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Oscillation criteria for second-order quasi-linear neutral functional differential equation
Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations
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 |
$ \mho_p $ |
$ p $ |
$ \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 $ |
$ t $ |
$ x(t) $ |
$ t\in [0, 1] $ |
$ \mathcal{D}^{\gamma}, \, $ |
$ \mathcal{D}^{\delta^*}, \mathcal{D}^{\kappa} $ |
$ \delta^*, \, \gamma\in(1, 2] $ |
$ n-1<\kappa\leq n, $ |
$ n\geq3 $ |
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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