Skip to main content
SpringerLink
Log in

Study on the integro-differential equations on \({C^1}({\mathbb {R}}_{+})\)

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

This paper provides sufficient conditions for the existence of a solution for some classes of integro-differential equations in unbounded domains. The investigation successfully applies the Darbo fixed point theorem by considering an appropriate measure of noncompactness on \(C^{1}({\mathbb {R}}_{+})\). Moreover, the paper contains some examples to illustrate the results in each type of the considered integro-differential equations. Further, some significant examples are solved by a numerical approach named the artificial small parameter method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Data Availability

Due to the nature of this research, there is no relevant data.

References

  • Aghajani A, Jalilian Y (2010) Existence and global attractivity of solutions of a nonlinear functional integral equation. Commun Nonlinear Sci Numer Simul 15:3306–3312

    Article  MathSciNet  MATH  Google Scholar 

  • Aghajani A, Banaś J, Sabzali N (2013) Some generalizations of Darbo fixed point theorem and applications. Bull Belg Math Soc Simon Stevin 20:345–358

    Article  MathSciNet  MATH  Google Scholar 

  • Aghajani A, O’Regan D, Shole Haghighi A (2015) Measure of noncompactness on \(L^{p}({\mathbb{R} }^{N})\) and applications. CUBO 17(1):85–97

    Article  MathSciNet  MATH  Google Scholar 

  • Arab R, Allahyari R, Shole Haghighi A (2017) Construction of measures of noncompactness of \(C^{k}(\Omega )\) and \(C_{0}^{k}\) and their application to functional integral-differential equations. Bull Iran Math Soc 43(1):53–67

    MATH  Google Scholar 

  • Banaś J, Goebel K (1980) Measures of noncompactness in Banach spaces. Lecture notes in pure and applied mathematics, vol 60. Marcel Dekker, New York

    MATH  Google Scholar 

  • Bungartz HJ, Zimmer S, Buchholz M, Pflüger D (2014) Global illumination in computer graphics. Modeling and simulation. Springer undergraduate texts in mathematics and technology. Springer, Berlin

    Google Scholar 

  • Darbo G (1955) Punti uniti in trasformazioni a codominio non compatto. Rend Sem Mat Univ Padova 24:84–92

    MathSciNet  MATH  Google Scholar 

  • Deep A, Deepmala, Hazarika B (2021) An existence result for Hadamard type two dimensional fractional functional integral equations via measure of noncompactness. Chaos Solitons Fractals 147:110874

    Article  MathSciNet  MATH  Google Scholar 

  • El-Sayed AMA, Ebead HR (2020) On the solvability of a self-reference functional and quadratic functional integral equations. Filomat 34:25

    Article  MathSciNet  MATH  Google Scholar 

  • Hazarika B, Srivastava HM, Arab R, Rabbani M (2018) Existence of solution for an infinite system of nonlinear integral equations via measure of noncompactness and homotopy perturbation method to solve it. J Comput Appl Math 343:341–352

    Article  MathSciNet  MATH  Google Scholar 

  • Hazarika B, Srivastava HM, Arab R, Rabbani M (2019) Application of simulation function and measure of noncompactness for solvability of nonlinear functional integral equations and introduction to an iteration algorithm to find solution. Appl Math Comput 360:131–146

    MathSciNet  MATH  Google Scholar 

  • Kijima M (2016) Stochastic processes with applications to finance. CRC Press, Boca Raton

    Book  MATH  Google Scholar 

  • Koyunbakan H, Shah K, Abdeljawad T (2023) Well-posedness of inverse Sturm–Liouville problem with fractional derivative. Qual Theory Dyn Syst 22:23

    Article  MathSciNet  MATH  Google Scholar 

  • Kuratowski K (1930) Sur les espaces complets. Fundam Math 15:301–309

    Article  MATH  Google Scholar 

  • Liao SJ (1995) An approximate solution technique not depending on small parameters: a special example. Int J Non-Linear Mech 30(3):371–380

    Article  MathSciNet  MATH  Google Scholar 

  • Liao S (2003) Beyond perturbation: introduction to the homotopy analysis method. Chapman and Hall/CRC, New York

    Book  Google Scholar 

  • Mishra LN, Pathak VK, Baleanu D (2022) Approximation of solutions for nonlinear functional integral equations. AIMS Math 7(9):17486–17506

    Article  MathSciNet  Google Scholar 

  • Muskhelishvili N (2011) Singular integral equations: boundary problems of function theory and their application to mathematical physics, 2nd edn. Dover Publications, New York

    Google Scholar 

  • Rabbani M, Deep A (2021) Deepmala, On some generalized non-linear functional integral equations of two variables via measures of noncompactness and numerical method to solve it. Math Sci 15:25

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Rabbani M, Das A, Hazarika B, Arab R (2020) Existence of solution for two dimensional nonlinear fractional integral equation by measure of noncompactness and iterative algorithm to solve it. J Comput Appl Math 370:112654

    Article  MathSciNet  MATH  Google Scholar 

  • Razani A, Figueiredo GM (2022) Existence of infinitely many solutions for an anisotropic equation using genus theory. Math Methods Appl Sci 20:20

    MathSciNet  Google Scholar 

  • Rubinstein A (1992) Stability of the numerical procedure for solution of singular integral equations on semi-infinite interval. application to fracture mechanics. Comput Struct 44(1–2):71–74

    Article  MATH  Google Scholar 

  • Saiedinezhad S (2019) On a measure of noncompactness in the Holder space \(C^{k,\gamma }(\Omega )\) and its application. J Comput Appl Math 346:566–571

    Article  MathSciNet  MATH  Google Scholar 

  • Shabbir S, Shah K, Abdeljawad T (2021) Stability analysis for a class of implicit fractional differential equations involving Atangana–Baleanu fractional derivative. Adv Differ Equ 20:20

    MathSciNet  MATH  Google Scholar 

  • Shah K, Abdalla B, Abdeljawad T, Gul R (2023) Analysis of multipoint impulsive problem of fractional-order differential equations. Bound Value Probl 20:20

    MathSciNet  Google Scholar 

  • Siraj Ahmad K, Shah K, Abdeljawad T, Abdalla B (2023) On the approximation of fractal-fractional differential equations using numerical inverse Laplace transform methods. Comput Model Eng Sci 20:20

    Google Scholar 

  • Tamimi H, Saiedinezhad S, Ghaemi MB (2022) The measure of noncompactness in a generalized coupled fixed point theorem and its application to an integro-differential system. J Comput Appl Math 413:114380

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

No funding was received to assist with the preparation of this manuscript. Moreover, The authors have no relevant financial or non-financial interests to disclose.

Author information

Author notes

  1. H. Tamimi and M. B. Ghaemi contributed equally to this work.

Authors and Affiliations

  1. School of Mathematics, Iran University of Science and Technology, 16846-13114, Tehran, Iran

    H. Tamimi, S. Saiedinezhad & M. B. Ghaemi

Authors
  1. H. Tamimi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Saiedinezhad
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. B. Ghaemi
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to the study conception and design. Detailed design of the problem were performed by SS and HT. The first draft of the manuscript was written by HT and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to S. Saiedinezhad.

Ethics declarations

Conflict of interest

The authors declare that they have no competing financial or non- financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Communicated by Hui Liang.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tamimi, H., Saiedinezhad, S. & Ghaemi, M.B. Study on the integro-differential equations on \({C^1}({\mathbb {R}}_{+})\). Comp. Appl. Math. 42, 93 (2023). https://doi.org/10.1007/s40314-023-02239-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-023-02239-4

Keywords

Mathematics Subject Classification

Search

Navigation