scholarly journals Singular Positone and Semipositone Boundary Value Problems of Nonlinear Fractional Differential Equations

2009 ◽  
Vol 2009 ◽  
pp. 1-17 ◽  
Author(s):  
Chengjun Yuan ◽  
Daqing Jiang ◽  
Xiaojie Xu
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Real Number ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Existence Results ◽  
Fractional Boundary Value Problem ◽  
Liouville Fractional Derivative ◽  
Fractional Differential

We present some new existence results for singular positone and semipositone nonlinear fractional boundary value problemD0+αu(t)=μa(t)f(t,u(t)), 0<t<1,u(0)=u(1)=u′(0)=u′(1)=0, whereμ>0,a,andfare continuous,α∈(3,4]is a real number, andD0+αis Riemann-Liouville fractional derivative. Throughout our nonlinearity may be singular in its dependent variable. Two examples are also given to illustrate the main results.

scholarly journals Multiple Positive Solutions for Nonlinear Semipositone Fractional Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Wen-Xue Zhou ◽  
Ji-Gen Peng ◽  
Yan-Dong Chu
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Real Number ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Multiple Positive Solutions ◽  
Fractional Boundary Value Problem ◽  
Fractional Differential ◽  
Multiplicity Of Positive Solutions

We present some new multiplicity of positive solutions results for nonlinear semipositone fractional boundary value problemD0+αu(t)=p(t)f(t,u(t))-q(t),0<t<1,u(0)=u(1)=u'(1)=0, where2<α≤3is a real number andD0+αis the standard Riemann-Liouville differentiation. One example is also given to illustrate the main result.

scholarly journals Integral presentations of the solution of a boundary value problem for impulsive fractional integro-differential equations with Riemann-Liouville derivatives

2021 ◽  
Vol 7 (2) ◽  
pp. 2973-2988
Author(s):  
Ravi Agarwal ◽  
◽  
Snezhana Hristova ◽  
Donal O'Regan ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Initial Time ◽  
Qualitative Properties ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
Time Point

<abstract><p>Riemann-Liouville fractional differential equations with impulses are useful in modeling the dynamics of many real world problems. It is very important that there are good and consistent theoretical proofs and meaningful results for appropriate problems. In this paper we consider a boundary value problem for integro-differential equations with Riemann-Liouville fractional derivative of orders from $ (1, 2) $. We consider both interpretations in the literature on the presence of impulses in fractional differential equations: With fixed lower limit of the fractional derivative at the initial time point and with lower limits changeable at each impulsive time point. In both cases we set up in an appropriate way impulsive conditions which are dependent on the Riemann-Liouville fractional derivative. We establish integral presentations of the solutions in both cases and we note that these presentations are useful for furure studies of existence, stability and other qualitative properties of the solutions.</p></abstract>

The method of upper and lower solutions for integral boundary value problem of semilinear fractional differential equations with non-instantaneous impulses

2020 ◽  
Vol 70 (3) ◽  
pp. 625-640 ◽  
Author(s):  
Mengrui Xu ◽  
Shurong Sun ◽  
Zhenlai Han
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Existence Results ◽  
Three Solutions ◽  
Integral Boundary ◽  
Integral Boundary Value Problem ◽  
Fractional Differential

AbstractIn this paper, we investigate a class of semilinear fractional differential equations with non-instantaneous impulses and integral boundary value conditions. By the method of upper and lower solutions combined with Amann three-solution theorem, existence results of at least three solutions are obtained.

scholarly journals Successive Approximation Technique in the Study of a Nonlinear Fractional Boundary Value Problem

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 724
Author(s):  
Kateryna Marynets
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Approximation Technique ◽  
Fractional Boundary Value Problem ◽  
Point Boundary ◽  
Fractional Differential ◽  
Fractional Boundary Value Problems

We studied one essentially nonlinear two–point boundary value problem for a system of fractional differential equations. An original parametrization technique and a dichotomy-type approach led to investigation of solutions of two “model”-type fractional boundary value problems, containing some artificially introduced parameters. The approximate solutions of these problems were constructed analytically, while the numerical values of the parameters were determined as solutions of the so-called “bifurcation” equations.

scholarly journals Existence Results for Singular Boundary Value Problem of Nonlinear Fractional Differential Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Yujun Cui
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Existence Results ◽  
Singular Boundary ◽  
Singular Boundary Value Problem ◽  
Fractional Boundary Value Problem ◽  
Nonlinear Fractional Differential Equation ◽  
Existence Of Positive Solutions ◽  
Liouville Fractional Derivative ◽  
Fractional Differential

By applying a fixed point theorem for mappings that are decreasing with respect to a cone, this paper investigates the existence of positive solutions for the nonlinear fractional boundary value problem: , , , where , is the Riemann-Liouville fractional derivative.

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