scholarly journals A Note on Fractional Equations of Volterra Type with Nonlocal Boundary Condition

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Zhenhai Liu ◽  
Rui Wang
Keyword(s):  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Monotone Iterative Technique ◽  
Banach Fixed Point Theorem ◽  
Weighted Norm ◽  
Uniqueness Of Solutions ◽  
Volterra Type ◽  
Monotone Iterative ◽  
Fractional Equations ◽  
Liouville Derivative

We deal with nonlocal boundary value problems of fractional equations of Volterra type involving Riemann-Liouville derivative. Firstly, by defining a weighted norm and using the Banach fixed point theorem, we show the existence and uniqueness of solutions. Then, we obtain the existence of extremal solutions by use of the monotone iterative technique. Finally, an example illustrates the results.

scholarly journals About Positivity of Green's Functions for Nonlocal Boundary Value Problems with Impulsive Delay Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Alexander Domoshnitsky ◽  
Irina Volinsky
Keyword(s):  
Differential Equation ◽  
Existence And Uniqueness ◽  
Green's Functions ◽  
Green’S Functions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Uniqueness Of Solutions ◽  
Impulsive Delay ◽  
Impulsive Delay Differential Equation ◽  
Delay Differential

The impulsive delay differential equation is considered(Lx)(t)=x′(t)+∑i=1mpi(t)x(t-τi(t))=f(t), t∈[a,b],  x(tj)=βjx(tj-0), j=1,…,k, a=t0<t1<t2<⋯<tk<tk+1=b, x(ζ)=0, ζ∉[a,b],with nonlocal boundary conditionlx=∫abφsx′sds+θxa=c,  φ∈L∞a,b;  θ, c∈R.Various results on existence and uniqueness of solutions and on positivity/negativity of the Green's functions for this equation are obtained.

scholarly journals On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations

2022 ◽  
Vol 6 (1) ◽  
pp. 41
Author(s):  
Ravshan Ashurov ◽  
Yusuf Fayziev
Keyword(s):  
Fractional Derivatives ◽  
Separable Hilbert Space ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Value Problem ◽  
Nonlocal Problems ◽  
Existence Of A Solution ◽  
Left Hand ◽  
Liouville Derivative ◽  
The Right

The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0<t≤T), u(ξ)=αu(0)+φ (α is a constant and 0<ξ≤T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann–Liouville derivative; naturally, in the case of the Riemann–Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant α on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function φ in the boundary conditions are investigated.

Analysis of a system of autonomous fractional differential equations

2017 ◽  
Vol 10 (07) ◽  
pp. 1750094 ◽  
Author(s):  
Xiaojun Zhou ◽  
Chuanju Xu
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Upper And Lower Solutions ◽  
Coupled System ◽  
Monotone Iterative Technique ◽  
Metapopulation Model ◽  
Uniqueness Of Solutions ◽  
Predator Prey ◽  
Monotone Iterative ◽  
Fractional Differential

In this work, we study a system of autonomous fractional differential equations. The differential operator is taken in the Caputo sense. Using the monotone iterative technique combined with the method of upper and lower solutions, we investigate the existence and uniqueness of solutions for coupled system which are nonlinear fractional differential equations, moreover, we obtain the dependence of the solution on the initial values. In addition, we give an important example that is a two-patch subdiffusive predator–prey metapopulation model, investigate the solvability and give the numerical results with this model. The numerical simulation indicates that the results of the subdiffusive model approximate to the two-patch predator–prey metapopulation model with the order [Formula: see text] approach to 1.

scholarly journals Extremal Solutions to Periodic Boundary Value Problem of Nabla Integrodifferential Equation of Volterra Type on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Yunlong Shi ◽  
Junfang Zhao
Keyword(s):  
Boundary Value Problem ◽  
Time Scales ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Monotone Iterative Technique ◽  
Periodic Boundary Value Problem ◽  
Extremal Solutions ◽  
Comparison Result ◽  
Volterra Type ◽  
Monotone Iterative

We firstly establish some new theorems on time scales, and then, by employing them together with a new comparison result and the monotone iterative technique, we show the existence of extremal solutions to the following nabla integrodifferential periodic boundary value problem:u∇(t)=f(t,u,∫0t‍g(t,s)∇s),  t∈[0,a]T,  u(0)=u(ρ(a)), whereTis a time scale.

scholarly journals Application of Lakshmikantham's monotone-iterative technique to the solution of the initial value problem for impulsive integro-differential equations

1993 ◽  
Vol 6 (1) ◽  
pp. 25-34 ◽  
Author(s):  
D. D. Bainov ◽  
S. G. Hristova
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Initial Value ◽  
Volterra Type ◽  
Monotone Iterative

In the present paper, a technique of V. Lakshmikantham is applied to approximate finding of extremal quasisolutions of an initial value problem for a system of impulsive integro-differential equations of Volterra type.

scholarly journals Existence and Numerical Simulation of Solutions for Fractional Equations Involving Two Fractional Orders with Nonlocal Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Jing Zhao ◽  
Peifen Lu ◽  
Yiliang Liu
Keyword(s):  
Numerical Simulation ◽  
Fixed Point ◽  
Existence And Uniqueness ◽  
Nonlocal Boundary ◽  
Sufficient Conditions ◽  
Nonlocal Boundary Conditions ◽  
Uniqueness Of Solutions ◽  
Fractional Equations ◽  
Fractional Langevin Equation ◽  
Fractional Orders

We study a boundary value problem for fractional equations involving two fractional orders. By means of a fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions for the fractional equations. In addition, we describe the dynamic behaviors of the fractional Langevin equation by using theG2algorithm.

scholarly journals On the Nonlocal Problems in Time for Time-fractional Subdiffusion Equations

2021 ◽  
Author(s):  
Ravshan Ashurov ◽  
Yusuf Fayziev
Keyword(s):  
Fractional Derivatives ◽  
Separable Hilbert Space ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Value Problem ◽  
Nonlocal Problems ◽  
Existence Of A Solution ◽  
Left Hand ◽  
Liouville Derivative ◽  
The Right

The nonlocal boundary value problem, dt&rho;u(t)+Au(t)=f(t) (0&amp;lt;&rho;&amp;lt;1, 0&amp;lt;t&le;T), u(&xi;)=&alpha;u(0)+&phi; (&alpha; is a constant and 0&amp;lt;&xi;&le;T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann-Liouville derivative; naturally, in the case of the Riemann - Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant &alpha; on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function &phi; in the boundary conditions are investigated.

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

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