Research Article
BibTex RIS Cite

Gecikmeli lineer olmayan bir Volterra integral denkleminin çözümü

Year 2017, Volume: 21 Issue: 6, 1367 - 1376, 01.12.2017
https://doi.org/10.16984/saufenbilder.305632

Abstract

Bu makalede, Ullah ve Arshad (SpringerPlus (2016)5:1616)
tarafından tanımlanan iterasyon metodunun basitleştirilmiş hali olan bir iteratif
dizisinin gecikmeli lineer olmayan bir Volterra integral denkleminin çözümüne kuvvetli
yakınsadığı gösterilmiştir. Dahası bu integral denklemin çözümü için bir veri
bağımlılığı sonucu ispatlanmıştır.

References

  • L. Cadariu, L. Gavruta and P. Gavruta. (2012). Weighted space method for the stability of some nonlinear equation, Appl. Anal. Discrete Math. 6(1), pp. 126-139.
  • L. P. Castro and R. C. Guerra. (2013). Hyers-Ulam-Rassias stability of Volterra integral equations within weighted spaces. Libertas Math. (new series), 33(2), pp. 21-35.
  • K. Ullah and M. Arshad. (2016). On different results for the new three step iteration process in Banach spaces. SpringerPlus, 5(1), pp. 1616.
  • M. Ertürk, F. Gürsoy, V. Karakaya, M. Başarır and A. Şahin. (2017). Some convergence and data dependence results by a simpler and faster iterative scheme. Appl. Comput. Math. submitted.
  • E. M. Picard. (1890). Memorie sur la theorie des equations aux derivees partielles et la methode des approximations successives. J. Math. Pure Appl. 6, pp. 145-210.
  • W. R. Mann. (1953). Mean value methods in iterations, Proc. Amer. Math. Soc. 4(3), pp. 506-510.
  • S. Ishikawa. (1974). Fixed point by a new iteration method. Proc. Amer. Math. Soc. 41(1), pp. 147-150.
  • M. A. Noor. (2000). New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 251, pp. 217-229.
  • R. P. Agarwal, D. O'Regan and D.R. Sahu. (2007). Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 8, pp. 61-79.
  • D. R. Sahu. (2011). Applications of S iteration process to constrained minimization problems and split feasibility problems. Fixed Point Theory, 12(1), pp. 187-204.
  • R. Chugh, V. Kumar and S. Kumar. (2012). Strong convergence of a new three step iterative scheme in Banach space. Amer. J. Comput. Math. 2, pp. 345-357.
  • F. Gürsoy and V. Karakaya. (2014). A Picard-S hybrid type iteration method for solving a differential equation with retarded argument. arXiv:1403.2546v2 [math. FA], https://arxiv.org/abs/1403.2546v2.
  • N. Kadıoğlu and I. Yıldırım. (2015) Approximating fixed points of nonexpansive mappings by a faster iteration process. J.Adv. Math. Stud. 8(2), pp. 257-264.
  • D. Thakur, B. S. Thakur and M. Postolache. (2014). New iteration schme for numerical reckoning fixed points of nonexpansive mapping. J. Inequal. Appl. 2014:1, 5 pages.
  • V. Karakaya, N. E. H. Bouzara, K. Doğan and Y. Atalan. (2015). On different results for a new two-step iteration method under weak-contraction mapping in Banach spaces. arXiv:1507.00200v1 [maths.FA], https://arxiv.org/abs/1507.00200vl.
  • M. Abbas and T. Nazır. (2014). A new faster iteration process applied to constrained minimization and feasibility problems. Mat. Vesnik, 66(2), pp. 223-234.
  • I. Karahan and M. Özdemir. (2013). A general iterative method for approximation of fixed points and their applications. Adv. Fixed Point Theory, 3(3), pp. 510-526.
  • Ş. M. Şoltuz and T. Grosan. (2008). Data dependence for Ishikawa iteration when dealing contractive like ope rators. Fixed Point Theory Appl. 2008: 242916, 7 pages.

On the solution of a nonlinear Volterra integral equation with delay

Year 2017, Volume: 21 Issue: 6, 1367 - 1376, 01.12.2017
https://doi.org/10.16984/saufenbilder.305632

Abstract

In this paper, we show that the
iterative sequence which is a simplified form of the iteration method
introduced by Ullah and Arshad (SpringerPlus, (2016)5:1616),
is
convergent strongly to the solution of a nonlinear Volterra integral equation
with delay in a complete metric space. Furthermore, we prove a data dependence
result for the solution of this integral equation.

References

  • L. Cadariu, L. Gavruta and P. Gavruta. (2012). Weighted space method for the stability of some nonlinear equation, Appl. Anal. Discrete Math. 6(1), pp. 126-139.
  • L. P. Castro and R. C. Guerra. (2013). Hyers-Ulam-Rassias stability of Volterra integral equations within weighted spaces. Libertas Math. (new series), 33(2), pp. 21-35.
  • K. Ullah and M. Arshad. (2016). On different results for the new three step iteration process in Banach spaces. SpringerPlus, 5(1), pp. 1616.
  • M. Ertürk, F. Gürsoy, V. Karakaya, M. Başarır and A. Şahin. (2017). Some convergence and data dependence results by a simpler and faster iterative scheme. Appl. Comput. Math. submitted.
  • E. M. Picard. (1890). Memorie sur la theorie des equations aux derivees partielles et la methode des approximations successives. J. Math. Pure Appl. 6, pp. 145-210.
  • W. R. Mann. (1953). Mean value methods in iterations, Proc. Amer. Math. Soc. 4(3), pp. 506-510.
  • S. Ishikawa. (1974). Fixed point by a new iteration method. Proc. Amer. Math. Soc. 41(1), pp. 147-150.
  • M. A. Noor. (2000). New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 251, pp. 217-229.
  • R. P. Agarwal, D. O'Regan and D.R. Sahu. (2007). Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 8, pp. 61-79.
  • D. R. Sahu. (2011). Applications of S iteration process to constrained minimization problems and split feasibility problems. Fixed Point Theory, 12(1), pp. 187-204.
  • R. Chugh, V. Kumar and S. Kumar. (2012). Strong convergence of a new three step iterative scheme in Banach space. Amer. J. Comput. Math. 2, pp. 345-357.
  • F. Gürsoy and V. Karakaya. (2014). A Picard-S hybrid type iteration method for solving a differential equation with retarded argument. arXiv:1403.2546v2 [math. FA], https://arxiv.org/abs/1403.2546v2.
  • N. Kadıoğlu and I. Yıldırım. (2015) Approximating fixed points of nonexpansive mappings by a faster iteration process. J.Adv. Math. Stud. 8(2), pp. 257-264.
  • D. Thakur, B. S. Thakur and M. Postolache. (2014). New iteration schme for numerical reckoning fixed points of nonexpansive mapping. J. Inequal. Appl. 2014:1, 5 pages.
  • V. Karakaya, N. E. H. Bouzara, K. Doğan and Y. Atalan. (2015). On different results for a new two-step iteration method under weak-contraction mapping in Banach spaces. arXiv:1507.00200v1 [maths.FA], https://arxiv.org/abs/1507.00200vl.
  • M. Abbas and T. Nazır. (2014). A new faster iteration process applied to constrained minimization and feasibility problems. Mat. Vesnik, 66(2), pp. 223-234.
  • I. Karahan and M. Özdemir. (2013). A general iterative method for approximation of fixed points and their applications. Adv. Fixed Point Theory, 3(3), pp. 510-526.
  • Ş. M. Şoltuz and T. Grosan. (2008). Data dependence for Ishikawa iteration when dealing contractive like ope rators. Fixed Point Theory Appl. 2008: 242916, 7 pages.
There are 18 citations in total.

Details

Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Aynur Şahin

Zeynep Kalkan

Hakan Arısoy This is me

Publication Date December 1, 2017
Submission Date April 11, 2017
Acceptance Date August 22, 2017
Published in Issue Year 2017 Volume: 21 Issue: 6

Cite

APA Şahin, A., Kalkan, Z., & Arısoy, H. (2017). On the solution of a nonlinear Volterra integral equation with delay. Sakarya University Journal of Science, 21(6), 1367-1376. https://doi.org/10.16984/saufenbilder.305632
AMA Şahin A, Kalkan Z, Arısoy H. On the solution of a nonlinear Volterra integral equation with delay. SAUJS. December 2017;21(6):1367-1376. doi:10.16984/saufenbilder.305632
Chicago Şahin, Aynur, Zeynep Kalkan, and Hakan Arısoy. “On the Solution of a Nonlinear Volterra Integral Equation With Delay”. Sakarya University Journal of Science 21, no. 6 (December 2017): 1367-76. https://doi.org/10.16984/saufenbilder.305632.
EndNote Şahin A, Kalkan Z, Arısoy H (December 1, 2017) On the solution of a nonlinear Volterra integral equation with delay. Sakarya University Journal of Science 21 6 1367–1376.
IEEE A. Şahin, Z. Kalkan, and H. Arısoy, “On the solution of a nonlinear Volterra integral equation with delay”, SAUJS, vol. 21, no. 6, pp. 1367–1376, 2017, doi: 10.16984/saufenbilder.305632.
ISNAD Şahin, Aynur et al. “On the Solution of a Nonlinear Volterra Integral Equation With Delay”. Sakarya University Journal of Science 21/6 (December 2017), 1367-1376. https://doi.org/10.16984/saufenbilder.305632.
JAMA Şahin A, Kalkan Z, Arısoy H. On the solution of a nonlinear Volterra integral equation with delay. SAUJS. 2017;21:1367–1376.
MLA Şahin, Aynur et al. “On the Solution of a Nonlinear Volterra Integral Equation With Delay”. Sakarya University Journal of Science, vol. 21, no. 6, 2017, pp. 1367-76, doi:10.16984/saufenbilder.305632.
Vancouver Şahin A, Kalkan Z, Arısoy H. On the solution of a nonlinear Volterra integral equation with delay. SAUJS. 2017;21(6):1367-76.