Existence of Solutions for a Class of Fourth-Order Multipoint Boundary-Value Problems on Time Scales

Abstract

The paper deals with the existence of solutions for the dynamic equation on time scaleswith the multipoint boundary conditionswhere T is a time scale, OE0; 1ç T D ft 2 T j 0  t  1g; a i > 0; i D 1; 2; : : : ; m � 2; b j > 0; j D D 1; 2; : : : ; n � 2; 0 < ⇠ 1 < ⇠ 2 < : : : < ⇠ m 2 < ⇢.1/; and 0 < ⌘ 1 < ⌘ 2 < : : : < ⌘ n 2 < ⇢.1/:The existence result is given by using Green's function, the method of upper and lower solutions, and the monotone iterative technique. We also give an example to illustrate our result.

Nonlinear fourth-order dynamic equations on unbounded time scales

“…At present, there have been some studies on fourth-order dynamic equations [6,7,11,12,[15][16][17]. To the authors' knowledge, there is no work on the existence of solutions for singular fourth-order dynamic problems in the lim-4 case so our problem is very different from the papers in the literature.…”

“…Further, we assume that there exists constant M > 0 such that Proof. Let T : L 2 ∆ [0, ∞) T → L 2 ∆ [0, ∞) T be the operator defined in (7). It follows from Theorems 2.2, 2.5, and (9) that T maps the set S M into itself.…”