Existence Results for a System of Third-Order Right Focal Boundary Value Problems

Abstract

We consider the following system of third-order three-point generalized right focal boundary value problems

$$\displaystyle{\begin{array}{c} u_{i}^{{\prime}{\prime}{\prime}}(t) = f_{i}(t,u_{1}(t),u_{2}(t),\ldots,u_{n}(t)),\ \ t \in [a,b] \\ u_{i}(a) = u^{\prime}_{i}(t_{i}) = 0,\qquad \gamma _{i}u_{i}(b) + \delta _{i}u^{{\prime}{\prime}}_{i}(b) = 0, \end{array} }$$

where $i = 1,2,\ldots,n,\ \gamma _{i} \geq 0,$ δ _i > 0 and $\frac{1} {2}(a + b) <t_{i} <b.$ By using a variety of tools like Leray–Schauder alternative and Krasnosel’skii’s fixed point theorem, we offer several criteria for the existence of fixed-sign solutions of the system. A solution $u = (u_{1},u_{2},\ldots,u_{n})$ is said to be of fixed sign if for each 1 ≤ i ≤ n, $\theta _{i}u_{i}(t) \geq 0$ for t ∈ [a,b] where θ _i ∈{−1,1} is fixed. We also consider a related eigenvalue problem

$$\displaystyle{\begin{array}{c} u_{i}^{{\prime}{\prime}{\prime}}(t) = \lambda f_{i}(t,u_{1}(t),u_{2}(t),\ldots,u_{n}(t)),\ \ t \in [a,b] \\ u_{i}(a) = u^{\prime}_{i}({t}^{{\ast}}) = 0,\qquad \gamma _{i}u_{i}(b) + \delta _{i}u^{{\prime}{\prime}}_{i}(b) = 0,\end{array} }$$

where $i = 1,2,\ldots,n,\ \lambda> 0,\ \gamma _{i} \geq 0,$ δ _i > 0 and $\frac{1} {2}(a + b) <{t}^{{\ast}} <b.$ Criteria will be established so that the above system has a fixed-sign solution for values of λ that form an interval (bounded or unbounded). Explicit intervals for such λ will also be presented. We include some examples to illustrate the results obtained.