This problem typifies the general one of iteration. Let g(x) be the &th order iterate of g [i.e. g°(x) = x, g(x) = g(g(x))]. The iteration problem is that of attaching a consistent meaning to this expression for fractional k (in the sense of preserving the additive law of exponents). A n / satisfying (1) is thus g(x). By ideas similar to those discussed herein, we can find the most general g an...

A theory of differential calculus for nonlinear maps between general locally convex spaces is developed. All convergence notions are topological, and only familiarity with basic results from point set topology, differential calculus in Banach spaces, and locally convex space theory is assumed. The chain rule for continuous kth order differentiability, smoothness of inverse functions, and functi...

The results from classical asymptotic theory typically require assumptions of pointwise differentiability of a criterion function with respect an unknown parameter. Taylor expansion about some " true " value in the parameter space then gives a quadratic approximation to the criterion function, within error terms that can be bounded using the remainder from the Taylor expansion. With appropriate...

In this paper, we consider homogeneous and non-homogeneous system of first order linear fuzzy boundary value problems (SFOLBVPs) under granular differentiability. Using the concept horizontal membership function, introduced notion differentiability for n-dimensional functions. We present integral its properties. Theorems on existence uniqueness solutions SFOLFBVPs are proved. develop an algorit...

Abstract In this paper, the dynamic behavior of a class switched systems with internally forced switching (IFS) is investigated. By introducing definitions continuous dependence and differentiability, differentiability solution relative to control function are obtained. past studies, optimal problem given by IFS mainly focused on special controlled (the piece affine system). Our results lay goo...

The Karush–Kuhn–Tucker (KKT) optimality conditions and saddle point optimality conditions in fuzzy programming problems have been studied in literature by various authors under different conditions. In this paper, by considering a partial order relation on the set of fuzzy numbers, and convexity with differentiability of fuzzy mappings, we have obtained the Fritz John (FJ) constraint qualificat...

Let I ⊆ R be an interval and let k : I2 → C be a reproducing kernel on I . We show that if k(x, y) is in the appropriate differentiability class, it satisfies a 2-parameter family of inequalities of which the diagonal dominance inequality for reproducing kernels is the 0th order case. We provide an application to integral operators: if k is a positive definite kernel on I (possibly unbounded) w...

In the theory of optimization an essential role is played by the differentiability of convex functions. In this paper we shall try to extend the results concerning differentiability to a larger class of functions called strongly α(·)-paraconvex. Let (X, ‖.‖) be a real Banach space. Let f(x) be a real valued strongly α(·)-paraconvex function defined on an open convex subset Ω ⊂ X , i.e. f ( tx+(...

We investigate optimality conditions for optimization problems constrained by a class of variational inequalities of the second kind. Based on a nonsmooth primal-dual reformulation of the governing inequality, the differentiability of the solution map is studied. Directional differentiability is proved both for finite-dimensional and function space problems, under suitable assumptions on the ac...