On the sign of Green’s function for an impulsive differential equation with periodic boundary conditions

In this article we investigate a formula for the Green’s function for the n-orderlinear differential equation with n additional conditions. We use this formula for calculatingthe Green’s function for problems with nonlocal boundary conditions.

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Nonlinear multi-order fractional differential equations with periodic/anti-periodic boundary conditions

Fractional Calculus and Applied Analysis ◽

10.2478/s13540-014-0172-6 ◽

2014 ◽

Vol 17 (2) ◽

Cited By ~ 4

Author(s):

Sangita Choudhary ◽

Varsha Daftardar-Gejji

Keyword(s):

Differential Equation ◽

Boundary Conditions ◽

Fractional Derivative ◽

Existence And Uniqueness ◽

Periodic Boundary ◽

Periodic Boundary Conditions ◽

Uniqueness Theorems ◽

Parametric Functions ◽

Fractional Differential ◽

Fractional Boundary Value Problems

AbstractIn the present manuscript we analyze non-linear multi-order fractional differential equation $$L\left( D \right)u\left( t \right) = f\left( {t,u\left( t \right)} \right), t \in \left[ {0,T} \right], T > 0,$$ where $$L\left( D \right) = \lambda _n ^c D^{\alpha _n } + \lambda _{n - 1} ^c D^{\alpha _{n - 1} } + \cdots + \lambda _1 ^c D^{\alpha _1 } + \lambda _0 ^c D^{\alpha _0 } ,\lambda _i \in \mathbb{R}\left( {i = 0,1, \cdots ,n} \right),\lambda _n \ne 0, 0 \leqslant \alpha _0 < \alpha _1 < \cdots < \alpha _{n - 1} < \alpha _n < 1,$$ and c D α denotes the Caputo fractional derivative of order α. We find the Greens functions for this equation corresponding to periodic/anti-periodic boundary conditions in terms of the two-parametric functions of Mittag-Leffler type. Further we prove existence and uniqueness theorems for these fractional boundary value problems.

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Green’s Function Related to a n-th Order Linear Differential Equation Coupled to Arbitrary Linear Non-Local Boundary Conditions

Mathematics ◽

10.3390/math9161948 ◽

2021 ◽

Vol 9 (16) ◽

pp. 1948

Author(s):

Alberto Cabada ◽

Lucía López-Somoza ◽

Mouhcine Yousfi

Keyword(s):

Differential Equation ◽

Boundary Conditions ◽

Green’S Function ◽

Explicit Expression ◽

Green's Function ◽

Homogeneous Problem ◽

Necessary Condition ◽

Local Conditions ◽

Uniqueness Of The Solution ◽

Non Local

In this paper, we obtain the explicit expression of the Green’s function related to a general n-th order differential equation coupled to non-local linear boundary conditions. In such boundary conditions, an n dimensional parameter dependence is also assumed. Moreover, some comparison principles are obtained. The explicit expression depends on the value of the Green’s function related to the two-point homogeneous problem; that is, we are assuming that when all the parameters involved on the boundary conditions take the value zero then the problem has a unique solution, which is characterized by the corresponding Green’s function g. The expression of the Green’s function G of the general problem is given as a function of g and the real parameters considered at the boundary conditions. It is important to note that, in order to ensure the uniqueness of the solution of the considered linear problem, we must assume a non-resonant additional condition on the considered problem, which depends on the non-local conditions and the corresponding parameters. We point out that the assumption of the uniqueness of the solution of the two-point homogeneous problem is not a necessary condition to ensure the existence of the solution of the general case. Of course, in this situation, the expression we are looking for must be obtained in a different manner. To show the applicability of the obtained results, a particular example is given.

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Klein's Oscillation Theorem for Periodic Boundary Conditions

Canadian Journal of Mathematics ◽

10.4153/cjm-1971-078-8 ◽

1971 ◽

Vol 23 (4) ◽

pp. 699-703 ◽

Cited By ~ 2

Author(s):

A. Howe

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Boundary Conditions ◽

Periodic Boundary ◽

Eigenvalue Problems ◽

Periodic Boundary Conditions ◽

Linear Differential Equations ◽

Oscillation Theorem ◽

Linear Differential ◽

Image Position

Multiparameter eigenvalue problems for systems of linear differential equations with homogeneous boundary conditions have been considered by Ince [4] and Richardson [5, 6], and more recently Faierman [3] has considered their completeness and expansion theorems. A survey of eigenvalue problems with several parameters, in mathematics, is given by Atkinson [1].We consider the two differential equations:1a1bwhere p1’(x), q1(x), A1(x), B1(x) and p2’(y), q2(y), A2(y), B2(y) are continuous for x ∈ [a1, b1] and y ∈ [a2, b2] respectively, and p1 (x) > 0(x ∈ [a1, b1]), p2(y) > 0 (y ∈ [a2, b2]), p1(a1) = p1(b1), p2(a2) = p2(b2). The differential equations (1) will be subjected to the periodic boundary conditions.2a2bLet us consider a single differential equation

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