Existence of solutions for Sturm-Liouville boundary value problems of higher-order coupled fractional differential equations at resonance

Xue, Tingting; Liu, Wenbin; Zhang, Wei

doi:10.1186/s13662-017-1345-5

Cited by 5 publications

(4 citation statements)

References 24 publications

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“…Furthermore, by Lemma 1, we know that problem (10) has a solution u(t) ∈ C 1− ( J, R) in the form as formula (5). Substituting the solution v(t) of problem (11) into the solution u(t) of problem (10), we obtain a unique solution u(t) ∈ X of problem (9). ∈ (a, b), so the assertion holds.…”

Section: Lemmamentioning

confidence: 87%

“…[1][2][3][4][5][6] For example, Anastasio 7 used fractional differential model to describe the neural control of the instantaneous motion of the eyeball with its head: derivative D 0+ x; see previous studies. [9][10][11][12][13][14][15][16] An interesting and effective method that is used to prove the existence results of nonlinear fractional differential problems is the monotone iterative method combined with lower and upper solutions; see previous studies. [17][18][19][20] Note that fractional differential equations with the right-handed Riemann-Liouville derivative D T x have been studied; see previous studies.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, fractional differential equations of various initial or boundary conditions have been widely discussed. For example, such problems have been studied for fractional differential equations with the left‐handed Riemann‐Liouville derivative

D_{0 +}^{α} x

; see previous studies . An interesting and effective method that is used to prove the existence results of nonlinear fractional differential problems is the monotone iterative method combined with lower and upper solutions; see previous studies .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Extremal solutions for p‐Laplacian boundary value problems with the right‐handed Riemann‐Liouville fractional derivative

Xue

Shen

2019

Math Methods in App Sciences

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The paper is concerned with the solvability for several nonlinear boundary value problems of fractional p‐Laplacian differential equation involving the right‐handed Riemann‐Liouville derivative. By applying monotone iterative technique, lower and upper solutions method and the Banach fixed point theorem, sufficient conditions for existence and uniqueness of extremal solutions are obtained and they extend existing results. At last, two examples are provided to illustrate the results.

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Section: Lemmamentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

D_{0 +}^{α} x

Section: Introductionmentioning

confidence: 99%

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Extremal solutions for p‐Laplacian boundary value problems with the right‐handed Riemann‐Liouville fractional derivative

Xue

Shen

2019

Math Methods in App Sciences

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“…Using fixed point theorem, the existence and approximation of solutions to initial value problems for nonlinear fractional differential equations of arbitrary order with Riemann-Liouville derivative have been considered in [36]. Authors in [39] examined the existence of solutions for a higher-order coupled system of fractional differential equations with Sturm-Liouville boundary value conditions at resonance by applying Mawhin continuation theorem.…”

Section: Introductionmentioning

confidence: 99%

Theory and numerical approaches of high order fractional Sturm–Liouville problems

Houlari¹,

Dehghan²,

Biazar³

et al. 2021

Turk J Math

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In this paper, fractional Sturm-Liouville problems of high-order are studied. A simple and efficient approach is presented to determine more eigenvalues and eigenfunctions than other approaches. Existence and uniqueness of solutions of a fractional high-order differential equation with initial conditions is addressed as well as the convergence of the proposed approach. This class of eigenvalue problems is important in finding solutions to linear fractional partial differential equations (LFPDE). This method is illustrated by three examples to signify the efficiency and reliability of the proposed numerical approach.

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Research on Sturm–Liouville boundary value problems of fractional p-Laplacian equation

2021

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In this work we investigate the following fractional p-Laplacian differential equation with Sturm–Liouville boundary value conditions: $$ \textstyle\begin{cases} {}_{t}D_{T}^{\alpha } ( { \frac{1}{{{{ ( {h ( t )} )}^{p - 2}}}}{\phi _{p}} ( {h ( t ){}_{0}^{C}D_{t}^{\alpha }u ( t )} )} ) + a ( t ){\phi _{p}} ( {u ( t )} ) = \lambda f (t,u(t) ),\quad \mbox{a.e. }t \in [ {0,T} ], \\ {\alpha _{1}} {\phi _{p}} ( {u ( 0 )} ) - { \alpha _{2}} {}_{t}D_{T}^{\alpha - 1} ( {{\phi _{p}} ( {{}_{0}^{C}D_{t}^{\alpha }u ( 0 )} )} ) = 0, \\ {\beta _{1}} { \phi _{p}} ( {u ( T )} ) + {\beta _{2}} {}_{t}D_{T}^{ \alpha - 1} ( {{\phi _{p}} ( {{}_{0}^{C}D_{t}^{\alpha }u ( T )} )} ) = 0, \end{cases} $$ { D T α t ( 1 ( h ( t ) ) p − 2 ϕ p ( h ( t ) 0 C D t α u ( t ) ) ) + a ( t ) ϕ p ( u ( t ) ) = λ f ( t , u ( t ) ) , a.e. t ∈ [ 0 , T ] , α 1 ϕ p ( u ( 0 ) ) − α 2 t D T α − 1 ( ϕ p ( 0 C D t α u ( 0 ) ) ) = 0 , β 1 ϕ p ( u ( T ) ) + β 2 t D T α − 1 ( ϕ p ( 0 C D t α u ( T ) ) ) = 0 , where ${}_{0}^{C}D_{t}^{\alpha }$ D t α 0 C , ${}_{t}D_{T}^{\alpha }$ D T α t are the left Caputo and right Riemann–Liouville fractional derivatives of order $\alpha \in ( {\frac{1}{2},1} ]$ α ∈ ( 1 2 , 1 ] , respectively. By using variational methods and critical point theory, some new results on the multiplicity of solutions are obtained.

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Existence of solutions for Sturm-Liouville boundary value problems of higher-order coupled fractional differential equations at resonance

Cited by 5 publications

References 24 publications

Extremal solutions for p‐Laplacian boundary value problems with the right‐handed Riemann‐Liouville fractional derivative

Extremal solutions for p‐Laplacian boundary value problems with the right‐handed Riemann‐Liouville fractional derivative

Theory and numerical approaches of high order fractional Sturm–Liouville problems

Research on Sturm–Liouville boundary value problems of fractional p-Laplacian equation

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