APPROXIMATION STRUCTURES AND APPLICATIONS TO EVOLUTION EQUATIONS

The purpose of this paper is to describe some approximation structures for nonlinear operators in Banach spaces as well as a number of applications to evolution equations. In Sections 2 and 3, we prove results concerning the nonlinear A-proper operators, such as the positive decomposition, the A-properness of the Dirac mass operator, and a method of approximating arbitrary continuous operators by A-proper mappings. We also prove a generalized Leray-Schauder Principle via the A-proper mapping theory. The third section is devoted to the existence of approximative solutions for linear evolution equations in Banach spaces. This is done via the same approximation schemes as in the case of A-proper operators. Our results complement somehow the classical results on C0-semigroups and show what happens beyond the standard hypotheses. Given a separable Banach space E with Schauder basis, we construct (via an approximation scheme) a linear operator A on E such that the differential equation


Introduction
The purpose of this paper is to describe some approximation structures for nonlinear operators in Banach spaces as well as a number of applications to evolution equations.In Sections 2 and 3, we prove results concerning the nonlinear A-proper operators, such as the positive decomposition, the A-properness of the Dirac mass operator, and a method of approximating arbitrary continuous operators by A-proper mappings.We also prove a generalized Leray-Schauder Principle via the A-proper mapping theory.
The third section is devoted to the existence of approximative solutions for linear evolution equations in Banach spaces.This is done via the same approximation schemes as in the case of A-proper operators.Our results complement somehow the classical results on C 0 -semigroups and show what happens beyond the standard hypotheses.
Given a separable Banach space E with Schauder basis, we construct (via an approximation scheme) a linear operator A on E such that the differential equation converges uniformly to a C ∞ function u and (du n /dt − Au n ) n converges uniformly to 0, but u is not a solution.A complementary phenomenon is described in Theorem 4. 2. For E = C([0,1] × [0,1]), one shows the existence of a wildly discontinuous linear operator A : E → E, for which the initial value problem has a generalized solution whatever are f ∈ C 1 ([0,T],E), u 0 ∈ E, and T > 0.

A-properness via approximation schemes
Let X and Y be two separable Banach spaces and let be an approximation scheme, where X n ⊂ X, Y n ⊂ Y are linear subspaces with dimX n = dimY n < ∞, P n : X n → X are the canonical isometries, and An operator T : X → Y is named A-proper with respect to Γ provided that the operators T n = Q n TP n are continuous, and any bounded sequence {x k ; x k ∈ X nk } such that T nk x k → f , where f ∈ Y , has a subsequence {x kj } so that x kj → x 0 and Tx 0 = f .Definition 2.1.The scheme Γ is called of type (C) if and only if it satisfies the following conditions: Then there exists a continuous noncompact operator S : X → Y , with bounded R(S), such that for any A-proper operator T : X → Y which is uniformly continuous on bounded subsets, the sum operator Proof.Let δ ∈ (0,1).We may construct the sequences {x i } ⊂ X and {z i } ⊂ Y such that (2.2) We define y i ∈ Y by A. Duma and C. Vladimirescu 687 Then (2.4) Now, choose µ ∈ (0,δ/2) and the sequence {ε n } ⊂ (0,µ] such that ε n → 0. We define the functions {ϕ i } ⊂ C(X,R) by (2.5) Clearly, and 0 ≤ ϕ i (x) ≤ 1 for all i ∈ N * and all x ∈ X.
We introduce the operator S : X → Y defined by We claim that S is continuous.Indeed, let u 0 ∈ X.We have the following two alternatives.
We obtain and thus m = n.Then we have (2.12) a contradiction.Consequently, there exists λ > 0 such that so that S is continuous at u 0 .Now, let T : X → Y be an A-proper operator which is uniformly continuous on bounded subsets.We claim that the operator T + S is A-proper too.Let {n k } ⊂ N * be such that n k → ∞ and let {w nk ; w nk ∈ X nk } be a bounded sequence such that which yields The following two cases are possible: (1) there exist {k l } ⊂ N * , k l → ∞ as l → ∞, and N * such that i kl < m (l ≥ 1).

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Now, since T is A-proper, it follows that {w nk(l) } l has no convergent subsequence and, since T + S is continuous, it will be A-proper too; (2) i k → ∞ (k → ∞).Now, if there exists a sequence {k j } ⊂ N * such that k j → +∞ ( j → +∞) and ϕ ik(j) (w nk(j) ) > 0 (j ≥ 1), then we obtain (2.21) We also have Now, since T is uniformly continuous on bounded subsets, we have We also have and, from (2.23), T ik(j) w nk(j) → f and, from (2.24), T ik(j) x ik(j) → f , which is impossible because T is A-proper and {x n } has no convergent subsequence.Then there exists k 0 ∈ N * such that k ≥ k 0 implies ϕ ik (w nk ) = 0. Consequently, S nk (w nk ) = 0, so that (2.26) Thus, {w nk } has convergent subsequences and, since T + S is continuous, it will be A-proper too.
Theorem 2.3.Let E be a separable Banach space such that E * is endowed with a sequence of finite-dimensional subspaces {X n } n≥1 , with the following properties: (a) for every n, there is a projection where S E * denotes the closed unit ball of E * , endowed with the weak-star topology.
Then the operator Proof.For every n ∈ N * , we choose µ n ∈ U(X n ) with µ n = n.We define the finite-dimensional subspace Y n ⊂ C(S E * ) * by Y n = U(X n ).Then, clearly, dimY n = dimX n < +∞.We also introduce the sequence of continuous operators Q n : C(S E * ) * → Y n given by (2.27) where j : E → C(S E * ) is the canonical isometrical embedding.
We claim that the operator T : S E * ⊂ E * → C(S E * ) * , Tx = ε x , is A-proper with respect to the approximation scheme (2.28) where for some µ ∈ C(S E * ) * .Then we obtain Therefore, there is an x ∈ S E * such that µ = δ x .It follows that and clearly Tx = µ, which completes the proof.

The generalized Leray-Schauder principle
Let X be a Banach space and let D be an open bounded subset of X with 0 ∈ D. Theorem 3.1.Let K : D → X be a compact operator.Suppose that there exists a pseudoboundary Γ of D such that Then K has at least one fixed point.

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Proof.We consider the A-proper homotopy H : [0,1] × D * → X given by Then H(t,z) = 0 with t ∈ (0,1) and z ∈ ∂D * implies Kx = λx with λ = t −1 and x = Az ∈ Γ, a contradiction.Clearly, 0 which assures the existence of a z ∈ D * such that (A − KA)(z) = 0. Consequently, Az ∈ D is a fixed point of K and the theorem is proved.
In what follows, X will be a Banach space and ∆ ⊂ X will be an open bounded subset satisfying the following property: and a sequence of linear subspaces X n ⊂ V such that (a) the sequence ∆ n = ∆ ∩ X n converges to {p} in the hyperspace 2 ∆V of ∆ V (see, e.g., [3]), (b) We have the following approximation result.
Theorem 3.2.For each continuous operator T : ∆ ⊂ X → 2 , there exists a sequence of operators {T n ; T n : ∆ → 2 } that are A-proper with respect to various approximation schemes for the pair (X, 2 ) such that Since T is continuous and ∆ V is compact, so is Q.We take ε > 0. We denote by {e n } n≥1 the standard orthonormal basis of 2 and by {Π n } n≥1 the associated sequence of orthoprojectors.Then there exists Now, we define the operator T : ∆ → 2 given by the formula We now choose f ∈ 2 \{0} and a sequence of subspaces and f ∈ Y n (n ∈ N * ).We then define a sequence of continuous nonlinear operators where We claim that the operator T is A-proper with respect to the approximation scheme (3.12) where P n : X n → X are the canonical isometries.Indeed, we observe that if x ∈ ∆ n , then Q n Tx = 0. On the other hand, let {x m ; x ∈ ∆ m } be a sequence such that for some y ∈ 2 .It follows that necessarily y = T p (= T p).Moreover, the property (ᏼ) implies that x m → p.The proof is complete.
Remark 3.3.The above argument can be easily modified in order to conclude that all the operators T n are A-proper with respect to the same approximation scheme.
In what follows, BCA denotes the class of all bounded continuous A-proper operators with respect to a given approximation scheme.Proposition 3.4.Let X and Y be two Banach spaces, Y being ordered by the cone Y + such that intY + = ∅.Let D ⊂ X be an open bounded nonempty subset.Then, for each operator A ∈ BCA(D,Y ), there exist two operators A ± ∈ BCA(D,Y + ) such that Let r * = min{r,(1/2) y 0 }.Since clearly y 0 = 0, it follows that r * > 0. Then we define the operators A ± by The proof is done.

Approximative solutions for evolution equations
Our first goal is to show that for every separable Banach space E with Schauder basis there exists a linear operator A : E → E with the following two properties: (A1) the problem has a solution u of class C ∞ for each choice of the initial datum u(0) = u 0 in E; (A2) for each u 0 ∈ E, u 0 = 0, there exist u and In order to prove this result, we will need the following construction.Let {e n } be a Schauder basis for E, let E 0 = span{e n }, and let {e n } ∪ {b x ; x ∈ E 0 } be a Hamel basis for E. We define the linear operator A : E → E by Ae n = 0 (n ∈ N * ) and Ab x = x (x ∈ E 0 ).It is easy to see that A 2 = 0.
Proof of (A2).We consider the sequence of linear continuous projections where {e * n } ⊂ E * is the associated sequence of coefficient functionals to {e n }.We choose a sequence of integers {k n } ⊂ N * , k n → ∞, such that and consider the functions for each t ≥ 0. Finally, we suppose that In the remainder of this section, we put I = [0,1], E = C(I 2 ), and keep fixed a number T > 0. It is known that E ≈ C(I)⊗ ε C(I) (the completion of the injective tensor product, see [1]).
We will need the following classical result.

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We consider the initial value problem u(0) = u 0 , (4.9) where A : D(A) = E → E is a linear operator and f : [0,T] → E, u 0 ∈ E are given.where P : E → C(I) ⊗ ε C(I) is an algebraic projection.Now, let u 0 ∈ E and f ∈ C 1 ([0,T],E).Since the linear subspace F = span{b n } ⊗ ε C(I) is dense in E (see [1, page 280]), there is a sequence {q n } ⊂ F such that q n → u 0 .We remark that F ⊂ Ker S.
We define {u k } ⊂ C 1 ([0,T],E) by Then, defining u(t) = u 0 (t ∈ [0,T]), we have u k → u uniformly because We also have We say that Γ ⊂ D is a pseudoboundary of D if there exist a Banach space Y , an open bounded subset D * ⊂ Y , and a continuous A-proper operator A : D * → D with Deg(A,D * ,0) = {0} such that Γ = A(∂D * ).