In this paper we examine a quasilinear periodic problem driven by the one- dimensional $p$-Laplacian and with discontinuous forcing term $f$. By filling in the gaps at the discontinuity points of $f$ we pass to a multivalued periodic problem. For this second order nonlinear periodic differential inclusion, using variational arguments, techniques from the theory of nonlinear operators of monotone type and the method of upper and lower solutions, we prove the existence of at least two non trivial solutions,...

We consider the boundary value problem $$\begin{array}{c}{u}^{\text{'}\text{'}\text{'}}\left(t\right)=f(t,u\left(t\right),u^{\text{'}}\left(t\right),u^{\text{'}\text{'}}\left(t\right)),0<t<1,\\ u\left(0\right)=u^{\text{'}}\left(0\right)=0,u\left(1\right)={\int }_0^{1}g\left(s\right)u\left(s\right)\mathrm{d}s,\end{array}$$ where $f:[0,1]\times {ℝ}^3\to {ℝ}^{+}$, $g:[0,1]\to {ℝ}^{+}$ are continuous functions. The case when $f=f\left(u\right(t\left)\right)$ was studied in 2018 by Guendouz et al. Using the fixed-point theory on cones they established the existence of positive solutions. Here, by the method developed by ourselves very recently, we establish the existence, uniqueness and positivity of the solution under easily verified conditions and propose an iterative method for finding the solution. Some examples demonstrate the validity of the obtained theoretical...

By using Mawhin’s continuation theorem, we provide some sufficient conditions for the existence of solution for a class of high order differential equations of the form $$x^{\left(n\right)}=f(t,x,x^{\text{'}},\cdots ,x^{(n-1)}),t\in [0,1],$$ associated with the integral boundary conditions at resonance. The interesting point is that we shall deal with the case of nontrivial kernel of arbitrary dimension corresponding to high order differential operator which will cause some difficulties in constructing the generalized inverse operator.

In this paper, we consider the following boundary value problem $$\left\{\begin{array}{c}{D}_{T^{-}}^{\alpha }\left(D_{0^{+}}^{\alpha }\left(D_{T^{-}}^{\alpha }\left(D_{0^{+}}^{\alpha }u\left(t\right)\right)\right)\right)=f(t,u\left(t\right)),t\in [0,T],\hfill \\ u\left(0\right)=u\left(T\right)=0\hfill \\ D_{T^{-}}^{\alpha }\left(D_{0^{+}}^{\alpha }u\left(0\right)\right)=D_{T^{-}}^{\alpha }\left(D_{0^{+}}^{\alpha }u\left(T\right)\right)=0,\hfill \end{array}\right.$$ where $0<\alpha \le 1$ and $f:[0,T]\times ℝ\to ℝ$ is a continuous function, $D_{0^{+}}^{\alpha }$, $D_{T^{-}}^{\alpha }$ are respectively the left and right fractional Riemann–Liouville derivatives and we prove the existence of at least one solution for this problem.

In this paper a fixed point theorem due to Covitz and Nadler for contraction multivalued maps, and the Schaefer’s theorem combined with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued operators with decomposables values, are used to investigate the existence of solutions for boundary value problems of fourth-order differential inclusions.

This paper presents existence results for initial and boundary value problems for nonlinear differential equations in Banach spaces.

This paper presents several sufficient conditions for the existence of at least one classical solution to impulsive fractional differential equations with a $p$-Laplacian and Dirichlet boundary conditions. Our technical approach is based on variational methods. Some recent results are extended and improved. Moreover, a concrete example of an application is presented.

In this article, we study the existence of solutions to systems of conformable fractional differential equations with periodic boundary value or initial value conditions. where the right member of the system is $L_{\alpha }^1$-carathéodory function. We employ the method of solution-tube and Schauder’s fixed-point theorem.