scholarly journals The method of upper and lower solutions for a class of fractional differential coupled systems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Huan Quan ◽  
Xiping Liu ◽  
Mei Jia
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Nonlocal Boundary ◽  
Coupled Systems ◽  
State Dependent ◽  
Fractional Differential ◽  
Functional Differential ◽  
Fractional Functional

AbstractIn this paper, we investigate a class of nonlocal boundary value problems of nonlinear fractional functional differential coupled systems with state dependent delays. The method of upper and lower solutions is established and some new results for the multiplicity of solutions of the boundary value problem are obtained. An example is also presented to illustrate our main results.

Existence of positive solutions for nonlocal boundary value problem of fractional differential equation

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Xiping Liu ◽  
Legang Lin ◽  
Haiqin Fang
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Nonlocal Boundary ◽  
Point Boundary ◽  
Nonlocal Boundary Value Problem ◽  
Existence Of Positive Solutions ◽  
Fractional Differential

AbstractIn this paper, we study a type of nonlinear fractional differential equations multi-point boundary value problem with fractional derivative in the boundary conditions. By using the upper and lower solutions method and fixed point theorems, some results for the existence of positive solutions for the boundary value problem are established. Some examples are also given to illustrate our results.

scholarly journals The Existence of Solutions for Boundary Value Problem of Fractional Functional Differential Equations with Delay

2018 ◽  
Vol 228 ◽  
pp. 01005
Author(s):  
Mengrui Xu ◽  
Yanan Li ◽  
Yige Zhao ◽  
Shurong Sun
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Functional Differential Equations ◽  
Contraction Mapping Principle ◽  
Alternative Theorem ◽  
Nonlinear Alternative ◽  
Banach Contraction ◽  
Functional Differential ◽  
Fractional Functional Differential Equations ◽  
Fractional Functional

A class of boundary value problem for fractional functional differential equation with delay $ \left\{ {\begin{array}{*{20}c} {^{C} D^{\sigma } \omega (t) = f(t,\omega _{t} ),t \in [0,\zeta ],} \\ {\omega (0) = 0,\,\omega ^{\prime}(0) = 0,\,\omega ^{\prime\prime}(\zeta ) = 1,} \\ \end{array} } \right. $ is studied, where $ 2 < \sigma \le 3,\,\,^{c} D^{\sigma } $ devote standard Caputo fractional derivative. In this article, three new criteria on existence and uniqueness of solution are obtained by Banach contraction mapping principle, Schauder fixed point theorem and nonlinear alternative theorem.

scholarly journals Wellposedness of Neumann boundary-value problems of space-fractional differential equations

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Hong Wang ◽  
Danping Yang
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Neumann Boundary Condition ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Fractional Differential ◽  
Neumann Boundary Value Problems ◽  
Well Posed

AbstractFractional differential equation (FDE) provides an accurate description of transport processes that exhibit anomalous diffusion but introduces new mathematical difficulties that have not been encountered in the context of integer-order differential equation. For example, the wellposedness of the Dirichlet boundary-value problem of one-dimensional variable-coefficient FDE is not fully resolved yet. In addition, Neumann boundary-value problem of FDE poses significant challenges, partly due to the fact that different forms of FDE and different types of Neumann boundary condition have been proposed in the literature depending on different applications.We conduct preliminary mathematical analysis of the wellposedness of different Neumann boundary-value problems of the FDEs. We prove that five out of the nine combinations of three different forms of FDEs that are closed by three types of Neumann boundary conditions are well posed and the remaining four do not admit a solution. In particular, for each form of the FDE there is at least one type of Neumann boundary condition such that the corresponding boundary-value problem is well posed, but there is also at least one type of Neumann boundary condition such that the corresponding boundary-value problem is ill posed. This fully demonstrates the subtlety of the study of FDE, and, in particular, the crucial mathematical modeling question: which combination of FDE and fractional Neumann boundary condition, rather than which form of FDE or fractional Neumann boundary condition, should be used and studied in applications.

Quasilinearization and boundary value problems at resonance

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Kareem Alanazi ◽  
Meshal Alshammari ◽  
Paul Eloe
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Approximate Solutions ◽  
Monotonicity Condition ◽  
Uniqueness Of Solutions ◽  
Shift Method ◽  
At Resonance ◽  
Problem At Resonance

Abstract A quasilinearization algorithm is developed for boundary value problems at resonance. To do so, a standard monotonicity condition is assumed to obtain the uniqueness of solutions for the boundary value problem at resonance. Then the method of upper and lower solutions and the shift method are applied to obtain the existence of solutions. A quasilinearization algorithm is developed and sequences of approximate solutions are constructed, which converge monotonically and quadratically to the unique solution of the boundary value problem at resonance. Two examples are provided in which explicit upper and lower solutions are exhibited.

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.

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