Some fixed point theorems for multi-valued mappings in graphical metric spaces
-
Satish Shukla
and
Hans-Peter A. Künzi
Abstract
In this paper, we discuss some topological properties of graphical metric spaces and introduce the G-set metric with respect to a graphical metric. Some fixed point results are introduced which generalize the famous Nadler’s fixed point theorem.
Communicated by Ľubica Holá
Acknowledgement
The authors are thankful to the Editor and Reviewer for their crucial suggestions on this paper. The first author is thankful to Professor Mahesh Kumar Dube for continuous motivation and encouragement for research. The second author would like to thank the South African National Research Foundation for partial financial support under grant 118517.
References
[1] Banach, S.: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922), 133–181.10.4064/fm-3-1-133-181Search in Google Scholar
[2] Berinde, V.—Pacurar, M.: The role of the Pompeiu-Hausdorff metric in fixed point theory, Creative Math. & Inf. 22(2) (2013), 143–150.10.37193/CMI.2013.02.13Search in Google Scholar
[3] Edelstein, M.: An extension of Banach’s contraction principle, Proc. Amer. Math. Soc. 12 (1961), 7–10.10.2307/2034113Search in Google Scholar
[4] Fréchet, M.: Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo 22 (1906), 1–74.10.1007/BF03018603Search in Google Scholar
[5] Hong, W. C.: Notes on Fréchet spaces, Internat. J. Math. Math. Sci. 22(3) (1999), 659–665.10.1155/S0161171299226592Search in Google Scholar
[6] Jachymski, J.: The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc. 136 (2008), 1359–1373.10.1090/S0002-9939-07-09110-1Search in Google Scholar
[7] Kirk, W. A.—Srinivasan, P. S.—Veeramani, P.: Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory 4 (2003), 79–89.Search in Google Scholar
[8] Nadler, S. B., Jr.: Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475–488.10.2140/pjm.1969.30.475Search in Google Scholar
[9] Nieto, J. J.—Rodríguez-López, R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005), 223–239.10.1007/s11083-005-9018-5Search in Google Scholar
[10] Ran, A. C. M.—Reurings, M. C. B.: A fixed point theorem in partially ordered sets and some application to matrix equations, Proc. Amer. Math. Soc. 132 (2004), 1435–1443.10.1090/S0002-9939-03-07220-4Search in Google Scholar
[11] Shukla, S.—Abbas, M.: Fixed point results of cyclic contractions in product spaces, Carpathian J. Math. 31 (2015), 119–126.10.37193/CJM.2015.01.14Search in Google Scholar
[12] Shukla, S.—Radenović, S.—Vetro, C.: Graphical metric space: a generalized setting in fixed point theory, RACSAM (2016), https://doi.org/10.1007/s13398-016-0316-0.10.1007/s13398-016-0316-0Search in Google Scholar
© 2020 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Regular papers
- Recurrences for the genus polynomials of linear sequences of graphs
- The existence of states on EQ-algebras
- On sidon sequences of farey sequences, square roots and reciprocals
- Repdigits as sums of three balancing numbers
- Lipschitz one sets modulo sets of measure zero
- Refinement of fejér inequality for convex and co-ordinated convex functions
- An extension of q-starlike and q-convex error functions endowed with the trigonometric polynomials
- Coefficients problems for families of holomorphic functions related to hyperbola
- Triebel-Lizorkin capacity and hausdorff measure in metric spaces
- The method of upper and lower solutions for integral boundary value problem of semilinear fractional differential equations with non-instantaneous impulses
- On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers
- Density of summable subsequences of a sequence and its applications
- Rough weighted 𝓘-limit points and weighted 𝓘-cluster points in θ-metric space
- A note on cosine series with coefficients of generalized bounded variation
- Some geometric properties of the non-Newtonian sequence spaces lp(N)
- On sequence spaces defined by the domain of a regular tribonacci matrix
- Jointly separating maps between vector-valued function spaces
- Some fixed point theorems for multi-valued mappings in graphical metric spaces
- Monotone transformations on the cone of all positive semidefinite real matrices
- Locally defined operators in the space of Ck,ω-functions
- Some properties of D-weak operator topology
- Estimating the distribution of a stochastic sum of IID random variables
- An internal characterization of complete regularity
Articles in the same Issue
- Regular papers
- Recurrences for the genus polynomials of linear sequences of graphs
- The existence of states on EQ-algebras
- On sidon sequences of farey sequences, square roots and reciprocals
- Repdigits as sums of three balancing numbers
- Lipschitz one sets modulo sets of measure zero
- Refinement of fejér inequality for convex and co-ordinated convex functions
- An extension of q-starlike and q-convex error functions endowed with the trigonometric polynomials
- Coefficients problems for families of holomorphic functions related to hyperbola
- Triebel-Lizorkin capacity and hausdorff measure in metric spaces
- The method of upper and lower solutions for integral boundary value problem of semilinear fractional differential equations with non-instantaneous impulses
- On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers
- Density of summable subsequences of a sequence and its applications
- Rough weighted 𝓘-limit points and weighted 𝓘-cluster points in θ-metric space
- A note on cosine series with coefficients of generalized bounded variation
- Some geometric properties of the non-Newtonian sequence spaces lp(N)
- On sequence spaces defined by the domain of a regular tribonacci matrix
- Jointly separating maps between vector-valued function spaces
- Some fixed point theorems for multi-valued mappings in graphical metric spaces
- Monotone transformations on the cone of all positive semidefinite real matrices
- Locally defined operators in the space of Ck,ω-functions
- Some properties of D-weak operator topology
- Estimating the distribution of a stochastic sum of IID random variables
- An internal characterization of complete regularity
