Presence and diversity of positive solutions for a Caputotype fractional order nonlinear differential equation with an advanced argument
Volume 23, Issue 3, pp 230244
http://dx.doi.org/10.22436/jmcs.023.03.06
Publication Date: November 11, 2020
Submission Date: July 10, 2020
Revision Date: September 07, 2020
Accteptance Date: October 09, 2020
Authors
Md. Asaduzzaman
 Department of Mathematics, Islamic University, Kushtia, Bangladesh.
Adem Kilicman
 Department of Mathematics, Universiti Putra Malaysia, Selangor, Malaysia.
Md. Zulfikar Ali
 Department of Mathematics, University of Rajshahi, Rajshahi, Bangladesh.
Abstract
This article aims to construct the presence and diversity principles of minimum one or two positive solutions for a Caputotype fractionalorder nonlinear differential equation (CFONLDE for short) with an advanced argument under threepoint boundary value conditions (BVCs for short). GuoKrasnoselskii's fixed point theorem and Fixedpoint index theory in cone spaces are used to analyze this article. First, the Green's function of the corresponding boundary value problem for a linear fractional differential equation with an advanced argument has been established. Next, several essential properties of that Green's function have been proved. Finally, in cone spaces, some novel presence and diversity principles of minimum of one or two positive solutions for a CFONLDE with an advanced argument are obtained. To support the analytic proof, some particular examples are included.
Share and Cite
ISRP Style
Md. Asaduzzaman, Adem Kilicman, Md. Zulfikar Ali, Presence and diversity of positive solutions for a Caputotype fractional order nonlinear differential equation with an advanced argument, Journal of Mathematics and Computer Science, 23 (2021), no. 3, 230244
AMA Style
Asaduzzaman Md., Kilicman Adem, Ali Md. Zulfikar, Presence and diversity of positive solutions for a Caputotype fractional order nonlinear differential equation with an advanced argument. J Math Comput SCIJM. (2021); 23(3):230244
Chicago/Turabian Style
Asaduzzaman, Md., Kilicman, Adem, Ali, Md. Zulfikar. "Presence and diversity of positive solutions for a Caputotype fractional order nonlinear differential equation with an advanced argument." Journal of Mathematics and Computer Science, 23, no. 3 (2021): 230244
Keywords
 CFONLDE with an advanced argument
 threepoint boundary value conditions
 GuoKrasnoselskii fixed point theorem
 fixedpoint index theory
 positive solution
MSC
References

[1]
H. Afshari, M. Sajjadmanesh, D. Baleanu, Existence and uniqueness of positive solutions for a new class of coupled system via fractional derivatives, Adv. Differ. Equ., 2020 (2020), 18 pages

[2]
R. P. Agarwal, D. O’Regan, S. Stanek, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, J. Math. Anal. Appl., 371 (2010), 5768

[3]
R. P. Agarwal, D. O’Regan, P. J. Y. Wong, Positive Solutions of Differential, Difference and Integral Equations,, Springer, Dordrecht (1999)

[4]
B. Ahmad, J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with threepoint boundary conditions, Comput. Math. Appl., 58 (2009), 18381843

[5]
M. Asaduzzaman, M. Z. Ali, Existence of positive solution to the boundary value problems for coupled system of nonlinear fractional differential equations, AIMS Math., 4 (2019), 880895

[6]
M. Asaduzzaman, A. Kilicman, M. Z. Ali, S. H. Sapar, Fixed point theorem based solvability of 2Dimensional dissipative cubic nonlinear KleinGordon equation, Math., 8 (2020), 1103

[7]
A. Augustynowicz, H. Leszczynski, W. Walter, On some nonlinear ordinary differential equations with advanced arguments, Nonlinear Anal., 53 (2003), 495505

[8]
Z. Bai, On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Anal., 72 (2010), 916924

[9]
S. Banaei, M. B. Ghaemi, R. Saadati, An extension of Darbo’s theorem and its application to system of neutral differential equations with deviating argument, Miskolc Math. Notes, 18 (2017), 8394

[10]
T. A. Burton, Differential inequalities for integral and delay differential equations, Comparison Methods and Stability Theory, Lecture Notes in Pure and Appl. Math., Dekker, , New York (1994)

[11]
Y.K. Chang, J. J. Nieto, Some new existence results for fractional differential inclusions with boundary conditions, Math. Comput. Model., 49 (2009), 605609

[12]
H. Chen, S. Kang, L. Kong, Y. Gao, Existence of Three Positive Solutions for a Class of Boundary Value Problems of Caputo Fractional qDifference Equation, Discrete Dyn. Nat. Soc., 2018 (2018), 9 pages

[13]
Y. Cu, W. Ma, Q. Sun, X. Su, New uniqueness results for boundary value problem of fractional differential equation, Nonlinear Anal. Model. Control, 23 (2018), 3139

[14]
F. H. Damag, A. Kilic¸man, A. T. AlArioi, On hybrid type nonlinear fractional integrodifferential equations, Mathematics, 8 (2020), 14 pages

[15]
F. H. Damag, A. Kilicman, H. Dutta, R. W. Ibrahim, A Note on the Lower and Upper Solutions of HybridType Iterative Fractional Differential Equations, Natl. Acad. Sci. Lett., 43 (2020), 277281

[16]
K. Deimling, Nonlinear Functional Analysis, Springer, Berlin (1985)

[17]
A. Devi, A. Kumar, D. Baleanu, A. Khan, On stability analysis and existence of positive solutions for a general nonlinear fractional differential equations, Adv. Differ. Equ., 2020 (2020), 16 pages

[18]
W. G. ElSayed, Solvability of a neutral differential equation with deviated argument, J. Math. Anal. Appl., 327 (2007), 342350

[19]
M. Frechet, Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo, 22 (1906), 172

[20]
C. S. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett., 23 (2010), 10501055

[21]
D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Notes and Reports in Math. Sci. Eng, Academic Press, Boston (1988)

[22]
C. P. Gupta, Solvability of a threepoint nonlinear boundary value problem for a second order ordinary differential equation, J. Math. Anal. Appl., 168 (1992), 540551

[23]
T. Jankowski, Solvability of threepoint boundary value problems for second order differential equations with deviating arguments, J. Math. Anal. Appl., 312 (2005), 620636

[24]
T. Jankowski, Fractional differential equations with deviating arguments, Dyn. Syst. Appl., 17 (2008), 677684

[25]
T. Jankowski, Positive solutions for threepoint onedimensional pLaplacian boundary value problems with advanced arguments, Appl. Math. Comput., 215 (2009), 125131

[26]
T. Jankowski, Positive solutions for fourthorder differential equations with deviating arguments and integral boundary conditions, Nonlinear Anal., 73 (2010), 12891299

[27]
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam (2006)

[28]
V. Lakshmikantham, S. Leela, J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge (2009)

[29]
R. Ma, Multiplicity of positive solutions for secondorder threepoint boundary value problems, Comput. Math. Appl., 40 (2000), 193204

[30]
J. J. Nieto, Maximum principles for fractional differential equations derived from MittagLeffler functions, Appl. Math. Lett., 23 (2010), 12481251

[31]
S. K. Ntouyas, G. Wang, L. Zhang, Positive solutions of arbitrary order nonlinear fractional differential equations with advanced arguments, Opuscula Math., 31 (2011), 433442

[32]
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999)

[33]
B. H. Rizqan, D. B. Dhaigude, Positive solutions of nonlinear fractional differential equations with an advanced argument under integral boundary value conditions, Indian J. Math., 60 (2018), 491507

[34]
Q. Sun, H. Ji, Y. Cui, Positive Solutions for Boundary Value Problems of Fractional Differential Equation with Integral Boundary Conditions, J. Funct. Spaces, 2018 (2018), 6 pages

[35]
Y. Sun, L. Liu, J. Zhang, R. P. Agarwal, Positive solutions of singular threepoint boundary value problems for secondorder differential equations, J. Comput. Appl. Math.,, 230 (2009), 738750

[36]
F. J. Torres, Existence of positive solutions for a boundary value problem of a nonlinear fractional differential equation, Bull. Iranian Math. Soc., 39 (2013), 307323

[37]
G. Wang, S. K. Ntouyas, L. Zhang, Positive solutions of the threepoint boundary value problem for fractionalorder differential equations with an advanced argument, Adv. Differ. Equ., 2011 (2011), 11 pages

[38]
J. R. L. Webb, Positive solutions of some threepoint boundary value problems via fixed point index theory, Nonlinear Anal., 47 (2001), 43194332

[39]
X. Xu, Multiplicity results for positive solutions of some semipositone threepoint boundary value problems, J. Math. Anal. Appl., 291 (2004), 673689

[40]
C. Yang, C. Zhai, J. Yan, Positive solutions of the threepoint boundary value problem for second order differential equations with an advanced argument, Nonlinear Anal., 65 (2006), 20132023