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The latest research articles published by Boundary Value Problems20150703T12:00:00Z Existence results for hybrid fractional integrodifferential equations In this paper we study existence results for initial value problems for hybrid fractional integrodifferential equations. A couple of hybrid fixed point theorems for the sum of three operators are used for proving the main results. Examples illustrating the results are also presented.MSC: 34A08, 34A12.
http://www.boundaryvalueproblems.com/content/2015/1/113
Surang SithoSotiris NtouyasJessada TariboonBoundary Value Problems 2015, null:11320150703T12:00:00Zdoi:10.1186/s1366101503767/content/figures/s1366101503767toc.gifBoundary Value Problems16872770${item.volume}11320150703T12:00:00ZXML The boundary value condition of an evolutionary <inlineformula> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" name="s1366101503776i1"> <m:mi>p</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:math> </inlineformula>Laplacian equation Consider an evolutionary equation related to the
p
(
x
)
Laplacian:
u
t
=
div
(
ρ
α

∇
u

p
(
x
)
−
2
∇
u
)
+
∂
b
i
(
u
,
x
,
t
)
∂
x
i
,
(
x
,
t
)
∈
Q
T
=
Ω
×
(
0
,
T
)
, which arises from electrorheological fluid mechanics. Since
ρ
(
x
)
=
dist
(
x
,
∂
Ω
)
, the equation is degenerate on the boundary, one may expect that there is not flux across the boundary. The paper shows that the facts may be unexpected. The paper reviews FicheraOleinik theory, then uses the theory to discuss the boundary value condition related to the equation. If
p
−
>
2
, the existence and the uniqueness of the solutions are researched. Finally, if
b
i
≡
0
, the behavior of the solutions near the boundary is studied by the comparison theorem.MSC: 35L65, 35K85, 35R35.
http://www.boundaryvalueproblems.com/content/2015/1/112
Huashui ZhanBoundary Value Problems 2015, null:11220150702T00:00:00Zdoi:10.1186/s1366101503776/content/figures/s1366101503776toc.gifBoundary Value Problems16872770${item.volume}11220150702T00:00:00ZXML New conditions on homoclinic solutions for a subquadratic second order Hamiltonian system In this paper, we deal with the secondorder Hamiltonian system(
∗
)
u
¨
−
L
(
t
)
u
+
∇
W
(
t
,
u
)
=
0
.We establish some criteria which guarantee that the above system has at least one or infinitely many homoclinic solutions under the assumption that
W
(
t
,
x
)
is subquadratic at infinity and
L
(
t
)
is a real symmetric matrix and satisfieslim inf

t

→
+
∞
[

t

ν
−
2
inf

x

=
1
(
L
(
t
)
x
,
x
)
]
>
0for some constant
ν
<
2
. In particular,
L
(
t
)
and
W
(
t
,
x
)
are allowed to be signchanging.MSC: 34C37, 58E05, 70H05.
http://www.boundaryvalueproblems.com/content/2015/1/111
Xiaoyan LinXianhua TangBoundary Value Problems 2015, null:11120150626T12:00:00Zdoi:10.1186/s1366101503513/content/figures/s1366101503513toc.gifBoundary Value Problems16872770${item.volume}11120150626T12:00:00ZXML Dynamics of competing systems in general heterogeneous environments In this paper, we address the question of the dynamics of the systems for two competing species in general heterogeneous environment with lethal boundary conditions. The existence and uniqueness of the positive steadystate solution are established under suitable conditions. Finally, we obtain global asymptotic stability of the positive steadystate solution for weak competition situation.MSC: 35K57, 92D25.
http://www.boundaryvalueproblems.com/content/2015/1/110
Benlong XuZhenzhang NiBoundary Value Problems 2015, null:11020150625T00:00:00Zdoi:10.1186/s1366101503687/content/figures/s1366101503687toc.gifBoundary Value Problems16872770${item.volume}11020150625T00:00:00ZXML Blow up of solutions for a class of fourth order nonlinear pseudoparabolic equation with a nonlocal source In this paper, we consider the initial boundary value problem for a fourth order nonlinear pseudoparabolic equation with a nonlocal source. By using the concavity method, we establish a blowup result of the solutions under suitable assumptions on the initial energy.MSC: 35B44, 35K30, 35K59.
http://www.boundaryvalueproblems.com/content/2015/1/109
Huafei DiYadong ShangBoundary Value Problems 2015, null:10920150624T00:00:00Zdoi:10.1186/s1366101503696/content/figures/s1366101503696toc.gifBoundary Value Problems16872770${item.volume}10920150624T00:00:00ZXML Existence of solutions to fourthorder differential equations with deviating arguments In this paper, we consider fourthorder differential equations on a halfline with deviating arguments of the form
u
(
4
)
(
t
)
+
q
(
t
)
f
(
t
,
[
u
(
t
)
]
,
[
u
′
(
t
)
]
,
[
u
″
(
t
)
]
,
u
‴
(
t
)
)
=
0
,
0
<
t
<
+
∞
, with the boundary conditions
u
(
0
)
=
A
,
u
′
(
0
)
=
B
,
u
″
(
t
)
−
a
u
‴
(
t
)
=
θ
(
t
)
,
−
τ
≤
t
≤
0
;
u
‴
(
+
∞
)
=
C
. We present sufficient conditions for the existence of a solution between a pair of lower and upper solutions by using Schäuder’s fixed point theorem. Also, we establish the existence of three solutions between two pairs of lower and upper solutions by using topological degree theory. An important feature of our existence criteria is that the obtained solutions may be unbounded. We illustrate the importance of our results through two simple examples.MSC: 34B15, 34B40.
http://www.boundaryvalueproblems.com/content/2015/1/108
Mostepha NaceriRavi AgarwalErbil ÇetinEl AmirBoundary Value Problems 2015, null:10820150624T00:00:00Zdoi:10.1186/s136610150373x/content/figures/s136610150373xtoc.gifBoundary Value Problems16872770${item.volume}10820150624T00:00:00ZXML Global existence and uniform boundedness of smooth solutions to a parabolicparabolic chemotaxis system with nonlinear diffusion This paper is devoted to the following quasilinear chemotaxis system:
{
u
t
=
∇
⋅
(
D
(
u
)
∇
u
)
−
∇
⋅
(
u
χ
(
v
)
∇
v
)
+
u
f
(
u
)
,
x
∈
Ω
,
t
>
0
,
v
t
=
Δ
v
−
u
g
(
v
)
,
x
∈
Ω
,
t
>
0
,
under homogeneous Neumann boundary conditions in a smooth bounded domain
Ω
⊂
R
N
. The given functions
D
(
s
)
,
χ
(
s
)
,
g
(
s
)
, and
f
(
s
)
are assumed to be sufficiently smooth for all
s
≥
0
and such that
f
(
s
)
≤
κ
−
μ
s
τ
. It is proved that the corresponding initial boundary value problem possesses a unique global classical solution for any
μ
>
0
and
τ
≥
1
, which is uniformly bounded in
Ω
×
(
0
,
+
∞
)
. Moreover, when
κ
=
0
, the decay property of the solution is also discussed in this paper.MSC: 35K55, 35Q92, 35Q35, 92C17.
http://www.boundaryvalueproblems.com/content/2015/1/107
Xie LiBoundary Value Problems 2015, null:10720150624T00:00:00Zdoi:10.1186/s136610150372y/content/figures/s136610150372ytoc.gifBoundary Value Problems16872770${item.volume}10720150624T00:00:00ZXML Dirichlet boundary value problem for differential equations involving dry friction Sufficient conditions in terms of growth restrictions are given for the solvability of the Dirichlet boundary value problem to forced nonlinear differential equations involving the combination of viscous and dry frictions. Explicit estimates of solutions and their derivatives allow us to restrict ourselves to a sufficiently large neighbourhood of the origin, when formulating these effective conditions. In this way, the behaviour of nonlinearities outside of this neighbourhood can be quite arbitrary. In order to get optimal solvability criteria, the problems with oneterm and complete linear differential operators will be treated separately by means of various Green’s functions. The obtained results are compared with some of their analogies of the other authors.MSC: 34A60, 34B15, 34B16, 34B27, 47H04.
http://www.boundaryvalueproblems.com/content/2015/1/106
Jan AndresHana Mach¿Boundary Value Problems 2015, null:10620150624T00:00:00Zdoi:10.1186/s136610150371z/content/figures/s136610150371ztoc.gifBoundary Value Problems16872770${item.volume}10620150624T00:00:00ZXML New existence and uniqueness results for an elastic beam equation with nonlinear boundary conditions In this article, we study the existence and uniqueness of monotone positive solutions for an elastic beam equation with nonlinear boundary conditions, and some sufficient conditions which guarantee the existence of unique monotone positive solution are established. The methods employed are two fixed point theorems for mixed monotone operators with perturbation. Our results can not only guarantee the existence of unique monotone positive solution, but also be applied to construct an iterative scheme for approximating it. Two examples are given to illustrate our main results.MSC: 34B18, 34B15.
http://www.boundaryvalueproblems.com/content/2015/1/104
Shunyong LiChengbo ZhaiBoundary Value Problems 2015, null:10420150619T00:00:00Zdoi:10.1186/s136610150365x/content/figures/s136610150365xtoc.gifBoundary Value Problems16872770${item.volume}10420150619T00:00:00ZXML An accurate Chebyshev pseudospectral scheme for multidimensional parabolic problems with time delays In this paper, the Chebyshev GaussLobatto pseudospectral scheme is investigated in spatial directions for solving onedimensional, coupled, and twodimensional parabolic partial differential equations with time delays. For the onedimensional problem, the spatial integration is discretized by the Chebyshev pseudospectral scheme with GaussLobatto quadrature nodes to provide a delay system of ordinary differential equations. The time integration of the reduced system in temporal direction is implemented by the continuous RungeKutta scheme. In addition, the present algorithm is extended to solve the coupled time delay parabolic equations. We also develop an efficient numerical algorithm based on the Chebyshev pseudospectral algorithm to obtain the two spatial variables in solving the twodimensional time delay parabolic equations. This algorithm possesses spectral accuracy in the spatial directions. The obtained numerical results show the effectiveness and highly accuracy of the present algorithms for solving onedimensional and twodimensional partial differential equations.
http://www.boundaryvalueproblems.com/content/2015/1/103
Ali BhrawyMohamed AbdelkawyFouad MallawiBoundary Value Problems 2015, null:10320150619T00:00:00Zdoi:10.1186/s136610150364y/content/figures/s136610150364ytoc.gifBoundary Value Problems16872770${item.volume}10320150619T00:00:00ZXML