Abstract
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.
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James, R.C.: Orthogonality and linear functionals in normed linear spaces. Trans. Am. Math. Soc. 61, 265–292 (1947)
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)
Petryshyn, W.V.: A characterization of strict convexity of Banach spaces and other uses of duality mappings. J. Funct. Anal. 6, 282–291 (1970)
Strawther, D., Gudder, S.: A characterization of strictly convex Banach spaces. Proc. Am. Math. Soc. 47, 268 (1975)
Gudder, S., Strawther, D.: Strictly convex normed linear spaces. Proc. Am. Math. Soc. 59(2), 263–267 (1976)
Clarkson, J.: Uniformly convex spaces. Trans. Am. Math. Soc. 40(3), 396–414 (1936)
Lumer, G.: Semi-inner-product spaces. Trans. Am. Math. Soc. 100, 29–43 (1961)
Giles, J.R.: Classes of semi-inner-product spaces. Trans. Am. Math. Soc. 129, 436–446 (1967)
Torrance, E.: Strictly convex spaces via semi-inner-product space orthogonality. Proc. Am. Math. Soc. 26, 108–110 (1970)
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)
Anthony Eldred, A., Veeramani, P.: Existence and convergence of best proximity points. J. Math. Anal. Appl. 323(2), 1001–1006 (2006)
Anthony Eldred, A., Sankar Raj, V., Veeramani, P.: On best proximity pair theorems for relatively u-continuous mappings. Nonlinear Anal. 74, 3870–3875 (2011)
Sankar Raj, V.: A best proximity point theorem for weakly contractive non-self mappings. Nonlinear Anal. 74, 4804–4808 (2011)
Espinola, R.: A new approach to relatively nonexpansive mappings. Proc. Am. Math. Soc. 136(6), 1987–1995 (2008)
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)
Sadiq Basha, S., Shahzad, N., Jeyaraj, R.: Optimal approximate solutions of fixed point equations. Abstr. Appl. Anal. 174560 (2011)
Shahzad, N., Sadiq Basha, S., Jeyaraj, R.: Common best proximity points: global optimal solutions. J. Optim. Theory Appl. 148(1), 69–78 (2011)
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)
Al-Thagafi, M.A., Shahzad, N.: Convergence and existence results for best proximity points. Nonlinear Anal. 70, 3665–3671 (2009)
Al-Thagafi, M., Shahzad, N.: Best proximity pairs and equilibrium pairs for Kakutani multimaps. Nonlinear Anal. 70(3), 1209–1216 (2009)
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)
Kim, W.K., Lee, K.H.: Existence of best proximity pairs and equilibrium pairs. J. Math. Anal. Appl. 316(2), 433–446 (2006)
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)
Srinivasan, P.S., Veeramani, P.: On existence of equilibrium pair for constrained generalized games. Fixed Point Theory Appl. 1, 21–29 (2004)
Kirk, W.A., Reich, S., Veeramani, P.: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 24(7–8), 851–862 (2003)
Sadiq Basha, S., Veeramani, P.: Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 103(1), 119–129 (2000)
Kirk, W.A.: A fixed point theorem for mappings which do not increase distances. Am. Math. Mon. 72, 1004–1006 (1965)
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The authors are grateful to the Editor and the Referees for their helpful suggestions and comments for the improvement of this manuscript.
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Communicated by Alfredo N. Iusem.
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Sankar Raj, V., Anthony Eldred, A. A Characterization of Strictly Convex Spaces and Applications. J Optim Theory Appl 160, 703–710 (2014). https://doi.org/10.1007/s10957-013-0357-x
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DOI: https://doi.org/10.1007/s10957-013-0357-x