Fourth order elliptic operator-differential equations with unbounded operator boundary conditions in the Sobolev-type spaces

Eman S Al-Aidarous,Elvin S Rzayev,Hassan A Zedan,Araz R Aliev

doi:10.1186/s13661-015-0453-y

Abstract

Conditions for well-posed and unique solvability of a non-homogeneous boundary value problem for a class of fourth order elliptic operator-differential equations with an unbounded operator in boundary conditions are found in this work. Note that these solvability conditions are sufficient, and they are expressed only in terms of the properties of operator coefficients of the boundary value problem. Besides, the estimates for the norms of intermediate derivative operators in a Sobolev-type space are obtained, and their close relationship with the solvability conditions is established.

Highlights

1 Introduction Many applied problems of mathematical physics require the study of spectral problems with a polynomial appearance of a parameter in the boundary conditions [ – ], while, in corresponding inverse problems, the unknown coefficients appearing in the equation and boundary conditions are found using known spectra [ – ]
Note that the well-posed and unique solvability and Fredholmness of the boundary value problems for second and third order operator-differential equations with operator boundary conditions have been widely studied both on a finite interval and on the half-axis
It should be noted that the solvability of the boundary value problems for operator-differential equations of fourth and higher orders in case where the coefficients in the boundary conditions are only complex numbers has been extensively studied in [ – ]

Summary

Introduction

Many applied problems of mathematical physics require the study of spectral problems with a polynomial appearance of a parameter in the boundary conditions [ – ], while, in corresponding inverse problems, the unknown coefficients appearing in the equation and boundary conditions are found using known spectra [ – ]. Consider the following boundary value problem in the space H: u( )(t) + A u(t) + Aju( –j)(t) = f (t), t ∈ R+, j=. ) under some restrictions on its operator coefficients To achieve this purpose, we use the estimates for the norms of intermediate derivative operators by the norm of an operator generated by the principal part of the considered equation and the given boundary conditions. : W ,K (R+; H) → L (R+; H), j = , , , , are continuous (see [ ]), the norms of these operators can be estimated through the norm P u L (R+;H) The need for these estimates arises when one tries to establish solvability conditions for the boundary value problem

Then we obtain or

Then ζ φ

The operator is defined