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The contraction principle for set valued mappings on a metric space with a graph. (English) Zbl 1201.54029

Summary: Let \((X,d)\) be a metric space and \(F:X\rightsquigarrow X\) be a set valued mapping. We obtain sufficient conditions for the existence of a fixed point of the mapping \(F\) in the metric space \(X\) endowed with a graph \(G\) such that the set \(V(G)\) of vertices of \(G\) coincides with \(X\) and the set of edges of \(G\) is \(E(G)=\{(x,y):(x,y)\in X\times X\}\).

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
65J15 Numerical solutions to equations with nonlinear operators
47H10 Fixed-point theorems
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References:

[1] Kirk, W. A.; Goebel, K., Topics in Metric Fixed Point Theory (1990), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0708.47031
[2] Granas, A.; Dugundji, J., Fixed Point Theory (2003), Springer-Verlag: Springer-Verlag New York · Zbl 1025.47002
[3] Tarski, A., A lattice theoretical fixed point and its application, Pacific J. Math., 5, 285-309 (1955) · Zbl 0064.26004
[4] Nadler, S. B., Multivalued contraction mappings, Pacific J. Math., 30, 475-488 (1969) · Zbl 0187.45002
[5] Beg, I.; Azam, A., Fixed points of asymptotically regular multivalued mappings, J. Aust. Math. Soc. (Series A), 53, 3, 313-326 (1992) · Zbl 0765.54036
[6] Daffer, P. Z., Fixed points of generalized contractive multivalued mappings, J. Math. Anal. Appl., 192, 655-666 (1995) · Zbl 0835.54028
[7] Daffer, P. Z.; Kaneko, H.; Li, W., On a conjecture of S. Reich, Proc. Amer. Math. Soc., 124, 3159-3162 (1996) · Zbl 0866.47040
[8] Feng, Y.; Liu, S., Fixed point theorems for multivalued contractive mappings and multivalued Caristi type mappings, J. Math. Anal. Appl., 317, 103-112 (2006) · Zbl 1094.47049
[9] Klim, D.; Wardowski, D., Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl., 334, 132-139 (2007) · Zbl 1133.54025
[10] Reich, S., Fixed points of contractive functions, Boll. Unione. Mat. Ital. (4), 5, 26-42 (1972) · Zbl 0249.54026
[11] Rhoades, B. E., A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc., 226, 257-290 (1977) · Zbl 0365.54023
[12] Qing, C. Y., On a fixed point problem of Reich, Proc. Amer. Math. Soc., 124, 3085-3088 (1996) · Zbl 0874.47027
[13] Zeidler, E., Nonlinear Functional Analysis and its Applications I: Fixed Point Theorems (1985), Springer-Verlag: Springer-Verlag New York
[14] Beg, I., Fixed points of fuzzy multivalued mappings with values in fuzzy ordered sets, J. Fuzzy Math., 6, 1, 127-131 (1998) · Zbl 0909.47055
[15] Echenique, F., A short and constructive proof of Tarski’s Fixed point theorem, Internat. J. Game Theory, 33, 2, 215-218 (2005) · Zbl 1071.91002
[16] Fujimoto, T., An extension of Tarski’s fixed point theorem and its application to isotone complementarity problems, Math. Program., 28, 116-118 (1984) · Zbl 0526.90084
[17] Ran, A. C.M.; Reurings, M. C.B., A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132, 1435-1443 (2003) · Zbl 1060.47056
[18] Altun, I.; Simsek, H., Some fixed point theorems on ordered metric spaces and application, Fixed Point Theory Appl., 2010, 17 (2010), Article ID 621469 · Zbl 1197.54053
[19] Altun, I.; Durmaz, G., Some fixed point theorems on ordered cone metric spaces, Rend. Circ. Mat. Palermo, 58, 319-325 (2009) · Zbl 1184.54038
[20] Altun, I.; Damjanovic, B.; Doric, D., Fixed point and common fixed point theorems on ordered cone metric spaces, Appl. Math. Lett. (2009)
[21] Beg, I.; Butt, A. R., Fixed point for set valued mappings satisfying an implicit relation in partially ordered metric spaces, Nonlinear Anal., 71, 3699-3704 (2009) · Zbl 1176.54028
[22] Beg, I.; Butt, A. R., Fixed point for weakly compatible mappings satisfying an implicit relation in partially ordered metric spaces, Carpathian J. Math., 25, 1, 1-12 (2009) · Zbl 1199.54207
[23] Drici, Z.; McRae, F. A.; Devi, J. V., Fixed point theorems in partially ordered metric space for operators with PPF dependence, Nonlinear Anal., 67, 641-647 (2007) · Zbl 1127.47049
[24] Harjania, J.; Sadarangani, K., Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Anal., 71, 3403-3410 (2009) · Zbl 1221.54058
[25] Harjani, J.; Sadarangani, K., Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal., 72, 1188-1197 (2010) · Zbl 1220.54025
[26] Kadelburg, Z.; Pavlovic, M.; Radenovic, S., Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces, Comput. Math. Appl., 59, 3148-3159 (2010) · Zbl 1193.54035
[27] Nieto, J. J.; Pouso, R. L.; Rodríguez-López, R., Fixed point theorems in ordered abstract spaces, Proc. Amer. Math. Soc., 135, 2505-2517 (2007) · Zbl 1126.47045
[28] Nieto, J. J.; Rodríguez-López, R., Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22, 223-239 (2005) · Zbl 1095.47013
[29] Nieto, J. J.; Rodríguez-López, R., Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. Engl. Ser., 23, 2205-2212 (2007) · Zbl 1140.47045
[30] O’Regan, D.; Petrusel, A., Fixed point theorems for generalized contraction in ordered metric spaces, J. Math. Anal. Appl., 341, 1241-1252 (2008) · Zbl 1142.47033
[31] Petrusel, A.; Rus, I. A., Fixed point theorems in ordered L-spaces, Proc. Amer. Math. Soc., 134, 411-418 (2005) · Zbl 1086.47026
[32] Jachymski, J., The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc., 1, 136, 1359-1373 (2008) · Zbl 1139.47040
[33] Lukawska, G. G.; Jachymski, J., IFS on a metric space with a graph structure and extension of the Kelisky-Rivlin theorem, J. Math. Anal. Appl., 356, 453-463 (2009) · Zbl 1171.28002
[34] Diestel, R., Graph Theory (2000), Springer-Verlag: Springer-Verlag new York · Zbl 0945.05002
[35] Assad, N. A.; Kirk, W. A., Fixed point theorems for set-valued mappings of contractive type, Pacific J. Math., 43, 3, 553-562 (1972) · Zbl 0239.54032
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