# A derivative-type operator and its application to the solvability of a nonlinear three point boundary value problem

• René Erlín Castillo
• Babar Sultan

## Abstract

In this paper we introduce an operator that can be thought as a derivative of variable order, i.e. the order of the derivative is a function. We prove several properties of this operator, for instance, we obtain a generalized Leibniz‘s formula, Rolle and Cauchy‘s mean theorems and a Taylor type polynomial. Moreover, we obtain its inverse operator. Also, with this derivative we analyze the existence of solutions of a nonlinear three-point boundary value problem of “variable order”.

## Keywords

• Universidad Nacional de Colombia, Departamento de Matemáticas, Bogotá, Colombia.
• Department of Mathematics, Quaid-I-Azam University, Islamabad 45320, Pakistan.
• Pages: 521–539
• Date Published: 2022-12-21
• Vol. 24 No. 3 (2022)

T. Abdeljawad, “On conformable fractional calculus”, J. Comput. Appl. Math, vol. 279, pp. 57–66, 2015.

H. Afshari, H. Marasi and H. Aydi, “Existence and uniqueness of positive solutions for boundary value problems of fractional differential equations”, Filomat, vol. 31, no. 9, pp. 2675–2682, 2017.

H. Afshari, H. Shojaat and M. S. Siahkali, “Existence of the positive solutions for a tripled system of fractional differential equations via integral boundary conditions”, Results in Nonlinear Analysis, vol. 4, no. 3, pp. 186–199, 2021.

B. Ahmad, A. Alsaedi, S. K. Ntouyas and J. Tariboon, Hadamard-type fractional differential equations, inclusions and inequalities, Cham: Springer, 2017.

B. Ahmad and S. Ntouyas, “A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations”, Fract. Calc. Appl. Anal., vol. 17, no. 2, pp. 348–360, 2014.

D. R. Anderson, “Taylor‘s formula and integral inequalities for conformable fractional derivatives” in Contributions in Mathematics and Engineering, Cham: Springer, 2016, pp. 25–43.

D. R. Anderson and D. J. Ulness, “Newly defined conformable derivatives”, Adv. Dyn. Syst. Appl., vol. 10, no. 2, pp. 109–137, 2015.

D. Baleanu, Z. B. Güvenç and J. A. Tenreiro Machado, New Trends in Nanotechnology and Fractional Calculus Applications, New York: Springer, 2010.

T. Bashiri, S. M. Vaezpour and J. J. Nieto, “Approximating solution of Fabrizio-Caputo Volterra‘s model for population growth in a closed system by homotopy analysis method”, J. Funct. Spaces, Art. ID 3152502, 10 pages, 2018.

H. Batarfi, J. Losada, J. J. Nieto and W. Shammakh, “Three-point boundary value problems for conformable fractional differential equations”, J. Funct. Spaces, Art. ID 706383, 6 pages, 2015.

M. Benchohra, J. R. Graef and S. Hamani, “Existence results for boundary value problems with non-linear fractional differential equations”, Appl. Anal., vol. 87, no. 7, pp. 851–863, 2008.

R. Caponetto, G. Dongola, L. Fortuna and I. Petras, Fractional order systems: modeling and control applications, World Scientific Series on Nonlinear Science Series A, vol. 72, Singapore: World Scientific Publishing Co, Pte. Ltd., 2010.

R. E. Castillo and S. A. Chapinz, “The fundamental theorem of calculus for the Riemann-Stieljes integral”, Lect. Mat., vol. 29, no. 2, pp. 115–122, 2008.

X. Dong, Z. Bai and S. Zhang, “Positive solutions to boundary value problems of p-Laplacian with fractional derivative”, Bound. Value Probl., Paper No. 5, 15 pages, 2017.

O. S. Iyiola and E. R. Nwaeze, “Some new results on the new conformable fractional calculus with applications using D‘Alambert approach”, Prog. Fract. Differ. Appl., vol. 2, no. 2, pp. 115–122, 2016.

R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, “A new definition of fractional derivative”, J. Comput. Appl. Math, vol. 264, pp. 65–70, 2014.

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Amsterdam: Elsevier B. V., 2006.

F. Mainardi, Fractional calculus and waves in linear viscoelesticity, London: Imperial College Press, 2010.

C. A. Monje, Y. Chen, B. M. Vinagre, D. Xue and V. Feliu, Fractional-order systems and controls, Advances in Industrial Control, London: Springer, 2010.

K. B. Oldham and J. Spanier, The fractional calculus, Mathematics in Science and Engineering 111, New York-London: Academic Press, 1974.

M. D. Ortigueira, Fractional calculus for scientists and engineers, Lecture Notes in Electrical Engineering 84, Dordrecht: Springer, 2011.

L.-E. Persson and H. Rafeiro, “On a Taylor remainder”, Acta Math. Acad. Paedagog. Nyha Ìzy, vol. 33, no. 2, pp. 195–198, 2017.

L.-E. Persson, H. Rafeiro and P. Wall, “Historical synopsis of the Taylor remainder”, Note Mat., vol. 37, no. 1, pp. 1–21, 2017.

C. Pinto and A. R. M. Carvalho, “New findings on the dynamics of HIV and TB coinfection models”, Appl. Math. and Comput., vol. 242, pp. 36–46, 2014.

C. Pinto and A. R. M Carvalho, “Fractional modeling of typical stages in HIV epidemics with drug-resistance”, Progr. Fract. Differ. Appl., vol. 1, no. 2, pp. 111–122, 2015.

I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering 198, San Diego: Academic Press, Inc., 1999.

H. Rafeiro and S. Kim, “Revisiting the first mean value theorem for integrals”, Teach. Math., vol. 25, no. 1, pp. 30–35, 2022.

M. N. Sahlan and H. Afshari, “Three new approaches for solving a class of strongly nonlinear two-point boundary value problems”, Bound. Value Probl., Paper No. 60, 21 pages, 2021.

M. N. Sahlan and H. Afshari, “Lucas polynomials based spectral methods for solving the fractional order electrohydrodynamics flow model”, Commun. Nonlinear Sci. Numer. Simul., vol. 107, Paper No. 106108, 21 pages, 2022.

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives: theory and applications, Switzerland: Gordon & Breach Science Publishers, 1993.

[pendiente]

E. Scalas, R. Gorenflo, F. Mainardi and M. Meerschaert, “Speculative option valuation and the fractional diffusion equation”, in Proceedings of the IFAC Workshop on Fractional Differentiation and its Applications, J. Sabatier and J. Tenreiro Machado, Bordeaux, 2004.

J. A. Tenreiro Machado, “And I say to myself: “What a fractional world!”, Fract. Calc. Appl. Anal., vol. 14, no. 4, Paper No. 635, 2011.

D. Zhao and M. Luo, “General conformable fractional derivative and its physical interpretation”, Calcolo, vol. 54, no. 3, pp. 903–917, 2017.

W. Zhong and L. Wang, “Positive solutions of conformable fractional differential equations with integral boundary conditions”, Bound. Value Probl., Paper No. 136, 12 pages, 2018.

2022-12-21

## How to Cite

[1]
R. E. Castillo and B. Sultan, “A derivative-type operator and its application to the solvability of a nonlinear three point boundary value problem”, CUBO, vol. 24, no. 3, pp. 521–539, Dec. 2022.

Articles