ACTA issues

Solvability of generalized third-order coupled systems with two-point boundary conditions

Feliz Minhós, Infeliz Coxe

Acta Sci. Math. (Szeged) 84:3-4(2018), 659-672

Abstract. In this paper we consider the nonlinear third-order coupled system composed by the differential equations \[\left\{{ -u^{\prime\prime\prime}(t)=f\left(t,u(t),u^{\prime }(t),u^{\prime\prime }(t),v(t),v^{\prime }(t),v^{\prime\prime}(t)\right), \atop -v^{\prime\prime\prime}(t) =h\left( t,u(t),u^{\prime }(t),u^{\prime\prime }(t),v(t),v^{\prime }(t),v^{\prime\prime }(t)\right ),}\right. \] with $f,h\colon[0,1] \times\mathbb {R}^{6}\rightarrow\mathbb {R}$ continuous functions, and the boundary conditions \[ \left\{{ u(0)=u^{\prime }(0) =u^{\prime}(1) =0, \atop v(0)=v^{\prime}(0) =v^{\prime}(1) =0. }\right.\] We remark that the nonlinearities can depend on all derivatives of both unknown functions, which is new in the literature, as far as we know. This is due to an adequate auxiliary integral problem with a truncature, applying lower and upper solutions method with bounded perturbations. The main theorem is an existence and localization result, which provides some qualitative data on the system solution, such as, sign, variation, bounds, etc., as it can be seen in the example.

DOI: 10.14232/actasm-017-785-0

AMS Subject Classification (1991): 34B15, 34B27, 34L30

Keyword(s): coupled systems, Green functions, Nagumo-type condition, coupled lower and upper solutions

Received May 29, 2017 and in final form June 10, 2018. (Registered under 35/2017.)