In this article, we considered the one-dimensional swelling problem in porous elastic soils with microtemperatures effects in the case of fluid saturation. First, we showed that the system is well-posed in the sens of semigroup. Then, we constructed a suitable Lyapunov functional based on the energy method and we proved that the dissipation given only by the microtemperatures is strong enough to provoke an exponential stability for the solution irrespective of the wave speeds of the system.

In this work, we concentrate on the existence of the solutions set of the following problem ${}^c{D}_q^{\alpha }\sigma \left(t\right)\in F(t,\sigma \left(t\right),{}^c{D}_q^{\alpha }\sigma \left(t\right)),t\in I=[0,T]\sigma \left(0\right)={\sigma }_0\in E$, as well as its topological structure in Banach space E. By transforming the problem posed into a fixed point problem, we provide the necessary conditions for the existence and compactness of solutions set. Finally, we present an example as an illustration of main results.

In this paper, we study a third-order differential inclusion with three-point boundary conditions. We prove the existence of a solution under convexity conditions on the multi-valued right-hand side; the proof is based on a nonlinear alternative of Leray-Schauder type. We also study the compactness of the set of solutions and establish some Filippov’s- type results for this problem.

Sponsored by the Environmental & Water Resources Institute (EWRI)

In this work, we examine a class of nonlinear neutral differential equations. Krasnoselskii’s fixed-point theorem is used to provide sufficient conditions for the existence of positive periodic solutions to this type of problem.