Some existence and uniqueness results for nonlinear fractional partial differential equations

Marasi, Hamidreza; Afshari, Hojjat; Zhai, Chengbo

doi:10.1216/rmj-2017-47-2-571

Cited by 22 publications

(13 citation statements)

References 28 publications

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“…So, recently much attention has been focused on the study of the existence and multiplicity of solutions of positive solutions for boundary and initial value problems of fractional differential equations. There are many techniques to deal with the existence of solutions of fractional differential equations such as fixed point theorems [5,9,13], upper and lower solutions method [8], fixed point index [4,12,13], coincidence theory [1], etc. In [2,3,7], the authors considered the existence of solutions of the following initial value problems…”

Section: Introductionmentioning

confidence: 99%

Existence and multiplicity of solutions for nonlinear fractional differential equations

Marasi¹,

Piri²,

Aydi³

2016

J. Nonlinear Sci. Appl.

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Section: Introductionmentioning

confidence: 99%

Existence and multiplicity of solutions for nonlinear fractional differential equations

Marasi¹,

Piri²,

Aydi³

2016

J. Nonlinear Sci. Appl.

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“…Next, we provide the definition of the measure of noncompactness and some auxiliary results; see for more details [11,13,15] and the references therein.…”

Section: For a Given Setmentioning

confidence: 99%

“…The qualitative theory of differential equations has significant applications, and the existence of solutions and of positive solutions of fractional differential equations, which respect the initial and boundary value, have also received considerable attention. In order to study such a type of problem, different kinds of techniques, such as fixed point theorems [9][10][11], the fixed point index [10,12], the upper and lower solution method [13], coincidence theory [14], etc., are in vogue. For instance, in [15][16][17], the authors investigated the existence of solutions of initial value problems.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Caputo Fractional Derivative with Nonlocal Riemann-Liouville Fractional Integral Condition Via Fixed Point Theorems

2019

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In this paper, we study and investigate an interesting Caputo fractional derivative and Riemann–Liouville integral boundary value problem (BVP): c D 0 + q u ( t ) = f ( t , u ( t ) ) , t ∈ [ 0 , T ] , u ( k ) ( 0 ) = ξ k , u ( T ) = ∑ i = 1 m β i R L I 0 + p i u ( η i ) , where n - 1 < q < n , n ≥ 2 , m , n ∈ N , ξ k , β i ∈ R , k = 0 , 1 , ⋯ , n - 2 , i = 1 , 2 , ⋯ , m , and c D 0 + q is the Caputo fractional derivatives, f : [ 0 , T ] × C ( [ 0 , T ] , E ) → E , where E is the Banach space. The space E is chosen as an arbitrary Banach space; it can also be R (with the absolute value) or C ( [ 0 , T ] , R ) with the supremum-norm. R L I 0 + p i is the Riemann–Liouville fractional integral of order p i > 0 , η i ∈ ( 0 , T ) , and ∑ i = 1 m β i η i p i + n - 1 Γ ( n ) Γ ( n + p i ) ≠ T n - 1 . Via the fixed point theorems of Krasnoselskii and Darbo, the authors study the existence of solutions to this problem. An example is included to illustrate the applicability of their results.

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“…To our knowledge, some remarkable results on the existence and multiplicity of solutions for fractional differential equations have been discussed widely on finite intervals [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. Instead, it is relatively rare for work to be done related to existence results on infinite intervals [34][35][36][37][38][39][40][41][42][43][44][45][46][47].…”

Section: Introductionmentioning

confidence: 99%

A coupled system of fractional differential equations on the half-line

Zhai

Ren

2019

Bound Value Probl

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In this paper, we consider a new fractional differential system on an unbounded domain D α u(t) + ϕ(t, v(t), D γ 1 v(t)) = 0, t ∈ [0, +∞), α ∈ (2, 3], D β v(t) + ψ(t, u(t), D γ 2 u(t)) = 0, t ∈ [0, +∞), β ∈ (2, 3], subject to the conditions I 3-α u(t)| t=0 = 0, D α-2 u(t)| t=0 = h 0 g 1 (s)u(s) ds, D α-1 u(+∞) = Mu(ξ) + a, I 3-β v(t)| t=0 = 0, D β-2 v(t)| t=0 = h 0 g 2 (s)v(s) ds, D β-1 v(+∞) = Nv(η) + b. The nonlinear terms ϕ and ψ are dependent on the fractional derivative of lower order γ i ∈ (0, 1), i = 1, 2, which creates additional complexity to verify the existence of solutions. Moreover, a proper choice of Banach space allows the solutions to be defined on the half-line. From some standard fixed point theorems, sufficient conditions for the existence and uniqueness of solutions to boundary value problems are developed. Finally, the main result is applied to an illustrative example.

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Some existence and uniqueness results for nonlinear fractional partial differential equations

Cited by 22 publications

References 28 publications

Existence and multiplicity of solutions for nonlinear fractional differential equations

Existence and multiplicity of solutions for nonlinear fractional differential equations

Nonlinear Caputo Fractional Derivative with Nonlocal Riemann-Liouville Fractional Integral Condition Via Fixed Point Theorems

A coupled system of fractional differential equations on the half-line

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