Symmetric positive solutions of fourth order boundary value problems for an increasing homeomorphism and homomorphism on time-scales

Let T ⊂ R be a symmetric bounded time-scale, with a = min T , b = max T . We consider the following fourth order boundary value problem ϕ ( − p x Δ ∇ ) Δ ∇ ( t ) + f ( t , x ( t ) ) = 0 , t ∈ T κ 2 κ 2 , x ( a ) = x ( b ) = 0 , x Δ ∇ ( σ ( a ) ) = x Δ ∇ ( ρ ( b ) ) = 0 for a suitable function p and an increasing homeomorphism and homomorphism ϕ . By using the Krasnosel’skii fixed point theorem, we present sufficient conditions for the existence of at least one or two symmetric positive solutions of the above problem on time-scales. As applications, two examples are given to illustrate the main results.