Analysis of mixed type nonlinear Volterra–Fredholm integral equations involving the Erdélyi–Kober fractional operator
This paper investigates the existence, uniqueness and stability of solutions to the nonlinear Volterra–Fredholm integral equations (NVFIE) involving the Erdélyi–Kober (E–K) fractional integral operator. We use the Leray–Schauder alternative and Banach’s fixed point theorem to examine the existence a...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Elsevier
2023-12-01
|
| Series: | Journal of King Saud University: Science |
| Subjects: | |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S1018364723004111 |
| _version_ | 1797615999345754112 |
|---|---|
| author | Supriya Kumar Paul Lakshmi Narayan Mishra Vishnu Narayan Mishra Dumitru Baleanu |
| author_facet | Supriya Kumar Paul Lakshmi Narayan Mishra Vishnu Narayan Mishra Dumitru Baleanu |
| author_sort | Supriya Kumar Paul |
| collection | DOAJ |
| description | This paper investigates the existence, uniqueness and stability of solutions to the nonlinear Volterra–Fredholm integral equations (NVFIE) involving the Erdélyi–Kober (E–K) fractional integral operator. We use the Leray–Schauder alternative and Banach’s fixed point theorem to examine the existence and uniqueness of solutions, and we also explore Hyers–Ulam (H–U) and Hyers–Ulam–Rassias (H–U–R) stability in the space C([0,β],R). Furthermore, three solution sets Uσ,λ, Uθ,1 and U1,1 are constructed for σ>0, λ>0, and θ∈(0,1), and then we obtain local stability of the solutions with some ideal conditions and by using Schauder fixed point theorem on these three sets, respectively. Also, to achieve the goal, we choose the parameters for the NVFIE as δ∈(12,1), ρ∈(0,1), γ>0. Three examples are provided to clarify the results. |
| first_indexed | 2024-03-11T07:33:56Z |
| format | Article |
| id | doaj.art-6a8630cef53540e6a443632709843fa1 |
| institution | Directory Open Access Journal |
| issn | 1018-3647 |
| language | English |
| last_indexed | 2024-03-11T07:33:56Z |
| publishDate | 2023-12-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Journal of King Saud University: Science |
| spelling | doaj.art-6a8630cef53540e6a443632709843fa12023-11-17T05:25:19ZengElsevierJournal of King Saud University: Science1018-36472023-12-013510102949Analysis of mixed type nonlinear Volterra–Fredholm integral equations involving the Erdélyi–Kober fractional operatorSupriya Kumar Paul0Lakshmi Narayan Mishra1Vishnu Narayan Mishra2Dumitru Baleanu3Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632 014, Tamil Nadu, IndiaDepartment of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632 014, Tamil Nadu, India; Corresponding author.Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Anuppur 484 887, Madhya Pradesh, IndiaDepartment of Mathematics, Faculty of Arts and Sciences, Cankaya University, Ankara 09790, Turkey; Institute of Space Sciences, 077125, Magurele, Ilfov, Romania; Department of Natural Sciences, School of Arts and Sciences, Lebanese American University, Beirut 11022801, LebanonThis paper investigates the existence, uniqueness and stability of solutions to the nonlinear Volterra–Fredholm integral equations (NVFIE) involving the Erdélyi–Kober (E–K) fractional integral operator. We use the Leray–Schauder alternative and Banach’s fixed point theorem to examine the existence and uniqueness of solutions, and we also explore Hyers–Ulam (H–U) and Hyers–Ulam–Rassias (H–U–R) stability in the space C([0,β],R). Furthermore, three solution sets Uσ,λ, Uθ,1 and U1,1 are constructed for σ>0, λ>0, and θ∈(0,1), and then we obtain local stability of the solutions with some ideal conditions and by using Schauder fixed point theorem on these three sets, respectively. Also, to achieve the goal, we choose the parameters for the NVFIE as δ∈(12,1), ρ∈(0,1), γ>0. Three examples are provided to clarify the results.http://www.sciencedirect.com/science/article/pii/S1018364723004111Erdélyi–Kober fractional integral operatorHyers–Ulam–Rassias stabilityHyers–Ulam stabilityLocal stabilityFixed point theorem |
| spellingShingle | Supriya Kumar Paul Lakshmi Narayan Mishra Vishnu Narayan Mishra Dumitru Baleanu Analysis of mixed type nonlinear Volterra–Fredholm integral equations involving the Erdélyi–Kober fractional operator Journal of King Saud University: Science Erdélyi–Kober fractional integral operator Hyers–Ulam–Rassias stability Hyers–Ulam stability Local stability Fixed point theorem |
| title | Analysis of mixed type nonlinear Volterra–Fredholm integral equations involving the Erdélyi–Kober fractional operator |
| title_full | Analysis of mixed type nonlinear Volterra–Fredholm integral equations involving the Erdélyi–Kober fractional operator |
| title_fullStr | Analysis of mixed type nonlinear Volterra–Fredholm integral equations involving the Erdélyi–Kober fractional operator |
| title_full_unstemmed | Analysis of mixed type nonlinear Volterra–Fredholm integral equations involving the Erdélyi–Kober fractional operator |
| title_short | Analysis of mixed type nonlinear Volterra–Fredholm integral equations involving the Erdélyi–Kober fractional operator |
| title_sort | analysis of mixed type nonlinear volterra fredholm integral equations involving the erdelyi kober fractional operator |
| topic | Erdélyi–Kober fractional integral operator Hyers–Ulam–Rassias stability Hyers–Ulam stability Local stability Fixed point theorem |
| url | http://www.sciencedirect.com/science/article/pii/S1018364723004111 |
| work_keys_str_mv | AT supriyakumarpaul analysisofmixedtypenonlinearvolterrafredholmintegralequationsinvolvingtheerdelyikoberfractionaloperator AT lakshminarayanmishra analysisofmixedtypenonlinearvolterrafredholmintegralequationsinvolvingtheerdelyikoberfractionaloperator AT vishnunarayanmishra analysisofmixedtypenonlinearvolterrafredholmintegralequationsinvolvingtheerdelyikoberfractionaloperator AT dumitrubaleanu analysisofmixedtypenonlinearvolterrafredholmintegralequationsinvolvingtheerdelyikoberfractionaloperator |