Let $f:\left[0,1\right]\times {ℝ}^2\to ℝ$ be a function satisfying Carathéodory′s conditions and $e\left(t\right)\in L^1\left[0,1\right]$. Let be given. This paper is concerned with the problem of existence of a solution for the $m$-point boundary value problem . This paper gives conditions for the existence of a solution for this boundary value problem using some new Poincaré type a priori estimates. This problem was studied earlier by Gupta, Ntouyas, and Tsamatos (1994) when all of the $a_i\in ℝ,i=1,2,\dots ,m-2$, had the same sign. The results of this paper give considerably better existence conditions even in the case when all of the $a_i\in ℝ,i=1,2,\dots ,m-2$, have the same sign. Some examples are given to illustrate this point.

We obtain new $A$-properness results for demicontinuous, dissipative type mappings defined only on closed convex subsets of a Banach space $X$ with uniformly convex dual and which satisfy a property called weakly inward. The method relies on a new property of the duality mapping in such spaces. New fixed point results are obtained by utilising a theory of fixed point index.

By using the spectral Galerkin method, we prove a result on global existence in time of strong solutions for the motion of magneto-micropolar fluid without assuming that the external forces decay with time. We also derive uniform in time estimates of the solution that are useful for obtaining error bounds for the approximate solutions.

We establish the global existence of mild solutions to a class of nonlocal Cauchy problems associated with semilinear Volterra integrodifferential equations in a Banach space.