EXISTENCE AND UNIQUENESS OF SOLUTION OF DIFFERENTIAL EQUATION OF FRACTIONAL ORDER VIA S-ITERATION
Abstract
In this paper, we study the existence, uniqueness and other properties of solutions of differential equation of fractional order involving the Caputo fractional derivative. The tool employed in the analysis is based on application of S− iteration method. The study of qualitative properties in general required differential and integral inequalities, and here S−iteration method itself has equally important contribution to study various properties such as dependence on initial data, closeness of solutions and dependence on parameters and functions involved therein. Finally, we present an example in support of all proved results.
Keywords
Full Text:
PDFReferences
bibitem{a1} {sc R. Agarwal, D. O'Regan, {rm and} D. Sahu}: textit{Iterative construction of fixed
points of nearly asymptotically nonexpansive mappings}. Journal of
Nonlinear and Convex Analysis {bf 8}(2007), 61--79.
bibitem{a2} {sc Y. Atalan {rm and} V. Karakaya}: textit{Iterative solution of functional Volterra-Fredholm integral
equation with deviating argument}. Journal of Nonlinear and Convex
Analysis {bf 18}(4)(2017), 675--684.
bibitem{ak} {sc Y. Atalan {rm and} V. Karakaya}: textit{Stability of Nonlinear
Volterra-Fredholm Integro Differential Equation: A Fixed Point
Approach}. Creative Mathematics and Informatics {bf 26}
(3)(2017), 247--254.
bibitem{a3} {sc Y. Atalan {rm and} V. Karakaya}: textit{An example of data dependence result for the class of
almost contraction mappings}. Sahand Communications in
Mathematical Analysis(SCMA) {bf 17}1(2020), 139--155.
bibitem{a4} {sc Y. Atalan, F. Gürsoy {rm and} A. R. Khan}: textit{Convergence of
S-iterative method to a solution of Fredholm integral equation and
data dependency}. Facta Universitatis, Ser. Math. Inform. {bf
}4(2021), 685--694.
bibitem{b1} {sc V. Berinde {rm and} M. Berinde}: textit{The fastest Krasnoselskij iteration for approximating
fixed points of strictly pseudo-contractive mappings}. Carpathian
J. Math. {bf21}(1-2)(2005), 13--20.
bibitem{b2} {sc V. Berinde}: textit{ Existence and approximation of solutions of some
first order iterative differential equations}. Miskolc
Mathematical Notes {bf 11} 1 (2010), 13--26.
bibitem{c2} {sc C. E. Chidume}: textit{Iterative approximation of fixed points of
Lipschitz pseudocontractivemaps}. Proceedings of the American
Mathematical Society {bf 129} 8 (2001), 2245--2251.
bibitem{c3} {sc R. Chugh, V. Kumar {rm and} S. Kumar}: textit{Strong Convergence of a new three
step iterative scheme in Banach spaces}. American Journal of
Computational Mathematics {bf 2}(2012), 345--357.
bibitem{d0} {sc M. B. Dhakne {rm and} H. L. Tidke}: textit{Existence and uniqueness of solutions of nonlinear mixed integrodifferential equations
with nonlocal condition in banach spaces}. Electronic Journal of
Differential Equations, {bf 2011} 31(2011), 1--10.
bibitem{d1} {sc M. Dobritoiu}: textit{An integral equation with modified argument}. Stud. Univ. Babes¸-Bolyai Math. XLIX
(3) (2004), 27--33.
bibitem{d3} {sc M. Dobritoiu}: textit{System of integral equations with modified argument}. Carpathian J.
Math. {bf 24} 2(2008), 26--36.
bibitem{d4} {sc M. Dobritoiu}: textit{A nonlinear Fredholm integral equation}. TJMM {bf 1}(1-2)(2009), 25--32.
bibitem{d5} {sc M. Dobritoiu}: textit{A class of nonlinear integral equations}. TJMM {bf 4} 2 (2012), 117--123.
bibitem{d6} {sc M. Dobritoiu}: textit{The approximate solution of a Fredholm integral
equation}. International Journal of Mathematical Models and
Methods in Applied Sciences, Vol.8(2014), pp. 173-180.
bibitem{d7} {sc M. Dobritoiu}: textit{The existence and uniqueness of the solution of a nonlinear
fredholmvolterra integral equation with modified argument via
geraghty contractions}. Mathematics 2020, 9, 29,
https://dx.doi.org/ 10.3390/math9010029.
bibitem{Geji} {sc V. Daftardar-Gejji {rm and} H. Jafari}: textit{Analysis of a system of nonautonomous fractional differential
equations involving Caputo derivatives}. J. Math. Anal. Appl. {bf
} (2007), 1026-1033.
bibitem{g1} {sc F. Gürsoy {rm and} V. Karakaya}: textit{Some Convergence and Stability Results
for Two New Kirk Type Hybrid Fixed Point Iterative Algorithms}.
Journal of Function Spaces doi:10.1155/2014/684191, 2014.
bibitem{g2} {sc F. Gürsoy, V. Karakaya {rm and} B. E. Rhoades}: textit{Some Convergence and Stability
Results for the Kirk Multistep and Kirk-SP Fixed Point Iterative
Algorithms}. Abstract and Applied Analysis
doi:10.1155/2014/806537, 2014.
bibitem{h} {sc E. Hacioglu, F. Gürsoy, S. Maldar, Y. Atalan, {rm and} G.
V. Milovanovic}: textit{Iterative approximation of fixed points
and applications to two-point second-order boundary value problems
and to machine learning}. Applied Numerical Mathematics, {bf 167}
(2021), 143--172.
bibitem{h1} {sc N. Hussain, A. Rafiq, B. Damjanovi$acute{mbox{c}}$ {rm and} R. Lazovi$acute{mbox{c}}$}: textit{On rate of convergence
of various iterative schemes}. Fixed Point Theory and Applications
{bf 1} (2011), 1--6.
bibitem{i1} {sc S. Ishikawa}: textit{Fixed points by a new iteration method}. Proceedings
of the American Mathematical Society {bf 449} (1974), 147--150.
bibitem{k1} {sc S. M. Kang, A. Rafiq {rm and} Y. C. Kwun}: textit{Strong convergence for hybrid $S-$ iteration
scheme}. Journal of Applied Mathematics (2013), Article ID
, 4 Pages, http://dx.doi.org/10.1155/2013/705814.
bibitem{k2} {sc S. H. Khan}:textit{ A Picard-Mann hybrid iterative process}. Fixed Point Theory
and Applications {bf 1} (2013), 1--10.
bibitem{KST} {sc A. A. Kilbas, H. M. Srivastava {rm and} J. J. Trujillo}:
textit{Theory and Applications of Fractional Differential
Equations}. {bf 204} Elsevier, Amsterdam, 2006.
bibitem{m1} {sc S. Maldar, F. Gürsoy, Y. Atalan {rm and} M. Abbas}: textit{On a three-step
iteration process for multivalued Reich- Suzuki type
a-nonexpansive and contractive mappings}. Journal of Applied
Mathematics and Computing, (2021), 1--21.
bibitem{m2} {sc S. Maldar}: textit{Iterative algorithms of generalized nonexpansive
mappings and monotone operators with application to convex
minimization problem}. Journal of Applied Mathematics and
Computing, (2021), 1--28.
bibitem{P} {sc I. Podlubny}: textit{Fractional Differential Equations}. Academic Press, New York, 1999.
bibitem{p1} {sc B. G. Pachpatte}: textit{On Fredholm type integrodifferential equation}. Amkang Journal
of Mathematics {bf 39} 1(2008), 85--94.
bibitem{p2} {sc B. G. Pachpatte}: textit{On higher order Volterra-Fredholm integrodifferential
equation}. Fasciculi Mathematici 47(2007), 35--48.
bibitem{s1} {sc D. R. Sahu}: textit{Applications of the S-iteration process to
constrained minimization problems and split feasibility problems}.
Fixed Point Theory {bf 12} 1(2011), 187--204.
bibitem{s2} {sc D. R. Sahu {rm and} A. Petrus¸el}: textit{Strong convergence of iterative
methods by strictly pseudocontractive mappings in Banach spaces}.
Nonlinear Analysis: Theory, Methods & Applications {bf 74}
(2011), 6012--6023.
bibitem{s3} {sc S. Soltuz {rm and} T. Grosan}: textit{Data dependence for Ishikawa iteration when dealing with
contractive-like operators}. Fixed Point Theory Appl, 2008,
(2008). https://doi.org/10.1155/2008/242916.
DOI: https://doi.org/10.22190/FUMI220118001T
Refbacks
- There are currently no refbacks.
ISSN 0352-9665 (Print)