New Existence of Fixed Point Results in Generalized Pseudodistance Functions with Its Application to Differential Equations
Abstract
:1. Introduction
2. Preliminaries
2.1. w-Distance and Generalized Pseudodistance
- (a)
- for all ;
- (b)
- for any : is lower semi-continuous. That is,
- (c)
- for each , there exists such that and imply .
- (J1)
- for all ,
- (J2)
- for any sequences and in X such that
- (A)
- there exists a generalized pseudodistance J on X which is not a metric.
- (B)
- Every metric d is a generalized pseudodistance on X.
- (C)
- Let and : ; then, .
- (D)
- For any with , then (see ([13]).
- (A)
- If we take , then the J-completeness reduces to the completeness.
- (B)
- If , when p is a w-distances, we call X a w-complete.
2.2. F-Contraction Mapping
- (F1)
- i.e., F is strictly increasing.
- (F2)
- for every sequence in .
- (F3)
- there exists a number such that
- (a)
- (b)
- Every F-contraction is a continuous mapping.
- (F2′)
- or, instead, by
- (F2″)
- there exists a sequence in such thatIn 2014, Piri and Kumam [7] changed the property by in the F-function as follows.
- (F3′)
- (F1)
- F is strictly increasing,
- (F3′)
- F is continuous on
3. Main Results
4. Applications
- (A)
- K is continuous and increasing function;
- (B)
- there exists such that
- (C)
- T: satisfies (33).
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
- Kada, O.; Suzuki, T.; Takahashi, W. Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math. Jpn. 1996, 44, 381–391. [Google Scholar]
- Włodarczyk, K.; Plebaniak, R. Maximality principle and general results of Ekeland and Caristi types without lower semicontinuity assumptions in cone uniform spaces with generalized pseudodistances. Fixed Point Theory Appl. 2010, 2010, 175453. [Google Scholar] [CrossRef]
- Włodarczyk, K.; Plebaniak, R. A fixed point theorem of Subrahmanyam type in uniform spaces with generalized pseudodistances. Appl. Math. Lett. 2011, 24, 325–328. [Google Scholar] [CrossRef]
- Włodarczyk, K.; Plebaniak, R. Contractions of Banach, Tarafdar, Meir-Keeler, irić-Jachymski-Matkowski and Suzuki types and fixed points in uniform spaces with generalized pseudodistances. J. Math. Anal. Appl. 2013, 404, 338–350. [Google Scholar] [CrossRef]
- Mongkolkeha, C.; Kumam, P. Some Existence of coincidence point and approximate solution method for generalized weak contraction in b-generalized pseudodistance functions. Filomat 2017, 31, 6185–6203. [Google Scholar] [CrossRef]
- Wardowski, D. Fixed points of new type of contractive mappings in complete metric space. Fixed Point Theory Appl. 2012, 2012, 94. [Google Scholar] [CrossRef]
- Piri, H.; Kumam, P. Some fixed point theorems concerning F-contraction in complete metric spaces. Fixed Point Theory Appl. 2014, 2014, 210. [Google Scholar] [CrossRef] [Green Version]
- Budhia, L.; Kumam, P.; Martinez-Moreno, J.; Gopal, D. Extensions of almost-F and F-Suzuki contractions with graph and some applications to fractional calculus. Fixed Point Theory Appl. 2016, 2016, 2. [Google Scholar] [CrossRef]
- Lukács, A.; Kájant, K. Fixed point theorems for various type of F-contractions in complete metric spaces. Fixed Point Theory 2018, 19, 321–334. [Google Scholar] [CrossRef]
- Secelean, N.A. Iterated function systems consisting of F-contractions. Fixed Point Theory Appl. 2013, 2013, 277. [Google Scholar] [CrossRef]
- Singh, D.; Joshi, V.; Imdad, M.; Kumam, P. Fixed point theorems via generalized F-contractions with applications to functional equations occurring in dynamic programming. J. Fixed Point Theory Appl. 2017, 19, 1453–1479. [Google Scholar] [CrossRef]
- Takahashi, W. Nonlinear Functional Analysis Fixed Point Theory and its Applications; Yokahama Publishers: Yokahama, Japan, 2000. [Google Scholar]
- Plebaniak, R. On best proximity points for set-valued contractions of Nadler type with respect to b-generalized pseudodistances in b-metric spaces. Fixed Point Theory Appl. 2014, 2014, 39. [Google Scholar] [CrossRef]
- Plebaniak, R. New generalized pseudodistance and coincidence point theorem in a b-metric spaces. Fixed Point Theory Appl. 2013, 2013, 270. [Google Scholar] [CrossRef]
© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Sanhan, S.; Sanhan, W.; Mongkolkeha, C. New Existence of Fixed Point Results in Generalized Pseudodistance Functions with Its Application to Differential Equations. Mathematics 2018, 6, 324. https://doi.org/10.3390/math6120324
Sanhan S, Sanhan W, Mongkolkeha C. New Existence of Fixed Point Results in Generalized Pseudodistance Functions with Its Application to Differential Equations. Mathematics. 2018; 6(12):324. https://doi.org/10.3390/math6120324
Chicago/Turabian StyleSanhan, Sujitra, Winate Sanhan, and Chirasak Mongkolkeha. 2018. "New Existence of Fixed Point Results in Generalized Pseudodistance Functions with Its Application to Differential Equations" Mathematics 6, no. 12: 324. https://doi.org/10.3390/math6120324