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

Boundary Value Problems, Jan 2013

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.

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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) + MT (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–(λMT–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. 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Wenjing Song, Wenjie Gao. Existence of solutions for nonlocal p-Laplacian thermistor problems on time scales, Boundary Value Problems, 2013, 1,