A Degenerate Two-point Problem

Abstract

This paper is concerned with the degenerate two point problem in a (complex) Hilbert space H, with inner product (·, ·) and norm ∣ · ∣,

$$ \left( {\frac{d}{{dt}} + \in } \right)({M_{0}}y) = - {l_{0}}y - {B_{0}}u + f(t),0 < t < \tau , $$

((1.1))

$$ \left( { - \frac{d}{{dt}} + \in } \right)\left( {{M_{1}}u} \right) = {B_{1}}y - {L_{1}}u + g(t),0 < t < \tau , $$

((1.2))

$$ {M_{0}}y\left( 0 \right) = {M_{0}}{y_{0}},{\text{ }}{M_{1}}u\left( \tau \right) = {M_{1}}{u_{r}}, $$

((1.3))

where ε is a non negative constant, B _i ] ∈ L(H), the space of all bounded linear operators from H into itself, L _i *, Mi are closed linear operators from if into itself, 0 ∈ p(L _i ), with domain D(L _i ) ⊂ D(M _i ),i = 0,1, f,g ∈ L ²(0, τ;H), yo ∈ D(L ₀),U _T ∈ D(L ₁) are given. No assumption is made on the invertibil-ity of the operators M _i . We shall say that the pair (y,u) is a solution to (1.1)-(1.3) if y(.) ∈ L ²(0,τ;D(L ₀ )), u(.) ∈ L ²(0,τ;D(L ₁)), M0y(.) ∈H ¹(0, τ; H), M ₁ u ∈H ¹(0, τ; H), the equations (1.1), (1.2) hold a.e. in (0, τ) and (1.3) is satisfied.

*

Work partially supported by the Italian M.I.U.R. and by University of Bologna Funds for selected research topics.

†

The second author is a member of G.N.A.M.P.A. of the Italian Istituto di Alta Maternatica (I.N.d.A.M.).