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Existence, Uniqueness and Stability of Solutions of a Variable-Order Nonlinear Integro-differential Equation in a Banach Space

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Abstract

This article studies some important results in a Banach space for non-discrete nonlinear integro-differential equations with variable order \(0<\sigma (\theta )<1\)

$$\begin{aligned}{} & {} D^{\sigma (\theta )}_{0,\theta } \vartheta (\theta ) =\eta (\theta ,\vartheta (\theta ))+\vartheta (\theta ) \int _{0}^{\theta } \kappa (\theta ,a,\vartheta (a)){\textrm{d}}a,\quad \theta \in \aleph =[0,\Theta ],\quad \Theta >0, \\{} & {} \vartheta (0)=\vartheta _0. \end{aligned}$$

The contraction mapping principle and Krasnoselskii fixed-point theorem are employed to investigate the results, and Ulam–Hyers definitions are used for stability theory. Further, we have discussed the maximal and minimal solutions with the continuation theorem for \(\sigma (\theta ) \rightarrow 1\).

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References

  1. Verma P, Tiwari S, Verma A (2023) Theoretical and numerical analysis of fractional order mathematical model on recent COVID-19 model using singular kernel. Proc Natl Acad Sci India Sect A 93:219–232

    Article  MathSciNet  MATH  Google Scholar 

  2. Abbasbandy S, Hashemi MS, Hashim I (2013) On convergence of homotopy analysis method and its application to fractional integro-differential equations. Quaest Math 36(1):93–105

    Article  MathSciNet  MATH  Google Scholar 

  3. Ghasemi M, Kajani Tavassoli M, Babolian E (2007) Application of He’s homotopy perturbation method to nonlinear integro-differential equations. Appl Math Comput 188(1):538–548

    MathSciNet  MATH  Google Scholar 

  4. MacCa RC (1977) An integro-differential equation with application in heat flow. Q Appl Math 35:1–19

    Article  MathSciNet  Google Scholar 

  5. Eslahchi MR, Dehghan M, Parvizi M (2014) Application of the collocation method for solving nonlinear fractional integro-differential equations. J Comput Appl Math 257:105–128

    Article  MathSciNet  MATH  Google Scholar 

  6. Malesza W, Macias M, Sierociuk D (2019) Analytical solution of fractional variable order differential equations. J Comput Appl Math 348:214–236

    Article  MathSciNet  MATH  Google Scholar 

  7. Lorenzo Carl F, Hartley Tom T (2002) Variable order and distributed order fractional operators. Nonlinear Dyn 29:57–98

    Article  MathSciNet  MATH  Google Scholar 

  8. Babaei A, Jafari H, Banihashemi S (2020) Numerical solution of variable order fractional nonlinear quadratic integro-differential equations based on the sixth-kind Chebyshev collocation method. J Comput Appl Math 377:112908

    Article  MathSciNet  MATH  Google Scholar 

  9. Hendi FA, Shammakh W, Al-badrani H (2019) Existence result and approximate solutions for quadratic integro-differential equations of fractional order. J King Saud Univ Sci 31(3):314–321

    Article  Google Scholar 

  10. Jiang J, Chen H, Guirao JLG, Cao D (2019) Existence of the solution and stability for a class of variable fractional order differential systems. Chaos Solitons Fractals 128:269–274

    Article  MathSciNet  MATH  ADS  Google Scholar 

  11. Bhrawy AH, Zaky MA (2016) Numerical algorithm for the variable-order Caputo fractional functional differential equation. Nonlinear Dyn 85:1815–1823

    Article  MathSciNet  MATH  Google Scholar 

  12. Xu Y, He Z (2013) Existence and uniqueness results for Cauchy problem of variable-order fractional differential equations. J Appl Math Comput 43:295–306

    Article  MathSciNet  MATH  Google Scholar 

  13. Das P, Rana S, Ramos H (2009) Homotopy perturbation method for solving Caputo-type fractional-order Volterra–Fredholm integro-differential equations. Comput Math Methods 1(5):e1047

    MathSciNet  Google Scholar 

  14. Das P, Ranaaand S, Ramos H (2020) A perturbation-based approach for solving fractional-order Volterra–Fredholm integro differential equations and its convergence analysis. Int J Comput Math 97(10):1994–2014

    Article  MathSciNet  MATH  Google Scholar 

  15. Kumar K, Chakravarthy Podila P, Das P, Ramos H (2021) A graded mesh refinement approach for boundary layer originated singularly perturbed time-delayed parabolic convection diffusion problems. Math Methods Appl Sci 44(16):12332–12350

    Article  MathSciNet  MATH  Google Scholar 

  16. Shakti D, Mohapatra J, Das P, Vigo-Aguiar J (2022) A moving mesh refinement based optimal accurate uniformly convergent computational method for a parabolic system of boundary layer originated reaction-diffusion problems with arbitrary small diffusion terms. J Comput Appl Math 404:113167

    Article  MathSciNet  MATH  Google Scholar 

  17. Das P (2019) An a posteriori based convergence analysis for a nonlinear singularly perturbed system of delay differential equations on an adaptive mesh. Numer Algorithms 81:465–487

    Article  MathSciNet  MATH  Google Scholar 

  18. Das P, Vigo-Aguiar J (2019) Parameter uniform optimal order numerical approximation of a class of singularly perturbed system of reaction diffusion problems involving a small perturbation parameter. J Comput Appl Math 354:533–544

    Article  MathSciNet  MATH  Google Scholar 

  19. Chandru M, Das P, Ramos H (2018) Numerical treatment of two-parameter singularly perturbed parabolic convection diffusion problems with non-smooth data. Math Methods Appl Sci 41(14):5359–5387

    Article  MathSciNet  MATH  Google Scholar 

  20. Das P, Das P, Ramos H (2015) Comparison of a priori and a posteriori meshes for singularly perturbed nonlinear parameterized problems. J Comput Appl Math 290:16–25

    Article  MathSciNet  MATH  Google Scholar 

  21. Das P, Natesan S (2014) Optimal error estimate using mesh equidistribution technique for singularly perturbed system of reaction–diffusion boundary-value problems. Appl Math Comput 249:265–277

    MathSciNet  MATH  Google Scholar 

  22. Das P, Natesan S (2013) A uniformly convergent hybrid scheme for singularly perturbed system of reaction–diffusion Robin type boundary-value problems. J Appl Math Comput 41:447–471

    Article  MathSciNet  MATH  Google Scholar 

  23. Das P, Natesan S (2012) Higher-order parameter uniform convergent schemes for Robin type reaction–diffusion problems using adaptively generated grid. Int J Comput Methods 09(04):1250052

    Article  MathSciNet  MATH  Google Scholar 

  24. Das P, Rana S, Vigo-Aguiar J (2020) Higher order accurate approximations on equidistributed meshes for boundary layer originated mixed type reaction diffusion systems with multiple scale nature. Appl Numer Math 148:79–97

    Article  MathSciNet  MATH  Google Scholar 

  25. Chandru M, Das P, Ramos H (2013) Numerical treatment of two-parameter singularly perturbed parabolic convection diffusion problems with non-smooth data. Math Methods Appl Sci 41(14):5359–5387

    Article  MathSciNet  MATH  Google Scholar 

  26. Das P (2018) A higher order difference method for singularly perturbed parabolic partial differential equations. J Differ Equa Appl 24(3):452–477

    Article  MathSciNet  MATH  Google Scholar 

  27. Chandru M, Prabha T, Das P, Shanthi V (2019) A numerical method for solving boundary and interior layers dominated parabolic problems with discontinuous convection coefficient and source terms. Differ Equa Dyn Syst 27:91–112

    Article  MathSciNet  MATH  Google Scholar 

  28. Das P, Rana S (2021) Theoretical prospects of fractional order weakly singular Volterra Integro differential equations and their approximations with convergence analysis. Math Methods Appl Sci 44(11):9419–9440

    Article  MathSciNet  MATH  Google Scholar 

  29. Das P, Rana S, Ramos H (2019) On the approximate solutions of a class of fractional order nonlinear Volterra integro-differential initial value problems and boundary value problems of first kind and their convergence analysis. J Comput Appl Math 404:113116

    Article  MathSciNet  MATH  Google Scholar 

  30. Das P, Rana S, Ramos H (2019) A perturbation-based approach for solving fractional-order Volterra–Fredholm integro differential equations and its convergence analysis. Int J Comput Math 97(10):1994–2014

    Article  MathSciNet  MATH  Google Scholar 

  31. Ahmadova A, Mahmudov NI (2021) Ulam–Hyers stability of Caputo type fractional stochastic neutral differential equations. Stat Probab Lett 168:108949

    Article  MathSciNet  MATH  Google Scholar 

  32. de Oliveira EC, da Sousa JVC (2018) Ulam–Hyers–Rassias stability for a class of fractional integro-differential equations. RM 73:111

    MathSciNet  MATH  Google Scholar 

  33. Srivastava HM, Mohammedd PO, Ryooe CS, Hamed YS (2021) Existence and uniqueness of a class of uncertain Liouville–Caputo fractional difference equations. J King Saud Univ Sci 33(6):101497

    Article  Google Scholar 

  34. Kvalheima MD, Revzen S (2021) Existence and uniqueness of global Koopman eigenfunctions for stable fixed points and periodic orbits. Physica D 425:132959

    Article  MathSciNet  MATH  Google Scholar 

  35. Moallem GR, Jafari H, Adem AR (2019) A numerical scheme to solve variable order diffusion-wave equations. Therm Sci 23(6):2063–207

    Article  Google Scholar 

  36. Ganji RM, Jafari H (2019) A numerical approach for multi-variable orders differential equations using Jacobi polynomials. Int J Appl Comput Math 5:34

    Article  MathSciNet  MATH  Google Scholar 

  37. Jafari H, Tajadodi H, Ganji RM (2019) A numerical approach for solving variable order differential equations based on Bernstein polynomials. Comput Math Methods 1(5):e1055

    Article  MathSciNet  Google Scholar 

  38. Ganji RM, Jafari H, Baleanu D (2020) A new approach for solving multi variable orders differential equations with Mittag–Leffler kernel. Chaos Solitons Fractals 130:109405

    Article  MathSciNet  MATH  Google Scholar 

  39. Tuan NH, Nemati S, Ganji RM, Jafari H (2020) Numerical solution of multi variable order fractional integro differential equations using the Bernstein polynomials. Eng Comput 38:139–147

    Article  Google Scholar 

  40. Ganji RM, Jafari H, Nemati S (2020) A new approach for solving integro-differential equations of variable order. J Comput Appl Math 379:112946

    Article  MathSciNet  MATH  Google Scholar 

  41. Ravichandran C, Logeswari K, Jarad F (2019) New results on existence in the framework of Atangana–Baleanu derivative for fractional integro-differential equations. Chaos Solitons Fractals 125:194–200

    Article  MathSciNet  MATH  ADS  Google Scholar 

  42. Ibrahim RW, Momani S (2017) On the existence and uniqueness of solutions of a class of fractional differential equations. J Math Anal Appl 334(1):1–10

    Article  MathSciNet  MATH  Google Scholar 

  43. Boulares H, Ardjouni A, Laskri Y (2017) Positive solutions for nonlinear fractional differential equations. Positivity 21:1201–1212

    Article  MathSciNet  MATH  Google Scholar 

  44. Jongen HT, Rückmann JJ, Shikhman V (2009) On stability of the feasible set of a mathematical problem with complementarity problems. SIAM J Optim. https://doi.org/10.1137/08072694X

    Article  MathSciNet  MATH  Google Scholar 

  45. Vivek D, Kanagarajan K, Elsayed EM, Laskri Y (2018) Existence and stability of fractional implicit differential equations with complex order. Results Fixed Point Theory Appl. https://api.semanticscholar.org/CorpusID:102504337

  46. Verma P, Kumar M (2020) An analytical solution with existence and uniqueness conditions for fractional integro differential equations. Int J Model Simul Sci Comput 11(05):2050045

    Article  Google Scholar 

  47. Verma P, Kumar M (2021) Analysis of a novel coronavirus (2019-nCOV) system with variable Caputo–Fabrizio fractional order. Chaos Solitons Fractals 142:110451

    Article  MathSciNet  Google Scholar 

  48. Verma P, Kumar M (2021) On the existence and stability of fuzzy CF variable fractional differential equation for COVID-19 epidemic. Eng Comput 38:1053–1064

    Article  Google Scholar 

  49. Verma P, Kumar M (2021) Hyers–Ulam stability and existence of solution for nonlinear variable fractional differential equations with singular kernel. Int J Appl Comput Math 7:147

    Article  MathSciNet  MATH  Google Scholar 

  50. Verma P, Kumar M, Shukla A (2022) Ulam–Hyers stability and analytical approach for m-dimensional Caputo space-time variable fractional order advection–dispersion equation. Int J Model Simul Sci Comput 13(01):2250004

  51. Verma P, Kumar M (2021) Positive solutions and stability of fuzzy Atangana–Baleanu variable fractional differential equation model for a novel coronavirus (COVID-19). Int J Model Simul Sci Comput 12(06):2150059

    Article  Google Scholar 

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Authors and Affiliations

  1. Faculty of Engineering & Technology, Siksha ‘O’ Anusandhan (Deemed to be University), Bhubaneswar, Odisha, 751030, India

    Pratibha Verma

  2. Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Prayagraj, Uttar Pradesh, 211004, India

    Surabhi Tiwari

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  1. Pratibha Verma
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Significant Statement: The main aim of this work is to provide an analysis of the nonlinear integro-differential equations using the Caputo derivative. This paper discusses theoretically the existence and uniqueness of solutions of such equations. Under some hypotheses, it has been proved that the unique solution of the equation exists. An important question is whether the solution we are getting is stable or not. We have taken this question into consideration and discussed the stability of the solution which is a very important aspect in many physical phenomena.

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Verma, P., Tiwari, S. Existence, Uniqueness and Stability of Solutions of a Variable-Order Nonlinear Integro-differential Equation in a Banach Space. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 93, 587–600 (2023). https://doi.org/10.1007/s40010-023-00852-w

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  • DOI: https://doi.org/10.1007/s40010-023-00852-w

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