Newly existence of solutions for pantograph a semipositone in $ \Psi $-Caputo sense

Abdelkader Moumen; Hamid Boulares; Tariq Alraqad; Hicham Saber; Ekram E. Ali; Abdelkader Moumen; Hamid Boulares; Tariq Alraqad; Hicham Saber; Ekram E. Ali

doi:10.3934/math.2023646

AIMS Mathematics

2023, Volume 8, Issue 6: 12830-12840. doi: 10.3934/math.2023646

Previous Article Next Article

Research article Special Issues

Newly existence of solutions for pantograph a semipositone in $ \Psi $-Caputo sense

1.
Department of Mathematics, Faculty of Science, University of Ha'il, Ha'il 55425, Saudi Arabia
2.
Laboratory of Analysis and Control of Differential Equations "ACED", Faculty MISM, Department of Mathematics, University of 8 May 1945 Guelma, P.O. Box 401, 24000 Guelma, Algeria

Received: 08 December 2022 Revised: 22 March 2023 Accepted: 28 March 2023 Published: 30 March 2023
MSC : 34A08, 34B10

In the present manuscript, the BVP problem of a semipostone multipoint $ \Psi $-Caputo fractional pantograph problem is addressed.

$ \mathcal{D}_{r}^{\nu;\psi}\varkappa(\varsigma)+\mathcal{F}(\varsigma , \varkappa(\varsigma), \varkappa(r+\lambda\varsigma)) = 0, \ \varsigma \mbox{ in }(r, \mathcal{\Im}), $

$ \varkappa(r) = \vartheta_{1}, \ \varkappa(\mathcal{\Im}) = \sum\limits_{i = 1}^{m-2} \zeta_{i}\varkappa(\mathfrak{\eta}_{i})+\vartheta_{2}, \ \vartheta_{i} \in\mathbb{R}, \ i\in\{1, 2\}, $

and $ \lambda $ in $ \left(0, \frac{\mathcal{\Im}\mathfrak{-}r}{\mathcal{\Im} }\right) $. The seriousness of this research is to prove the existence of the solution of this problem by using Schauder's fixed point theorem (SFPT). We have developed our results in our research compared to some recent research in this field. We end our work by listing an example to demonstrate the result reached.
- $ \Psi $-Caputo derivative,
- BVP,
- pantograph problem,
- changing sign nonlinearity,
- Schauder fixed point theorem
Citation: Abdelkader Moumen, Hamid Boulares, Tariq Alraqad, Hicham Saber, Ekram E. Ali. Newly existence of solutions for pantograph a semipositone in $ \Psi $-Caputo sense[J]. AIMS Mathematics, 2023, 8(6): 12830-12840. doi: 10.3934/math.2023646

Related Papers:

Abstract

In the present manuscript, the BVP problem of a semipostone multipoint $ \Psi $-Caputo fractional pantograph problem is addressed.

$ \mathcal{D}_{r}^{\nu;\psi}\varkappa(\varsigma)+\mathcal{F}(\varsigma , \varkappa(\varsigma), \varkappa(r+\lambda\varsigma)) = 0, \ \varsigma \mbox{ in }(r, \mathcal{\Im}), $

$ \varkappa(r) = \vartheta_{1}, \ \varkappa(\mathcal{\Im}) = \sum\limits_{i = 1}^{m-2} \zeta_{i}\varkappa(\mathfrak{\eta}_{i})+\vartheta_{2}, \ \vartheta_{i} \in\mathbb{R}, \ i\in\{1, 2\}, $

and $ \lambda $ in $ \left(0, \frac{\mathcal{\Im}\mathfrak{-}r}{\mathcal{\Im} }\right) $. The seriousness of this research is to prove the existence of the solution of this problem by using Schauder's fixed point theorem (SFPT). We have developed our results in our research compared to some recent research in this field. We end our work by listing an example to demonstrate the result reached.

References

[1]	B. Ahmad, Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations, Appl. Math. Lett., 23 (2010), 390–394. http://dx.doi.org/10.1016/j.aml.2009.11.004 doi: 10.1016/j.aml.2009.11.004
[2]	Z. Bai, On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Anal.-Theor., 72 (2010), 916–924. http://dx.doi.org/10.1016/j.na.2009.07.033 doi: 10.1016/j.na.2009.07.033
[3]	Z. Bai, H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311 (2005), 495–505. http://dx.doi.org/10.1016/j.jmaa.2005.02.052 doi: 10.1016/j.jmaa.2005.02.052
[4]	R. Almeida, A. Malinowska, M. Teresa, T. Monteiro, Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications, Math. Method. Appl. Sci., 41 (2018), 336–352. http://dx.doi.org/10.1002/mma.4617 doi: 10.1002/mma.4617
[5]	A. Kilbas, H. Srivastava, J. Trujillo, Theory and applications of fractional differential equations, Boston: Elsevier, 2006. http://dx.doi.org/10.1016/S0304-0208(06)80001-0
[6]	K. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, New York: Wiley, 1993.
[7]	I. Podlubny, Fractional differential equations, mathematics in science and engineering, New York: Academic Press, 1999.
[8]	H. Boulares, A. Benchaabane, N. Pakkaranang, R. Shafqat, B. Panyanak, Qualitative properties of positive solutions of a kind for fractional pantograph problems using technique fixed point theory, Fractal Fract., 6 (2022), 593. http://dx.doi.org/10.3390/fractalfract6100593 doi: 10.3390/fractalfract6100593
[9]	A. Hallaci, H. Boulares, A. Ardjouni, Existence and uniqueness for delay fractional differential equations with mixed fractional derivatives, Open J. Math. Anal., 4 (2020), 26–31. http://dx.doi.org/10.30538/psrp-oma2020.0059 doi: 10.30538/psrp-oma2020.0059
[10]	A. Hallaci, H. Boulares, M. Kurulay, On the study of nonlinear fractional differential equations on unbounded interval, General Letters in Mathematics, 5 (2018), 111–117. http://dx.doi.org/10.31559/glm2018.5.3.1 doi: 10.31559/glm2018.5.3.1
[11]	A. Ardjouni, H. Boulares, Y. Laskri, Stability in higher-order nonlinear fractional differential equations, Acta Comment. Univ. Ta., 22 (2018), 37–47. http://dx.doi.org/10.12697/ACUTM.2018.22.04 doi: 10.12697/ACUTM.2018.22.04
[12]	S. Liang, J. Zhang, Positive solutions for boundary value problems of nonlinear fractional differential equation, Nonlinear Anal.-Theor., 71 (2009), 5545–5550. http://dx.doi.org/10.1016/j.na.2009.04.045 doi: 10.1016/j.na.2009.04.045
[13]	S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electron. J. Differ. Eq., 2006 (2006), 36.
[14]	W. Zhong, W. Lin, Nonlocal and multiple-point boundary value problem for fractional differential equations, Comput. Math. Appl., 59 (2010), 1345–1351. http://dx.doi.org/10.1016/j.camwa.2009.06.032 doi: 10.1016/j.camwa.2009.06.032
[15]	E. Doha, A. Bhrawy, S. Ezz-Eldien, A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order, Comput. Math. Appl., 62 (2011), 2364–2373. http://dx.doi.org/10.1016/j.camwa.2011.07.024 doi: 10.1016/j.camwa.2011.07.024
[16]	M. Alsuyuti, E. Doha, S. Ezz-Eldien, Modified Galerkin algorithm for solving multitype fractional differential equations, Math. Meth. Appl. Sci., 42 (2019), 1389–1412. http://dx.doi.org/10.1002/mma.5431 doi: 10.1002/mma.5431
[17]	S. Ezz-Eldien, Y. Wang, M. Abdelkawy, M. Zaky, A. Aldraiweesh, J. Tenreiro Machado, Chebyshev spectral methods for multi-order fractional neutral pantograph equations, Nonlinear Dyn., 100 (2020), 3785–3797. http://dx.doi.org/10.1007/s11071-020-05728-x doi: 10.1007/s11071-020-05728-x
[18]	M. Alsuyuti, E. Doha, S. Ezz-Eldien, I. Youssef, Spectral Galerkin schemes for a class of multi-order fractional pantograph equations, J. Comput. Appl. Math., 384 (2021), 113157. http://dx.doi.org/10.1016/j.cam.2020.113157 doi: 10.1016/j.cam.2020.113157
[19]	J. Hale, Retarded functional differential equations: basic theory, New York: Springer, 1977. http://dx.doi.org/10.1007/978-1-4612-9892-2_3
[20]	K. Mahler, On a special functional equation, J. Lond. Math. Soc., 1 (1940), 115–123. http://dx.doi.org/10.1112/JLMS/S1-15.2.115 doi: 10.1112/JLMS/S1-15.2.115
[21]	L. Fox, D. Mayers, J. Ockendon, A. Tayler, On a functional differential equation, IMA J. Appl. Math., 8 (1971), 271–307. http://dx.doi.org/10.1093/imamat/8.3.271 doi: 10.1093/imamat/8.3.271
[22]	J. Ockendon, A. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. R. Soc. Lond. A, 322 (1971), 447–468. http://dx.doi.org/10.1098/rspa.1971.0078 doi: 10.1098/rspa.1971.0078
[23]	D. Smart, Fixed point theorems, Cambridge: Cambridge University Press, 1980.
[24]	J. Wang, Y. Zhou, M. Feckan, Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comput. Math. Appl., 64 (2012), 3389–3405. http://dx.doi.org/10.1016/j.camwa.2012.02.021 doi: 10.1016/j.camwa.2012.02.021
[25]	F. Li, Y. Zhang, Y. Li, Sign-changing solutions on a kind of fourth-order Neumann boundary value problem, J. Math. Anal. Appl., 344 (2008), 417–428. http://dx.doi.org/10.1016/j.jmaa.2008.02.050 doi: 10.1016/j.jmaa.2008.02.050
[26]	Y. Li, F. Li, Sign-changing solutions to second-order integral boundary value problems, Nonlinear Anal.-Theor., 69 (2008), 1179–1187. http://dx.doi.org/10.1016/j.na.2007.06.024 doi: 10.1016/j.na.2007.06.024
[27]	Z. Liu, Y. Ding, C. Liu, C. Zhao, Existence and uniqueness of solutions for singular fractional differential equation boundary value problem with p-Laplacian, Adv. Differ. Equ., 2020 (2020), 83. http://dx.doi.org/10.1186/s13662-019-2482-9 doi: 10.1186/s13662-019-2482-9
[28]	A. Tychonoff, Ein fixpunktsatz, Math. Ann., 111 (1935), 767–776. http://dx.doi.org/10.1007/BF01472256
[29]	X. Xu, Multiple sign-changing solutions for some m-point boundary-value problems, Electron. J. Differ. Eq., 2004 (2004), 1–14.
[30]	B. Ahmad, Sharp estimates for the unique solution of two-point fractional-order boundary value problems, Appl. Math. Lett., 65 (2017), 77–82. http://dx.doi.org/10.1016/j.aml.2016.10.008 doi: 10.1016/j.aml.2016.10.008

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)