It is shown that a result of Chidume, involving the strong convergence of the Mann iteration process for continuous strongly accretive operators, is actually a corollary to a result by Nevanlinna and Reich. It is then shown that the Nevanlinna and Reich result can be extended to the case of an Ishikawa iteration process.

This paper is concerned with the existence and stability of solutions of a class of semilinear nonautonomous evolution equations. A procedure is discussed which associates to each nonautonomous equation the so-called evolution semigroup of (possibly nonlinear) operators. Sufficient conditions for the existence and stability of solutions and the existence of periodic oscillations are given in terms of the accretiveness of the corresponding infinitesimal generator. Furthermore, through the existence of integral manifolds for abstract evolutionary processes we obtain a reduction principle for stability questions of mild solutions. The results are applied to a class of partial functional differential equations.

This paper is devoted to the development of a local degree for multi-valued vector fields of the form $f-F$. Here, $f$ is a single-valued, proper, nonlinear, Fredholm, $C^1$-mapping of index zero and $F$ is a multi-valued upper semicontinuous, admissible, compact mapping with compact images. The mappings $f$ and $F$ are acting from a subset of a Banach space $E$ into another Banach space $E_1$. This local degree is used to investigate the existence of solutions of a certain class of operator inclusions.

Recently, Z. Ditzian gave an interesting direct estimate for Bernstein polynomials. In this paper we give direct and inverse results of this type for linear combinations of Bernstein polynomials.

In this note we give a result for the operator $p$-Laplacian complementing a theorem by Brézis and Kamin concerning a necessary and sufficient condition for the equation $-\Delta u=h\left(x\right)u^q$ in ${ℝ}^N$, where , to have a bounded positive solution. While Brézis and Kamin use the method of sub and super solutions, we employ variational arguments for the existence of solutions.

Let $X$ be a real Banach space, $J=\left[0,a\right]\subset \mathbf{R}$, $A:D\left(A\right)\subset X\to 2^X\backslash \phi$ an $m$-accretive operator and $f:J\times X\to X$ continuous. In this paper we obtain necessary and sufficient conditions for weak positive invariance (also called viability) of closed sets $K\subset X$ for the evolution system $$u^{\prime }+Au∍f\left(t,u\right)\text{on}J=\left[0,a\right]$$. More generally, we provide conditions under which this evolution system has mild solutions satisfying time-dependent constraints $u\left(t\right)\in K\left(t\right)$ on $J$. This result is then applied to obtain global solutions of reaction-diffusion systems with nonlinear diffusion, e.g. of type $$u_t=\Delta \Phi \left(u\right)+g\left(u\right)\text{in}\left(0,\infty \right)\times \Omega ,\Phi \left(u\left(t,\cdot \right)\right){|}_{\partial \Omega }=0,u\left(0,\cdot \right)=u_0$$ under certain assumptions on the set$\Omega \subset {\mathbf{R}}^n$ the function $\Phi \left(u_1,\dots ,u_m\right)=\left({\varphi }_1\left(u_1\right),\dots ,{\varphi }_m\left(u_m\right)\right)$ and $g:{\mathbf{R}}_{+}^m\to {\mathbf{R}}^m$.

We study, by the variational method, the Differential Riccati Equation which arises in the theory of quadratic optimal control problems for ‘abstract hyperbolic’ equations (which encompass hyperbolic and Petrowski-type partial differential equations (P.D.E.) with boundary control). We markedly relax, at the abstract level, the original assumption of smoothing required of the observation operator by the direct method of [D-L-T.1]. This is achieved, by imposing additional higher level regularity requirements on the dynamics, which, however, are always satisfied by the class of hyperbolic and Petrowski-type mixed P.D.E. problems which we seek to cover. To appreciate the additional level of generality, and related technical difficulties associate with it, it suffices to point out that in the present treatment—unlike in [D-L-T.1]—the gain operator $B^{*}P\left(t\right)$ is no longer bounded between the state space $Y$ and the control space $U$. The abstract theory is illustrated by its application to a Kirchoff equation with one boundary control. This requires establishing new higher level interior and boundary regularity results.