Abstract
Abstract. Let 𝑋 be a space with ∞-metric 𝜌 (a metric with possibly infinite value) and 𝑌 a space with ∞-distance 𝑑 satisfying the identity axiom. We consider the problem of coincidence point for mappings 𝐹,𝐺:𝑋→𝑌, i.e. the problem of existence of a solution for the equation 𝐹(𝑥)=𝐺(𝑥). We provide conditions of the existence of coincidence points in terms of a covering set for the mapping 𝐹 and a Lipschitz set for the mapping 𝐺 in the space 𝑋×𝑌. An 𝛼-covering set (𝛼>0) of the mapping 𝐹 is a set of (𝑥,𝑦) such that ∃𝑢∈𝑋 𝐹(𝑢)=𝑦, 𝜌(𝑥,𝑢)≤𝛼−1𝑑(𝐹(𝑥),𝑦), 𝜌(𝑥,𝑢)<∞, and a 𝛽 - Lipschitz set (𝛽≥0) for the mapping 𝐺 is a set of (𝑥,𝑦) such that ∀𝑢∈𝑋 𝐺(𝑢)=𝑦⇒𝑑(𝑦,𝐺(𝑥))≤𝛽𝜌(𝑢,𝑥). The new results are compared with the known theorems about coincidence points.
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