HIGHER FRACTIONAL ORDER -LAPLACIAN BOUNDARY VALUE PROBLEM AT RESONANCE ON AN UNBOUNDED DOMAIN

Abstract

In this work, we use the Ge and Ren extension of Mawhin's coincidence degree theory to investigate the solvability of the $p$-Laplacian fractional order boundary value problem of the form $$ \begin{aligned} & \left(\phi_p\left(D_{0+}^\alpha x(t)\right)\right)^{\prime} \\ = & f\left(t, x(t), D_{0+}^{\alpha-3} x(t), D_{0+}^{\alpha-2} x(t), D_{0+}^{\alpha-1} x(t), D_{0+}^\alpha x(t)\right), \quad t \in(0,+\infty), \\ & x(0)=0=D_{0+}^{\alpha-3} x(0), \quad D_{0+}^{\alpha-2} x(0)=\int_0^1 D_{0+}^{\alpha-2} x(t) d A(t), \\ & \lim _{t \rightarrow+\infty} D_{0+}^{\alpha-1} x(t)=\sum_{i=1}^m \mu_i D_{0+}^{\alpha-1} x\left(\xi_i\right), \quad D_{0+}^\alpha x(\infty)=0, \end{aligned} $$ where $3<\alpha \leq 4$. The conditions $\int_0^1 d A(t)=1, \quad \int_0^1 t d A(t)=0$, $\sum_{i=1}^m \mu_i=1$ and $\sum_{i=1}^m \mu_i \xi_i^{-1}=0$ are critical for resonance.