Existence and Uniqueness of Mild Solutions for Fractional Integrodifferential Equations

V. Dhanapalan; M. Thamilselvan; M. Chandrasekaran

doi:doi:10.11648/j.acm.20140301.15

| Peer-Reviewed

Existence and Uniqueness of Mild Solutions for Fractional Integrodifferential Equations

V. Dhanapalan, M. Thamilselvan, M. Chandrasekaran

Published in Applied and Computational Mathematics (Volume 3, Issue 1)

Received: 30 January 2014 Accepted: Published: 10 March 2014

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

The aim of this paper is to prove the existence and uniqueness of mild solution of a class of l nonlinear fractional integrodifferential equations {█((d^q u(t))/(dt^q )+Au(t)=∫_0^t▒f(t,s,x(s) )ds+∫_0^t▒〖a(t-s)g(s,y(s) )ds, t∈[0,T],〗@u(0)=u_(o.) )┤ in a Banach space X, where 0

Published in	Applied and Computational Mathematics (Volume 3, Issue 1)
DOI	10.11648/j.acm.20140301.15
Page(s)	32-37
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Integrodifferential Equation, Fractional Equation, Mild Solution, Compact Semigroup, Krasnoselskii Theorem, Semi Group of Linear Operators

References

[1]	M. M. El-Borai and A. Debbouche, "On some fractional integrodifferential equations with analytic semigroups", International Journal of Contemporary Mathematical Sciences, vol.4, no.25-28, 2009, pp.1361-1371.
[2]	H-S. Ding, J. Liang and T-J. Xiao,"Positive almost automorphic solutions for a class of non-linear integral equations", Applicable Analysis, vol.88, no.2, 2009, pp.231-244.
[3]	H-S. Ding, T-J. Xiao and J. Liang., "Existence of positive almost automorphic solutions for a class of nonlinear integral equations", Nonlinear Analysis: Theory Methods & Applications, vol.70, no.6, 2009, pp.2216-2231.
[4]	Fang Li and G.M.N'Gue ́re ́kata, "Existence of uniqueness of mild solution for fractional integrodifferential equations", Advances of Difference Equations, vol.2010, 2010, Article ID 158789, 10 pages.
[5]	J. Liang, J.H. Liu and T-J.Xiao, "Nonlocal problems for integrodifferential equations", Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol.15, no.6, 2008, pp.815-824.
[6]	J. Liang, R. Nagel and T-J. Xiao, "Approximation theorems for the propagators of higher order abstract Cauchy problems", Transactions of the American Mathematical society, vol.360, no.4, 2008, pp. 1723-1739.
[7]	J. Liang and T-J. Xiao, "Semilinear integrodifferential equations with nonlocal initial conditions", Computers & Mathematics with Applications, vol.74, no.67, 2004, pp.863- 875.
[8]	T-J. Xiao and J. Liang, "The Cauchy Problem for Higher Order Abstract Differential Equations", Lecture Notes in Mathematics, vol.1701, 1998 , Springer, Berlin, Germany.
[9]	T-J. Xiao and J. Liang, "Approximations of Laplace transforms and integral semigroups", Journal of Functional Analysis, vol.172, 2000, pp.202-220.
[10]	T-J. Xiao and J. Liang, "Second order differential operators with Feller-Wentzell type boundary conditions", Journal of Functional Analysis, vol.254, 2008, pp.1467-1486.
[11]	T-J. Xiao and J. Liang, "Blow-up and global existence of solutions to integral equations with infinite delay in Banach spaces", Nonlinear Analysis: Theory, Methods & Applications, vol.71, no.12, 2009, pp.1442-1447.
[12]	T-J. Xiao, J. Liang and J. van Casteren, "Time dependent Desch-Schappacher type perturbations of Volterra integral equations", Integral Equations and Operator Theory, vol.44, no.4, 2002, pp.494 -506.
[13]	R.Hilfer, Ed., Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA. 2000.
[14]	J. Henderson and A. Ouahab,"Fractional functional differential inclusions with finite delay", Nonlinear Analysis : Theory, Methods & Applications, vol.70, no.5, 2009, pp.2091-2105.
[15]	V. Lakshmikantham, "Theory of fractional functional differential equations", Nonlinear Analysis: Theory, Methods & Applications, vol.69, no.10, 2008, pp.3337-3343.
[16]	V. Lakshmikantham and A.S. Vatsala, "Basic theory of fractional differential equations", Nonlinear Analysis: Theory, Methods & Applications, vol.69, no.8, 2008, pp.2677-2682.
[17]	H. Liu and J-C. Chang, "Existence for a class of partial differential equations with nonlocal conditions", Nonlinear Analysis: Theory, Methods & Applications, vol.70, no.9, 2009, pp.3076-3083.
[18]	K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley Interscience publication, John Wiley & Sons, New York, 1993.
[19]	G.M. N'Gue ́re ́kata, "A Cauchy problem for some fractional abstract differential equation with nonlocal conditions", Nonlinear Analysis :Theory, Methods & Applications, vol.70, no.5, 2009, pp.1873-1876.
[20]	G.M. Mophou and G.M. N'Gue ́re ́kata, "Existence of the mild solution for some fractional differential equations with nonlocal conditions", Semigroup Forum, vol.79, no.2, 2009, pp. 315-322.
[21]	G.M. Mophou and G.M. N'Gue ́re ́kata, "A note on a semi linear fractional differential equation of neutral type with infinite delay", Advances in Difference Equations, vol.2010, 2010, Article ID 674630, 8 pages.
[22]	X.-X. Zhu, "A Cauchy problem for abstract fractional differential equations with infinite delay", Communications in Mathematical Analysis, vol.6, no.1, 2009, pp.94-100.
[23]	M. M. El-Borai, "Some probability densities and fundamental solutions of fractional evolution equations", Chaos, Solitons and Fractals, vol.14, no.3, 2002, pp.433-440.
[24]	M. M. El-Borai, "On some stochastic fractional integrodifferential equations", Advances in Dynamical Systems and Applications, vol.1, no.1, 2006, pp.49-57.
[25]	J. Liang, J. van Casteren and T-J. Xiao, "Nonlocal Cauchy problems for semilinear evolution equations", Nonlinear Analysis: Theory, Methods & Applications, vol.50, no.2, 2002, pp.173-189.
[26]	J. Liang and T-J. Xiao, "Solvability of the Cauchy problems for infinite delay equations", Nonlinear Analysis: Theory, Methods & Applications, vol.58, no.3-4, 2004, pp.271-297.
[27]	A.Pazy, Semigroups of Linear Operators and Applications to partial Differential Equations, vol.44 of Applied Mathematical Sciences, Springer, New York, USA.1983.
[28]	T-J. Xiao and J. Liang, Existence of classical solutions to non autonomous nonlocal parabolic problems, Nonlinear Analysis: Theory, Methods & Applications, vol.63, no.5-7, 2005, pp.225-232.

Cite This Article

Plain Text BibTeX RIS

APA Style

V. Dhanapalan, M. Thamilselvan, M. Chandrasekaran. (2014). Existence and Uniqueness of Mild Solutions for Fractional Integrodifferential Equations. Applied and Computational Mathematics, 3(1), 32-37. https://doi.org/10.11648/j.acm.20140301.15

Copy | Download

ACS Style

V. Dhanapalan; M. Thamilselvan; M. Chandrasekaran. Existence and Uniqueness of Mild Solutions for Fractional Integrodifferential Equations. Appl. Comput. Math. 2014, 3(1), 32-37. doi: 10.11648/j.acm.20140301.15

Copy | Download

AMA Style

V. Dhanapalan, M. Thamilselvan, M. Chandrasekaran. Existence and Uniqueness of Mild Solutions for Fractional Integrodifferential Equations. Appl Comput Math. 2014;3(1):32-37. doi: 10.11648/j.acm.20140301.15

Copy | Download

@article{10.11648/j.acm.20140301.15,
  author = {V. Dhanapalan and M. Thamilselvan and M. Chandrasekaran},
  title = {Existence and Uniqueness of Mild Solutions for Fractional Integrodifferential Equations},
  journal = {Applied and Computational Mathematics},
  volume = {3},
  number = {1},
  pages = {32-37},
  doi = {10.11648/j.acm.20140301.15},
  url = {https://doi.org/10.11648/j.acm.20140301.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140301.15},
  abstract = {The aim of this paper is to prove the existence and uniqueness of mild solution of a class of l nonlinear fractional integrodifferential equations {█((d^q u(t))/(dt^q )+Au(t)=∫_0^t▒f(t,s,x(s) )ds+∫_0^t▒〖a(t-s)g(s,y(s) )ds,    t∈[0,T],〗@u(0)=u_(o.) )┤ in a Banach space X, where 0
											
												Copy
												|
												Download

TY  - JOUR
T1  - Existence and Uniqueness of Mild Solutions for Fractional Integrodifferential Equations
AU  - V. Dhanapalan
AU  - M. Thamilselvan
AU  - M. Chandrasekaran
Y1  - 2014/03/10
PY  - 2014
N1  - https://doi.org/10.11648/j.acm.20140301.15
DO  - 10.11648/j.acm.20140301.15
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 32
EP  - 37
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20140301.15
AB  - The aim of this paper is to prove the existence and uniqueness of mild solution of a class of l nonlinear fractional integrodifferential equations {█((d^q u(t))/(dt^q )+Au(t)=∫_0^t▒f(t,s,x(s) )ds+∫_0^t▒〖a(t-s)g(s,y(s) )ds,    t∈[0,T],〗@u(0)=u_(o.) )┤ in a Banach space X, where 0
											
												Copy
												|
												Download

Author Information

V. Dhanapalan

Government College of Technology, Coimbatore-641013.Tamilnadu, India
M. Thamilselvan

Thanthai Periyar Government Institute of Technology, Vellore-632002.Tamilnadu, India
M. Chandrasekaran

Higher College of Technology, Muscat, the Sultanate of Oman

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

V. Dhanapalan, M. Thamilselvan, M. Chandrasekaran. (2014). Existence and Uniqueness of Mild Solutions for Fractional Integrodifferential Equations. Applied and Computational Mathematics, 3(1), 32-37. https://doi.org/10.11648/j.acm.20140301.15

Copy | Download

ACS Style

V. Dhanapalan; M. Thamilselvan; M. Chandrasekaran. Existence and Uniqueness of Mild Solutions for Fractional Integrodifferential Equations. Appl. Comput. Math. 2014, 3(1), 32-37. doi: 10.11648/j.acm.20140301.15

Copy | Download

AMA Style

V. Dhanapalan, M. Thamilselvan, M. Chandrasekaran. Existence and Uniqueness of Mild Solutions for Fractional Integrodifferential Equations. Appl Comput Math. 2014;3(1):32-37. doi: 10.11648/j.acm.20140301.15

Copy | Download

@article{10.11648/j.acm.20140301.15,
  author = {V. Dhanapalan and M. Thamilselvan and M. Chandrasekaran},
  title = {Existence and Uniqueness of Mild Solutions for Fractional Integrodifferential Equations},
  journal = {Applied and Computational Mathematics},
  volume = {3},
  number = {1},
  pages = {32-37},
  doi = {10.11648/j.acm.20140301.15},
  url = {https://doi.org/10.11648/j.acm.20140301.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140301.15},
  abstract = {The aim of this paper is to prove the existence and uniqueness of mild solution of a class of l nonlinear fractional integrodifferential equations {█((d^q u(t))/(dt^q )+Au(t)=∫_0^t▒f(t,s,x(s) )ds+∫_0^t▒〖a(t-s)g(s,y(s) )ds,    t∈[0,T],〗@u(0)=u_(o.) )┤ in a Banach space X, where 0
									
										Copy
										|
										Download

TY  - JOUR
T1  - Existence and Uniqueness of Mild Solutions for Fractional Integrodifferential Equations
AU  - V. Dhanapalan
AU  - M. Thamilselvan
AU  - M. Chandrasekaran
Y1  - 2014/03/10
PY  - 2014
N1  - https://doi.org/10.11648/j.acm.20140301.15
DO  - 10.11648/j.acm.20140301.15
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 32
EP  - 37
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20140301.15
AB  - The aim of this paper is to prove the existence and uniqueness of mild solution of a class of l nonlinear fractional integrodifferential equations {█((d^q u(t))/(dt^q )+Au(t)=∫_0^t▒f(t,s,x(s) )ds+∫_0^t▒〖a(t-s)g(s,y(s) )ds,    t∈[0,T],〗@u(0)=u_(o.) )┤ in a Banach space X, where 0
									
										Copy
										|
										Download