Riv. Mat. Univ. Parma, Vol. 6, No. 2, 2015

Satish Shukla[1]

Generalized Nadler \(G\)-contraction in cone metric spaces over Banach algebras endowed with a graph

Pages: 331-343
Received:7 September 2015   
Accepted in revised form: 20 November 2015
Mathematics Subject Classification (2010): 47H10, 54H25.
Keywords:Cone metric space, set-valued mapping, Nadler \(G\)-contraction, fixed point.
Author address:
[1] : Department of Applied Mathematics, Shri Vaishnav Institute of Technology & Science , Gram Baroli, Sanwer Road, Indore, 453331, (M.P.) India

Abstract: In this paper, we introduce the generalized Nadler \(G\)-contractions in cone metric spaces endowed with a graph and defined over a Banach algebra. A fixed point result for such mappings is proved. Our result generalizes some known results in metric and cone metric spaces. An example is presented which verifies the significance and usability of the result proved herein.

References

[1] M. Arshad and J. Ahmad, On multivalued contractions in cone metric spaces without normality, The Scientific World Journal 2013 (2013), Art. ID 481601, 3 pp. DOI 10.1155/2013/481601
[2] H. Çakalli, A. Sönmez and Ç. Genç, On an equivalence of topological vector space valued cone metric spaces and metric spaces, Appl. Math. Lett. 25 (2012), 429-433. MR2856000
[3] W.-S. Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal. 72 (2010), 2259-2261. MR2577793
[4] Y. Feng and W. Mao, The equivalence of cone metric spaces and metric spaces, Fixed Point Theory 11 (2010), 259-264. MR2743780
[5] L.-G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007), 1468-1476. MR2324351
[6] J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc. 136 (2008), 1359-1373. MR2367109
[7] R. Johnsonbaugh, Discrete Mathematics, Prentice Hall, New Jersey, 1997. Zbl 0860.68078
[8] G. Jungck, S. Radenović, S. Radojević and V. Rakočević, Common fixed point theorems for weakly compatible pairs on cone metric spaces, Fixed Point Theory Appl. 2009, Art. ID 643840, 13 pp. MR2501490
[9] Z. Kadelburg, S. Radenović and V. Rakočević, A note on the equivalence of some metric and cone metric fixed point results, Appl. Math. Lett. 24 (2011) 370-374. MR2741048
[10] Z. Kadelburg, S. Radenović and V. Rakočević, Topological vector space-valued cone metric spaces and fixed point theorems, Fixed Point Theory Appl. 2010, Art. ID 170253, 17 pp. MR2684111
[11] D. Klim and D. Wardowski, Dynamic processes and fixed points of set-valued nonlinear contractions in cone metric spaces, Nonlinear Anal. 71 (2009), 5170-5175. MR2560187
[12] H. Liu and S.-Y. Xu, Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory Appl. 2013, 2013:320, 10 pp. MR3213135
[13] H. Liu and S.-Y. Xu, Fixed point theorems of quasicontractions on cone metric spaces with Banach algebras, Abstr. Appl. Anal. 2013, Art. ID 187348, 5 pp. MR3129333
[14] S. B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475-488. MR0254828
[15] Sh. Rezapour and R. Hamlbarani, Some notes on the paper "Cone metric spaces and fixed point theorems of contractive mappings", J. Math. Anal. Appl. 345 (2008), 719-724. MR2429171
[16] W. Rudin, Functional Analysis, 2nd ed., McGraw-Hill, New York 1991. MR1157815
[17] A. Sultana and V. Vetrivel, Fixed points of Mizoguchi–Takahashi contraction on a metric space with a graph and applications, J. Math. Anal. Appl. 417 (2014), 336–344. MR3191430
[18] D. Wardowski, Endpoints and fixed points of set-valued contractions in cone metric spaces, Nonlinear Anal. 71 (2009), 512-516. MR2518056
[19] D. Wardowski, On set-valued contraction of Nadler type in cone metric spaces, Appl. Math. Lett. 24 (2011), 275-278. MR2741028
[20] S.-Y. Xu and S. Radenović, Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality, Fixed Point Theory Appl. 2014, 2014:102, 12 pp. MR3347826


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