A brief survey on the development and applications of Goebel’s coincidence point theorem in differential and integral equations

Bisht, Ravindra K.

doi:10.1515/jaa-2023-0114

Artykuł - szczegóły

Tytuł artykułu

A brief survey on the development and applications of Goebel’s coincidence point theorem in differential and integral equations

Autorzy

Bisht Ravindra K.

Identyfikatory

DOI

10.1515/jaa-2023-0114

Warianty tytułu

Języki publikacji

Abstrakty

Goebel’s coincidence theorem, a remarkably simple extension of Banach’s contraction principle widely applied in analysis, has found significant utility in the theory of differential and integral equations. Over the past fifty years, researchers have endeavored to generalize the definition of a metric space, thereby extending the scope of Goebel’s coincidence theorem to diverse settings. This survey paper overviews valuable insights into Goebel’s coincidence theorem’s historical background, its relevance in the field of fixed point theory, and its practical implications in solving problems related to differential and integral equations.

Słowa kluczowe

coincidence point contraction mapping fixed points integral equations

Wydawca

De Gruyter

Czasopismo

Journal of Applied Analysis

Rocznik

2024

Tom

Vol. 30, nr 2

Strony

311--324

Opis fizyczny

Bibliogr. 84 poz.

Twórcy

autor

Bisht Ravindra K.

ravindra.bisht@yahoo.com

Department of Mathematics, National Defence Academy, Khadakwasla, Pune, India

https://orcid.org/0000-0002-6333-8127

Bibliografia

[1] R. P. Agarwal, R. K. Bisht and N. Shahzad, A comparison of various noncommuting conditions in metric fixed point theory and their applications, Fixed Point Theory Appl. 2014 (2014), Paper No. 38.
[2] A. Alam and M. Imdad, Relation-theoretic contraction principle, J. Fixed Point Theory Appl. 17 (2015), no. 4, 693-702.
[3] A. Alam and M. Imdad, Relation-theoretic metrical coincidence theorems, Filomat 31 (2017), no. 14, 4421-4439.
[4] V. G. Angelov, A coincidence theorem in uniform spaces and applications, Math. Balkanica (N. S.) 5 (1991), no. 1, 47-65.
[5] V. G. Angelov and T. Y. Tsachev, A coincidence theorem for densifying mappings and applications, Demonstr. Math. 31 (1998), no. 2, 365-376.
[6] D. Ariza-Ruiz and J. Garcia-Falset, Existence and uniqueness of solution to several kinds of differential equations using the coincidence theory, Taiwanese J. Math. 19 (2015), no. 6, 1661-1692.
[7] D. Ariza Ruiz, J. Garcia Falset and K. Sadarangani, Wardowski conditions to the coincidence problem, Front. Appl. Math. Stat. 1 (2015), DOI 10.3389/fams.2015.00009.
[8] M. I. Ayari, M. Berzig and I. Kédim, Coincidence and common fixed point results for β-quasi contractive mappings on metric spaces endowed with binary relation, Math. Sci. 10 (2016), no. 3, 105-114.
[9] H. Ben-El-Mechaiekh, The Ran-Reurings fixed point theorem without partial order: A simple proof, J. Fixed Point Theory Appl. 16 (2014), no. 1-2, 373-383.
[10] V. Berinde, Approximating fixed points of implicit almost contractions, Hacet. J. Math. Stat. 41 (2012), no. 1, 93-102.
[11] V. Berinde and F. Vetro, Common fixed points of mappings satisfying implicit contractive conditions, Fixed Point Theory Appl. 2012 (2012), Paper No. 105.
[12] M. Berzig, Coincidence and common fixed point results on metric spaces endowed with an arbitrary binary relation and applications, J. Fixed Point Theory Appl. 12 (2012), no. 1-2, 221-238.
[13] M. Berzig and M. Bouali, A coincidence point theorem and its applications to fractional differential equations, J. Fixed Point Theory Appl. 22 (2020), no. 3, Paper No. 56.
[14] M. Berzig and E. Karapınar, Fixed point results for (αψ, βφ)-contractive mappings for a generalized altering distance, Fixed Point Theory Appl. 2013 (2013), Paper No. 205.
[15] M. Berzig, E. Karapınar and A.-F. Roldán-López-de Hierro, Discussion on generalized-(αψ, βϕ)-contractive mappings via generalized altering distance function and related fixed point theorems, Abstr. Appl. Anal. 2014 (2014), Article ID 259768.
[16] A. Bielecki, Une remarque sur la méthode de Banach-Cacciopoli-Tikhonov dans la théorie des équations différentielles ordinaires, Bull. Acad. Polon. Sci. Cl. III. 4 (1956), 261-264.
[17] R. K. Bisht, An overview of the emergence of weaker continuity notions, various classes of contractive mappings and related fixed point theorems, J. Fixed Point Theory Appl. 25 (2023), no. 1, Paper No. 11.
[18] D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458-464.
[19] F. E. Browder, Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Natl. Acad. Sci. USA 53 (1965), 1272-1276.
[20] F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Natl. Acad. Sci. USA 54 (1965), 1041-1044.
[21] C. Carathéodory, Vorlesungen über reelle Funktionen, 3rd ed., B. G. Teubner, Berlin, 1918.
[22] S. Chandok and S. Radenović, R type functions and coincidence points, Appl. Math. E-Notes 19 (2019), 250-256.
[23] L. Ćirić, A new fixed-point theorem for contractive mappings, Publ. Inst. Math. (Beograd) (N. S.) 30(44) (1981), 25-27.
[24] L. B. Ćirić, A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc. 45 (1974), 267-273.
[25] W.-S. Du and F. Khojasteh, New results and generalizations for approximate fixed point property and their applications, Abstr. Appl. Anal. 2014 (2014), Article ID 581267.
[26] M. Gabeleh, O. Olela Otafudu and N. Shahzad, Coincidence best proximity points in convex metric spaces, Filomat 32 (2018), no. 7, 2451-2463.
[27] J. Garcia Falset and O. Mleşnițe, Coincidence problems for generalized contractions, Appl. Anal. Discrete Math. 8 (2014), no. 1, 1-15.
[28] S. Ghods, M. E. Gordji, M. Ghods and M. Hadian, Comment on “Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces” [Lakshmikantham and Ćirić, Nonlinear Anal. TMA 70 (2009) 4341-4349] [mr2514765], J. Comput. Anal. Appl. 14 (2012), no. 5, 958-966.
[29] K. Goebel, A coincidence theorem, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 733-735.
[30] D. Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251-258.
[31] D. Gopal, C. Vetro, M. Abbas and D. K. Patel, Some coincidence and periodic points results in a metric space endowed with a graph and applications, Banach J. Math. Anal. 9 (2015), no. 3, 128-139.
[32] K. Gopalsamy, Nonoscillation in systems of linear differential equations with delayed and advanced arguments, J. Math. Anal. Appl. 140 (1989), no. 2, 374-380.
[33] L. Górniewicz and A. Granas, Some general theorems in coincidence theory. I, J. Math. Pures Appl. (9) 60 (1981), no. 4, 361-373.
[34] L. Górniewicz and Z. Kucharski, Coincidence of k-set contraction pairs, J. Math. Anal. Appl. 107 (1985), no. 1, 1-15.
[35] R. H. Haghi, S. Rezapour and N. Shahzad, Some fixed point generalizations are not real generalizations, Nonlinear Anal. 74 (2011), no. 5, 1799-1803.
[36] G. Jungck, Commuting mappings and fixed points, Amer. Math. Monthly 83 (1976), no. 4, 261-263.
[37] M. S. Khan, M. Berzig and S. Chandok, Fixed point theorems in bimetric space endowed with binary relation and applications, Miskolc Math. Notes 16 (2015), no. 2, 939-951.
[38] F. Khojasteh, S. Shukla and S. Radenović, A new approach to the study of fixed point theory for simulation functions, Filomat 29 (2015), no. 6, 1189-1194.
[39] K. Lan and J. R. L. Webb, Positive solutions of semilinear differential equations with singularities, J. Differential Equations 148 (1998), no. 2, 407-421.
[40] X.-L. Liu, A. H. Ansari, S. Chandok and S. Radenović, On some results in metric spaces using auxiliary simulation functions via new functions, J. Comput. Anal. Appl. 24 (2018), no. 6, 1103-1114.
[41] R. Machuca, Mathematical notes: A coincidence theorem, Amer. Math. Monthly 74 (1967), no. 5, 569-572.
[42] J. Matkowski, Integrable solutions of functional equations, Dissertationes Math. (Rozprawy Mat.) 127 (1975), 1-6868.
[43] S. G. Matthews, Partial metric topology, in: Papers on General Topology and Applications (Flushing 1992), Ann. New York Acad. Sci. 728, New York Acad. Sci., New York (1994), 183-197.
[44] J. Mawhin, Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces, J. Differential Equations 12 (1972), 610-636.
[45] A. Meir and E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl. 28 (1969), 326-329.
[46] O. Mleşnițe, Existence and Ulam-Hyers stability results for coincidence problems, J. Nonlinear Sci. Appl. 6 (2013), no. 2, 108-116.
[47] J. R. Morales and E. M. Rojas, Generalized Ćirić’s pairs of maps and some systems of nonlinear integral equations, Thai J. Math. 20 (2022), no. 2, 611-628.
[48] J. R. Morales, E. M. Rojas and R. K. Bisht, Common fixed points for pairs of mappings with variable contractive parameters, Abstr. Appl. Anal. 2014 (2014), Article ID 209234.
[49] J. R. Morales, E. M. Rojas and R. K. Bisht, Generalized contraction mappings of rational type and applications to nonlinear integral equations, Bol. Soc. Parana. Mat. (3) 38 (2020), no. 1, 131-149.
[50] J. J. Nieto and R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005), no. 3, 223-239.
[51] J. J. Nieto and R. Rodríguez-López, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 12, 2205-2212.
[52] L. Nikolova and S. Varošanec, Properties of some functionals associated with h-concave and quasilinear functions with applications to inequalities, J. Inequal. Appl. 2014 (2014), Paper No. 30.
[53] D. O’Regan and M. Meehan, Existence Theory for Nonlinear Integral and Integrodifferential Equations, Math. Appl. 445, Kluwer Academic, Dordrecht, 1998.
[54] R. P. Pant, Common fixed points of noncommuting mappings, J. Math. Anal. Appl. 188 (1994), no. 2, 436-440.
[55] S. Park, A coincidence theorem, Bull. Acad. Polon. Sci. Sér. Sci. Math. 29 (1981), no. 9-10, 487-489.
[56] V. Popa, Fixed point theorems for implicit contractive mappings, Stud. Cercet. Ştiinț. Ser. Mat. Univ. Bacău (1997), no. 7, 127-133.
[57] V. Popa, Some fixed point theorems for compatible mappings satisfying an implicit relation, Demonstr. Math. 32 (1999), no. 1, 157-163.
[58] T. Pruszko, Topological degree methods in multivalued boundary value problems, Nonlinear Anal. 5 (1981), no. 9, 959-973.
[59] T. Pruszko, Some applications of the topological degree theory to multivalued boundary value problems, Dissertationes Math. (Rozprawy Mat.) 229 (1984), 4-48.
[60] S. Radenović and S. Chandok, Simulation type functions and coincidence points, Filomat 32 (2018), no. 1, 141-147.
[61] A. C. M. Ran and M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1435-1443.
[62] A. Roldán and E. Karapınar, Some multidimensional fixed point theorems on partially preordered G∗-metric spaces under (ψ, φ)-contractivity conditions, Fixed Point Theory Appl. 2013 (2013), Paper No. 158.
[63] A.-F. Roldán-López-de Hierro, A unified version of Ran and Reuring’s theorem and Nieto and Rodríguez-López’s theorem and low-dimensional generalizations, Appl. Math. Inf. Sci. 10 (2016), no. 2, 383-393.
[64] A.-F. Roldán-López-de Hierro, E. Karapınar and M. de la Sen, Coincidence point theorems in quasi-metric spaces without assuming the mixed monotone property and consequences in G-metric spaces, Fixed Point Theory Appl. 2014 (2014), Paper No. 184.
[65] A.-F. Roldán-López-de Hierro, E. Karapınar, C. Roldán-López-de Hierro and J. Martínez-Moreno, Coincidence point theorems on metric spaces via simulation functions, J. Comput. Appl. Math. 275 (2015), 345-355.
[66] A.-F. Roldán-López-de Hierro and N. Shahzad, Some fixed/coincidence point theorems under (ψ, φ)-contractivity conditions without an underlying metric structure, Fixed Point Theory Appl. 2014 (2014), Paper No. 218.
[67] A. F. Roldán-López-de Hierro and N. Shahzad, New fixed point theorem under R-contractions, Fixed Point Theory Appl. 2015 (2015), Paper No. 98.
[68] A.-F. Roldán-López-de Hierro and N. Shahzad, Common fixed point theorems under (R, S)-contractivity conditions, Fixed Point Theory Appl. 2016 (2016), Paper No. 55.
[69] A. F. Roldán-López-de Hierro and N. Shahzad, Fixed point theorems by combining Jleli and Samet’s, and Branciari’s inequalities, J. Nonlinear Sci. Appl. 9 (2016), no. 6, 3822-3849.
[70] I. A. Rus, A. Petruşel and G. Petruşel, Fixed Point Theory, Cluj University, Cluj-Napoca, 2008.
[71] B. Rzepecki, On the Banach principle and its application to the theory of differential equations, Comment. Math. Prace Mat. 19 (1976/77), no. 2, 355-363.
[72] B. Samet and M. Turinici, Fixed point theorems on a metric space endowed with an arbitrary binary relation and applications, Commun. Math. Anal. 13 (2012), no. 2, 82-97.
[73] W. Segiet, Local coincidence index for morphisms, Bull. Acad. Polon. Sci. Sér. Sci. Math. 30 (1982), no. 5-6, 261-267.
[74] N. Shahzad, E. Karapınar and A.-F. Roldán-López-de Hierro, On some fixed point theorems under (α, ψ, ϕ)-contractivity conditions in metric spaces endowed with transitive binary relations, Fixed Point Theory Appl. 2015 (2015), Paper No. 124.
[75] N. Shahzad, A. F. Roldán-López-de Hierro and F. Khojasteh, Some new fixed point theorems under (A, S)-contractivity conditions, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 111 (2017), no. 2, 307-324.
[76] S. L. Singh and S. N. Mishra, Remarks on recent fixed point theorems, Fixed Point Theory Appl. 2010 (2010), Article ID 452905.
[77] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008), no. 2, 1088-1095.
[78] T. Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear Anal. 71 (2009), no. 11, 5313-5317.
[79] M. Turinici, Fixed points of implicit contraction mappings, An. Şti. Univ. “Al. I. Cuza” Iaşi Secț. I Mat. (N. S.) 22 (1976), no. 2, 177-180.
[80] M. Turinici, Ran-Reurings fixed point results in ordered metric spaces, Libertas Math. 31 (2011), 49-55.
[81] M. Turinici, Nieto-Lopez theorems in ordered metric spaces, Math. Student 81 (2012), no. 1-4, 219-229.
[82] M. Turinici, Linear contractions in product ordered metric spaces, Ann. Univ. Ferrara Sez. VII Sci. Mat. 59 (2013), no. 1, 187-198.
[83] C. Vetro and F. Vetro, Common fixed points of mappings satisfying implicit relations in partial metric spaces, J. Nonlinear Sci. Appl. 6 (2013), no. 3, 152-161.
[84] D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl. 2012 (2012), Paper No. 94.

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