Modification of the Lyapunov-Schmidt method and the stability of solutions of differential equations with a singular operator of finite index multiplying the derivative. (English. Russian original) Zbl 0815.34056

Loginov, B. V.; Kim-Tyan, L. R.; Rusak, Yu. B.

Russ. Acad. Sci., Dokl., Math. 47, No. 3, 599-603 (1993); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 330, No. 6, 687-690 (1993).

Let \(E_ 1\) and \(E_ 2\) be Banach spaces. We consider the equation (1) \(A {dx \over dt} = Bx + f(x,t)\); \(\| f(x,t) \| = 0(\| x \|)\), \(\| x \| \to 0\), where \(A,B\): \(E_ 1 \to E_ 2\) are closed linear operators such that \(A\) is an operator of finite index which has a complete generalized \(B\)-Jordan system and the spectrum \(\sigma_ A (B)\) of the generalized eigenvalue problem \((B - \mu A) \varphi = 0\) lies in the left half-plane. This paper studies the stability of the trival solution of equation (1) by using a modification of the well-known Lyapunov-Schmidt method. A stability principle for a stationary solution of the linearized problem is also given. No proofs are given in the paper.

Reviewer: A.Petrusel (Cluj-Napoca)

MSC:

34G20	Nonlinear differential equations in abstract spaces
34D20	Stability of solutions to ordinary differential equations
34D05	Asymptotic properties of solutions to ordinary differential equations
34C23	Bifurcation theory for ordinary differential equations
47A53	(Semi-) Fredholm operators; index theories

Keywords:

Banach spaces; operator of finite index; stability; Lyapunov-Schmidt method