Existance of solutions for some class of system quasilinear differentials with nonlocal boundary conditions and with nonlinearity depandant on the first derivate in time scales

Abstract

This work is conserned with the construction of minimal and maximal solution for some equation quasilinear differentials with nonlocal boundary condition with nonlinearity is continuous function dependent on the first derivate in the time scale. In a previous study, see[11] it was found that there exists the extrimals solutions of a quasilinear elliptic system with intergral boundary conditions in the continuous case, and from here we asked the following question using the upper and lower solutions method coupled with monotone iterative technique and we asked the following question: do we have the existence of extrimals solutions for certain classes of systems of differential equations with nonlocal boundary conditions in time scales by same method. We can classify the problem following the monotonicity of the two functions and in the true types.Type 1: Increasing qasimonotonic systems.Type 2: Decreasing qasimonotonic systems. Type 3: Mexed qasimonotonic systems. In this paper we show the existence minimuls-maximuls solutions if the problem is of the type 1. If the problem is of the type 2 , we show the existence of maximuls-minimuls solutions. If the problem is of the type 3 , we show existence of least quasilear-solution. In the end, we gave examples of the results presented in this work to prove their validity through two cases: the continuous case, represented by the set of real numbers, and the discontinuous case, represented by the set of integers.