whenever ‖~v‖V < δ. In this case the bounded linear operator dF~v0 is called the (Frechet) derivative of F at ~v0. Prove that if ‖ • ‖V is a norm on V which is equivalent to ‖ • ‖V , then F is differentiable at ~v9 with respect to ‖ • ‖V if and only if it is differentiable at ~v0 with respect to ‖ • ‖V , and in this case the derivatives are equal. Similarly, if ‖ • ‖W is a norm on W which is eq...

This paper aims at solving an optimal control problem governed by anisotropic Allen-Cahn equation numerically. Therefor we first prove the Frechet differentiability of in time discretized parabolic under certain assumptions on involved quasilinearity and formulate order necessary conditions. As a next step, since anisotropies are general not smooth enough, convergence behavior controls is studi...

It is shown that if E is a Frechet space with the strong dual E∗ then Hb(E ∗), the space of holomorphic functions on E∗ which are bounded on every bounded set in E∗, has the property (DN) when E ∈ (DN) and that Hb(E∗) ∈ (Ω) when E ∈ (Ω) and either E∗ has an absolute basis or E is a Hilbert-Frechet-Montel space. Moreover the complementness of ideals J(V ) consisting of holomorphic functions on E...

In this short note, we study the properties of the weighted Frechet mean as a convex combination operator on an arbitrary metric space (Y, d). We show that this binary operator is commutative, non-associative, idempotent, invariant to multiplication by a constant weight and possesses an identity element. We also cover the properties of the weighted cumulative Frechet mean. These tools allow us ...

Received (received date) Revised (revised date) Communicated by (Name) The article analyzes similarity of closed polygonal curves in Frechet metric, which is stronger than the well-known Hausdorff metric and therefore is more appropriate in some applications. An algorithm that determines whether the Frechet distance between two closed polygonal curves with m and n vertices is less than a given ...

In this paper, the concept of lacunary almost summability of sequences in locally convex spaces has been defined and investigated. It is also proved Kojima-Schur and SilvarmanToeplitz type theorems for lacunary almost conservatively and lacunary almost regularity of the which transform sequences in a Frechet space into a sequence in an other Frechet space. We also stated that the same results h...

Today I start a series of lectures on the optimal control of systems with distributed parameters, that is to say, optimal control of systems described by partial differential equations. Optimal Control theory is the generalization of the classical calculus of variations where minimization of the functional is pursued in the class of non-smooth functions. In XX century optimal control theory mad...

This defines an operator MK̄ , which we will call a Minkowski operator. Denote by A(C) the Frechet space of entire functions in n variables with the usual topology of the uniform convergence on compact sets in C, and C(R) the Frechet space of r times differentiable functions on R with the topology of the uniform convergence on compact sets in R of all partial derivatives up to the order r (1 ≤ r...