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Iterated Cauchy and Dirichlet problems with the Bessel operator in Banach space. (English. Russian original) Zbl 0988.34046
Russ. Math. 43, No. 8, 1-8 (1999); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1999, No. 8, 3-10 (1999).
Consider the operator-differential equation of order $$2n>2$$, $(B_k-A)^nu(t)=A_0^nu(t),\quad t>0,\tag{1}$ in a Banach space $$E$$. Here, $$A_0\in {\mathcal B}(E)$$, $$B_ku(t)=u''(t)+\frac kt u'(t)$$ for $$t>0$$ and $$k>0$$, and $$A$$ satisfies the following condition: the Cauchy problem $$B_ku(t)=Au(t)$$, $$u(0)=x_0$$, $$u'(0)=0,$$ has a unique exponentially bounded solution.
Here, the Cauchy problem $\lim_{t\to 0}(B_k-A)^ju(t)=x_{j+1},\qquad \lim_{t\to 0}((B_k-A)^ ju(t))'=0\tag{2}$ for the equation (1) is investigated. As is well known, the solvability of a higher-order differential equation cannot be stated in the general case. For this special kind of equation, conditions for (1),(2) to be well posed are obtained, and a formula for the solution is given.
Moreover, the iterated Dirichlet problem $(B_m+A)^nw(t)=A_0^nw(t)\text{ for }t>0,\;w_i(0)=x_i,\;\sup_{i\geq 0}|w_i(0)|\leq M,\tag{3}$ with $$w_1(t)=w(t)$$, $$w_i(t)=(B_m+A)w_{i-1}(t)$$, $$i=2,\dots,n$$, is studied. Existence and uniqueness conditions for (3) are found, and the form of the solution is given.

##### MSC:
 34G10 Linear differential equations in abstract spaces
##### Keywords:
existence; uniqueness