Abstract
Abstract The paper studies the question of solution existence to a nonlinear equation in the degenerate case. This question is studied for three particular boundary value problems for ordinary and partial second-order differential equations. The so-called p-regularity theory is applied to these purposes as an effective apparatus to investigate many nonlinear mathematical, physical and numerical problems. All results obtained in the paper are based on the constructions of this theory whose basic concepts were described by Tret’yakov. We recall the main definitions and theorems of p-regularity theory and illustrate the results by examples including singular boundary value problems. In the first and second ones, the description of solutions by a tangent cone at an initial point are given. In the third example, we formulate a sufficient condition for p-regularity ( p = 2 ), which can be tested using the notion of resultant. MSC: 47J05, 34B15, 34B16.
Highlights
In the paper, we study the question of solution existence to a nonlinear equation
The authors consider the bifurcation problem F(x, λ) = Lx + (λ – λ )x + R(x) =, where L is the elliptic selfadjoint operator on a suitable Banach space Y of functions, with another suitable Banach space of function X - the domain of L ⊂ Y, R : X → Y is a smooth map with R( ) = and R ( ) =, λ - is an eigenvalue of L of multiplicity n, x ∈ X and λ ∈ R
Our examples are special cases of the above problem, and we use p-regularity theory to prove the existence of solutions and give an approximative description of the solution set
Summary
The tangent cone to the solution set coincides with the kernel of the first derivative of the map F. Definition A mapping F is called p-regular at x∗ along h, p > , if Im p(x∗, h) = Y (i.e., the operator p(x∗, h) is surjection).
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