## Abstract and Applied Analysis

### The Best Approximation Theorems and Fixed Point Theorems for Discontinuous Increasing Mappings in Banach Spaces

#### Abstract

We prove that Fan’s theorem is true for discontinuous increasing mappings $f$ in a real partially ordered reflexive, strictly convex, and smooth Banach space $X$. The main tools of analysis are the variational characterizations of the generalized projection operator and order-theoretic fixed point theory. Moreover, we get some properties of the generalized projection operator in Banach spaces. As applications of our best approximation theorems, the fixed point theorems for non-self-maps are established and proved under some conditions. Our results are generalizations and improvements of the recent results obtained by many authors.

#### Article information

Source
Abstr. Appl. Anal., Volume 2015, Special Issue (2015), Article ID 165053, 7 pages.

Dates
First available in Project Euclid: 17 August 2015

https://projecteuclid.org/euclid.aaa/1439816292

Digital Object Identifier
doi:10.1155/2015/165053

Mathematical Reviews number (MathSciNet)
MR3378555

Zentralblatt MATH identifier
06929069

#### Citation

Kong, Dezhou; Liu, Lishan; Wu, Yonghong. The Best Approximation Theorems and Fixed Point Theorems for Discontinuous Increasing Mappings in Banach Spaces. Abstr. Appl. Anal. 2015, Special Issue (2015), Article ID 165053, 7 pages. doi:10.1155/2015/165053. https://projecteuclid.org/euclid.aaa/1439816292

#### References

• Y. Alber, “Generalized projection operators in Banach spaces: properties and applications,” in Proceedings of the Israel Seminar Ariel, Israel, Function Differential Equation, vol. 1, pp. 1–21, 1994.
• J. Li, “The generalized projection operator on reflexive Banach spaces and its applications,” Journal of Mathematical Analysis and Applications, vol. 306, no. 1, pp. 55–71, 2005.
• G. Isac, “On the order monotonicity of the metric projection operator,” in Approximation Theory, Wavelets and Applications, S. P. Singh, Ed., vol. 454 of NATO Science Series, pp. 365–379, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1995.
• H. Nishimura and E. A. Ok, “Solvability of variational inequalities on Hilbert lattices,” Mathematics of Operations Research, vol. 37, no. 4, pp. 608–625, 2012.
• J. Li and E. A. Ok, “Optimal solutions to variational inequalities on Banach lattices,” Journal of Mathematical Analysis and Applications, vol. 388, no. 2, pp. 1157–1165, 2012.
• K. Fan, “Extensions of two fixed point theorems of F. E. Browder,” Mathematische Zeitschrift, vol. 112, pp. 234–240, 1969.
• T. Lin and S. Park, “Approximation and fixed-point theorems for condensing composites of multifunctions,” Journal of Mathematical Analysis and Applications, vol. 223, no. 1, pp. 1–8, 1998.
• D. O'Regan and N. Shahzad, “Approximation and fixed point theorems for countable condensing composite maps,” Bulletin of the Australian Mathematical Society, vol. 68, no. 1, pp. 161–168, 2003.
• K. Tan and X. Yuan, “Random fixed-point theorems and approximation in cones,” Journal of Mathematical Analysis and Applications, vol. 185, no. 2, pp. 378–390, 1994.
• L. Liu, “Approximation theorems and fixed point theorems for various classes of 1-set-contractive mappings in Banach spaces,” Acta Mathematica Sinica, vol. 17, no. 1, pp. 103–112, 2001.
• L. Liu, “Random approximations and random fixed point theorems for random 1-set-contractive non-self-maps in abstract cones,” Stochastic Analysis and Applications, vol. 18, no. 1, pp. 125–144, 2000.
• L. Liu, “Some random approximations and random fixed point theorems for $1$-set-contractive random operators,” Proceedings of the American Mathematical Society, vol. 125, no. 2, pp. 515–521, 1997.
• L. Liu, “Random approximations and random fixed point theorems in infinite-dimensional Banach spaces,” Indian Journal of Pure and Applied Mathematics, vol. 28, no. 2, pp. 139–150, 1997.
• I. Beg and N. Shahzad, “Random fixed points of random multivalued operators on Polish spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 20, no. 7, pp. 835–847, 1993.
• N. Shahzad, “Fixed point and approximation results for multimaps in S-KKM class,” Nonlinear Analysis: Theory, Methods & Applications, vol. 56, no. 6, pp. 905–918, 2004.
• J. Markin and N. Shahzad, “Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 6, pp. 2435–2441, 2009.
• A. Amini-Harandi, “Best and coupled best approximation theorems in abstract convex metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 3, pp. 922–926, 2011.
• D. Roux and S. P. Singh, “On some fixed point theorems,” International Journal of Mathematics and Mathematical Sciences, vol. 12, no. 1, pp. 61–64, 1989.
• L. Liu, “On approximation theorems and fixed point theorems for non-self-mappings in infinite-dimensional Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 188, no. 2, pp. 541–551, 1994.
• L. Liu and X. Li, “On approximation theorems and fixed point theorems for non-self-mappings in uniformly convex Banach spaces,” Banyan Mathematical Journal, vol. 4, pp. 11–20, 1997.
• D. O'Regan, “Existence and approximation of fixed points for multivalued maps,” Applied Mathematics Letters, vol. 12, no. 6, pp. 37–43, 1999.
• D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, NY, USA, 1988.
• D. Guo, Y. Cho, and J. Zhu, Partial Ordering Methods in Nonlinear Problems, Nova Science Publishers, New York, NY, USA, 2004.
• P. Meyer-Nieberg, Banach Lattices, Universitext, Springer, New York, NY, USA, 1991.
• I. Cioranescu, “Geometry of Banach spaces,” in Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990.
• W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and its Applications, Yokohama, Yokohama, Japan, 2000.
• D. Kong, L. Liu, and Y. Wu, “Best approximation and fixed-point theorems for discontinuous increasing maps in Banach lattices,” Fixed Point Theory and Applications, vol. 2014, article 18, 2014. \endinput