EXISTENCE OF A POSITIVE SOLUTION FOR AN NTH ORDER BOUNDARY VALUE PROBLEM FOR NONLINEAR DIFFERENCE EQUATIONS

The nth order eigenvalue problem: ∆x(t) = (−1)n−kλf(t, x(t)), t ∈ [0, T ], x(0) = x(1) = · · · = x(k − 1) = x(T + k + 1) = · · · = x(T + n) = 0, is considered, where n ≥ 2 and k ∈ {1, 2, . . . , n − 1} are given. Eigenvalues λ are determined for f continuous and the case where the limits f0(t) = lim n→0+ f(t,u) u and f∞(t) = lim n→∞ f(t,u) u exist for all t ∈ [0, T ]. Guo’s fixed point theorem is applied to operators defined on annular regions in a cone.

A key to applying this fixed point theorem involves discrete concavity of solutions of the boundary value problem in conjunction with a lower bound on an appropriate Green's function.Extensive use of the results by Eloe [8] concerning a lower bound for the Green's function will be made.Related results for nth order differential equation may be found in Agarwal and Wong [3,4], Eloe and Henderson [9,10], and Fang [13].

Preliminaries
Let G(t, s) be the Green's funtion for the disconjugate boundary value problem and satisfying (2), where, as shown in Kelly and Peterson [18], G(t, s) is the unique function satisfying: (a) G(t, s) is defined for all t ∈ [0, G(t, s) satisfies the boundary conditions (2) in t.We will use G(t, s) as the kernel of an integral operator preserving a cone in a Banach space.This is the setting for our fixed point theorem.
Let B be a Banach space and let P ⊂ B be such that P is closed and non-empty.Then P is a cone provided (i) au + bv ∈ P for all u, v ∈ P and for all a, b ≥ 0, and (ii) u, −u ∈ P implies u = 0.
Eloe [8] employed these three lemmas to arrive at the following theorem that gives a lower bound for the solution to the class of boundary value problems studied by Hartman.

Theorem 2. Assume that u satisfies the difference inequality
and the homogeneous boundary conditions, (2).
We remark that Agarwal and Wong [5] have recently sharpened the inequality of Theorem 2. However, this sharper inequality is of little consequence for this work.
Eloe also contributed the following corollary.

Corollary 1. Let G(t, s) denote the Green's function for the boundary value problem, (3), (2). Then for all
where To fulfill the hypotheses of Theorem 1, let and define a cone

Main Results
Theorem 3. Assume conditions (A) and (B) are satisfied.Then, for each λ satisfying .
there exists at least one solution of ( 1), (2) in P.
Proof.Let λ be given as in Theorem 3. Let > 0 be such that .
Define a summation operator H : P → B by We seek a fixed point of H in the cone P. By the nonnegativity of f and (−1) n−k G, Hx(t) ≥ 0 on [0, T + n], and from the properties of G, Hx satisfies the boundary conditions.Now if we choose x ∈ P, we have Thus H : P → P. Additionally, H is completely continuous.Now consider f 0 (t).For each t ∈ [0, T ], there exists k t > 0 such that f (t, u) ≤ (f 0 (t) + )u for 0 < u ≤ k t .Let K 1 = min t∈[0,T ] k t .So, for x ∈ P with x = K 1 , we have Therefore, H(x) ≤ x .Hence, if we set x(t) ≥ σ x ≥ K2 , and Thus, Hx ≥ x , and so Hx ≥ x for all x ∈ P ∩ ∂Ω 2 (9) So with ( 8) and ( 9) we have shown that H satisfies the first condition of Theorem 1. Thus we can conclude that H has a fixed point u(t) ∈ P ∩ ( Ω2 \Ω 1 ).This fixed point, u(t), is a solution of (1), (2) corresponding to the given value of λ.

Theorem 4. Assume conditions (A) and (B) are satisfied. Then, for each
, there exists at least solution of ( 1), ( 2) in P.
Proof.Let λ be given as stated above.Let > 0 be such that Let H be the cone preserving, completely continuous operator defined in (7).Consider f 0 (t).For each t ∈ [0, T ] there exists Therefore, Hx ≥ x .Hence, if we set such that for all t ∈ I, f(t, u) is bounded as a function of u, and for all t ∈ J, f (t, u) is unbounded as a function of u.
Choose M > 0 such that for all positive u and for all t ∈ I, f(t, u) ≤ M .Let By the continuity of f , for all t ∈ J there exists µ t , where κ t ≤ µ t ≤ K 2 , such that f (t, u) ≤ f (t, µ t ) for all 0 < u ≤ K 2 .Now for x ∈ P with x = K 2 .Now if we take Thus, with (10) and (11), we have shown that H satisfies the hypotheses to Theorem 1(ii), which yields a fixed point of H belonging to P ∩ ( Ω2 \Ω 1 ).this fixed point is a solution of (1), (2) corresponding to the given λ.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: