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 ) = λ f i ( t , u 1 ( φ 1 ( t ) ) , u 2 ( φ 2 ( t ) ) , … , u n ( φ n ( t ) ) ) , t ∈ [ a , b ] , u i ( a ) = u i ′ ( z ) = 0 , γ i u i ( b ) + δ i u i ″ ( b ) = 0 , where λ > 0 , i = 1 , 2 , … , n , 1 2 ( a + b ) < z < b , γ i ⩾ 0 , δ i > 0 , and φ i are deviating arguments. Criteria will be developed so that for values of λ that form an interval (bounded or unbounded), the above system has a fixed-sign solution u = ( u 1 , u 2 , … , u n ) , i.e., for each 1 ⩽ i ⩽ n , θ i u i ( t ) ⩾ 0 for t ∈ [ a , b ] , where θ i ∈ { 1 ,- 1 } is fixed. Explicit intervals for such values of λ will also be presented. A numerical example is included to illustrate the results obtained.