Abstract
In the fixed point theory of contractive mappings, that is, mappings satisfying contractive inequalities, in partially ordered metric spaces the fixed point results are obtained under the assumption that the contractive inequality condition holds for pairs of points which are related by partial order rather than for arbitrary pairs of points. Alternatives to partial order for the purpose of restricting the contraction conditions in the fixed point results are the admissibility conditions. In this work we define a multivalued hybrid inequality by generalizing and combining two types of contractive inequalities and establish end point results for those operators which satisfy the hybrid inequality defined here under the two separate environments described above. For our purpose we define a new admissibility condition. The results are established in the most general structure of a metric space. The main theorems proved here are in the domain of setvalued analysis. The corresponding singlevalued cases are discussed. There is nowhere any assumption of continuity. The methodology is either a combination of analytic and order theoretic methods or purely analytic. Moreover the newly introduced method of proof in fixed point theory through Pata-type results are followed in the multivalued case. There are supporting examples of the main results.
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Alizadeh, S., Moradlou, F., Salimi, P.: Some fixed point results for \((\alpha - \beta )\) - \((\psi - \varphi )\)—contractive mappings. Filomat 28(3), 635–647 (2014)
Altun, I., Turkoglu, D.: Some fixed point theorems for weakly compatible multivalued mappings satisfying an implicit relation. Filomat 22, 13–21 (2008)
Banach, S.: Sur les oprations dans les ensembles abstraits et leurs applications aux quations intgrales. Fund Math. 3, 133–181 (1922)
Beg, I., Butt, A.R.: Common fixed point for generalized set valued contractions satisfying an implicit relation in partially ordered metric spaces. Math. Commun. 15, 65–76 (2010)
Chakraborty, M., Samanta, S.K.: A fixed point theorem for Kannan-type maps in metric spaces, 7331v2 [math. GN] 16 December 2012. arXiv:1211
Chatterjea, S.K.: Fixed-point theorems. C.R. Acad. Bulgare Sci. 25, 727–730 (1972)
Cho, S.H.: A fixed point theorem for weakly \(\alpha \)-contractive mappings with application. Appl. Math. Sci. 7, 2953–2965 (2013)
Choudhury, B.S., Metiya, N.: Fixed point theorems for almost contractions in partially ordered metric spaces. Ann Univ Ferrara 58, 21–36 (2012)
Choudhury, B.S., Metiya, N.: Coincidence point theorems for a family of multivalued mappings in partially ordered metric spaces. Acta Universitatis Matthiae Belii, series Mathematics 21, 13–26 (2013)
Choudhury, B.S., Metiya, N., Postolache, M.: A generalized weak contraction principle with applications to coupled coincidence point problems. Fixed Point Theory Appl. 2013, 152 (2013)
Choudhury, B.S., Kundu, A., Metiya, N.: Fixed point results for Ćirić type weak contraction in metric spaces with applications to partial metric spaces. Filomat 28(7), 1505–1516 (2014)
Choudhury, B.S., Metiya, N., Som, T., Bandyopadhyay, C.: Multivalued fixed point results and stability of fixed point sets in metric spaces. Facta Universitatis (NIS) Ser. Math. Inform. 30, 501–512 (2015)
Choudhury, B.S., Metiya, N., Bandyopadhyay, C.: Fixed points of multivalued \(\alpha \)-admissible mappings and stability of fixed point sets in metric spaces. Rend. Circ. Mat. Palermo 64, 43–55 (2015)
Ćirić, L.B., Ume, J.S.: Some common fixed point theorems for weakly compatible mappings. J. Math. Anal. Appl. 314, 488–499 (2006)
Damjanović, B., Samet, B., Vetro, C.: Common fixed point theorems for multi-valued maps. Acta Mathematica Scientia 32B(2), 818–824 (2012)
Dorić, D.: Common fixed point for generalized \((\psi, \varphi )\)-weak contractions. Appl. Math. Lett. 22, 1896–1900 (2009)
Fisher, B.: Common fixed points of mappings and setvalued mappings. Rostock Math. Colloq. 18, 69–77 (1981)
Gnana Bhaskar, T., Lakshmikantham, V.: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65, 1379–1393 (2006)
Gordji, M.E., Baghani, H., Khodaei, H., Ramezani, M.: A generalization of Nadler’s fixed point theorem. J. Nonlinear Sci. Appl. 3(2), 148–151 (2010)
Harjani, J., Sadarangani, K.: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal. 71, 3403–3410 (2009)
Hussain, N., Karapinar, E., Salimi, P., Akbar, F.: \(\alpha \)-admissible mappings and related fixed point theorems. J. Inequal. Appl. 2013, 114 (2013)
Kadelburg, Z., Radenović, S.: A note on pata-type cyclic contractions. Sarajevo J. Math. 11(2), 235–245 (2015)
Kadelburg, Z., Radenović, S.: Fixed point theorems under Pata-type conditions in metric spaces. J. Egypt. Math. Soc. 24, 77–82 (2016)
Kannan, R.: Some results on fixed points. Bull. Cal. Math. Soc. 60, 71–76 (1968)
Karapinar, E., Samet, B.: Generalized \(\alpha - \psi \) contractive type mappings and related fixed point theorems with applications. Abstr. Appl. Anal. 2012 (2012), Article ID 793486
Nadler Jr., S.B.: Multivalued contraction mappings. Pac. J. Math. 30, 475–488 (1969)
Nashine, H.K., Samet, B., Vetro, C.: Monotone generalized nonlinear contractions and fixed point theorems in ordered metric spaces. Math. Comput. Modell. 54, 712–720 (2011)
Nieto, J.J., Rodrguez-López, R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22, 223–239 (2005)
Pata, V.: A fixed point theorem in metric spaces. J. Fixed Point Theory Appl. 10, 299–305 (2011)
Ran, A.C.M., Reurings, M.C.B.: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 132, 1435–1443 (2004)
Reich, S.: Fixed points of contractive functions. Boll. Un. Mat. Ital. 5, 26–42 (1972)
Samet, B., Vetro, C., Vetro, P.: Fixed point theorems for \(\alpha -\psi \)-contractive type mappings. Nonlinear Anal. 75, 2154–2165 (2012)
Shen, M., Hong, S.: Common fixed points for generalized contractive multivalued operators in complete metric spaces. Appl. Math. Lett. 22, 1864–1869 (2009)
Turinici, M.: Abstract comparison principles and multivariable Gronwall–Bellman inequalities. J. Math. Anal. Appl. 117, 100–127 (1986)
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The authors gratefully acknowledge the suggestions made by the learned referee.
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Choudhury, B.S., Metiya, N. & Kundu, S. End point theorems of multivalued operators without continuity satisfying hybrid inequality under two different sets of conditions. Rend. Circ. Mat. Palermo, II. Ser 68, 65–81 (2019). https://doi.org/10.1007/s12215-018-0344-z
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DOI: https://doi.org/10.1007/s12215-018-0344-z
Keywords
- Metric space
- Partial order
- Multivalued cyclic \((\alpha , \beta )\)-admissible mapping
- \(\delta \)-distance
- End point