On nonlinear implicit fractional differential equations without compactness
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Authors
Samia Bushnaq
- Department of Science, Princess Sumaya University for Technology, Amman 11941, Jordan.
Wajid Hussain
- Department of Mathematics, University of Qurtuba, Peshawar, Khyber Pakhtunkhwa, Pakistan.
Kamal Shah
- Department of Mathematics, University of Malakand, Chakdara, Dir(L), Khyber Pakhtunkhwa, Pakistan.
Abstract
The main purpose of this research paper is to develop some sufficient
conditions for the existence of solution of a nonlinear problem of
implicit fractional differential equations (IFDEs) with boundary conditions, using
prior estimate method. The distinction of the method applied here
is, it does not require compactness of the operator. This idea is
the result of motivation from the book of O'Regan and other [R. P. Agarwal, D. O'Regan, Y. J. Cho, Y.-Q. Chen, Taylor and Francis Group, New York, (2006)].
Devising the respective conditions, we also developed some conditions
for Hyers-Ulam type stability to the solution of the said problem.
To justify the relevant results a suitable example is provided.
Share and Cite
ISRP Style
Samia Bushnaq, Wajid Hussain, Kamal Shah, On nonlinear implicit fractional differential equations without compactness, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 10, 5528--5539
AMA Style
Bushnaq Samia, Hussain Wajid, Shah Kamal, On nonlinear implicit fractional differential equations without compactness. J. Nonlinear Sci. Appl. (2017); 10(10):5528--5539
Chicago/Turabian Style
Bushnaq, Samia, Hussain, Wajid, Shah, Kamal. "On nonlinear implicit fractional differential equations without compactness." Journal of Nonlinear Sciences and Applications, 10, no. 10 (2017): 5528--5539
Keywords
- Implicit fractional differential equations
- Brouwer degree
- topological degree theory
- boundary conditions
- Banach contraction theorem
- Lebesgue dominated convergence theorem
MSC
References
-
[1]
S. Abbas, M. Benchohra, G. M. N'Guérékata, Topics in fractional differential equations, Springer, New York (2012)
-
[2]
R. P. Agarwal, D. Baleanu, S. Rezapour, S. Salehi , The existence of solutions for some fractional finite difference equations via sum boundary conditions, Adv. Difference Equ., 2014 (2014), 16 pages.
-
[3]
R. P. Agarwal, M. Benchohra, S. Hamani , Boundary value problems for fractional differential equations, Georgian Math. J., 16 (2009), 401–411.
-
[4]
R. P. Agarwal, D. O’Regan, Y. J. Cho, Y.-Q. Chen, Topological Degree Theory and its Applications, Taylor and Francis Group, New York (2006)
-
[5]
Z. Bai , On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Anal., 72 (2010), 916–924.
-
[6]
D. Baleanu, R. P. Agarwal, H. Khan, R. A. Khan, H. Jafari , On the existence of solution for fractional differential equations of order \(3 < \delta\leq 4\) , Adv. Difference Equ., 2015 (2015), 9 pages.
-
[7]
D. Baleanu, R. P. Agarwal, H. Mohammad, S. Rezapour, Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces, Bound. Value Probl., 2013 (2013), 8 pages.
-
[8]
M. Benchohra, J. R. Graef, S. Hamani, Existence results for boundary value problems with non-linear fractional differential equations, Appl. Anal., 87 (2008), 851–863.
-
[9]
M. Benchohra, S. Hamani, S. K. Ntouyas, Boundary value problems for differential equations with fractional order, Surv. Math. Appl., 3 (2008), 1–12.
-
[10]
M. Benchohra, J. E. Lazreg, Existence results for nonlinear implicit fractional differential equations, Surv. Math. Appl., 9 (2014), 79–92.
-
[11]
K. Diethelm, N. J. Ford, Analysis of Fractional Differential Equations, J. Math. Anal. Appl., 265 (2002), 229–248.
-
[12]
C. S. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett., 23 (2010), 1050–1055.
-
[13]
D. H. Hyers , On the stability of the linear functional equations, Proc. Nat. Acad. Sci., 27 (1941), 222–224.
-
[14]
F. Isaia, On a nonlinear integral equation without compactness, Acta Math. Univ. Comenian., 75 (2006), 233–240.
-
[15]
S.-M. Jung , Hyers-Ulam Stability of linear differential equations of first order , Appl. Math. Lett., 17 (2004), 1135–1140.
-
[16]
R. A. Khan, M. Rehman, J. Henderson , Existence and uniqueness of solutions for nonlinear fractional differential equations with integral boundary conditions, Fract. Differ. Calc., 1 (2011), 29–43.
-
[17]
R. A. Khan, K. Shah, Existence and uniqueness of solutions to fractional order multi-point boundary value problems, Commun. Appl. Anal., 19 (2015), 515–526.
-
[18]
A. A. Kilbas, O. I. Marichev, S. G. Samko, Fractional Integrals and Derivatives , (Theory and Applications), Gordon and Breach, Switzerland (1993)
-
[19]
V. Lakshmikantham, S. Leela, J. V. Devi , Theory of Fractional Dynamic Systems, CSP, UK (2009)
-
[20]
L. Lv, J.-R. Wang, W. Wei, Existence and uniqueness results for fractional differential equations with boundary value conditions, Opuscula Math., 31 (2011), 629–643.
-
[21]
R. L. Magin, Fractional calculus in bioengineering-part 2, Crit. Rev. Biomed. Eng., 32 (2004), 105–193.
-
[22]
J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, Am. Math. Soc., United States of America (1979)
-
[23]
K. S. Miller, B. Ross, An Introduction to the fractional Calculus and fractional Differential Equations, Wiley, New York (1993)
-
[24]
N. Nyamoradi, D. Baleanu, R. P. Agarwal , Existence and uniqueness of positive solutions to fractional boundary value problems with nonlinear boundary conditions, Adv. Difference Equ., 2013 (2013), 11 pages.
-
[25]
N. Nyamoradi, D. Baleanu, R. P. Agarwal, On a Multipoint Boundary Value Problem for a Fractional Order Differential Inclusion on an Infinite Interval , Adv. Math. Phys., 2013 (2013), 9 pages.
-
[26]
I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, , San Diego (1999)
-
[27]
T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300.
-
[28]
T. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math., 62 (2000), 23–130.
-
[29]
W. Soedel , Vibrations of Shells and Plates, Dekker, New York (2004)
-
[30]
I. M. Stamova, Mittag-Leffler stability of impulsive differential equations of fractional order, Quart. Appl. Math., 73 (2015), 525–535.
-
[31]
S. P. Timoshenko, Theory of Elastic Stability, McGraw-Hill Book Co., New York (1961)
-
[32]
J. C. Trigeassou, N. Maamri, J. Sabatier, A. Oustaloup, A Lyapunov approach to the stability of fractional differential equations, Signal Processing, 91 (2011), 437–445.
-
[33]
S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publishers, New York (1960)
-
[34]
J.-R. Wang, X.-Z. Li , Ulam-Hyers stability of fractional Langevin equations, Appl. Math. Comput., 258 (2015), 72–83.
-
[35]
J.-R. Wang, L. Lv, Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ., 2011 (2011), 10 pages.
-
[36]
J.-R. Wang, Y. Zhou, W. Wei , Study in fractional differential equations by means of topological degree methods , Numer. Func. Anal. Optim., 33 (2012), 216–238.
-
[37]
Y. Zhao, S. Sun, Z. Han, Q.-P. Li , The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 2086–2097.