Fixed Points and Cyclic Contraction Mappings Under Implicit Relations and Applications to Integral Equations
DOI:
https://doi.org/10.5644/SJM.10.2.11Keywords:
Fixed point, cyclic contraction, implicit relation, integral equationAbstract
Inspired by the fact that the discontinuous mappings cannot be (Banach type) contractions and cyclic contractions need not be continuous, and taking into account that there are applications to integral and differential equations based on cyclic contractions, a new type of cyclic contraction mappings satisfying an implicit relation that involves a control function for a map in a metric space is originated. As a result, we derive existence and uniqueness results of fixed points for such mappings. We furnish suitable examples to demonstrate the validity of the hypotheses of our results. The results will be applied to the study of the existence and uniqueness of solutions for a class of nonlinear integral equations.
2010 Mathematics Subject Classification. Primary: 54H25; Secondary: 47H10.
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References
S. Banach, Sur les op´erations dans les ensembles abstraits et leur application aux equations integrales, Fund. Math., 3 (1922) 133–181.
H. K. Nashine, New fixed point theorems for mappings satisfying generalized weakly contractive condition with weaker control functions, Ann. Polonici Math., 104 (2012), 109–119.
W. A. Kirk, P. S. Srinivasan and P. Veeramani, Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory, 4 (1) (2003), 79–89.
M. Pacurar and I. A. Rus, Fixed point theory for cyclic ϕ-contractions, Nonlinear Anal., 72 (2010), 1181–1187.
R. P. Agarwal, M. A. Alghamd and, N. Shahzad, Fixed point theory for cyclic generalized contractions in partial metric spaces, Fixed Point Theory Appl., 2012 (40) (2012), doi:10.1186/1687-1812-2012-40.
E. Karapınar, Fixed point theory for cyclic weak φ-contraction, Appl. Math. Lett., 24 (6) (2011), 822–825.
E. Karapınar and K. Sadaranagni, Fixed point theory for cyclic (φ-ψ)-contractions, Fixed Point Theory Appl., 2011, 69 (2011), doi:10.1186/1687-1812-2011-69.
M. A. Petric, Some results concerning cyclical contractive mappings, General Math., 18 (4) (2010), 213–226.
I. A. Rus, Cyclic representations and fixed points, Ann. T. Popoviciu, Seminar Funct. Eq. Approx. Convexity, 3 (2005), 171–178.
C. M. Chen, Fixed point theory for the cyclic weaker Meir-Keeler function in complete metric spaces, Fixed Point Theory Appl., 2012, 2012:17, doi:10.1186/1687-1812-2012-17.
H. K. Nashine, Z. Kadelburg and P. Kumam, Implicit-relation-type cyclic contractive mappings and applications to integral equations, Abstr. Appl. Anal., Volume 2012, Article ID 386253, 15 pages.
V. Popa, A fixed point theorem for mapping in d-complete topological spaces, Math. Moravica, 3 (1999), 43-48.
I. Altun and D. Turkoglu, Some fixed point theorems for weakly compatible multivalued mappings satisfying an implicit relation, Filomat, 22 (1) (2008), 13-23.
I. Altun and D. Turkoglu, Some fixed point theorems for weakly compatible mappings satisfying an implicit relation, Taiwanese J. Math., 13 (4) (2009), 1291–1304.
I. Altun and H. Simsek, Some fixed point theorems on ordered metric spaces and application, Fixed Point Theory Appl., 2010 (2010), Article ID 621492, 17 pages.
M. Imdad, S. Kumar and M. S. Khan, Remarks on some fixed point theorems satisfying implicit relations, Rad. Math., 11 (1) (2002), 135-143.
V. Popa, A general coincidence theorem for compatible multivalued mappings satisfying an implicit relation, Demontsratio Math., 33 (1) (2000), 159-164.
V. Popa and M. Mocanu, Altering distance and common fixed points under implicit relations, Hacet. J. Math. Stat., 38 (3) (2009), 329–337.
A. Aliouche and A. Djoudi, Common fixed point theorems for mappings satisfying an implicit relation without decreasing assumption, Hacet. J. Math. Stat., 36 (1) (2007), 11-18.
S. Sharma and B. Desphande, On compatible mappings satisfying an implicit relation in common fixed point consideration, Tamkang J. Math., 33 (3) (2002), 245–252.
M. S. Khan, M. Swaleh and S. Sessa, Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc., 30 (1984), 1–9.
D. W. Boyd and J. S. Wong, On nonlinear contractions, Proc. Amer. Math. Soc., 20 (1969), 458–469.
G. E. Hardy and T. D. Rogers, A Generalization of a fixed point theorem of Reich, Canad. Math. Bull., 16 (2) (1973), 201–206.
