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The General Differential Operators Generated by a Quasi-Differential Expressions with their Interior Singular Points

El-sayed Ibrahim, Sobhy — 1999

Serdica Mathematical Journal

The general ordinary quasi-differential expression M of n-th order with complex coefficients and its formal adjoint M + are considered over a regoin (a, b) on the real line, −∞ ≤ a < b ≤ ∞, on which the operator may have a finite number of singular points. By considering M over various subintervals on which singularities occur only at the ends, restrictions of the maximal operator generated by M in L2|w (a, b) which are regularly solvable with respect to the minimal operators T0 (M ) and T0...

On L w 2 -quasi-derivatives for solutions of perturbed general quasi-differential equations

Sobhy El-sayed Ibrahim — 1999

Czechoslovak Mathematical Journal

This paper is concerned with square integrable quasi-derivatives for any solution of a general quasi-differential equation of n th order with complex coefficients M [ y ] - λ w y = w f ( t , y [ 0 ] , ... , y [ n - 1 ] ) , t [ a , b ) provided that all r th quasi-derivatives of solutions of M [ y ] - λ w y = 0 and all solutions of its normal adjoint M + [ z ] - λ ¯ w z = 0 are in L w 2 ( a , b ) and under suitable conditions on the function f .

The spectra of general differential operators in the direct sum spaces

Sobhy El-sayed Ibrahim — 2004

Czechoslovak Mathematical Journal

In this paper, the general ordinary quasi-differential expression M p of n -th order with complex coefficients and its formal adjoint M p + on any finite number of intervals I p = ( a p , b p ) , p = 1 , , N , are considered in the setting of the direct sums of L w p 2 ( a p , b p ) -spaces of functions defined on each of the separate intervals, and a number of results concerning the location of the point spectra and the regularity fields of general differential operators generated by such expressions are obtained. Some of these are extensions or generalizations...

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