Fractional Boundary Value Problem on the Half-Line
DOI:
https://doi.org/10.15407/mag16.01.027Анотація
У статті розглядається напівлінійна крайова задача з дробовими похідними \begin{equation*} D^{\beta}\left(\frac{1}{b(x)}D^{\alpha}u\right)=a(x)u^{\sigma} \quad\text{на } (0,\infty) \end{equation*} з умовами $\lim_{x\rightarrow 0} x^{2-\beta} \frac{1}{b(x)}D^{\alpha}u(x) =\lim_{x\rightarrow \infty} x^{1-\beta}\frac{1}{b(x)}D^{\alpha}u(x)=0$ та $\lim_{x\rightarrow 0} x^{2-\alpha}u(x)= \lim_{x\rightarrow \infty} x^{1-\alpha}u(x)=0$, де $\beta,\alpha \in (1,2)$, $\sigma\in(-1,1)$ і $D^{\beta}, D^{\alpha}$ означають стандартні дробові похідні Рімана--Ліувілля. Функції $ a,b : (0,\infty)\longrightarrow \mathbb{R}$ є невід'ємними неперервними функціями, які задовольняють деякі відповідні умови. Встановлено існування та єдиність позитивного розв'язку. Також надано опис глобальної поведінки цього розв'язку.Mathematics Subject Classification: 34A08, 35B09, 47H10.
Ключові слова:
рівняння з дробовими похідними, позитивний розв'язок, теорема Шаудера про нерухому точку.Посилання
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