In the research literature, one can find distinct notions for higher order averaged functions of regularly perturbed non-autonomous
T-periodic differential equations of the kind x′=ε
F(t,x,ε
). By one hand, the classical (stroboscopic) averaging method provides asymptotic estimates for its solutions in terms of some uniquely defined functions gi's, called averaged functions, which are obtained through near-identity stroboscopic transformations and by solving homological equations. On the other hand, a Melnikov procedure is employed to obtain bifurcation functions fi's which controls in some sense the existence of isolated T-periodic solutions of the differential equation above. In the research literature, the bifurcation functions fi's are sometimes likewise called averaged functions, nevertheless, they also receive the name of Poincaré–Pontryagin–Melnikov functions or just Melnikov functions. While it is known that
f1=Tg1, a general relationship between gi and fi is not known so far for i≥
2. Here, such a general relationship between these two distinct notions of averaged functions is provided, which allows the computation of the stroboscopic averaged functions of any order avoiding the necessity of dealing with near-identity transformations and homological equations. In addition, an Appendix is provided with implemented Mathematica algorithms for computing both higher order averaging functions.
In this paper we consider the following
Schrödinger–Kirchhoff–Poisson-type system
{
−
(
a
+
b
∫
Ω
|
∇
u
|
2
d
x
)
Δ
u
+
λ
ϕ
u
=
Q
(
x
)
|
u
|
p
−
2
u
in
Ω
,
−
Δ
ϕ
=
u
2
in
Ω
,
u
=
ϕ
=
0
on
∂
Ω
,
where
Ω
is a
bounded smooth domain of
R
3
,
a
>
0
,
b
≥
0
are constants
and
λ
is a positive parameter. Under suitable conditions on
Q
(
x
)
and combining the method of invariant sets of descending flow, we establish
the existence and multiplicity of sign-changing solutions to this problem
for the case that
2
<
p
<
4
as
λ
sufficient small. Furthermore,
for
λ
=
1
and the above assumptions on
Q
(
x
)
, we obtain the same
conclusions with
2
<
p
<
12
5
.
We investigate existence of solutions for a fractional Klein–Gordon
coupled with Maxwell's equation. On the basis of overcoming the lack of
compactness, we obtain that there is a radially symmetric solution for the
critical system by means of variational methods.
In this paper, we focus on the global dynamics of a neoclassical
growth system incorporating patch structure and multiple pairs of
time-varying delays. Firstly, we prove the global existence, positiveness
and boundedness of solutions for the addressed system. Secondly, by
employing some novel differential inequality analyses and the fluctuation
lemma, both delay-independent and delay-dependent criteria are established
to ensure that all solutions are convergent to the unique positive
equilibrium point, which supplement and improve some existing results.
Finally, some numerical examples are afforded to illustrate the
effectiveness and feasibility of the theoretical findings.
Let
Ω⊂Rn n>1 and let
p,q≥2. We consider the system of nonlinear Dirichlet problems
equation* brace(Au)(x)=Nu′(x,u(x),v(x)),x∈Ω,r-(Bv)(x)=Nv′(x,u(x),v(x)),x∈Ω,ru(x)=0,x∈∂Ω,rv(x)=0,x∈∂Ω,endequation* where
N:R×R→R is
C1 and is partially convex-concave and
A:W01,p(Ω)→(W01,p(Ω))*
B:W01,p(Ω)→(W01,p(Ω))* are monotone and potential operators. The solvability of this system is reached via the Ky–Fan minimax theorem.
Stability index of linear random dynamical systems