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}).
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).
Moreover, in [15] the representation of the space X as a direct sum X=mathcal{N}(A)oplus X_{1}=mathcal{R}(A)oplus X_{2}, where M is assumed to have a closed kernel (mathcal{N}(M)) and a closed range (mathcal{R}(M)), and (X_{1}), (X_{2}) are closed subspaces of X, is essential to guarantee a unique strict solution.
(2) More precisely, we are concerned with the determination of the conditions under which we can identify (fin mathcal{C}([0,tau ]; mathbb{R} )) such that v is a strict solution to the above problem, i.e., Mvin mathcal{C}^{1}bigl([0,tau ];Xbigr), qquad Lvin mathcal{C} bigl([0, tau ];Xbigr).
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.
Cyprus's difficulty in overcoming recession while its main bank is in such a mess should serve as a warning against strict solutions that smack of puritanism rather than pragmatism.
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.
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 ).
