Existence of a solution in the Holder space for a nonlinear functional integral equation

Dipankar Saha; Mausumi Sen; Nimai Sarkar; Subhankar Saha

doi:10.52737/18291163-2020.12.7-1-8

Authors

Dipankar Saha National Institute of Technology, Silchar
Mausumi Sen National Institute of Technology, Silchar
Nimai Sarkar National Institute of Technology, Silchar
Subhankar Saha National Institute of Technology, Silchar

DOI:

https://doi.org/10.52737/18291163-2020.12.7-1-8

Keywords:

Functional integral equation, Fixed point theory, Holder space

Abstract

This article is entirely devoted to the application of the measure of noncompactness defined in the Holder space. Here the emphasis is on the study of the nonlinear functional integral equation with changed arguments. Precisely, the existence of a solution is obtained by employing the Darbo fixed point theorem under certain hypotheses. Finally, we provide a tangible example which supports our results.

References

A. Aghajani and M. Aliaskari, Measure of noncompactness in Banach algebra and application to the solvability of integral equations in $BC(R_{+})$, Inf. Sci. Lett., 4 (2015), no. 2, 93-99.

R. Arab, R. Allahyari, and A. S. Haghighi, Construction of measures of noncompactness of $C^{k}(Omega)$ and $C_{0}^{k}$ and their application to functional integral differential equations, Bull. Iran. Math. Soc, 43 (2017), no. 1, 53-67.

J. Banas, On measures of noncompactness in Banach spaces, Comment. Math. Univ. Carolin., 21 (1980), no. 1, 131-143.

J. Banas and R. Nalepa, On the space of functions with growths tempered by a modulus of continuity and its applications, J.Funct. Spaces Appl., 2013 (2013), ARTICLE ID 820437, 13 pages. https://doi.org/10.1155/2013/820437

J. Banas and B. Rzepka, On existence and asymptotic stability of solutions of a nonlinear integral equation, J. Math. Anal. Appl., 284 (2003), no. 1, 165-173. https://doi.org/10.1016/s0022-247x(03)00300-7

J. Caballero, M. A. Darwish, and K. Sadarangani, Solvability of a quadratic integral equation of Fredholm type in Holder spaces, Electron. J. Differential Equations, 2014 (2014), no. 31, 1-10.

G. Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend. Semin. Mat. Univ. Padova, 24 (1955), 84-92.

L. N. Mishra, R. P. Agarwal, and M. Sen, Solvability and asymptotic behaviour for some nonlinear quadratic integral equation involving Erdelyi-Kober fractional integrals on the unbounded interval, Progress in Fractional Differentiation and Applications, 2 (2016), no. 3, 153-168. https://doi.org/10.18576/pfda/020301

L. N. Mishra and M. Sen, On the concept of existence and local attractivity of solutions for some quadratic Volterra integral equation of fractional order, Appl. Math. Comput., 285 (2016), 174-183. https://doi.org/10.1016/j.amc.2016.03.002

S.Saiedinezhad, On a measure of noncompactness in the Holder space $C^{k,gamma}(Omega)$ and its application, J. Comput. Appl. Math., 346 (2019), 566-571.

M. Sen, D. Saha, and R. P. Agarwal, A Darbo fixed point theory approach towards the existence of a functional integral equation in a Banach algebra, Appl. Math. Comput., 358 (2019), 111-118. https://doi.org/10.1016/j.amc.2019.04.021