scholarly journals Existence of Solutions for Fractional Boundary Value Problems with a Quadratic Growth of Fractional Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Yaning Li ◽  
Quanguo Zhang ◽  
Baoyan Sun
Keyword(s):  
Fractional Derivative ◽  
Variational Methods ◽  
Boundary Value Problems ◽  
Boundary Problem ◽  
Linear Growth ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Existence Results ◽  
Quadratic Growth ◽  
Fractional Boundary Value Problems

In this paper, we deal with two fractional boundary value problems which have linear growth and quadratic growth about the fractional derivative in the nonlinearity term. By using variational methods coupled with the iterative methods, we obtain the existence results of solutions. To the best of the authors’ knowledge, there are no results on the solutions to the fractional boundary problem which have quadratic growth about the fractional derivative in the nonlinearity term.

scholarly journals Solvability of Some Fractional Boundary Value Problems with a Convection Term

2019 ◽  
Vol 2019 ◽  
pp. 1-6 ◽  
Author(s):  
Yongfang Wei ◽  
Zhanbing Bai
Keyword(s):  
Fixed Point ◽  
Green Function ◽  
Fractional Derivative ◽  
Boundary Value Problems ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Existence Results ◽  
Convection Term ◽  
Triple Positive Solutions ◽  
Fractional Boundary Value Problems

This paper is devoted to the research of some Caputo’s fractional derivative boundary value problems with a convection term. By the use of some fixed-point theorems and the properties of Green function, the existence results of at least one or triple positive solutions are presented. Finally, two examples are given to illustrate the main 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 Existence of solutions for some two-point fractional boundary value problems under barrier strip conditions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Zhiyu Li ◽  
Zhanbing Bai
Keyword(s):  
Boundary Value Problems ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Existence Results ◽  
Nonlinear Alternative ◽  
Sign Conditions ◽  
Fractional Boundary Value Problems

AbstractIn this paper, we are dedicated to researching the boundary value problems (BVPs) for equation $D^{\alpha }x(t)=f(t,x(t),D^{\alpha -1}x(t))$Dαx(t)=f(t,x(t),Dα−1x(t)), with the boundary value conditions to be either: $x(0)=A$x(0)=A, $D^{\alpha -1}x(1)=B$Dα−1x(1)=B or $D^{\alpha -1}x(0)=A$Dα−1x(0)=A, $x(1)=B$x(1)=B. Let the nonlinear term f satisfy some sign conditions, then by making use of the Leray–Schauder nonlinear alternative, some existence results are obtained. In the end, an example is given to verify the main results.

scholarly journals On solvability of some $ p $-Laplacian boundary value problems with Caputo fractional derivative

2021 ◽  
Vol 6 (12) ◽  
pp. 13622-13633
Author(s):  
Xiaoping Li ◽  
◽  
Dexin Chen ◽  
Keyword(s):  
Fixed Point ◽  
Fractional Derivative ◽  
Boundary Value Problems ◽  
Caputo Fractional Derivative ◽  
Fixed Point Theory ◽  
Boundary Value ◽  
Point Theory ◽  
Existence Results ◽  
Analysis Techniques

<abstract><p>The solvability of some $ p $-Laplace boundary value problems with Caputo fractional derivative are discussed. By using the fixed-point theory and analysis techniques, some existence results of one or three non-negative solutions are obtained. Two examples showed that the conditions used in this paper are somewhat easy to check.</p></abstract>

Existence results for a second order nonlocal boundary value problem at resonance

2016 ◽  
Vol 53 (1) ◽  
pp. 42-52
Author(s):  
Katarzyna Szymańska-Dȩbowska
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Existence Results ◽  
Function Of Bounded Variation ◽  
Nonlocal Boundary Value Problem ◽  
At Resonance ◽  
Problem At Resonance

The paper focuses on existence of solutions of a system of nonlocal resonant boundary value problems , where f : [0, 1] × ℝk → ℝk is continuous and g : [0, 1] → ℝk is a function of bounded variation. Imposing on the function f the following condition: the limit limλ→∞f(t, λ a) exists uniformly in a ∈ Sk−1, we have shown that the problem has at least one solution.

scholarly journals On the Numerical Solution of Fractional Boundary Value Problems by a Spline Quasi-Interpolant Operator

Axioms ◽  
2020 ◽  
Vol 9 (2) ◽  
pp. 61
Author(s):  
Francesca Pitolli
Keyword(s):  
Fractional Derivative ◽  
Boundary Value Problems ◽  
Numerical Experiments ◽  
Boundary Value ◽  
Bernstein Operator ◽  
Explicit Formulas ◽  
Shape Preserving ◽  
Engineering Control ◽  
Shape Preserving Properties ◽  
Fractional Boundary Value Problems

Boundary value problems having fractional derivative in space are used in several fields, like biology, mechanical engineering, control theory, just to cite a few. In this paper we present a new numerical method for the solution of boundary value problems having Caputo derivative in space. We approximate the solution by the Schoenberg-Bernstein operator, which is a spline positive operator having shape-preserving properties. The unknown coefficients of the approximating operator are determined by a collocation method whose collocation matrices can be constructed efficiently by explicit formulas. The numerical experiments we conducted show that the proposed method is efficient and accurate.

scholarly journals Existence of solutions for a class of nonlinear boundary value problems on the hexasilinane graph

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ali Turab ◽  
Zoran D. Mitrović ◽  
Ana Savić
Keyword(s):  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Chemical Formula ◽  
Nonlinear Boundary Value Problems ◽  
Star Graphs ◽  
Chemical Graph ◽  
Chemical Graph Theory ◽  
Fractional Boundary Value Problems

AbstractChemical graph theory is a field of mathematics that studies ramifications of chemical network interactions. Using the concept of star graphs, several investigators have looked into the solutions to certain boundary value problems. Their choice to utilize star graphs was based on including a common point connected to other nodes. Our aim is to expand the range of the method by incorporating the graph of hexasilinane compound, which has a chemical formula $\mathrm{H}_{12} \mathrm{Si}_{6}$ H 12 Si 6 . In this paper, we examine the existence of solutions to fractional boundary value problems on such graphs, where the fractional derivative is in the Caputo sense. Finally, we include an example to support our significant findings.

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