Nonlocal Sturm-Liouville problems with integral terms in the boundary conditions

Abstract

We consider a new type Sturm-Liouville problems whose main feature is the nature of boundary conditions. Namely, we study the nonhomogeneous Sturm-Liouville equation $$ p(x)u''(x)+(q(x)-\lambda )u=f(x) $$ on two disjoint intervals $[-1,0)$ and $(0,1]$, subject to the nonlocal boundary-transmission conditions $$\eqalign{ \alpha _ku^{(m_k)}(-1)+\beta _ku^{(m_k)}(-0)+\eta _ku^{(m_k)}(+0)+\gamma _ku^{(m_k)}(1) \cr +\sum_{j=1}^{n_k}\delta _{kj}u^{(m_k)}(x_{kj})+\sum_{\upsilon =1}^{2}\sum_{j=0}^{m_k}\int_{\Omega _{\upsilon }}\mathcal{K} _{k\upsilon j}(t)u^{(j)}(t)dt=f_k,\quad k=1,2,3,4. }$$ where $\Omega _1:=[-1,0)$, $\Omega _2:=(0,1]$ and $x_{kj}\in (-1,0)\cup (0,1)$ are internal points. By using our own approaches we establish such important properties as Fredholmness, coercive solvability and isomorphism with respect to the spectral parameter $\lambda$.

Keywords

• Sturm-Liouville problem
• nonlocal boundary conditions
• coercive
• solvability
• Fredholmness