Boundary value problems for higher order linear impulsive differential equations

Abstract

In this paper higher order linear impulsive differential equations with fixed moments of impulses subject to linear boundary conditions are studied. Green's formula is defined for piecewise differentiable functions. Properties of Green's functions for higher order impulsive boundary value problems are introduced. An appropriate example of the Green's function for a boundary value problem is provided. Furthermore, eigenvalue problems and basic properties of eigensolutions are considered.

The Sturm-Liouville operator on the space of functions with discontinuity conditions

“…provides us with many theorems of operator theory, concerning the properties of eigenvalues and eigenfunctions, ranging from the spectrum of C0 to the orthogonality relations of its eigensolutions. However, it must be noted that the Green's function for such an operator is significantly different from that of its classical counterparts [14][15][16].…”

Sturm-Liouville Problems with finitely many point $\delta-$interactions and eigen-parameter in boundary condition

“…We note that Sturm-Liouville problems with eigen-dependent boundary conditions and with transmission conditions have been investigated in [1], [3], [9], [13], [11], [12]. Furthermore Green's formula for impulsive differential equation has been studied in [16] and [17].…”

A novel approach to construct the adjoint problem for a first-order functional integro-differential equation with general nonlocal condition

“…The third one is an improved method, which is substantially based on the method of variation of parameters, for the fundamental solution of the problems with discontinuities. By means of this third method, Green's function for some problems has been constructed [15,21,34]. But, as far as we know, the problems with nonlocal condition have been omitted with a few exceptions in this respect.…”