Notice of RetractionNontrivial solution of second-order m-point boundary value problem with sign changing coefficient on time scales

Notice of Retraction

After careful and considered review of the content of this paper by a duly constituted expert committee, this paper has been found to be in violation of IEEE´s Publication Principles.

We hereby retract the content of this paper. Reasonable effort should be made to remove all past references to this paper.

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Let T be a time scale. In this paper, we study the existence of positive solutions for the following nonlinear second-order m-point boundary value problem with sign changing coefficient on time scales {u^Δ∇(t)+f(t,u(t))=0, 0<;t<;T, u^Δ(0) = Σ_i=1^m-2 a_iu^Δ(ξ_i), u(T)=Σ_i=1^k b_iu(ξ_i)- Σ_i=k+1^s b_iu(ξ_i) - Σ_i=s+1^m-2 b_iu^Δ(ξ_i), where 1 ≤ k ≤ s ≤ m - 2, a_i, b_i ∈ (0, +∞) with 0 <; Σ_i=1^k b_i- Σ_i=k+1^s b_i<;1, 0<;Σ_i=1^m-2 a_i<;1, 0<;ξ₁ <;ξ₂ ⋯<; ξm-2 <; ρ(T), f ∈ C([0,1] × R,R), R = (-∞,+∞). Under certain growth conditions on the nonlinearity, several sufficient conditions for the existence of nontrivial solution are obtained by using Leray-Schauder nonlinear alternative. As an application, some examples to demonstrate our results are given. In particular, our criteria extend and improve some known results.