Boundedness of Positive Solutions of Second-order Rational Difference Equations

2004 ◽  
Vol 10 (11) ◽  
pp. 935-940 ◽  
Author(s):  
Y.S. Huang ◽  
P.M. Knopf
Keyword(s):  
Positive Solutions ◽  
Difference Equations ◽  
Second Order ◽  
Rational Difference Equations

scholarly journals Multiple positive solutions of second-order nonlinear difference equations with discrete singular ϕ-Laplacian

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaoxiao Su ◽  
Ruyun Ma
Keyword(s):  
Dirichlet Problem ◽  
Difference Equation ◽  
Positive Solutions ◽  
Difference Equations ◽  
Second Order ◽  
Topological Method ◽  
Multiple Positive Solutions ◽  
Nonlinear Difference Equations ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

AbstractWe consider the existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear difference equation $$ \textstyle\begin{cases} -\nabla [\phi (\triangle u(t))]=\lambda a(t,u(t))+\mu b(t,u(t)), \quad t\in \mathbb{T}, \\ u(1)=u(N)=0, \end{cases} $$ { − ∇ [ ϕ ( △ u ( t ) ) ] = λ a ( t , u ( t ) ) + μ b ( t , u ( t ) ) , t ∈ T , u ( 1 ) = u ( N ) = 0 , where $\lambda ,\mu \geq 0$ λ , μ ≥ 0 , $\mathbb{T}=\{2,\ldots ,N-1\}$ T = { 2 , … , N − 1 } with $N>3$ N > 3 , $\phi (s)=s/\sqrt{1-s^{2}}$ ϕ ( s ) = s / 1 − s 2 . The function $f:=\lambda a(t,s)+\mu b(t,s)$ f : = λ a ( t , s ) + μ b ( t , s ) is either sublinear, or superlinear, or sub-superlinear near $s=0$ s = 0 . Applying the topological method, we prove the existence of either one or two, or three positive solutions.

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