A general fixed point theorem for two pairs of absorbing mappings in Gp-metric spaces
Abstract
A general fixed point theorem for two pairs of absorbing mappings satisfying a new type of implicit relation ([37]), without weak compatibility in Gp-metric spaces is proved. As applications, new results for mappings satisfying contractive conditions of integral type and for ϕ-contractive mappings are obtained.
Keywords
fixed point; Gp-metric space; absorbing mapping; limit range property; almost altering distance; implicit relation
References
1. M. Aamri and D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270 (2002), no. 1, 181–188.
2. M.R. Ahmadi Zand and A. Dehghan Nezhad, A generalization of partial metric spaces, J. Contemp. Appl. Math. 1 (2011), no. 1, 86–93.
3. J. Ali and M. Imdad, An implicit function implies several contractive conditions, Sarajevo J. Math. 4(17) (2008), no. 2, 269–285.
4. I. Altun, F. Sola and H. Simsek, Generalized contractions on partial metric spaces, Topology Appl. 157 (2010), no. 18, 2778–2785.
5. H. Aydi, E. Karapınar and P. Salimi, Some fixed point results in GP-metric spaces, J. Appl. Math. 2012, Art. ID 891713, 16 pp.
6. M.A. Barakat and A.M. Zidan, A common fixed point theorem for weak contractive maps in Gp-metric spaces, J. Egyptian Math. Soc. 23 (2015), no. 2, 309–314.
7. N. Bilgili, E. Karapınar and P. Salimi, Fixed point theorems for generalized contractions on GP-metric spaces, J. Inequal. Appl. 2013, 2013:39, 13 pp.
8. A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. 29 (2002), no. 9, 531–536.
9. K.P. Chi, E. Karapınar and T.D. Thanh, A generalized contraction principle in partial metric spaces, Math. Comput. Modelling 55 (2012), no. 5–6, 1673–1681.
10. L. Ćirić, B. Samet, H. Aydi and C. Vetro, Common fixed points of generalized contractions on partial metric spaces and an application, Appl. Math. Comput. 218 (2011), no. 6, 2398–2406.
11. B.C. Dhage, Generalised metric spaces and mappings with fixed point, Bull. Calcutta Math. Soc. 8 (1992), no. 4, 329–336.
12. B.C. Dhage, Generalized metric spaces and topological structure. I, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 46 (2000), no. 1, 3–24.
13. D. Gopal, A.S. Ranadive and U. Mishra, On some open problems of common fixed point theorems of noncompatible mappings, Proc. Math. Soc. BHU 20 (2004), 135–141.
14. D. Gopal, A.S. Ranadive and R.P. Pant, Common fixed points of absorbing maps, Bull. Marathwada Math. Soc. 9 (2008), no. 1, 43–48.
15. M. Imdad and S. Chauhan, Employing common limit range property to prove unified metrical common fixed point theorems, Int. J. Anal. 2013, Art. ID 763261, 10 pp.
16. M. Imdad, S. Chauhan and Z. Kadelburg, Fixed point theorem for mappings with common limit range property satisfying generalized (ψ,φ)-weak contractive conditions, Math. Sci. (Springer) 7 (2013), Art. 16, 8 pp.
17. M. Imdad, B.D. Pant and S. Chauhan, Fixed point theorems in Menger spaces using CLR(S,T) property and applications, J. Nonlinear Anal. Optim. 3 (2012), no. 2, 225–237.
18. M. Imdad, A. Sharma and S. Chauhan, Unifying a multitude of common fixed point theorems in symmetric spaces, Filomat 28 (2014), no. 6, 1113–1132.
19. G. Jungck, Compatible mappings and common fixed points, Internat. J. Math. Math. Sci. 9 (1986), no. 4, 771–779.
20. Z. Kadelburg, H.K. Nashine and S. Radenović, Fixed point results under various contractive conditions in partial metric spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 107 (2013), no. 2, 241–256.
21. E. Karapınar and U. Yüksel, Some common fixed point theorems in partial metric spaces, J. Appl. Math. 2011, Art. ID 263621, 16 pp.
22. M.S. Khan, M. Swaleh and S. Sessa, Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc. 30 (1984), no. 1, 1–9.
23. Y. Liu, J. Wu and Z. Li, Common fixed points of single-valued and multivalued maps, Int. J. Math. Math. Sci. 2005, no. 19, 3045–3055.
24. J. Matkowski, Fixed point theorems for mappings with a contractive iterate at a point, Proc. Amer. Math. Soc. 62 (1977), no. 2, 344–348.
25. S.G Matthews, Partial metric topology, in: S. Andima et al. (eds.), Papers on General Topology and Applications, Eighth Summer Conference at Queens College, Annals of the New York Academy of Sciences, 728, New York, 1994, pp. 183–197.
26. U. Mishra and A.S. Ranadive, Common fixed point of absorbing mappings satisfying implicit relations. Preprint.
27. Z. Mustafa, H. Obiedat and F. Awawdeh, Some fixed point theorem for mapping on complete G-metric spaces, Fixed Point Theory Appl. 2008, Art. ID 189870, 12 pp.
28. Z. Mustafa and B. Sims, Some remarks concerning D-metric spaces, in: J.G. Falset et al. (eds.), International Conference on Fixed Point Theory and Applications, Proceedings of the conference held in Valencia, July 13-19, 2003, Yokohama Publishers, Yokohama, 2004, pp. 189–198.
29. Z. Mustafa and B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal. 7 (2006), no. 2, 289–297.
30. R.P. Pant, Common fixed point theorems for contractive maps, J. Math. Anal. Appl. 226 (1998), no. 1, 251–258.
31. R.P. Pant, R-weak commutativity and common fixed points of noncompatible maps, Ganita 49 (1998), no. 1, 19–27.
32. R.P. Pant, R-weak commutativity and common fixed points, Soochow J. Math. 25 (1999), no. 1, 37–42.
33. V. Parvanieh, J.R. Roshan and Z. Kadelburg, On generalized weakly GP-contractive mappings in ordered GP-metric spaces, Gulf J. Math. 1 (2013), no. 1, 78–97.
34. V. Popa, Fixed point theorems for implicit contractive mappings, Stud. Cercet. Ştiinţ. Ser. Mat. Univ. Bacău No. 7 (1997), 127–133.
35. V. Popa, Some fixed point theorems for compatible mappings satisfying an implicit relation, Demonstratio Math. 32 (1999), no. 1, 157–163.
36. V. Popa, A general fixed point theorem for occasionally weakly compatible mappings and applications, Sci. Stud. Res. Ser. Math. Inform. 22 (2012), no. 1, 77–91.
37. V. Popa, Fixed point theorems for two pairs of mappings satisfying a new type of common limit range property, Filomat 31 (2017), no. 11, 3181–3192.
38. V. Popa and M. Mocanu, Altering distance and common fixed points under implicit relations, Hacet. J. Math. Stat. 38 (2009), no. 3, 329–337.
39. V. Popa and A.-M. Patriciu, A general fixed point theorem for pairs of weakly compatible mappings in G-metric spaces, J. Nonlinear Sci. Appl. 5 (2012), no. 2, 151–160.
40. V. Popa and A.-M. Patriciu, Fixed point theorems for mappings satisfying an implicit relation in complete G-metric spaces, Bul. Inst. Politeh. Iaşi. Secţ. Mat. Mec. Teor. Fiz. 59(63) (2013), no. 2, 97–123.
41. V. Popa and A.-M. Patriciu, A general fixed point theorem for a pair of self mappings with common limit range property in G-metric spaces, Facta Univ. Ser. Math. Inform. 29 (2014), no. 4, 351–370.
42. V. Popa and A.-M. Patriciu, Two general fixed point theorems for a sequence of mappings satisfying implicit relations in Gp-metric spaces, Appl. Gen. Topol. 16 (2015), no. 2, 225–231.
43. V. Popa and A.-M. Patriciu, Well posedness of fixed point problem for mappings satisfying an implicit relation in Gp-metric spaces, Math. Sci. Appl. E-Notes 3 (2015), no. 1, 108–117.
44. V. Popa and A.-M. Patriciu, Fixed point theorems for two pairs of mappings satisfying common limit range property in G-metric spaces, Bul. Inst. Politeh. Iaşi. Secţ. Mat. Mec. Teor. Fiz. 62(66) (2016), no. 2, 19–42.
45. V. Popa and A.-M. Patriciu, Fixed point results for pairs of absorbing mappings in partial metric spaces, Acta Univ. Apulensis Math. Inform. No. 50 (2017), 97–109.
46. V. Popa and A.-M. Patriciu, Fixed point theorems for two pairs of mappings satisfying a new type of common limit range property in Gp-metric spaces, Ann. Math. Sil. 32 (2018), 295–312.
47. W. Shatanawi, Fixed point theory for contractive mappings satisfying Φ-maps in G-metric spaces, Fixed Point Theory Appl. 2010, Art. ID 181650, 9 pp.
48. W. Sintunavarat and P. Kumam, Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces, J. Appl. Math. 2011, Art. ID 637958, 14 pp.
49. 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.
2. M.R. Ahmadi Zand and A. Dehghan Nezhad, A generalization of partial metric spaces, J. Contemp. Appl. Math. 1 (2011), no. 1, 86–93.
3. J. Ali and M. Imdad, An implicit function implies several contractive conditions, Sarajevo J. Math. 4(17) (2008), no. 2, 269–285.
4. I. Altun, F. Sola and H. Simsek, Generalized contractions on partial metric spaces, Topology Appl. 157 (2010), no. 18, 2778–2785.
5. H. Aydi, E. Karapınar and P. Salimi, Some fixed point results in GP-metric spaces, J. Appl. Math. 2012, Art. ID 891713, 16 pp.
6. M.A. Barakat and A.M. Zidan, A common fixed point theorem for weak contractive maps in Gp-metric spaces, J. Egyptian Math. Soc. 23 (2015), no. 2, 309–314.
7. N. Bilgili, E. Karapınar and P. Salimi, Fixed point theorems for generalized contractions on GP-metric spaces, J. Inequal. Appl. 2013, 2013:39, 13 pp.
8. A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. 29 (2002), no. 9, 531–536.
9. K.P. Chi, E. Karapınar and T.D. Thanh, A generalized contraction principle in partial metric spaces, Math. Comput. Modelling 55 (2012), no. 5–6, 1673–1681.
10. L. Ćirić, B. Samet, H. Aydi and C. Vetro, Common fixed points of generalized contractions on partial metric spaces and an application, Appl. Math. Comput. 218 (2011), no. 6, 2398–2406.
11. B.C. Dhage, Generalised metric spaces and mappings with fixed point, Bull. Calcutta Math. Soc. 8 (1992), no. 4, 329–336.
12. B.C. Dhage, Generalized metric spaces and topological structure. I, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 46 (2000), no. 1, 3–24.
13. D. Gopal, A.S. Ranadive and U. Mishra, On some open problems of common fixed point theorems of noncompatible mappings, Proc. Math. Soc. BHU 20 (2004), 135–141.
14. D. Gopal, A.S. Ranadive and R.P. Pant, Common fixed points of absorbing maps, Bull. Marathwada Math. Soc. 9 (2008), no. 1, 43–48.
15. M. Imdad and S. Chauhan, Employing common limit range property to prove unified metrical common fixed point theorems, Int. J. Anal. 2013, Art. ID 763261, 10 pp.
16. M. Imdad, S. Chauhan and Z. Kadelburg, Fixed point theorem for mappings with common limit range property satisfying generalized (ψ,φ)-weak contractive conditions, Math. Sci. (Springer) 7 (2013), Art. 16, 8 pp.
17. M. Imdad, B.D. Pant and S. Chauhan, Fixed point theorems in Menger spaces using CLR(S,T) property and applications, J. Nonlinear Anal. Optim. 3 (2012), no. 2, 225–237.
18. M. Imdad, A. Sharma and S. Chauhan, Unifying a multitude of common fixed point theorems in symmetric spaces, Filomat 28 (2014), no. 6, 1113–1132.
19. G. Jungck, Compatible mappings and common fixed points, Internat. J. Math. Math. Sci. 9 (1986), no. 4, 771–779.
20. Z. Kadelburg, H.K. Nashine and S. Radenović, Fixed point results under various contractive conditions in partial metric spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 107 (2013), no. 2, 241–256.
21. E. Karapınar and U. Yüksel, Some common fixed point theorems in partial metric spaces, J. Appl. Math. 2011, Art. ID 263621, 16 pp.
22. M.S. Khan, M. Swaleh and S. Sessa, Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc. 30 (1984), no. 1, 1–9.
23. Y. Liu, J. Wu and Z. Li, Common fixed points of single-valued and multivalued maps, Int. J. Math. Math. Sci. 2005, no. 19, 3045–3055.
24. J. Matkowski, Fixed point theorems for mappings with a contractive iterate at a point, Proc. Amer. Math. Soc. 62 (1977), no. 2, 344–348.
25. S.G Matthews, Partial metric topology, in: S. Andima et al. (eds.), Papers on General Topology and Applications, Eighth Summer Conference at Queens College, Annals of the New York Academy of Sciences, 728, New York, 1994, pp. 183–197.
26. U. Mishra and A.S. Ranadive, Common fixed point of absorbing mappings satisfying implicit relations. Preprint.
27. Z. Mustafa, H. Obiedat and F. Awawdeh, Some fixed point theorem for mapping on complete G-metric spaces, Fixed Point Theory Appl. 2008, Art. ID 189870, 12 pp.
28. Z. Mustafa and B. Sims, Some remarks concerning D-metric spaces, in: J.G. Falset et al. (eds.), International Conference on Fixed Point Theory and Applications, Proceedings of the conference held in Valencia, July 13-19, 2003, Yokohama Publishers, Yokohama, 2004, pp. 189–198.
29. Z. Mustafa and B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal. 7 (2006), no. 2, 289–297.
30. R.P. Pant, Common fixed point theorems for contractive maps, J. Math. Anal. Appl. 226 (1998), no. 1, 251–258.
31. R.P. Pant, R-weak commutativity and common fixed points of noncompatible maps, Ganita 49 (1998), no. 1, 19–27.
32. R.P. Pant, R-weak commutativity and common fixed points, Soochow J. Math. 25 (1999), no. 1, 37–42.
33. V. Parvanieh, J.R. Roshan and Z. Kadelburg, On generalized weakly GP-contractive mappings in ordered GP-metric spaces, Gulf J. Math. 1 (2013), no. 1, 78–97.
34. V. Popa, Fixed point theorems for implicit contractive mappings, Stud. Cercet. Ştiinţ. Ser. Mat. Univ. Bacău No. 7 (1997), 127–133.
35. V. Popa, Some fixed point theorems for compatible mappings satisfying an implicit relation, Demonstratio Math. 32 (1999), no. 1, 157–163.
36. V. Popa, A general fixed point theorem for occasionally weakly compatible mappings and applications, Sci. Stud. Res. Ser. Math. Inform. 22 (2012), no. 1, 77–91.
37. V. Popa, Fixed point theorems for two pairs of mappings satisfying a new type of common limit range property, Filomat 31 (2017), no. 11, 3181–3192.
38. V. Popa and M. Mocanu, Altering distance and common fixed points under implicit relations, Hacet. J. Math. Stat. 38 (2009), no. 3, 329–337.
39. V. Popa and A.-M. Patriciu, A general fixed point theorem for pairs of weakly compatible mappings in G-metric spaces, J. Nonlinear Sci. Appl. 5 (2012), no. 2, 151–160.
40. V. Popa and A.-M. Patriciu, Fixed point theorems for mappings satisfying an implicit relation in complete G-metric spaces, Bul. Inst. Politeh. Iaşi. Secţ. Mat. Mec. Teor. Fiz. 59(63) (2013), no. 2, 97–123.
41. V. Popa and A.-M. Patriciu, A general fixed point theorem for a pair of self mappings with common limit range property in G-metric spaces, Facta Univ. Ser. Math. Inform. 29 (2014), no. 4, 351–370.
42. V. Popa and A.-M. Patriciu, Two general fixed point theorems for a sequence of mappings satisfying implicit relations in Gp-metric spaces, Appl. Gen. Topol. 16 (2015), no. 2, 225–231.
43. V. Popa and A.-M. Patriciu, Well posedness of fixed point problem for mappings satisfying an implicit relation in Gp-metric spaces, Math. Sci. Appl. E-Notes 3 (2015), no. 1, 108–117.
44. V. Popa and A.-M. Patriciu, Fixed point theorems for two pairs of mappings satisfying common limit range property in G-metric spaces, Bul. Inst. Politeh. Iaşi. Secţ. Mat. Mec. Teor. Fiz. 62(66) (2016), no. 2, 19–42.
45. V. Popa and A.-M. Patriciu, Fixed point results for pairs of absorbing mappings in partial metric spaces, Acta Univ. Apulensis Math. Inform. No. 50 (2017), 97–109.
46. V. Popa and A.-M. Patriciu, Fixed point theorems for two pairs of mappings satisfying a new type of common limit range property in Gp-metric spaces, Ann. Math. Sil. 32 (2018), 295–312.
47. W. Shatanawi, Fixed point theory for contractive mappings satisfying Φ-maps in G-metric spaces, Fixed Point Theory Appl. 2010, Art. ID 181650, 9 pp.
48. W. Sintunavarat and P. Kumam, Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces, J. Appl. Math. 2011, Art. ID 637958, 14 pp.
49. 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.
PopaV. (2020). A general fixed point theorem for two pairs of absorbing mappings in Gp-metric spaces. Annales Mathematicae Silesianae, 34(2), 268-285. Retrieved from https://www.journals.us.edu.pl/index.php/AMSIL/article/view/13617
Valeriu Popa
vpopa@ub.ro
“Vasile Alecsandri” University of Bacau, Romania Romania
“Vasile Alecsandri” University of Bacau, Romania Romania
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