# Some Results on Fixed Point Theorems in Banach Algebras

## Main Article Content

### Abstract

Let X be a Banach algebra and D be a nonempty subset of X. Let (T 1, T 2) be a pair of self mappings on D satisfying some specific conditions. Here we discuss different situations for existence of solution of the operator equation u = T 1 uT 2 u in D. Similar results are established in case of reflexive Banach algebra X with the subset D. Again, considering a bounded, open and convex subset B in a uniformly convex Banach algebra X with three self mappings T 1 ,T 2 ,T 3 on B, we derive the conditions for existence of solution of the operator equation u = T 1 uT 2 u + T 3 u in B. Application of some of these results to the tensor product is also shown here with some examples.

## Article Details

### References

- Afif Ben Amar, Soufiene Chouayekh, Aref Jeribi, New fixed point theorems in Banach algebras under weak topology features and applications to nonlinear integral equations, Journal of Functional Analysis 259 (2010), 2215-2237.
- F. F. Bonsal and J. Duncan, Complete Normed algebras, Springer-Verlag, Berlin Heidelberg New York, 1973.
- Felix. E. Browder On a Generalization of the Schauder Fixed Point Theorem, Duke Mathematical Journal, 26 (2) (1959), 291-303.
- B.C. Dhage On a fixed point theorem in Banach algebras with applications, Applied Mathematics Letters 18 (2005), 273-280.
- B.C. Dhage, On some variants of Schauders fixed point principle and applications to nonlinear integral equations, J. Math. Phys. Sci. 25 (1988) 603-611.
- B.C. Dhage, On Existance Theorems for Nonlinear Integral Equations in Banach Algebra via Fixed Point Tech- niques, East Asian Math. J. 17(2001), 33-45.
- D. Das, N. Goswami, Some Fixed Point Theorems on the Sum and Product of Operators in Tensor Product Spaces, IJPAM, 109(2016) 651-663.
- Deepmala and H.K. Pathak, On solutions of some functional-integral equations in Banach algebra, Research J. Science and Tech. 5 (3) (2013), 358-362.
- Deepmala, A Study on Fixed Point Theorems for Nonlinear Contractions and its Applications, Ph.D. Thesis, Pt. Ravishankar Shukla University, Raipur 492 010, Chhatisgarh, India, 2014.
- K. Goebeli and W. A. Kirk, A Fixed Point Theorem for Asymptotically Nonexpansive Mappings, Proc. of the American Math. Soc. 35 (1) (1972), 171-174.
- Olga Hadzic, A Fixed Point Theorem For The Sum of Two Mappings, Proc. Amer. Math. Soc. 85 (1) (1982), 37-41.
- A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North- Holland Math. Std., vol. 204, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2006.
- L.N. Mishra, H.M. Srivastava, M. Sen, On existence results for some nonlinear functional-integral equations in Banach algebra with applications, Int. J. Anal. Appl., 11 (1) (2016), 1-10.
- L.N. Mishra, S.K. Tiwari, V.N. Mishra, Fixed point theorems for generalized weakly S-contractive mappings in partial metric spaces, J. of App. Anal. and Comp., 5 (4) 2015, 600-612.
- L.N. Mishra, M. Sen, On the concept of existence and local attractivity of solutions for some quadratic Volterra integral equation of fractional order, Applied Mathematics and Computation 285 (2016), 174-183.
- L. N. Mishra, R. P. Agarwal, M. Sen, Solvability and asymptotic behavior for some nonlinear quadratic integral equa- tion involving Erd ´ elyi-Kober fractional integrals on the unbounded interval, Progress in Fractional Differentiation and Applications 2 (3) (2016), 153-168.
- L.N. Mishra, M. Sen, R.N. Mohapatra, On existence theorems for some generalized nonlinear functional-integral equations with applications, Filomat, in press.
- L.N. Mishra, S.K. Tiwari, V.N. Mishra, I.A. Khan, Unique Fixed Point Theorems for Generalized Contractive Mappings in Partial Metric Spaces, Journal of Function Spaces, 2015 (2015), Article ID 960827.
- V.N. Mishra, Some Problems on Approximations of Functions in Banach Spaces, Ph.D. Thesis, Indian Institute of Technology, Roorkee 247 667, Uttarakhand, India, 2007.
- M. Mursaleen, H.M. Srivastava, S.K. Sharma, Generalized statistically convergent sequences of fuzzy numbers, J. Intelligent Fuzzy Systems 30 (2016), 1511-1518.
- W. V. Petryshyn and T. S. Tucker, On the Functional Equations Involving Nonlinear Generalized P-Compact Operators, Transactions of the American Mathematical Society, 135 (1969), 343-373
- W. V. Petryshyn and T. E. Williamson, JR. Strong and Weak Convergence of the Sequence of Successive Approxi- mations for Quasi-Nonexpansive Mappings, J. of Math. Ana. and Appl. 43 (1973), 459-497.
- J. Schu, Iterative Construction of Fixed Points of Asymptotically Nonexpansive Mappings, J. of Math. Ana. and Appl. 158 (1991), 407-413.
- H.K. Pathak and Deepmala, Remarks on some fixed point theorems of Dhage, Applied Mathematics Letters, 25 (11) (2012), 1969-1975.
- P. Vijayaraju, A fixed point theorem for a sum of two mappings in reflexive Banach spaces, Math. J. Toyoma Univ. 14 (1991), 41-50
- P. Vijayaraju, Iterative Construction of Fixed Points of Asymptotic 1-Set Contraction in Banach Spaces, Taiwanese J. of Math. 1 (3) (1997), 315-325.
- P. Vijayraju, Fixed Point Theorems For a Sum of Two Mappings in Locally Convex Spaces, Int. J. Math. and Math. Sci., 17(4) (1994), 681-686.
- A. Raymond Ryan, Introduction to Tensor Product of Banach Spaces, London, Springer -Verlag, 2002.