Manifolds of mappings on Cartesian products

Given smooth manifolds $$M_1,\ldots , M_n$$ M 1 , … , M n (which may have a boundary or corners), a smooth manifold N modeled on locally convex spaces and $$\alpha \in ({{\mathbb {N}}}_0\cup \{\infty \})^n$$ α ∈ ( N 0 ∪ { ∞ } ) n , we consider the set $$C^\alpha (M_1\times \cdots \times M_n,N)$$ C α ( M 1 × ⋯ × M n , N ) of all mappings $$f:M_1\times \cdots \times M_n\rightarrow N$$ f : M 1 × ⋯ × M n → N which are $$C^\alpha $$ C α in the sense of Alzaareer. Such mappings admit, simultaneously, continuous iterated directional derivatives of orders $$\le \alpha _j$$ ≤ α j in the jth variable for $$j\in \{1,\ldots , n\}$$ j ∈ { 1 , … , n } , in local charts. We show that $$C^\alpha (M_1\times \cdots \times M_n,N)$$ C α ( M 1 × ⋯ × M n , N ) admits a canonical smooth manifold structure whenever each $$M_j$$ M j is compact and N admits a local addition. The case of non-compact domains is also considered.

An Implicit Algorithm for Finding a Fixed Point of a Q -Nonexpansive Mapping in Locally Convex Spaces

In this paper, the notion of the q -duality mappings in locally convex spaces is introduced. An implicit method for finding a fixed point of a Q -nonexpansive mapping is provided. Finally, the convergence of the proposed implicit scheme is investigated. Some examples in order to illustrate of the main results are presented.

Inductive limits of quasi locally Baire spaces

Quasi-locally complete locally convex spaces are generalized to quasi-locally Baire locally convex spaces. It is shown that an inductive limit of strictly webbed spaces is regular if it is quasi-locally Baire. This extends Qiu’s theorem on regularity. Additionally, if each step is strictly webbed and quasi-locally Baire, then the inductive limit is quasi-locally Baire if it is regular. Distinguishing examples are provided.