Summation equations with sign changing kernels and applications to discrete fractional boundary value problems

Christopher S. Goodrich

Commentationes Mathematicae Universitatis Carolinae (2016)

  • Volume: 57, Issue: 2, page 201-229
  • ISSN: 0010-2628

Abstract

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We consider the summation equation, for t ∈ [ μ - 2 , μ + b ] ℕ μ - 2 , y ( t ) = γ 1 ( t ) H 1 ∑ i = 1 n a i y ξ i + γ 2 ( t ) H 2 ∑ i = 1 m b i y ζ i + λ ∑ s = 0 b G ( t , s ) f ( s + μ - 1 , y ( s + μ - 1 ) ) in the case where the map ( t , s ) ↦ G ( t , s ) may change sign; here μ ∈ ( 1 , 2 ] is a parameter, which may be understood as the order of an associated discrete fractional boundary value problem. In spite of the fact that G is allowed to change sign, by introducing a new cone we are able to establish the existence of at least one positive solution to this problem by imposing some growth conditions on the functions H 1 and H 2 . Finally, as an application of the abstract existence result, we demonstrate that by choosing the maps t ↦ γ 1 ( t ) , γ 2 ( t ) in particular ways, we can recover the existence of at least one positive solution to various discrete fractional- or integer-order boundary value problems possessing Green’s functions that change sign.

How to cite

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Goodrich, Christopher S.. "Summation equations with sign changing kernels and applications to discrete fractional boundary value problems." Commentationes Mathematicae Universitatis Carolinae 57.2 (2016): 201-229. <http://eudml.org/doc/280127>.

@article{Goodrich2016,
abstract = {We consider the summation equation, for $t\in [\mu -2,\mu +b]_\{\mathbb \{N\}_\{\mu -2\}\}$, \begin\{align*\} y(t)=\gamma \_1(t)H\_1\left(\sum \_\{i=1\}^\{n\}a\_iy\left(\xi \_i\right)\right) & + \gamma \_2(t)H\_2\left(\sum \_\{i=1\}^\{m\}b\_iy\left(\zeta \_i\right)\right) &+ \lambda \sum \_\{s=0\}^\{b\}G(t,s)f(s+\mu -1,y(s+\mu -1)) \end\{align*\} in the case where the map $(t,s)\mapsto G(t,s)$ may change sign; here $\mu \in (1,2]$ is a parameter, which may be understood as the order of an associated discrete fractional boundary value problem. In spite of the fact that $G$ is allowed to change sign, by introducing a new cone we are able to establish the existence of at least one positive solution to this problem by imposing some growth conditions on the functions $H_1$ and $H_2$. Finally, as an application of the abstract existence result, we demonstrate that by choosing the maps $t\mapsto \gamma _1(t)$, $\gamma _2(t)$ in particular ways, we can recover the existence of at least one positive solution to various discrete fractional- or integer-order boundary value problems possessing Green’s functions that change sign.},
author = {Goodrich, Christopher S.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {summation equation; sign-changing kernel; discrete fractional calculus; positive solution; nonlocal boundary condition},
language = {eng},
number = {2},
pages = {201-229},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Summation equations with sign changing kernels and applications to discrete fractional boundary value problems},
url = {http://eudml.org/doc/280127},
volume = {57},
year = {2016},
}

TY - JOUR
AU - Goodrich, Christopher S.
TI - Summation equations with sign changing kernels and applications to discrete fractional boundary value problems
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2016
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 57
IS - 2
SP - 201
EP - 229
AB - We consider the summation equation, for $t\in [\mu -2,\mu +b]_{\mathbb {N}_{\mu -2}}$, \begin{align*} y(t)=\gamma _1(t)H_1\left(\sum _{i=1}^{n}a_iy\left(\xi _i\right)\right) & + \gamma _2(t)H_2\left(\sum _{i=1}^{m}b_iy\left(\zeta _i\right)\right) &+ \lambda \sum _{s=0}^{b}G(t,s)f(s+\mu -1,y(s+\mu -1)) \end{align*} in the case where the map $(t,s)\mapsto G(t,s)$ may change sign; here $\mu \in (1,2]$ is a parameter, which may be understood as the order of an associated discrete fractional boundary value problem. In spite of the fact that $G$ is allowed to change sign, by introducing a new cone we are able to establish the existence of at least one positive solution to this problem by imposing some growth conditions on the functions $H_1$ and $H_2$. Finally, as an application of the abstract existence result, we demonstrate that by choosing the maps $t\mapsto \gamma _1(t)$, $\gamma _2(t)$ in particular ways, we can recover the existence of at least one positive solution to various discrete fractional- or integer-order boundary value problems possessing Green’s functions that change sign.
LA - eng
KW - summation equation; sign-changing kernel; discrete fractional calculus; positive solution; nonlocal boundary condition
UR - http://eudml.org/doc/280127
ER -

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