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.

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.

scholarly journals A nonlinear second order problem with a nonlocal boundary condition

2006 ◽  
Vol 2006 ◽  
pp. 1-11 ◽  
Author(s):  
P. Amster ◽  
P. De Nápoli
Keyword(s):  
Nonlocal Boundary Condition ◽  
Upper And Lower Solutions ◽  
Nonlocal Boundary ◽  
Type Condition ◽  
Existence Of A Solution ◽  
Mawhin’S Continuation Theorem ◽  
Order Problem ◽  
Nonlinear Operators ◽  
Pendulum Equation ◽  
Type Boundary

We study a nonlinear problem of pendulum-type for ap-Laplacian with nonlinear periodic-type boundary conditions. Using an extension of Mawhin's continuation theorem for nonlinear operators, we prove the existence of a solution under a Landesman-Lazer type condition. Moreover, using the method of upper and lower solutions, we generalize a celebrated result by Castro for the classical pendulum equation.

scholarly journals Existence of solutions for a fractional nonlocal boundary value problem

2020 ◽  
Vol 36 (3) ◽  
pp. 453-462
Author(s):  
RODICA LUCA
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Fractional Integrals ◽  
Nonlocal Boundary Value Problem ◽  
Fractional Differential

We investigate the existence of solutions for a Riemann-Liouville fractional differential equation with a nonlinearity dependent of fractional integrals, subject to nonlocal boundary conditions which contain various fractional derivatives and Riemann-Stieltjes integrals. In the proof of our main results we use different fixed point theorems.

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 On a boundary value problem for differential equation with p-Laplacian

2011 ◽  
Vol 48 (1) ◽  
pp. 189-195
Author(s):  
Boris Rudolf
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Boundary Value Problem ◽  
Nonlocal Boundary Condition ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Lower And Upper Solutions ◽  
Existence Of A Solution

Abstract The existence of a solution of a boundary value problem for differential equation with p-Laplacian is proved by the technique of lower and upper solutions. A nonlocal boundary condition and a derivative dependent nonlinearity is assumed.

scholarly journals An inverse source problem for a two terms time-fractional diffusion equation

2021 ◽  
Vol 40 ◽  
pp. 1-15
Author(s):  
Fatima Dib ◽  
Mokhtar Kirane
Keyword(s):  
Boundary Condition ◽  
Fractional Derivatives ◽  
Source Term ◽  
Space Variable ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Fractional Diffusion ◽  
Inverse Source Problem ◽  
Integral Type ◽  
Time Fractional Diffusion Equation

In this paper, we consider an inverse problem for a linear heat equation involving two time-fractional derivatives, subject to a nonlocal boundary condition. We determine a source term independent of the space variable, and the temperature distribution with an over- determining function of integral type.

scholarly journals ASYMPTOTIC DISTRIBUTION OF EIGENVALUES AND EIGENFUNCTIONS OF A NONLOCAL BOUNDARY VALUE PROBLEM

2021 ◽  
Vol 26 (2) ◽  
pp. 253-266
Author(s):  
Erdoğan Şen ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problem ◽  
Asymptotic Distribution ◽  
Nonlocal Boundary Condition ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Asymptotic Formulas ◽  
Nonlocal Boundary Value Problem ◽  
Eigenvalues And Eigenfunctions ◽  
Distribution Of Eigenvalues

In this work, we obtain asymptotic formulas for eigenvalues and eigenfunctions of the second order boundary-value problem with a Bitsadze–Samarskii type nonlocal boundary condition.

scholarly journals Existence of solution for Hilfer fractional differential problem with nonlocal boundary condition in Banach spaces

2021 ◽  
Vol 66 (3) ◽  
pp. 521-536
Author(s):  
Hanan A. Wahash ◽  
Mohammed S. Abdo ◽  
Satish K. Panchal ◽  
Sandeep P. Bhairat
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Integral Equation ◽  
Banach Spaces ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Existence Of Solution ◽  
Existence Of A Solution ◽  
Differential Problem ◽  
Fractional Differential

"This paper is devoted to study the existence of a solution to Hilfer fractional differential equation with nonlocal boundary condition in Banach spaces. We use the equivalent integral equation to study the considered Hilfer differential problem with nonlocal boundary condition. The Monch type fixed point theorem and the measure of the noncompactness technique are the main tools in this study. We demonstrate the existence of a solution with a suitable illustrative example."

scholarly journals Nonlinear Integro-Differential Equations Involving Mixed Right and Left Fractional Derivatives and Integrals with Nonlocal Boundary Data

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 336 ◽  
Author(s):  
Bashir Ahmad ◽  
Abrar Broom ◽  
Ahmed Alsaedi ◽  
Sotiris K. Ntouyas
Keyword(s):  
Functional Analysis ◽  
Differential Equations ◽  
Boundary Data ◽  
Fractional Derivatives ◽  
Existence Of Solutions ◽  
Nonlocal Boundary ◽  
Fractional Integrals ◽  
Nonlocal Boundary Value Problem ◽  
Fractional Derivatives And Integrals ◽  
Left And Right

In this paper, we study the existence of solutions for a new nonlocal boundary value problem of integro-differential equations involving mixed left and right Caputo and Riemann–Liouville fractional derivatives and Riemann–Liouville fractional integrals of different orders. Our results rely on the standard tools of functional analysis. Examples are constructed to demonstrate the application of the derived results.

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