Existence of solution and positive solution of BVP for nonlinear third-order dynamic equation. (English) Zbl 1099.34022

Summary: We are concerned with the following third-order two-point boundary value problem on time scale \(\mathbb{T}\) \[ \begin{cases} u^{\Delta\Delta\Delta} (t)+f\bigl(t,u(t), u^{\Delta\Delta}(t)\bigr)=0,\;t\in\bigl[a,\rho(b)\bigr],\\ u(a)=A,\;u\bigl( \sigma^2(b)\bigr)=B,\;u^{\Delta\Delta}(a)=C. \end{cases} \] Some existence criteria for a solution and for a positive solution are established by using Leray-Schauder’s fixed-point theorem. Our main conditions are local. An example is included, too.


34B15 Nonlinear boundary value problems for ordinary differential equations
39A10 Additive difference equations
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