Perov Type Results in Gauge Spaces and its Applications to Systems of Integral Equations

Rajesh Shrivastava1 and Sachin Mokhle1

1Government Dr. Shyama Prashad Mukherjee Science \& Commerce College, Old Benajir, Bhopal, Madhya Pradesh, India.

Abstract: In this paper we present Perov type fixed point theorems for contractive mappings in Gheorghiu’s sense on spaces endowed with a family of vector valued pseudo-metrics. Applications to systems of integral equations are given to illustrate the theory. The examples also prove the advantage of using vector valued pseudo-metrics and matrices that are convergent to zero, for the study of systems of equations.
Keywords: Integral equation, Gauge space, Fixed point.

Cite this article as: Rajesh Shrivastava and Sachin Mokhle, Perov Type Results in Gauge Spaces and its Applications to Systems of Integral Equations, Int. J. Math. And Appl., vol. 10, no. 1, 2022, pp. 85-96.

  1. I. Colojoar\u{a}, Sur un th\'{e}or\`{e}me de point fixe dans les espaces uniformes complets, Com. Acad. R. P. Rom., 11(1961), 281-283.
  2. K. L. Cooke and J. L. Kaplan, A periodicity threshold theorem for epidemics and population growth, Math. Biosci., 31(1976), 87-104.
  3. Deepmala, A Study on Fixed Point Theorems for Nonlinear Contractions and its Applications, Ph.D. Thesis, Pt. Ravishankar Shukla University, Raipur, Chhatisgarh, India, (2014).
  4. M. Frigon, Fixed point and continuation results for contractions in metric and gauge spaces, In: Fixed Point Theory and Its Applications. Banach Center Publ. 77, Polish Acad. Sci., Warsaw, (2007), 89-114.
  5. N. Gheorghiu, Contraction theorem in uniform spaces, Stud. Cerc. Mat., 19 (1967), 119-122.
  6. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, (2006).
  7. R. J. Knill, Fixed points of uniform contractions, J. Math. Anal. Appl., 12(1965), 449-455.
  8. G. Marinescu, Topological and Pseudo topological Vector Spaces, Editura Acad. R. P. Rom., Bucharest, (1959).
  9. L. N. Mishra, H. M. Srivastava and M. Sen, Existence Results for Some Nonlinear Functional-Integral Equations In Banach Algebra With Applications, Int. J. of Anal. and Appl., 11(1)(2016), 1-10.
  10. L. N. Mishra and R. P. Agrawal, On existence theorems for some nonlinear functional-integral equations, Dynamic Systems and Applications, 25(2016), 303-320.
  11. L. N. Mishra, On existence and behavior of solutions to some nonlinear integral equations with Applications, Ph.D. Thesis, National Institute of Technology, Silchar, Assam, India, (2017).
  12. L. N. Mishra, M. Sen and R. N. Mohapatra, On existence theorems for some generalized nonlinear functional-integral equations with applications, Filomat, 31(7)(2017), 2081-2091.
  13. V. N. Mishra, Some Problems on Approximations of Functions in Banach Spaces, Ph.D. Thesis, Indian Institute of Technology, Roorkee, Uttarakhand, India, (2007).
  14. L. N. Mishra and M. Sen, On the concept of existence and local attractivity of solutions for some quadratic Volterra integral equation of fractional order, Applied Mathematics and Computation, 285(2016), 174-183.
  15. A. I. Perov and A. V. Kibenko, On a certain general method for investigation of boundary value problems, Izv. Akad. Nauk SSSR, 30(1966), 249-264.
  16. R. Precup, Methods in Nonlinear Integral Equations, Kluwer, Dordrecht, (2002).
  17. R. Precup, The role of matrices that are convergent to zero in the study of semilinear operator systems, Math. Comput. Modelling, 49(2009), 703-708.
  18. R. Precup, Existence, localization and multiplicity results for positive radial solutions of semilinear elliptic systems, J. Math. Anal. Appl., 352(2009), 48-56.
  19. R. Precup and A. Viorel, Existence results for systems of nonlinear evolution inclusions, Fixed Point Theory, 11(2010), 337-346.
  20. Qinghua Xu, Yongfa Tang, Ting Yang and Hari Mohan Srivastava, Schwarz lemma involving the boundary fixed point, Fixed Point Theory and Applications, 2016(2016), DOI 10.1186/s13663-016-0574-8.
  21. H. M. Srivastava, S. V. Bedre, S. M. Khairnar and B. S. Desale, Krasnosel'skii type hybrid fixed point theorems and their applications to fractional integral equations, Abstr. Appl. Anal., 2014 (2014), 1-9.
  22. H. M. Srivastava and R. G. Bushman, Theory and Applications of Convolution Integral Equations, Vol. 79 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, (1992).
  23. E. Tarafdar, An approach to fixed-point theorems on uniform spaces, Trans. Amer. Math. Soc., 191(1974), 209-225.