An application of Perov type results in gauge spaces. (English) Zbl 1431.54031

Summary: 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.


54H25 Fixed-point and coincidence theorems (topological aspects)
54E50 Complete metric spaces
47J05 Equations involving nonlinear operators (general)
45G15 Systems of nonlinear integral equations
Full Text: DOI Euclid


[1] COLOJOAR \(\breve{A} \), I.: Sur un th \(\acute{e}or\grave{e}\) me de point fixe dans les espaces uniformes complets, Com. Acad. R. P. Rom. 11 (1961), 281-283.
[2] COOKE, K. L.-KAPLAN, J. L.: A periodicity threshold theorem for epidemics and population growth, Math. Biosci. 31 (1976), 87-104. · Zbl 0341.92012
[3] DEEPMALA, A Study on Fixed Point Theorems for Nonlinear Contractions and its Applications, Ph.D. Thesis (2014), Pt. Ravishankar Shukla University, Raipur 492 010, Chhatisgarh, India. · Zbl 1286.39018
[4] FRIGON, M.: 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, pp. 89-114. · Zbl 1122.47045
[5] GHEORGHIU, N.: Contraction theorem in uniform spaces, Stud. Cerc. Mat. 19 (1967), 119-122 (Romanian). · Zbl 0179.28202
[6] KILBAS A. A., SRIVASTAVA H. M., and TRUJILLO J. J., Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006 · Zbl 1092.45003
[7] KNILL, R. J.: Fixed points of uniform contractions, J. Math. Anal. Appl. 12 (1965), 449-455. · Zbl 0132.18902
[8] MARINESCU, G.: Topological and Pseudo topological Vector Spaces, Editura Acad. R. P. Rom., Bucharest, 1959 (Romanian).
[9] MISHRA L.N., SRIVASTAVA H. M., and SEN M., “Existence Results for Some Nonlinear Functional-Integral Equations In Banach Algebra With Applications ”, Int. J. of Anal. and Appl., 11(1) 2016, 1-10. · Zbl 1379.45005
[10] MISHRA L.N., AGRAWAL R.P., On existence theorems for some nonlinear functional-integral equations, Dynamic Systems and Applications, Vol. 25, (2016), pp. 303-320. · Zbl 1415.45003
[11] MISHRA L.N., On existence and behavior of solutions to some nonlinear integral equations with Applications, Ph.D. Thesis (2017), National Institute of Technology, Silchar 788 010, Assam, India.
[12] MISHRA L.N., SEN M., MOHAPATRA R.N., On existence theorems for some generalized nonlinear functional-integral equations with applications, Filomat, 31:7 (2017), 2081-2091.
[13] MISHRA V.N., Some Problems on Approximations of Functions in Banach Spaces, Ph.D. Thesis (2007), Indian Institute of Technology, Roorkee 247 667, Uttarakhand, India.
[14] MISHRA L.N., SEN M., On the concept of existence and local attractivity of solutions for some quadratic Volterra integral equation of fractional order, Applied Mathematics and Computation Vol. 285, (2016), 174-183. DOI: 10.1016/j.amc.2016.03.002 · Zbl 1410.45008
[15] PEROV, A. I.-KIBENKO, A. V.: On a certain general method for investigation of boundary value problems, Izv. Akad. Nauk SSSR 30 (1966), 249-264 (Russian).
[16] PRECUP, R.: Methods in Nonlinear Integral Equations, Kluwer, Dordrecht, 2002. · Zbl 1060.65136
[17] PRECUP, R.: The role of matrices that are convergent to zero in the study of semilinear operator systems, Math. Comput. Modelling 49 (2009), 703-708. · Zbl 1165.65336
[18] PRECUP, R.: Existence, localization and multiplicity results for positive radial solutions of semilinear elliptic systems, J. Math. Anal. Appl. 352 (2009), 48-56. · Zbl 1178.35162
[19] PRECUP, R.-VIOREL, A.: Existence results for systems of nonlinear evolution inclusions, Fixed Point Theory 11 (2010), 337-346. · Zbl 1207.35202
[20] QINGHUA XU, YONGFA TANG, TING YANG and HARI MOHAN SRIVASTAVA, “Schwarz lemma involving the boundary fixed point,” Fixed Point Theory and Applications (2016) 2016:84 DOI 10.1186/s13663-016-0574-8 · Zbl 1370.30013
[21] SRIVASTAVA H. M.,BEDRE S. V.,KHAIRNAR S. M., and DESALE B. S., Krasnosel’skii type hybrid fixed point theorems and their applications to fractional integral equations, Abstr. Appl. Anal. 2014 (2014), Article ID 710746, 1-9; see also Corrigendum, Abstr. Appl. Anal. 2015 (2015), Article ID 467569, 1-2. · Zbl 1473.47019
[22] SRIVASTAVA H. M. and BUSHMAN R. G., Theory and Applications of Convolution Integral Equations, vol. 79 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.
[23] TARAFDAR, E.: An approach to fixed-point theorems on uniform spaces, Trans. Amer. Math. Soc. 191 (1974), 209-225. · Zbl 0287.54048
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.