A study of some properties of an n-order functional inclusion

Volume 9, Issue 2, pp 350--356 http://dx.doi.org/10.22436/jnsa.009.02.01

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Authors

Tania Angelica Lazăr - Department of Mathematics, Technical University of Cluj-Napoca, Memorandumului St.28, 400114, Cluj-Napoca, Romania. Vasile Lucian Lazăr - The Faculty of Economics, Western University of Arad, Mihai Eminescu St.15, 310086, Arad, Romania.

Abstract

The purpose of this paper is to study the solution set of the functional inclusion of n-th order of the following form: \[x(t) \in G(t; x(f_1(t)); ...; x(f_n(t))); t \in X;\quad (1)\] where the function \(G: X\times Y^n\rightarrow P_{cl,cv}(Y)\) and \(f_1; f_2; ...; f_n : X \rightarrow X\) are given. The approach is based on some fixed point theorems for multivalued operators, satisfying the nonlinear contraction condition, see [V. L. Lazăr, Fixed Point Theory Appl., 2011 (2011), 12 pages].

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Keywords

• Functional inclusion
• multivalued weakly Picard operator
• fixed point
• \(\varphi\)-contraction
• data dependence
• well-posedness
• Ulam-Hyers stability.

MSC

•  47H10
•  54H25
•  54C60

References

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• [2] I. R. Petre, Fixed point theorems in E-b-metric spaces, J. Nonlinear Sci. Appl., 7 (2014), 264-271.


• [3] A. Petruşel , Operatorial Inclusions, House of the Book of Science, Cluj-Napoca (2002)


• [4] A. Petruşel , Multivalued weakly Picard operators and applications, Scientiae Math. Jpn., 59 (2004), 167-202.


• [5] A. Petruşel, I. A. Rus , Multivalued Picard and weakly Picard operators, International Conference on Fixed Point Theory and Applications, Yokohama Publ., Yokohama, (2004), 207-226.


• [6] A. Petruşel, I. A. Rus, The theory of a metric fixed point theorem for multivalued operators, In: L.J. Lin, A. Petruşel, H.K. Xu, Fixed Point Theory and its Applications, Yokohama Publ., (2010), 161-175.


• [7] A. Petruşel, I. A. Rus, J. C. Yao , Well-posedness in the generalized sense of the fixed point problems, Taiwan. J. Math., 11 (2007), 903-914.


• [8] R. Węgrzyk , Fixed point theorems for multivalued functions and their applications to functional equations , Dissertationes Math. (Rozprawy Mat.), 201 (1982), 28 pages.