Existence and multiplicity of homoclinic solutions for second-order nonlinear difference equations with Jacobi operators

2016 ◽  
Vol 39 (18) ◽  
pp. 5705-5719 ◽  
Author(s):  
Peng Chen ◽  
Xiaofei He
Keyword(s):  
Difference Equations ◽  
Second Order ◽  
Homoclinic Solutions ◽  
Nonlinear Difference Equations ◽  
Jacobi Operators ◽  
Existence And Multiplicity

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.

scholarly journals Infinitely Many Homoclinic Solutions for Second Order Nonlinear Difference Equations withp-Laplacian

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Guowei Sun ◽  
Ali Mai
Keyword(s):  
Difference Equations ◽  
Critical Point Theory ◽  
Point Theory ◽  
Nehari Manifold ◽  
Homoclinic Solutions ◽  
Nonlinear Difference Equations ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Superlinear Nonlinearity ◽  
Laplacian Equations

We employ Nehari manifold methods and critical point theory to study the existence of nontrivial homoclinic solutions of discretep-Laplacian equations with a coercive weight function and superlinear nonlinearity. Without assuming the classical Ambrosetti-Rabinowitz condition and without any periodicity assumptions, we prove the existence and multiplicity results of the equations.

scholarly journals A boundary value problem for second-order nonlinear difference equations on the integers

2005 ◽  
Vol 47 (2) ◽  
pp. 237-248
Author(s):  
F. Dal ◽  
G. Sh. Guseinov
Keyword(s):  
Boundary Value Problem ◽  
Difference Equations ◽  
Boundary Value ◽  
Second Order ◽  
Nonlinear Difference Equations ◽  
Left Hand ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Limit Circle Case ◽  
Difference Expression

AbstractIn this study, we are concerned with a boundary value problem (BVP) for nonlinear difference equations on the set of all integers Z. under the assumption that the left-hand side is a second-order linear difference expression which belongs to the so-called Weyl-Hamburger limit-circle case. The BVP is considered in the Hilbert space l2 and includes boundary conditions at infinity. Existence and uniqueness results for solution of the considered BVP are established.

MODIFIED OSCILLATION CRITERIA OF SECOND-ORDER NEUTRAL DIFFERENCE EQUATIONS

2021 ◽  
Vol 28 (1-2) ◽  
pp. 19-30
Author(s):  
G. CHATZARAKIS G. CHATZARAKIS ◽  
R. KANAGASABAPATHI R. KANAGASABAPATHI ◽  
S. SELVARANGAM S. SELVARANGAM ◽  
E. THANDAPANI E. THANDAPANI
Keyword(s):  
Difference Equations ◽  
Second Order ◽  
Riccati Transformation ◽  
Nonlinear Difference Equations ◽  
Transformation Technique ◽  
Neutral Difference Equations ◽  
Oscillation Criteria ◽  
Neutral Term

In this paper we shall consider a class of second-order nonlinear difference equations with a negative neutral term. Some new oscillation criteria are obtained via Riccati transformation technique. These criteria improve and modify the existing results mentioned in the literature. Some examples are given to show the applicability and significance of the main results.

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