Fractional Boundary Value Problem on the Half-Line

Автор(и)

  • Bilel Khamessi Department of Mathematics, College of Science, Taibah University, Al-Madinah Al-Munawwarah, Saudi-Arabia
    Université Tunis El Manar, Faculté des sciences de Tunis, LR18ES09 Modélisation mathématique, analyse harmonique et théorie du potentiel, 2092 Tunis, Tunisia

DOI:

https://doi.org/10.15407/mag16.01.027

Ключові слова:

рівняння з дробовими похідними, позитивний розв'язок, теорема Шаудера про нерухому точку.

Анотація

У статті розглядається напівлінійна крайова задача з дробовими похідними \begin{equation*} D^{\beta}\left(\frac{1}{b(x)}D^{\alpha}u\right)=a(x)u^{\sigma} \quad\text{на } (0,\infty) \end{equation*} з умовами $\lim_{x\rightarrow 0} x^{2-\beta} \frac{1}{b(x)}D^{\alpha}u(x) =\lim_{x\rightarrow \infty} x^{1-\beta}\frac{1}{b(x)}D^{\alpha}u(x)=0$ та $\lim_{x\rightarrow 0} x^{2-\alpha}u(x)= \lim_{x\rightarrow \infty} x^{1-\alpha}u(x)=0$, де $\beta,\alpha \in (1,2)$, $\sigma\in(-1,1)$ і $D^{\beta}, D^{\alpha}$ означають стандартні дробові похідні Рімана--Ліувілля. Функції $ a,b : (0,\infty)\longrightarrow \mathbb{R}$ є невід'ємними неперервними функціями, які задовольняють деякі відповідні умови. Встановлено існування та єдиність позитивного розв'язку. Також надано опис глобальної поведінки цього розв'язку.

Mathematics Subject Classification: 34A08, 35B09, 47H10.

Посилання

R.P. Agarwal and D. O’Regan, Boundary value problems of nonsingular type on the semi-infinite interval, Tohoku Math. J. (2) 51 (1999), 391–397. https://doi.org/10.2748/tmj/1178224769

R.P. Agarwal, and D. O’Regan, Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht,The Netherlands, 2001. https://doi.org/10.1007/978-94-010-0718-4

R.P. Agarwal, D. O’Regan, and S. Staněk, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, J. Math. Anal. Appl. 371 (2010), 57–68. https://doi.org/10.1016/j.jmaa.2010.04.034

R.P. Agarwal, M. Benchohra, S. Hamani, and S. Pinelas, Boundary value problems for differential equations involving Riemann–Liouville fractional derivative on the half line, Dyn. Contin. Discrete Impulsive Syst. A 18 (2011), 235–244.

B. Ahmad, Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations, Appl. Math. Lett. 23 (2010), 390–394. https://doi.org/10.1016/j.aml.2009.11.004

B. Ahmad and J.J. Nieto, Rieman–Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions, Bound. Value Probl. 2011 (2011), Art. No.: 36 (2011). https://doi.org/10.1186/1687-2770-2011-36

A. Arara, M. Benchohra, N. Hamidi, and J.J. Nieto, Fractional order differential equations on an unbounded domain, Nonlinear Anal. 72 (2010), 580–586. https://doi.org/10.1016/j.na.2009.06.106

I. Bachar and H. Mâagli, Existence and global asymptotic behavior of positive solutions for nonlinear fractional Dirichlet problems on the half-line, Abstract. Appl. Anal. (2014), Art. ID 537971. https://doi.org/10.1155/2014/537971

Z. Bai and H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equations, J. Math. Anal. Appl. 311 (2005), 495–505. https://doi.org/10.1016/j.jmaa.2005.02.052

S. Ben Othman, S. Dridi, and B. Khamessi, Combined effects in fractional boundary value problem, Int. J. Nonlinear Sc. 20 (2015), 154–165.

R. Chemmam, A. Dhifli, and H. Mâagli, Asymptotic behavior of ground state solutions for sublinear and singular nonlinear Dirichlet problems, Electron. J. Differ. Equ., 88 (2011), 1–12.

R. Chemmam, H. Mâagli, S. Masmoudi, and M. Zribi, Combined effects in nonlinear singular elliptic problems in a bounded domain, Adv. Nonlinear Anal. 1 (2012), 301– 318. https://doi.org/10.1515/anona-2012-0008

Y. Chen and X. Tang, Positive solutions of fractional differential equations at resonance on the half-line, Bound. Value Probl. 2012 (2012), Art. No.: 64 (2012). https://doi.org/10.1186/1687-2770-2012-64

A. Dhifli and B. Khamessi, Existence and boundary behavior of positive solution for a Sturm–Liouville fractional problem with p-laplacian, J. Fix. Point Theory A 19 (2017), 2763-2784. https://doi.org/10.1007/s11784-017-0455-8

K. Diethelm and A.D. Freed, On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, (eds. F. Keil, W. Mackens, H. Voss, and J. Werther), Springer, Heidelberg, 1999, 217–307. https://doi.org/10.1007/978-3-642-60185-9_24

B. Khamessi and A. Hamiaz, Existence and exact asymptotic behaviour of positive solutions for fractional boundary value problem with p-Laplacian operator, J. Taibah University Sci. 13 (2019), 370–376. https://doi.org/10.1080/16583655.2019.1579953

A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam 2006.

C. Kou, H. Zhou, and Y. Yan, Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis, Nonlinear Anal. 74 (2011), 5975–5986. https://doi.org/10.1016/j.na.2011.05.074

Y. Liu, W. Zhang, and X. Liu, A sufficient condition for the existence of a positive solution for a nonlinear fractional differential equation with the Riemann–Liouville derivative, Appl. Math. Lett. 25 (2012), 1986–1992. https://doi.org/10.1016/j.aml.2011.08.016

Y. Liu, Existence of solutions of periodic-type boundary value problems for multiterm fractional differential equations, Math. Meth. Appl. Sci. 36 (2013), 2187–2207. https://doi.org/10.1002/mma.2700

H. Mâagli, Existence of positive solutions for a nonlinear fractional differential equation, Electron. J. Differ. Equ. 2013 (2013), No. 29, 1–5.

H. Mâagli and A. Dhifli, Positive solutions to a nonlinear fractional Dirichlet problem on the half-line, Electron. J. Differ. Equ. 2014 (2014), No. 50, 1–7. https://doi.org/10.1155/2014/537971

V. Marić, Regular Variation and Differential Equations, 1726, Lecture Notes in Mathematics, Springer, Berlin, 2000. https://doi.org/10.1007/BFb0103952

I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal. 5 (2002), 367–386.

T. Qiu and Z. Bai, Existence of positive solutions for singular fractional differential equations, Electron. J. Differ. Equ. 146 (2008), 1–9.

R. Seneta, Regular Varying Functions, 508, Lectures Notes in Mathematics Springer-Verlag, Berlin, 1976. https://doi.org/10.1007/BFb0079658

X. Su and S. Zhang, Unbounded solutions to a boundary value problem of fractional order on the half-line, Comput. Math. Appl. 61 (2011), 1079–1087. https://doi.org/10.1016/j.camwa.2010.12.058

X. Zhao and W. Ge, Unbounded solutions for a fractional boundary value problems on the infinite interval, Acta Appl. Math. 109 (2010), 495–505. https://doi.org/10.1007/s10440-008-9329-9

Y. Zhao, S. Sun, Z. Han, and Q. Li, Positive solutions to boundary value problems of nonlinear fractional differential equations, Abstract. Appl. Anal. (2011), Article ID 390543. https://doi.org/10.1155/2011/390543

Downloads

Як цитувати

(1)
Khamessi, B. Fractional Boundary Value Problem on the Half-Line. Журн. мат. фіз. анал. геом. 2020, 16, 27-45.

Номер

Розділ

Статті

Завантаження

Дані завантаження ще не доступні.