Some aspects of a coupled system of nonlinear integral equations

Binayak S. Choudhury, Nikhilesh Metiya, Sunirmal Kundu

Abstract


In the present work we take a system of two integral equations and prove the existence and uniqueness of their solution. We investigate four aspects of the problem, namely, error estimation and rate of convergence of the iteration leading to the solution, Ulam-Hyers stability, well-posedness and data dependence of the solution sets. We give some new definitions pertaining to the system we analyze here. In order to establish our results we utilize the coupled contraction mapping principle due to Bhaskar and Lakshmikantham (Nonlinear Anal. TMA 65 (2006), 1379-1393) and several related results which we deduce here.

Keywords


Fixed point; integral equation;stability; well-posedness; data dependence

Full Text:

PDF

References


Abbas, M., Aydi, H.,

Well-posedness of a common coupled fixed point problem,

Communications in Mathematics and Applications, textbf{9}(2018), no. 1, 27--40.

Babu, G.V.R., Kameswari, M.V.R.,

Coupled fixed points for generalized contractive maps with rational expressions in partially ordered metric spaces,

Journal of Advanced Research in Pure Mathematics, textbf{6}(2014), 43--57.

Baker, M., Zachariasen, F.,

Coupled integral equations for the nucleon and pion electromagnetic form factors,

Phys. Rev., textbf{119}(1960), 438--448.

Berinde, V.,

Error estimates for approximating fixed points of quasi contractions,

Gen. Math., textbf{13}(2005), no. 2, 23--34.

Bota, M.F., Karap{i}nar, E., Mlec{s}nic{t}e, O.,

Ulam-Hyers stability results for fixed point problems via $alpha-psi$-contractive mapping in $b$-metric space,

Abstr. Appl. Anal., textbf{2013}(2013), Article ID 825293, 6 pages.

Chifu, C., Petruc{s}el, G.,

Coupled fixed point results for $(varphi,G)$-contractions of type (b) in b-metric spaces endowed with a graph,

J. Nonlinear Sci. Appl., textbf{10}(2017), 671--683.

Choudhury, B.S., Gnana Bhaskar, T., Metiya, N., Kundu, S.,

Existence and stability of coupled fixed point sets for multi-valued mappings,

Fixed Point Theory, textbf{22}(2021), no. 2, 571--586.

Choudhury, B.S., Metiya, N., Kundu, S.,

Existence, data-dependence and stability of coupled fixed point sets of some multivalued operators,

Chaos, Solitons and Fractals, textbf{133}(2020), 109678.

Ciepli'{n}ski, K.,

Applications of fixed point theorems to the Hyers-Ulam stability of functional equations -- A suvey,

Ann. Funct. Anal., textbf{3}(2012), no. 1, 151--164.

Esp'{i}nola, R., Petruc{s}el, A.,

Existence and data dependence of fixed points for multivalued operators on gauge spaces,

J. Math. Anal. Appl., textbf{309}(2005), no. 2, 420--432.

Friedman, M., Colonias, J.,

On the coupled differential-integral equations for the solution of the general magnetostatic problem,

IEEE Transactions on Magnetics, textbf{18}(1982), no. 2, 336--339.

Gauthier, A., Knight, P.A., McKee, S.,

The Hertz contact problem, coupled Volterra integral equations and a linear complementarity problem,

J. Comput. Appl. Math., textbf{206}(2007), 322--340.

Gnana Bhaskar, T., Lakshmikantham, V.,

Fixed point theorems in partially ordered metric spaces and applications,

Nonlinear Anal., textbf{65}(2006), 1379--1393.

Guo, D., Lakshmikantham, V.,

Coupled fixed points of nonlinear operators with applications,

Nonlinear Anal., textbf{11}(1987), 623--632.

Harjani, J., L'{o}pez, B., Sadarangani, K.,

Fixed point theorems for mixed monotone operators and applications to integral equations,

Nonlinear Anal., textbf{74}(2011), 1749--1760.

Hazarika, B., Arab, R., Kumam, P.,

Coupled fixed point theorems in partially ordered metric spaces via mixed $g$-monotone property,

J. Fixed Point Theory Appl., textbf{21}(2019), no. 1.

Hyers, D.H.,

On the stability of the linear functional equation,

Proc. Natl. Acad. Sci. USA, textbf{27}(1941), no. 4, 222--224.

Kim, G.H., Shin, H.Y.,

Hyers-Ulam stability of quadratic functional euations on divisible square-symmetric groupoid,

Int. J. Pure Appl. Math., textbf{112}(2017), no. 1, 189--201.

Kutbi, M.A., Sintunavarat, W.,

Ulam-Hyers stability and well-posedness of fixed point problems for $alpha - lambda$-contraction

mapping in metric spaces,

Abstr. Appl. Anal., textbf{2014}(2014), Article ID 268230, 6 pages.

Mennig, J., "{O}zic{s}ik, M.N.,

Coupled integral equation approach for solving melting or solidification,

Int. J. Heat Mass Transfer, textbf{28}(1985), no. 8, 1481--1485.

Phiangsungnoen, S., Kumam, P.,

Generalized Ulam-Hyers stability and well-posedness for fixed point equation via $alpha$-admissibility,

J. Inequal. Appl., textbf{2014}(2014), no. 418.

Popa, V.,

Well-posedness of fixed point problem in orbitally complete metric spaces,

Stud. Cercet. Stiint. Ser. Mat., textbf{16}(2006), 209--214.

Rassias, T.M.,

On the stability of the linear mappings in Banach spaces,

Proc. Amer. Math. Soc., textbf{72}(1978), 297--300.

Rassias, T.M.,

Isometries and approximate isometries, IJMMS, textbf{25}(2001), no. 2, 73--91.

Rus, I.A.,

Remarks on Ulam stability of the operatorial equations,

Fixed Point Theory, textbf{10}(2009), no. 2, 305--320.

Sintunavarat, W.,

Generalized Ulam-Hyers stability, well-posedness and limit shadowing of fixed point problems

for $alpha-beta$-contraction mapping in metric spaces,

The Scientific World Journal, textbf{2014}(2014), Article ID 569174, 7.

Ulam, S.M.,

Problems in Modern Mathematics, Wiley, New York, 1964.




DOI: http://dx.doi.org/10.24193/subbmath.2024.4.16

Refbacks

  • There are currently no refbacks.