Ukrainian Mathematical Journal

, Volume 69, Issue 11, pp 1784–1804 | Cite as

Common Fixed-Point Theorems for Hybrid Generalized (F, 𝜑)-Contractions Under the Common Limit Range Property with Applications

  • H. K. Nashine
  • M. Imdad
  • M. Ahmadullah

We consider a relatively new hybrid generalized F-contraction involving a pair of mappings and use this contraction to prove a common fixed-point theorem for a hybrid pair of occasionally coincidentally idempotent mappings satisfying the generalized (F, 𝜑)-contraction condition with the common limit range property in complete metric spaces. A similar result involving a hybrid pair of mappings satisfying the rational-type Hardy–Rogers (F, 𝜑)-contractive condition is also proved. We also generalize and improve several results available from the existing literature. As applications of our results, we prove two theorems on the existence of solutions of certain systems of functional equations encountered in dynamic programming and the Volterra integral inclusion. Moreover, we present an illustrative example.


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  1. 1.
    M. Aamri and D. El Moutawakil, “Some new common fixed point theorems under strict contractive conditions,” J. Math. Anal. Appl., 270, No. 1, 181–188 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    J. Ali and M. Imdad, “Common fixed points of nonlinear hybrid mappings under strict contractions in semimetric spaces,” Nonlin. Anal. Hybrid Syst., 4, No. 4, 830–837 (2010).CrossRefzbMATHGoogle Scholar
  3. 3.
    H. H. Alsulami, E. Karapinar, and H. Piri, “Fixed points of modified F-contractive mappings in complete metric-like spaces,” J. Funct. Spaces, Article ID 270971 (2014).Google Scholar
  4. 4.
    J. P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin (1984).Google Scholar
  5. 5.
    R. Bellman and E. S. Lee, “Functional equations in dynamic programming,” Aequationes Math., 17, 1–18 (1978).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    T. C. Bhakta and S. Mitra, “Some existence theorems for functional equations arising in dynamic programming,” J. Math. Anal. Appl., 98, 348–362 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    S. Chauhan, M. A. Khan, Z. Kadelburg, and M. Imdad, “Unified common fixed point theorem for a hybrid pair of mappings via an implicit relation involving altering distance function,” Abstr. Appl. Anal., Article ID 718040, 8 p. (2014).Google Scholar
  8. 8.
    K. Deimling, Multivalued Differential Equations, de Gruyter, Berlin (1992).Google Scholar
  9. 9.
    B. C. Dhage, “A functional integral inclusion involving Carathéodories,” Electron. J. Qual. Theory Differ. Equat., 14, 1–18 (2003).Google Scholar
  10. 10.
    S. Dhompongsa and H. Yingtaweesittikul, “Fixed points for multivalued mappings and the metric completeness,” Fixed Point Theory Appl., Article ID 972395 (2009).Google Scholar
  11. 11.
    B. Fisher, “Common fixed point theorems for mappings and set-valued mappings,” Rostock Math. Kolloq., 18, 69–77 (1981).zbMATHGoogle Scholar
  12. 12.
    T. Hu, “Fixed point theorems for multivalued mappings,” Canad. Math. Bull., 23, 193–197 (1980).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    M. Imdad, A. Ahmad, and S. Kumar, “On nonlinear nonself-hybrid contractions,” Rad. Mat., 10, No. 2, 233–244 (2001).MathSciNetzbMATHGoogle Scholar
  14. 14.
    M. Imdad, S. Chauhan, A. H. Soliman, and M. A. Ahmed, “Hybrid fixed point theorems in symmetric spaces via common limit range property,” Demonstr. Math., 47, No. 4, 949–962 (2014).MathSciNetzbMATHGoogle Scholar
  15. 15.
    M. Imdad, M. S. Khan, and S. Sessa, “On some weak conditions of commutativity in common fixed point theorems,” Int. J. Math. Math. Sci., 11, No. 2, 289–296 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    M. Imdad and A. H. Soliman, “Some common fixed point theorems for a pair of tangential mappings in symmetric spaces,” Appl. Math. Lett., 23, No. 4, 351–355 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    G. Jungck and B. E. Rhoades, “Fixed points for set-valued functions without continuity,” J. Indian Pure Appl. Math., 29, No. 3, 227–248 (1998).MathSciNetzbMATHGoogle Scholar
  18. 18.
    Z. Kadelburg, S. Chauhan, and M. Imdad, “A hybrid common fixed point theorem under certain recent properties,” Sci. World J., Article ID 860436, 6 p. (2014).Google Scholar
  19. 19.
    T. Kamran, “Coincidence and fixed points for hybrid strict contractions,” J. Math. Anal. Appl., 299, No. 1, 235–241 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    H. Kaneko, “Single-valued and multi-valued f-contractions,” Boll. Unione Mat. Ital., 4, 29–33 (1985).MathSciNetzbMATHGoogle Scholar
  21. 21.
    H. Kaneko, “A common fixed point of weakly commuting multivalued mappings,” Math. Japan, 33, No. 5, 741–744 (1988).zbMATHGoogle Scholar
  22. 22.
    H. Kaneko and S. Sessa, “Fixed point theorems for compatible multivalued and single-valued mappings,” Int. J. Math. Math. Sci., 12, No. 2, 257–262 (1989).CrossRefzbMATHGoogle Scholar
  23. 23.
    E. Karapinar, H. Piri, and H. Al-Sulami, “Fixed points of generalized F-Suzuki type contraction in complete b-metric spaces,” Discrete Dyn. Nat. Soc., Article ID 969726, 1–8 (2015).Google Scholar
  24. 24.
    T. Kubiak, “Fixed point theorems for contractive-type multivalued mappings,” Math. Japan, 30, No. 1, 89–101 (1985).MathSciNetzbMATHGoogle Scholar
  25. 25.
    A. Lasota and Z. Opial, “An application of the Kakutani–Ky-Fan theorem in the theory of ordinary differential equations,” Bull. Acad. Polon. Sci. Sér. Sci. Math., Astoronom. Phys., 13, 781–786 (1965).MathSciNetzbMATHGoogle Scholar
  26. 26.
    Z. Liu and J. S. Ume, “On properties of solutions for a class of functional equations arising in dynamic programming,” J. Optim. Theory Appl., 117, 533 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Z. Liu, L. Wong, Kug H. Kim, and S. M. Kang, “Common fixed point theorems for contractive mappings and their applications in dynamic programming,” Bull. Korean Math. Soc., 45, No. 3, 573–585 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    S. B. Nadler, Jr., “Multivalued contraction mappings,” Pacific J. Math., 20, No. 2, 457–488 (1969).Google Scholar
  29. 29.
    S. A. Naimpally, S. L. Singh, and J. H. M. Whitfield, “Coincidence theorems for hybrid contractions,” Math. Nachr., 127, 177–180 (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    H. K. Pathak, “Fixed point theorems for weak compatible multivalued and single-valued mappings,” Acta Math. Hungar., 67, No. 1-2, 69–78 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    H. K. Pathak, Y. J. Cho, S. M. Kang, and B. S. Lee, “Fixed point theorems for compatible mappings of type (P) and applications to dynamic programming,” Matematiche, 50, 15–33 (1995).MathSciNetzbMATHGoogle Scholar
  32. 32.
    H. K. Pathak and Deepmala, “Some existing theorems for solvability of certain functional equations arising in dynamic programming,” Bull. Calcutta Math. Soc., 104, No. 3, 237–244 (2012).Google Scholar
  33. 33.
    H. K. Pathak and B. Fisher, “Common fixed point theorems with applications in dynamic programming,” Glas. Mat., Ser. III, 31, 321–328 (1996).Google Scholar
  34. 34.
    H. K. Pathak, S. M. Kang, and Y. J. Cho, “Coincidence and fixed point theorems for nonlinear hybrid generalized contractions,” Czechoslovak Math. J., 48(123), 341–357 (1998).Google Scholar
  35. 35.
    H. K. Pathak and M. S. Khan, “Fixed and coincidence points of hybrid mappings,” Arch. Math. (Brno), 38, 201–208 (2002).MathSciNetzbMATHGoogle Scholar
  36. 36.
    H. K. Pathak and R. Rodríguez-López, “Noncommutativity of mappings in hybrid fixed point results,” Boundary Value Probl., 145, 21 p. (2013).Google Scholar
  37. 37.
    H. K. Pathak and R. Tiwari, “Common fixed points for weakly compatible mappings and applications in dynamic programming,” Ital. J. Pure Appl. Math., 30, 253–268 (2013).Google Scholar
  38. 38.
    D. O’Regan, “Integral inclusions of upper semicontinuous or lower semicontinuous type,” Proc. Amer. Math. Soc., 124, 2391–2399 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    M. Sgroi and C. Vetro, “Multivalued F-contractions and the solution of certain functional and integral equations,” Filomat, 27, No. 7, 1259–1268 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    S. L. Singh, K. S. Ha, and Y. J. Cho, “Coincidence and fixed points of nonlinear hybrid contractions,” Int. J. Math. Math. Sci., 12, No. 2, 247–256 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    S. L. Singh and A. M. Hashim, “New coincidence and fixed point theorems for strictly contractive hybrid maps,” Austr. J. Math. Anal. Appl., 2, No. 1, Article 12, 7 p. (2005).Google Scholar
  42. 42.
    S. L. Singh and S. N. Mishra, “Coincidence and fixed points of nonself-hybrid contractions,” J. Math. Anal. Appl., 256, 486–497 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    S. L. Singh and S. N. Mishra, “Coincidence theorems for certain classes of hybrid contractions,” Fixed Point Theory Appl., Article ID 898109 (2010).Google Scholar
  44. 44.
    W. Sintunavarat and P. Kumam, “Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces,” J. Appl. Math., Article ID 637958, 14 p. (2011).Google Scholar
  45. 45.
    S. Sessa, M. S. Khan, and M. Imdad, “Common fixed point theorem with a weak commutativity condition,” Glas. Mat., Ser. III, 21(41), 225–235 (1986).MathSciNetzbMATHGoogle Scholar
  46. 46.
    D. Türkoğlu and I. Altun, “A fixed point theorem for multivalued mappings and its applications to integral inclusions,” Appl. Math. Lett., 20, 563–570 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    D. Wardowski, “Fixed points of a new type of contractive mappings in complete metric spaces,” Fixed Point Theory Appl., 2012, No. 94 (2012).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • H. K. Nashine
    • 1
  • M. Imdad
    • 2
  • M. Ahmadullah
    • 2
  1. 1.Texas A & M University-KingsvilleKingsvilleUSA
  2. 2.Aligarh Muslim UniversityAligarhIndia

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