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 problemsui‴(t)=λfi(t,u1(φ1(t)),u2(φ2(t)),…,un(φn(t))),t∈[a,b],ui(a)=ui′(z)=0,γiui(b)+δiui″(b)=0,where λ>0,i=1,2,…,n, 12(a+b)0δi>0, and φiφi are deviating arguments. Criteria will be developed so that for values of λλ that form an interval (bounded or unbounded), the above system has a fixed-sign   solution u=(u1,u2,…,un)u=(u1,u2,…,un), i.e., for each 1⩽i⩽n1⩽i⩽n, θiui(t)⩾0θiui(t)⩾0 for t∈[a,b]t∈[a,b], where θi∈{1,-1}θi∈{1,-1} is fixed. Explicit intervals for such values of λλ will also be presented. A numerical example is included to illustrate the results obtained.