Approximating Fixed Points of Nonself Contractive Type Mappings in Banach Spaces Endowed with a Graph

Balog, Laszlo; Berinde, Vasile; Păcurar, Mădălina

doi:10.1515/auom-2016-0026

Abstract

Let K be a non-empty closed subset of a Banach space X endowed with a graph G. We obtain fixed point theorems for nonself G-contractions of Chatterjea type. Our new results complement and extend recent related results [Berinde, V., Păcurar, M., The contraction principle for nonself mappings on Banach spaces endowed with a graph, J. Nonlinear Convex Anal. 16 (2015), no. 9, 1925-1936; Balog, L., Berinde, V., Fixed point theorems for nonself Kannan type contractions in Banach spaces endowed with a graph, Carpathian J. Math. 32 (2016), no. 3 (in press)] and thus provide more general and flexible tools for studying nonlinear functional equations.

References

[1] Abbas, M., Ali, B., Petruşel, G., Fixed points of set-valued contractions in partial metric spaces endowed with a graph. Carpathian J. Math. 30 (2014), no. 2, 129-137.
Search in Google Scholar Back to article
[2] Agarwal, R.P., El-Gebeily, M.A. and O'Regan, D., Generalized contrac- tions in partially ordered metric spaces, Appl. Anal. 87 (2008) 1-810.1080/00036810701714164
Search in Google Scholar Back to article
[3] Alghamdi, Maryam A., Berinde, V. and Shahzad, N., Fixed points of multi-valued non-self almost contractions, J. Appl. Math. Volume 2013, Article ID 621614, 6 pages10.1155/2013/621614
Search in Google Scholar Back to article
[4] Alghamdi, Maryam A., Berinde, V. and Shahzad, N., Fixed points of non-self almost contractions, Carpathian J. Math. 33 (2014), No. 1, 1-8
Search in Google Scholar Back to article
[5] Ariza-Ruiz, D., Jimfienez-Melado, A., A continuation method for weakly Kannan maps. Fixed Point Theory Appl. 2010, Art. ID 321594, 12 pp.10.1155/2010/321594
Search in Google Scholar Back to article
[6] Assad, N. A. On a fixed point theorem of Isfieki, Tamkang J. Math. 7 (1976), no. 1, 19-22
Search in Google Scholar Back to article
[7] Assad, N. A. On a fixed point theorem of Kannan in Banach spaces, Tamkang J. Math. 7 (1976), no. 1, 91-94
Search in Google Scholar Back to article
[8] Assad, N. A., On some nonself nonlinear contractions, Math. Japon. 33 (1988), no. 1, 17-26
Search in Google Scholar Back to article
[9] Assad, N. A., On some nonself mappings in Banach spaces, Math. Japon. 33 (1988), no. 4, 501-515
Search in Google Scholar Back to article
[10] Assad, N. A., Approximation for fixed points of multivalued contractive mappings, Math. Nachr. 139 (1988), 207-21310.1002/mana.19881390119
Search in Google Scholar Back to article
[11] Assad, N. A., A fixed point theorem in Banach space, Publ. Inst. Math. (Beograd) (N.S.) 47(61) (1990), 137-140
Search in Google Scholar Back to article
[12] Assad, N. A., A fixed point theorem for some non-self-mappings, Tamkang J. Math. 21 (1990), no. 4, 387-393
Search in Google Scholar Back to article
[13] Assad, N. A. and Kirk, W. A., Fixed point theorems for set-valued map- pings of contractive type, Pacific J. Math. 43 (1972), 553-562 10.2140/pjm.1972.43.553
Search in Google Scholar Back to article
[14] Assad, N. A. Sessa, S., Common fixed points for nonself compatible maps on compacta, Southeast Asian Bull. Math. 16 (1992), No.2, 91-95
Search in Google Scholar Back to article
[15] Balog, L., Berinde, V., Fixed point theorems for nonself Kannan type contractions in Banach spaces endowed with a graph, Carpathian J. Math. 32 (2016), no. 3 (in press).10.37193/CJM.2016.03.05
Search in Google Scholar Back to article
[16] Berinde, V., A common fixed point theorem for nonself mappings. Miskolc Math. Notes 5 (2004), no. 2, 137-144
Search in Google Scholar Back to article
[17] Berinde, V., Approximation of fixed points of some nonself generalized fi-contractions. Math. Balkanica (N.S.) 18 (2004), no. 1-2, 85-93
Search in Google Scholar Back to article
[18] Berinde, V., Iterative Approximation of Fixed Points, 2nd Ed., Springer Verlag, Berlin Heidelberg New York, 200710.1109/SYNASC.2007.49
Search in Google Scholar Back to article
[19] Berinde, V., Păcurar, Mfiadfialina, Fixed point theorems for nonself single- valued almost contractions, Fixed Point Theory, 14 (2013), No. 2, 301-312
Search in Google Scholar Back to article
[20] Berinde, V., Păcurar, Mfiadfialina, The contraction principle for nonself mappings on Banach spaces endowed with a graph, J. Nonlinear Convex Anal. 16 (2015), no. 9, 1925-1936.
Search in Google Scholar Back to article
[21] Berinde, V., Păcurar, M., A constructive approach to coupled fixed point theorems in metric spaces, Carpathian J. Math. 31 (2015), no. 3, 277-287.
Search in Google Scholar Back to article
[22] Berinde, V., Petric, M. A., Fixed point theorems for cyclic non-self single- valued almost contractions, Carpathian J. Math. 31 (2015), no. 3, 289-296.
Search in Google Scholar Back to article
[23] Bojor F., Fixed point of '-contraction in metric spaces endowed with a graph, Ann. Univ. Craiova, Math. Comput. Sci. Ser., 37 (2010), no. 4, 85-92.
Search in Google Scholar Back to article
[24] Bojor, F., Fixed points of Bianchini mappings in metric spaces endowed with a graph, Carpathian J. Math. 28 (2012), no. 2, 207-214
Search in Google Scholar Back to article
[25] Bojor, F., Fixed points of Kannan mappings in metric spaces endowed with a graph, An. Stiint. Univ. "Ovidius" Constanta Ser. Mat. 20 (2012), no. 1, 31-40
Search in Google Scholar Back to article
[26] Bojor, F., Fixed point theorems for Reich type contractions on metric spaces with a graph, Nonlinear Anal. 75 (2012), no. 9, 3895-3901
Search in Google Scholar Back to article
[27] Bojor, F., Fixed point theorems in in metric spaces endowed with a graph (in Romanian), PhD Thesis, North University of Baia Mare, 201210.1186/1687-1812-2012-161
Search in Google Scholar Back to article
[28] Caristi, J., Fixed point theorems for mappings satisfying inwardness con- ditions, Trans. Amer. Math. Soc. 215 (1976), 241-25110.1090/S0002-9947-1976-0394329-4
Search in Google Scholar Back to article
[29] Caristi, J., Fixed point theory and inwardness conditions. Applied nonlinear analysis (Proc. Third Internat. Conf., Univ. Texas, Arlington, Tex., 1978), pp. 479-483, Academic Press, New York-London, 1979.10.1016/B978-0-12-434180-7.50047-4
Search in Google Scholar Back to article
[30] Chatterjea, S.K., Fixed-point theorems, C.R. Acad. Bulgare Sci. 25 (1972) 727-730
Search in Google Scholar Back to article
[31] Chifu, C. Petruşel, Gabriela., Generalized contractions in metric spaces endowed with a graph, Fixed Point Theory Appl. 2012, 2012:161, 9 pp.10.1186/1687-1812-2012-161
Open DOI Search in Google Scholar Back to article
[32] Cho, S.-H., A fixed point theorem for a fi Cirific-Berinde type mapping in orbitally complete metric spaces. Carpathian J. Math. 30 (2014), no. 1, 63-70.
Search in Google Scholar Back to article
[33] Choudhury, B. S, Das, K., Bhandari, S. K., Cyclic contraction of Kan- nan type mappings in generalized Menger space using a control function. Azerb. J. Math. 2 (2012), no. 2, 43-55.
Search in Google Scholar Back to article
[34] Ćirifić, Lj. B., A remark on Rhoades' fixed point theorem for non-self map- pings, Internat. J. Math. Math. Sci. 16 (1993), no. 2, 397-400 10.1155/S016117129300047X
Open DOI Search in Google Scholar Back to article
[35] Ćirifić, Lj. B., Quasi contraction non-self mappings on Banach spaces, Bull. Cl. Sci. Math. Nat. Sci. Math. No. 23 (1998), 25-31
Search in Google Scholar Back to article
[36] Ćirifić, Lj. B., Ume, J. S., Khan, M. S. and Pathak, H. K., On some nonself mappings, Math. Nachr. 251 (2003), 28-3310.1002/mana.200310028
Search in Google Scholar Back to article
[37] Eisenfeld, J. and Lakshmikantham, V., Fixed point theorems on closed sets through abstract cones. Appl. Math. Comput. 3 (1977), no. 2, 155-167.
Search in Google Scholar Back to article
[38] Filip, A.-D., Fixed point theorems for multivalued contractions in Kasa- hara spaces. Carpathian J. Math. 31 (2015), no. 2, 189-196.
Search in Google Scholar Back to article
[39] Gabeleh, M., Existence and uniqueness results for best proximity points. Miskolc Math. Notes 16 (2015), no. 1, 123-131.
Search in Google Scholar Back to article
[40] Hussain, N.; Salimi, P.; Vetro, P., Fixed points for fi- -Suzuki contrac- tions with applications to integral equations. Carpathian J. Math. 30 (2014), no. 2, 197-207.
Search in Google Scholar Back to article
[41] Jachymski, J., The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc. 136 (2008), no. 4, 1359-1373
Search in Google Scholar Back to article
[42] Kannan, R., Some results on fixed points, Bull. Calcutta Math. Soc. 10 (1968) 71-76
Search in Google Scholar Back to article
[43] Kikkawa, M., Suzuki, T., Some similarity between contractions and Kan- nan mappings. II. Bull. Kyushu Inst. Technol. Pure Appl. Math. No. 55 (2008), 1-13.
Search in Google Scholar Back to article
[44] Kikkawa, M., Suzuki, T., Some similarity between contractions and Kan- nan mappings. Fixed Point Theory Appl. 2008, Art. ID 649749, 8 pp.10.1155/2008/649749
Search in Google Scholar Back to article
[45] Kirk W.A., Srinivasan P.S. and Veeramani P., Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory, 4 (2003), no. 1, 79-89
Search in Google Scholar Back to article
[46] Meszaros, J., A comparison of various definitions of contractive type map- pings, Bull. Calcutta Math. Soc. 84 (2) (1992) 167-194
Search in Google Scholar Back to article
[47] Nicolae, Adriana, O'Regan, D. and Petruşel, A., Fixed point theorems for singlevalued and multivalued generalized contractions in metric spaces endowed with a graph, Georgian Math. J. 18 (2011), no. 2, 307-327
Search in Google Scholar Back to article
[48] Nieto, J. J., Rodriguez-Lopez, R., Contractive mapping theorems in par- tially ordered sets and applications to ordinary difierential equations, Order 22 (2005), no. 3, 223-239 (2006)10.1007/s11083-005-9018-5
Open DOI Search in Google Scholar Back to article
[49] Nieto, J. J., Rodriguez-Lopez, R., Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary difierential equa- tions, Acta. Math. Sin., (Engl. Ser.) 23(2007), no. 12, 2205-221210.1007/s10114-005-0769-0
Open DOI Search in Google Scholar Back to article
[50] Panja, C., Samanta, S. K., On determination of a common fixed point. Indian J. Pure Appl. Math. 11 (1980), no. 1, 120-127.
Search in Google Scholar Back to article
[51] Nieto, Juan J.; Pouso, Rodrigo L.; Rodriguez-Lopez, Rosana, Fixed point theorems in ordered abstract spaces, Proc. Amer. Math. Soc. 135 (2007), no. 8, 2505-2517
Search in Google Scholar Back to article
[52] Păcurar, M., Approximating common fixed points of Prefisific-Kannan type operators by a multi-step iterative method, An. Sfitiintfi,. Univ. "Ovidius" Constantfia Ser. Mat. 17 (2009), no. 1, 153-168
Search in Google Scholar Back to article
[53] Păcurar, M., Iterative Methods for Fixed Point Approximation, Risoprint, Cluj-Napoca, 2010
Search in Google Scholar Back to article
[54] Păcurar, M., A multi-step iterative method for approximating fixed points of Prefisific-Kannan operators, Acta Math. Univ. Comen. New Ser., 79 (2010), No. 1, 77-88
Search in Google Scholar Back to article
[55] Păcurar, M., A multi-step iterative method for approximating common fixed points of Prefisific-Rus type operators on metric spaces, Stud. Univ. Babefis-Bolyai Math. 55 (2010), no. 1, 149-162.
Search in Google Scholar Back to article
[56] Păcurar, M., Fixed points of almost Prefisific operators by a k-step iterative method, An. Sfitiint,. Univ. Al. I. Cuza Iafisi, Ser. Noua, Mat. 57 (2011), Supliment 199-210 10.2478/v10157-011-0014-3
Search in Google Scholar Back to article
[57] Petric, M., Some results concerning cyclical contractive mappings, Gen. Math. 18 (2010), no. 4, 213-226
Search in Google Scholar Back to article
[58] Petric, M., Best proximity point theorems for weak cyclic Kannan con- tractions, Filomat 25 (2011), no. 1, 145-154
Search in Google Scholar Back to article
[59] Petruşel, Adrian; Rus, Ioan A., Fixed point theorems in ordered L-spaces, Proc. Amer. Math. Soc. 134 (2006), no. 2, 411-418
Search in Google Scholar Back to article
[60] Ran, A. C. M., Reurings, M. C. B., A fixed point theorem in partially or- dered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1435-1443
Search in Google Scholar Back to article
[61] Rhoades, B. E., A comparison of various definitions of contractive map- pings, Trans. Amer. Math. Soc. 226 (1977) 257-29010.1090/S0002-9947-1977-0433430-4
Search in Google Scholar Back to article
[62] Rhoades, B. E., A fixed point theorem for some non-self-mappings, Math. Japon. 23 (1978/79), no. 4, 457-459
Search in Google Scholar Back to article
[63] Rhoades, B. E., Contractive definitions revisited, Contemporary Mathematics 21 (1983) 189-20510.1090/conm/021/729516
Search in Google Scholar Back to article
[64] Rhoades, B. E., Contractive definitions and continuity, Contemporary Mathematics 72 (1988) 233-24510.1090/conm/072/956495
Search in Google Scholar Back to article
[65] Rus, I. A., Principles and Applications of the Fixed Point Theory (in Romanian), Editura Dacia, Cluj-Napoca, 1979
Search in Google Scholar Back to article
[66] Rus, I. A., Generalized contractions, Seminar on Fixed Point Theory 3(1983) 1-130
Search in Google Scholar Back to article
[67] Rus, I. A., Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, 2001
Search in Google Scholar Back to article
[68] Rus, I. A., Picard operators and applications, Sci. Math. Jpn. 58 (2003), No. 1, 191-219
Search in Google Scholar Back to article
[69] Rus, I. A., Private communication (2015)
Search in Google Scholar Back to article
[70] Rus, I. A., Petruşel, A. and Petruşel, G., Fixed Point Theory, Cluj University Press, Cluj-Napoca, 2008
Search in Google Scholar Back to article
[71] Samanta, S. K., Fixed point theorems for non-self-mappings. Indian J. Pure Appl. Math. 15 (1984), no. 3, 221-232.
Search in Google Scholar Back to article
[72] Samanta, S. K., Fixed point theorems for Kannan maps in a metric space with some convexity structure. Bull. Calcutta Math. Soc. 80 (1988), no. 1, 58-64.
Search in Google Scholar Back to article
[73] Samanta, C., Samanta, S. K., Fixed point theorems for multivalued non- self mappings. Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 22 (1992), no. 1, 11-22.
Search in Google Scholar Back to article
[74] Shukla, S., Abbas, M., Fixed point results of cyclic contractions in product spaces. Carpathian J. Math. 31 (2015), no. 1, 119-126.
Search in Google Scholar Back to article
[75] Sun, Y. I., Su, Y. F., Zhang, J. L., A new method for the research of best proximity point theorems of nonlinear mappings. Fixed Point Theory Appl. 2014, 2014:116, 18 pp.10.1186/1687-1812-2014-116
Open DOI Search in Google Scholar Back to article
[76] Ume, J. S., Fixed point theorems for Kannan-type maps. Fixed Point Theory Appl. 2015, 2015:38, 13 pp.10.1186/s13663-015-0286-5
Search in Google Scholar Back to article
[77] Zhang, J. L., Su, Y. F., Best proximity point theorems for weakly contrac- tive mapping and weakly Kannan mapping in partial metric spaces. Fixed Point Theory Appl. 2014, 2014:50, 8 pp.10.1186/1687-1812-2014-50
Search in Google Scholar Back to article

Approximating Fixed Points of Nonself Contractive Type Mappings in Banach Spaces Endowed with a Graph

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