Heteroclinic Solutions of Singular Φ−Laplacian Boundary Value Problems on Infinite Time Scales

In this paper, we derive sufficient conditions for the existence of heteroclinic solutions to the singular �-Laplacian boundary value problem, � �(y � (t)) �� = f(t, y(t), y � (t)), t 2 T y(−1 ) = −1, y(+1 ) = +1, on infinite time scales by using the Brouwer invariance domain theorem. As an application we demonstrate our result with an example.


Introduction
In the study of qualitative theory of differential equations, a prominent role is played by special class of solutions, like positive solutions, periodic solutions, symmetric solutions, chaotic solutions.When a system of dynamic equations has equilibria, i.e. constant solution, whose stability properties are known, it is significant to study the connections between them by trajectories of solutions for the given system.These are called homoclinic or heteroclinic solutions, sometimes pulse or kinks, according to whether they describe a loop based at one stable equilibrium or two stable equilibriums.
Recently, heteroclinic solutions for boundary value problems on the whole real line had a certain impulse, motivated by applications in various disciplines of biological, physical, mechanical and chemical models, such as phase transition, physical processes in which the variable transits from an unstable equilibrium to a stable one, and front propagation in reaction diffusion equation.EJQTDE, 2012 No. 86, p. 1 Indeed, heteroclinic solutions are often called transitional solutions.These connections provide important information on the dynamics of the system.For autonomous second order system, for instance, homoclinic and heteroclinics separate, in the phase portrait, region where solutions behave differently.When they appear as connection between saddles they prevent structural stability.Due to the importance in both theory and applications, the study of heteroclinic solutions gained momentum on real intervals, we list a few; Avrameseu and Vladimirecu [1], Cabada and Cid [3,4], Cabada and Tersion [5], and Marcelli and Papalini [7,8].
A time scale is an arbitrary nonempty closed subset of the real numbers and we denote the time scale by the symbol T. A time scale T is said to be infinite time scale, we mean, if it has no infimum and no supremum.(i.e., inf T = −∞ and sup T = +∞.)A time scale T is said to be semi infinite time scale, we mean, if it has either infimum but no supremum or supremum but no infimum.For basic definitions and results on time scales we refer to the text books by Bohner and Peterson [2] and Lakshmikantham, Sivasundaram and Kaymakcalan [6].
In this paper, we are dealing with some special class of time scales namely infinite and semi infinite time scales because continuous time orbits and discrete time orbits are topologically different.We consider the following dynamical equation on infinite time scales satisfying the boundary conditions where Φ(y) is singular Φ-Laplacian operator and f : T × R 2 → R is continuous function.We derive sufficient conditions for the existence of heteroclinic solutions to the singular Φ-Laplacian boundary value problem on infinite time scales.
The main idea is that the existence of loops or connections at some rest points can be prove by analyzing the intersection properties of the stable and unstable manifolds through these equilibrium points.
We assume the following conditions throughout the paper.
(A2).f : T × R 2 → R is continuous and satisfies the symmetric type condition (A5). for every compact set of the form K = [−r, r] × [−ǫ, ǫ], where 0 < r < 1 and 0 < ǫ < 1, there exist t l ≥ 0 and a continuous function h l : [t l , ∞) → R such that f (t, x, y) ≤ h l (t) for all t ≥ t l and (x, y) ∈ K, and (A6). if {y n (t)} is a sequence of solutions of (1) for which for each n ∈ T + 0 an arbitrary compact interval [0, σ 2 (n)] T and there exists an M > 0 such that y n (t) ≤ M for all t ∈ [0, σ 2 (n)] T and for all t ∈ N, then there exists a subsequence {y n j (t)}, such that {y The rest of the paper is organized as follows, in Section 2, we prove some lemmas which are needed in the main result.In Section 3, we derive sufficient conditions for the existence of heteroclinic solutions to the singular Φ-Laplacian boundary value problem on infinite time scales.Finally, as an application, we give an example to demonstrate our result.EJQTDE, 2012 No. 86, p. 3

Existence of solution on semi infinite time scales
In this section, we establish a lemma and theorem on semi infinite time scales which are needed to establish heteroclinic solution on infinite time scales.
Proof: We can construct a time scale integral equation equivalent to the boundary value problem (3)-( 4), τ g being a solution of the following equation, The fact that the function F g : R → R has a unique real root is deduced from the following facts: 1. the continuity of Φ −1 implies the continuity of the function F g 2. since Φ −1 is strictly increasing implies the same hold for function F g .

lim
Thus we can conclude that τ g is unique, and hence the boundary value problem (3)-( 4) has a unique solution y g .We have that for all t ∈ [0, and hence y ∆ ∈ (−a, a) and also |y ∆ (t)| < a.

2
Theorem 2.2 Suppose that Φ : (−a, a) → R satisfies (A1) and (A6), then the problem Φ(y has at least one solution, and this solution satisfies y ∆ ∞ < a, where f : with τ y the unique solution of the following equation From Lemma 2.1, the operator A satisfies the problem ( 7) and ( 8), [i.e.A is well defined and that the fixed points of the operator A are the solutions of the problem ( 7) and ( 8)] First we prove that operator A is continuous.Suppose y m → y in [0, σ 2 (n)]× R and let τ m be the corresponding value for y m given by ( 8) and τ y associated to y.Let us see that lim By construction of τ m and τ y we have that for all m ∈ N.
Since {y m } is a convergent sequence to y we have that As consequence f (t, y m (t), y ∆ m (t)) m∈N is a bounded sequence in R, from ( 9) and (A6) the sequence {τ m } is bounded too, and we conclude that there exists a subsequence {τ m k } converging to a real number γ = lim sup{τ m }.Thus, from the continuity of Φ −1 and f , we arrive that and since Φ −1 is a strictly increasing function, we conclude that τ y = γ, Analogously, τ y = lim inf{τ m }.Now, by the continuity of f it follows that for each y ∈ R and then Brouwer fixed point theorem ensures the existence of at least one fixed point of the operator A and as a consequence, the existence of at least one solution of the problem ( 5), (6), by using equations ( 7),(8), we have that for all t ∈ [0, σ(n)] T that and hence y ∆ ∈ (−a, a). 2
On the other hand, from the uniform continuity of function Φ −1 on compact set it follows that the sequence {y ∆ n } is an equicontinuous family, and as consequence it is easy to verify that Φ(y ∆ (t)) ∆ = f (t, y(t), y ∆ (t)) ≤ 0.
So we deduce that y ∆ is nonincreasing.If y ∆ (t 0 ) < 0 at some point t 0 ≥ 0 then y ∆ (t) ≤ y ∆ (t 0 ) < 0 for all t ≥ t 0 and consequently lim t→∞ y(t) = −∞ a contradiction.Thus y ∆ (t) ≥ 0 for all t ≥ 0 and hence y is nondecreasing.Since y ∆ is decreasing there exists, But as y is bounded we deduce that Suppose that l < 1.From (A4), (A5) and the facts that 0 ≤ y(t) ≤ l < 1 and lim t→+∞ y ∆ (t) = 0 it follows that there exist a suitable compact set K ⊂ (−1, 1) × R, t k > 0 for which 0 < y ∆ (t k ) < 1 and a continuous function h k such that Φ(y ∆ (t)) ∆ = f (t, y(t), y ∆ (t)) ≤ h k (t) for all t ≥ t k , and ∞ t k h k (t) = −∞.But in this case Φ(y ∆ ) → −∞ and then y ∆ (t) → −a < 0, which is a contradiction.Thus l = 1 and the proof is over.
(i.e., inf T = a and sup T = +∞ or inf T = −∞ and sup T = b)and we denote as T + a = [a, +∞) and T − b = (−∞, b] respectively.For example if we consider time scale T = {−(a) n } n∈N ∪ {0} ∪ {b n } n∈N , (a, b > 1) then it is an infinite time scale and if we remove either negative terms or positive terms then it is an example of semi infinite time scale.By an interval [a, b] T means the intersection of the real interval with a given time scale.i.e [a, b] T = [a, b] ∩ T.

7 Since
y is concave and bounded then there exists l ∈ (0, 1] such that lim t→+∞ y(t) = l.