We characterize a broad class of semilinear dense range operatorsGH:W→Zgiven by the following formula,GHw=Gw+H(w),w∈W, whereZ,Ware Hilbert spaces,G∈L(W,Z), andH:W→Zis a suitable nonlinear operator. First, we give a necessary and sufficient condition for the linear operatorGto have dense range. Second, under some condition on the nonlinear termH, we prove the following statement: IfRang(G)¯=Z, thenRang(GH)¯=Zand for allz∈Zthere exists a sequence{wα∈Z:0<α≤1}given bywα=G*(αI+GG*)-1(z-H(wα)), such that limα→0+{Guα+H(uα)}=z. Finally, we apply this result to prove the approximate controllability of the following semilinear evolution equation:z′=Az+Bu(t)+F(t,z,u(t)),z∈Z,u∈U,t>0, whereZ,Uare Hilbert spaces,A:D(A)⊂Z→Zis the infinitesimal generator of strongly continuous compact semigroup{T(t)}t≥0inZ,B∈L(U,Z), the control functionubelongs toL2(0,τ;U), andF:[0,τ]×Z×U→Zis a suitable function. As a particular case we consider the controlled semilinear heat equation.
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