Positive solutions for nonlinear three-point boundary value problems on time scales

Abstract

Let T be a time scale such that 0, T ∈ T. Consider the following three-point boundary value problem on time scales:where β, γ 0, β + γ > 0, η ∈ (0, ρ(T )), 0 < α < T /η, and d = β(T − αη) + γ (1 − α) > 0. By using fixed point theorems in cones, some new and general results are obtained for the existence of single and multiple positive solutions of the above problem. In particular, our criteria generalize and improve some known results.  2004 Published by Elsevier Inc.

Second-order n-point eigenvalue problems on time scales

“…Problem (1.1), (1.2) is a generalization to time scales of the problem when T is restricted to R on the unit interval in Ma and Thompson [19], and extends the type of time-scale boundary value problem found in Anderson [2], Atici and Guseinov [6], Kaufmann [15], Kaufmann and Raffoul [16], and Sun and Li [21,22]. Other related three-point problems on time scales include Anderson and Avery [4], Anderson et al [5], Peterson et al [20], and a singular problem in DaCunha et al [12].…”

The formulation of second-order boundary value problems on time scales

“…This consideration motivates the domain and range of K chosen above. Again, the above formulation of K is different to that typically used in the previous literature (see, e.g., [3,17,18]), but is again motivated by the desire to produce a precise and unambiguous definition of K, and to ensure that derivatives are only evaluated at points at which they are defined, for all classes of time scales (we observe that in, e.g., [3,18], the problem of evaluating derivatives at points outwith their normal domain of definition can again occur for certain time scales).…”

Existence and multiplicity for positive solutions of a system of higher-order multi-point boundary value problems

“…Some particular cases of the above problems have been studied in [6][7][8][9]12,13,15]. In recent years, the multi-point boundary value problems for secondorder or higher-order differential or difference equations/systems have been investigated by many authors, by using different methods such as fixed point theorems in cones, the Leray-Schauder continuation theorem and its nonlinear alternatives and the coincidence degree theory.…”