THE EXISTENCE OF POSITIVE SOLUTIONS AND A LYAPUNOV TYPE INEQUALITY FOR BOUNDARY VALUE PROBLEMS OF THE FRACTIONAL CAPUTO-FABRIZIO DIFFERENTIAL EQUATIONS

In this paper, a Lyapunov-type inequality and the existence of the positive solutions for boundary value problems of the nonlinear fractional Caputo-Fabrizio differential equation have been presented. By using the Guo Krasnoselskii’s fixed point theorem on cone and the properties of the associated Green`s function, we prove the existence of the positive solution. Finally, we gave some numerical examples to validate the theoretical findings.

Keywords:

Lyapunov-type inequality, fractional Caputo-Fabrizio derivative, Green’s function, positive solution fixed point.,

PDF

___

[1] Miller KS., Ross B., (Eds.) (1993) An introduction to the fractional cxalculus and fractional differential equations, John Wiley, NY.
[2] Podlubny I., (1999) Fractional differential equations, Academic Press, New York.
[3] Kilbas A.A., Srivastava H. M. and Trujillo J. J., (2006) Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, Amsterdam.
[4] Mamedov F.I., Zeren Y., (2016) On boundedness of fractıonal maxımal operator ın weıghted Lp(.) spaces. Mathematıcal Inequalıtıes & Applıcatıons 19, 1-14.
[5] Sabatier J., Lanusse P., Melchior P., Oustaloup A., (2015) Fractional order Differentiation and Robust Control Design. Springer.
[6] Mainardi F., (2010) Fractional Calculus and Waves in linear Viscoelasticity: an Introduction to Mathematical Models, World Scientific.
[7] Reid W.T., (1973) A generalized Liapunov inequality. J. Differential Equations 13, 182-196.
[8] Usta F., Sarikaya M.Z., (2019) The analytical solution of Van der Pol and Lienard differential equations within conformable fractional operator by retarded integral inequalities. Demonstr. Math. 52(1), 204-2012.
[9] Sarikaya M.Z., Usta F., (2016) On Comparison Theorems for Conformable Fractional Differential Equations. International Journal of Analysis and Applications 12 (2), 207-214.
[10] Usta F., (2018) A Conformable Calculus of Radial Basis Functions and its Applications. An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 8 (2), 176-182.
[11] Usta F., (2016) Mesh-Free Technique of Numerical Solution of Newly Defined Conformable Differential Equations. Konuralp Journal of Mathematics 4 (2), 149-157.
[12] Caputo M., Fabrizio M., (2015) A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1, 73–85.
[13] Caputo M., Fabrizio M., (2017) On the notion of fractional derivative and applicatios to the hysteresis phenomena. Mecc. 52, No 13, 3043-3052.
[14] Toprakseven Ş., (2019) The Existence and Uniqueness of Initial-Boundary Value Problems of the Fractional Caputo-Fabrizio Differential Equations. Universal Journal of Mathematics and Applications 2 (2), 100-106.
[15] Das K.M., Vatsala A.S., (1975) Green's function for n–n boundary value problem and an analogue of Hartman's result J. Math. Anal. 51, 670-677.
[16] Pachpatte B.G., (1995) On Lyapunov-type inequalities for certain higher order differential equations J. Math. Anal. Appl. 195, 527-536.
[17] Ferreira R.A.C., (2013) A Lyapunov-type inequality for a fractional boundary value problem Fract. Calc. Appl. Anal., 16 (4), 978-984.
[18] Ferreira R.A.C., (2014) On a Lyapunov-type inequality and the zeros of a certain Mittag-Leffler function. J. Math. Anal. Appl. 412, 2, 1058–1063.
[19] Jleli M., Samet B.,(2015) Lyapunov-type inequalities for a fractional differential equation with mixed boundary conditions. Math. Inequal. Appl. 18, 2, 443–451.
[20] O’Regan D., Samet B., (2015) Lyapunov-type inequalities for a class of fractional differential equations. J. Inequal. Appl. 2015, 247, 10.
[21] Kirane M., Torebek B. T., (2018) A Lyapunov-type inequality for a fractional boundary value problem with Caputo-Fabrizio derivative, J. Math. Inequal. 12, 4, 1005–1012.
[22] Hu L., Zhang S.Q., (2017) Existence results for a coupled system of fractional differential equations with p–Laplacian operator and infinite–point boundary conditions Bound. Value Probl. 2017, 88.
[23] Bai Z.B., Sun W.C., (2012) Existence and multiplicity of positive solutions for singular fractional boundary value problems Comput. Math. Appl. 63 , 1369-1381.
[24] Al-Refai M., Hajji M.A., (2011) Monotone iterative sequences for nonlinear boundary value problems of fractional order Nonlinear Anal. Theory Methods Appl. 74 , 3531-3539.
[25] Ahmad B., Nieto J.J., (2009) Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations Abstr. Appl. Anal. 2009, 1-9
[26] Nieto J.J., Pimentel J., (2013) Positive solutions of a fractional thermostat model Bound. Value Probl. 5, 1-11.
[27] Zhang X., Liu L., Wu Y., (2013) The uniqueness of positive solution for a singular fractional differential system involving derivatives Commun. Nonlinear Sci. Numer. Simul. 18, 1400-1409.

Sigma Mühendislik ve Fen Bilimleri Dergisi-Cover

ISSN: 1304-7191
Yayın Aralığı: Yılda 4 Sayı
Yayıncı: Yıldız Teknik Üniversitesi

Arşiv