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On Fixed Points of Asymptotically Regular Mappings

Part of: Connections with other structures, applications

Published online by Cambridge University Press:  09 April 2009

B. E. Rhoades ,
S. sessa ,
M. S. Khan  and
M. Swaleh
B. E. Rhoades
Affiliation:
Department of MathematicsIndiana UniversityBloomington, Indiana 47405, U.S.A.
S. sessa
Affiliation:
Università di AnpoliFacoltà di Architettura Istituto di Matematica Via Monteoliveto 3 80134 Napoli, Italy
M. S. Khan
Affiliation:
Department of Mathematics Faculty of ScienceKing Abdul Aziz UniversityP.O. Box 9028 21413 Jeddah, Saudi Arabia
M. Swaleh
Affiliation:
P. G. T. MathematicsMussorie Public SchoolMussorie, India
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Some results on fixed points of asymptotically regular mappings are obtained in complete metric spaces and normed linear spaces.

The structure of the set of common fixed points is also discussed in Banach spaces. Our work generalizes essentially known results of Das and Naik, Fisher, Jaggi, Jungck, Rhoades, Singh and Tiwari and several others.

Keywords

asymptotically regular sequenceasymptotically regular mappingorbitally continuous mappingweakly commuting mappingsnormed linear spaceconvex setstrictly convex Banach spaceuniformly convex Banach space.

MSC classification

Secondary: 54H25: Fixed-point and coincidence theorems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 43 , Issue 3 , December 1987 , pp. 328 - 346
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Bose, R. K. and Mukherjee, R. N., ‘Approximating fixed points of some mappings‘, Proc. Amer. Math. Soc. 82 (1981), 603–606.CrossRefGoogle Scholar
[2]Browder, F. E. and Petryshyn, W. V., ‘The solution by iteration of nonlinear functional equations in Banach spaces’, Bull. Amer. Math. Soc. 72 (1966), 571–575.CrossRefGoogle Scholar
[3]Chang, C. C., ‘On a fixed point theorem of contractive type‘, Comment. Math. Univ. St. Paul 32 (1983), 15–19.Google Scholar
[4]Chang, S. S., ‘A common fixed point theorem for commuting mappings’, Proc. Amer. Math. Soc. 83 (1981), 645–652.CrossRefGoogle Scholar
[5]Yeh, Cheh-Chih, ‘Common fixed point of continuous mappings in metric spaces’, Publ. Inst. Math. 27 (41) (1980), 21–25.Google Scholar
[6]Yeh, Cheh-Chih, ‘Remark on common fixed point of commuting mappings in L-spaces’, Matematiche (Catania) 34 (1979), 81–85.Google Scholar
[7]Ciric, L., ‘Fixed and periodic points of almost contractive operators’, Math. Balkanica 3 (1973), 33–44.Google Scholar
[8]Ciric, L., ‘On some maps with a non-unique fixed point’, Publ. Inst. Math. 17(31) (1974), 52–58.Google Scholar
[9]Conserva, V., ‘Common fixed point theorems for commuting maps on a metric space’, Publ. Inst. Math. 32 (46) (1982), 37–43.Google Scholar
[10]Das, K. M. and Naik, K. V., ‘Common fixed point theorems for commuting maps on a metric space’, Proc. Amer. Math. Soc. 77 (1979), 369–373.Google Scholar
[11]Diaz, J. B. and Metcalf, F. T., ‘On the set of subsequential limits of successive approximations’, Trans. Amer. Math. Soc. 135 (1969), 459–465.Google Scholar
[12]Dotson, W. G. Jr, ‘Fixed points of quasi-nonexpansive mappings’, J. Austral. Math. Soc. 13 (1972), 167–170.CrossRefGoogle Scholar
[13]Edelstein, M., ‘A remark on a theorem of M. A. Krasnoselskii‘, Amer. Math. Monthly 73 (1966), 509–510.CrossRefGoogle Scholar
[14]Emmanuele, G., ‘Fixed point theorems in complete metric spaces’, Nonlinear Anal. 5(3) (1981), 287–292.CrossRefGoogle Scholar
[15]Engl, H. W., ‘Weak convergence of asymptotically regular sequences for non-expansive mappings and connections with certain Chebyshef-centers’, Nonlinear Anal. 1(5) (1977), 495–501.CrossRefGoogle Scholar
[16]Fisher, B., ‘Results on common fixed points’, Math. Japon. 22 (1977) 335–338.Google Scholar
[17]Fisher, B., ‘Mappings with a common fixed point’, Mat. Sem. Notes Kobe Univ. 7 (1979), 81–84; addendum 8 (1980), 513–514.Google Scholar
[18]Fisher, B., ‘Common fixed points of commuting mappings’, Bull. Inst. Math. Acad. Sinica 9 (1981), 399–406.Google Scholar
[19]Fisher, B., ‘Three mappings with a common fixed point’, Math. Sem. Notes Kobe Univ. 10 (1982), 293–302.Google Scholar
[20]Fisher, B., ‘Common fixed points of four mappings‘, Bull. Inst. Math. Acad. Sinica 11 (1983), 103–113.Google Scholar
[21]Garegnani, G. and Zanco, C., ‘Fixed points of somehow contractive multivalued mappings’, 1st. LombardoAccad. Set. Lett. Rend. A 114 (1980), 138–148.Google Scholar
[22]Goebel, K., and Massa, S., ‘Some remarks on nonexpansive mappings in Hilbert spaces’, Boll. Un. Mat. Ital. (6) 3-A (1984), 139–145.Google Scholar
[23]Guay, M. D. and Singh, K. L., ‘Fixed points of asymptotically regular mappings’, Mat. Vesnik 35 (1983), 101–106.Google Scholar
[24]Hardy, G. E. and Rogers, T. D., ‘A generalization of a fixed point theorem of Reich’, Canad. Math. Bull. 16(1973), 201–206.CrossRefGoogle Scholar
[25]Kannan, R., ‘Construction of fixed points of a class of nonlinear mappings’, J. Math. Anal. Appl. 41 (1973), 430–438.CrossRefGoogle Scholar
[26]Karlovitz, L. A., ‘On non-expansive mappings’, Proc. Amer. Math. Soc. 55 (1976), 321–325.CrossRefGoogle Scholar
[27]Khan, M. S., ‘Remarks on some fixed point theorems‘, C. R. Bulg. Sci. 33(12) (1980), 1581–1583.Google Scholar
[28]Khan, M. S., ‘Commuting mappings and fixed points in uniform spaces’, Bull. Acad. Pol. Sci. Ser. Sci. Mat. XXIX, 9–10 (1981), 499–507.Google Scholar
[29]Khan, M. S. and Imdad, M., ‘Fixed point theorems for a class of mappings’, Indian J. Pure Appl. Math. 14 (1983), 1220–1227.Google Scholar
[30]Khan, M. S. and Imdad, M., ‘Some common fixed point theorems’, Glasnik Mat. Sci. III 18 (1983), 321–326.Google Scholar
[31]Krasnoselskii, M. A., ‘Two remarks about the method of successive approximations’, Uspehi Math. Nauk. 10 (1955), 123–127.Google Scholar
[32]Kuhn, G., ‘Generalized non-expansive mappings: approximation of fixed points’, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 114 (1980), 93–101.Google Scholar
[32]Jaggi, D. S., ‘Fixed point theorems for orbitally continuous functions’, Mat. Vesnik 1 (14) (1977), 129–135.Google Scholar
[34]Jaggi, D. S., ‘Fixed point theorems for orbitally continuous functions II’, Mat. Vesnik 19 (2) (1977), 113–118.Google Scholar
[35]Jungck, G., ‘Commuting maps and fixed points’, Amer. Math. Monthly 83 (1976), 261–263.Google Scholar
[36]Maluta, E., ‘Generalized non-expansive maps on the real axis: existence and approximation of fixed points’, 1st. Lombardo Accad. Sci. Lett. Rend. A 112 (1978), 213–221.Google Scholar
[37]Mann, W. R., ‘Mean value method in iteration’, Proc. Amer. Math. Soc. 4 (1953), 506–510.Google Scholar
[38]Massa, S., ‘Generalized contractions in metric spaces’, Boll. Un. Mat. Ital. (4) 10 (1974), 689–694.Google Scholar
[39]Massa, S., ‘Approximation of fixed points by Cesaro's means of iterates’, Rend. 1st. Mat. Univ. Trieste 9 (1977), 127–133.Google Scholar
[40]Massa,, S. ‘On the approximation of fixed points for quasi nonexpansive mappings’, 1st. Lombardo Accad. Sci. Lett. Rend. A 111 (1977), 188–193.Google Scholar
[41]Massa, S., ‘Convergence of an iterative process for a class of quasi-nonexpansive mappings’, Boll. Un. Mat. Ital. (5) 15-A (1978), 154–158.Google Scholar
[42]Massa, S., ‘Fixed point approximation for quasi non-expansive mappings’, Matematiche (Catania) 37 (1982), 3–7.Google Scholar
[43]Massa, S. and Roux, D., ‘A fixed point theorem for generalized nonexpansive mappings’, Boll. Un. Mat. Ital. (5) 15-A (1978), 624–634.Google Scholar
[44]Meir, A. and Keeler, E., ‘A theorem on contraction mappings’, J. Math. Anal. Appl. 28 (1969), 326–329.CrossRefGoogle Scholar
[45]Papini, P. L., Una bibliografia italiana sui punti fissi 1970–1983 (Dipartimento di Matematica, Univ. di Bologna, 1983).Google Scholar
[46]Park, S., ‘Fixed points of f-contractive maps’, Rocky Mountain J. Math. 8 (1978), 743–750.Google Scholar
[47]Park, S., ‘A unified approach to fixed points of contractive type’, J. Korean Math. Soc. 16 (1980), 95–105.Google Scholar
[48]Park, S. and Rhoades, B. E., ‘Extension of some fixed point theorems of Hegedüs and Kasahara’, Math. Sem. Notes Kobe Univ. 9 (1981), 113–118.Google Scholar
[49]Petryshyn, W. V. and Williamson, T. E. Jr, ‘Strong and weak convergence of the sequence of successive approximation for quasi-nonexpansive mappings’, J. Math. Anal. Appl. 43 (1973), 459–497.CrossRefGoogle Scholar
[50]Rhoades, B. E., ‘A comparison of various definitions of contractive mappings’, Trans. Amer. Math. Soc. 226 (1977), 257–290.CrossRefGoogle Scholar
[51]Rhoades, B. E., ‘Contractive definitions revisited, topological methods in nonlinear functional analysis’, Contemporary Math. AMS 21 (1983), 189–205.Google Scholar
[52]Rhoades, B. E., ‘Some fixed point theorems for generalized non-expansive mappings’, Nonlinear analysis and applications, ed. Singh/Burry, , pp. 223–228 (Lecture Notes Pure Appl. Math. 80 (1982), Marcel-Dekker).Google Scholar
[53]Rhoades, B. E., Sessa, S., Khan, M. S. and Khan, M. D., ‘Some fixed point theorems for Hardy-Rogers type mappings’, Internal. J. Math. Math. Sci., 7 (1) (1984), 75–87.CrossRefGoogle Scholar
[54]Roux, D., ‘Applicazioni quasi non-espansive: approssimazione dei punti fissi’, Rend. Mat. (6) 10 (1977), 597–605.Google Scholar
[55]Roux, D., ‘Teoremi di punto fisso per applicazioni contrattive‘, Atti del Convegno Linceo “Applicazioni del teorema di punto fisso all'Analisi Economica”, pp. 89–110 (Roma 1977 (1978)).Google Scholar
[56]Roux, D., ‘Applicazioni del teorema di Brouwer a spazi a infinite dimensioni‘, Atti del Convegno Linceo “Applicazioni del teorema di punto fisso all'A nalisi Economica” (Roma 1977 (1978)).Google Scholar
[57]Roux, D. and Zanco, C., ‘Quasi non-expansive mappings: strong and weak convergence to a fixed point of the sequence of iterates’, Matematiche (Catania) 32 (1977), 307–315.Google Scholar
[58]Schaeffer, M., ‘Uber die method suksessiver approximation’. Jber. Deutsh Math. 59 (1957), 131–140.Google Scholar
[59]Senter, M. F. and Dotson, W. G. Jr, ‘Approximating fixed points of non-expansive mappings’, Proc. Amer. Math. Soc. 44 (1974), 375–379.Google Scholar
[60]Sessa, S., ‘On a weak commutativity condition in fixed point considerations’, Publ. Inst. Math. 32 (46) (1982), 149–153.Google Scholar
[61]Singh, S. L., ‘On common fixed points of commuting mappings’, Math. Seminar Notes Kobe Univ. 5 (1977), 131–134.Google Scholar
[62]Singh, S. L. and Tiwari, B. M. L., ‘Common fixed points of mappings in complete metric spaces’, Proc. Nat. Acad. Sci. India 51 (A) I (1981), 41–44.Google Scholar
[63]Soardi, P., ‘Existence of fixed points of non-expansive mappings in certain Banach lattices’, Proc. Amer. Math. Soc. 73 (1979), 25–29.Google Scholar
[64]Taskovic, M. R., ‘On common fixed points of mappings’, Math. Balk. 8 (1978), 213–219.Google Scholar
[65]Yen, C. L., ‘Some fixed point theorems’, Chung Juan J. 4 (1975), 1–8.Google Scholar