scholarly journals Existence Result and Uniqueness for Some Fractional Problem

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 516 ◽  
Author(s):  
Guotao Wang ◽  
Abdeljabbar Ghanmi ◽  
Samah Horrigue ◽  
Samar Madian
Keyword(s):  
Boundary Value Problem ◽  
Positive Solution ◽  
Existence Result ◽  
Boundary Value ◽  
Lower And Upper Solutions ◽  
Fractional Boundary Value Problem

In this article, by the use of the lower and upper solutions method, we prove the existence of a positive solution for a Riemann–Liouville fractional boundary value problem. Furthermore, the uniqueness of the positive solution is given. To demonstrate the serviceability of the main results, some examples are presented.

scholarly journals Existence and Exact Asymptotic Behavior of Positive Solutions for a Fractional Boundary Value Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Habib Mâagli ◽  
Noureddine Mhadhebi ◽  
Noureddine Zeddini
Keyword(s):  
Asymptotic Behavior ◽  
Continuous Function ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Exact Asymptotic Behavior ◽  
Nonnegative Continuous Function

We establish the existence and uniqueness of a positive solution for the fractional boundary value problem , with the condition , where , and is a nonnegative continuous function on that may be singular at or .

Unique positive solution for a fractional boundary value problem

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Keyu Zhang ◽  
Jiafa Xu
Keyword(s):  
Boundary Value Problem ◽  
Positive Solution ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Fractional Boundary Value Problem ◽  
Unique Positive Solution ◽  
Monotone Iterative ◽  
Liouville Fractional Derivative

AbstractIn this work we consider the unique positive solution for the following fractional boundary value problem $\left\{ \begin{gathered} D_{0 + }^\alpha u(t) = - f(t,u(t)),t \in [0,1], \hfill \\ u(0) = u'(0) = u'(1) = 0. \hfill \\ \end{gathered} \right. $ Here α ∈ (2, 3] is a real number, D 0+α is the standard Riemann-Liouville fractional derivative of order α. By using the method of upper and lower solutions and monotone iterative technique, we also obtain that there exists a sequence of iterations uniformly converges to the unique solution.

scholarly journals Upper and Lower Solutions Method for a Class of Singular Fractional Boundary Value Problems withp-Laplacian Operator

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Jinhua Wang ◽  
Hongjun Xiang
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Laplacian Operator ◽  
Fractional Boundary Value Problem ◽  
Fractional Boundary Value Problems

The upper and lower solutions method is used to study thep-Laplacian fractional boundary value problemD0+γ(ϕp(D0+αu(t)))=f(t,u(t)),0<t<1,u(0)=0,u(1)=au(ξ),D0+αu(0)=0, andD0+αu(1)=bD0+αu(η), where1<α,γ⩽2,0⩽a,b⩽1,0<ξ,η<1. Some new results on the existence of at least one positive solution are obtained. It is valuable to point out that the nonlinearityfcan be singular att=0,1oru=0.

scholarly journals On positive solutions to nonlocal fractional and integer-order difference equations

2011 ◽  
Vol 5 (1) ◽  
pp. 122-132 ◽  
Author(s):  
Christopher Goodrich
Keyword(s):  
Boundary Value Problem ◽  
Positive Solution ◽  
Recent Work ◽  
Positive Solutions ◽  
Difference Equations ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Linear Functionals ◽  
Order Difference ◽  
Integer Order

In this paper, we consider a discrete fractional boundary value problem of the form -??y(t) = f(t+? - 1; y(t + ? - 1)), y(? - 2) = ?(y), y(? + b) = ?(y), where t ? [0,b]N0, f : [? ? 1,... , ? + b - 1]N?-2 x R ? [0,+?) is continuous, ?, ? : C([? - 2; ? + b]N?-2 ) ? R are given functionals, and 1 < ? ? 2. We show that provided that both ? and ? are linear functionals, then under certain conditions the fractional boundary value problem will have at least one positive solution even if neither ? nor ? is nonnegative for all y ? 0. This provides new results not only for the fractional boundary value problem but also in the case when ? = 2. Our results also generalize some recent work on the conjugate fractional boundary value problem. We conclude with two examples to illustrate our results.

scholarly journals Existence and Estimates of Positive Solutions for Some Singular Fractional Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Habib Mâagli ◽  
Noureddine Mhadhebi ◽  
Noureddine Zeddini
Keyword(s):  
Continuous Function ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Global Behavior ◽  
Fractional Boundary Value Problem ◽  
Nonnegative Continuous Function ◽  
Fractional Boundary Value Problems

We establish the existence and uniqueness of a positive solutionufor the following fractional boundary value problem:Dαu(x)=−a(x)uσ(x),x∈(0,1)with the conditionslimx→0+⁡x2−αu(x)=0,u(1)=0, where1<α≤2,σ∈(−1,1), andais a nonnegative continuous function on(0,1)that may be singular atx=0orx=1. We also give the global behavior of such a solution.

scholarly journals Lower and Upper Solutions Method for Positive Solutions of Fractional Boundary Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
R. Darzi ◽  
B. Mohammadzadeh ◽  
A. Neamaty ◽  
D. Bǎleanu
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Fixed Point Theorems ◽  
Positive Real Number ◽  
Boundary Value ◽  
Lower And Upper Solutions ◽  
Fractional Boundary Value Problem ◽  
Positive Real ◽  
Liouville Fractional Derivative ◽  
Fractional Boundary Value Problems

We apply the lower and upper solutions method and fixed-point theorems to prove the existence of positive solution to fractional boundary value problemD0+αut+ft,ut=0,0<t<1,2<α≤3,u0=u′0=0,D0+α−1u1=βuξ,0<ξ<1, whereD0+αdenotes Riemann-Liouville fractional derivative,βis positive real number,βξα−1≥2Γα, andfis continuous on0,1×0,∞. As an application, one example is given to illustrate the main result.

scholarly journals Existence of a unique positive solution for a singular fractional boundary value problem

2018 ◽  
Vol 34 (1) ◽  
pp. 57-64
Author(s):  
E. T. KARIMOV ◽  
◽  
K. SADARANGANI ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Positive Solution ◽  
Fractional Order ◽  
Boundary Value ◽  
Order Equation ◽  
Fractional Boundary Value Problem ◽  
Unique Positive Solution

In the present work, we discuss the existence of a unique positive solution of a boundary value problem for a nonlinear fractional order equation with singularity. Precisely, order of equation Dα 0+u(t) = f(t, u(t)) belongs to (3, 4] and f has a singularity at t = 0 and as a boundary conditions we use... Using a fixed point theorem, we prove the existence of unique positive solution of the considered problem.

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