Best proximity point theorems for $ \alpha^{+} F, (\theta-\phi )$-proximal contraction

In this paper, inspired by the idea of Suzuki type $ \alpha^{+} F$-proximal contraction in metric spaces, we prove a new existence of best proximity point for Suzuki type $ \alpha^{+} F$-proximal contraction and $ \alpha^{+} (\theta-\phi )$-proximal contraction defined on a closed subset of a complete metric space. Our theorems extend, generalize, and improve many existing results.

Compact reduction in Lipschitz-free spaces

We prove a general principle satisfied by weakly precompact sets of Lipschitz-free spaces. By this principle, certain infinite dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in free spaces over their compact subsets. As easy consequences we derive several new and some known results. The main new results are: $\mathcal {F}(X)$ is weakly sequentially complete for every superreflexive Banach space $X$, and $\mathcal {F}(M)$ has the Schur property and the approximation property for every scattered complete metric space $M$.

A New Approach to Proinov-Type Fixed-Point Results in Non-Archimedean Fuzzy Metric Spaces

Very recently, by considering a self-mapping T on a complete metric space satisfying a general contractivity condition of the form ψ(d(Tx,Ty))≤φ(d(x,y)), Proinov proved some fixed-point theorems, which extended and unified many existing results in the literature. Accordingly, inspired by Proinov-type contraction conditions, Roldán López de Hierro et al. introduced a novel family of contractions in fuzzy metric spaces (in the sense of George and Veeramani), whose main advantage is the very weak constraints imposed on the auxiliary functions that appear in the contractivity condition. They also proved the existence and uniqueness of fixed points for the discussed family of fuzzy contractions in the setting of non-Archimedean fuzzy metric spaces. In this paper, we introduce a new family of fuzzy contractions based on Proinov-type contractions for which the involved auxiliary functions are not supposed to satisfy any monotonicity assumptions; further, we establish some new results about the existence and uniqueness of fixed points. Furthermore, we show how the main results in the above-mentioned paper can be deduced from our main statements. In this way, our conclusions provide a positive partial solution to one of the open problems posed by such authors for deleting or weakening the hypothesis of the nondecreasingness character of the auxiliary functions.

A Relation-Theoretic Metrical Fixed Point Theorem for Rational Type Contraction Mapping with an Application

In this article, we discuss the relation theoretic aspect of rational type contractive mapping to obtain fixed point results in a complete metric space under arbitrary binary relation. Furthermore, we provide an application to find a solution to a non-linear integral equation.

Nonunique Fixed Point Results via Kannan F -Contraction on Quasi-Partial b -Metric Space

This paper is aimed at acquainting with a new Kannan F -expanding type mapping by the approach of Wardowski in the complete metric space. We establish some fixed point results for Kannan F -expanding type mapping and F -contractive type mappings which satisfy F -contraction conditions. Additionally, some new results are given which generalize several results present in the literature. Moreover, some applications and examples are provided to show the practicality of our results.

Fixed point theorem with contractive mapping on C*-Algebra valued G-Metric Space

Banach-Caccioppoli Fixed Point Theorem is an interesting theorem in metric space theory. This theorem states that if T : X → X is a contractive mapping on complete metric space, then T has a unique fixed point. In 2018, the notion of C *-algebra valued G-metric space was introduced by Congcong Shen, Lining Jiang, and Zhenhua Ma. The C *-algebra valued G-metric space is a generalization of the G-metric space and the C*-algebra valued metric space, meanwhile the G-metric space and the C *-algebra valued metric space itself is a generalization of known metric space. The G-metric generalized the domain of metric from X × X into X × X × X, the C *-algebra valued metric generalized the codomain from real number into C *-algebra, and the C *-algebra valued G-metric space generalized both the domain and the codomain. In C *-algebra valued G-metric space, there is one theorem that is similar to the Banach-Caccioppoli Fixed Point Theorem, called by fixed point theorem with contractive mapping on C *-algebra valued G-metric space. This theorem is already proven by Congcong Shen, Lining Jiang, Zhenhua Ma (2018). In this paper, we discuss another new proof of this theorem by using the metric function d(x, y) = max{G(x, x, y),G(y, x, x)}.

Variational Principles for Maximization Problems with Lower-semicontinuous Goal Functions

Let X be a completely regular topological space and f a real-valued bounded from above lower semicontinuous function in it. Let C(X) be the space of all bounded continuous real-valued functions in X endowed with the usual sup-norm. We show that the following two properties are equivalent: X is α-favourable (in the sense of the Banach-Mazur game); The set of functions h in C(X) for which f + h attains its supremum in X contains a dense and Gδ-subset of the space C(X). In particular, property (b) has place if X is a compact space or, more generally, if X is homeomorphic to a dense Gδ subset of a compact space.We show also the equivalence of the following stronger properties: X contains some dense completely metrizable subset; the set of functions h in C(X) for which f + h has strong maximum in X contains a dense and Gδ-subset of the space C(X). If X is a complete metric space and f is bounded, then the set of functions h from C(X) for which f + h has both strong maximum and strong minimum in X contains a dense Gδ-subset of C(X).

(ρ,η,μ)-Interpolative Kannan Contractions I

We point out a vital error in the paper of Gaba et al. (2019), showing that a (ρ,η,μ) interpolative Kannan contraction in a complete metric space need not have a fixed point. Then we give an appropriate restriction on a (ρ,η,μ)-interpolative Kannan contraction that guarantees the existence of a fixed point and provide an equivalent formulation. Moreover, we show that this formulation can be extended to the interpolative Reich-Rus-Ćirić type contraction.