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In this work we study the problem of the existence of bifurcation in the solution set of the equation F(x, λ)=0, where F: X×Rk→Y is a C2-smooth operator, X and Y are Banach spaces such that X⊂Y. Moreover, there is given a scalar product 〈·,·〉: Y×Y→R1 that is continuous with respect to the norms in X and Y. We show that under some conditions there is bifurcation at a point (0, λ0)∈X×Rk and we describe the solution set of the studied equation in a small neighbourhood of this point.

Local properties of the solution set of the operator equation in Banach spaces in a neighbourhood of a bifurcation point