Given (v_{0}in X) and (gin mathcal{C}^{1}([0, tau ];mathbb{R} )), find (fin mathcal{C}([0,tau ];mathbb{R} )) and a strict solution (vin mathcal{C}^{1}([0,tau ];X)) to the degenerate Cauchy problem textstylebegin{cases} frac{dMv}{dt}=Lv(t)+f(t z,&0leq tleq tau, Mv(0)=Mv_{0}, end{cases} (1) satisfying the additional condition phi bigl[ Mv(t bigr]=g(t),quad 0leq tleq tau.
Then (omega_in W_) is a strict solution of the inverse problem (1.5 - 1.6 1.5 - 1.6nly if (vandhi(x, y,t;onlya_) equif0), a.e. on (Omega_{T}).
In order to establish our multiplicity result, we introduce the notion of a strict solution-tube of (1.1) which will permit one to obtain solutions satisfying (|x t -v(t)|x t -v)) for all (t in[0,1]).
The fact that ((v_{j},r_{j})) is a strict solution-tube of (1.1), when (j in{1,2}), permits us to get more precision on the localization of the solutions of (3.1 j ).
To this aim, we introduce the notion of a strict solution-tube.
Obviously, a strict solution-tube is a solution-tube of (1.1).
In order to obtain the existence of at least three solutions of (1.1), we introduce the notion of a strict solution-tube of (1.1).
We say that ((v,r)) is a strict solution-tube of ( 1.1 ) if the following conditions hold: (i) there exists a l.s.c.
Observe that the periodic boundary condition and the fact that ((v_{j},r_{j})) is a strict solution-tube of (1.1) imply that begin{aligned} bigl| x(a -v_{j}(a -v_{j &lebigl| x(b)-v_{j}(b)bigr| - r_{j}(b) + bigl| v_{j}(b)-v_{j}(a bigr| + r_{j}(b) &lebigl|(a) - delta_{j}.
Finally, let us assume the initial value (w_{0}) of the form (25) and (w_{0}in mathcal{R}(B^{k+1})). Then there exists a unique strict solution ((w,f)) to the identification problem ((mathcal{IP}2)) such that win C^{1}bigl([0,tau ];Xbigr),qquad fin mathcal{C}^{k} bigl([0,tau ]; mathbb{R}bigr).
