scholarly journals Uniqueness for linear integro-differential equations in the real line and applications

2021 ◽  
Vol 60 (6) ◽  
Author(s):  
Juan-Carlos Felipe-Navarro
Keyword(s):  
Differential Operator ◽  
Linear Equation ◽  
Positive Solution ◽  
Real Line ◽  
Nonlinear Problems ◽  
Liouville Type ◽  
Uniqueness Of Solutions ◽  
Type Method ◽  
Monotone Solutions ◽  
Semilinear Problem

AbstractIn this work we prove the uniqueness of solutions to the nonlocal linear equation $$L \varphi - c(x)\varphi = 0$$ L φ - c ( x ) φ = 0 in $$\mathbb {R}$$ R , where L is an elliptic integro-differential operator, in the presence of a positive solution or of an odd solution vanishing only at zero. As an application, we deduce the nondegeneracy of layer solutions (bounded and monotone solutions) to the semilinear problem $$L u = f(u)$$ L u = f ( u ) in $$\mathbb {R}$$ R when the nonlinearity is of Allen–Cahn type. To our knowledge, this is the first work where such uniqueness and nondegeneracy results are proven in the nonlocal framework when the Caffarelli–Silvestre extension technique is not available. Our proofs are based on a nonlocal Liouville-type method developed by Hamel, Ros-Oton, Sire, and Valdinoci for nonlinear problems in dimension two.

An Existence Theorem for Nonlinear Boundary Value Problems

1975 ◽  
Vol 17 (5) ◽  
pp. 693-701
Author(s):  
W. L. McCandless
Keyword(s):  
Boundary Value Problems ◽  
Fixed Point Theorems ◽  
Standard Procedure ◽  
Boundary Value ◽  
Nonlinear Problems ◽  
Nonlinear Boundary Value Problems ◽  
Uniqueness Of Solutions ◽  
Research Activity ◽  
Approximation Techniques ◽  
The Subject

Boundary value problems for ordinary differential equations have long been the subject of extensive research activity. In particular, questions concerning the existence and uniqueness of solutions for these problems have received much attention, and algebraic fixed-point theorems have served as important tools in such investigations. For example Picard [8] based his pioneering work in this area on the use of successive approximation techniques, and recently his classical methods have been refined and extended to more general nonlinear problems (see [1] and [4]). The standard procedure for applying these techniques requires that the boundary value problem under consideration first be converted into an equivalent integral equation through the choice of a suitable Green’s function. The resulting theory is consequently limited to problems for which such a formulation is possible.

scholarly journals Semilinear fourth order boundary value problems

1990 ◽  
Vol 42 (1) ◽  
pp. 101-114 ◽  
Author(s):  
Gerhard Metzen
Keyword(s):  
Maximum Principle ◽  
Differential Operator ◽  
Linear Operator ◽  
Fourth Order ◽  
Existence Of Solutions ◽  
Principal Eigenvalue ◽  
Nonlinear Problems ◽  
Order Differential Operator ◽  
Positive Eigenfunction ◽  
Fourth Order Differential Operator

We study a certain linear fourth order differential operator and show the existence of solutions to corresponding nonlinear problems. It will be shown that a maximum principle holds and that under certain conditions the linear operator has a positive principal eigenvalue with corresponding positive eigenfunction.

scholarly journals Multipoint Fractional Iterative Methods with (2α + 1)th-Order of Convergence for Solving Nonlinear Problems

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 452
Author(s):  
Giro Candelario ◽  
Alicia Cordero ◽  
Juan R. Torregrosa
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Newton Method ◽  
Fractional Derivatives ◽  
Order Of Convergence ◽  
Recent Literature ◽  
Nonlinear Problems ◽  
Type Method ◽  
Numerical Tests ◽  
Multipoint Method

In the recent literature, some fractional one-point Newton-type methods have been proposed in order to find roots of nonlinear equations using fractional derivatives. In this paper, we introduce a new fractional Newton-type method with order of convergence α + 1 and compare it with the existing fractional Newton method with order 2 α . Moreover, we also introduce a multipoint fractional Traub-type method with order 2 α + 1 and compare its performance with that of its first step. Some numerical tests and analysis of the dependence on the initial estimations are made for each case, including a comparison with classical Newton ( α = 1 of the first step of the class) and classical Traub’s scheme ( α = 1 of fractional proposed multipoint method). In this comparison, some cases are found where classical Newton and Traub’s methods do not converge and the proposed methods do, among other advantages.

scholarly journals Existence Results for a Nonlocal Coupled System of Differential Equations Involving Mixed Right and Left Fractional Derivatives and Integrals

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 578
Author(s):  
Sotiris K. Ntouyas ◽  
Abrar Broom ◽  
Ahmed Alsaedi ◽  
Tareq Saeed ◽  
Bashir Ahmad
Keyword(s):  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Coupled System ◽  
Uniqueness Result ◽  
Existence Results ◽  
System Of Differential Equations ◽  
Liouville Type ◽  
Uniqueness Of Solutions ◽  
Fractional Derivatives And Integrals

In this paper, we study the existence and uniqueness of solutions for a new kind of nonlocal four-point fractional integro-differential system involving both left Caputo and right Riemann–Liouville fractional derivatives, and Riemann–Liouville type mixed integrals. The Banach and Schaefer fixed point theorems are used to obtain the desired results. An example illustrating the existence and uniqueness result is presented.

On a quadratic integral inequality

1978 ◽  
Vol 78 (3-4) ◽  
pp. 241-256 ◽  
Author(s):  
R. J. Amos ◽  
W. N. Everitt
Keyword(s):  
Differential Operator ◽  
Real Number ◽  
Function Space ◽  
Real Line ◽  
Integral Inequality ◽  
Hilbert Function ◽  
Linear Manifold ◽  
Quadratic Integral ◽  
Image Position ◽  
Hilbert Function Space

SynopsisThe inequality considered iswhere p and q are given real-valued coefficients on the interval [a, b), with b ≦ ∝, of the real line; here D is a linear manifold of the Hilbert function space L2(a, b), and μ is a real number characterised in terms of the spectrum of a uniquely determined self-adjoint differential operator in L2(a, b).

Global bifurcation analysis and uniqueness for a semilinear problem

1989 ◽  
Vol 111 (3-4) ◽  
pp. 265-284 ◽  
Author(s):  
Achilles Tertikas
Keyword(s):  
Bifurcation Diagram ◽  
Bifurcation Analysis ◽  
Implicit Function Theorem ◽  
Global Bifurcation ◽  
Implicit Function ◽  
Bifurcation Diagrams ◽  
Uniqueness Of Solutions ◽  
Image Position ◽  
Semilinear Problem

SynopsisWe study the bifurcation diagram and uniqueness of solutions ofBy using a rescaling technique and the Implicit Function Theorem, we establish the global bifurcation diagram. Uniqueness is proved by a separation argument to complete the bifurcation picture of the problem. Our study suggests that the bifurcation diagrams have different behaviour at λ = 0, depending on whether g(∞) > 0 or g(∞) < 0 in L∞ norm, but quite similar behaviour in Lp or W2,1.

scholarly journals System of fractional differential equations with Erdélyi-Kober fractional integral conditions

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Natthaphong Thongsalee ◽  
Sorasak Laoprasittichok ◽  
Sotiris K. Ntouyas ◽  
Jessada Tariboon
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fractional Integral ◽  
Existence Of Solutions ◽  
Integral Conditions ◽  
Liouville Type ◽  
Uniqueness Of Solutions ◽  
Fractional Differential ◽  
Banach’S Contraction Principle

AbstractIn this paper we study existence and uniqueness of solutions for a system consisting from fractional differential equations of Riemann-Liouville type subject to nonlocal Erdélyi-Kober fractional integral conditions. The existence and uniqueness of solutions is established by Banach’s contraction principle, while the existence of solutions is derived by using Leray-Schauder’s alternative. Examples illustrating our results are also presented.

Sign in / Sign up

Export Citation Format

Share Document