| Title:
|
Some surjectivity theorems with applications (English) |
| Author:
|
Pathak, H. K. |
| Author:
|
Mishra, S. N. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
49 |
| Issue:
|
1 |
| Year:
|
2013 |
| Pages:
|
17-27 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper a new class of mappings, known as locally $\lambda $-strongly $\phi $-accretive mappings, where $\lambda $ and $\phi $ have special meanings, is introduced. This class of mappings constitutes a generalization of the well-known monotone mappings, accretive mappings and strongly $\phi $-accretive mappings. Subsequently, the above notion is used to extend the results of Park and Park, Browder and Ray to locally $\lambda $-strongly $\phi $-accretive mappings by using Caristi-Kirk fixed point theorem. In the sequel, we introduce the notion of generalized directional contractor and prove a surjectivity theorem which is used to solve certain functional equations in Banach spaces. (English) |
| Keyword:
|
strongly $\phi $-accretive |
| Keyword:
|
locally strongly $\phi $-accretive |
| Keyword:
|
locally $\lambda $-strongly $\phi $-accretive |
| Keyword:
|
fixed point theorem |
| MSC:
|
47H05 |
| MSC:
|
47H15 |
| idZBL:
|
Zbl 06321144 |
| idMR:
|
MR3073012 |
| DOI:
|
10.5817/AM2013-1-17 |
| . |
| Date available:
|
2013-05-28T13:26:21Z |
| Last updated:
|
2014-07-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143296 |
| . |
| Reference:
|
[1] Altman, M.: Contractor directions, directional contractors and directional contractions for solving equations.Pacific J. Math. 62 (1976), 1–18. Zbl 0352.47027, MR 0473939, 10.2140/pjm.1976.62.1 |
| Reference:
|
[2] Altman, M.: Contractors and contractor directions theory and applications.Marcel Dekker, New York, 1977. Zbl 0363.65045, MR 0451686 |
| Reference:
|
[3] Altman, M.: Weak contractor directions and weak directional contractions.Nonlinear Anal. 7 (1983), 1043–1049. Zbl 0545.47034, MR 0713214 |
| Reference:
|
[4] Browder, F. E.: Normal solvability and existence theorems for nonlinear mappings in Banach spaces.Problems in Nonlinear Analysis (C.I.M.E., IV Ciclo, Varenna, 1970), pp. 17–35, Edizioni Cremones, Rome, Italy, 1971. Zbl 0234.47056, MR 0467430 |
| Reference:
|
[5] Browder, F. E.: Normal solvability for nonlinear mappings and the geometry of Banach spaces.Problems in Nonlinear Analysis,C.I.M.E., IV Ciclo, Varenna, 1970, pp. 37–66, Edizioni Cremonese, Rome, Italy, 1971. Zbl 0234.47055, MR 0438201 |
| Reference:
|
[6] Browder, F. E.: Normal solvability $\phi $–accretive mappings of Banach spaces.Bull. Amer. Math. Soc. 78 (1972), 186–192. MR 0306992, 10.1090/S0002-9904-1972-12907-0 |
| Reference:
|
[7] Browder, F. E.: Nonlinear operators and nonlinear equations of evolution in Banach spaces.Proc. Sympos. Pure Math., vol. 18, Amer. Math. Soc., Providence, 1976. Zbl 0327.47022, MR 0405188 |
| Reference:
|
[8] Caristi, J.: Fixed point theorems for mappings satisfying inwardness conditions.Trans. Amer. Math. Soc. 215 (1976), 241–251. Zbl 0305.47029, MR 0394329, 10.1090/S0002-9947-1976-0394329-4 |
| Reference:
|
[9] Ekeland, I.: Sur les problems variationnels.C. R. Acad. Sci. Paris Sér. I Math. 275 (1972), 1057–1059. MR 0310670 |
| Reference:
|
[10] Goebel, K., Kirk, W. A.: Topics in metric fixed point theory.Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, 1990. Zbl 0708.47031, MR 1074005 |
| Reference:
|
[11] Kirk, W. A.: Caristi’s fixed point theorem and the theory of normal solvability.Proc. Conf. Fixed Point Theory and its Applications (Dalhousie Univ., June 1975), Academic Press, 1976, pp. 109–120. Zbl 0377.47042, MR 0454754 |
| Reference:
|
[12] Park, J. A., Park, S.: Surjectivity of $\phi $–accretive operators.Proc. Amer. Math. Soc. 90 (2) (1984), 289–292. MR 0727252 |
| Reference:
|
[13] Ray, W. O.: Phi–accretive operators and Ekeland’s theorem.J. Math. Anal. Appl. 88 (1982), 566–571. Zbl 0497.47034, MR 0667080, 10.1016/0022-247X(82)90215-3 |
| Reference:
|
[14] Ray, W. O., Walker, A. M.: Mapping theorems for Gâteaux differentiable and accretive operators.Nonlinear Anal. 6 (5) (1982), 423–433. Zbl 0488.47031, MR 0661709, 10.1016/0362-546X(82)90057-8 |
| . |
