Electron. J. Differential Equations, Vol. 2017 (2017), No. 11, pp. 1-12.

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

Mustafa Kandemir, Oktay Sh. Mukhtarov

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 $\lambda$.

Submitted October 11, 2016. Published January 12, 2017.
Math Subject Classifications: 34A36, 34B08, 34B24.
Key Words: Sturm-Liouville problem; nonlocal boundary conditions; coercive; solvability; Fredholmness.

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Mustafa Kandemir
Department of Mathematics
Education Faculty, Amasya University
Amasya, Turkey
email: mkandemir5@yahoo.com
Oktay Sh. Mukhtarov
Department of Mathematics
Faculty of Science and Arts
Gaziosmanpasa University
60100 Tokat, Turkey
email: omukhtarov@yahoo.com

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