On the Nonlinear Impulsive Volterra-Fredholm Integrodifferential Equations

Document Type : Research Paper

Authors

1 Department of Mathematics, Shivaji University, Kolhapur-416 004, Maharashtra, India.

2 Shivaji University, Kolhapur.

3 Department of Applied Mathematics, Imecc-Unicamp, 13083-859, Campinas, SP, Brazil.

Abstract

In this paper, we investigate existence and uniqueness of solutions of nonlinear Volterra-Fredholm impulsive integrodifferential equations. Utilizing theory of Picard operators we examine data dependence of solutions on initial conditions and on nonlinear functions involved in integrodifferential equations. Further, we extend the integral inequality for piece-wise continuous functions to mixed case and apply it to investigate the dependence of solution on initial data through $\epsilon$-approximate solutions. It is seen that the uniqueness and dependency results got by means of integral inequity requires less restrictions on the functions involved in the equations than that required through Picard operators theory.

Keywords

[1] V. Lakshmikantham, D.D. Bainov and P.S. Simeonov, Theory of impulsive differential equations, World Scientific Singapore, 1989.
[2] A. M. Samoilenko and N. A. Perestyuk, Impulsive differential equations, World Scientific Singapore, 1995.
[3] J.H. Liu, Nonlinear impulsive evolution equations, Dyn. Contin. Discrete Impul. Syst. 6(1) (1999) 77–85.
[4] J. Wang, Y. Zhou and M. Medved, Picard and weakly Picard operators technique for nonlinear differential equations in Banach spaces, J. Math. Anal. Appl. 389(1) (2012) 261–274.
[5] E. Hern´andez and D. O’regan, Existence results for a class of abstract impulsive differential equations, Bull. Aust. Math. Soc. 87 (2013) 366–385.
[6] J. Wang, Mi. Feckan and Y. Zhoud, On the stability of first order impulsive evolution equations, Opuscula Mathematica, 34(3) (2014) 639–657.
[7] X. Haoa and L. Liua, Mild solution of semilinear impulsive integrodifferential evolution equation in Banach spaces, Math. Meth. Appl. Sci. 40(13) (2017) 4832–4841.
[8] A. Zada, U. Riaz and F.U. Khan, Hyers–Ulam stability of impulsive integral equations, Bollettino dell’Unione Matematica Italiana (2018) 1–15.
[9] J. Wang and Y. Zhang, Existence and stability of solutions to nonlinear impulsive differential equations in βnormed spaces, Elect. J. Diff. Equ. 2014(83) (2014) 1–10.
[10] K.D. Kucche and P.U. Shikhare, On impulsive delay integrodifferential equations with integral impulses, Mediterranean Journal of Mathematics, 17(4) (2020) 1–22.
[11] M. Frigon and D. O’regan, Existence results for first-order impulsive differential equations, J. Math. Anal. Appl. 193 (1995) 96–113.
[12] J. Wang, Mi. Feckan and Y. Zhoud, Ulam’s type stability of impulsive ordinary differential equations, J. Math. Anal. Appl. 395 (2012) 258–264.
[13] A. Anguraj and M.M. Arjunan, Existence and uniqueness of mild and classical solutions of impulsive evolution equations, Elect. J. Diff. Equ. 2005(111) (2005), 1–8.
[14] V. Muresan, Existence, uniqueness and data dependence for the solutions of some integro-differential equations of mixed type in Banach space, Zeits. Anal. Anwend. 23(1) (2004) 205–216.
[15] M. Campiti, Second-order differential operators with non-local Ventcel’s boundary conditions, Constr. Math. Anal. 2 (4) (2019) 144–152.
[16] C. Park, S. Yun, J.R. Lee and D.Y. Shin, Set-valued additive functional equations, Constr. Math. Anal. 2(2) (2019) 89–97.
[17] I. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math. 26(1) (2010) 103–107.
[18] I. Rus, Ulam stability of ordinary differential equations, Stud. Univ. Babe¸s-Bolyai Math. 54(4) (2009) 125–133.
[19] K. D. Kucche and M. B. Dhakne, On existence results and qualitative properties of mild solution of semilinear mixed Volterra–Fredholm functional integrodifferential equations in Banach spaces, Appl. Math. Comput. 219 (2013) 10806–10816.
[20] K. D. Kucche and M. B. Dhakne, Existence of solution via integral inequality of Volferra-Fredholm neutral functional integrodifferential equations with infinite delay, Int. J. Diff. Eq. Article ID 784956 (2014) 13 pages.
[21] K. D. Kucche and P. U. Shikhare, Ulam–Hyers stability of integrodifferential equations in Banach spaces via Pachpatte’s inequality, Asian–European J. Math. 11(2) 1850062 (2018) 19 pages.
[22] K. D. Kucche and P. U. Shikhare, Ulam stabilities for nonlinear Volterra–Fredholm delay integrodifferential equations, Int. J. Nonlinear Anal. Appl. 9(2) (2018) 145–159.
[23] K. D. Kucche and P. U. Shikhare, Ulam stabilities via Pachpatte’s inequality for Volterra–Fredholm delay integrodifferential equations in Banach spaces, Note Mate. 38(1) (2018) 67–82.
[24] K.D. Kucche and P.U. Shikhare, Ulam stabilities for nonlinear Volterra delay integrodifferential equations, J. Contemp. Math. Anal. 54(5) 276–287.
[25] B.G. Pachpatte, Inequalities for Differential and Integral Equations, Academic Press, New York, 1998.
[26] B.G. Pachpatte, Integral and Finite Difference Inequalities and Applications, North-Holland Mathematics Studies, 205, Elsevier Science, B.V., Amsterdam, 2006.
[27] D.D. Bainov and S.G. Hristova, Integral inequalities of Gronwall type for piece-wise continuous functions, J. Appl. Math. Stoch. Anal. 10(1) (1997) 89–94.
[28] I.A. Rus, Picard operators and well-posedness of fixed point problems, Stud. Univ. Babe¸s-Bolyai Math. 52(3) (2007) 147–156.
[29] D. Otrocol and V. Ilea, Qualitative properties of a functional differential equation, Elect. J. Qual. Theory Diff. Equ. 47 (2014) 1–8.
[30] A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, 1983.
Volume 13, Issue 1
March 2022
Pages 523-537
  • Receive Date: 10 March 2020
  • Revise Date: 13 May 2020
  • Accept Date: 14 May 2020