Positive solutions of two-point boundary value problems of nonlinear fractional differential equation at resonance

This paper is concerned with a kind of nonlinear fractional differential boundary value problem at resonance with Caputo's fractional derivative. Our main approach is the recent Leggett-Williams norm-type theorem for coincidences due to O'Regan and Zima. The most interesting point is the acquisition of positive solutions for fractional differential boundary value problem at resonance. Moreover, an example is constructed to show that our result here is valid

Positive solutions for higher-order nonlinear fractional differential equation with integral boundary condition

In this paper, we study a kind of higher-order nonlinear fractional differential equation with integral boundary condition. The fractional differential operator here is the Caputo's fractional derivative. By means of fixed point theorems, the existence and multiplicity results of positive solutions are obtained. Furthermore, some examples given here illustrate that the results are almost sharp

The study of higher-order resonant and non-resonant boundary value problems

The existence of at least one solution to a nonlinear $n^\textrm{th}$ order differential equation $x^{(n)} = f(t,x,x',\ldots,x^{(n-1)}), \ 0 <t<1$, under both non-resonant and resonant boundary conditions, is proved. The methods involve the characterization of the $R_\delta$-set and an application of a new generalization for a multi-valued version of the Miranda Theorem

Solvability for second-order nonlocal boundary value problems with a p-Laplacian at resonance on a half-line

This paper investigates the solvability of the second-order boundary value problems with the one-dimensional $p$-Laplacian at resonance on a half-line $$\left\{\begin{array}{llll} (c(t)\phi_{p}(x'(t)))'=f(t,x(t),x'(t)),~~~~0<t<\infty,\\ x(0)=\sum\limits_{i=1}^{n}\mu_ix(\xi_{i}), ~~\lim\limits_{t\rightarrow +\infty}c(t)\phi_{p}(x'(t))=0 \end{array}\right.$$ and $$\left\{\begin{array}{llll} (c(t)\phi_{p}(x'(t)))'+g(t)h(t,x(t),x'(t))=0,~~~~0<t<\infty,\\ x(0)=\int_{0}^{\infty}g(s)x(s)ds,~~\lim\limits_{t\rightarrow +\infty}c(t)\phi_{p}(x'(t))=0 \end{array}\right.$$ with multi-point and integral boundary conditions, respectively, where $\phi_{p}(s)=|s|^{p-2}s$, $p>1$. The arguments are based upon an extension of Mawhin's continuation theorem due to Ge. And examples are given to illustrate our results