Approximation of solutions for nonlinear functional integral equations

Lakshmi Narayan Mishra; Vijai Kumar Pathak; Dumitru Baleanu; Lakshmi Narayan Mishra; Vijai Kumar Pathak; Dumitru Baleanu

doi:10.3934/math.2022964

AIMS Mathematics

2022, Volume 7, Issue 9: 17486-17506. doi: 10.3934/math.2022964

Previous Article Next Article

Research article

Approximation of solutions for nonlinear functional integral equations

1.
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, Tamil Nadu, India
2.
Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, Ankara, Turkey
3.
Institute of Space Sciences, Magurele-Bucharest, Romania
4.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

Received: 28 April 2022 Revised: 11 July 2022 Accepted: 18 July 2022 Published: 28 July 2022
MSC : 45G10, 45M99, 47H08, 47H10

In this article, we consider a class of nonlinear functional integral equations, motivated by an equation that offers increasing evidence to the extant literature through replication studies. We investigate the existence of solution for nonlinear functional integral equations on Banach space $ C[0, 1] $. We use the technique of the generalized Darbo's fixed-point theorem associated with the measure of noncompactness (MNC) to prove our existence result. Also, we have given two examples of the applicability of established existence result in the theory of functional integral equations. Further, we construct an efficient iterative algorithm to compute the solution of the first example, by employing the modified homotopy perturbation (MHP) method associated with Adomian decomposition. Moreover, the condition of convergence and an upper bound of errors are presented.
- measure of noncompactness,
- nonlinear functional integral equation,
- fixed point theorem,
- modified homotopy perturbation
Citation: Lakshmi Narayan Mishra, Vijai Kumar Pathak, Dumitru Baleanu. Approximation of solutions for nonlinear functional integral equations[J]. AIMS Mathematics, 2022, 7(9): 17486-17506. doi: 10.3934/math.2022964

Related Papers:

Abstract

In this article, we consider a class of nonlinear functional integral equations, motivated by an equation that offers increasing evidence to the extant literature through replication studies. We investigate the existence of solution for nonlinear functional integral equations on Banach space $ C[0, 1] $. We use the technique of the generalized Darbo's fixed-point theorem associated with the measure of noncompactness (MNC) to prove our existence result. Also, we have given two examples of the applicability of established existence result in the theory of functional integral equations. Further, we construct an efficient iterative algorithm to compute the solution of the first example, by employing the modified homotopy perturbation (MHP) method associated with Adomian decomposition. Moreover, the condition of convergence and an upper bound of errors are presented.

References

[1]	M. A. Abdou, On the solution of linear and nonlinear integral equation, Appl. Math. Comput., 146 (2003), 857–871. https://doi.org/10.1016/S0096-3003(02)00643-4 doi: 10.1016/S0096-3003(02)00643-4
[2]	I. K. Argyros, Quadratic equations and applications to Chandrasekhar's and related equations, Bull. Aust. Math. Soc., 32 (1985), 275–292. https://doi.org/10.1017/S0004972700009953 doi: 10.1017/S0004972700009953
[3]	A. Aghajani, J. Banaś, Y. Jalilian, Existence of solutions for a class of nonlinear Volterra singular integral equations, Comput. Math. Appl., 62 (2011), 1215–1227. https://doi.org/10.1016/j.camwa.2011.03.049 doi: 10.1016/j.camwa.2011.03.049
[4]	A. Aghajani, Y. Jalilian, Existence and global attractivity of solutions of a nonlinear functional integral equation, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 3306–3312. https://doi.org/10.1016/j.cnsns.2009.12.035 doi: 10.1016/j.cnsns.2009.12.035
[5]	E. Alvarez, C. Lizama, Attractivity for functional Volterra integral equations of convolution type, J. Comput. Appl. Math., 301 (2016), 230–240. https://doi.org/10.1016/j.cam.2016.01.048 doi: 10.1016/j.cam.2016.01.048
[6]	J. Banaś, K. Balachandran, D. Julie, Existence and global attractivity of solutions of a nonlinear functional integral equation, Appl. Math. Comput., 216 (2010), 261–268. https://doi.org/10.1016/j.amc.2010.01.049 doi: 10.1016/j.amc.2010.01.049
[7]	J. Banaś, K. Goebel, Measure of noncompactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics, Vol. 60, New York: Dekker, 1980.
[8]	J. Banaś, L. Olszowy, Measures of noncompactness related to monotonicity, In: Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae, 2001, 13–23.
[9]	J. Banaś, B. Rzepka, On existence and asymptotic stability of solutions of a nonlinear integral equation, J. Math. Anal. Appl., 284 (2003), 165–173. https://doi.org/10.1016/S0022-247X(03)00300-7 doi: 10.1016/S0022-247X(03)00300-7
[10]	J. Banaś, B. Rzepka, On local attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation, Appl. Math. Comput., 213 (2009), 102–111. https://doi.org/10.1016/j.amc.2009.02.048 doi: 10.1016/j.amc.2009.02.048
[11]	D. Chalishajar, C. Ravichandran, S. Dhanalakshmi, R. Murugesu, Existence of fractional impulsive functional integro-differential equations in Banach spaces, Appl. Syst. Innov., 2 (2019), 1–17. https://doi.org/10.3390/asi2020018 doi: 10.3390/asi2020018
[12]	B. C. Dhage, Global attractivity results for nonlinear functional integral equations via a Krasnoselskii type fixed point theorem, Nonlinear Anal., 70 (2009), 2485–2493. https://doi.org/10.1016/j.na.2008.03.033 doi: 10.1016/j.na.2008.03.033
[13]	B. C. Dhage, Local asymptotic attractivity for nonlinear quadratic functional integral equations, Nonlinear Anal., 70 (2009), 1912–1922. https://doi.org/10.1016/j.na.2008.02.109 doi: 10.1016/j.na.2008.02.109
[14]	A. Deep, Deepmala, J. R. Roshan, Solvability for generalized nonlinear functional integral equations in Banach spaces with applications, J. Integral Equ. Appl., 33 (2021), 19–30. https://doi.org/10.1216/jie.2021.33.19 doi: 10.1216/jie.2021.33.19
[15]	A. Deep, Deepmala, C. Tunç, On the existence of solutions of some non-linear functional integral equations in Banach algebra with applications, Arab J. Basic Appl. Sci., 27 (2020), 279–286. https://doi.org/10.1080/25765299.2020.1796199 doi: 10.1080/25765299.2020.1796199
[16]	B. C. Dhage, S. B. Dhage, H. K. Pathak, A generalization of Darbo's fixed point theorem and local attractivity of generalized nonlinear functional integral equations, Differ. Equ. Appl., 7 (2015), 57–77. https://doi.org/10.7153/dea-07-05 doi: 10.7153/dea-07-05
[17]	B. C. Dhage, V. Lakshmikantham, On global existence and attractivity results for nonlinear functional integral equations, Nonlinear Anal., 72 (2010), 2219–2227. https://doi.org/10.1016/j.na.2009.10.021 doi: 10.1016/j.na.2009.10.021
[18]	A. M. A. El-Sayed, H. R. Ebead, On the solvability of a self-reference functional and quadratic functional integral equations, Filomat, 34 (2020), 129–141. https://doi.org/10.2298/FIL2001129E doi: 10.2298/FIL2001129E
[19]	R. C. Guerra, On the solution of a class of integral equations using new weighted convolutions, J. Integral Equ. Appl., 34 (2022), 39–58. https://doi.org/10.1216/jie.2022.34.39 doi: 10.1216/jie.2022.34.39
[20]	X. L. Hu, J. R. Yan, The global attractivity and asymptotic stability of solution of a nonlinear integral equation, J. Math. Anal. Appl., 321 (2006), 147–156. https://doi.org/10.1016/j.jmaa.2005.08.010 doi: 10.1016/j.jmaa.2005.08.010
[21]	K. Jothimani, K. Kaliraj, S. K. Panda, K. S. Nisar, C. Ravichandran, Results on controllability of non-densely characterized neutral fractional delay differential system, Evol. Equ. Control Theory, 10 (2021), 619–631. https://doi.org/10.3934/eect.2020083 doi: 10.3934/eect.2020083
[22]	K. Jangid, S. D. Purohit, R. Agarwal, On Gruss type inequality involving a fractional integral operator with a multi-index Mittag-Leffler function as a kernel, Appl. Math. Inf. Sci., 16 (2022), 269–276. https://doi.org/10.18576/amis/160214 doi: 10.18576/amis/160214
[23]	A. Karimi, K. Maleknejad, R. Ezzati, Numerical solutions of system of two-dimensional Volterra integral equations via Legendre wavelets and convergence, Appl. Numer. Math., 156 (2020), 228–241. https://doi.org/10.1016/j.apnum.2020.05.003 doi: 10.1016/j.apnum.2020.05.003
[24]	S. Karmakar, H. Garai, L. K. Dey, A. Chanda, Existence of solutions to non-linear quadratic integral equations via measure of non-compactness, Filomat, 36 (2022), 73–87. https://doi.org/10.2298/FIL2201073K doi: 10.2298/FIL2201073K
[25]	A. Karoui, A. Jawahdou, Existence and approximate $L^{p}$ and continuous solutions of nonlinear integral equations of the Hammerstein and Volterra types, Appl. Math. Comput., 216 (2010), 2077–2091. https://doi.org/10.1016/j.amc.2010.03.042 doi: 10.1016/j.amc.2010.03.042
[26]	K. Logeswari, C. Ravichandran, K. S. Nisar, Mathematical model for spreading of COVID-19 virus with the Mittag–Leffler kernel, Numer. Methods Partial Differ. Equ., 2020, 1–16. https://doi.org/10.1002/num.22652
[27]	L. N. Mishra, R. P. Agarwal, On existence theorems for some nonlinear functional-integral equations, Dyn. Syst. Appl., 25 (2016), 303–320.
[28]	L. N. Mishra, R. P. Agarwal, M. Sen, Solvability and asymptotic behavior for some nonlinear quadratic integral equation involving Erd$\acute{ e }$lyi-Kober fractional integrals on the unbounded interval, Progr. Fract. Differ. Appl., 2 (2016), 153–168. https://doi.org/10.18576/pfda/020301 doi: 10.18576/pfda/020301
[29]	K. Maleknejad, K. Nouri, R. Mollapourasl, Investigation on the existence of solutions for some nonlinear functional-integral equations, Nonlinear Anal., 71 (2009), e1575–e1578. https://doi.org/10.1016/j.na.2009.01.207 doi: 10.1016/j.na.2009.01.207
[30]	B. Matani, J. R. Roshan, Multivariate generalized Meir-Keeler condensing operators and their applications to systems of integral equations, J. Fixed Point Theory Appl., 22 (2020), 1–28. https://doi.org/10.1007/s11784-020-00820-6 doi: 10.1007/s11784-020-00820-6
[31]	B. Matani, J. R. Roshan, N. Hussain, An extension of Darbo's theorem via measure of non-compactness with its application in the solvability of a system of integral equations, Filomat, 33 (2019), 6315–6334. https://doi.org/10.2298/FIL1919315M doi: 10.2298/FIL1919315M
[32]	L. N. Mishra, M. Sen, On the concept of existence and local attractivity of solutions for some quadratic Volterra integral equation of fractional order, Appl. Math. Comput., 285 (2016), 174–183. https://doi.org/10.1016/j.amc.2016.03.002 doi: 10.1016/j.amc.2016.03.002
[33]	L. N. Mishra, M. Sen, R. N. Mohapatra, On existence theorems for some generalized nonlinear functional-integral equations with applications, Filomat, 31 (2017), 2081–2091. https://doi.org/10.2298/FIL1707081N doi: 10.2298/FIL1707081N
[34]	L. N. Mishra, H. M. Srivastava, M. Sen, Existence results for some nonlinear functional-integral equations in Banach algebra with applications, Int. J. Anal. Appl., 11 (2016), 1–10.
[35]	E. Najafi, Nyström-quasilinearization method and smoothing transformation for the numerical solution of nonlinear weakly singular Fredholm integral equations, J. Comput. Appl. Math., 368 (2020), 112538. https://doi.org/10.1016/j.cam.2019.112538 doi: 10.1016/j.cam.2019.112538
[36]	H. Nasiri, J. R. Roshan, M. Mursaleen, Solvability of system of Volterra integral equations via measure of noncompactness, Comput. Appl. Math., 40 (2021), 1–25. https://doi.org/10.1007/s40314-021-01552-0 doi: 10.1007/s40314-021-01552-0
[37]	K. S. Nisar, K. Jothimani, K. Kaliraj, C. Ravichandran, An analysis of controllability results for nonlinear Hilfer neutral fractional derivatives with non-dense domain, Chaos Solitons Fract., 146 (2021), 110915. https://doi.org/10.1016/j.chaos.2021.110915 doi: 10.1016/j.chaos.2021.110915
[38]	S. Noeiaghdam, M. A. F. Araghi, D. Sidorov, Dynamical strategy on homotopy perturbation method for solving second kind integral equations using the CESTAC method, J. Comput. Appl. Math., 411 (2022), 114226. https://doi.org/10.1016/j.cam.2022.114226 doi: 10.1016/j.cam.2022.114226
[39]	D. O'Regan, Existence results for nonlinear integral equations, J. Math. Anal. Appl., 192 (1995), 705–726. https://doi.org/10.1006/jmaa.1995.1199 doi: 10.1006/jmaa.1995.1199
[40]	Y. B. Pan, J. Huang, Extrapolation method for solving two-dimensional Volterral integral equations of the second kind, Appl. Math. Comput., 367 (2020), 124784. https://doi.org/10.1016/j.amc.2019.124784 doi: 10.1016/j.amc.2019.124784
[41]	V. K. Pathak, L. N. Mishra, Application of fixed point theorem to solvability for non-linear fractional Hadamard functional integral equations, Mathematics, 10 (2022), 1–16. https://doi.org/10.3390/math10142400 doi: 10.3390/math10142400
[42]	M. Rabbani, B. Zarali, Solution of Fredholm integro-differential equations system by modified decomposition method, J. Math. Comput. Sci., 5 (2012), 258–264.
[43]	M. Rabbani, New homotopy perturbation method to solve non-linear problems, J. Math. Comput. Sci., 7 (2013), 272–275.
[44]	M. Rabbani, Modified homotopy method to solve non-linear integral equations, Int. J. Nonlinear Anal. Appl., 6 (2015), 133–136. https://doi.org/10.22075/IJNAA.2015.262 doi: 10.22075/IJNAA.2015.262
[45]	M. Rabbani, R. Arab, Extension of some theorems to find solution of nonlinear integral equation and homotopy perturbation method to solve it, Math. Sci., 11 (2017), 87–94. https://doi.org/10.1007/s40096-017-0206-4 doi: 10.1007/s40096-017-0206-4
[46]	M. Rabbani, A. Deep, Deepmala, On some generalized non‑linear functional integral equations of two variables via measures of noncompactness and numerical method to solve it, Math. Sci., 15 (2021), 317–324. https://doi.org/10.1007/s40096-020-00367-0 doi: 10.1007/s40096-020-00367-0
[47]	J. R. Roshan, Existence of solutions for a class of system of functional integral equation via measure of noncompactness, J. Comput. Appl. Math., 313 (2017), 129–141. https://doi.org/10.1016/j.cam.2016.09.011 doi: 10.1016/j.cam.2016.09.011
[48]	S. K. Sahoo, P. O. Mohammed, B. Kodamasingh, M. Tariq, Y. S. Hamed, New fractional integral inequalities for convex functions pertaining to Caputo-Fabrizio operator, Fractal Fract., 6 (2022), 1–17. https://doi.org/10.3390/fractalfract6030171 doi: 10.3390/fractalfract6030171
[49]	H. M. Srivastava, S. K. Sahoo, P. O. Mohammed, B. Kodamasingh, Y. S. Hamed, New Riemann-Liouville fractional-order inclusions for convex functions via integral-valued setting associated with pseudo-order relations, Fractal Fract., 6 (2022), 1–17. https://doi.org/10.3390/fractalfract6040212 doi: 10.3390/fractalfract6040212
[50]	N. Valliammal, C. Ravichandran, Results on fractional neutral integro-differential systems with state-dependent delay in Banach spaces, Nonlinear Stud., 25 (2018), 159–171.
[51]	J. R. Wang, X. W. Dong, Y. Zhou, Existence, attractiveness and stability of solutions for quadratic Urysohn fractional integral equations, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 545–554. https://doi.org/10.1016/j.cnsns.2011.05.034 doi: 10.1016/j.cnsns.2011.05.034
[52]	X. Y. Zhang, A new strategy for the numerical solution of nonlinear Volterra integral equations with vanishing delays, Appl. Math. Comput., 365 (2020), 124608. https://doi.org/10.1016/j.amc.2019.124608 doi: 10.1016/j.amc.2019.124608
[53]	J. K. Xu, H. X. Wu, Z. Tan, Radial symmetry and asymptotic behaviors of positive solutions for certain nonlinear integral equations, J. Math. Anal. Appl., 427 (2015), 307–319. https://doi.org/10.1016/j.jmaa.2015.02.043 doi: 10.1016/j.jmaa.2015.02.043

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)