Multi-valued fixed point results with Feng-Liu type integral inequality and application to dynamical programming

Nilakshi Goswami, Md Shahruz Zaman

Abstract


In this paper, we obtain a fixed point result for multi-valued mappings satisfying Feng-Liu type integral inequality in the framework of partial metric space without using Pompeiu-Hausdorff distance. Our result extends an existing result in the integral setting for partial metric space. A common fixed point result and a coupled fixed point result of integral type are also obtained. The results are supported by suitable examples. An application is given to functional equations arising in dynamical programming.

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References


I. Altun, G. Minak, On fixed point theorems for multivalued mappings of Feng-Liu type, Journal of the Korean Mathematical Society 52 (2015), no. 6, 1901-1910.

I. Altun, F. Sola, H. Simsek, Generalized contractions on partial metric spaces, Topology and its Applications 157 (2010), 2778-2785.

H. Aydi, M. Abbas, C. Vetro, Partial Hausdor metric and Nadler's fixed point theorem on partial metric spaces, Topology and its Applications 159 (2012), no. 14, 3234-3242.

S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundamenta mathematicae 3 (1922), no. 1, 133-181.

R. Baskaran, P.V. Subrahmanyam, A note on the solution of a class of functional equations, Applicable Analysis 22 (1986), no. 3-4, 235-241.

S.A. Belbas, Dynamic programming and maximum principle for discrete Goursat systems, Journal of Mathematical Analysis and Applications 116 (1991), no. 1, 57-77.

R. Bellman, Dynamic programming and stochastic control processes, Information and control 1 (1958), no. 3, 228-239.

R. Bellman, E.S. Lee, Functional equations in dynamic programming, Aequationes mathematicae 17 (1978), no. 1, 1-18.

P.C. Bhakta, S. Mitra, Some existence theorems for functional equations arising in dynamic programming, Journal of Mathematical Analysis and Applications 98 (1984), no. 2, 348-362.

P.C. Bhakta, S.R. Choudhury, Some existence theorems for functional equations arising in dynamic programming, II, Journal of Mathematical Analysis and Applications 131 (1988), no. , 217-231.

M. Bukatin, R. Kopperman, S.G. Matthews, H. Pajoohesh, Partial metric spaces, The American Mathematical Monthly 116 (2009), no. 8, 708-718.

S. Chauhan, H. Aydi, W. Shatanawi, C. Vetro, Some integral type fixed-point theorems and an application to systems of functional equations, Vietnam Journal of Mathematics 42 (2014), 17-37.

Y. Feng, S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, Journal of Mathematical Analysis and Applications 317 (2006), no. 1, 103-112.

N. Goswami, R. Roy, V.N. Mishra, L.M. Sánchez Ruiz, Common Best Proximity Point Results for T-GKT Cyclic ϕ-Contraction Mappings in Partial Metric Spaces with Some Applications, Symmetry 13 (2021), no. 6, 1098.

D. Guo, V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications, Nonlinear analysis: theory, methods & applications 11 (1987), no. 5, 623-632.

R. Heckmann, Approximation of metric spaces by partial metric spaces, Applied Categorical Structures 7 (1999), no.1, 71-83.

M. Jleli, B. Samet, A new generalization of the Banach contraction principle, Journal of inequalities and applications 2014 (2014), 1-8.

S.B. Kaliaj, A functional equation arising in dynamic programming, Aequationes mathematicae 91 (2017), 635-645.

E. Karapinar, Generalizations of Caristi Kirk's theorem on partial metric spaces, Fixed Point Theory and Applications 2011 (2011), no. 1, 1-7.

E. Karapınar, K. Taș, V. Rakočević, Advances on fixed point results on partial metric spaces, Mathematical Methods in Engineering: Theoretical Aspects (2019), 3-66.

Z. Liu, J.S. Ume, S.M. Kang, Some existence theorems for functional equations and system of functional equations arising in dynamic programming, Taiwanese Journal of Mathematics 14 (2010), no. 4, 1517-1536.

N.A. Majid, Z.C. Ng, S.K. Lee, Some applications of fixed point results for monotone multi-Valued and integral type contractive mappings, Fixed Point Theory Algorithms Sci Eng 2023 (2023), no. 11.

S.G. Matthews, Partial metric topology, Annals of the New York Academy of Sciences 728 (1994), no. 1, 183-197.

S.B. Nadler Jr, Multi-valued contraction mappings, Pacific Journal of Mathematics 30 (1969), no. 2, 475-488.

L.V. Nguyen, L.T. Phuong, Fixed point theorem for set-valued mappings with new type of inequalities, Asian-European Journal of Mathematics 14 (2021), no. 2, 2150024.

K.P.R. Rao, G.N.V. Kishore, K. Tas, S. Satyanaraya, D.R. Prasad, Applications and common coupled fixed point results in ordered partial metric spaces, Fixed Point Theory and Applications 2017 (2017), 1-20.

T. Rasham, M. Nazam, P. Agarwal, A. Hussain, H.H. Al Sulmi, Existence results for the families of multi-mappings with applications to integral and functional equations, Journal of Inequalities and Applications 2023 (2023), no. 82.




DOI: https://doi.org/10.52846/ami.v52i1.1934