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