# Existence and Ulam-Hyers stability results for coincidence problems

Volume 6, Issue 2, pp 108--116 Publication Date: May 08, 2013
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### Authors

Oana Mleşniţe - Department of Mathematics, Babeş-Bolyai University Cluj-Napoca, Kogălniceanu Street No.1, 400084, Cluj-Napoca, Romania.

### Abstract

Let $X, Y$ be two nonempty sets and $s, t : X \rightarrow Y$ be two single-valued operators. By definition, a solution of the coincidence problem for s and $t$ is a pair $(x^*; y^*) \in X \times Y$ such that $s(x^*) = t(x^*) = y^*.$ It is well-known that a coincidence problem is, under appropriate conditions, equivalent to a fixed point problem for a single-valued operator generated by s and t. Using this approach, we will present some existence, uniqueness and Ulam - Hyers stability theorems for the coincidence problem mentioned above. Some examples illustrating the main results of the paper are also given.

### Keywords

• metric space
• coincidence problem
• singlevalued contraction
• vector-valued metric
• fixed point
• Ulam-Hyers stability.

•  47H10
•  54H25

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