A Stirling-like method with Hölder continuous first derivative in Banach spaces

In this paper, the convergence of a Stirling-like method used for finding a solution for a nonlinear operator in a Banach space is examined under the relaxed assumption that the first Fréchet derivative of the involved operator satisfies the Hölder continuity condition. Many results exist already in the literature to cover the stronger case when the second Fréchet derivative of the involved operator satisfies the Lipschitz/Hölder continuity condition. Our convergence analysis is done by using recurrence relations. The error bounds and the existence and uniqueness regions for the solution are obtained. Finally, two numerical examples are worked out to show that our convergence analysis leads to better error bounds and existence and uniqueness regions for the fixed points.     

Semilocal convergence for the Super-Halley’s method

The semilocal convergence of Super-Halley’s method for solving nonlinear equations in Banach spaces is established under the assumption that the second Frëchet derivative satisfies the ω-continuity condition. This condition is milder than the well-known Lipschitz and Hölder continuity conditions. The importance of our work lies in the fact that numerical examples can be given to show that our approach is successful even in cases where the Lipschitz and the Hölder continuity conditions fail. The difficult computation of second Frëchet derivative is also avoided by replacing it with the divided difference containing only the first Frëchet derivatives. A number of recurrence relations based on two parameters are derived. A convergence theorem is established to estimate a priori error bounds along with the domains of existence and uniqueness of the solutions. The R-order convergence of the method is shown to be at least three. Two numerical examples are worked out to demonstrate the efficacy of our method. It is observed that in both examples the existence and uniqueness regions of solution are improved when compared with those obtained in [7].     

Semilocal convergence of a continuation method with Hölder continuous second derivative in Banach spaces

In this paper, the semilocal convergence of a continuation method combining the Chebyshev method and the convex acceleration of Newton’s method used for solving nonlinear equations in Banach spaces is established by using recurrence relations under the assumption that the second Frëchet derivative satisfies the Hölder continuity condition. This condition is mild and works for problems in which the second Frëchet derivative fails to satisfy Lipschitz continuity condition. A new family of recurrence relations are defined based on two constants which depend on the operator. The existence and uniqueness regions along with a closed form of the error bounds in terms of a real parameter α∈[0,1]$\alpha \in [0,1]$ for the solution x^∗$x^{\ast }$ is given. Two numerical examples are worked out to demonstrate the efficacy of our approach. On comparing the existence and uniqueness regions for the solution obtained by our analysis with those obtained by using majorizing sequences under Hölder continuity condition on F^″$F^{″}$, it is found that our analysis gives improved results. Further, we have observed that for particular values of the α$\alpha$, our analysis reduces to those for the Chebyshev method (α=0$\alpha =0$) and the convex acceleration of Newton’s method (α=1)$(\alpha =1)$ respectively with improved results.     

Convergence of a continuation method under the Hölder condition of the first derivative

The aim of this paper is to describe a continuation method combining the Chebyshev’s method and the convex acceleration of Newton’s method to solve nonlinear equations in Banach spaces. The semilocal convergence analysis of the method is established using recurrence relations under the assumption that the first Fréchet derivative satisfies the Hölder continuity condition. This condition is milder than the usual Lipschitz condition. The computation of second Fréchet derivative is also avoided. Two real valued functions and a real sequence are used to establish a convergence criterion of R-order (1+p), where p∈(0,1] is the order of the Hölder condition. An existence and uniqueness theorem along with the closed form of error bounds is derived in terms of a real parameter α∈[0,1]. Two numerical examples are worked out to demonstrate the efficacy of our convergence analysis. For both the examples, the convergence conditions hold for the Chebyshev’s method (α=0). However, for the convex acceleration of Newton’s method (α=1), these convergence conditions hold for the first example but fail for the second example. For particular values of α, our method reduces to the Chebyshev’s method (α=0) and the convex acceleration of Newton’s method (α=1).     

Convergence of a third order method for fixed points in Banach spaces

In this paper, the semilocal convergence of a third order Stirling-like method used to find fixed points of nonlinear operator equations in Banach spaces is established under the assumption that the first Fréchet derivative of the involved operator satisfies ??-continuity condition. It turns out that this convergence condition is weaker than the Lipschitz and the H?lder continuity conditions on first Fréchet derivative of the involved operator. The importance of our work lies in the fact that numerical examples can be given to show that our approach is successful even in cases where Lipschitz and H?lder continuity conditions on first Fréchet derivative fail. It also avoids the evaluation of second order Fréchet derivative which is difficult to compute at times. A priori error bounds along with the domains of existence and uniqueness of a fixed point are derived. The R-order of the method is shown to be equal to (2p?+?1) for p????(0,1]. Finally, two numerical examples involving nonlinear integral equations are worked out to show the efficacy of our approach.     

Semilocal convergence and R-order for modified Chebyshev-Halley methods

In this paper, we study the semilocal convergence and R-order for a class of modified Chebyshev-Halley methods for solving non-linear equations in Banach spaces. To solve the problem that the third-order derivative of an operator is neither Lipschitz continuous nor Hölder continuous, the condition of Lipschitz continuity of third-order Fréchet derivative considered in Wang et al. (Numer Algor 56:497–516, 2011) is replaced by its general continuity condition, and the latter is weaker than the former. Furthermore, the R-order of these methods is also improved under the same condition. By using the recurrence relations, a convergence theorem is proved to show the existence-uniqueness of the solution and give a priori error bounds. We also analyze the R-order of these methods with the third-order Fréchet derivative of an operator under different continuity conditions. Especially, when the third-order Fréchet derivative is Lipschitz continuous, the R-order of the methods is at least six, which is higher than the one of the method considered in Wang et al. (Numer Algor 56:497–516, 2011) under the same condition.     

A third order method for fixed points in Banach spaces

This paper deals with a third order Stirling-like method used for finding fixed points of nonlinear operator equations in Banach spaces. The semilocal convergence of the method is established by using recurrence relations under the assumption that the first Fréchet derivative of the involved operator satisfies the Hölder continuity condition. A theorem is given to establish the error bounds and the existence and uniqueness regions for fixed points. The R-order of the method is also shown to be equal to at least (2p+1) for p∈(0,1]. The efficacy of our approach is shown by solving three nonlinear elementary scalar functions and two nonlinear integral equations by using both Stirling-like method and Newton-like method. It is observed that our convergence analysis is more effective and give better results.     

Semilocal convergence of a family of third-order methods in Banach spaces under Hölder continuous second derivative

In this paper, the semilocal convergence of a family of multipoint third-order methods used for solving F(x)=0$F\left(x\right)=0$ in Banach spaces is established. It is done by using recurrence relations under the assumption that the second Fréchet derivative of F$F$ satisfies Hölder continuity condition. Based on two parameters depending upon F$F$, a new family of recurrence relations is defined. Using these recurrence relations, an existence–uniqueness theorem is established to prove that the R$R$-order convergence of the method is (2+p)$(2+p)$. A priori error bounds for the method are also derived. Two numerical examples are worked out to demonstrate the efficacy of our approach.     

A uniparametric family of iterative processes for solving nondifferentiable equations

In this work we study a class of secant-like iterations for solving nonlinear equations in Banach spaces. We consider a condition for divided differences which generalizes the usual ones, i.e., Lipschitz and Hölder continuous conditions. A semilocal convergence result is obtained for nondifferentiable operators. For that, we use a technique based on a new system of recurrence relations to obtain domains of existence and uniqueness of the solution. Finally, we apply our results to the numerical solution of several examples.     

Convergence analysis for a family of improved super-Halley methods under general convergence condition

In this paper, we focus on the semilocal convergence for a family of improved super-Halley methods for solving non-linear equations in Banach spaces. Different from the results in Wang et al. (J Optim Theory Appl 153:779–793, 2012), the condition of Hölder continuity of third-order Fréchet derivative is replaced by its general continuity condition, and the latter is weaker than former. Moreover, the R-order of the methods is also improved. By using the recurrence relations, we prove a convergence theorem to show the existence-uniqueness of the solution. The R-order of these methods is analyzed with the third-order Fréchet derivative of the operator satisfies general continuity condition and Hölder continuity condition.     

Complexity analysis of an interior-point algorithm for linear optimization based on a new proximity function

In this paper, we focus on a family of modified Chebyshev methods and study the semilocal convergence for these methods. Different from the results in reference (Hernández and Salanova, J. Comput. Appl. Math. 126:131–143, 2000), the Hölder continuity of the second derivative is replaced by its generalized continuity condition, and the latter is weaker than the former. Using the recurrence relations, we establish the semilocal convergence of these methods and prove a convergence theorem to show the existence-uniqueness of the solution. The R-order of these methods is also analyzed. Especially, when the second derivative of the operator is Hölder continuous, the R-order of these methods is at least 3 + 2p, which is higher than the one of Chebyshev method considered in reference (Hernández and Salanova, J. Comput. Appl. Math. 126:131–143, 2000) under the same condition. Finally, we give some numerical results to show our approach.     

Global Minimization Algorithms for Holder Functions

This paper deals with the one-dimensional global optimization problem where the objective function satisfies a Hölder condition over a closed interval. A direct extension of the popular Piyavskii method proposed for Lipschitz functions to Hölder optimization requires an a priori estimate of the Hölder constant and solution to an equation of degree N at each iteration. In this paper a new scheme is introduced. Three algorithms are proposed for solving one-dimensional Hölder global optimization problems. All of them work without solving equations of degree N. The case (very often arising in applications) when a Hölder constant is not given a priori is considered. It is shown that local information about the objective function used inside the global procedure can accelerate the search signicantly. Numerical experiments show quite promising performance of the new algorithms.     

Properties of the Bellman Function in Time-Optimal Control Problems

In this paper, time-optimal control problems with closed terminal sets are considered. We give conditions which guarantee the Bellman function to be Hölder and Lipschitz continuous. We then prove that the condition for Lipschitz continuity is also necessary.     

Semilocal convergence for a family of Chebyshev-Halley like iterations under a mild differentiability condition

The semilocal convergence of a family of Chebyshev-Halley like iterations for nonlinear operator equations is studied under the hypothesis that the first derivative satisfies a mild differentiability condition. This condition includes the usual Lipschitz condition and the H?lder condition as special cases. The method employed in the present paper is based on a family of recurrence relations. The R-order of convergence of the methods is also analyzed. As well, an application to a nonlinear Hammerstein integral equation of the second kind is provided. Furthermore, two numerical examples are presented to demonstrate the applicability and efficiency of the convergence results.     

Remarks on Hölder continuity for solutions of the p-Laplace evolution equations

We study the interior Hölder regularity problem for the gradient of solutions of the p-Laplace evolution equations with the external forces. Misawa gave some conditions for the Hölder continuity of the gradient of solutions. We show Hölder estimates of the solutions with weaker condition as for Misawa.     

Newton's method under mild differentiability conditions with error analysis

Summary The concept of majorizing sequences introduced by Rheinboldt (SIAM J.N.A. 1968) is used to prove convergence for Newton's method for operator equations of the formT f= when the operator satisfied the condition that the Fréchet derivative is Hölder continuous.A detailed analysis of computational errors is given for Newton's method applied to operators with Hölder continuous derivatives. This analysis is shown to reduce the analysis of Lancaster (Num. Math. 1968) when the operator has a continuous second derivative.The above analysis is applied to an example of a second order differential equation.The author is grateful to National Research Council of Canada and National Defency Research Board of Canada for financial support.     

On stability of minimizers in convex programming

In this paper, we establish conditions ensuring Hölder and Lipschitz continuity of minimizers in convex programming. Lipschitz continuity is proved by establishing and applying a generalized version of the implicit function theorem.     

A priori estimate for non-uniform elliptic equations

A priori estimate for non-uniform elliptic equations with periodic boundary conditions is concerned. The domain considered consists of two sub-regions, a connected high permeability region and a disconnected matrix block region with low permeability. Let ? denote the size ratio of one matrix block to the whole domain. It is shown that in the connected high permeability sub-region, the Hölder and the Lipschitz estimates of the non-uniform elliptic solutions are bounded uniformly in ?. But Hölder gradient estimate and Lp estimate of the second order derivatives of the solutions in general are not bounded uniformly in ?.     

Convergence radius of the modified Newton method for multiple zeros under Hölder continuous derivative

In recent years, a lot of iterative methods for finding multiple zeros of nonlinear equations have been presented and analyzed. However, almost all these studies give no information for the convergence radius of the corresponding method. In this paper, we give an estimate of the convergence radius of the well-known modified Newton’s method for multiple zeros, when the involved function satisfies a Hölder and center-Hölder continuity condition.     

Semilocal convergence of a k-step iterative process and its application for solving a special kind of conservative problems

In this paper, we analyze the semilocal convergence of k-steps Newton’s method with frozen first derivative in Banach spaces. The method reaches order of convergence k + 1. By imposing only the assumption that the Fréchet derivative satisfies the Lipschitz continuity, we define appropriate recurrence relations for obtaining the domains of convergence and uniqueness. We also define the accessibility regions for this iterative process in order to guarantee the semilocal convergence and perform a complete study of their efficiency. Our final aim is to apply these theoretical results to solve a special kind of conservative systems.