Existence and Ulam-Hyers stability results for coincidence problems

Volume 6, Issue 2, pp 108--116 http://dx.doi.org/10.22436/jnsa.006.02.06

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1606 Downloads
• 2889 Views

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.

Share and Cite

Keywords

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

MSC

•  47H10
•  54H25

References

• [1] M. Bota, A. Petruşel , Ulam-Hyers stability for operatorial equations, Analele Univ. Al.I. Cuza Iaşi, 57 (2011), 65-74.


• [2] A. Buică , Coincidence Principles and Applications, Cluj University Press, in Romanian (2001)


• [3] L. P. Castro, A. Ramos, Hyers-Ulam-Rassias stability for a class of Volterra integral equations, Banach J. Math. Anal., 3 (2009), 36-43.


• [4] K. Goebel, A coincidence theorem, Bull. de L'Acad. Pol. des Sciences, 16 (1968), 733-735.


• [5] 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.


• [6] A. I. Perov, On Cauchy problem for a system of ordinary differential equations, Pviblizhen. Met. Reshen. Differ. Uravn., 2 (1964), 115-134.


• [7] 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.


• [8] I. A. Rus, Remarks on Ulam stability of the operatorial equations, Fixed Point Theory, 10 (2009), 305-320.


• [9] I. A. Rus, Ulam stability of ordinary differential equations, Studia Univ. Babeş-Bolyai Math., 54 (2009), 125-133.


• [10] 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.