Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations

Hasib Khan; Cemil Tunc; Aziz Khan

doi:10.3934/dcdss.2020139

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September 2020, 13(9): 2475-2487. doi: 10.3934/dcdss.2020139

Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations

Hasib Khan ^1, , Cemil Tunc ^2,, and Aziz Khan ^3,

1.	Department of Mathematics, Shaheed BB University, Sheringal, Dir Upper 18000, Khybar Pakhtunkhwa, Pakistan
2.	Department of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, 65080 Van, Turkey
3.	Prince Sultan University, P.O. Box 66833, 11586 Riyadh, Saudi Arabia

^* Corresponding author: Cemil Tunc

Received January 2019 Revised February 2019 Published September 2020 Early access November 2019

In this paper, we are dealing with singular fractional differential equations (DEs) having delay and

$ \mho_p $

(

$ p $

-Laplacian operator). In our problem, we Contemplate two fractional order differential operators that is Riemann–Liouville and Caputo's with fractional integral and fractional differential initial boundary conditions.The SFDE is given by

$ \begin{equation*} \left\{\begin{split} &\mathcal{D}^{\gamma}\big[\mho^*_p[\mathcal{D}^{\kappa}x(t)]\big]+\mathcal{Q}(t)\zeta_1(t, x(t-\varrho^*)) = 0, \\& \mathcal{I}_0^{1-\gamma}\big(\mho^*_p[\mathcal{D}^{\kappa}x(t)]\big)|_{t = 0} = 0 = \mathcal{I}_0^{2-\gamma}\big(\mho^*_p[\mathcal{D}^{\kappa}x(t)]\big)|_{t = 0}, \\& \mathcal{D}^{\delta^*}x(1) = 0, \, \, x(1) = x'(0), \, \, x^{(k)}(0) = 0\text{ for $k = 2, 3, \ldots, n-1$}, \end{split}\right. \end{equation*} $

$ \zeta_1 $

is a continuous function and singular at

$ t $

and

$ x(t) $

for some values of

$ t\in [0, 1] $

. The operator

$ \mathcal{D}^{\gamma}, \, $

is Riemann–Liouville fractional derivative while

$ \mathcal{D}^{\delta^*}, \mathcal{D}^{\kappa} $

stand for Caputo fractional derivatives and

$ \delta^*, \, \gamma\in(1, 2] $

$ n-1<\kappa\leq n, $

where

$ n\geq3 $

. For the study of the EUS, fixed point approach is followed in this paper and an application is given to explain the findings.

Keywords: Fractional differential equations with singularity, existence of positive solution, Hyers-Ulam stability, Caputo's fractional derivative.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation: Hasib Khan, Cemil Tunc, Aziz Khan. Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2475-2487. doi: 10.3934/dcdss.2020139

References:

[1]	B. Ahmad, A. Alsaedi, R. P. Agarwal and A. Alsharif, On sequential fractional integro-differential equations with nonlocal integral boundary conditions, Bull. Malays. Math. Sci. Soc., 41 (2018), 1725-1737. doi: 10.1007/s40840-016-0421-4.
[2]	A. Atangana and J. F. Gómez-Aguilar, Hyperchaotic behaviour obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws, Chaos Solitons Fractals, 102 (2017), 285-294. doi: 10.1016/j.chaos.2017.03.022.
[3]	A. Atangana and J. F. Gómez-Aguilar, A new derivative with normal distribution kernel: Theory, methods and applications, Phys. A, 476 (2017), 1-14. doi: 10.1016/j.physa.2017.02.016.
[4]	A. Atangana and J. F. Gómez-Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, The European Physical Journal Plus, 133 (2018), 1-22. doi: 10.1140/epjp/i2018-12021-3.
[5]	A. Atangana and J. F. Gómez-Aguilar, Numerical approximation of Riemann-Liouville definition of fractional derivative: From Riemann-L iouville to Atangana-Baleanu, Numer. Methods Partial Differential Equations, 34 (2018), 1502-1523. doi: 10.1002/num.22195.
[6]	T. Abdeljawad, F. Jarad and D. Baleanu, On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives, Sci. China Ser. A, 51 (2008), 1775-1786. doi: 10.1007/s11425-008-0068-1.
[7]	T. Abdeljawad, D. Baleanu and F. Jarad, Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives, J. Math. Phys., 49 (2008), 083507, 11 pp. doi: 10.1063/1.2970709.
[8]	T. Abdeljawad and Q. M. Al-Mdallal, Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall's inequality, J. Comput. Appl. Math., 339 (2018), 218-230. doi: 10.1016/j.cam.2017.10.021.
[9]	T. Abdeljawad and J. Alzabut, On Riemann-Liouville fractional q–difference equations and their application to retarded logistic type model, Math. Methods Appl. Sci., 41 (2018), 8953-8962. doi: 10.1002/mma.4743.
[10]	B. Ahmad and R. Luca, Existence of solutions for sequential fractional integro-differential equations and inclusions with nonlocal boundary conditions, Appl. Math. Comput., 339 (2018), 516-534. doi: 10.1016/j.amc.2018.07.025.
[11]	J. Alzabut, T. Abdeljawad and D. Baleanu, Nonlinear delay fractional difference equations with application on discrete fractional Lotka-Volterra model, J. Comput. Anal. Appl., 25 (2018), 889-898.
[12]	J. Alzabut, T. Abdeljawad and D. Baleanu, Nonlinear delay fractional difference equations with application on discrete fractional Lotka-Volterra model, J. Comput. Anal. Appl., 25 (2018), 889-898.
[13]	T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Adv. Difference Equ., 2017 (2017), 11 pp. doi: 10.1186/s13662-017-1285-0.
[14]	A. Babakhani and T. Abdeljawad, A Caputo Fractional Order Boundary Value Problem with Integral Boundary Conditions, J. Comput. Anal. Appl., 15 (2013), 753-763.
[15]	Y. K. Chang and R. Ponce, Uniform exponential stability and applications to bounded solutions of integro-differential equations in Banach spaces, J. Integral Equations Appl., 30 (2018), 347-369. doi: 10.1216/JIE-2018-30-3-347.
[16]	A. Coronel-Escamilla, J. F. Gómez-Aguilar, M. G. López-López, V. M. Alvarado-Martínez and G. V. Guerrero-Ramírez, Triple pendulum model involving fractional derivatives with different kernels, Chaos Solitons Fractals, 91 (2016), 248-261. doi: 10.1016/j.chaos.2016.06.007.
[17]	J. Henderson and R. Luca, Systems of Riemann–Liouville fractional equations with multi-point boundary conditions, Appl. Math. Comput., 309 (2017), 303-323. doi: 10.1016/j.amc.2017.03.044.
[18]	L. Guo, L. Liu and Y. Wu, Iterative unique positive solutions for singular p-Laplacian fractional differential equation system with several parameters, Nonlinear Anal., Model. Control, 23 (2018), 182-203. doi: 10.15388/NA.2018.2.3.
[19]	A. Ghanmia, M. Kratoub and K. Saoudib, A Multiplicity Results for a Singular Problem Involving a Riemann-Liouville Fractional Derivative, Filomat, 32 (2018), 653-669. doi: 10.2298/FIL1802653G.
[20]	J. F. Gómez-Aguilar and A. Atangana, New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applications, The European Physical Journal Plus, 132 (2017), 13pp.
[21]	J. F. Gómez-Aguilar, L. Torres, H. Yépez-Martínez, D. Baleanu, J. M. Reyes and I. O. Sosa, Fractional Liénard type model of a pipeline within the fractional derivative without singular kernel, Adv. Difference Equ., 2016 (2016), Paper No. 173, 13 pp. doi: 10.1186/s13662-016-0908-1.
[22]	R. Hilfer, Application of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747.
[23]	S. Hristova and C. Tunc, Stability of nonlinear volterra integro-differential equations with caputo fractional derivative and bounded delays, Electron. J. Differential Equations, 2019 (2019), Paper No. 30, 11 pp.
[24]	D. Ji, Positive Solutions of Singular Fractional Boundary Value Problem with p-Laplacian., Bull. Malays. Math. Sci. Soc., 41 (2018), 249-263. doi: 10.1007/s40840-015-0276-0.
[25]	E. T. Karimov and K. Sadarangani, Existence of a unique positive solution for a singular fractional boundary value problem, Carpathian J. Math., 34 (2018), 57-64.
[26]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.
[27]	A. Khan, Y. Li, K. Shah and T. S. Khan, On coupled p-Laplacian fractional differential equations with nonlinear boundary conditions, Complexity, 2017 (2017), Art. ID 8197610, 9 pp. doi: 10.1155/2017/8197610.
[28]	H. Khan, C. Tunc, W. Chen and A. Khan, Existence theorems and Hyers-Ulam stability for a class of hybrid fractional differential equations with p-Laplacian operator, J. Appl. Anal. Comput., 8 (2018), 1211-1226.
[29]	H. Khan, W. Chen and H. Sun, Analysis of positive solution and Hyers–Ulam stability for a class of singular fractional differential equations with p–Laplacian in Banach space, Math. Methods Appl. Sci., 41 (2018), 3430-3440. doi: 10.1002/mma.4835.
[30]	B. López, J. Harjani and K. Sadarangani, Existence of positive solutions in the space of Lipschitz functions to a class of fractional differential equations of arbitrary order, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 112 (2018), 1281-1294. doi: 10.1007/s13398-017-0426-3.
[31]	R. Luca, On a class of nonlinear singular Riemann-Liouville fractional differential equations, Results Math., 73 (2018), Art. 125, 15 pp. doi: 10.1007/s00025-018-0887-5.
[32]	I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.
[33]	S. G. Samko, A. A. Kilbas and O. I Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.
[34]	K. Saoudi, A critical fractional elliptic equation with singular nonlinearities, Fract. Calc. Appl. Anal., 20 (2017), 1507-1530. doi: 10.1515/fca-2017-0079.
[35]	H. Srivastava, A. El-Sayed and F. Gaafar, A Class of Nonlinear Boundary Value Problems for an Arbitrary Fractional-Order Differential Equation with the Riemann-Stieltjes Functional Integral and Infinite-Point Boundary Conditions, Symmetry, 2018. doi: 10.3390/sym10100508.
[36]	S. Xie and Y. Xie, Nonlinear solutions of non local boundary value problems for nonlinear higher-order singular fractional differential equations, J. Appl. Anal. Comput., 8 (2018), 938-953.
[37]	F. Yan, M. Zuo and X. Hao, Positive solution for a fractional singular boundary value problem with p-Laplacian operator, Bound. Value Probl., 2018 (2018), Paper No. 51, 10 pp. doi: 10.1186/s13661-018-0972-4.
[38]	H. Yépez-Martínez, J. F. Gómez-Aguilar, I. O. Sosa, J. M. Reyes and J. Torres-Jiménez, The Feng's first integral method applied to the nonlinear mKdV space-time fractional partial differential equation, Rev. Mexicana Fís., 62 (2016), 310-316.
[39]	X. Zhang and Q. Zhong, Triple positive solutions for nonlocal fractional differential equations with singularities both on time and space variables, Appl. Math. Lett., 80 (2028), 12-19. doi: 10.1016/j.aml.2017.12.022.
[40]	L. Zhang, Z. Sun and X. Hao, Positive solutions for a singular fractional nonlocal boundary value problem, Adv. Difference Equ., 2018 (2018), Paper No. 381, 8 pp. doi: 10.1186/s13662-018-1844-z.
[41]	C. J. Zuñiga-Aguilar, J. F. Gómez-Aguilar and R. F. Escobar-Jiménez, Romero-Ugalde HM. Robust control for fractional variable-order chaotic systems with non-singular kernel, The European Physical Journal Plus, 133 (2018), 13pp.

show all references

References:

[1]	B. Ahmad, A. Alsaedi, R. P. Agarwal and A. Alsharif, On sequential fractional integro-differential equations with nonlocal integral boundary conditions, Bull. Malays. Math. Sci. Soc., 41 (2018), 1725-1737. doi: 10.1007/s40840-016-0421-4.
[2]	A. Atangana and J. F. Gómez-Aguilar, Hyperchaotic behaviour obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws, Chaos Solitons Fractals, 102 (2017), 285-294. doi: 10.1016/j.chaos.2017.03.022.
[3]	A. Atangana and J. F. Gómez-Aguilar, A new derivative with normal distribution kernel: Theory, methods and applications, Phys. A, 476 (2017), 1-14. doi: 10.1016/j.physa.2017.02.016.
[4]	A. Atangana and J. F. Gómez-Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, The European Physical Journal Plus, 133 (2018), 1-22. doi: 10.1140/epjp/i2018-12021-3.
[5]	A. Atangana and J. F. Gómez-Aguilar, Numerical approximation of Riemann-Liouville definition of fractional derivative: From Riemann-L iouville to Atangana-Baleanu, Numer. Methods Partial Differential Equations, 34 (2018), 1502-1523. doi: 10.1002/num.22195.
[6]	T. Abdeljawad, F. Jarad and D. Baleanu, On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives, Sci. China Ser. A, 51 (2008), 1775-1786. doi: 10.1007/s11425-008-0068-1.
[7]	T. Abdeljawad, D. Baleanu and F. Jarad, Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives, J. Math. Phys., 49 (2008), 083507, 11 pp. doi: 10.1063/1.2970709.
[8]	T. Abdeljawad and Q. M. Al-Mdallal, Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall's inequality, J. Comput. Appl. Math., 339 (2018), 218-230. doi: 10.1016/j.cam.2017.10.021.
[9]	T. Abdeljawad and J. Alzabut, On Riemann-Liouville fractional q–difference equations and their application to retarded logistic type model, Math. Methods Appl. Sci., 41 (2018), 8953-8962. doi: 10.1002/mma.4743.
[10]	B. Ahmad and R. Luca, Existence of solutions for sequential fractional integro-differential equations and inclusions with nonlocal boundary conditions, Appl. Math. Comput., 339 (2018), 516-534. doi: 10.1016/j.amc.2018.07.025.
[11]	J. Alzabut, T. Abdeljawad and D. Baleanu, Nonlinear delay fractional difference equations with application on discrete fractional Lotka-Volterra model, J. Comput. Anal. Appl., 25 (2018), 889-898.
[12]	J. Alzabut, T. Abdeljawad and D. Baleanu, Nonlinear delay fractional difference equations with application on discrete fractional Lotka-Volterra model, J. Comput. Anal. Appl., 25 (2018), 889-898.
[13]	T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Adv. Difference Equ., 2017 (2017), 11 pp. doi: 10.1186/s13662-017-1285-0.
[14]	A. Babakhani and T. Abdeljawad, A Caputo Fractional Order Boundary Value Problem with Integral Boundary Conditions, J. Comput. Anal. Appl., 15 (2013), 753-763.
[15]	Y. K. Chang and R. Ponce, Uniform exponential stability and applications to bounded solutions of integro-differential equations in Banach spaces, J. Integral Equations Appl., 30 (2018), 347-369. doi: 10.1216/JIE-2018-30-3-347.
[16]	A. Coronel-Escamilla, J. F. Gómez-Aguilar, M. G. López-López, V. M. Alvarado-Martínez and G. V. Guerrero-Ramírez, Triple pendulum model involving fractional derivatives with different kernels, Chaos Solitons Fractals, 91 (2016), 248-261. doi: 10.1016/j.chaos.2016.06.007.
[17]	J. Henderson and R. Luca, Systems of Riemann–Liouville fractional equations with multi-point boundary conditions, Appl. Math. Comput., 309 (2017), 303-323. doi: 10.1016/j.amc.2017.03.044.
[18]	L. Guo, L. Liu and Y. Wu, Iterative unique positive solutions for singular p-Laplacian fractional differential equation system with several parameters, Nonlinear Anal., Model. Control, 23 (2018), 182-203. doi: 10.15388/NA.2018.2.3.
[19]	A. Ghanmia, M. Kratoub and K. Saoudib, A Multiplicity Results for a Singular Problem Involving a Riemann-Liouville Fractional Derivative, Filomat, 32 (2018), 653-669. doi: 10.2298/FIL1802653G.
[20]	J. F. Gómez-Aguilar and A. Atangana, New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applications, The European Physical Journal Plus, 132 (2017), 13pp.
[21]	J. F. Gómez-Aguilar, L. Torres, H. Yépez-Martínez, D. Baleanu, J. M. Reyes and I. O. Sosa, Fractional Liénard type model of a pipeline within the fractional derivative without singular kernel, Adv. Difference Equ., 2016 (2016), Paper No. 173, 13 pp. doi: 10.1186/s13662-016-0908-1.
[22]	R. Hilfer, Application of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747.
[23]	S. Hristova and C. Tunc, Stability of nonlinear volterra integro-differential equations with caputo fractional derivative and bounded delays, Electron. J. Differential Equations, 2019 (2019), Paper No. 30, 11 pp.
[24]	D. Ji, Positive Solutions of Singular Fractional Boundary Value Problem with p-Laplacian., Bull. Malays. Math. Sci. Soc., 41 (2018), 249-263. doi: 10.1007/s40840-015-0276-0.
[25]	E. T. Karimov and K. Sadarangani, Existence of a unique positive solution for a singular fractional boundary value problem, Carpathian J. Math., 34 (2018), 57-64.
[26]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.
[27]	A. Khan, Y. Li, K. Shah and T. S. Khan, On coupled p-Laplacian fractional differential equations with nonlinear boundary conditions, Complexity, 2017 (2017), Art. ID 8197610, 9 pp. doi: 10.1155/2017/8197610.
[28]	H. Khan, C. Tunc, W. Chen and A. Khan, Existence theorems and Hyers-Ulam stability for a class of hybrid fractional differential equations with p-Laplacian operator, J. Appl. Anal. Comput., 8 (2018), 1211-1226.
[29]	H. Khan, W. Chen and H. Sun, Analysis of positive solution and Hyers–Ulam stability for a class of singular fractional differential equations with p–Laplacian in Banach space, Math. Methods Appl. Sci., 41 (2018), 3430-3440. doi: 10.1002/mma.4835.
[30]	B. López, J. Harjani and K. Sadarangani, Existence of positive solutions in the space of Lipschitz functions to a class of fractional differential equations of arbitrary order, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 112 (2018), 1281-1294. doi: 10.1007/s13398-017-0426-3.
[31]	R. Luca, On a class of nonlinear singular Riemann-Liouville fractional differential equations, Results Math., 73 (2018), Art. 125, 15 pp. doi: 10.1007/s00025-018-0887-5.
[32]	I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.
[33]	S. G. Samko, A. A. Kilbas and O. I Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.
[34]	K. Saoudi, A critical fractional elliptic equation with singular nonlinearities, Fract. Calc. Appl. Anal., 20 (2017), 1507-1530. doi: 10.1515/fca-2017-0079.
[35]	H. Srivastava, A. El-Sayed and F. Gaafar, A Class of Nonlinear Boundary Value Problems for an Arbitrary Fractional-Order Differential Equation with the Riemann-Stieltjes Functional Integral and Infinite-Point Boundary Conditions, Symmetry, 2018. doi: 10.3390/sym10100508.
[36]	S. Xie and Y. Xie, Nonlinear solutions of non local boundary value problems for nonlinear higher-order singular fractional differential equations, J. Appl. Anal. Comput., 8 (2018), 938-953.
[37]	F. Yan, M. Zuo and X. Hao, Positive solution for a fractional singular boundary value problem with p-Laplacian operator, Bound. Value Probl., 2018 (2018), Paper No. 51, 10 pp. doi: 10.1186/s13661-018-0972-4.
[38]	H. Yépez-Martínez, J. F. Gómez-Aguilar, I. O. Sosa, J. M. Reyes and J. Torres-Jiménez, The Feng's first integral method applied to the nonlinear mKdV space-time fractional partial differential equation, Rev. Mexicana Fís., 62 (2016), 310-316.
[39]	X. Zhang and Q. Zhong, Triple positive solutions for nonlocal fractional differential equations with singularities both on time and space variables, Appl. Math. Lett., 80 (2028), 12-19. doi: 10.1016/j.aml.2017.12.022.
[40]	L. Zhang, Z. Sun and X. Hao, Positive solutions for a singular fractional nonlocal boundary value problem, Adv. Difference Equ., 2018 (2018), Paper No. 381, 8 pp. doi: 10.1186/s13662-018-1844-z.
[41]	C. J. Zuñiga-Aguilar, J. F. Gómez-Aguilar and R. F. Escobar-Jiménez, Romero-Ugalde HM. Robust control for fractional variable-order chaotic systems with non-singular kernel, The European Physical Journal Plus, 133 (2018), 13pp.

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Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations

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Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations

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