Eigenvalues of a system of generalized right focal problems with deviating arguments
We consider the following system of third-order three-point generalized right focal boundary value problems \[ u'''_i(t)=\lambda f_i(t,u_1(\varphi_1(t)),u_2(\varphi_2(t)),\dots,u_n(\varphi_n(t))),\quad t\in [a,b], \] \[ u_i(a)=u_i'(z)=0,\quad \gamma_iu_i(b)+\delta_iu''_i(b)=0, \] where \(\lambda>0\), \(i=1,2,\dots,n\), \(\frac12(a+b)<z<b\), \(\gamma_i\geq 0\), \(\delta_i>0\), and \(\varphi_i\) are deviating arguments. Criteria will be developed so that for values of \(\lambda\) that form an interval (bounded or unbounded), the above system has a fixed-sign solution \(u=(u_1,u_2,\dots,u_n)\), i.e., for each \(1\leq i\leq n\), \(\theta_iu_i(t)\geq 0\) for \(t\in [a,b]\), where \(\theta_i\in\{1,-1\}\) is fixed. Explicit intervals for such values of \(\lambda\) will also be presented. A numerical example is included to illustrate the results obtained.
scientific article; zbMATH DE number 5300714