A generalization of some results on multi-valued weakly Picard mappings in b-metric space

• Autorzy: Olatinwo M. O. , Imoru C. O.
• Języki publikacji: EN

Abstrakty

EN

In this paper, we establish some convergence results in a complete b-metric space for the Picard iteration associated to two multi-valued weak contractions by employing the concepts of monotone and comparison functions. Our results generalize and extend those of Berinde and Berinde [8], Daffer and Kaneko [15] and Nadler [27]. Theorem 2.1 in our paper generalizes Theorem 5 of Nadler [27] and a recent result of Berinde and Berinde [8], it also extends, improves and unifies several classical results pertainning to single and multi-valued contractive mappings in the fixed point theory. Also, Theorem 2.3 is a generalization and extension of Theorem 5 of Nadler [27] as well as Theorem 4 of Berinde and Berinde [8].

Słowa kluczowe

EN

multi-valued weak contraction; single-valued contractive mappings

• Wydawca: Wydawnictwo Politechniki Poznańskiej
• Czasopismo: Fasciculi Mathematici
• Rocznik: 2008
• Tom: Nr 40
• Strony: 45--56
• Opis fizyczny: Bibliogr. 36 poz.

Twórcy

autor

Olatinwo M. O.

autor

Imoru C. O.

• Department of Mathematics, Obafemi Awolwo University, Ile-Ife, Nigeria,

Bibliografia

• [1]AGARWAL R.P., MECHAN M., O'REGAN D., Fixed Point Theory and Applications, Cambridge University Press 2001.
• [2]BANACH S., Sur les Operations dans Ies Ensembles Abstraits et Ieur Applications aux Equations Integrales, Fund. Math., 3(1922), 133-181.
• [3]BERINDE V., A priori and a posteriori Error Estimates for a Class of (φ-Contractions, Bulletins for Applied & Computing Math., (1999), 183-192.
• [4]BERINDE V., Iterative Approximation of Fixed Points, Editura Efemeride (2002).
• [5]BERINDE V., On the Approximation of Fixed Points of Weak (^-Contractive Operators, Fixed Point Theory, 4(2)(2003), 131-142.
• [6]BERINDE V., On the Approximation of Fixed Points of Weak Contractive Mappings, Carpathian J. Math., 19(1)(2003), 7-22.
• [7]BERINDE V., Approximating Fixed Points of Weak Contractions Using Picard Iteration, Nonlinear Analysis Forum, 9(1)(2004), 43-53.
• [8]BERINDE M., BERINDE V., On A General Class of Multi-valued Weakly Picard Mappings, J. Math. Anal Appl., 326(2007), 772-782.
• [9]BOYD D.W., WONG J.S.W., On Linear Contractions, Proc. Amer. Math.Soc., 20(1969), 458-464.
• [10]BROWDER F., Nonexpansive Nonlinear Operators in a Banach Space, Proc.Nat. Acad. Scl U.S.A., 54(1965), 1041-1044.
• [11]CZERWLK S., Nonlinear Set-valued Contraction Mappings in b—Metric Spaces, Atti Sem. Mat. Fis. Univ. Modena, 46(2)(1998), 263-276. MR1665883 (99j :54043).
• [12]CiRic L.B., Fixed Point Theory, Contraction Mapping Principle, FME Press, Beograd 2003.
• [13]CiRic L.B., UME J.S., Common Fixed Point Theorems for Multi-valued Non-self Mappings, Publ Math. Debrecen, 60(3-4)(2002), 359-371.
• [14]CIRIC L.B., UME J.S., On the Convergence of Ishikawa Iterates to A Common Fixed Point of Multi-valued Mappings, Demonstratio Math., 36(4)(2003), 951-956.
• [15]DAFFER P.Z., KANEKO H., Fixed Points of Generalized Contractive Multi-valued Mappings, J. Math. Anal Appl, 192(1995), 655-666.
• [16]GERAGHTY M.A., On Contractive Mappings, Proc. Amer. Math. Soc.,40(1973), 604-608.
• [17]GOFFMAN C., PEDRICK G., First Course in Functional Analysis, Prentice Hall of India, Private Limited, New Delhi-11000 (1993).
• [18]ITOH S., Multi-valued Generalized Contractions and Fixed Point Theorems, Comment. Math. Univ. Carolin., 18(1977), 247-258.
• [19]JOSHI M.C., BOSE R.K., Some Topics in Nonlinear Functional Analysis, Wiley Eastern Limited (1985).
• [20]KANEKO H., A General Principle for Fixed Points of Contractive Multi-valued Mappings, Math. Japon., 31(1986), 407-411.
• [21]KANEKO H., Generalized Contractive Multi-valued Mappings and their Fixed Points, Math. Japon., 33(1988), 57-64.
• [22]KHAMSI M.A., KIRK W.A., An Introduction to Metric Spaces and Fixed Point Theory, John Wiley & Sons, Inc. (2001).
• [23]KUBIACZYK L, ALI N.M., On the Convergence of the Ishikawa Iterates to A Common Fixed Point for A Pair of Multi-valued Mappings, Acta Math.Hungar., 75(3)(1997), 253-257.
• [24]LlM T.C., On Fixed Point Stability for Set-valued Contractive Mappings with Applications to Generalized Differential Equations, J. Math. Anal. Appl.,110(2)(1985), 436-441.
• [25]MARKINS J.T., A Fixed Point Theorem for Set-valued Mappings, Bull Amer.Math. Soc., 74(1968), 639-640.
• [26]MIZOGUCHI N., TAKAHASHI W., Fixed Point Theorems for Multi-valued Mappings on Complete Metric Spaces, J. Math. Anal. Appl, 141(1989), 177-188.
• [27]NADLER S.B., Multi-valued Contraction Mappings, Pacific J. Math.,30(1969), 282-291.
• [28]PlCARD E., Memoire sur la Theorie des Equations aux Derivees partielles et Ia Methode des Approximations Successives, J. Math. Pures et AppL, 6(1890), 145-210.
• [29]RHOADES B.E., A Comparison of Various Definitions of Contractive Mappings, Trans. Amer. Math. Soc., 226(1977), 257-290.
• [30]RHOADES B.E., A Fixed Point Theorem for A Multi-valued Non-self Mapping, Comment. Math. Univ. Carolin., 37(1996), 401-404.
• [31]RHOADES B.E., WATSON B., Fixed Points for Set-valued Mappings on Metric Spaces, Math. Japon., 35(4)(1990), 735-743.
• [32]Rus I.A., Fixed Point Theorems for Multi-valued Mappings in Complete Metric Spaces, Math. Japon., 20(1975), 21-24.
• [33]Rus I.A., Generalized Contractions and Applications, Cluj Univ. Press, Cluj Napoca (2001).
• [34]Rus I.Α., PETRUSEL Α., PETRUSEL G., Fixed Point Theory, 1950-2000, Romanian Contributions, House of the Book of Science, Cluj Napoca (2002).
• [35]SINGH S.L., BHATNAGAR C., HASHIM A.M., Round-off Stability of Picard Iterative Procedure for Multi-valued Operators, Nonlinear Anal. Forum,10(2005), 13-19.
• [36]ZEIDLER E., Nonlinear Functional Analysis and its Applications-Fixed Point Theorems, Springer-Verlag, New York, Inc. (1986).