Abstract
In this paper, we establish the existence of countably infinitely many positive solutions for a certain even order two-point boundary value problem with integral boundary conditions on time scales by using Holder’s inequality and Krasnoselskii’s fixed point theorem for operators on a cone.
Highlights
The study of dynamic equations on time scales unifies existing results in differential and finite difference equations, and provides powerful new tools for exploring connections between the traditionally separated fields
The boundary value problems with integral boundary conditions occur in the study of nonlocal phenomena in many different areas of applied mathematics, physics and engineering, in particular, in heat conduction, chemical engineering, underground waterflow, thermo-elasticity, plasma physics, [2], [10], [11], [21], [22], [27], [33], [36] and reference therin
Authors established the existence of positive solutions to boundary value problems with integral boundary conditions on time scales; for details, see [9], [12], [13], [18], [26], [28], [32], [34] and reference therein
Summary
The study of dynamic equations on time scales unifies existing results in differential and finite difference equations, and provides powerful new tools for exploring connections between the traditionally separated fields. Authors established the existence of positive solutions to boundary value problems with integral boundary conditions on time scales; for details, see [9], [12], [13], [18], [26], [28], [32], [34] and reference therein. In 2013, Karasa and Tokmak [20] established the existence of a positive solution of the following third order boundary value problem with integral boundary conditions, φ(−u∆∆(t)) ∆ + q(t)f (t, u(t), u∆(t)) = 0, t ∈ [0, 1]T , au(0) − bu∆(0) = g1(s)u(s)∆s, cu(1) + du∆(1) = g2(s)u(s)∆s, u∆∆(1) = 0,. Motivated by the work mentioned above, in this paper we investigate the existence of infinitely many positive solutions for the even order boundary value problem on time scales given by (−1)nu(∆∇)n (t) = ω(t)f u(t) , t ∈ [0, 1]T, satisfying the Sturm-Liouville type integral boundary conditions (1.6). The key tool in our approach is the Holder’s inequality and Krasnoselskii’s fixed point theorem for operators on a cone
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