Acceleration of Convergence in Dontchev’s Iterative Method for Solving Variational Inclusions

2000 Mathematics Subject Classification: 47H04, 65K10.In this paper we investigate the existence of a sequence (xk ) satisfying 0 ∈ f (xk )+ ∇f (xk )(xk+1 − xk )+ 1/2 ∇2 f (xk )(xk+1 − xk )^2 + G(xk+1 ) and converging to a solution x∗ of the generalized equation 0 ∈ f (x) + G(x); where f is a function and G is a set-valued map acting in Banach spaces

Superquadratic method for generalized equations under relaxed conditions

We present a new approach to study the convergence of some superquadratic iterative method in Banach space for solving variational inclusions under different assumptions used in [12, 14, 2]. Here, we relax Lipschitz, Holder or center–H ¨ older type conditions by introducing ¨ ω–type–conditioned second order Frechet derivative. Under this condi- ´ tions, we show that the sequence is locally superquadratically convergent if some Aubin continuity property is satisfied. In particular, we recover a quadratic and a cubic convergence.&nbsp

Enlarging The convergence domain of secant-like methods for equations

We present two new semilocal convergence analyses for secant-like methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. These methods include the secant, Newtons method and other popular methods as special cases. The convergence analysis is based on our idea of recurrent functions. Using more precise majorizing sequences than before we obtain weaker convergence criteria. These advantages are obtained because we use more precise estimates for the upper bounds on the norm of the inverse of the linear operators involved than in earlier studies. Numerical examples are given to illustrate the advantages of the new approaches. © 2015, Mathematical Society of the Rep. of China. All rights reserved

Enlarging the convergence domain of secant-like methods for equations

We present two new semilocal convergence analyses for secant-like methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. These methods include the secant, Newton's method and other popular methods as special cases. The convergence analysis is based on our idea of recurrent functions. Using more precise majorizing sequences than before we obtain weaker convergence criteria. These advantages are obtained because we use more precise estimates for the upper bounds on the norm of the inverse of the linear operators involved than in earlier studies. Numerical examples are given to illustrate the advantages of the new approaches

Role for Dynamin in Late Endosome Dynamics and Trafficking of the Cation-independent Mannose 6-Phosphate Receptor

It is well established that dynamin is involved in clathrin-dependent endocytosis, but relatively little is known about possible intracellular functions of this GTPase. Using confocal imaging, we found that endogenous dynamin was associated with the plasma membrane, the trans-Golgi network, and a perinuclear cluster of cation-independent mannose 6-phosphate receptor (CI-MPR)–containing structures. By electron microscopy (EM), it was shown that these structures were late endosomes and that the endogenous dynamin was preferentially localized to tubulo-vesicular appendices on these late endosomes. Upon induction of the dominant-negative dynK44A mutant, confocal microscopy demonstrated a redistribution of the CI-MPR in mutant-expressing cells. Quantitative EM analysis of the ratio of CI-MPR to lysosome-associated membrane protein-1 in endosome profiles revealed a higher colocalization of the two markers in dynK44A-expressing cells than in control cells. Western blot analysis showed that dynK44A-expressing cells had an increased cellular procathepsin D content. Finally, EM revealed that in dynK44A-expressing cells, endosomal tubules containing CI-MPR were formed. These results are in contrast to recent reports that dynamin-2 is exclusively associated with endocytic structures at the plasma membrane. They suggest instead that endogenous dynamin also plays an important role in the molecular machinery behind the recycling of the CI-MPR from endosomes to the trans-Golgi network, and we propose that dynamin is required for the final scission of vesicles budding from endosome tubules

On the semilocal convergence of efficient Chebyshev–Secant-type methods

AbstractWe introduce a three-step Chebyshev–Secant-type method (CSTM) with high efficiency index for solving nonlinear equations in a Banach space setting. We provide a semilocal convergence analysis for (CSTM) using recurrence relations. Numerical examples validating our theoretical results are also provided in this study