# American Institute of Mathematical Sciences

October  2021, 14(10): 3497-3528. doi: 10.3934/dcdss.2020442

## Nonlinear singular $p$ -Laplacian boundary value problems in the frame of conformable derivative

 1 Department of Mathematics, Faculty of Sciences University of Bougara, U.M.B.B., Algeria 2 Dynamic of Engines and Vibroacoustic Laboratory, U.M.B.B., Algeria 3 Department of Mathematical Sciences, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia 4 Department of Mathematics, Çankaya University 06790, Ankara, Turkey 5 Department of Mathematics, Faculty of Sciences University of M'hamed Bougara, UMBB, Algeria 6 Laboratory of Dynamical Systems, Faculty of Mathematics, U.S.T.H.B., Algeria 7 Department of Mathematics and Physical Sciences, Prince Sultan University, P. O. Box 66833, Riyadh, 11586, KSA 8 Department of Medical Research, China Medical University, 40402, Taichung, Taiwan 9 Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan

* Corresponding author: tabdeljawad@psu.edu.sa

Received  November 2019 Revised  April 2020 Published  October 2021 Early access  November 2020

This paper studies a class of fourth point singular boundary value problem of $p$-Laplacian operator in the setting of a specific kind of conformable derivatives. By using the upper and lower solutions method and fixed point theorems on cones., necessary and sufficient conditions for the existence of positive solutions are obtained. In addition, we investigate the dependence of the solution on the order of the conformable differential equation and on the initial conditions.

Citation: Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $p$ -Laplacian boundary value problems in the frame of conformable derivative. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3497-3528. doi: 10.3934/dcdss.2020442
##### References:
 [1] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66.  doi: 10.1016/j.cam.2014.10.016. [2] A. R. Aftabizadeh, Existence and uniqueness theorems for fourth order boundary value problems, J. Math. Anal. Appl., 116 (1986), 415-426.  doi: 10.1016/S0022-247X(86)80006-3. [3] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel. Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A. [4] Z. Bai and H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311 (2005), 495-505.  doi: 10.1016/j.jmaa.2005.02.052. [5] A. Cabada, P. Habets and R. L. Pouso, Optimal existence conditions for $\varphi$ -Laplacian equations with upper and lower solutions in the reversed order, J. Differential Equations, 166 (2000), 385-401.  doi: 10.1006/jdeq.2000.3803. [6] A. Cabada and G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl., 389 (2012), 403-411.  doi: 10.1016/j.jmaa.2011.11.065. [7] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 2 (2015), 1-13. [8] G. Chai and S. Hu, Existence of positive solutions for a fractional high-order three-point boundary value problem, Adv. Difference Equ., 2014 (2014), 17 pp. doi: 10.1186/1687-1847-2014-90. [9] W. Chen, Time-space fabric underlying anomalous diffusion, Chaos Soliton Fract., 28 (2006), 923-929.  doi: 10.1016/j.chaos.2005.08.199. [10] W. Chen and Y. Liang, New methodologies in fractional and fractal derivatives modeling, Chaos Solitons Fractals, 102 (2017), 72-77.  doi: 10.1016/j.chaos.2017.03.066. [11] Y. Chen and Y. Li, The existence of positive solutions for boundary value problem of nonlinear fractional differential equations, Abst. Appl. Anal. (2014), Art. ID 681513, 7 pp. doi: 10.1155/2014/681513. [12] K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, vol 2004, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2. [13] X. Dong, Z. Bai and S. Zhang, Positive solutions to boundary value problems of $p$-Laplacian with fractional derivative, Bound. Value Probl., (2017), Paper No. 5, 15 pp. doi: 10.1186/s13661-016-0735-z. [14] D. J. Guo and V. Lakshmikantham, Nonlinear problems in abstract cones, Notes and Reports Math. Sci. Eng., vol. 5, Academic Press, Inc., Boston, MA, 1988. [15] D. Jiang and W. Gao, Upper and lower solution method and a singular boundary value problem for the one-dimensional $p$ -Laplacian, J. Math. Anal. Appl., 252 (2000), 631-648.  doi: 10.1006/jmaa.2000.7012. [16] D. Ji, Z. Bai and W. Ge, The existence of countably many positive solutions for singular multipoint boundary value problems, Nonlinear Anal., 72 (2010), 955-964.  doi: 10.1016/j.na.2009.07.031. [17] R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002. [18] A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, B.V., Amsterdam, 2006. [19] L. S. Leibenson, General problem of the movement of a compressible fluid in a porous medium, Izv. Akad. Nauk Kirg. SSSR, 9 (1983), 7-10. [20] S. J. Linz and J. C. Sprott, Elementary chaotic flow, Phys. Lett. A, 259 (1999), 240-245.  doi: 10.1016/S0375-9601(99)00450-8. [21] X. Liu and M. Jia, Multiple solutions for fractional differential equations with nonlinear boundary conditions, Comput. Math. Appl., 59 (2010), 2880-2886.  doi: 10.1016/j.camwa.2010.02.005. [22] D. O'Regan, Theory of Singular Boundary Value Problems, World Scientific, Singapore, 1994. doi: 10.1142/2352. [23] D. O'Regan, Solvability of some fourth (and higher) order singular boundary value problems, J. Math. Anal. Appl., 161 (1991), 78-116.  doi: 10.1016/0022-247X(91)90363-5. [24] I. Podlubny, Fractional Differential Equations. Academic Press, Inc., San Diego, CA, 1999. [25] T. Ren and X. Chen, Positive solutions of fractional differential equation with $p$ -Laplacian operator, Abstr. Appl. Anal., 2013, Art ID 789836, 7 pp. doi: 10.1155/2013/789836. [26] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. [27] J. Schroder, Fourth order two-point boundary value problems; estimates by two-sided bounds, Nonlinear Anal., 8 (1984), 107-114.  doi: 10.1016/0362-546X(84)90063-4. [28] K. Sheng, W. Zhang and Z. Bai, Positive solutions to fractional boundary-value problems with $p$ -Laplacian on time scales, Bound. Value Prob., 2018 (2018), Paper No. 70, 15 pp. doi: 10.1186/s13661-018-0990-2. [29] J. Wang and H. Xiang, Upper and lower solutions method for a class of singular fractional boundary value problems with $p$ -Laplacian operator, Abstr. Appl. Anal., 2010, Art. ID 971824, 12 pp. doi: 10.1155/2010/971824. [30] Z. Wei, A class of fourth order singular boundary value problems, Appl. Math. Comput., 153 (2004), 865-884.  doi: 10.1016/S0096-3003(03)00683-0. [31] Y. Wei, Z. Bai and S. Sun, On positive solutions for some second-order three-point boundary value problems with convection term, J. Inequal. Appl., 2019 (2019), Paper No. 72, 11 pp. doi: 10.1186/s13660-019-2029-3. [32] P. Yan, Nonresonance for one-dimensional$p$ -Laplacian with regular restoring, J. Math. Anal. Appl., 285 (2003), 141-154.  doi: 10.1016/S0022-247X(03)00383-4. [33] Y. Zhang, Existence results for a coupled system of nonlinear fractional multi-point boundary value problems at resonance, J. Inequal. Appl., 2018 (2018), Paper No. 198, 17 pp. doi: 10.1186/s13660-018-1792-x. [34] X. Zhang and L. Liu, Positive solutions of fourth-order four-point boundary value problems with$p$-Laplacian operator, J. Math. Anal. Appl., 336 (2007), 1414-–1423. doi: 10.1016/j.jmaa.2007.03.015. [35] X. Zhang and L. Liu, A necessary and sufficient condition for positive solutions for fourth-order multi-point boundary value problems with $p$ -Laplacian, Nonlinear Anal., 68 (2008), 3127-3137.  doi: 10.1016/j.na.2007.03.006.

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##### References:
 [1] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66.  doi: 10.1016/j.cam.2014.10.016. [2] A. R. Aftabizadeh, Existence and uniqueness theorems for fourth order boundary value problems, J. Math. Anal. Appl., 116 (1986), 415-426.  doi: 10.1016/S0022-247X(86)80006-3. [3] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel. Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A. [4] Z. Bai and H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311 (2005), 495-505.  doi: 10.1016/j.jmaa.2005.02.052. [5] A. Cabada, P. Habets and R. L. Pouso, Optimal existence conditions for $\varphi$ -Laplacian equations with upper and lower solutions in the reversed order, J. Differential Equations, 166 (2000), 385-401.  doi: 10.1006/jdeq.2000.3803. [6] A. Cabada and G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl., 389 (2012), 403-411.  doi: 10.1016/j.jmaa.2011.11.065. [7] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 2 (2015), 1-13. [8] G. Chai and S. Hu, Existence of positive solutions for a fractional high-order three-point boundary value problem, Adv. Difference Equ., 2014 (2014), 17 pp. doi: 10.1186/1687-1847-2014-90. [9] W. Chen, Time-space fabric underlying anomalous diffusion, Chaos Soliton Fract., 28 (2006), 923-929.  doi: 10.1016/j.chaos.2005.08.199. [10] W. Chen and Y. Liang, New methodologies in fractional and fractal derivatives modeling, Chaos Solitons Fractals, 102 (2017), 72-77.  doi: 10.1016/j.chaos.2017.03.066. [11] Y. Chen and Y. Li, The existence of positive solutions for boundary value problem of nonlinear fractional differential equations, Abst. Appl. Anal. (2014), Art. ID 681513, 7 pp. doi: 10.1155/2014/681513. [12] K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, vol 2004, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2. [13] X. Dong, Z. Bai and S. Zhang, Positive solutions to boundary value problems of $p$-Laplacian with fractional derivative, Bound. Value Probl., (2017), Paper No. 5, 15 pp. doi: 10.1186/s13661-016-0735-z. [14] D. J. Guo and V. Lakshmikantham, Nonlinear problems in abstract cones, Notes and Reports Math. Sci. Eng., vol. 5, Academic Press, Inc., Boston, MA, 1988. [15] D. Jiang and W. Gao, Upper and lower solution method and a singular boundary value problem for the one-dimensional $p$ -Laplacian, J. Math. Anal. Appl., 252 (2000), 631-648.  doi: 10.1006/jmaa.2000.7012. [16] D. Ji, Z. Bai and W. Ge, The existence of countably many positive solutions for singular multipoint boundary value problems, Nonlinear Anal., 72 (2010), 955-964.  doi: 10.1016/j.na.2009.07.031. [17] R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002. [18] A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, B.V., Amsterdam, 2006. [19] L. S. Leibenson, General problem of the movement of a compressible fluid in a porous medium, Izv. Akad. Nauk Kirg. SSSR, 9 (1983), 7-10. [20] S. J. Linz and J. C. Sprott, Elementary chaotic flow, Phys. Lett. A, 259 (1999), 240-245.  doi: 10.1016/S0375-9601(99)00450-8. [21] X. Liu and M. Jia, Multiple solutions for fractional differential equations with nonlinear boundary conditions, Comput. Math. Appl., 59 (2010), 2880-2886.  doi: 10.1016/j.camwa.2010.02.005. [22] D. O'Regan, Theory of Singular Boundary Value Problems, World Scientific, Singapore, 1994. doi: 10.1142/2352. [23] D. O'Regan, Solvability of some fourth (and higher) order singular boundary value problems, J. Math. Anal. Appl., 161 (1991), 78-116.  doi: 10.1016/0022-247X(91)90363-5. [24] I. Podlubny, Fractional Differential Equations. Academic Press, Inc., San Diego, CA, 1999. [25] T. Ren and X. Chen, Positive solutions of fractional differential equation with $p$ -Laplacian operator, Abstr. Appl. Anal., 2013, Art ID 789836, 7 pp. doi: 10.1155/2013/789836. [26] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. [27] J. Schroder, Fourth order two-point boundary value problems; estimates by two-sided bounds, Nonlinear Anal., 8 (1984), 107-114.  doi: 10.1016/0362-546X(84)90063-4. [28] K. Sheng, W. Zhang and Z. Bai, Positive solutions to fractional boundary-value problems with $p$ -Laplacian on time scales, Bound. Value Prob., 2018 (2018), Paper No. 70, 15 pp. doi: 10.1186/s13661-018-0990-2. [29] J. Wang and H. Xiang, Upper and lower solutions method for a class of singular fractional boundary value problems with $p$ -Laplacian operator, Abstr. Appl. Anal., 2010, Art. ID 971824, 12 pp. doi: 10.1155/2010/971824. [30] Z. Wei, A class of fourth order singular boundary value problems, Appl. Math. Comput., 153 (2004), 865-884.  doi: 10.1016/S0096-3003(03)00683-0. [31] Y. Wei, Z. Bai and S. Sun, On positive solutions for some second-order three-point boundary value problems with convection term, J. Inequal. Appl., 2019 (2019), Paper No. 72, 11 pp. doi: 10.1186/s13660-019-2029-3. [32] P. Yan, Nonresonance for one-dimensional$p$ -Laplacian with regular restoring, J. Math. Anal. Appl., 285 (2003), 141-154.  doi: 10.1016/S0022-247X(03)00383-4. [33] Y. Zhang, Existence results for a coupled system of nonlinear fractional multi-point boundary value problems at resonance, J. Inequal. Appl., 2018 (2018), Paper No. 198, 17 pp. doi: 10.1186/s13660-018-1792-x. [34] X. Zhang and L. Liu, Positive solutions of fourth-order four-point boundary value problems with$p$-Laplacian operator, J. Math. Anal. Appl., 336 (2007), 1414-–1423. doi: 10.1016/j.jmaa.2007.03.015. [35] X. Zhang and L. Liu, A necessary and sufficient condition for positive solutions for fourth-order multi-point boundary value problems with $p$ -Laplacian, Nonlinear Anal., 68 (2008), 3127-3137.  doi: 10.1016/j.na.2007.03.006.
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