# Existence of a bounded variation solution of a nonlinear integral equation in L1(R+)

## DOI:

https://doi.org/10.24297/jam.v21i.9317## Keywords:

Darbo fixed point theorem, Functions of bounded variation, Hausdorff measure of noncompactness, Volterra integral operator, Nemytskii operator## Abstract

In this paper we study the existence of a unique solution of a nonlinear integral equation in the space of bounded variation on an unbounded interval by using measure of noncompactness and Darbo fixed point theorem.

### Downloads

## References

R. P. Agarwal, D. O’Regan and P. J. Y. Wong, Positive solutions of differential, difference and integral equations, Kluwer Academic Publishers, Dordrecht, 1999.

J. Appell and P. P. Zabrejko, Continuity properties of the superposition operator, Preprint No. 131, Univ. Augsburg, 1986.

I. K. Argros, Quadratic equations and applications to Chandrasekhar’s and related equations, Bull. Austral. Math. Soc., 32, (1985), 275-292.

J. Banas and W. G. El-Sayed, Measures of noncompactness and solvability of an integral equation in the class of functions of locally bounded variation, J. Math. Anal. Appl. 167 (1) (1992), 133-151.

J. Banas and K. Goebel, Measures of noncompactness in Banach spaces, Lect. Notes in Math. 60, M. Dekker, New york and Basel 1980.

D. Bugajewski, On BV-solutions of some nonlinear integralequations,Integral Equations and Operator Theory, vol. 46, no. 4, pp. 387-398, 2003.

J. Caballero, J. Rocha and K. Sadarangani, Solvability of a Volterra integral equation of convolution type in the class of monotonic functions, Intern. Math. Journal, 4, no.1, (2001), 69-77.

G. Darbo, Punti untiti in transformazioni a condominio noncompatto, Rend. Sem. Mat. Univ. Padora 24 (1955) 84-92.

F. S. De Blasi, On a property of the unit sphere in Banach spaces, Bull. Math. Soc. Sci. Math. R. S. Roum. 21 (1977), 259–262.

M. M. El-Borai, W. G. El-Sayed & R. M. Bayomi, Solvability of non-linear integro-differential equation, Inter. J. Sc. & Eng. Res., Vol. 10, Issu. 7, July-2019, ISSN 2229-5518, pp. 1085-1093.

M. M. El-Borai, W. G. El-Sayed & F. N. Ghaffoori, On the solvability of nonlinear integral functional equation, Inter. J. Math. Tren. & Tech. (IJMTT), Vol. 34, No. 1, June 2016, 39-44.

M. M. El-Borai, W. G. El-Sayed & F. N. Ghaffoori, Existence Solution For a Fractional Nonlinear Integral Equation of Volterra Type, Aryabhatta J. M. & Inform., Vol. 08, Iss.-02, (Jul.-Dec. 2016), 1-15.

W. G. El-Sayed, M. M. El-Borai, M. M. metwali and N. I. Shemais, On the solvability of a nonlinear functional integral equations via measure of noncompactness in L P (RN ), J. Adv. Math. (JAM), Vol 19 (2020) ISSN: 2347-1921, 74-88.

W. G. El-Sayed, M. M. El-Borai, M. M. metwali and N. I. Shemais, An existence theorem for a nonlinear integral equation of Urysohn type in L P (RN ), Adv. Math. Sci. J. 9 (2020), no. 11, ISSN: 1857-8365, 9995-10005.

W. G. El-Sayed, M. M. El-Borai, M. M. metwali and N. I. Shemais, On Monotonic Solutions of Nonlinear Quadratic Integral Equation of Convolution Type, Case Studies J., ISSN (2305-509X)- Vol. 9, Issue 10-oct-2020, 78-87.

M. M. El-Borai, W. G. El-Sayed, A. M. Moter, Continuous Solutions of a Quadratic Integral Equation, Inter. J. Life Science and Math. (IJLSM), Vol. 2 (5)-4, (2015), 21-30.

W. G. El-Sayed, Nonlinear functional integral equations of convolution type, Portugaliae Mathematica 54 (4) (1997) 449-456.

W.G. El-Sayed, A.A. El-Bary, M.A. Darwish, Solvability of Urysohn integral equation, Appl. Math. Comput. 145 (2003) 487-493.

W. G. El-Sayed, On the Solvability of a Functional Integral Equation, East-West J. Math. Vol. 10 (2),( 2008) pp. 153-160.

W. G. El-Sayed, R. O. Abd El-Rahman, S. A. Abd El-Salam, A. A. El Shahawy, Bounded variation solutions of a functional integral equation in L1(R+), Int. J. Mech. Eng., ISSN: 0974-5823, Vol. 7 No. 2 February 2022, 2600-2605.

W. G. El-Sayed, R. O. Abd El-Rahman, S. A. Abd El-Salam, A. A. El Shahawy, On the existence of a bounded variation solution of a fractional integral equation in L1[0, T] due to the spread of COVID 19, J. Adv. Math. (JAM),Vol. 21 (2022) ISSN: 2347-1921, 107-115.

S. Hu, M. Khavanin and W. Zhuang, Integral equations arising in the kinetic theory of gases, Appl. Analysis, 34, (1989), 261-266.

M. A. krasnosel’ski, P. P. Zabrejko, J. I. Pustyl’nik and P. J. Sobolevskii, Integral operators in spaces of summable functions, Noordhoff, Leyden, (1976).

D. O’Regan and M. Meehan, Existence theory for nonlinear integral and integrodifferential equations, Kluwer Academic, Dordrecht, 1998.

P. P. Zabrejko, A. I. Koshelev, M. A. krasnosel’ski, S.G. Mikhlin, L. S. Rakovshchik and V. J. Stecenco, Integeral Equations, Noordhoff, Leyden, (1975).

## Downloads

## Published

## How to Cite

*JOURNAL OF ADVANCES IN MATHEMATICS*,

*21*, 182–191. https://doi.org/10.24297/jam.v21i.9317

## Issue

## Section

## License

Copyright (c) 2022 Wagdy El-Sayed, Ragab O. Abd El-Rahman, Sheren A. Abd El-Salam, Asmaa El Shahawy

This work is licensed under a Creative Commons Attribution 4.0 International License.

All articles published in *Journal of Advances in Linguistics* are licensed under a Creative Commons Attribution 4.0 International License.