Sufficient conditions for regular solvability of an arbitrary order operator-differential equation with initial-boundary conditions

Nashat Faried,Mohamed A Labeeb,Abdel Baset I Ahmed

doi:10.1186/s13662-020-02557-5

Nashat Faried, Mohamed A Labeeb + Show 1 more

Open Access

https://doi.org/10.1186/s13662-020-02557-5

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Abstract

On this paper, for an arbitrary order operator-differential equation with the weight e^{frac{-alpha t}{2}}, alpha in (-infty ,+ infty ), in the space W^{n+m}_{2}(R_{+};H), we attain sufficient conditions for the well-posedness of a regular solvable of the boundary value problem. These conditions are provided only by the operator coefficients of the investigated equation where the leading part of the equation has multiple characteristics. We prove the connection between the lower bound of the spectrum of the higher-order differential operator in the main part and the exponential weight and also obtain estimations of the norms of operator intermediate derivatives. We apply the results of this paper to a mixed problem for higher-order partial differential equations (HOPDs).

Highlights

The theory of initial-boundary value problems of operator-differential equations in a Banach or Hilbert space is of great value and secures the possibility of looking at ordinary and partial differential operators [27]
We prove the connection between the lower bound of the spectrum of the higher-order differential operator in the main part and the exponential weight and obtain estimations of the norms of operator intermediate derivatives
1 Introduction The theory of initial-boundary value problems of operator-differential equations in a Banach or Hilbert space is of great value and secures the possibility of looking at ordinary and partial differential operators [27]

Summary

Introduction

The theory of initial-boundary value problems of operator-differential equations in a Banach or Hilbert space is of great value and secures the possibility of looking at ordinary and partial differential operators [27]. It is worth mentioning that the principal parts of the investigated equations have multiple characteristics and appear in applications, for instance, by modeling the stability of the plates from the plastic and in particular, in the dynamics problems of arches and rings [25, 26]. The solvability of initial boundary value problems for higher-order operator-differential equations has been researched by many authors, for example, A. In a separable Hilbert space H, we study the following initial-boundary value problem: md nd n+m dn+m–j. All derivatives here are perceived in the sense of distributions theory

H e–αt dt

Conclusion