In this paper we develop the monotone method in the presence of upper and lower solutions for the $2$nd order Lidstone boundary value problem $$u^{\left(2n\right)}\left(t\right)=f(t,u\left(t\right),u^{\text{'}\text{'}}\left(t\right),\cdots ,u^{\left(2\right(n-1\left)\right)}\left(t\right)),0<t<1,u^{\left(2i\right)}\left(0\right)=u^{\left(2i\right)}\left(1\right)=0,0\le i\le n-1,$$ where $f[0,1]\times {ℝ}^n\to ℝ$ is continuous. We obtain sufficient conditions on $f$ to guarantee the existence of solutions between a lower solution and an upper solution for the higher order boundary value problem.

In this paper we use the upper and lower solutions method to investigate the existence of solutions of a class of impulsive partial hyperbolic differential inclusions at fixed moments of impulse involving the Caputo fractional derivative. These results are obtained upon suitable fixed point theorems.

In this paper, an existence theorem for nth order perturbed differential inclusion is proved under the mixed Lipschitz and Carathéodory conditions. The existence of extremal solutions is also obtained under certain monotonicity conditions on the multi-functions involved in the inclusion. Our results extend the existence results of Dhage et al. [7,8] and Agarwal et al. [1].

We consider the initial-boundary-value problem for the one-dimensional fast diffusion equation $u_t=[sign\left(u_x\right)log|u_x{\left|\right]}_x$ on $Q_T=[0,T]\times [0,l]$. For monotone initial data the existence of classical solutions is known. The case of non-monotone initial data is delicate since the equation is singular at $u_x=0$. We ‘explicitly’ construct infinitely many weak Lipschitz solutions to non-monotone initial data following an approach to the Perona-Malik equation. For this construction we rephrase the problem as a differential inclusion which enables us...

In this paper, we develop monotone iterative technique to obtain the extremal solutions of a second order periodic boundary value problem (PBVP) with impulsive effects. We present a maximum principle for ``impulsive functions'' and then we use it to develop the monotone iterative method. Finally, we consider the monotone iterates as orbits of a (discrete) dynamical system.

We discuss the spectral properties of the operator ${}_{ℳ}\left(\alpha \right):=-d²/dt²+(1/2t²-\alpha )²$ on the line. We first briefly describe how this operator appears in various problems in the analysis of operators on nilpotent Lie groups, in the spectral properties of a Schrödinger operator with magnetic field and in superconductivity. We then give a new proof that the minimum over α of the groundstate energy is attained at a unique point and also prove that the minimum is non-degenerate. Our study can also be seen as a refinement for a specific...

The article concerns the symmetries of differential equations with short digressions to the underdetermined case and the relevant differential equations with delay. It may be regarded as an introduction into the method of moving frames relieved of the geometrical aspects: the stress is made on the technique of calculations employing only the most fundamental properties of differential forms. The present Part I is devoted to a single ordinary differential equation subjected to the change of the independent...