Modified Sturm-Liouville systems

Two-point boundary conditions linear in a parameter

“…If the boundary conditions are written in vector form (1) s Theorem 2.1 of [7] established that if Condition (A) holds then jP(λ) and Q(\) must necessarily be linear in λ. It is to be noted that the boundary conditions associated with the problems discussed in [1], [2], [5], and [6] are each linear in λ and their coefficients satisfy Condition (A).…”

“…A necessary and sufficient condition that Condition (A) hold for a set of linearly independent boundary conditions (1) is that the 2n x 2n matrix \\M 0 NΛ (2) U N\\ have rank n + p, where p is the rank of the n x 2n matrix || MjiViH. Moreover, in this case P(X) and Q(X) must be linear in λ.…”

“…n^ \\ has rank p -r and its rows are linearly independent 388 of the rows of | | Λf 0 JVo| | , and m'f and n [ 2) are each r x n matrices such that the r x 2n matrix ||m{ 2) n[ 2) || has rank r and its rows are linearly independent of the p -r rows of ||mi 1} fίj"!! and, furthermore, linearly dependent only on the rows of |!M 0 ΛΓ 0 ||.…”

Asymptotic Approximations of Eigen-Functions for Regular Sturm-Liouville Problems with Eigenvalue Parameter in the Boundary Condition for Integrable Potential

“…In this paper we consider the boundary value problem (1) τy:= −y + qy = λy, q(t) is integrable on [a, b],…”

Regularity results and exponential growth for pullback attractors of a non-autonomous reaction–diffusion model with dynamical boundary conditions

“…Its use dates back at least to 1901 in the context of heat transfer in a solid in contact with a moving fluid (see the pioneering work by Peddie [36] in 1901, by March and Weaver [34] in 1928, by Peek [37] in 1929, by Langer [28] in 1932, and by Bauer [7] in 1953).…”