Let q > 1 and E be a real q-uniformly smooth Banach space, K be a nonempty closed convex subset of E and T : K → K be a Lipschitz continuous mapping. Let {un} and {vn} be bounded sequences in K and {αn} and {βn} be real sequences in [0, 1] satisfying some restrictions. Let {xn} be the sequence generated from an arbitrary x1 ∈ K by the Ishikawa iteration process with errors: yn = (1− βn)xn + βnT...

We introduce a new iterative scheme for finding the common element of the set of solutions of the generalized equilibrium problems, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of the variational inequality problems for a relaxed u, v cocoercive and ξ-Lipschitz continuous mapping in a real Hilbert space. Then, we prove the strong convergence o...

We introduce a new approximation scheme combining the viscosity method with extragradient method for finding a common element of the set of solutions of a mixed equilibrium problem and the set of fixed points of a finite family of nonexpansive mappings and the set of the variational inequality for a monotone, Lipschitz continuous mapping. We obtain a strong convergence theorem for the sequences...

ABSTRACT. For nonlinear parabolic evolution equations, it is proved that, under the assumptions oflocal Lipschitz continuity of nonlinearity and the dissipativity of semiflows, there exist approximate inertial manifolds (AIM) in the energy space and that the approximate inertial manifolds are constructed as the graph of the steady-state determining mapping based on the spectral decomposition. I...

Using results from parametric optimization, we derive for chance-constrained stochastic programs quantitative stability properties for locally optimal values and sets of local minimizers when the underlying probability distribution is subjected to perturbations in a metric space of probability measures. Emphasis is placed on verifiable sufficient conditions for the constraint-set mapping to ful...

Let T : K → H be a mapping from a nonempty closed convex subset K of a finitedimensional Hilbert space H into H . Let f : K → R be proper, convex, and lower semicontinuous on K and let h : K → R be continuously Frećhet-differentiable on K with h′ (gradient of h), α-strongly monotone, and β-Lipschitz continuous on K . Then the sequence {xk} generated by the general auxiliary problem principle co...

Many theoretical results in the machine learning domain stand only for functions that are Lipschitz continuous. Lipschitz continuity is a strong form of continuity that linearly bounds the variations of a function. In this paper, we derive tight Lipschitz constants for two families of metrics: Mahalanobis distances and bounded-space bilinear forms. To our knowledge , this is the first time the ...

We present in this paper some properties of k-Lipschitz quasi-arithmetic means. The Lipschitz aggregation operations are stable with respect to input inaccuracies, what is a very important property for applications. Moreover, we provide sufficient conditions to determine when a quasi–arithemetic mean holds the k-Lipschitz property and allow us to calculate the Lipschitz constant k. Keywords— k-...

The unique solvability in the sense of classical solutions for nonlinear inverse problems to differential equations, solved oldest Dzhrbashyan–Nersesyan fractional derivative, is studied. linear part equation contains a bounded operator, continuous operator that depends on lower-order derivatives, and an unknown element. problem given by equation, special initial value conditions lower overdete...