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strongly $\phi $-accretive; locally strongly $\phi $-accretive; locally $\lambda $-strongly $\phi $-accretive; fixed point theorem
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.
[1] Altman, M.: Contractor directions, directional contractors and directional contractions for solving equations. Pacific J. Math. 62 (1976), 1–18. DOI 10.2140/pjm.1976.62.1 | MR 0473939 | Zbl 0352.47027
[2] Altman, M.: Contractors and contractor directions theory and applications. Marcel Dekker, New York, 1977. MR 0451686 | Zbl 0363.65045
[3] Altman, M.: Weak contractor directions and weak directional contractions. Nonlinear Anal. 7 (1983), 1043–1049. MR 0713214 | Zbl 0545.47034
[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. MR 0467430 | Zbl 0234.47056
[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. MR 0438201 | Zbl 0234.47055
[6] Browder, F. E.: Normal solvability $\phi $–accretive mappings of Banach spaces. Bull. Amer. Math. Soc. 78 (1972), 186–192. DOI 10.1090/S0002-9904-1972-12907-0 | MR 0306992
[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. MR 0405188 | Zbl 0327.47022
[8] Caristi, J.: Fixed point theorems for mappings satisfying inwardness conditions. Trans. Amer. Math. Soc. 215 (1976), 241–251. DOI 10.1090/S0002-9947-1976-0394329-4 | MR 0394329 | Zbl 0305.47029
[9] Ekeland, I.: Sur les problems variationnels. C. R. Acad. Sci. Paris Sér. I Math. 275 (1972), 1057–1059. MR 0310670
[10] Goebel, K., Kirk, W. A.: Topics in metric fixed point theory. Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, 1990. MR 1074005 | Zbl 0708.47031
[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. MR 0454754 | Zbl 0377.47042
[12] Park, J. A., Park, S.: Surjectivity of $\phi $–accretive operators. Proc. Amer. Math. Soc. 90 (2) (1984), 289–292. MR 0727252
[13] Ray, W. O.: Phi–accretive operators and Ekeland’s theorem. J. Math. Anal. Appl. 88 (1982), 566–571. DOI 10.1016/0022-247X(82)90215-3 | MR 0667080 | Zbl 0497.47034
[14] Ray, W. O., Walker, A. M.: Mapping theorems for Gâteaux differentiable and accretive operators. Nonlinear Anal. 6 (5) (1982), 423–433. DOI 10.1016/0362-546X(82)90057-8 | MR 0661709 | Zbl 0488.47031
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