Volume 4, Issue 3
Triple Positive Solutions for a Class of Fractional Boundary Value Problem System

Zhanbing Bai & Dongmei Shi

J. Nonl. Mod. Anal., 4 (2022), pp. 454-464.

Published online: 2022-06

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  • Abstract

In this paper, the solvability for the following fractional boundary value problem system $$^CD^{σ_1}_{0+}v_1(t) = f_1(t, v_2(t), D^{\mu_1}_{0+}v_2(t)), \ 0 < t < 1,$$ $$^CD^{σ_2}_{0+}v_2(t) = f_2(t, v_1(t), D^{\mu_2}_{0+}v_1(t)), \ 0 < t < 1,$$ $$v_1' (0) = bv_1(0), \ v_1''(0) = 0, \ ^CD^{θ_1}_{0+}v_1(1) = a · ^C D^{θ_2} _{0+}v_1(η),$$ $$v_2'(0) = bv_2(0), \ v_2''(0) = 0, \ ^CD^{θ_1}_{0+}v_2(1) = a · ^CD^{θ_2}_{0+}v_2(η),$$ is studied, where $a > 0,$ $−1 < b < 0,$ $2 < σ_1, σ_2 ≤ 3,$ $0 < η < 1,$ $0 < \mu_1, \mu_2 ≤ 1,$ $0 < θ_2 ≤ θ_1 ≤ 1,$ $f_1, f_2:$ $[0, 1] × \mathbb{R}^+ \times \mathbb{R} → \mathbb{R}^+$ are continuous, $^CD^{σ_1} _{0+}v_1(t),$ $^CD^{σ_2}_{0+}v_2(t)$ are the Caputo fractional derivatives, and $D^{\mu_1}_{0+}v_2(t),$ $D^{\mu_2}_{0+}v_1(t)$ are the Riemann-Liouville fractional derivatives. The fixed point theorem is used to prove that there are three positive solutions to problems.

  • AMS Subject Headings

34A08, 34A37, 34A12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JNMA-4-454, author = {Bai , Zhanbing and Shi , Dongmei}, title = {Triple Positive Solutions for a Class of Fractional Boundary Value Problem System}, journal = {Journal of Nonlinear Modeling and Analysis}, year = {2022}, volume = {4}, number = {3}, pages = {454--464}, abstract = {

In this paper, the solvability for the following fractional boundary value problem system $$^CD^{σ_1}_{0+}v_1(t) = f_1(t, v_2(t), D^{\mu_1}_{0+}v_2(t)), \ 0 < t < 1,$$ $$^CD^{σ_2}_{0+}v_2(t) = f_2(t, v_1(t), D^{\mu_2}_{0+}v_1(t)), \ 0 < t < 1,$$ $$v_1' (0) = bv_1(0), \ v_1''(0) = 0, \ ^CD^{θ_1}_{0+}v_1(1) = a · ^C D^{θ_2} _{0+}v_1(η),$$ $$v_2'(0) = bv_2(0), \ v_2''(0) = 0, \ ^CD^{θ_1}_{0+}v_2(1) = a · ^CD^{θ_2}_{0+}v_2(η),$$ is studied, where $a > 0,$ $−1 < b < 0,$ $2 < σ_1, σ_2 ≤ 3,$ $0 < η < 1,$ $0 < \mu_1, \mu_2 ≤ 1,$ $0 < θ_2 ≤ θ_1 ≤ 1,$ $f_1, f_2:$ $[0, 1] × \mathbb{R}^+ \times \mathbb{R} → \mathbb{R}^+$ are continuous, $^CD^{σ_1} _{0+}v_1(t),$ $^CD^{σ_2}_{0+}v_2(t)$ are the Caputo fractional derivatives, and $D^{\mu_1}_{0+}v_2(t),$ $D^{\mu_2}_{0+}v_1(t)$ are the Riemann-Liouville fractional derivatives. The fixed point theorem is used to prove that there are three positive solutions to problems.

}, issn = {2562-2862}, doi = {https://doi.org/10.12150/jnma.2022.454}, url = {http://global-sci.org/intro/article_detail/jnma/20717.html} }
TY - JOUR T1 - Triple Positive Solutions for a Class of Fractional Boundary Value Problem System AU - Bai , Zhanbing AU - Shi , Dongmei JO - Journal of Nonlinear Modeling and Analysis VL - 3 SP - 454 EP - 464 PY - 2022 DA - 2022/06 SN - 4 DO - http://doi.org/10.12150/jnma.2022.454 UR - https://global-sci.org/intro/article_detail/jnma/20717.html KW - Fractional derivative, Boundary value problems, Fixed point theorem, Positive solutions. AB -

In this paper, the solvability for the following fractional boundary value problem system $$^CD^{σ_1}_{0+}v_1(t) = f_1(t, v_2(t), D^{\mu_1}_{0+}v_2(t)), \ 0 < t < 1,$$ $$^CD^{σ_2}_{0+}v_2(t) = f_2(t, v_1(t), D^{\mu_2}_{0+}v_1(t)), \ 0 < t < 1,$$ $$v_1' (0) = bv_1(0), \ v_1''(0) = 0, \ ^CD^{θ_1}_{0+}v_1(1) = a · ^C D^{θ_2} _{0+}v_1(η),$$ $$v_2'(0) = bv_2(0), \ v_2''(0) = 0, \ ^CD^{θ_1}_{0+}v_2(1) = a · ^CD^{θ_2}_{0+}v_2(η),$$ is studied, where $a > 0,$ $−1 < b < 0,$ $2 < σ_1, σ_2 ≤ 3,$ $0 < η < 1,$ $0 < \mu_1, \mu_2 ≤ 1,$ $0 < θ_2 ≤ θ_1 ≤ 1,$ $f_1, f_2:$ $[0, 1] × \mathbb{R}^+ \times \mathbb{R} → \mathbb{R}^+$ are continuous, $^CD^{σ_1} _{0+}v_1(t),$ $^CD^{σ_2}_{0+}v_2(t)$ are the Caputo fractional derivatives, and $D^{\mu_1}_{0+}v_2(t),$ $D^{\mu_2}_{0+}v_1(t)$ are the Riemann-Liouville fractional derivatives. The fixed point theorem is used to prove that there are three positive solutions to problems.

Zhanbing Bai & Dongmei Shi. (2022). Triple Positive Solutions for a Class of Fractional Boundary Value Problem System. Journal of Nonlinear Modeling and Analysis. 4 (3). 454-464. doi:10.12150/jnma.2022.454
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