Positive solutions of 2mth-order boundary value problems

Chyan, Chuan Jen; Henderson, Johnny

doi:10.1016/s0893-9659(02)00040-x

Cited by 18 publications

(16 citation statements)

References 6 publications

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“…As pointed out in the proceeding section, Equation (5) is equivalent to Equation (9). Thus for u, v ∈ K we have…”

Section: Unique Existence Of Solutionmentioning

confidence: 93%

“…In References [5] and [6] the existence of solutions for nonlinear fractional differential equations involving the standard Riemann-Liouville fractional derivatives has been presented. Existence of positive solutions of differential equations is well studied in case of ordinary differential equations [7][8][9]. In contrast its fractional analog has received attention only recently.…”

Section: Introductionmentioning

confidence: 99%

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Existence of Positive Solutions for N-term Non-autonomous Fractional Differential Equations

Babakhani

Daftardar-Gejji

2005

Positivity

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Existence of positive solutions for the nonlinear fractional differential equation D α u = f (x, u), 0 < α < 1 has been given (S. Zhang, J. Math. Anal. Appl. 252 (2000), [804][805][806][807][808][809][810][811][812] where D α denotes Riemann-Liouville fractional derivative. In the present work we extend this analysis for n-term non autonomous fractional differential equations. We investigate existence of positive solutions for the following initial value problem:with initial conditions u(0) = 0, [D α−n+1 u(x)] x=0 = b n−1 0, [D α−n+j u(x)] x=0 = b n−j , b n−j j −1 k=1 a k b k+n−j , j = 2, 3, . . . , n − 1, n − 1 < α < n,n ∈ IN where L(D) = D α − n−1 j =1 a j D α−j , a j > 0, ∀j, D α−j is the standard Riemann-Liouville fractional derivative. Further the conditions on a j 's and f, under which the solution is (i) unique and (ii) unique and positive as well, are given.

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“…As pointed out in the proceeding section, Equation (5) is equivalent to Equation (9). Thus for u, v ∈ K we have…”

Section: Unique Existence Of Solutionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Existence of Positive Solutions for N-term Non-autonomous Fractional Differential Equations

Babakhani

Daftardar-Gejji

2005

Positivity

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“…Recently, the study of the existence of positive solutions of boundary value problems for second-order or higher-order ordinary differential equations has gained prominence and it is a rapidly growing field, since it arises, especially for fourth-order differential equations [1], in many applications. We refer the reader to [2][3][4][5][6][7][8][9][10] and the monograph [11][12][13].…”

Section: Introductionmentioning

confidence: 99%

“…x (0) = 0 = x (1) has received much attention in searches for conditions on the nonlinearity f for which there are at least one, at least two or at least three positive solutions; [14][15][16][17][18] provide examples. However, the existence of positive solutions of the following differential equation:…”

Section: Introductionmentioning

confidence: 99%

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Positive solutions of two-point boundary value problems for2n-order differential equations dependent on all derivatives

Liu¹,

Ge²

2005

Applied Mathematics Letters

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In this work, we consider the higher-order differential equationsubject to one of the following multi-point boundary value conditions:orwhere f (t, x 0 , x 1 , . . . , x 2n−1 ) is continuous with f (t, x 0 , x 1 , . . . , x 2n−1 ) ≥ 0 for all t ∈ [0, 1] and (x 0 , x 1 , . . . , x 2n−1 ) ∈ R 2n . Sufficient conditions for the existence of at least one positive solution of the BVP (1) and (2) and BVP (1) and (3) are established, respectively. The emphasis in this work is on f depending on all higher-order derivatives. Examples are given to illustrate the main results.

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Existence of Solutions of Impulsive Lidstone BVPs of Singular Higher Order Fractional Differential Equations

Liu

2017

Vietnam J. Math.

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Positive solutions of 2mth-order boundary value problems

Cited by 18 publications

References 6 publications

Existence of Positive Solutions for N-term Non-autonomous Fractional Differential Equations

Existence of Positive Solutions for N-term Non-autonomous Fractional Differential Equations

Positive solutions of two-point boundary value problems for2n-order differential equations dependent on all derivatives

Existence of Solutions of Impulsive Lidstone BVPs of Singular Higher Order Fractional Differential Equations

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