Existence and Ulam-Hyers stability results for coincidence problems
- Department of Mathematics, Babeş-Bolyai University Cluj-Napoca, Kogălniceanu Street No.1, 400084, Cluj-Napoca, Romania.
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.
Share and Cite
Oana Mleşniţe, Existence and Ulam-Hyers stability results for coincidence problems, Journal of Nonlinear Sciences and Applications, 6 (2013), no. 2, 108--116
Mleşniţe Oana, Existence and Ulam-Hyers stability results for coincidence problems. J. Nonlinear Sci. Appl. (2013); 6(2):108--116
Mleşniţe, Oana. "Existence and Ulam-Hyers stability results for coincidence problems." Journal of Nonlinear Sciences and Applications, 6, no. 2 (2013): 108--116
- metric space
- coincidence problem
- singlevalued contraction
- vector-valued metric
- fixed point
- Ulam-Hyers stability.
M. Bota, A. Petruşel , Ulam-Hyers stability for operatorial equations, Analele Univ. Al.I. Cuza Iaşi, 57 (2011), 65-74.
A. Buică , Coincidence Principles and Applications, Cluj University Press, in Romanian (2001)
L. P. Castro, A. Ramos, Hyers-Ulam-Rassias stability for a class of Volterra integral equations, Banach J. Math. Anal., 3 (2009), 36-43.
K. Goebel, A coincidence theorem, Bull. de L'Acad. Pol. des Sciences, 16 (1968), 733-735.
S.-M. Jung, A fixed point approach to the stability of a Volterra integral equation, Fixed Point Theory and Applications, Article ID 57064, 2007 (2007), 9 pages.
A. I. Perov, On Cauchy problem for a system of ordinary differential equations, Pviblizhen. Met. Reshen. Differ. Uravn., 2 (1964), 115-134.
P. T. Petru, A. Petruşel, J. C. Yao, Ulam-Hyers stability for operatorial equations and inclusions via nonself operators, Taiwanese J. Math., 15 (2011), 2195-2212.
I. A. Rus, Remarks on Ulam stability of the operatorial equations, Fixed Point Theory, 10 (2009), 305-320.
I. A. Rus, Ulam stability of ordinary differential equations, Studia Univ. Babeş-Bolyai Math., 54 (2009), 125-133.
I. A. Rus, Gronwall lemma approach to the Ulam-Hyers-Rassias stability of an integral equation, Nonlinear Analysis and Variational Problems (P.M. Pardalos et al. (eds.)), 147 Springer Optimization and Its Applications, New York, 35 (), 147-152.