Systems of Riemann–Liouville fractional equations with multi-point boundary conditions

Henderson, Johnny; Luca, Rodica

doi:10.1016/j.amc.2017.03.044

Cited by 43 publications

(34 citation statements)

References 9 publications

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“…In this section we present some auxiliary results from [1] that will be used to prove our main theorems. We consider the fractional differential system { 0+ ( ) + ( ) = 0, ∈ (0,1),…”

Section: Auxiliary Resultsmentioning

confidence: 99%

“…We present now the assumptions that we will use in the sequel. (H1) , ∈ ℝ, ∈ ( − 1, ], ∈ ( − 1, ], , ∈ ℕ, , ≥ 3, 1 , 2 , 1 , 2 ∈ ℝ, 1…”

Section: Resultsmentioning

confidence: 99%

“…By using (H2) and the definitions of 0 and 0 , we deduce that there exists 1 So, ‖ 1 ( , )‖ ≥ 1 ( , )( 1 ) ≥ ‖( , )‖ .…”

Section: International Journal Of Applied Physics and Mathematicsmentioning

confidence: 96%

“…The nonexistence of positive solutions for (S)-(BC) is also investigated. The system (S) without parameters , with the boundary conditions (BC) was investigated in [1], where f and g are non-singular or singular functions. The existence of positive solutions of the system (S) without parameters subject to the uncoupled multi-point boundary conditions , was studied in [2], [3].…”

Section: Introductionmentioning

confidence: 99%

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Positive Solutions for a System of Fractional Differential Equations with Parameters and Coupled Multi-point Boundary Conditions

Henderson¹,

Luca²,

Tudorache³

2018

IJAPM

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In this paper, we study a system of nonlinear Riemann-Liouville fractional ordinary differential equations with parameters, subject to coupled multi-point boundary conditions which contain fractional derivatives. By using some properties of the associated Green's functions and the Guo-Krasnosel'skii fixed point theorem, we prove the existence of positive solutions for this problem when the parameters belong to various intervals. Then, we present sufficient conditions for the nonexistence of positive solutions.Key words: Fractional differential equations, multi-point boundary conditions, positive solutions, existence, nonexistence.for all = 1, … , ( ∈ ℕ), 0 < 1 < ⋯ < ≤ 1, , > 0, and 0+ denotes the Riemann-Liouville derivative of order (for = , , 1 , 2 , 1 , 2 ). Under some assumptions on the nonnegative functions and , we present intervals for the parameters and such that positive solutions of (S)-(BC) exist. By a positive solution of problem (S)-(BC)we mean a pair of functions ( , ) ∈ ( ([0,1]), [0, ∞))) 2 satisfying ( ) and ( ), with ( ) > 0 for all Lemma 2.1 If ≠ 0 and , ∈ (0,1) ∩ 1 (0,1), then the pair of functions ( , ) ∈ [0,1] × [0,1] given by

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“…In this section we present some auxiliary results from [1] that will be used to prove our main theorems. We consider the fractional differential system { 0+ ( ) + ( ) = 0, ∈ (0,1),…”

Section: Auxiliary Resultsmentioning

confidence: 99%

“…We present now the assumptions that we will use in the sequel. (H1) , ∈ ℝ, ∈ ( − 1, ], ∈ ( − 1, ], , ∈ ℕ, , ≥ 3, 1 , 2 , 1 , 2 ∈ ℝ, 1…”

Section: Resultsmentioning

confidence: 99%

“…By using (H2) and the definitions of 0 and 0 , we deduce that there exists 1 So, ‖ 1 ( , )‖ ≥ 1 ( , )( 1 ) ≥ ‖( , )‖ .…”

Section: International Journal Of Applied Physics and Mathematicsmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Positive Solutions for a System of Fractional Differential Equations with Parameters and Coupled Multi-point Boundary Conditions

Henderson¹,

Luca²,

Tudorache³

2018

IJAPM

Self Cite

View full text Add to dashboard Cite

show abstract

“…The research on fractional di erential equations is very important in both theory and applications. By using nonlinear analysis tools, some scholars established the existence, uniqueness, multiplicity and qualitative properties of solutions, we refer the readers to [7][8][9][10][11][12][13][14][15][16][17][18][19][20] and the references therein for fractional di erential equations, and [21][22][23][24][25][26][27][28][29][30][31][32][33] for fractional di erential systems.…”

Section: Introductionmentioning

confidence: 99%

Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions

Hao

Wang

2018

Open Mathematics

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System of Riemann‐Liouville fractional differential equations with nonlocal boundary conditions: Existence, uniqueness, and multiplicity of solutions

Padhi

B.S.R.V.

Mahendru

2019

Math Methods in App Sciences

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Systems of Riemann–Liouville fractional equations with multi-point boundary conditions

Cited by 43 publications

References 9 publications

Positive Solutions for a System of Fractional Differential Equations with Parameters and Coupled Multi-point Boundary Conditions

Positive Solutions for a System of Fractional Differential Equations with Parameters and Coupled Multi-point Boundary Conditions

Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions

System of Riemann‐Liouville fractional differential equations with nonlocal boundary conditions: Existence, uniqueness, and multiplicity of solutions

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