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.     

Mysovskii-type theorem for the Secant method under Hölder continuous Fréchet derivative

The Mysovskii-type condition is considered in this study for the Secant method in Banach spaces to solve a nonlinear operator equation. We suppose the inverse of divided difference of order one is bounded and the Fréchet derivative of the nonlinear operator is Hölder continuous. By use of Fibonacci generalized sequence, a semilocal convergence theorem is established which matches with the convergence order of the method. Finally, two simple examples are provided to show that our results apply, where earlier ones fail.     

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.     

On convergence of the modified Newton’s method under Hölder continuous Fréchet derivative

In this paper, the upper and lower estimates of the radius of the convergence ball of the modified Newton’s method in Banach space are provided under the hypotheses that the Fréchet derivative of the nonlinear operator are center Hölder continuous for the initial point and the solution of the operator. The error analysis is given which matches the convergence order of the modified Newton’s method. The uniqueness ball of solution is also established. Numerical examples for validating the results are also provided, including a two point boundary value problem.     

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 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.     

On a continuous form of Hölder inequality

A continuous form of Hölder inequality is considered here extending some known results.     

Hölder Estimates for A Class of Degenerate Elliptic Equations

In this paper we study the regularity theory of the solutions of a class of degenerate elliptic equations in divergence form. By introducing a proper distance and applying the compactness method we establish the Hölder type estimates for the weak solutions.     

Weakly hyperbolic systems with Hölder continuous coefficients

We study the Cauchy Problem for a hyperbolic system with multiple characteristics and non-smooth coefficients depending on time. We prove in particular that, if the leading coefficients are α-Hölder continuous, and the system has size m?3, then the Problem is well posed in each Gevrey class of exponent s<1+α/m.     

Uniformly continuous superposition operators in the Banach space of Hölder functions

Let I,J⊂R be intervals. One of the main results says that if a superposition operator H generated by a two place ,

H(φ)(x):=h(x,φ(x)),     

Hölder continuity in time for SG hyperbolic systems

We investigate well posedness of the Cauchy problem for SG hyperbolic systems with non-smooth coefficients with respect to time. By assuming the coefficients to be Hölder continuous we show that this low regularity has a considerable influence on the behavior at infinity of the solution as well as on its regularity. This leads to well posedness in suitable Gelfand-Shilov classes of functions on Rn. A simple example shows the sharpness of our results.     

A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients

We provide a rate for the strong convergence of Euler approximations for stochastic differential equations (SDEs) whose diffusion coefficient is not Lipschitz but only (1/2+α)-Hölder continuous for some α≥0.     

Positive solutions of the generalized Emden-Fowler equation in Hölder spaces

A singular sublinear BVP related to the Emden-Fowler equation is considered. Existence, nonexistence, and regularity of positive solutions in Hölder spaces is obtained.     

The Cauchy problem for hyperbolic systems with Hölder continuous coefficients with respect to time

We discuss the local existance and uniqueness of solutions of certain nonstrictly hyperbolic systems, with Hölder continuous coefficients with respect to time variable. We reduce the nonstrictly hyperbolic systems to the parabolic ones, then we shall prove them by use of Tanabe-Sobolevski’s method.     

The Hölder continuity of solutions to generalized vector equilibrium problems

In this paper, by using a weaker assumption, we discuss the Hölder continuity of solution maps for two cases of parametric generalized vector equilibrium problems under the case that the solution map is a general set-valued one, but not a single-valued one. These results extend the recent ones in the literature. Several examples are given for the illustration of our results.     

Convexity with respect to Hölder mean involving zero-balanced hypergeometric functions

In this note we investigate the convexity of zero-balanced hypergeometric functions with respect to Hölder mean.     

On the Hölder continuity of solutions to parametric multivalued vector equilibrium problems

We consider parametric multivalued vector equilibrium problems of both weak and strong types in metric linear spaces. Sufficient conditions for the local uniqueness and Hölder continuity of the solutions are established. As consequences some new results for variational inequalities are derived and compared with recent papers on the subject.     

Concavity with respect to Hölder means involving the generalized Grötzsch function

The generalized Grötzsch function has numerous applications in geometric function theory and analytic number theory and its properties have been investigated by many authors. In this paper we study the concavity of the generalized Grötzsch function with respect to Hölder means.     

Hölder continuity and upper estimates of solutions to vector quasiequilibrium problems

In this paper, we establish the Hölder continuity of solution mappings to parametric vector quasiequilibrium problems in metric spaces under the case that solution mappings are set-valued. Our main assumptions are weaker than those in the literature, and the results extend and improve the recent ones. Furthermore, as an application of Hölder continuity, we derive upper bounds for the distance between an approximate solution and a solution set of a vector quasiequilibrium problem with fixed parameters.