 1) − (f 3) and (M ), respectively.
Proof.Problem (??) can be substituted with the following problem where f : It is easy to verify that problem (??) satisfies all conditions in Theorem ??, therefore it, and hence problem (??), has a solution.
In order to illustrate the result, we consider two integro-differential equations.The first one is considered on an open bounded domain Ω ⊂ R k (k ≥ 2) with Lipschitz boundary (Example ??), whereas the second one is defined on a given interval (Example ??).Example 5.1.Consider the periodic problem for a.e.t ∈ [0, 1] and all ξ ∈ Ω, where a, b > 0 and f : By a solution to (??) we mean a continuous function u : [0, 1]×Ω → R whose partial derivative ∂u(t,ξ) ∂t exists and satisfies (??).Moreover, we can consider relation (??) as the law of evolution of a dynamical system with the state function u(t, ξ).Our goal can be formulated as finding of the dynamics of the system as the continuous function u(t, ξ) such that at every value t the function u(t, •) belongs to the Sobolev space W 1,2 (Ω).Theorem 5.2.Let conditions (f 1) ′ − (f 2) ′ be satisfied.Then problem (??) has a solution.Moreover, if f (t, 0) ̸ = 0 for all t ∈ [0, 1], then the solution is non-zero.
Proof.Let H = W 1,2 (Ω) and E = L 2 (Ω).It is clear that H is a separable Hilbert space which is compactly embedded in E and for every w ∈ H: where ∥w∥ ∫ Ω w 2 (ξ) dξ and D denotes the derivative (i.e.gradient) of a function with several variables.For each t ∈ [0, 1], set x(t) = u(t, •).Then we can substitute (??) with the following problem x(0) = x(1), (5.4) where A : H → H, Aw = a ∫ Ω w(ξ)dξ, and Notice that the map f is well-defined since where f for all (t, z) ∈ [0, 1] × R, where η is a number between 0 and z.
It is easy to verify that A is a linear bounded operator which is E − E continuous.Let us show that the map f (t, •) satisfies condition (f 2).Let {w n } in H be such that w n E → w 0 .By (f 1) ′ for every (t, ξ) ∈ (0, 1) × Ω we have where η is a number between w n (ξ) and w 0 (ξ).Therefore, Let U ⊂ H be a bounded set.For every w ∈ U we have where for each ξ ∈ Ω, Dw(ξ) is a vector in R k and Consequently, condition (f 3) is satisfied.
To verify condition (f 1) we will apply the Pettis Measurability Theorem (see, e.g. [?]).Notice that H can be identified with its dual space H * .So, in order to apply Pettis Theorem, we have to prove that for every φ ∈ H the map is measurable.To this aim, we will prove that f φ is a Carathéodory map.Fix w ∈ H and consider the map f φ (•, w) : [0, 1] → R. Assume that there is a sequence {t n } ⊂ [0, 1] such that t n → t 0 ∈ [0, 1].Let r n = f φ (t n , w) and r 0 = f φ (t 0 , w).From the continuity property of f and f ′ 2 we have r n → r 0 .Therefore, f φ (•, w) is continuous, and hence, it is measurable.
We will prove now that for every t ∈ [0, 1] the map f φ (t, •) : Then there are subsequences {w n k } and {w m k } of {w n } such that The sets {f (t, w n k )} and {f (t, w m k )} are bounded in H, and so they are weakly relatively compact.W.l.o.g.assume that Therefore, Consequently, f 0 = f 1 , and hence λ 0 = Λ 0 , i.e. f (t, •) is continuous.Thus, the map f φ is Carathéodory, and so, condition (f 1) is satisfied.Now for w ∈ H and for a.e.t ∈ [0, 1] we have By virtue of (f 1) ′ − (f 2) ′ and (??) the following estimation is true Applying Theorem ?? we obtain the existence of a solution of (??), and therefore, problem (??) has a solution.
Example 5.2.Consider the mean value problem for an integro-differential equation of the form: Notice that the map f is well-defined since where k ′ 1 = ∂k ∂s .It is clear that A is a linear bounded operator which is E − E continuous and the operator M satisfies condition (M ).The map f can be written as and g : H → H is defined by Notice that the map f differs from the one in Example ?? only by the linear term bw, b ∈ R, here missing.Hence, as in the cited example it is possible to prove that satisfies conditions (f 1) − (f 3).Let us show that the map g satisfies conditions (f 1) − (f 3).

At first, let {w
), and hence, condition (f 2) is satisfied.
Let D ⊂ H be a bounded set, for any w ∈ D we have where To verify condition (f 1), we prove that the map g is H − H continuous.For this, let {w n } ⊂ H be such that w n H → w 0 .We have where N is the constant from (f 2) ′ and β = max t∈[0,1] |f (t, 0)|.

BLMO An approximation solvability method for nonlocal differential problems in Hilbert spaces 19
Remark 5.1.Under similar assumptions we can consider problems (??) and (??) with various boundary conditions (periodic, anti-periodic, mean value or multi-point conditions).
According to (f 3) there is By applying Theorem ?? in the interval [0, 1] we obtain a subsequence By an induction argument, for every p > 1 it is possible to get a sequence { x From the compact embedding H → E we have Fix t and take n > t.Notice that From (f 2), (??) and since By virtue of (??) we have for a.e.τ ∈ [0, t].Therefore, passing to the limit we obtain Thus, x 0 is a solution to problem (??) with ∥x 0 (t)∥ H ≤ R 0 for all t ∈ [0, ∞).

Systems of differential equations
Let H i (i = 1, 2) be separable Hilbert spaces which are compactly embedded in Banach spaces E i , respectively.Consider the product spaces H = H 1 × H 2 and E = E 1 × E 2 with the norms: and It is clear that the embedding H → E is compact and H is a separable Hilbert space with the inner product ⟨ w, w Consider now a nonlocal b.v.p. associated to a system of differential equations where Assume that, for i = 1, 2: (h1) the maps f i are (globally) measurable; (h2) for a.e.
With z = (x, y), problem (??) can be rewritten as where f : I × H → H and M : C(I, H) → H are defined by and By using conditions (h1) − (h4), it is easy to verify that the maps f and M satisfy (f 1) − (f 3) and (M ), respectively.Applying Theorem ?? we easily obtain the following assertion.
To illustrate the result let us consider the following system of integro-differential equations for a.e.t ∈ [0, 1] and for all ξ ∈ Ω, where By a solution to problem (??) we mean a pair (u, v) consisting of continuous functions u, v : [0, 1] × Ω → R whose partial derivatives ∂u(t,ξ) ∂t and ∂v(t,ξ) ∂t exist and satisfy (??).Moreover, we can consider relations (??) as the law of evolution of a feedback control system with the state function u(t, ξ) and the control function v(t, ξ).Our goal can be formulated as the finding of the state and control as continuous functions u(t, ξ) and v(t, ξ) such that at every value t the functions u(t, •) and v(t, •) belong to the Sobolev space W 1,2 (Ω).
We assume the following conditions.
Then we can reduce (??) to BLMO An approximation solvability method for nonlocal differential problems in Hilbert spaces 23 the following problem where and M 1 , M 2 : C(I, H 1 ) → H 1 are given by It is clear that M 1 and M 2 are linear bounded maps with Notice that for every (t, ξ) and for all z = (x, y) ∈ C(I, H); x, y ∈ C(I, H 1 ).Following the lines of the proof of Theorem ??, we obtain that the maps f 1 and f 2 satisfy conditions (h1) − (h3).Now for w = (w 1 , w 2 ) ∈ H and for t ∈ [0, 1] we have ⟨ w, f (t, w) On the other hand, } .

BLMO
An approximation solvability method for nonlocal differential problems in Hilbert spaces 25 Applying Theorem ?? we obtain the existence of a solution to (??), and so, we conclude that problem (??) has a solution.

Uniqueness results
We now examine the uniqueness of the solutions for the problems seen in the previous sections.To this aim we need to consider stronger H-regularity assumptions on the term f, however we are able to weaken the assumption (f 1).Precisely, we introduce the following condition Proof.The existence follows by Theorem ??.Assume by contradiction the existence of two solutions y 1 , y 2 to problem (??)-(??) in B H (0, R 0 ).We have that Hence for any t ∈ I, Hence, by Theorem ??, under condition (f 1 ′ ), (f 2), (f 4), (M ) and assuming ∥M ∥e ∥A∥T e ∥η∥1 < 1, Problem (??) admits a unique solution.Moreover, under the same hypotheses (f 1 ′ ), (f 2), (f 4) and (M ) it is possible to prove the existence of a unique solution in B H (0, R 0 ) for Problem (??) too.Indeed, assume by contradiction the existence of two solutions x 1 , x 2 of Problem (??) in B H (0, R 0 ).Define x 1 (n) for t ≥ n, and x 2 (n) for t ≥ n.
Hence, we have that for every n ∈ N the functions x(0) = 0.

BLMO
An approximation solvability method for nonlocal differential problems in Hilbert spaces 27 Finally, as in Section ??, let H i (i = 1, 2) be separable Hilbert spaces which are compactly embedded in Banach spaces E i , respectively.Consider the product spaces for each ω 1 , ω 2 ∈ B H (0, R 0 ) and for a.e.t ∈ I, with R 0 as in (f 4).

Non-smooth bounding function in a Hilbert space
Let H be a separable Hilbert space which is compactly embedded in a Banach space E. We consider again problem (??)-(??): A function V is said to be locally Lipschitzian if for every x ∈ U there exists ε > 0 such that B H (x, ε) ⊂ U and the restriction V | B H (x,ε) is Lipschitzian.It is easy to see that if V is locally Lipschitzian, then for every x ∈ U and for all w ∈ H the following limit lim inf exists and is finite.
) means that there exists a subsequence of spaces {H nm } such that for all n m the relation lim inf V is continuously differentiable, then for every w ∈ H the Frétcher derivative ∇V (w) of V at w can be identified with an element in H. Hence, for a.e.t ∈ I and for every n ∈ N: Therefore, in case V ∈ C 1 (I, H), condition (V 2) can be written as  (∂K).Notice that a trivial example of projectively homogenous potential ∇V (w) = w, w ∈ H, was used in Section 4 to obtain our main result.Since for every bounding function V of equation (??) there exist an open bounded convex set K and a number ε > 0 such that (V 1) − (V 2) are satisfied, in the sequel (for short) we will call V a (K, ε)−bounding function.
The following assertion illustrates the geometric sense of a bounding function.has a fixed point x nm , i.e. x nm = T nm (x nm , λ) for some λ ∈ (0, 1), such that x nm (0) / ∈ ∂K nm , then x nm (t) ∈ K nm for all t ∈ I.
Proof.From the definition of a bounding function it follows that there exists a subsequence of spaces {H nm } such that for a.e.(∂K nm ).Now, assume that the fixed point x nm touches the boundary ∂K nm .Since x nm (0) / ∈ ∂K nm we can choose t 0 ∈ (0, T ] such that x nm (t 0 ) ∈ ∂K nm and x nm (t) ∈ K nm for sufficiently small t < t 0 .From the locally Lipschitz property of V it follows that there exists δ > 0 such that the restriction of V on B H ( x nm (t 0 ), δ ) is Lipschitzian with constant L > 0. We can choose δ ∈ (0, ε) such that x nm (t) ∈ B H ( x nm (t 0 ), δ ) ∩ K nm for all t ∈ (t 0 − δ, t 0 ).It is easy to see that the function g nm (t) = V (x nm (t)) is absolutely continuous in t ∈ (t 0 − δ, t 0 ), and so g ′ nm (t) exists for a.e.t ∈ (t 0 − δ, t 0 ).Hence, ∫ t0 t0−δ g ′ nm (s)ds = V (x nm (t 0 )) − V (x nm (t 0 − δ)) = −V (x nm (t 0 − δ)) ≥ 0.

Proof.
Let assumption (i) holds.Assume that x n = T n (x n , λ) and x n (0) ∈ ∂K n .Therefore, λP n M x n ∈ ∂K n .Since M x n ∈ K we have P n M x n ∈ K n .From the convexity of the set K and the assumption that 0 ∈ K it follows that λP n M x n = x n (0) ∈ K n , for λ ∈ (0, 1).That is the contradiction.

Remark 4 . 1 .
(a) Condition (f 1) can be easily obtained if f is a Carathéodory map, i.e., for every w ∈ H the function f (•, w) : I → H is measurable and for a.e.t ∈ I the map f (t, •) : H → H is continuous.(b) The class of boundary value problems with the operator M satisfying condition (M ) is sufficiently large.In particular, it includes the following well-known problems: provided ∥w∥ H > β|Ω| b − N , where β = max [0,1] |f (t, 0)| and |Ω| denotes the Lebesgue measure of Ω.
for a.e.t ∈ I, x(0) = M x, where f : I × H → H satisfies conditions (f 1) − (f 3) and M : C(I, H) → H is a bounded linear map.Let U ⊂ H be an open subset.A function V : U → R is said to be Lipschitzian with constant L > 0 if

Definition 9 . 1 .
A locally Lipschitzian functional V : H → R is said to be a bounding function for equation (??), if there exist ε > 0 and an open bounded convex subset

′Proposition 9 . 1 .
nm (s)ds < 0, giving the contradiction.Assume that the operator M satisfies the following condition:(M ) ′ For every sufficiently large n ∈ N, if x n is a fixed point of the solution mapT n : C(I, K n ) × (0, 1) → C(I, H n ) of the linearized problem { x ′ (t) = λP n f (t, y(t)) for a.e.t ∈ I, x(0) = λP n M y, then x n (0) / ∈ ∂K n .Let us consider some sufficient conditions that provide condition (M ) ′ .The operator M satisfies condition (M ) ′ if the set K contains 0 and at least one of the following assumptions is fulfilled.
t. the state variable.Concerning the most important developments of this method: Mawhin and Thompson [?] for the introduction of Hartman-type conditions (see (ii) above) which are strictly located on ∂K; Taddei [?] for nonsmooth bounding functions in finite-dimensional spaces; Loi and Obukhovskii [?] for application of this method to generalized periodic problem in finite-dimensional spaces; Zanolin [?] for generalized definition of bound set; Loi, Kornev, Obukhovskii and Zecca for the extensions of this method to equations and inclusions in Hilbert spaces (see, [?,?,?]); Andres, Malaguti and Taddei [?] for bounding functions in Banach spaces; Benedetti, Malaguti and Taddei [?] for bounding functions in Banach spaces with weak topology; Benedetti, Taddei and Väth [?] for sufficient conditions for invariant sets for nonlocal semilinear differential inclusions.
.1) Let {e n } ∞ n=1 be the orthonormal basis of H and for every n ∈ N, let H n be the n−dimensional subspaces of H with the bases {e k } n k=1 and P n be the natural projections of H onto H n .
for all t ∈ I, we can repeat previous reasoning also for {P n f n = y ′n } and obtain that the set {y ′ n } is weakly relatively compact in L 1 (I, H).Again w.l.o.g.we assume that From (??) we obtain y 0 (0) = M y 0 .On the other hand, the weak convergence y n (t) ⇀ y 0 (t) in H for all t implies and hence, condition (f 2) is satisfied.