Abstract
In this paper, a class of third-order three-point boundary value problem on time scales is considered. Using monotone iterative technique and cone expansion and compression fixed point theorem of norm type, we do not only obtain the existence and uniqueness of positive solutions of the problem, but also establish the iterative schemes for approximating the solutions.
Highlights
We are interested in the existence and uniqueness of positive solutions and establish the corresponding iterative schemes for the following third-order three-point boundary value problem (BVP) on time scales∇(t) + f (t, x(t)) = 0, t ∈ [t1, t3]T, x(ρ(t1)) = 0 = x∆(ρ(t1)), x∆(σ(t3)) = αx∆(t2), (1.1)
In [16], Anderson and Hoffacker were concerned with the existence and form of solutions to the following nonlinear third-order three-point boundary value problem on time scales:∇(t) + a(t)f (x(t)) = 0, t ∈ [t1, t3]T, x(ρ(t1)) = 0 = x∆(ρ(t1)), x∆(σ(t3)) = αx∆(t2)
In order to obtain the uniqueness of positive solutions for BVP (1.1), we adopt some ideas established in [22]
Summary
We are interested in the existence and uniqueness of positive solutions and establish the corresponding iterative schemes for the following third-order three-point boundary value problem (BVP) on time scales (px∆∆)∇(t) + f (t, x(t)) = 0, t ∈ [t1, t3]T, x(ρ(t1)) = 0 = x∆(ρ(t1)), x∆(σ(t3)) = αx∆(t2),. In [16], Anderson and Hoffacker were concerned with the existence and form of solutions to the following nonlinear third-order three-point boundary value problem on time scales:. In [17], Anderson and Smyrlis applied Leray-Schauder nonlinear alternative to study the following third-order three-point boundary value problem on time scales:. By considering the ′′heights′′ of the nonlinear term f on some bounded sets and applying monotone iterative techniques on a Banach space, we do obtain the existence and uniqueness of positive solutions for BVP (1.1), and give the iterative schemes for approximating the solutions. In order to obtain the uniqueness of positive solutions for BVP (1.1), we adopt some ideas established in [22]
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