Let α be an orientation-preserving homeomorphism of [0, ∞] onto itself with only two fixed points at 0 and ∞, whose restriction to R + = (0, ∞) is a diffeomorphism, and let U α be the isometric shift operator acting on the Lebesgue spaceWe establish criteria of the two-sided and one-sided invertibility of functional operators of the formon the spaces L p (R + ) under the assumptions that the functions log α and a k for all k ∈ Z are bounded and continuous on R + and may have slowly oscillating discontinuities at 0 and ∞. The unital Banach algebraObtained criteria are of two types: in terms of the two-sided or one-sided invertibility of so-called discrete operators on the spaces l p and in terms of conditions related to the fixed points of α and the orbits {α n (t) : n ∈ Z} of points t ∈ R + .

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We present new modifications to Newton's method for solving nonlinear equations. The analysis of convergence shows that these methods have fourth-order convergence. Each of the three methods uses three functional evaluations. Thus, according to Kung-Traub's conjecture, these are optimal methods. With the previous ideas, we extend the analysis to functions with multiple roots. Several numerical examples are given to illustrate that the presented methods have better performance compared with Newton's classical method and other methods of fourth-order convergence recently published.