Araştırma Makalesi
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Yıl 2019, Cilt: 48 Sayı: 4, 1092 - 1109, 08.08.2019

Öz

Kaynakça

  • [1] B. Ahmad and J.J. Nieto, Riemann–Laiouville fractional differential equations with fractional boundary conditions, Fixed Point Theor, 13 (2), 329–336, 2012.
  • [2] B. Ahmad and S. Sivasundaram, On four–point nonlocal boundary value problems of nonlinear integro–differential equations of fractional order, Appl. Math. Comput. 217, 480–487, 2010.
  • [3] N. Ahmad, Z. Ali, K. Shah, A. Zada and G. Ur Rahman, Analysis of implicit type nonlinear dynamical problem of impulsive fractional differential equations, Complexity, 2018, 1–15, 2018.
  • [4] Z. Ali, A. Zada and K. Shah, Existence and stability analysis of three point boundary value problem, Int. J. Appl. Comput. Math. 2017, DOI:10.1007/s40819–017–0375–8.
  • [5] G.A. Anastassiou, On right fractional calculus, Chaos Solitons Fractals, 42 (1), 365– 376, 2009.
  • [6] D. Baleanu, Z.B. Güvenc and J.A.T. Machado, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York, 2010.
  • [7] M. Benchohra and S. Bouriah, Existence and Stability Results for Nonlinear Boundary Value Problem for Implicit Differential Equations of Fractional Order, Moroccan J. Pure Appl. Anal. 1 (1), 22–37, 2015.
  • [8] M. Benchohra, S. Hamani and S.K. Ntouyas, Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal. 71, 2391– 2396, 2009.
  • [9] A. Browder, Mathematical Analysis: An Introduction, New York, Springer-Verlag, 1996.
  • [10] M. El-Shahed and J.J. Nieto, Nontrivial solutions for a nonlinear multi–point boundary value problem of fractional order, Comput. Math. Appl. 59, 3438–3443, 2010.
  • [11] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
  • [12] E. Hiffer (ed.), Application of Fractional Calculus in Physics, Word Scientific, Singapore, 2000.
  • [13] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A, 27 (4), 222–224, 1941.
  • [14] D.H. Hyers, G. Isac and T.M. Rassias, Stability of Functional Equations in Several Variables, Birkhäiuser, Boston, 1998.
  • [15] J.K. Hale and S.M.V. Lunel, Introduction to Functional Differential Equations, in: Applied Mathematicals Sciences series, Springer–Verlag, New York, 99, 1993.
  • [16] R.W. Ibrahim, Generalized Ulam–Hyers stability for fractional differential equations, Int. J. Math. 23 (5), 9 pages, 2012.
  • [17] S.M. Jung, Hyers–Ulam stability of linear differential equations of first order, Appl. Math. Lett. 19, 854–858, 2006.
  • [18] R.A. Khan and K. Shah, K. Existence and uniqueness of solutions to fractional order multi-point boundary value problems, Commun. Appl. Anal. 19, 515–526, 2015.
  • [19] A.A. Kilbas, H.M. Srivasta and J.J. Trujilllo, Theory and Application of Fractional Differential Equations, North-Holland Mathematics Studies, 24, North-Holand, Amsterdam, 2006.
  • [20] Y.H. Lee and K.W. Jun, A Generalization of the Hyers–Ulam–Rassias stability of Pexider equation, J. Math. Anal. Appl. 246, 627–638, 2000.
  • [21] T. Li and A. Zada, Connections between Hyers–Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces, Adv. Difference Equ. 2016 (1), 1–8.
  • [22] T. Li, A. Zada and S. Faisal, Hyers–Ulam stability of nth order linear differential equations, J. Nonlinear Sci. Appl. 9, 2070–2075, 2016.
  • [23] J.T. Machado, V. Kiryakova and F. Mainardi, Recent history of fractional calculus, Commun. Nonlin. Sci. Numer. Simul. 16 (3), 1140–1153, 2011.
  • [24] R. Metzler and K. Joseph, Boundary value problems for fractional diffusion equations, Phys. A, 278 (1), 107–125, 2000.
  • [25] M. Obloza, Hyers stability of the linear differential equation, Rocznik Nauk-Dydakt. Prace Mat. 13, 259–270, 1993.
  • [26] K.B. Oldham, Fractional differential equations in electrochemistry, Adv. Eng. Softw. 41, 9–12, 2010.
  • [27] M.D. Ortigueira, Fractional Calculus for Scientists and Engineers, Lecture Notes in Electrical Engineering, 84, Springer, Dordrecht, 2011.
  • [28] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • [29] T.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (2), 297–300, 1978.
  • [30] T.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta. Appl. Math. 62, 23–130, 2000.
  • [31] F.A. Rihan, Numerical Modeling of Fractional-Order Biological Systems, Abstr. Appl. Anal. 2013, 11 pages, 2013.
  • [32] I.A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math. 26, 103-107, 2010.
  • [33] J. Sabatier, O.P. Agrawal and J.A.T. Machado, Advances in fractional calculus, Dordrecht, Springer, 2007.
  • [34] K. Shah and R.A. Khan, Existence and uniqueness of positive solutions to a coupled system of nonlinear fractional order differential equations with anti periodic boundary conditions, Differ. Equ. Appl. 7 (2), 245–262, 2015.
  • [35] K. Shah, N. Ali and R.A. Khan, Existence of positive solution to a class of fractional differential equations with three point boundary conditions, Math. Sci. Lett. 5 (3), 291–296, 2016.
  • [36] K. Shah,H. Khalil and R.A. Khan, Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations, Chaos Solitons Fractals, 77, 240–246, 2015.
  • [37] K. Shah, H. Khalil and R.A. Khan, Upper and lower solutions to a coupled system of nonlinear fractional differential equations, Prog. Fract. Differ. Appl. 1 (1), 1–8, 2016.
  • [38] X. Su and L. Liu, Existence of solution for boundary value problem of nonlinear fractional differential equation, Appl. Math. 22 (3), 291–298, 2007.
  • [39] S. Tang, A. Zada, S. Faisal, M.M.A. El-Sheikh and T. Li, Stability of higher order nonlinear impulsive differential equations, J. Nonlinear Sci. Appl. 9, 4713–4721, 2016.
  • [40] V.E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of particles, Fields and Media, Springer, Heidelberg, Higher Education Press, Beijing, 2010.
  • [41] S.M. Ulam, A Collection of the Mathematical Problems, Interscience, New York, 1960.
  • [42] B.M. Vintagre, I. Podlybni, A. Hernandez and V. Feliu, Some approximations of fractional order operators used in control theory and applications, Fract. Calc. Appl. Anal. 3 (3), 231–248, 2000.
  • [43] J. Wang, L. Lv and W. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ. 63, 1–10, 2011.
  • [44] J.R. Wang, Y. Zhou and M. Feckan, Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comput. Math. Appl. 64 (10), 3389–3405, 2012.
  • [45] B. Xu, J. Brzdek and W. Zhang, Fixed point results and the Hyers–Ulam stability of linear equations of higher orders, Pacific J. Math. 273, 483–498, 2015.
  • [46] A. Zada, S. Faisal and Y. Li, On the Hyers-Ulam Stability of First Order Impulsive Delay Differential Equations, J. Func. Spac. 2016, 6 pages, 2016.
  • [47] A. Zada, O. Shah and R. Shah, Hyers-Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems, Appl. Math. Comput. 271, 512–518, 2015.

Ulam stability results for the solutions of nonlinear implicit fractional order differential equations

Yıl 2019, Cilt: 48 Sayı: 4, 1092 - 1109, 08.08.2019

Öz

In this manuscript, we study the existence and uniqueness of solution for a class of fractional order boundary value problem (FBVP) for implicit fractional differential equations with Riemann-Liouville derivative. Furthermore, we investigate different kinds of Ulam stability such as Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability for the proposed problem. The concerned analysis is carried out through using classical technique of nonlinear functional analysis. The main results are illustrated by providing a couple of examples

Kaynakça

  • [1] B. Ahmad and J.J. Nieto, Riemann–Laiouville fractional differential equations with fractional boundary conditions, Fixed Point Theor, 13 (2), 329–336, 2012.
  • [2] B. Ahmad and S. Sivasundaram, On four–point nonlocal boundary value problems of nonlinear integro–differential equations of fractional order, Appl. Math. Comput. 217, 480–487, 2010.
  • [3] N. Ahmad, Z. Ali, K. Shah, A. Zada and G. Ur Rahman, Analysis of implicit type nonlinear dynamical problem of impulsive fractional differential equations, Complexity, 2018, 1–15, 2018.
  • [4] Z. Ali, A. Zada and K. Shah, Existence and stability analysis of three point boundary value problem, Int. J. Appl. Comput. Math. 2017, DOI:10.1007/s40819–017–0375–8.
  • [5] G.A. Anastassiou, On right fractional calculus, Chaos Solitons Fractals, 42 (1), 365– 376, 2009.
  • [6] D. Baleanu, Z.B. Güvenc and J.A.T. Machado, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York, 2010.
  • [7] M. Benchohra and S. Bouriah, Existence and Stability Results for Nonlinear Boundary Value Problem for Implicit Differential Equations of Fractional Order, Moroccan J. Pure Appl. Anal. 1 (1), 22–37, 2015.
  • [8] M. Benchohra, S. Hamani and S.K. Ntouyas, Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal. 71, 2391– 2396, 2009.
  • [9] A. Browder, Mathematical Analysis: An Introduction, New York, Springer-Verlag, 1996.
  • [10] M. El-Shahed and J.J. Nieto, Nontrivial solutions for a nonlinear multi–point boundary value problem of fractional order, Comput. Math. Appl. 59, 3438–3443, 2010.
  • [11] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
  • [12] E. Hiffer (ed.), Application of Fractional Calculus in Physics, Word Scientific, Singapore, 2000.
  • [13] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A, 27 (4), 222–224, 1941.
  • [14] D.H. Hyers, G. Isac and T.M. Rassias, Stability of Functional Equations in Several Variables, Birkhäiuser, Boston, 1998.
  • [15] J.K. Hale and S.M.V. Lunel, Introduction to Functional Differential Equations, in: Applied Mathematicals Sciences series, Springer–Verlag, New York, 99, 1993.
  • [16] R.W. Ibrahim, Generalized Ulam–Hyers stability for fractional differential equations, Int. J. Math. 23 (5), 9 pages, 2012.
  • [17] S.M. Jung, Hyers–Ulam stability of linear differential equations of first order, Appl. Math. Lett. 19, 854–858, 2006.
  • [18] R.A. Khan and K. Shah, K. Existence and uniqueness of solutions to fractional order multi-point boundary value problems, Commun. Appl. Anal. 19, 515–526, 2015.
  • [19] A.A. Kilbas, H.M. Srivasta and J.J. Trujilllo, Theory and Application of Fractional Differential Equations, North-Holland Mathematics Studies, 24, North-Holand, Amsterdam, 2006.
  • [20] Y.H. Lee and K.W. Jun, A Generalization of the Hyers–Ulam–Rassias stability of Pexider equation, J. Math. Anal. Appl. 246, 627–638, 2000.
  • [21] T. Li and A. Zada, Connections between Hyers–Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces, Adv. Difference Equ. 2016 (1), 1–8.
  • [22] T. Li, A. Zada and S. Faisal, Hyers–Ulam stability of nth order linear differential equations, J. Nonlinear Sci. Appl. 9, 2070–2075, 2016.
  • [23] J.T. Machado, V. Kiryakova and F. Mainardi, Recent history of fractional calculus, Commun. Nonlin. Sci. Numer. Simul. 16 (3), 1140–1153, 2011.
  • [24] R. Metzler and K. Joseph, Boundary value problems for fractional diffusion equations, Phys. A, 278 (1), 107–125, 2000.
  • [25] M. Obloza, Hyers stability of the linear differential equation, Rocznik Nauk-Dydakt. Prace Mat. 13, 259–270, 1993.
  • [26] K.B. Oldham, Fractional differential equations in electrochemistry, Adv. Eng. Softw. 41, 9–12, 2010.
  • [27] M.D. Ortigueira, Fractional Calculus for Scientists and Engineers, Lecture Notes in Electrical Engineering, 84, Springer, Dordrecht, 2011.
  • [28] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • [29] T.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (2), 297–300, 1978.
  • [30] T.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta. Appl. Math. 62, 23–130, 2000.
  • [31] F.A. Rihan, Numerical Modeling of Fractional-Order Biological Systems, Abstr. Appl. Anal. 2013, 11 pages, 2013.
  • [32] I.A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math. 26, 103-107, 2010.
  • [33] J. Sabatier, O.P. Agrawal and J.A.T. Machado, Advances in fractional calculus, Dordrecht, Springer, 2007.
  • [34] K. Shah and R.A. Khan, Existence and uniqueness of positive solutions to a coupled system of nonlinear fractional order differential equations with anti periodic boundary conditions, Differ. Equ. Appl. 7 (2), 245–262, 2015.
  • [35] K. Shah, N. Ali and R.A. Khan, Existence of positive solution to a class of fractional differential equations with three point boundary conditions, Math. Sci. Lett. 5 (3), 291–296, 2016.
  • [36] K. Shah,H. Khalil and R.A. Khan, Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations, Chaos Solitons Fractals, 77, 240–246, 2015.
  • [37] K. Shah, H. Khalil and R.A. Khan, Upper and lower solutions to a coupled system of nonlinear fractional differential equations, Prog. Fract. Differ. Appl. 1 (1), 1–8, 2016.
  • [38] X. Su and L. Liu, Existence of solution for boundary value problem of nonlinear fractional differential equation, Appl. Math. 22 (3), 291–298, 2007.
  • [39] S. Tang, A. Zada, S. Faisal, M.M.A. El-Sheikh and T. Li, Stability of higher order nonlinear impulsive differential equations, J. Nonlinear Sci. Appl. 9, 4713–4721, 2016.
  • [40] V.E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of particles, Fields and Media, Springer, Heidelberg, Higher Education Press, Beijing, 2010.
  • [41] S.M. Ulam, A Collection of the Mathematical Problems, Interscience, New York, 1960.
  • [42] B.M. Vintagre, I. Podlybni, A. Hernandez and V. Feliu, Some approximations of fractional order operators used in control theory and applications, Fract. Calc. Appl. Anal. 3 (3), 231–248, 2000.
  • [43] J. Wang, L. Lv and W. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ. 63, 1–10, 2011.
  • [44] J.R. Wang, Y. Zhou and M. Feckan, Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comput. Math. Appl. 64 (10), 3389–3405, 2012.
  • [45] B. Xu, J. Brzdek and W. Zhang, Fixed point results and the Hyers–Ulam stability of linear equations of higher orders, Pacific J. Math. 273, 483–498, 2015.
  • [46] A. Zada, S. Faisal and Y. Li, On the Hyers-Ulam Stability of First Order Impulsive Delay Differential Equations, J. Func. Spac. 2016, 6 pages, 2016.
  • [47] A. Zada, O. Shah and R. Shah, Hyers-Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems, Appl. Math. Comput. 271, 512–518, 2015.
Toplam 47 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Matematik
Yazarlar

Zeeshan Ali 0000-0001-6846-9516

Akbar Zada 0000-0002-2556-2806

Kamal Shah 0000-0002-8851-4844

Yayımlanma Tarihi 8 Ağustos 2019
Yayımlandığı Sayı Yıl 2019 Cilt: 48 Sayı: 4

Kaynak Göster

APA Ali, Z., Zada, A., & Shah, K. (2019). Ulam stability results for the solutions of nonlinear implicit fractional order differential equations. Hacettepe Journal of Mathematics and Statistics, 48(4), 1092-1109.
AMA Ali Z, Zada A, Shah K. Ulam stability results for the solutions of nonlinear implicit fractional order differential equations. Hacettepe Journal of Mathematics and Statistics. Ağustos 2019;48(4):1092-1109.
Chicago Ali, Zeeshan, Akbar Zada, ve Kamal Shah. “Ulam Stability Results for the Solutions of Nonlinear Implicit Fractional Order Differential Equations”. Hacettepe Journal of Mathematics and Statistics 48, sy. 4 (Ağustos 2019): 1092-1109.
EndNote Ali Z, Zada A, Shah K (01 Ağustos 2019) Ulam stability results for the solutions of nonlinear implicit fractional order differential equations. Hacettepe Journal of Mathematics and Statistics 48 4 1092–1109.
IEEE Z. Ali, A. Zada, ve K. Shah, “Ulam stability results for the solutions of nonlinear implicit fractional order differential equations”, Hacettepe Journal of Mathematics and Statistics, c. 48, sy. 4, ss. 1092–1109, 2019.
ISNAD Ali, Zeeshan vd. “Ulam Stability Results for the Solutions of Nonlinear Implicit Fractional Order Differential Equations”. Hacettepe Journal of Mathematics and Statistics 48/4 (Ağustos 2019), 1092-1109.
JAMA Ali Z, Zada A, Shah K. Ulam stability results for the solutions of nonlinear implicit fractional order differential equations. Hacettepe Journal of Mathematics and Statistics. 2019;48:1092–1109.
MLA Ali, Zeeshan vd. “Ulam Stability Results for the Solutions of Nonlinear Implicit Fractional Order Differential Equations”. Hacettepe Journal of Mathematics and Statistics, c. 48, sy. 4, 2019, ss. 1092-09.
Vancouver Ali Z, Zada A, Shah K. Ulam stability results for the solutions of nonlinear implicit fractional order differential equations. Hacettepe Journal of Mathematics and Statistics. 2019;48(4):1092-109.