Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables

Abstract

: We prove that homogeneous problem for PDE of second order in time variable, and generally infinite order in spatial variables with local two-point conditions with respect to time variable, has only trivial solution in the case when the characteristic determinant of the problem is nonzero. In another, opposite case, we prove the existence of nontrivial solutions of the problem, and we propose a differential-symbol method of constructing them.

Analytical method to study a mathematical model of wave processes under two­point time conditions

“…Studying a set of non-trivial solutions to the homogeneous problem (6), (7) was addressed in paper [10] for s=1, in work [11] for s=1 and conditions (9), as well as in article [12] for s≥1.…”

“…are also pairwise different and belong to the set M, Set M is to be selected so that class ,M K C is the class of unique solvability of problem (10),(7).Similarly to papers[45,46], we consider an entire function of vector-parameter well as the fact that function (12) is a quasipolynomial with respect to τ for any . s ν ∈C Let L be the set of zeros of function(12), then put \ , in equation(10), which predetermines a wave process, is to be considered a quasi-polynomial, which belongs to , ,…”

Two-point nonlocal problem for a weak nonlinear differential-operator equation

“…for any u ∈ G 0 , i.e., H 0 (G 0 ) ⊂ G 0 . Using inequality (12), property (P 3) and formula H 0 (u)−H 0 (u ′ ) = εL −1 P N 0 (f (u)−f (u ′ )) for any u, u ′ ∈ G 0 , we get the estimate…”

“…Introduction. Problems with nonlocal conditions for partial differential equations represent an important part of the modern theory of differential equations [11,12,15]. Such problems are mainly ill posed in the Hadamard sence, and their solvability is connected with the problem of small denominators that arise in the construction of solutions [1,5,6,15].…”

Two-Point Problem for Homogeneous Partial Differential Equation in Unbounded Strip

“…Задачі з двоточковими умовами для рівнянь із частинними похідними у необмежених областях вивчено в роботах [10][11]. У цих роботах використано узагальнений метод відокремлення змінних і на його підставі встановлено умови розв'язності задач (1), (2) у класах квазіполіномів або у класах цілих функцій з експоненційним зростанням на…”