Journal of Optimization Theory and Applications

, Volume 160, Issue 2, pp 703–710 | Cite as

A Characterization of Strictly Convex Spaces and Applications

  • V. Sankar Raj
  • A. Anthony Eldred


In this article, we establish a new characterization of strictly convex normed linear spaces. Using this characterization, we obtain an extended version of Banach’s Contraction Principle in a best proximity point setting.


Strictly convex spaces d-Property Best proximity points Non-self contraction mapping Nonexpansive mapping Fixed points Metric projection Normal structure 



The authors are grateful to the Editor and the Referees for their helpful suggestions and comments for the improvement of this manuscript.


  1. 1.
    James, R.C.: Orthogonality and linear functionals in normed linear spaces. Trans. Am. Math. Soc. 61, 265–292 (1947) CrossRefGoogle Scholar
  2. 2.
    Šmulian, V.: On some geometrical properties of the unit sphere in the space of the type (B). Rec. Math. N. S. [Mat. Sb.] 6(48), 77–94 (1939) Google Scholar
  3. 3.
    Petryshyn, W.V.: A characterization of strict convexity of Banach spaces and other uses of duality mappings. J. Funct. Anal. 6, 282–291 (1970) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Strawther, D., Gudder, S.: A characterization of strictly convex Banach spaces. Proc. Am. Math. Soc. 47, 268 (1975) zbMATHMathSciNetGoogle Scholar
  5. 5.
    Gudder, S., Strawther, D.: Strictly convex normed linear spaces. Proc. Am. Math. Soc. 59(2), 263–267 (1976) CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Clarkson, J.: Uniformly convex spaces. Trans. Am. Math. Soc. 40(3), 396–414 (1936) CrossRefMathSciNetGoogle Scholar
  7. 7.
    Lumer, G.: Semi-inner-product spaces. Trans. Am. Math. Soc. 100, 29–43 (1961) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Giles, J.R.: Classes of semi-inner-product spaces. Trans. Am. Math. Soc. 129, 436–446 (1967) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Torrance, E.: Strictly convex spaces via semi-inner-product space orthogonality. Proc. Am. Math. Soc. 26, 108–110 (1970) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    DePrima, C.R., Petryshyn, W.V.: Remarks on strict monotonicity and surjectivity properties of duality mappings defined on real normed linear spaces. Math. Z. 123, 49–55 (1971) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Anthony Eldred, A., Veeramani, P.: Existence and convergence of best proximity points. J. Math. Anal. Appl. 323(2), 1001–1006 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Anthony Eldred, A., Sankar Raj, V., Veeramani, P.: On best proximity pair theorems for relatively u-continuous mappings. Nonlinear Anal. 74, 3870–3875 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Sankar Raj, V.: A best proximity point theorem for weakly contractive non-self mappings. Nonlinear Anal. 74, 4804–4808 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Espinola, R.: A new approach to relatively nonexpansive mappings. Proc. Am. Math. Soc. 136(6), 1987–1995 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Haghi, R.H., Rakoc̆ević, V., Rezapour, S., Shahzad, N.: Best proximity results in regular cone metric spaces. Rend. Circ. Mat. Palermo 60, 323–327 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Sadiq Basha, S., Shahzad, N., Jeyaraj, R.: Optimal approximate solutions of fixed point equations. Abstr. Appl. Anal. 174560 (2011) Google Scholar
  17. 17.
    Shahzad, N., Sadiq Basha, S., Jeyaraj, R.: Common best proximity points: global optimal solutions. J. Optim. Theory Appl. 148(1), 69–78 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Suzuki, T., Kikkawa, M., Vetro, C.: The existence of best proximity points in metric spaces with the property UC. Nonlinear Anal. 71(7–8), 2918–2926 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Al-Thagafi, M.A., Shahzad, N.: Convergence and existence results for best proximity points. Nonlinear Anal. 70, 3665–3671 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Al-Thagafi, M., Shahzad, N.: Best proximity pairs and equilibrium pairs for Kakutani multimaps. Nonlinear Anal. 70(3), 1209–1216 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Anthony Eldred, A., Veeramani, P.: On best proximity pair solutions with applications to differential equations. In: The Indian Math. Soc., Special Centenary Volume (1907–2007), pp. 51–62 (2008) Google Scholar
  22. 22.
    Kim, W.K., Lee, K.H.: Existence of best proximity pairs and equilibrium pairs. J. Math. Anal. Appl. 316(2), 433–446 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Kim, W.K., Kum, S., Lee, K.H.: On general best proximity pairs and equilibrium pairs in free abstract economies. Nonlinear Anal. 68(8), 2216–2227 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Srinivasan, P.S., Veeramani, P.: On existence of equilibrium pair for constrained generalized games. Fixed Point Theory Appl. 1, 21–29 (2004) MathSciNetGoogle Scholar
  25. 25.
    Kirk, W.A., Reich, S., Veeramani, P.: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 24(7–8), 851–862 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Sadiq Basha, S., Veeramani, P.: Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 103(1), 119–129 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Kirk, W.A.: A fixed point theorem for mappings which do not increase distances. Am. Math. Mon. 72, 1004–1006 (1965) CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsManonmaniam Sundaranar UniversityTirunelveliIndia
  2. 2.Department of MathematicsSt. Joseph’s CollegeTiruchirappalliIndia

Personalised recommendations