TAILIEUCHUNG - Solutions for 2nth order lidstone BVP on time scales
In this paper, we prove the existence of solutions for nonlinear Lidstone boundary value problems by using the monotone method on time scale and also we show the existence of at least one positive solution if f is either superlinear or sublinear by the fixed point theorem in a Banach space. | Turk J Math 33 (2009) , 359 – 373. ¨ ITAK ˙ c TUB doi: Solutions for 2nth order lidstone BVP on time scales Erbil C ¸ etin and S. G¨ ul¸san Topal Abstract In this paper, we prove the existence of solutions for nonlinear Lidstone boundary value problems by using the monotone method on time scale and also we show the existence of at least one positive solution if f is either superlinear or sublinear by the fixed point theorem in a Banach space. Key Words: Lidstone boundary value problem, upper and lower solutions, fixed point theorem, positive solution. 1. Introduction Let T be any time scale (nonempty closed subset of R ) and [0, 1] is subset of T such that [0, 1] = {t ∈ T : 0 ≤ t ≤ 1} . In this paper, we shall consider the nonlinear Lidstone boundary value problem (LBVP), 2n (−1)n y (t) = 2i y (0) = f(t, yσ (t), y (t), ., y 2i y (σ(1)) = 0, 2(n−1) (t)), t ∈ [0, 1] 0 ≤ i ≤ n − 1, () () where n ≥ 1 and f : [0, σ(1)] ×Rn → R is continuous. We assume that σ(1) is right dense so that σ j (1) = σ(1) for j ≥ 1 . In this section we give some inequalities for certain Green’s function which are proved in the reference [5]. In Section 2 we give the existence and uniqueness theorem for solution using the method of upper and lower solutions when they are given in the well order. This method is generally used to obtain the existence of solutions within specified bounds determined by the upper and lower solutions. Also we obtain a unique solution within the appropriate bounds. Then we develop the monotone method which yields the solution of the LBVP (), (). The method of upper and lower solutions have been applied by several authors in [4, 7, 10, 13] and the references therein. In [9] Ehme, Eloe and Henderson applied this method to 2nth order problems. Cone theory techniques have been applied by several authors for ordinary differential equations and dynamic equations on time scales including two-point, three-point and Lidstone .
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