Positive solutions for three-point boundary value problem of fractional differential equation with \(p\)-Laplacian operator
Summary: We investigate the existence of multiple positive solutions for three-point boundary value problem of fractional differential equation with \(p\)-Laplacian operator \(-\mathcal D^\beta_{\mathbf{t}}(\varphi_p(\mathcal D^\alpha_{\mathbf{t}}x))(t) = h(t)f(t, x(t)), t \in (0, 1), x(0) = 0, \mathcal D^\gamma_{\mathbf{t}}x(1) = a\mathcal D^\gamma_{\mathbf{t}}x(\xi), \mathcal D^\alpha_{\mathbf{t}}x(0) = 0\), where \(\mathcal D^\beta_{\mathbf{t}}, \mathcal D^\alpha_{\mathbf{t}}, \mathcal D^\gamma_{\mathbf{t}}\) are the standard Riemann-Liouville derivatives with \(1 < \alpha \leq 2, 0 < \beta \leq 1, 0 < \gamma \leq 1, 0 \leq \alpha - \gamma- 1, \xi \in (0, 1)\) and the constant \(a\) is a positive number satisfying \(a\xi^{\alpha-\gamma-2} \leq 1 - \gamma\); \(p\)-Laplacian operator is defined as \(\varphi_p(s) = |s|^{p-2}s\), \(p > 1\). By applying monotone iterative technique, some sufficient conditions for the existence of multiple positive solutions are established; moreover iterative schemes for approximating these solutions are also obtained, which start off a known simple linear function. In the end, an example is worked out to illustrate our main results.
scientific article; zbMATH DE number 6175073