Some properties of the fractal convolution of functions

We consider the fractal convolution of two maps f and g defined on a real interval as a way of generating a new function by means of a suitable iterated function system linked to a partition of the interval. Based on this binary operation, we consider the left and right partial convolutions, and study their properties. Though the operation is not commutative, the one-sided convolutions have similar (but not equal) characteristics. The operators defined by the lateral convolutions are both nonlinear, bi-Lipschitz and homeomorphic. Along with their self-compositions, they are Fréchet differentiable. They are also quasi-isometries under certain conditions of the scale factors of the iterated function system. We also prove some topological properties of the convolution of two sets of functions. In the last part of the paper, we study stability conditions of the dynamical systems associated with the one-sided convolution operators.

General Norm Inequalities of Trapezoid Type for Fréchet Differentiable Functions in Banach Spaces

In this paper we establish some error bounds in approximating the integral by general trapezoid type rules for Fréchet differentiable functions with values in Banach spaces.

Global bifurcation at isolated singular points of the Hadamard derivative

Consider F ∈ C ( R × X , Y ) such that F ( λ , 0) = 0 for all λ ∈ R , where X and Y are Banach spaces. Bifurcation from the line R × { 0 } of trivial solutions is investigated in cases where F ( λ , · ) need not be Fréchet differentiable at 0. The main results provide sufficient conditions for μ to be a bifurcation point and yield global information about the connected component of { ( λ , u ) : F ( λ , u ) = 0 and u ≠ 0 } ∪ { ( μ , 0 ) } containing ( μ , 0). Some necessary conditions for bifurcation are also formulated. The abstract results are used to treat several singular boundary value problems for which Fréchet differentiability is not available. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

Some Grüss Type Inequalities for Fréchet Differentiable Mappings

Let X be a Hilbert C∗-module on C∗-algebra A and p ∈ A. We denote by Dp(A,X) the set of all continuous functions f : A → X, which are Fréchet differentiable on a open neighborhood U of p. Then, we introduce some generalized semi-inner products on Dp(A,X), and using them some Grüss type inequalities in semi-inner product C∗-module Dp(A,X) and Dp(A,Xn) are established.

Extending the Convexity of Nonlinear Image of a Ball Appearing in Optimization

Let X, Y be Hilbert spaces and F : X → Y be Frechet differentiable. Suppose that F′ is center-Lipschitz on U(w, r) and F′(w) be a surjection. Then, S1 = F(U(w, ε1)) is convex where ε1 ≤ r. The set S1 contains the corresponding set given in [18] under the Lipschitz condition. Numerical examples where the old conditions are not satisfied but the new conditions are satisfied are provided in this paper.

On the Convergence of Newton-like Method for Variational Inclusions under Pseudo-Lipschitz Mapping

In the present paper, we study a Newton-like method for solving the variational inclusion defined by the sums of a Frechet differentiable function, divided difference admissible function and a set-valued mapping with closed graph. Under some suitable assumptions on the Frechet derivative of the differentiable function and divided difference admissible function, we establish the existence of any sequence generated by the Newton-like method and prove that the sequence generated by this method converges linearly and superlinearly to a solution of the variational inclusion. Specifically, when the Frechet derivative of the differentiable function is continuous, Lipschitz continuous, divided difference admissible function admits first order divided di_erence and the setvalued mapping is pseudo-Lipschitz continuous, we show the linear and superlinear convergence of the method. GANIT J. Bangladesh Math. Soc.Vol. 40 (2020) 43-53

Strong Convergence of Mann’s Iteration Process in Banach Spaces

Mann’s iteration process for finding a fixed point of a nonexpansive mapping in a Banach space is considered. This process is known to converge weakly in some class of infinite-dimensional Banach spaces (e.g., uniformly convex Banach spaces with a Fréchet differentiable norm), but not strongly even in a Hilbert space. Strong convergence is therefore a nontrivial problem. In this paper we provide certain conditions either on the underlying space or on the mapping under investigation so as to guarantee the strong convergence of Mann’s iteration process and its variants.

Continuous second order minimization method with variable metric projection operator

The paper examines a new continuous projection second order method of minimization of continuously Frechet differentiable convex functions on the convex closed simple set in separable, normed Hilbert space with variable metric. This method accelerates common continuous projection minimization method by means of quasi-Newton matrices. In the method, apart from variable metric operator, vector of search direction for motion to minimum, constructed in auxiliary extrapolated point, is used. By other word, complex continuous extragradient variable metric method is investigated. Short review of allied methods is presented and their connections with given method are indicated. Also some auxiliary inequalities are presented which are used for theoretical reasoning of the method. With their help, under given supplemental conditions, including requirements on operator of metric and on method parameters, convergence of the method for convex smooth functions is proved. Under conditions completely identical to those in convergence theorem, without additional requirements to the function, estimates of the method's convergence rate are obtained for convex smooth functions. It is pointed out, that one must execute computational implementation of the method by means of numerical methods for ODEs solution and by taking into account the conditions of proved theorems.