Solvability of k-fractional Hilfer integral equations via Darbo’s fixed point theorem
Abstract
In this article we extended the Darbo's fixed point theorem using $\mathcal{U}$-class mapping. Using the Darbo type theorem, we give a solvability result for a k-fractional Hilfer integral equation along with an appropriate illustration
Downloads
References
Agarwal, R. P., and O’regan, D., Fixed point theory and applications, Cambridge university press 141, (2001).
Ansari, A. H., Note on φ − ψ-contractive type mappings and related fixed point, In The 2nd regional conference on mathematics and applications, Payame Noor University 11, 377-380, (2014).
Banas, J., and Goebel, K., Measure of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York 60, (1980).
Caponetti, D., Trombetta, A., and Trombetta, G., Regular measures of noncompactness and Ascoli–Arzela type compactness criteria in spaces of vector-valued functions, Banach J. Math. Anal. 17(3), 48, (2023).
Chandra Deuri, B., V. Paunovic, M., Das, A., and Parvaneh, V., Solution of a fractional integral equation using the Darbo fixed point theorem, J. Math. 2022(1), 8415616, (2022).
Chen, X., and Cheng, L., Representation of measures of noncompactness and its applications related to an initial value problem in Banach spaces, Science China Mathematics, 66(4), 745-776, (2023).
Darbo, G., Punti uniti in trasformazioni a codominio non compatto, Rendiconti del Seminario matematico della Universit a di Padova 24, 84-92, (1955).
Das, A., Hazarika, B., Parvaneh, V., and Mursaleen, M., Solvability of generalized fractional order integral equations via measures of noncompactness, Mathematical Sciences 15(3), 241-251, (2021).
Das, A., Hazarika, B., Arab, R., and Mursaleen, M., Applications of a fixed point theorem to the existence of solutions to the nonlinear functional integral equations in two variables, Rendiconti del Circolo Matematico di Palermo Series 2 68(1), 139-152, (2019).
Das, A., Application of Measure of Noncompactness on Infinite System of Functional Integro-differential Equations with Integral Initial Conditions, In Sequence Space Theory with Applications, Chapman and Hall/CRC, 45-62, (2022).
Deb, S., Jafari, H., Das, A., and Parvaneh, V., New fixed point theorems via measure of noncompactness and its application on fractional integral equation involving an operator with iterative relations, J. Inequa. Appl. 2023(1), 106, (2023).
Dorrego, G. A., and Cerutti, R. A., The k-fractional Hilfer derivative, Int. J. Math. Anal. 7(11), 543-550, (2013).
Gabeleh, M., Malkowsky, E., Mursaleen, M., and Rakoˇcevic, V., A new survey of measures of noncompactness and their applications, Axioms 11(6), 299, (2022).
Golshan, H. M., On solution of fractional integral equation via measure of noncompactness and Petryshyn’s fixed point theorem, (2024). doi: 10.22541/au.170669940.03355969/v1
Hammad, H. A., Aydi, H., and De la Sen, M., Solving a Nonlinear Fractional Integral Equation by Fixed Point Approaches Using Auxiliary Functions Under Measure of Noncompactness, Inter. J. Anal. Appl. 22, 53-53, (2024).
Hazarika, B., Arab, R., and Mursaleen, M., Applications of measure of noncompactness and operator type contraction for existence of solution of functional integral equations, Complex Analysis and Operator Theory 13(8), 3837-3851, (2019).
Khan, M. S., Swaleh, M., and Sessa, S., Fixed point theorems by altering distances between the points, Bull. Aust. Math. Soc. 30(1), 1-9, (1984).
Karakaya, V., and Sekman, D., A new type of contraction via measure of non-compactness with an application to Volterra integral equation, Publications de l’Institut Mathematique 111(125), 111-121, (2022).
Khokhar, G. K., Patel, D. K., Patle, P. R., and Samei, M. E., Optimum solution of (k, ג)-Hilfer FDEs by A-condensing operators and the incorporated measure of noncompactness, J. Inequa. Appl. 2024(1), 1-31, (2024).
Kuratowski, K., Sur les espaces complets, Fundamenta Mathematicae 1(15), 301-309, (1930).
Masty lo, M., and da Silva, E. B., Measures of noncompactness of interpolated polynomials, In Forum Mathematicum 35(2), 487-505, (2023).
Mesmouli, M. B., Hamza, A. E., and Rizk, D., A Study of an IBVP of Fractional Differential Equations in Banach Space via the Measure of Noncompactness, Fractal and Fractional 8(1), 30, (2023).
Metwali, M. M., On measure of noncompactness in variable exponent Lebesgue spaces and applications to integral equations, J. Inequa. Appl. 2023(1), 157, (2023).
Nashine, H. K., and Das, A., Extension of Darbo’s fixed point theorem via shifting distance functions and its application, Nonlinear Analysis: Modelling and Control 27(2), 275-288, (2022).
Nikam, V., Shukla, A. K., Gopal, D., and Sumalai, P., Some Darbo-type fixed-point theorems in the modular space and existence of solution for fractional ordered 2019-nCoV mathematical model, Mathematical Methods in the Applied Sciences 47(18), 13923-13947, (2024).
Olszowy, L., and Zajac, T., On Darbo-and Sadovskii-Type Fixed Point Theorems in Banach Spaces, Symmetry 16(4), 392, (2024).
Rabbani, M., Das, A., Hazarika, B., and Arab, R., Measure of noncompactness of a new space of tempered sequences and its application on fractional differential equations, Chaos, Solitons & Fractals, 140, 110221, (2020).
Salem, A., Malaikah, H., and Kamel, E. S., An Infinite System of Fractional Sturm–Liouville Operator with Measure of Noncompactness Technique in Banach Space, Mathematics 11(6), 1444, (2023).
Srivastava, H. M., Das, A., Hazarika, B., and Mohiuddine, S., Existence of solution for non-linear functional integral equations of two variables in Banach algebra, Symmetry, 11(5), 674, (2019).
Taoudi, M. A., On Darbo’s fixed point principle, Moroccan Journal of Pure and Applied Analysis 9(3), 304-310, (2023).
Telli, B., Souid, M. S., and Stamova, I., Boundary-value problem for nonlinear fractional differential equations of variable order with finite delay via Kuratowski measure of noncompactness, Axioms 12(1), 80, (2023).
Copyright (c) 2025 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
