Abstract
In this paper, a nonlocal initial value problem to a pLaplacian equation on time scales is studied. The existence of solutions for such a problem is obtained by using the topological degree method.
Keywords:
existence; pLaplacian; time scales; topological degree1 Introduction
In this paper, we are concerned with the existence of solutions of the following nonlocal
pLaplacian dynamic equation on a time scale
with integral initial value
where
This model arises in ohmic heating phenomena, which occur in shear bands of metals which are deformed at high strain rates [1,2], in the theory of gravitational equilibrium of polytropic stars [3], in the investigation of the fully turbulent behavior of real flows, using invariant measures for the Euler equation [4], in modeling aggregation of cells via interaction with a chemical substance (chemotaxis) [5]. For the onedimensional case, problems with the nonlocal initial condition appear in the investigation of diffusion phenomena for a small amount of gas in a transparent tube [6,7]; 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 sewage [8].
The study of dynamic equations on time scales has led to some important applications [911], and an amount of literature has been devoted to the study the existence of solutions of secondorder nonlinear boundary value problems (e.g., see [1218]).
Motivated by the above works, in this paper, we study the existence of solutions to Problem (1.1), (1.2). 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 (1.1), (1.2) 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 2, we provide some necessary preliminaries, and in Section 3, the main results are stated and proved.
Definition 1.1 For
for all
Definition 1.2 For
for all
for all
Definition 1.3 If
If
Throughout this paper, we assume that
Lemma 1.1 (Alternative theorem)
Suppose thatXis a Banach space andAis a completely continuous operator fromXto X. Then for any
(i) For any
(ii) There exists an
2 Preliminaries
Let
Consider the following problem:
where
Integrating Eq. (2.1) from 0 to t, one obtains
Using the initial condition (2.2), we have
Integrating the above equality from 0 to t again, we obtain
Let
Define an operator
then (2.3) can be rewritten as
Thus,
Lemma 2.1
Proof To prove that
It is easy to see from the definition of K that K is a bounded linear operator from
Lemma 2.2Problem (2.1), (2.2) admits a unique solution.
Proof Since Problem (2.1), (2.2) is equivalent to Problem (2.4), we need only to show that Problem (2.4) has a unique solution.
Using Lemma 2.1 and the alternative theorem, it is sufficient to prove that
has a trivial solution
On the contrary, suppose (2.5) 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
Thus, we complete the proof. □
3 Main results
Throughout this section, we assume that the following conditions hold.
(H1)
(H2)
(H3)
(H4)
(H5)
From Lemma 2.2 we know that
Define an operator
then (3.1) can be rewritten as
In order to prove the existence of solutions to (3.1), we need the following lemmas.
Lemma 3.1Fis completely continuous.
Proof Let
This shows that
Moreover, for any
Thus, it is easy to prove that
Therefore, F is completely continuous. The proof of Lemma 3.1 is completed. □
Theorem 3.1Assume that conditions (H1)(H5) hold. Then Problem (1.1), (1.2) has at least one solution.
Proof Lemma 2.1 and Lemma 3.1 imply that the operator
has at least one solution.
Define
and it is clear that H is completely continuous.
Set
To apply the LeraySchauder degree to
If
From (H4), we have
If
If
This implies
Since
Competing interests
All authors declare that they have no competing interests.
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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