scholarly journals Nonlocal q-fractional boundary value problem with Stieltjes integral conditions

2019 ◽  
Vol 24 (4) ◽  
Author(s):  
Jing Ren ◽  
Chengbo Zhai
Keyword(s):  
Linear Operator ◽  
Boundary Value Problem ◽  
Green's Functions ◽  
Green’S Functions ◽  
Boundary Value ◽  
Positive Linear Operator ◽  
First Eigenvalue ◽  
Multiple Positive Solutions ◽  
Stieltjes Integral ◽  
Integral Boundary

In this paper, we are dedicated to investigating a new class of one-dimensional lower-order fractional q-differential equations involving integral boundary conditions supplemented with Stieltjes integral. This condition is more general as it contains an arbitrary order derivative. It should be pointed out that the problem discussed in the current setting provides further insight into the research on nonlocal and integral boundary value problems. We first give the Green's functions of the boundary value problem and then develop some properties of the Green's functions that are conductive to our main results. Our main aim is to present two results: one considering the uniqueness of nontrivial solutions is given by virtue of contraction mapping principle associated with properties of u0-positive linear operator in which Lipschitz constant is associated with the first eigenvalue corresponding to related linear operator, while the other one aims to obtain the existence of multiple positive solutions under some appropriate conditions via standard fixed point theorems due to Krasnoselskii and Leggett–Williams. Finally, we give an example to illustrate the main results.

Sign Properties of Green's Functions For Two Classes of Boundary Value Problems

1987 ◽  
Vol 30 (1) ◽  
pp. 28-35 ◽  
Author(s):  
P. W. Eloe
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Green’S Function ◽  
Boundary Value Problems ◽  
Green's Function ◽  
Difference Equations ◽  
Green's Functions ◽  
Green’S Functions ◽  
Boundary Value ◽  
Partial Derivatives

AbstractLet G(x,s) be the Green's function for the boundary value problem y(n) = 0, Ty = 0, where Ty = 0 represents boundary conditions at two points. The signs of G(x,s) and certain of its partial derivatives with respect to x are determined for two classes of boundary value problems. The results are also carried over to analogous classes of boundary value problems for difference equations.

scholarly journals Multiple Positive Solutions of Third-Order BVP with Advanced Arguments and Stieltjes Integral Conditions

2016 ◽  
Vol 2016 ◽  
pp. 1-12
Author(s):  
Jian Chang ◽  
Jian-Ping Sun ◽  
Ya-Hong Zhao
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Integral Boundary Conditions ◽  
Multiple Positive Solutions ◽  
Stieltjes Integral ◽  
Integral Conditions ◽  
Third Order ◽  
Integral Boundary

We consider the following third-order boundary value problem with advanced arguments and Stieltjes integral boundary conditions:u′′′t+ft,uαt=0,  t∈0,1,  u0=γuη1+λ1uandu′′0=0,  u1=βuη2+λ2u, where0<η1<η2<1,0≤γ,β≤1,α:[0,1]→[0,1]is continuous,α(t)≥tfort∈[0,1], andα(t)≤η2fort∈[η1,η2]. Under some suitable conditions, by applying a fixed point theorem due to Avery and Peterson, we obtain the existence of multiple positive solutions to the above problem. An example is also included to illustrate the main results obtained.

Multiple positive solutions of nonlinear fractional differential equations with integral boundary value conditions

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Wei Sun ◽  
Youyu Wang
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Integral Boundary Conditions ◽  
Multiple Positive Solutions ◽  
Integral Boundary ◽  
Fractional Differential

AbstractIn this paper, we consider the multiplicity of positive solutions for a class of nonlinear boundary-value problem of fractional differential equations with integral boundary conditions. By means of a fixed point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of multiple positive solutions to the integral boundary value problem.

scholarly journals Multiple Positive Solutions of a Singular Semipositone Integral Boundary Value Problem for Fractionalq-Derivatives Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Yulin Zhao ◽  
Guobing Ye ◽  
Haibo Chen
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fixed Point Index ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Index Theorem ◽  
Multiple Positive Solutions ◽  
Integral Boundary ◽  
Integral Boundary Value Problem ◽  
Fixed Point Index Theorem

By using the fixed point index theorem, this paper investigates a class of singular semipositone integral boundary value problem for fractionalq-derivatives equations and obtains sufficient conditions for the existence of at least two and at least three positive solutions. Further, an example is given to illustrate the applications of our main results.

scholarly journals Eigenvalues, eigenfunctions and Green's functions on a path via Chebyshev polynomials

2009 ◽  
Vol 3 (2) ◽  
pp. 282-302 ◽  
Author(s):  
E. Bendito ◽  
A.M. Encinas ◽  
A. Carmona
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Poisson Equation ◽  
Chebyshev Polynomials ◽  
Ground State ◽  
Green's Functions ◽  
Periodic Boundary ◽  
Green’S Functions ◽  
Boundary Value ◽  
The Poisson Equation

In this work we analyze the boundary value problems on a path associated with Schr?dinger operators with constant ground state. These problems include the cases in which the boundary has two, one or none vertices. In addition, we study the periodic boundary value problem that corresponds to the Poisson equation in a cycle. Moreover, we obtain the Green's function for each regular problem and the eigenvalues and their corresponding eigenfunctions otherwise. In each case, the Green's functions, the eigenvalues and the eigenfunctions are given in terms of first, second and third kind Chebyshev polynomials.

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