Monotone iterative technique for sequential \(\delta\)-Caputo fractional differential equations with nonlinear boundary conditions

Zidane Baitiche, Choukri Derbazi, Abdelkrim Salim, Mouffak Benchohra

Abstract


In this article, we discuss the existence of extremal solutions for a class of nonlinear sequential $\delta $--Caputo fractional differential equations involving nonlinear boundary conditions. Our results are founded on advanced functional analysis methods. To be more specific, we use the monotone iterative approach in conjunction with the upper and lower solution method to create adequate requirements for the existence of extremal solutions. As an application, we give an example to illustrate our results.

Keywords


Sequential $\delta $--Caputo derivative; nonlinear boundary conditions; monotone iterative technique; upper and lower solutions

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References


Abbas, M.I., On the nonlinear sequential Hilfer fractional differential equations, International Journal of Mathematical Analysis, 14(2020), 77{90.

Abbas, S., Benchohra, M., Graef, J.R., Henderson, J., Implicit Fractional Differential and Integral Equations: Existence and Stability, De Gruyter, Berlin, 2018.

Abbas, S., Benchohra, M., N'Guerekata, G.M., Advanced Fractional Differential and Integral Equations, Nova Sci. Publ., New York, 2014.

Abbas, S., Benchohra, M., N'Guerekata, G.M., Topics in Fractional Differential Equations, Dev. Math., 27, Springer, New York, 2015.

Abdo, M.S., Panchal, S.K., Saeed, A.M., Fractional boundary value problem with Caputo fractional derivative, Proc. Indian Acad. Sci. (Math. Sci), 65(2019), 129.

Agarwal, R.P., Benchohra, M., Hamani, S., A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109(2010), 973{1033.

Aghajani, A., Pourhadi, E., Trujillo, J.J., Application of measure of noncompactness to a Cauchy problem for fractional differential equations in Banach spaces, Fract. Calc. Appl. Anal., 16(2013), 962-977.

Ahmad, B., Alghamdi, N., Alsaedi, A., Ntouyas, S.K., Multi-term fractional differential equations with nonlocal boundary conditions, Open Math., 16(2018), 1519{1536.

Ahmad, B., Nieto, J.J., Boundary value problems for a class of sequential integrodifferential equations of fractional order, Journal of Function Spaces and Applications, 2013(2013), Article ID 149659, 8 pages.

Almeida, R., A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul., 44(2017), 460{481.

Almeida, R., Fractional differential equations with mixed boundary conditions, Bull. Malays. Math. Sci. Soc., 42(2019), 1687{1697.

Almeida, R., Malinowska, A.B., Monteiro, M.T.T., Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications, Math. Meth. Appl. Sci., 41(2018), 336{352.

Al-Refai, M., Ali Hajji, M., Monotone iterative sequences for nonlinear boundary value problems of fractional order, Nonlinear Anal., 74(2011), 3531{3539.

Baitiche, Z., Guerbati, K., Benchohra, M., Zhou, Y., Solvability of fractional multi-point BVP with nonlinear growth at resonance, J. Contemp. Math. Anal., 55(2020), 126-142.

Bouriah, S., Salim, A., Benchohra, M., On nonlinear implicit neutral generalized Hilfer fractional differential equations with terminal conditions and delay, Topol. Algebra Appl., 10(2022), 77-93.

Chen, C., Bohner, M., Jia, B., Method of upper and lower solutions for nonlinear Caputo fractional difference equations and its applications, Fract. Calc. Appl. Anal., 22(2019), 1307{1320.

Derbazi, C., Hammouche, H., Salim, A., Benchohra, M., Measure of noncompactness and fractional hybrid differential equations with hybrid conditions, Di er. Equ. Appl., 14(2022), 145-161.

Fazli, H., Sun, H., Aghchi, S., Existence of extremal solutions of fractional Langevin equation involving nonlinear boundary conditions, International Journal of Computer Mathematics, (2020).

Hilfer, R., Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.

Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies Elsevier Science B.V. Amsterdam the Netherlands, 2006.

Kucche, K.D., Mali, A., Vanterler da C. Sousa, J., On the nonlinear -Hilfer fractional differential equations, Comput. Appl. Math., 38(2019), no. 2, Art. 73, 25 pp.

Ladde, G.S., Lakshmikantham, V., Vatsala, A.S., Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston, 1985.

Lazreg, J.E., Benchohra, M., Salim, A., Existence and Ulam stability of |-generalized Hilfer fractional problem, J. Innov. Appl. Math. Comput. Sci., 2(2022), 1-13.

Lin, X., Zhao, Z., Iterative technique for a third-order differential equation with three-point nonlinear boundary value conditions, Electron. J. Qual. Theory Di er. Equ., 2016, Paper No. 12, 10 pp.

Matar, M.M., Solution of sequential Hadamard fractional di erential equations by variation of parameter technique, Abstr. Appl. Anal., 2018(2018), Article ID 9605353, 7 pages.

Miller, K.S., Ross, B., An Introduction to Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.

Nieto, J.J., An abstract monotone iterative technique, Nonlinear Analysis, Theory, Methods and Applications, 28(1997), 1923-1933.

Oldham, K.B., Fractional differential equations in electrochemistry, Adv. Eng. Softw., 41(2010), 9{12.

Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, 1999.

Royden, H.L., Real Analysis, Macmillan Publishing Company, New York, NY, USA, 3rd edition, 1988.

Sabatier, J., Agrawal, O.P., Machado, J.A.T., Advances in Fractional Calculus. Theoretical Developments and Applications in Physics and Engineering, Dordrecht: Springer, 2007.

Salim, A., Ahmad, B., Benchohra, M., Lazreg, J.E., Boundary value problem for hybrid generalized Hilfer fractional differential equations, Di er. Equ. Appl., 14(2022), 379-391.

Salim, A., Benchohra, M., Graef, J.R., Lazreg, J.E., Initial value problem for hybrid Hilfer fractional implicit differential equations, J. Fixed Point Theory Appl., 24(2022), 14 pp.

Salim, A., Benchohra, M., Lazreg, J.E., Nonlocal k-generalized Hilfer impulsive initial value problem with retarded and advanced arguments, Appl. Anal. Optim., 6(2022), 21-47.

Salim, A., Lazreg, J.E., Ahmad, B., Benchohra, M., Nieto, J.J., A study on k-generalized Hilfer derivative operator, Vietnam J. Math., (2022).

Tarasov, V.E., Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Heidelberg & Higher Education Press, Beijing, 2010.

Vanterler da C. Sousa, J., Capelas de Oliveira, E., On the Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60(2018), 72-91.

Wang, G., Sudsutad, W., Zhang, L., Tariboon, J., Monotone iterative technique for a nonlinear fractional q-difference equation of Caputo type, Adv. Difference Equ., 2016, Paper No. 211, 11 pp.

Yang, W., Monotone iterative technique for a coupled system of nonlinear Hadamard fractional differential equations, J. Appl. Math. Comput., 59(2019), no. 1-2, 585{596.

Zhang, S., Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives, Nonlinear Anal., 71(2009), no. 5-6, 2087{2093.

Zhou, Y., Basic Theory of Fractional Differential Equations,World Scientific, Singapore, 2014.

Zhou, Y., Fractional Evolution Equations and Inclusions: Analysis and Control, Elsevier, Acad. Press, 2016.




DOI: http://dx.doi.org/10.24193/subbmath.2024.3.06

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