Fixed points of mappings over a locally convex topological vector space and Ulam-Hyers stability of fixed point problems

Cantor's intersection theorem and some generalized fixed point theorems over a locally convex topological vector space

“…Denition 2.1. [13] Let X be a real vector space and C be a subset of X. Then C is said to be convex if for any two elements x, y \in C and for any scalar 0 \leq \alpha \leq 1, \alpha x + (1 -\alpha )y \in C that is the line segment joining two points x, y must lie in the set C. Equivalently, \alpha C + (1 -\alpha )C \subset C for all scalars \alpha satisfying 0 \leq \alpha \leq 1.…”

“…Denition 2.4. [13] A subset B of a vector space X is said to be balanced if \alpha B \subset B for all scalars \alpha , whenever | \alpha | \leq 1.…”

“…x \in X, there corresponds a scalar \beta \not = 0 such that \beta x \in B. Denition 2.8. [13] A vector space X over \BbbR or \BbbC equipped with a T 1 topology \tau is said to be a topological vector space(TVS) if the following conditions are satised.…”

“…Denition 2.9. (Local base) [13] By local base of a TVS (X, \tau ) we mean a neighborhood base \BbbB of \theta \in X (\theta is the zero vector in X) that is for every neighborhood V of \theta there exists a member B \in \BbbB such that \theta \in B \subset V. Denition 2.10. [13] A TVS X is said to be locally convex if X has a local base whose members are all convex sets.…”

“…Attempts have now been in progress to establish xed point Theorems with Frechet Topological Vector space as underlying space [1]. Very recently Roy and Saha in 2019 (See [13]) have developed xed point theory in Topological Vector spaces, specially in Locally Convex Topological Vector spaces so far unexplored, and they have proved xed point theorems involving Kannan type contractive mappings. Since not necessarily continuous mappings are also now welcome in xed point theory as developed by Roy and Saha [13], we have been prompted to look into problems involving like-Kannan mappings and our investigations in the concerned area have shown that we can derive some interesting and useful too xed point theorems by taking Reich and Caccioppoli-type mappings over Locally Convex Topological Vector space where Cantor-Intersection Theorem has been proved.…”