Existence of Solution for a Nonlinear Fractional Order Differential Equation with a Quadratic Perturbations

Ahmed Kajounı; Najat Chefnaj; Khalid Hilal

doi:10.53006/rna.1124961

Araştırma Makalesi

Yıl 2022, Cilt: 5 Sayı: 3, 360 - 371, 30.09.2022

Ahmed Kajounı Najat Chefnaj Khalid Hilal

https://doi.org/10.53006/rna.1124961

Öz

Kaynakça

[1] R.P. Agarwal, V. Lakshmikantham, J. J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlinear Analysis, 72 (2010)2859-2862.
[2] M. Asadi,On Ekeland’s variational principle in M-metric spaces, Journal of nonlinear and convex analysis 17 (6)(2016) 1151-1158.
[3] M. Asadi, M. Gabeleh, C. Vetro, A New Approach to the Generalization of Darbo’s Fixed Point Problem by Using Simulation Functions with Application to Integral Equations, Results Math 86 (2019).
[4] M. Asadi, P. Salimi, Fixed Point and Common Fixed Point Theorems on G-Metric Spaces, Functional Analysis and Applications 21 (3)(2016) 523-530.
[5] A. Babakhanil, D. Baleanu, Existence and Uniqueness of Solution for a Class of Nonlinear Fractional Order Differential Equations, Hindawi Publishing Corporation, Abstract and Applied Analysis,( 2012).
[6] Z. Bai and H. Lu, Positive solutions for boundary value problem of nonlinear fractional differential equation, Journal of Mathematical Analysis and Applications, 311 (2005) 495-505.
[7] L. Byszewski and V. Lakshmikantham, Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Appl. Anal. 40 (1991) 11-19.
[8] M. Caputo, Linear models of dissipation whose Q is almost frequency independent, part II, Journal of the Royal Society of Western Australia, 13( 1967) 529-539.
[9] B.C. Dhage, On a condensing mappings in Banach algebras, Math. Student 63 (1994) 146-152.
[10] B.C. Dhage, On a fixed point theorem in Banach algebras with applications, Appl. Math. Lett. 18 (2005) 273-280.
[11] B.C. Dhage, V. Lakshmikantham, Basic results on hybrid differential equations, Nonlinear Anal. Hybrid 4 (2010)414-424.
[12] B.C. Dhage, V. Lakshmikantham, Quadratic perturbations of periodic boundary value problems of second order ordinary differential equations, Diff. Eq. et App. 2 (2010).
[13] A. El Mfadel, S. Melliani and M. Elomari, Existence and uniqueness results for Caputo fractional boundary value problems involving the p-Laplacian operator, U.P.B. Sci. Bull. Series A. 84(1) (2022)37-46 .
[14] A. El Mfadel, S. Melliani and M. Elomari, New existence results for nonlinear functional hybrid differential equations involving the Ψ- Caputo fractional derivative, Results in Nonlinear Analysis. 5(1)(2022) 78-86.
[15] A. El Mfadel, S. Melliani and M. Elomari, Existence results for nonlocal Cauchy problem of nonlinear Ψ-Caputo type fractional differ- ential equations via topological degree methods, Advances in the Theory of Nonlinear Analysisand its Application, 6(2) (2022)270-279.
[16] A.M.A. El-Sayed, Fractional order evolution equations, J. Fract. Calc. 7 (1995).
[17] K. Hilal, A. Kajouni, Boundary value problems for hybrid differential equations with fractional order, Adv Differ Equ , 183 (2015).
[18] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier, 204( 2006).
[19] Ü. Lepik, Solving fractional integral equations by the Haar wavelet method, Applied Mathematics and Computation, 214( 2009)468-478.
[20] H. Monfared, M. Asadi, M. Azhini, Coupled fixed point theorems for generalized contractions in ordered M -metric spaces, Results in Fixed Point Theory and Applications, (2018).
[21] H. Monfared, M. Asadi, M. Azhini, D. O’Regan, F(ψ,ϕ)-Contractions for α-admissible mappings on M-metric spaces , Fixed Point Theory and Applications,1 (2018) 22.
[22] H. Monfared, M. Azhini, M. Asadi, A generalized Contraction Principe with Control Function on M-Metric spaces, Results in Fixed Point Theory and Applications, (2018).
[23] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Application, Mathematics in Science and Engineering, 198 (1999).
[24] Y. Zhao, S. Sun, Z. Han, Q. Li, Theory of fractional hybrid differential equations, Computers and Mathematics with Appl. 62 (2011) 1312-1324.

Existence of Solution for a Nonlinear Fractional Order Differential Equation with a Quadratic Perturbations

Yıl 2022, Cilt: 5 Sayı: 3, 360 - 371, 30.09.2022

Ahmed Kajounı Najat Chefnaj Khalid Hilal

https://doi.org/10.53006/rna.1124961

Öz

In this work, we prove the existence of a solution for the initial value problem of nonlinear fractional differential equation with quadratic perturbations involving the Caputo fractional derivative
$(^{c} D_{0^{+}}^{α} - ρ t^{c} D_{0^{+}}^{β}) (\frac{x (t)}{f (t, x (t))}) = g (t, x (t)), t \in J = [0, 1], 1 < α < 2, 0 < β < α$
with conditions $x_{0} = \frac{x (0)}{f (0, x (0))}$ and \\ $x_{1} = \frac{x (1)}{f (1, x (1))}$ . Dhage's fixed-point
the theorem was used to establish this existence. As an application, we have given
example to demonstrate the effectiveness of our main result.

Anahtar Kelimeler

Fractional differential equation, Quadratic perturbations, Dhage fixed point.

Kaynakça

[1] R.P. Agarwal, V. Lakshmikantham, J. J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlinear Analysis, 72 (2010)2859-2862.
[2] M. Asadi,On Ekeland’s variational principle in M-metric spaces, Journal of nonlinear and convex analysis 17 (6)(2016) 1151-1158.
[3] M. Asadi, M. Gabeleh, C. Vetro, A New Approach to the Generalization of Darbo’s Fixed Point Problem by Using Simulation Functions with Application to Integral Equations, Results Math 86 (2019).
[4] M. Asadi, P. Salimi, Fixed Point and Common Fixed Point Theorems on G-Metric Spaces, Functional Analysis and Applications 21 (3)(2016) 523-530.
[5] A. Babakhanil, D. Baleanu, Existence and Uniqueness of Solution for a Class of Nonlinear Fractional Order Differential Equations, Hindawi Publishing Corporation, Abstract and Applied Analysis,( 2012).
[6] Z. Bai and H. Lu, Positive solutions for boundary value problem of nonlinear fractional differential equation, Journal of Mathematical Analysis and Applications, 311 (2005) 495-505.
[7] L. Byszewski and V. Lakshmikantham, Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Appl. Anal. 40 (1991) 11-19.
[8] M. Caputo, Linear models of dissipation whose Q is almost frequency independent, part II, Journal of the Royal Society of Western Australia, 13( 1967) 529-539.
[9] B.C. Dhage, On a condensing mappings in Banach algebras, Math. Student 63 (1994) 146-152.
[10] B.C. Dhage, On a fixed point theorem in Banach algebras with applications, Appl. Math. Lett. 18 (2005) 273-280.
[11] B.C. Dhage, V. Lakshmikantham, Basic results on hybrid differential equations, Nonlinear Anal. Hybrid 4 (2010)414-424.
[12] B.C. Dhage, V. Lakshmikantham, Quadratic perturbations of periodic boundary value problems of second order ordinary differential equations, Diff. Eq. et App. 2 (2010).
[13] A. El Mfadel, S. Melliani and M. Elomari, Existence and uniqueness results for Caputo fractional boundary value problems involving the p-Laplacian operator, U.P.B. Sci. Bull. Series A. 84(1) (2022)37-46 .
[14] A. El Mfadel, S. Melliani and M. Elomari, New existence results for nonlinear functional hybrid differential equations involving the Ψ- Caputo fractional derivative, Results in Nonlinear Analysis. 5(1)(2022) 78-86.
[15] A. El Mfadel, S. Melliani and M. Elomari, Existence results for nonlocal Cauchy problem of nonlinear Ψ-Caputo type fractional differ- ential equations via topological degree methods, Advances in the Theory of Nonlinear Analysisand its Application, 6(2) (2022)270-279.
[16] A.M.A. El-Sayed, Fractional order evolution equations, J. Fract. Calc. 7 (1995).
[17] K. Hilal, A. Kajouni, Boundary value problems for hybrid differential equations with fractional order, Adv Differ Equ , 183 (2015).
[18] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier, 204( 2006).
[19] Ü. Lepik, Solving fractional integral equations by the Haar wavelet method, Applied Mathematics and Computation, 214( 2009)468-478.
[20] H. Monfared, M. Asadi, M. Azhini, Coupled fixed point theorems for generalized contractions in ordered M -metric spaces, Results in Fixed Point Theory and Applications, (2018).
[21] H. Monfared, M. Asadi, M. Azhini, D. O’Regan, F(ψ,ϕ)-Contractions for α-admissible mappings on M-metric spaces , Fixed Point Theory and Applications,1 (2018) 22.
[22] H. Monfared, M. Azhini, M. Asadi, A generalized Contraction Principe with Control Function on M-Metric spaces, Results in Fixed Point Theory and Applications, (2018).
[23] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Application, Mathematics in Science and Engineering, 198 (1999).
[24] Y. Zhao, S. Sun, Z. Han, Q. Li, Theory of fractional hybrid differential equations, Computers and Mathematics with Appl. 62 (2011) 1312-1324.

Toplam 24 adet kaynakça vardır.

Ayrıntılar

Birincil Dil	İngilizce
Konular	Matematik
Bölüm	Articles
Yazarlar	Ahmed Kajounı Najat Chefnaj Khalid Hilal
Yayımlanma Tarihi	30 Eylül 2022
Yayımlandığı Sayı	Yıl 2022 Cilt: 5 Sayı: 3

Kaynak Göster

APA	Kajounı, A., Chefnaj, N., & Hilal, K. (2022). Existence of Solution for a Nonlinear Fractional Order Differential Equation with a Quadratic Perturbations. Results in Nonlinear Analysis, 5(3), 360-371. https://doi.org/10.53006/rna.1124961
AMA	Kajounı A, Chefnaj N, Hilal K. Existence of Solution for a Nonlinear Fractional Order Differential Equation with a Quadratic Perturbations. RNA. Eylül 2022;5(3):360-371. doi:10.53006/rna.1124961
Chicago	Kajounı, Ahmed, Najat Chefnaj, ve Khalid Hilal. “Existence of Solution for a Nonlinear Fractional Order Differential Equation With a Quadratic Perturbations”. Results in Nonlinear Analysis 5, sy. 3 (Eylül 2022): 360-71. https://doi.org/10.53006/rna.1124961.
EndNote	Kajounı A, Chefnaj N, Hilal K (01 Eylül 2022) Existence of Solution for a Nonlinear Fractional Order Differential Equation with a Quadratic Perturbations. Results in Nonlinear Analysis 5 3 360–371.
IEEE	A. Kajounı, N. Chefnaj, ve K. Hilal, “Existence of Solution for a Nonlinear Fractional Order Differential Equation with a Quadratic Perturbations”, RNA, c. 5, sy. 3, ss. 360–371, 2022, doi: 10.53006/rna.1124961.
ISNAD	Kajounı, Ahmed vd. “Existence of Solution for a Nonlinear Fractional Order Differential Equation With a Quadratic Perturbations”. Results in Nonlinear Analysis 5/3 (Eylül 2022), 360-371. https://doi.org/10.53006/rna.1124961.
JAMA	Kajounı A, Chefnaj N, Hilal K. Existence of Solution for a Nonlinear Fractional Order Differential Equation with a Quadratic Perturbations. RNA. 2022;5:360–371.
MLA	Kajounı, Ahmed vd. “Existence of Solution for a Nonlinear Fractional Order Differential Equation With a Quadratic Perturbations”. Results in Nonlinear Analysis, c. 5, sy. 3, 2022, ss. 360-71, doi:10.53006/rna.1124961.
Vancouver	Kajounı A, Chefnaj N, Hilal K. Existence of Solution for a Nonlinear Fractional Order Differential Equation with a Quadratic Perturbations. RNA. 2022;5(3):360-71.

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