Abstract
The paper deals with the inverse problem for the Sobolev type equation of the second order in Banach spaces. The introduction contains a problem statement and the historiography of Sobolev type equations. The second part includes preliminary information based on the results of the theory of higher-order Sobolev type equations. In the third part the initial problem is reduced to the inverse regular and singular problems. A theorem of unique solvability of regular problem is formulated and proved. Using the results of the third part, the solution for the singular problem is obtained in the fourth part. The sum of regular and singular solutions is a solution to the original problem, thus a theorem on the unique solvability of the inverse problem for Sobolev type equation of the second order is stated and proved
Highlights
Let U, F, Y be Banach spaces, operator χ :[0,T ] → L(Y; F ), functions M :[0,T ] → F,Ψ :[0,T ] → Y, operator C ∈ L(U ; F )
The inverse is a problem of finding a pair of functions v ∈C2 ([0,T ];U ) and q ∈C2 ([0,T ];Y ) from relations (1)–(3)
Keldysh equation (4) was reduced to the equivalent first-order Sobolev-type equation, which has been studied by methods described in [10]
Summary
The paper deals with the inverse problem for the Sobolev type equation of the second order in Banach spaces. The introduction contains a problem statement and the historiography of Sobolev type equations. The second part includes preliminary information based on the results of the theory of higher-order Sobolev type equations. In the third part the initial problem is reduced to the inverse regular and singular problems. A theorem of unique solvability of regular problem is formulated and proved. Using the results of the third part, the solution for the singular problem is obtained in the fourth part. The sum of regular and singular solutions is a solution to the original problem, a theorem on the unique solvability of the inverse problem for Sobolev type equation of the second order is stated and proved
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