American Institute of Mathematical Sciences

Advanced
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 & 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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[22]

R. Hilfer, Application of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[22]

R. Hilfer, Application of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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