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Approximation of *-nonexpansive random multivalued operators on Banach spaces

Part of: Nonlinear operators and their properties Approximations and expansions Stochastic analysis

Published online by Cambridge University Press:  09 April 2009

Ismat Beg ,
A. R. Khan  and
N. Hussain
Ismat Beg
Affiliation:
Lahore University of Management Sciences, Lahore, Pakistan, e-mail: [email protected]
A. R. Khan
Affiliation:
KFUPM Dhahran, Saudi Arabia
N. Hussain
Affiliation:
Bhauddin Zakariya UniversityMultan, Pakistan
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Abstract

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We establish the existence and approximation of solutions to the operator inclusion y ∈ Ty for deterministic and random cases for a nonexpansive and *-nonexpansive multivalued mapping T defined on a closed bounded (not necessarily convex) subset C of a Banach space. We also prover random fixed points and approximation results for*-nonexpansive random operators defined on an unbounded subject C of a uniformly convex Banach space.

Keywords

random fixed point*-nonexpansive random multivalued mapweakly inward orperatorLeray-Schuder conditionOpial conditionrandom approximationBanach space

MSC classification

Secondary: 47H40: Random operators 47H10: Fixed-point theorems 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 47H09: Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc. 60H25: Random operators and equations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 76 , Issue 1 , February 2004 , pp. 51 - 66
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Anderson, D. E., Nelson, J. L. and Singh, K. L., ‘Fixed point for single and multivalued mappings in locally convex spaces’, Math. Japon. 31 (1986), 665–672.Google Scholar
[2]Beg, I., ‘Random fixed points of non-self maps and random approximations’, J. Appl. Math. Stochastic Anal. 102 (1997), 127–130.CrossRefGoogle Scholar
[3]Beg, I. and Shahzad, N., ‘Random fixed point theorems for nonexpansive and contractive-type random operators on Bancah spcaes’, J. Appl. Math. Stochastic Anal. 7 (1994), 569–580.CrossRefGoogle Scholar
[4]Beg, I., ‘Random fixed point theorems for multivalued inward random operators of Banach spaces’, Adv. Math. Sci. Appl. 5 (1995), 31–37.Google Scholar
[5]Beg, I., ‘Applications of the proximity map to random fixed point theorems in Hilbert spaces’, J. Math. Anal. Appl. 196 (1995), 606–613.CrossRefGoogle Scholar
[6]Beg, I., ‘On random approiximations and a random fixed theorem for multivalued mappings defined on unbounded sets in Hilbert spaces’, Stochastic Anal. Appl. 14 (1996), 507–511.CrossRefGoogle Scholar
[7]Canetti, A., Marino, G. and Pietramala, P., ‘Fixed point theorems for multivalued mappings in Banach spaces’, Nonlinear Anal. 17 (1991), 11–20.CrossRefGoogle Scholar
[8]Himmelberg, C. J., ‘Measurable ralations’, Fund Math. 87 (1975), 53–72.CrossRefGoogle Scholar
[9]Husain, T. and Latif, A., ‘Foxed points of multivalued nonexpansice maps’, Math. Japon. 33 (1988), 385–391.Google Scholar
[10]Itoh, S., ‘Random fixed point theorems with applications to random differential equations in Banach spaces’, J. Math. Anal. Appl. 67 (1979), 261–273.CrossRefGoogle Scholar
[11]Khan, A. R. and Hussain, N., ‘Random fixed points for *-nonexpansive random operators’, J. Appl. Math. Stochastic Anal. 14 (2002), 341–349.CrossRefGoogle Scholar
[12]Khan, A. R., Hussain, N. and Thaheem, A. B., ‘Applications of fixed theorems to invariant approximation’, Approx. Theory and its Appl. 16 (2000), 48–55.CrossRefGoogle Scholar
[13]Kirk, W. A. and Ray, W. O., ‘Fixed point theorems for mappings defined on unbounded sets in Banach spaces’, Studia Math. 64 (1979), 127–138.CrossRefGoogle Scholar
[14]Lin, T. C., ‘Random approximations and random fixed point theorems for non-self maps’, Proc. Amer. Math. Sco. 103 (1988), 1129–1135.CrossRefGoogle Scholar
[15]Lin, T. C., ‘Random approximations and random fixed point theroems for continuous 1-set constructive random maps’, Proc. Amer. Math. Soc. 123 (1995), 1167–1176.Google Scholar
[16]O'Regan, D., ‘Nonlinear operator approximation theory’, Neumer. Funct. Anal. Optimiz. 19 (1998), 587–592.CrossRefGoogle Scholar
[17]Papageorgiou, N. S., ‘Random fixed point theroems for measurable multifunctions in Banach spaces’, Proc. Amer. Math. Soc. 97 (1986), 507–514.CrossRefGoogle Scholar
[18]Park, S., ‘Best approximations and fixed points of nunexpansive maps in Hilbert spaces’, Nemer. Funct. Anal. Optimiz. 18 (1997), 649–657.Google Scholar
[19]Sehgal, V. M. and Singh, S. P., ‘On random approximation and a random fixed theorem for set-valued mappints’, Proc. Amer. Math. Soc. 95 (1985), 91–94.CrossRefGoogle Scholar
[20]Sehgal, V. M., ‘A generalization to multifunctions of Fan'best approximation theroem’, Proc. Amer. Math. Soc. 102 (1988), 534–537.Google Scholar
[21]Shahzad, N., ‘Random fixed point theorems forvarious classes of 1-set-contractive maps in Banach spaces’, J. Math. Anal. Appl. 203 (1996), 712–718.CrossRefGoogle Scholar
[22]Shahzad, N. and Latif, S., ‘Random fixed points for several classes of 1-ball-contractive and 1-set-contractive random maps’, J. Math. Anal. Appl. 237 (1999), 83–92.CrossRefGoogle Scholar
[23]Tan, K. K. and Yuan, X. X., ‘On deterministic and random fixed points’, Proc. Amer. Math. Soc. 119 (1993), 849–856.CrossRefGoogle Scholar
[24]Tan, K. K., ‘Random fixed point theorems and approximation in Cones’, J. Math. Anal. Appl. 185 (1994), 378–390.CrossRefGoogle Scholar
[25]Tan, K. K., ‘Random fixed point theorems and approximation’, Stochastic Anal. Apal. 15 (1997), 103–123.CrossRefGoogle Scholar
[26]Xu, H. K., ‘Some random fixed point theorems for condensing and nonexpansive operators’, Proc. Amer. Math. Soc. 110 (1990), 395–400.CrossRefGoogle Scholar
[27]Xu, H. K., ‘On weakly nonexpansive and *-nonexpansive multivalued mappings’, Math. Japon. 36 (1991), 441–445.Google Scholar
[28]Yi, H. W. and Zhao, Y. C., ‘Fixed point theorems for weakly inward multivalued mappings and theie randomizations’, J. Math. Anal. Appl. 183 (1994), 613–619.CrossRefGoogle Scholar