Existence and Uniqueness of Solutions for BVP of Nonlinear Fractional Differential Equation

Su, C. H.; Sun, Jian-Ping; Zhao, Yue

doi:10.1155/2017/4683581

Cited by 8 publications

(8 citation statements)

References 19 publications

(12 reference statements)

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“…Therefore, it follows from Theorem 1.1, Lemma 2.7, (15), and (17) that T has a fixed point u ∈ K ∩ (Ω 4 \ Ω 3 ), which is a desired positive solution of BVP (4). Therefore, it follows from Theorem 3.1 that BVP (18) has at least one positive solution.…”

Section: This Indicates Thatmentioning

confidence: 92%

Existence of positive solution for BVP of nonlinear fractional differential equation with integral boundary conditions

Sun

Zhao

2020

Adv Differ Equ

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This paper is concerned with the following boundary value problem of nonlinear fractional differential equation with integral boundary conditions: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ (C D q 0+ u)(t) + f (t, u(t)) = 0, t ∈ [0, 1], u (0) = 0, αu(0)-βu (0) = 1 0 h 1 (s)u(s) ds, γ u(1) + δ(C D σ 0+ u)(1) = 1 0 h 2 (s)u(s) ds, where 2 < q ≤ 3, 0 < σ ≤ 1, α, γ , δ ≥ 0, and β > 0 satisfying 0 < ρ := (α + β)γ + αδ Γ (2-σ) < β[γ + δΓ (q) Γ (q-σ) ]. C D q 0+ denotes the standard Caputo fractional derivative. First, Green's function of the corresponding linear boundary value problem is constructed. Next, some useful properties of the Green's function are obtained. Finally, existence results of at least one positive solution for the above problem are established by imposing some suitable conditions on f and h i (i = 1, 2). The method employed is Guo-Krasnoselskii's fixed point theorem. An example is also included to illustrate the main results of this paper.

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Section: This Indicates Thatmentioning

confidence: 92%

Existence of positive solution for BVP of nonlinear fractional differential equation with integral boundary conditions

Sun

Zhao

2020

Adv Differ Equ

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“…In the past decades, the existence of solutions or positive solutions for boundary value problems (BVPs for short) of nonlinear fractional differential equations attracted considerable attention from many authors, see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Positive Solutions for BVP of Fractional Differential Equation with Integral Boundary Conditions

Sun

Zhao

2020

Discrete Dynamics in Nature and Society

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In this paper, we consider a class of boundary value problems of nonlinear fractional differential equation with integral boundary conditions. By applying the monotone iterative method and some inequalities associated with Green’s function, we obtain the existence of minimal and maximal positive solutions and establish two iterative sequences for approximating the solutions to the above problem. It is worth mentioning that these iterative sequences start off with zero function or linear function, which is useful and feasible for computational purpose. An example is also included to illustrate the main result of this paper.

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“…Different kind of fixed point theorems are widely used as fundamental tools to prove the existence and uniqueness of solutions for various classes of fractional differential equations; for example, we refer the reader to [1][2][3][4]9,12,14,15,17,18,21,25,27] and the references cited therein. Moreover, some mathematicians considered Caputo fractional differential equations with a nonlinear term depending on the Caputo derivative (see Benchohra and Souid [5], Benchohra and Lazreg [6], Benchohra et al [7], El-Sayed and Bin-Taher [10,11], Guezane-Lakoud and Khaldi [12], Guezane-Lakoud and Bensebaa [13], Houas and Benbachir [14], Nieto et al [20], and Yan et al [27]).…”

Section: Introductionmentioning

confidence: 99%

Boundary value problems for Caputo fractional differential equations with nonlocal and fractional integral boundary conditions

Derbazi

Hammouche

2020

Arab. J. Math.

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In this paper, we study the existence and uniqueness of solutions for fractional differential equations with nonlocal and fractional integral boundary conditions. New existence and uniqueness results are established using the Banach contraction principle. Other existence results are obtained using O’Regan fixed point theorem and Burton and Kirk fixed point. In addition, an example is given to demonstrate the application of our main results.

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Existence and Uniqueness of Solutions for BVP of Nonlinear Fractional Differential Equation

Cited by 8 publications

References 19 publications

Existence of positive solution for BVP of nonlinear fractional differential equation with integral boundary conditions

Existence of positive solution for BVP of nonlinear fractional differential equation with integral boundary conditions

Positive Solutions for BVP of Fractional Differential Equation with Integral Boundary Conditions

Boundary value problems for Caputo fractional differential equations with nonlocal and fractional integral boundary conditions

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