#### Existence of solutions for nonlocal p-Laplacian thermistor problems on time scales

Boundary Value Problems
Existence of solutions for nonlocal p-Laplacian thermistor problems on time scales
Wenjing Song 1
Wenjie Gao 0
0 Institute of Mathematics, Jilin University , Changchun, 130012 , P.R. China
1 Institute of Applied Mathematics, Jilin University of Finance and Economics , Changchun, 130117 , P.R. China
In this paper, a nonlocal initial value problem to a p-Laplacian equation on time scales is studied. The existence of solutions for such a problem is obtained by using the topological degree method.
existence; p-Laplacian; time scales; topological degree
1 Introduction
In this paper, we are concerned with the existence of solutions of the following nonlocal
p-Laplacian dynamic equation on a time scale T:
( T f (u(s))∇s)k
∀t ∈ (, T )T,
with integral initial value
g(s)u(s)∇s,
u () = A,
where φp(·) is the p-Laplace operator defined by φp(s) = |s|p–s, p > , φp– = φq with q the
Hölder conjugate of p, i.e., p + q = , λ > , k > , f : [, T ]T –→ R+∗ is continuous (R+∗
denotes positive real numbers), a : [, T ]T –→ R+ is left dense continuous, g(s) ∈ L([, T ]T)
and A is a real constant.
This model arises in ohmic heating phenomena, which occur in shear bands of metals
which are deformed at high strain rates [, ], in the theory of gravitational equilibrium of
polytropic stars [], in the investigation of the fully turbulent behavior of real flows, using
invariant measures for the Euler equation [], in modeling aggregation of cells via
interaction with a chemical substance (chemotaxis) []. For the one-dimensional case, problems
with the nonlocal initial condition appear in the investigation of diffusion phenomena
for a small amount of gas in a transparent tube [, ]; nonlocal initial value problems in
higher dimension are important from the point of view of their practical applications to
modeling and investigating of pollution processes in rivers and seas, which are caused by
sew-age [].
The study of dynamic equations on time scales has led to some important applications
[–], and an amount of literature has been devoted to the study the existence of solutions
of second-order nonlinear boundary value problems (e.g., see [–]).
Motivated by the above works, in this paper, we study the existence of solutions to
Problem (.), (.). Compared with the works mentioned above, this article has the
following new features: firstly, the main technique used in this paper is the topological degree
method; secondly, Problem (.), (.) involves the integral initial condition.
The paper is organized as follows. We introduce some necessary definitions and lemmas
in the rest of this section. In Section , we provide some necessary preliminaries, and in
Section , the main results are stated and proved.
Definition . For t < sup T and r > inf T, define the forward jump operator σ and the
backward jump operator ρ, respectively,
σ (t) = inf{τ ∈ T | τ > t} ∈ T,
ρ(r) = sup{τ ∈ T | τ < r} ∈ T
for all t, r ∈ T. If σ (t) > t, t is said to be right scattered, and if ρ(r) < r, r is said to be left
scattered. If σ (t) = t, t is said to be right dense, and if ρ(r) = r, r is said to be left dense. If
T has a right scattered minimum m, define Tk = T – {m}; otherwise, set Tk = T. If T has a
left scattered maximum M, define Tk = T – {M}; otherwise, set Tk = T.
x ρ(t) – x(s) – x∇ (t) ρ(t) – s < ε ρ(t) – s
for all s ∈ V .
f (s) s = F(t) – F(a).
f (s)∇s = (t) – (a).
Throughout this paper, we assume that T is a nonempty closed subset of R with ∈ Tk ,
T ∈ Tk .
u =
(ii) There exists an x ∈ X, x = , such that
2 Preliminaries
Let E = Cld([, T ]T , R) be a Banach space equipped with the maximum norm
max lim[,T]T |u(t)|.
Consider the following problem:
∀t ∈ (, T )T,
g(s)x(s)∇s,
x () = A,
where y ∈ C([, T ]T), T g(s)∇s = .
Integrating Eq. (.) from to t, one obtains
φp x (t) – φp x () = –
y(s)∇s.
Using the initial condition (.), we have
y(s)∇s .
Integrating the above equality from to t again, we obtain
g(s)x(s)∇s =
y(s)∇s
g(s)x(s)∇s,
then (.) can be rewritten as
(I – K )x(t) = F(t).
Let F(t) := t φp–(φp(A) – τ y(s)∇s) τ .
Define an operator K : Cld([, T ]T) –→ Cld([, T ]T) by
Thus, x(t) is a solution to (.), (.) if and only if it is a solution to (.).
Lemma . I – K is a Fredholm operator.
Lemma . Problem (.), (.) admits a unique solution.
Proof Since Problem (.), (.) is equivalent to Problem (.), we need only to show that
Problem (.) has a unique solution.
Using Lemma . and the alternative theorem, it is sufficient to prove that
(I – K )u(t) =
∇s
Define an operator F : Cld([, T ]T) –→ Cld([, T ]T) by
∇s
(I – K )x(t) =
has a trivial solution x ≡ only.
On the contrary, suppose (.) has a nontrivial solution μ, then μ is a constant, and we
have
The definition of K and the above equality yield
which is a contradiction to the assumptions T g(s)∇s = and μ ≡ .
Thus, we complete the proof.
3 Main results
then (.) can be rewritten as
(I – K )u(t) = (Fu)(t).
In order to prove the existence of solutions to (.), we need the following lemmas.
Lemma . F is completely continuous.
Proof Let R be an arbitrary positive real number and denote B = {u ∈ Cld([, T ]T); u ≤
R}. Then we have for any u ∈ B,
(Fu)(t) ≤
∇s
This shows that F(B) is uniformly bounded.
Moreover, for any t ∈ [, T ]T, we have
(Fu) (t) = φ– φp(A) –
p
( T f (u(s))∇s)k
λ supu∈B f
φp(A) + MT (T infu∈B f )k .
∇s
Thus, it is easy to prove that F(B) is equicontinuous. This together with the
AscoliArzelà theorem guarantees that F(B) is relatively compact in Cld([, T ]T).
Therefore, F is completely continuous. The proof of Lemma . is completed.
Theorem . Assume that conditions (H)-(H) hold. Then Problem (.), (.) has at least
one solution.
Proof Lemma . and Lemma . imply that the operator K + F is completely continuous.
It suffices for us to prove that the equation
I – (K + F) u =
h(u) = (I – K )u,
h(u) = I – (K + F) u.
has at least one solution.
Define H : [, ] × Cld([, T ]T) → Cld([, T ]T) as
and it is clear that H is completely continuous.
Set hσ (u) = u – H(σ , u), then we have
f u(s) ∇s
= u(t) –
From (H), we have
≥ u(t) –
g(s)u(s)∇s –
≥ ( – M)R –
∇s
∇s
∇s
≥ ( – M)R –
≥ ( – M)R –
Competing interests
All authors declare that they have no competing interests.
If k > , choosing R >
q–|A|T
–M–q–φp–(λMT–kc)T , then for any fixed u ∈ ∂BR(θ ), there exists a
Authors’ contributions
WS dfafted this paper and WG checked and corrected the manuscript.
Acknowledgements
This work was supported by NSFC (11271154) and by Key Lab of Symbolic Computation and Knowledge Engineering of
Ministry of Education and by the 985 program of Jilin University, and the first author is also supported by the Youth
Studies Program of Jilin University of Finance and Economics (XJ2012006).
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