On solvability of quadratic Erdélyi-Köber fractional integral equations in Orlicz space

Pallav  Bora; Bipan Hazarika

doi:10.5269/bspm.70062

Pallav Bora Gauhati University
Bipan Hazarika Rajiv Gandhi University

Abstract

In this paper, we focus on providing some properties of Erdélyi-Köber fractional operators and prove fixed point results using the operator type condensing map to find the existence of solution of Erdélyi-Köber fractional Integral equation in Orlicz space. The main tool in our considerations is the technique associated to measure of non-compactness. Lastly, we provide some examples to illustrate our main results.

Downloads

Download data is not yet available.

Author Biography

Bipan Hazarika, Rajiv Gandhi University

Mathematics

Professor

References

Abdalla, A.M., Salem, H.A.H., On the Monotonic Solutions of Quadratic Integral Equations in Orlicz Space. J. Advances Math. Computer Sci. 30, 1–11, (2019). https://doi.org/10.9734/JAMCS/2019/46641

Aghajani, A., Pourhadi, E., Trujillo, J.J., Application of measure of noncompactness to a Cauchy problem for fractional differential equation in Banach spaces. Frac. Cal. Appl. Anal. 16(4), 962-977, (2003).

Aghajani, A., Allahyari, R., Mursaleen, M., A generalization of Darbo’s theorem with application to the solvability of systems of integral equations. J. Comput. Appl. Math. 260, 68-77, (2014).

Alamo, J.A., Rodriguez, J., Operational calculus for modified Eedelyi-Kober operators. Serdica: Bulgaricae mathematicae publicationes. 20, 351-363, (1994).

Arab, R., Nashine, H.K., Can, N.H., Binh, T.T., Solvability of functional-integral equations (fractional order) using measure of noncompactness. Advances in Diff. Equ. 2020; Article number: 12. doi.org/10.1186/s13662-019-2487-4.

Banas, J., Leszek, O., Measure of noncompactness related to monotonicity. Commentationes Mathematicae. 41, 13-23, (2001).

Banas, J., Goebel, K., Measures of Noncompactness in Banach Spaces, Lect. Notes in Math., 60, M. Dekker, New York - Basel, (1980).

Benchohra, M., Henderson, J., Ntouyas, S.K., Ouahab, A., Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 332, 1340–1350, (2008).

Chandrasekhar, S., Radiative Transfer. Dover Publications, New York, (1960).

Cichon, M., Metwali, M., Existence of monotonic Lϕ-solutions for quadratic Volterra functionl integral equations. Electron. J. Qual. Theory Differ. Equ. 13, 1–16, (2015).

Cichon, M., Metwali, M., On solutions of quadratic integral equations in Orlicz spaces. Mediterr. J. Math. 12, 901–920, (2015).

Cichon, M., Metwali, M., On the existence of solutions for quadratic integral equations in Orlicz space. Math. Slovaca 66, 1413–1426, (2016).

Cichon, M., Salem, H.A.H., On the solutions of Caputo-Hadamard Pettis-type fractional differential equations. Revista de la Real Academia de Ciencias Exactas, Fısicas y Naturales. Serie A. Matematicas 1–23, (2019).

Darbo, G., Punti uniti in trasformazioni a codominio non compatto (Italian). Rendiconti del Seminario Matematico della Universita di Padova. 24, 84-92, (1995).

Darwish, M.A., Sadarangani, K., On a quadratic integral equation with supremum involving Erdelyi-Kober fractional order. Mathematische Nachrichten. 288(5-6), 566-576, (2015).

MA Darwish and K. Sadarangani, On Erdelyi-Kober type quadratic integral equation with linear modification of the argument. Appl. Math. Comput. 238, 30-42, (2014).

D. Delboso and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 204, 609–625, (1996).

Diethelm, K., Ford, N.J., Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248, (2002).

N. Erzakova, Compactness in measure and measure of noncompactness. Siberian Math. J. 38, 926–928, (1997).

Furati, K.M., Tatar, N.-eddine, An existence result for a nonlocal fractional differential problem. J. Fractional Calc. 26, 43–51, (2004).

Hazarika, B., Arab, R., 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, 3837-3851, (2019).

Hilfer, R., Applications of fractional calculus in physics. World Scientific, Singapore, (2000).

Katugampola, U.N., A New Approach To Generalized Fractional Derivatives. Bull. Math. Anal. Appl. 6(4), 1-15, (2014).

Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional Differential Equations. North Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, (2006).

Krasnosel’skii, M.A., Rutitskii, Yu., Convex Functions and Orlicz Spaces. Groningen, (1961).

Kuratowski, K., Sur les espaces completes. Fundamenta Mathematicae. 1, 301-309, (1930).

Maligranda, L., Orlicz spaces and interpolation. Campinas SP Brazil: Departamento de Matematica, Universidade Estadual de Campinas, 1989.

Metwali, M., Solvability of quadratic Hadamard-type fractional integral equation in orlicz spaces. Rocky Mountain J. Math. 2022.

Metwali, M., On perturbed quadratic integral equations and initial value problem with nonlocal conditions in Orlicz spaces. Demonstratio Mathematica 53, 86–94, (2020).

Metwali, M., Cichon, K., On solutions of some delay Volterra integral problems on a half-line. Nonlinear Analysis: Modelling and Control 26, 661–677, (2021).

Miller, K.S., Ross, B., An Introduction to the Fractional Calculus and Differential Equations. John Wiley, New York, 1993.

Momani, S.M., Hadid, S.B., Some comparison results for integro-fractional differential inequalities. J. Fract. Calc. 24, 37–44, (2003).

Nashine, H.K., Arab, R., Agarwal, R.P., Haghigh, A.S., Darbo type fixed and coupled fixed point results and its application to integral equation. Periodica Mathematica Hungarica. 77, 94-107, (2018).

O’Neil, R., Convolution operators and L(p, q) spaces. Duke Math. J. 30, 129–142, (1963).

O’Neil, R., Fractional integration in Orlicz spaces-I. Trans. Am. Math. Soc. 115, 300–328, (1965).

Rabbani, M., Das, A., Hazarika, B., Arab, R., Existence of solution for two dimensional nonlinear fractional integral equation by measure of noncompactness and iterative algorithm to solve it. J. Comput. Appl. Math.. 370, 1-17, (2020).

Rabbani, M., Das, A., Hazarika, B., Arab, R., Measure of noncompactness of a new space of tempered sequences and its application on fractional differential equations. Chaos, Solitons and Fractals. 140(2020), article no 110221, doi:10.1016/j.chaos.2020.110221.

Rabbani, M., Arab R., Hazarika, B., Solvability of nonlinear quadratic integral equation by using simulation type condensing operator and measure of noncompactness. Appl. Math. Comput. 349 (15), 102-117, (2019).