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.     

A stability version of Hölder's inequality

We present a stability version of Hölder's inequality, incorporating an extra term that measures the deviation from equality. Applications are given.     

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.     

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.     

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.     

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.     

On the two-parameter fractional Brownian motion and Stieltjes integrals for Hölder functions

We extend the Stieltjes integral to Hölder functions of two variables and prove an existence and uniqueness result for the corresponding deterministic ordinary differential equations and also for stochastic equations driven by a two-parameter fractional Brownian motion.     

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.     

Hölder regularity for weak solutions to divergence form degenerate quasilinear parabolic systems

In this paper, we establish Hölder regularity of weak solutions to the diagonal divergence form degenerate quasilinear parabolic system related to Hörmander type vector fields by deriving a parabolic Poincaré inequality and the higher integrability of gradients of weak solutions.     

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 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.     

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.     

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.     

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.     

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.     

Generalization of a sharp Hölder's inequality and its application

In this paper, we generalize Hu Ke's sharpness of Hölder's inequality. As application, the obtained result is used to improve the well-known Opial-Olech inequality.     

The equivalence of generalized Hölder and Cesáro matrices

We show that the generalized Hölder and Cesáro matrices of order α > −1 are equivalent. We also show that the corresponding is true for doubly infinite generalized Hölder and Cesáro matrices.     

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.     

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.