scholarly journals Hartman-Wintner and Lyapunov-type inequalities for high order fractional boundary value problems

Filomat ◽  
2020 ◽  
Vol 34 (7) ◽  
pp. 2273-2281
Author(s):  
Şuayip Toprakseven
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Nontrivial Solution ◽  
Eigenvalue Problems ◽  
Boundary Value ◽  
High Order ◽  
Fractional Boundary Value Problem ◽  
Sturm Liouville ◽  
Fractional Boundary Value Problems

In this paper, we obtain Hartman-Wintner and Lyapunov-type inequalities for the three-point fractional boundary value problem of the fractional Liouville-Caputo differential equation of order ? 2 (2; 3]. The results presented in this work are sharper than the existing results in the literature. As an application of the results, the fractional Sturm-Liouville eigenvalue problems have also been presented. Moreover, we examine the nonexistence of the nontrivial solution of the fractional boundary value problem.

scholarly journals Blow-Up Solutions for a Singular Nonlinear Hadamard Fractional Boundary Value Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Imed Bachar ◽  
Said Mesloub
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Point Theorem ◽  
Blow Up ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Fractional Boundary Value Problems

We consider singular nonlinear Hadamard fractional boundary value problems. Using properties of Green’s function and a fixed point theorem, we show that the problem has positive solutions which blow up. Finally, some examples are provided to explain the applications of the results.

Eigenvalue comparison for fractional boundary value problems with the Caputo derivative

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Johnny Henderson ◽  
Nickolai Kosmatov
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Caputo Derivative ◽  
Boundary Value ◽  
Positive Operators ◽  
Fractional Boundary Value Problem ◽  
Comparison Results ◽  
Eigenvalue Comparison ◽  
Fractional Boundary Value Problems

AbstractWe apply the theory for u 0-positive operators to obtain eigenvalue comparison results for a fractional boundary value problem with the Caputo derivative.

A Discreate Calculus with Applications of High-Order Discretizations to Boundary-Value Problems

2004 ◽  
Vol 4 (2) ◽  
pp. 228-261 ◽  
Author(s):  
Stanly Steinberg
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Energy Inequality ◽  
Numerical Test ◽  
Mixed Boundary Conditions ◽  
Discrete Analog ◽  
High Order ◽  
Sturm Liouville

AbstractWe develop a discrete analog of the differential calculus and use this to develop arbitrarily high-order approximations to Sturm–Liouville boundary-value problems with general mixed boundary conditions. An important feature of the method is that we obtain a discrete exact analog of the energy inequality for the continuum boundary-value problem. As a consequence, the discrete and continuum problems have exactly the same solvability conditions. We call such discretizations mimetic. Numerical test confirm the accuracy of the discretization. We prove the solvability and convergence for the discrete boundary-value problem modulo the invertibility of a matrix that appears in the discretization being positive definite. Numerical experiments indicate that the spectrum of this matrix is real, greater than one, and bounded above by a number smaller than three.

scholarly journals A New Proof for Lyapunov-Type Inequality on the Fractional Boundary Value Problem

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 29
Author(s):  
Yumei Zou ◽  
Xin Zhang ◽  
Hongyu Li
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Green’S Function ◽  
Boundary Value Problems ◽  
Green's Function ◽  
Boundary Value ◽  
Type Inequality ◽  
Fractional Boundary Value Problem ◽  
Lyapunov Type Inequality ◽  
Fractional Boundary Value Problems

In this article, some new Lyapunov-type inequalities for a class of fractional boundary value problems are established by use of the nonsymmetry property of Green’s function corresponding to appropriate boundary conditions.

Uniqueness of positive solutions of fractional boundary value problems with non-homogeneous integral boundary conditions

2012 ◽  
Vol 15 (3) ◽  
Author(s):  
John Graef ◽  
Lingju Kong ◽  
Qingkai Kong ◽  
Min Wang
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Integral Boundary Conditions ◽  
Fractional Boundary Value Problem ◽  
Integral Boundary ◽  
Fractional Boundary Value Problems

AbstractThe authors study a type of nonlinear fractional boundary value problem with non-homogeneous integral boundary conditions. The existence and uniqueness of positive solutions are discussed. An example is given as the application of the results.

scholarly journals On solutions of a class of three-point fractional boundary value problems

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Zhanbing Bai ◽  
Yu Cheng ◽  
Sujing Sun
Keyword(s):  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Existence Results ◽  
Conformable Fractional Derivative ◽  
Fractional Boundary Value Problem ◽  
Nonlinear Alternative ◽  
Fractional Boundary Value Problems

AbstractExistence results for the three-point fractional boundary value problem $$\begin{aligned}& D^{\alpha}x(t)= f \bigl(t, x(t), D^{\alpha-1} x(t) \bigr),\quad 0< t< 1, \\& x(0)=A, \qquad x(\eta)-x(1)=(\eta-1)B, \end{aligned}$$ Dαx(t)=f(t,x(t),Dα−1x(t)),0<t<1,x(0)=A,x(η)−x(1)=(η−1)B, are presented, where $A, B\in\mathbb{R}$A,B∈R, $0<\eta<1$0<η<1, $1<\alpha\leq2$1<α≤2. $D^{\alpha}x(t)$Dαx(t) is the conformable fractional derivative, and $f: [0, 1]\times\mathbb{R}^{2}\to\mathbb{R}$f:[0,1]×R2→R is continuous. The analysis is based on the nonlinear alternative of Leray–Schauder.

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 Numerical Results for Linear Sequential Caputo Fractional Boundary Value Problems

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 910
Author(s):  
Bhuvaneswari Sambandham ◽  
Aghalaya S. Vatsala ◽  
Vinodh K. Chellamuthu
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Mixed Boundary Conditions ◽  
Numerical Code ◽  
Fractional Boundary Value Problem ◽  
Monotone Method ◽  
Fractional Boundary Value Problems

The generalized monotone iterative technique for sequential 2 q order Caputo fractional boundary value problems, which is sequential of order q, with mixed boundary conditions have been developed in our earlier paper. We used Green’s function representation form to obtain the linear iterates as well as the existence of the solution of the nonlinear problem. In this work, the numerical simulations for a linear nonhomogeneous sequential Caputo fractional boundary value problem for a few specific nonhomogeneous terms with mixed boundary conditions have been developed. This in turn will be used as a tool to develop the accurate numerical code for the linear nonhomogeneous sequential Caputo fractional boundary value problem for any nonhomogeneous terms with mixed boundary conditions. This numerical result will be essential to solving a nonlinear sequential boundary value problem, which arises from applications of the generalized monotone method.

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 Existence of Positive Solutions for a Kind of Fractional Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Hongjie Liu ◽  
Xiao Fu ◽  
Liangping Qi
Keyword(s):  
Fixed Point ◽  
Green Function ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Existence Of Positive Solutions ◽  
Liouville Fractional Derivative ◽  
Fractional Boundary Value Problems

We are concerned with the following nonlinear three-point fractional boundary value problem:D0+αut+λatft,ut=0,0<t<1,u0=0, andu1=βuη, where1<α≤2,0<β<1,0<η<1,D0+αis the standard Riemann-Liouville fractional derivative,at>0is continuous for0≤t≤1, andf≥0is continuous on0,1×0,∞. By using Krasnoesel'skii's fixed-point theorem and the corresponding Green function, we obtain some results for the existence of positive solutions. At the end of this paper, we give an example to illustrate our main results.

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