Unified Convergence Analysis of Two-Step Iterative Methods for Solving Equations

We present a local convergence analysis of two methods in order to approximate a locally unique solution of an equation in a Banach space setting. Earlier studies have shown convergence using Taylor expansions and hypotheses reaching up to the sixth derivative (or even higher order derivatives) although only the first derivative appears in these methods. We only show convergence using hypotheses on the first derivative. We also provide computable: error bounds, radii of convergence as well as uniqueness results for the solution based on Lipschitz constants not given in earlier studies. The computational order of convergence is also used to determine the order of convergence. Finally, numerical examples are also provided to show that our results apply to solve equations in cases where earlier studies cannot apply.

In this paper, we give a necessary and sufficient condition under which a variational inequality defined on unbounded sets in a Banach space has a solution. Furthermore, we establish a necessary and sufficient condition under which the minimax equality on unbounded sets is true.

A "purely geometric" proof of the Lindenstrauss-Troyanski result ([2], [6]) on strongly exposed points of weakly compact sets in Banach spaces is given.I would like to thank the referee for his suggestions.We consider a Banach-space X, \\ ||.If G X and if e > 0, then B(,e) = {x G X; \\x -|| < e}.The closed convex hull of a set A G X, is denoted c(A).Let C be a convex subset of X, then Ce is the set of the extremal points of C. If F is a convex, weakly compact, F a convex, closed, bounded set of X and J a closed subinterval of [0,1 ], (K, B,J) denotes the closed, convex setWe have to introduce a few geometrical definitions, referring to [4].Suppose that C is a nonempty, bounded, closed and convex subset of X.Let"strongly exposed" if there exists / G X* (\\f\\ = 1) such that Ve > 0, 3a > 0 with G S(fa, C) G B(e).Let S be the set of all/ G X* such that 11/11 = 1 and/strongly exposes some point of C. Proposition I.If K is a nonempty, convex, weakly-compact subset of X and if B is convex, closed and bounded with K fl B = 0, then the set D = c(K U F) = (K,B, [0,1]) has the following property: Ve > 0, 3 G K such that c(D\B(,e)).First we observe that it is sufficient to prove the proposition for X separable.Indeed, suppose D does not have the required property.Then there exists e > 0 with Vx G K: x G c(D\B(x,e)) and hence, Vx G K, 3A(x) G D with the properties: x G c(A(x)), A(x) n B(x, e) = 0 and A(x) is countable.By induction we construct a sequence (Kn,Bn)n where Kn is countable in K and Bn is countable in F. Let (K0,B0) = ({x},0), where x is some element of K. Suppose we already found (Kn,Bn).Consider Vy G UxeKnA(x) an element ky in K and by in B, with y G c({ky,by)).Let

We present local and semilocal convergence results for some iterative algorithms in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. In earlier studies to operator involved is assumed to be at least once Frechet-differentiable.

A Steffensen type method of two steps in Banach spaces with applications

On the semi-local convergence of Halley’s method under a center-Lipschitz condition on the second Fréchet derivative

The aim of this paper is to expand the applicability of a fast iterative method in a Banach space setting. Moreover, we provide computable radius of convergence, error bounds on the distances involved and a uniqueness of the solution result based on Lipschitz-type functions not given before. Furthermore, we avoid hypotheses on high order derivatives which limit the applicability of the method. Instead, we only use hypotheses on the first derivative. The convegence order is determined using the computational order of convergence or the approximate order of convergence. Numerical examples where earlier results cannot be applied to solve equations but our results can be applied are also given in this study.

In this paper, the convergence and dynamics of improved Chebyshev-Secant-type iterative methods are studied for solving nonlinear equations in Banach space settings. Their semilocal convergence is established using recurrence relations under weaker continuity conditions on first-order divided differences. Convergence theorems are established for the existence-uniqueness of the solutions. Next, center-Lipschitz condition is defined on the first-order divided differences and its influence on the domain of starting iterates is compared with those corresponding to the domain of Lipschitz conditions. Several numerical examples including Automotive Steering problems and nonlinear mixed Hammerstein-type integral equations are analyzed, and the output results are compared with those obtained by some of similar existing iterative methods. It is found that improved results are obtained for all the numerical examples. Further, the dynamical analysis of the iterative method is carried out. It confirms that the proposed iterative method has better stability properties than its competitors.

This paper is concerned with a class of problem of best simultaneous approximations from $RS$-sets in Banach spaces $X$. It is shown that the best simultaneous approximations from an $RS$-set is strongly unique in the case when $X$ is a real Banach space, and strongly unique of order $\alpha \geq 2$ in the case when $X$ is a complex Banach space.

In this paper, we first introduce a Bregman prox-convexity notion as the generalization of the convexity for Bregman prox-convexity in Banach spaces. We verify the single-valudeness and Bregman firm nonexpansiveness of the Bregman proximity operator of the functions and provide examples of the notion. Using this notion, we show that the Bregman proximal point scheme remains convergent under appropriate conditions on the functions to be minimized in the setting of Banach spaces.

In this paper, we define the prox-regularity for functions on Banach spaces by adapting the original definition in R n . In this context, we establish a subdifferential characterization and show that qualified convexly C 1,+-composite functions and primal lower nice functions belong to this class, as already known in the setting of Hilbert spaces. We also study, in a geometrical point of view, the epigraphs of prox-regular functions. The subdifferential characterization allows us to show that some Moreau-envelope-like regularizations of such functions are of class C 1 in the context of certain uniformly convex spaces.

ABSTRACTFor a bounded closed convex subset C of a Banach space X with the origin as an interior point, we study the convexity issue of proximinal sets G in Banach spaces in the sense of generalized best approximation determined by the Minkowski functional generated by C. Some sufficient conditions such as the smoothness and compactly locally uniform convexity of C, or the weakly uniform convexity of are provided for a proximinal set G to be convex.

We study the notion of (I)-generating introduced by V. Fonf and J. Lindendstrauss and a related notion of (I)-envelope. As a consequence of our results we get an easy proof of the James characterization of weak* compactness in Banach spaces with weak angelic dual unit ball and an easy proof of the James characterization of reflexivity within a large class of spaces. We also show by an example that the general James theorem cannot be proved by this method.

In this paper, we propose a local convergence analysis of an eighth order three-step method to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Further, we also study the dynamic behaviour of that scheme. In an earlier study, Sharma and Arora (2015) did not discuss these properties. Furthermore, the order of convergence was shown using Taylor series expansions and hypotheses up to the fourth order derivative or even higher of the function involved which restrict the applicability of the proposed scheme. However, only the first order derivatives appear in the proposed scheme. To overcome this problem, we present the hypotheses for the proposed scheme maximum up to first order derivative. In this way, we not only expand the applicability of the methods but also suggest convergence domain. Finally, a variety of concrete numerical examples are proposed where earlier studies can not be applied to obtain the solutions of nonlinear equations on the other hand our study does not exhibit this type of problem/restriction.

Weak convergence of random sets in Banach spaces