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Multivalued and random version of Perov fixed point theorem in generalized gauge spaces

A. Laadjel, Juan J. Nieto ORCID logo, Abdelghani Ouahab ORCID logo and Rosana Rodríguez-López ORCID logo

Abstract

In this paper, we present some random fixed point theorems in complete gauge spaces. We establish then a multivalued version of a Perov–Gheorghiu’s fixed point theorem in generalized gauge spaces. Finally, some examples are given to illustrate the results.

MSC 2010: 47H10; 47H40; 54H25

Communicated by Vyacheslav L. Girko


Funding source: Xunta de Galicia

Award Identifier / Grant number: ED431C 2019/02

Funding statement: The research of J. J. Nieto and R. Rodríguez-López has been partially supported by the Agencia Estatal de Investigación (AEI) of Spain, cofinanced by the European Fund for Regional Development (FEDER) corresponding to the 2014–2020 multiyear financial framework, project MTM2016-75140-P; and by Xunta de Galicia under grant ED431C 2019/02. The research of A. Laadjel and A. Ouahab has been partially supported by the General Direction of Scientific Research and Technological Development (DGRSDT), Algeria.

References

[1] S. Almezel, Q. Hasan Ansari and M. A. Khamsi, Topics in Fixed Point Theory, Springer, Cham, 2014. 10.1007/978-3-319-01586-6Search in Google Scholar

[2] J. Andres and L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems, Kluwer Academic, Dordrecht, 2003. 10.1007/978-94-017-0407-6Search in Google Scholar

[3] V. G. Angelov, Fixed Points in Uniform Spaces and Applications, Cluj University, Cluj-Napoca, 2009. Search in Google Scholar

[4] A. Arunchai and S. Plubtieng, Random fixed point theorem of Krasnoselskii type for the sum of two operators, Fixed Point Theory Appl. 2013 (2013), Paper No. 142. 10.1186/1687-1812-2013-142Search in Google Scholar

[5] A. Baliki, J. J. Nieto, A. Ouahab and M. L. Sinacer, Random semilinear system of differential equations with impulses, Fixed Point Theory Appl. 2017 (2017), Paper No. 27. 10.1186/s13663-017-0622-zSearch in Google Scholar

[6] I. Beg and N. Shahzad, Applications of the proximity map to random fixed point theorems in Hilbert spaces, J. Math. Anal. Appl. 196 (1995), no. 2, 606–613. 10.1006/jmaa.1995.1428Search in Google Scholar

[7] I. Beg and N. Shahzad, Some random approximation theorems with applications, Nonlinear Anal. 35 (1999), no. 5, 609–616. 10.1016/S0362-546X(98)00020-0Search in Google Scholar

[8] A. Ben Amar and D. O’Regan, Topological Fixed Point Theory for Singlevalued and Multivalued Mappings and Applications, Springer, Cham, 2016. 10.1007/978-3-319-31948-3Search in Google Scholar

[9] A. T. Bharucha-Reid, Random Integral Equations, Math. Sci. Eng. 96, Academic Press, New York, 1972. Search in Google Scholar

[10] G. L. Cain, Jr. and M. Z. Nashed, Fixed points and stability for a sum of two operators in locally convex spaces, Pacific J. Math. 39 (1971), 581–592. 10.2140/pjm.1971.39.581Search in Google Scholar

[11] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer, Berlin, 1977. 10.1007/BFb0087685Search in Google Scholar

[12] A. Chiş and R. Precup, Continuation theory for general contractions in gauge spaces, Fixed Point Theory Appl. 2004 (2004), no. 3, 173–185. 10.1155/S1687182004403027Search in Google Scholar

[13] J. Dugundji, Topology, Allyn and Bacon, Boston, 1966. Search in Google Scholar

[14] M. Frigon, Fixed point results for generalized contractions in gauge spaces and applications, Proc. Amer. Math. Soc. 128 (2000), no. 10, 2957–2965. 10.1090/S0002-9939-00-05838-XSearch in Google Scholar

[15] M. Frigon, Fixed point results for multivalued contractions on gauge spaces, Set Valued Mappings with Applications in Nonlinear Analysis, Ser. Math. Anal. Appl. 4, Taylor & Francis, London (2002), 175–181. Search in Google Scholar

[16] N. Gheorghiu, Contraction theorem in uniform spaces, Stud. Cerc. Mat. 19 (1967), 119–122. Search in Google Scholar

[17] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Stud. Adv. Math. 28, Cambridge University, Cambridge, 1990. 10.1017/CBO9780511526152Search in Google Scholar

[18] L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, Math. Appl. 495, Kluwer Academic, Dordrecht, 1999. 10.1007/978-94-015-9195-9Search in Google Scholar

[19] J. R. Graef, J. Henderson and A. Ouahab, Some Krasnosel’skii type random fixed point theorems, J. Nonlinear Funct. Anal. 2017 (2017), Article ID 46. Search in Google Scholar

[20] J. R. Graef, J. Henderson and A. Ouahab, Topological Methods for Differential Equations and Inclusions, Monogr. Res. Notes Math., CRC Press, Boca Raton, 2019. 10.1201/9780429446740Search in Google Scholar

[21] J. R. Graef, H. Kadari, A. Ouahab and A. Oumansour, Existence results for systems of second-order impulsive differential equations, Acta Math. Univ. Comenian. (N. S.) 88 (2019), no. 1, 51–66. Search in Google Scholar

[22] O. Hanš, Random fixed point theorems, Transactions of the First Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, Publishing House of the Czechoslovak Academy of Sciences, Prague (1957), 105–125. Search in Google Scholar

[23] S. Itoh, Random fixed-point theorems with an application to random differential equations in Banach spaces, J. Math. Anal. Appl. 67 (1979), no. 2, 261–273. 10.1016/0022-247X(79)90023-4Search in Google Scholar

[24] A. Jeribi and B. Krichen, Nonlinear Functional Analysis in Banach Spaces and Banach Algebras. Fixed Point Theory Under Weak Topology for Nonlinear Operators and Block Operator Matrices with Applications, Monogr. Res. Notes Math., CRC Press, Boca Raton, 2016. 10.1201/b18790Search in Google Scholar

[25] W. Kirk and N. Shahzad, Fixed Point Theory in Distance Spaces, Springer, Cham, 2014. 10.1007/978-3-319-10927-5Search in Google Scholar

[26] D. R. Kurepa, Tableaux ramifiés d’ensembles. Espaces pseudo-distanciés, C. R. Acad. Sci. Paris 198 (1934), 1563–1565. Search in Google Scholar

[27] O. Nica, Initial-value problems for first-order differential systems with general nonlocal conditions, Electron. J. Differential Equations 2012 (2012), Paper No. 74. Search in Google Scholar

[28] O. Nica and R. Precup, On the nonlocal initial value problem for first order differential systems, Stud. Univ. Babeş-Bolyai Math. 56 (2011), no. 3, 113–125. Search in Google Scholar

[29] J. J. Nieto, A. Ouahab and R. Rodríguez-López, Random fixed point theorems in partially ordered metric spaces, Fixed Point Theory Appl. 2016 (2016), Paper No. 98. 10.1186/s13663-016-0590-8Search in Google Scholar

[30] A. Novac and R. Precup, Perov type results in gauge spaces and their applications to integral systems on semi-axis, Math. Slovaca 64 (2014), no. 4, 961–972. 10.2478/s12175-014-0251-5Search in Google Scholar

[31] N. S. Papageorgiou, Random fixed point theorems for measurable multifunctions in Banach spaces, Proc. Amer. Math. Soc. 97 (1986), no. 3, 507–514. 10.1090/S0002-9939-1986-0840638-3Search in Google Scholar

[32] A. I. Perov, On the Cauchy problem for a system of ordinary differential equations, Pviblizhen. Met. Reshen. Differ. Uvavn. 2 (1964), 115–134. Search in Google Scholar

[33] A. I. Perov and A. V. Kibenko, On a certain general method for investigation of boundary value problems, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), 249–264. Search in Google Scholar

[34] I.-R. Petre, A multivalued version of Krasnoselskii’s theorem in generalized Banach spaces, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 22 (2014), no. 2, 177–192. 10.2478/auom-2014-0041Search in Google Scholar

[35] I.-R. Petre and A. Petruşel, Krasnoselskii’s theorem in generalized Banach spaces and applications, Electron. J. Qual. Theory Differ. Equ. 2012 (2012), Paper No. 85. 10.14232/ejqtde.2012.1.85Search in Google Scholar

[36] A. Petruşel, Vector-valued Metrics in Fixed Point Theory, Babeş-Bolyai University, Cluj-Napoca, 2012. Search in Google Scholar

[37] B. L. S. Prakasa Rao, Stochastic integral equations of mixed type. II, J. Math. Phys. Sci. 7 (1973), 245–260. Search in Google Scholar

[38] R. Precup, Methods in Nonlinear Integral Equations, Kluwer Academic, Dordrecht, 2002. 10.1007/978-94-015-9986-3Search in Google Scholar

[39] R. Precup, The role of matrices that are convergent to zero in the study of semilinear operator systems, Math. Comput. Modelling 49 (2009), no. 3–4, 703–708. 10.1016/j.mcm.2008.04.006Search in Google Scholar

[40] I. A. Rus, The theory of a metrical fixed point theorem: theoretical and applicative relevances, Fixed Point Theory 9 (2008), no. 2, 541–559. Search in Google Scholar

[41] L. E. Rybiński, Random fixed points and viable random solutions of functional-differential inclusions, J. Math. Anal. Appl. 142 (1989), no. 1, 53–61. 10.1016/0022-247X(89)90163-7Search in Google Scholar

[42] V. M. Sehgal and S. P. Singh, On random approximations and a random fixed point theorem for set valued mappings, Proc. Amer. Math. Soc. 95 (1985), no. 1, 91–94. 10.1090/S0002-9939-1985-0796453-1Search in Google Scholar

[43] N. Shahzad, Random fixed point theorems for various classes of 1-set-contractive maps in Banach spaces, J. Math. Anal. Appl. 203 (1996), no. 3, 712–718. 10.1006/jmaa.1996.0407Search in Google Scholar

[44] N. Shahzad and S. Latif, Random fixed points for several classes of 1-ball-contractive and 1-set-contractive random maps, J. Math. Anal. Appl. 237 (1999), no. 1, 83–92. 10.1006/jmaa.1999.6454Search in Google Scholar

[45] M. L. Sinacer, J. J. Nieto and A. Ouahab, Random fixed point theorem in generalized Banach space and applications, Random Oper. Stoch. Equ. 24 (2016), no. 2, 93–112. 10.1515/rose-2016-0007Search in Google Scholar

[46] D. H. Thang and P. T. Anh, Random fixed points of completely random operators, Random Oper. Stoch. Equ. 21 (2013), no. 1, 1–20. 10.1515/rose-2013-0001Search in Google Scholar

[47] R. S. Varga, Matrix Iterative Analysis, Springer Ser. Comput. Math. 27, Springer, Berlin, 2000. 10.1007/978-3-642-05156-2Search in Google Scholar

Received: 2020-12-07
Accepted: 2021-09-15
Published Online: 2022-01-06
Published in Print: 2022-03-01

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