Analysis of the solvability and stability of the operator-valued Fredholm integral equation in Hölder space

Manalisha Bhujel; Bipan Hazarika; Sumati Kumari Panda; Dimplekumar Chalishajar; Manalisha Bhujel; Bipan Hazarika; Sumati Kumari Panda; Dimplekumar Chalishajar

doi:10.3934/math.20231334

AIMS Mathematics

2023, Volume 8, Issue 11: 26168-26187. doi: 10.3934/math.20231334

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Analysis of the solvability and stability of the operator-valued Fredholm integral equation in Hölder space

1.
Department of Mathematics, Gauhati University, Guwahati-781014, Assam, India
2.
Department of Mathematics, GMR Institute of Technology, Rajam-532127, Andhra Pradesh, India
3.
Department of Applied Mathematics, Mallory Hall, Virginia Military Institute (VMI), Lexington, VA 24450, USA

Received: 07 July 2023 Revised: 25 August 2023 Accepted: 04 September 2023 Published: 12 September 2023
MSC : 26B35, 45B05, 47H10

In this paper, the solvability of an operator-valued integral equation in Hölder spaces, i.e.,

$ \begin{equation*} \label{fredholm} w(\zeta_1) = y(\zeta_1)+w(\zeta_1)\int_{\bf J}\kappa(\zeta_1, \varphi)(T_1w)(\varphi)d\varphi+z(\zeta_1)\int_{\bf J}h(\varphi, (T_2w)(\varphi))d\varphi, \end{equation*} $

for $ \zeta_1\in{\bf J} = [0, 1], $ is studied by using Darbo's fixed point theorem (FPT). The process of the measure of noncompactness of the operators which constitute an intermediary of contraction and compact mappings can be explained with the help of Darbo's FPT. The greater effectiveness of Darbo's FPT due to its non-involvement of the compactness property gives a better scope when dealing with the Schauder FPT, where compactness is an essential property. To obtain a unique solution, we apply the Banach fixed point theorem and discuss the Hyers-Ulam stability of the integral equation. We also give some important examples to illustrate the existence and uniqueness of the results.
- Hölder function space,
- measure of noncompactness,
- Fredholm integral equation,
- fixed-point theorem
Citation: Manalisha Bhujel, Bipan Hazarika, Sumati Kumari Panda, Dimplekumar Chalishajar. Analysis of the solvability and stability of the operator-valued Fredholm integral equation in Hölder space[J]. AIMS Mathematics, 2023, 8(11): 26168-26187. doi: 10.3934/math.20231334

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Abstract

In this paper, the solvability of an operator-valued integral equation in Hölder spaces, i.e.,

$ \begin{equation*} \label{fredholm} w(\zeta_1) = y(\zeta_1)+w(\zeta_1)\int_{\bf J}\kappa(\zeta_1, \varphi)(T_1w)(\varphi)d\varphi+z(\zeta_1)\int_{\bf J}h(\varphi, (T_2w)(\varphi))d\varphi, \end{equation*} $

for $ \zeta_1\in{\bf J} = [0, 1], $ is studied by using Darbo's fixed point theorem (FPT). The process of the measure of noncompactness of the operators which constitute an intermediary of contraction and compact mappings can be explained with the help of Darbo's FPT. The greater effectiveness of Darbo's FPT due to its non-involvement of the compactness property gives a better scope when dealing with the Schauder FPT, where compactness is an essential property. To obtain a unique solution, we apply the Banach fixed point theorem and discuss the Hyers-Ulam stability of the integral equation. We also give some important examples to illustrate the existence and uniqueness of the results.

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