On a solvability of one non-linear operator equation

Author: jemal rogava
Co-authors: Mikheil Tsiklauri
Keywords: Non-linear operator equation, Kirchhoff equation
Annotation:

The following operator equation is considered in the Hilbert space H: T(u)=Iu+a(||B||)Au=f, where A is a self-adjoint, positively defined (generally unbounded) operator with the domain D(A) which is everywhere dense in H; B is square root of A; I is identity operator; a(s) is continuous, increasing function defined on positive half-axis (including an arbitrary point), besides there exists the positive constant c such that a(s) is greater than or equal to c; f is known function from H, u is unknown function from D(A). It is true the following fact: if the range of operator T(.) is dense in H, then for any f from H the equation T(u)=f has a unique solution. We note that the considered equation is related to an abstract analogue of the classical Kirchhoff equation for a string.