Existence of solutions for a third-order boundary value problem with integral boundary conditions

Öz

In this study, the existence of bounded or unbounded solutions for the following third order three-point integral conditional boundary value problem on a half line

${\vartheta }^{\prime \prime \prime }\left(t\right)+r\left(t\right)g(t,\vartheta (t),{\vartheta }^{\prime }(t),{\vartheta }^{\prime \prime }(t\left)\right)=0,t\in (0,\infty ),$
$\vartheta \left(0\right)={\int }_0^{\eta }\vartheta \left(s\right)ds,{\vartheta }^{\prime }\left(0\right)=A,{\vartheta }^{\prime \prime }\left(\infty \right)={lim}_{t\to \infty }\vartheta \left(t\right)=B$

where $\eta \in (0,\infty )$ fixed and $g:(0,\infty )\times R^3\to R$ satisfies Nagumo’s condition is proved. The expected result is obtained by applying Schäuder’s fixed point theorem and the lower and upper solutions method. The existence of at least one solution of the problem has been shown by determining suitable and sufficient conditions in our problem. Since it is difficult to work on the semi-infinite interval, this study with integral conditions on the semi-infinite interval will contribute to the literature for studies on this subject. Also, the solutions to this boundary value problem can be unbounded.

Anahtar Kelimeler

• Lower and upper solutions
• Nagumo’s condition
• Schäuder’s fixed point
• infinite interval

İntegral sınır koşullu üçüncü mertebeden sınır değer probleminin çözümlerinin varlığı

Öz

Bu çalışmada, yarı sonsuz aralık üzerinde tanımlı üçüncü mertebeden üç noktalı integral koşullu, $\eta \in (0,\infty )$ sabit ve $g:(0,\infty )\times R^3⟶R$ Nagumo koşullarını sağlayan bir fonksiyon olmak üzere 
${\vartheta }^{\prime \prime \prime }\left(t\right)+r\left(t\right)g(t,\vartheta (t),{\vartheta }^{\prime }(t),{\vartheta }^{\prime \prime }(t\left)\right)=0,t\in (0,\infty ),$
$\vartheta \left(0\right)={\int }_0^{\eta }\vartheta \left(s\right)ds,{\vartheta }^{\prime }\left(0\right)=A,{\vartheta }^{\prime \prime }\left(\infty \right)={lim}_{t\to \infty }\vartheta \left(t\right)=B$
sınır değer probleminin sınırlı veya sınırlı olmayan çözümlerinin varlığı ispatlanmıştır. Schäuder sabit nokta teoremi ve alt ve üst çözümler yöntemi uygulanarak istenilen sonuca ulaşılmıştır. Problemimizde uygun ve yeterli koşullar belirlenerek problemin en az bir çözümünün varlığı gösterilmiştir. Yarı sonsuz aralık üzerinde çalışılması zor olduğundan yarı sonsuz aralık üzerinde integral koşullu bu çalışma bu konuda yapılacak çalışmalar için litaratüre katkı sağlamış olacaktır. Ayrıca, bu sınır değer probleminin çözümleri sınırsız olabilir.

Anahtar Kelimeler

• Alt ve üst çözümler
• Naguma koşulu
• Schäuder sabit nokta teoremi
• yarı sonsuz aralık

Kaynakça

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