Title: Cyclic and Noncyclic Geraghty Type Condensing Operators and Optimal Solutions of Nonlocal Integro-Differential Equations
Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814
Article ID: MTJPAM-D-20-00061; Volume 3 / Issue 3 / Year 2021 (Special Issue), Pages 344-357
Document Type: Research Paper
Author(s): Pradip Ramesh Patle a , Dhananjay Gopal b , Rabha Waell Ibrahim c
aDepartment of Mathematics, Amity School of Engineering and Technology Amity University, Madhya Pradesh Gwalior 474020, India
bDepartment of Mathematics, Guru ghasidas Vishvavidyalaya, Koni- Bilaspur-495009, India
cIEEE: 94086547, Kuala Lumpur, 59200, Malaysia
Received: 7 January 2021, Accepted: 6 April 2021, Published: 25 April 2021.
Corresponding Author: Rabha Waell Ibrahim (Email address: rabhaibrahim@yahoo.com)
Full Text: PDF
Abstract
The present work considers a family of cyclic (non-cyclic) relatively Geraghty type condensing functions and primarily aims to study the existence of (coupled) points and pairs of best proximity in Banach spaces. The occurrence of optimal solutions for a system of non-local integro-differential equations is demonstrated as an application. As numerical illustrations, we present the optimal solution of integro-differential systems (type cell growth), which can be optimized by using fractal entropy (the measurement of complexity). The fractal power is playing an important role to state the stability and maximization of solutions.
Keywords: oupled point (pair) of best proximity, Cyclic (non-cyclic) relatively Geraghty condensing operator, Optimal solution, Non-local integro-differential equation, Fractal, Entropy, Fractional calculus
References:- A. Abkar and M. Gabeleh, Best proximity points for asymptotic cyclic contraction mappings, Nonlinear Anal. 74, 7261-7268, 2011.
- G. C. Ahuja, T. D. Narang and S. Trehan, Best approximation on convex sets in a metric space, J. Approx. Theory 12, 94-97, 1974.
- R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina and B. N. Sadovskii, Measures of Noncompactness and Condensing Operators, vol. 55, Birkhauser, Basel, 1992.
- M. A. Al-Thagafi and N. Shahzad, Convergence and existence results for best proximity points, Nonlinear Anal. 70, 3665-3671, 2009.
- J. Anuradha and P. Veeramani, Proximal pointwise contraction, Topology Appl. 156, 2942-2948, 2009.
- M. Ayerbe Toledano, T. Dominguez Benavides and G. Lopez Acedo, Measures of Noncompactness in Metric Fixed Point Theory. Operator Theory: Advances and Applications, vol. 99, Birkhäuser, Basel, 1997.
- A. Eldred and P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323, 1001-1006, 2006.
- A. A. Eldred, W. A. Kirk and P. Veeramani, Proximal normal structure and relatively nonexpansive mappings, Studia Math. 171, 283-293, 2005.
- A. M. A. El-Sayed and R. G. Ahmed,Solvability of a coupled system of functional integro-differential equations with infinite point and Riemann-Stieltjes integral conditions, Appl. Math. Comput. https: //doi.org/10.1016/j.amc.2019.124918
- A. M. A. El-Sayed and R. G. Ahmed, Existence of solutions for a functional integro-differential equation with infinite point and integral conditions, Int. J. Appl. Comput.Math. 5 (4), 108, 2019. doi:10.1007/s40819-019-0691-2
- R. Espínola, A new approach to relatively nonexpansive mappings, Proc. Amer. Math. Soc. 136, 1987-1996, 2008.
- R. Espínola, O. Madiedo and A. Nicolae, Borsuk-Dugundji type extensions theorems with Busemann convex target spaces, Annales Academiae Scientiarum Fennicae Mathematica, 43, 225-238, 2018.
- R. Espinola and A. Nicolae, Mutually nearest and farthest points of sets and the drop theorem in geodesic spaces, Monatsh. Math. 165, 173-197, 2012.
- R. Espinola and B. Piatek, Fixed point property and unbounded sets in CAT(0) spaces, J. Math. Anal. Appl. 408, 638-654, 2013.
- K. Fan, Extensions of two fixed point theorems of F.E. Browder, Math. Z. 122, 234-240, 1969.
- A. Fernandez Leon, A. Nicolae, Best proximity pair results relatively nonexpansive mappings in geodesic spaces, Numer. Funct. Anal. Optim. 35, 1399-1418, 2014.
- M. Gabeleh, Characterization of proximal normal structures via proximal diametral sequences, J. Fixed Point Theory Appl. 19, 2909–2925, 2017.
- M. Gabeleh, Best proximity points for cyclic mappings, Ph.D Thesis, 2012.
- M. Gabeleh and J. Markin, Optimum solutions for a system of differential equations via measure of noncompactness, Indagationes Mathematicae 29, 895-906, 2018.
- M. Gabeleha, S. P. Moshokoab and C. Vetroc, Cyclic (noncyclic) ’-condensing operator and its application to a system of differential equations, Nonlinear Analysis: Modelling and Control, 24 (6), 985-1000, 2019.
- M. Geraghty, On contractive mappings, Proc. Amer. Math. Soc 40, 604-608, 1973.
- C. Horvath, Applications of measure of noncompactness to coupled fixed points and system of integral equations, Miskolc Mathematical Notes 19, 537-553, 2018.
- K. Kuratowski, Sur les espaces complets, Fund. Math. 15, 301-309, 1930.
- V. Lakshmikantham and M. Rama Mohana Rao, Theory of Integro-Differential Equations, CRC Press, 1995.
- S. Momani, R.W. Ibrahim and S. B. Hadid, Susceptible-infected-susceptible epidemic discrete dynamic system based on tsallis entropy. Entropy 22 (7), 769, 2020.
- P. R. Patle, D. K. Patel and R. Arab, Darbo type best proximity point results via simulation function with application, Afrika Mathematika 2020. DOI 10.1007/s13370-020-00764-7
- H. Rehman. D. Gopal and P. Kumam, Generalizations of Darbo’s fixed point theorem for a new condensing operators with application to a functional integral equation, Demonstr. Mat. 52, 166-182, 2019.
- B. Samet, C. Vetro and P. Vetro, Fixed point theorems for α, ψ -contractive type mappings, Nonlinear Anal. 75 (4), 2154-2165, 2012.
- C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52 (1-2), 479-487, 1988.
- A. Visintin, Strong convergence results related to strict convexity, Commun. Partial Differ. Equations 9 (5), 439-466, 1984.