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.
Keywords:existence; p-Laplacian; time scales; topological degree
with integral initial value
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 , 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 [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 sew-age .
The study of dynamic equations on time scales has led to some important applications [9-11], 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 [12-18]).
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.
for all . If , t is said to be right scattered, and if , r is said to be left scattered. If , t is said to be right dense, and if , r is said to be left dense. If has a right scattered minimum m, define ; otherwise, set . If has a left scattered maximum M, define ; otherwise, set .
Lemma 1.1 (Alternative theorem)
Consider the following problem:
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
then (2.3) can be rewritten as
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
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
Thus, we complete the proof. □
3 Main results
Throughout this section, we assume that the following conditions hold.
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.
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.
has at least one solution.
and it is clear that H is completely continuous.
From (H4), we have
All authors declare that they have no competing interests.
WS dfafted this paper and WG checked and corrected the manuscript.
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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