### A Construction of Monotonically Convergent Sequences from Successive Approximations in Certain Banach Spaces.

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The Newton-Kantorovich approach and the majorant principle are used to provide new local and semilocal convergence results for Newton-like methods using outer or generalized inverses in a Banach space setting. Using the same conditions as before, we provide more precise information on the location of the solution and on the error bounds on the distances involved. Moreover since our Newton-Kantorovich-type hypothesis is weaker than before, we can cover cases where the original Newton-Kantorovich...

We study the convergence of the iterations in a Hilbert space $V,{x}_{k+1}=W\left(P\right){x}_{k},W\left(P\right)z=w=T(Pw+(I-P)z)$, where $T$ maps $V$ into itself and $P$ is a linear projection operator. The iterations converge to the unique fixed point of $T$, if the operator $W\left(P\right)$ is continuous and the Lipschitz constant $\u2225(I-P)W\left(P\right)\u2225<1$. If an operator $W\left({P}_{1}\right)$ satisfies these assumptions and ${P}_{2}$ is an orthogonal projection such that ${P}_{1}{P}_{2}={P}_{2}{P}_{1}={P}_{1}$, then the operator $W\left({P}_{2}\right)$ is defined and continuous in $V$ and satisfies $\u2225(I-{P}_{2})W\left({P}_{2}\right)\u2225\le \u2225(I-{P}_{1})W\left({P}_{1}\right)\u2225$.

From Kantorovich’s theory we present a semilocal convergence result for Newton’s method which is based mainly on a modification of the condition required to the second derivative of the operator involved. In particular, instead of requiring that the second derivative is bounded, we demand that it is centered. As a consequence, we obtain a modification of the starting points for Newton’s method. We illustrate this study with applications to nonlinear integral equations of mixed Hammerstein type.

From Kantorovich’s theory we present a semilocal convergence result for Newton’s method which is based mainly on a modification of the condition required to the second derivative of the operator involved. In particular, instead of requiring that the second derivative is bounded, we demand that it is centered. As a consequence, we obtain a modification of the starting points for Newton’s method. We illustrate this study with applications to nonlinear...