Coercive resolvability of weakly degenerate differential equations with an unbounded operator coefficient (Q1390419)

The paper is devoted to the problem \[ t^{2\alpha}v''(t)- Av(t)= f(t),\quad 0< t\leq 1,\quad v(0)= v(1)= 0,\tag{1} \] where \(A\) is an unbounded operator in a Banach space \(E\) and \(\alpha\in(0, 1)\) (this case is called a weak degeneracy). Moreover, \(A\) is assumed to be weakly positive in \(E\) which means that it is densely defined in \(E\) and for all nonpositive \(\lambda\) the operator \(\lambda I-A\) has a bounded inverse such that \(\|(\lambda I- A)^{-1}\|\leq M(1+|\lambda|)^{-1}\). The problem (1) is considered in the Bochner space \(B_p= B_p((0, 1),E)\), \(p\in (1, \infty)\). The problem (1) is coercively resolvable (c.r.) if it is uniquely resolvable for any \(f\in B_p\) and satisfies some additional estimate. The main result asserts that if the nondegenerate problem \(u''(t)- Au(t)= \varphi(t)\), \(u(0)= u(T)= 0\) is c.r. in \(B_p((0, T),E)\) for \(p\in ((1-\alpha)^{-1}, \infty)\) then the problem (1) is c.r. in \(B_p\).