We obtain results of existence of weak solutions in the Hopf sense of the initial-boundary value problem for the generalized Navier-Stokes equations containing perturbations of retarded type. The degree theory for maps $A-g$, where $A$ is invertible and $g$ is $𝒜$-condensing, is used.

In this paper we consider one-dimensional two-phase Stefan problems for a class of parabolic equations with nonlinear heat source terms and with nonlinear flux conditions on the fixed boundary. Here, both time-dependent and time-independent source terms and boundary conditions are treated. We investigate the large time behavior of solutions to our problems by using the theory for dynamical systems. First, we show the existence of a global attractor $𝒜$ of autonomous Stefan problem. The main purpose in the present paper is to prove that the set $𝒜$ attracts all solutions of non-autonomous Stefan problems as time tends to infinity under the assumption that time-dependent data converge to time-independent ones as time goes to infinity.

A convergence result is given for discrete descent based on Sobolev gradients arising from differential equations which may be expressed as quadratic forms. The argument is an extension of the result of David G. Luenberger on Euclidean descent and compliments the work of John W. Neuberger on Sobolev descent.

In this paper, we characterize wellposedness of nonautonomous, linear Cauchy problems $$\left(NCP\right)\{\begin{array}{l}\dot{u}\left(t\right)=A\left(t\right)u\left(t\right)\\ u\left(s\right)=x\in X\end{array}$$ on a Banach space $X$ by the existence of certain evolution semigroups.

Then, we use these generation results for evolution semigroups to derive wellposedness for nonautonomous Cauchy problems under some “concrete” conditions. As a typical example, we discuss the so called “parabolic” case.

In this paper we show the weak convergence and stability of the proximal point method when applied to the constrained convex optimization problem in uniformly convex and uniformly smooth Banach spaces. In addition, we establish a nonasymptotic estimate of convergence rate of the sequence of functional values for the unconstrained case. This estimate depends on a geometric characteristic of the dual Banach space, namely its modulus of convexity. We apply a new technique which includes Banach space geometry, estimates of duality mappings, nonstandard Lyapunov functionals and generalized projection operators in Banach spaces.

Let $i{A}_j\left(1\le j\le n\right)$ be generators of commuting bounded strongly continuous groups, $A\equiv \left(A_1,A_2,\dots ,A_n\right)$. We show that, when $f$ has sufficiently many polynomially bounded derivatives, then there exist $k,r>0$ such that $f\left(A\right)$ has a ${\left(1+{\left|A\right|}^2\right)}^{-r}$-regularized $B{C}^k\left(f\left({\mathbf{R}}^n\right)\right)$ functional calculus. This immediately produces regularized semigroups and cosine functions with an explicit representation; in particular, when $f\left({\mathbf{R}}^n\right)\subseteq \mathbf{R}$, then, for appropriate $k,r$, $t\mapsto {\left(1-it\right)}^{-k}{e}^{-itf\left(A\right)}{\left(1+{\left|A\right|}^2\right)}^{-r}$ is a Fourier-Stieltjes transform, and when $f\left({\mathbf{R}}^n\right)\subseteq \left[0,\infty \right)$, then $t\mapsto {\left(1+t\right)}^{-k}{e}^{-tf\left(A\right)}{\left(1+{\left|A\right|}^2\right)}^{-r}$ is a Laplace-Stieltjes transform. With $A\equiv i\left(D_1,\dots ,D_n\right),f\left(A\right)$ is a pseudodifferential operator on or $BUC\left({\mathbf{R}}^n\right)$.

In this paper we develop a new approach for the stable approximation of a minimal surface problem associated with a relaxed Dirichlet problem in the space $BV\left(\Omega \right)$ of functions of bounded variation. The problem can be reformulated as an unconstrained minimization problem of a functional $𝒥$ on $BV\left(\Omega \right)$ defined by $𝒥\left(u\right)=𝒜\left(u\right)+{\int }_{\partial \Omega }\left|Tu-\Phi \right|$, where $𝒜\left(u\right)$ is the “area integral” of $u$ with respect to $\Omega ,T$ is the “trace operator” from $BV\left(\Omega \right)$ into $L^i\left(\partial \Omega \right)$, and $\phi$ is the prescribed data on the boundary of $\Omega$. We establish convergence and stability of approximate regularized solutions which are solutions of a family of variational inequalities. We also prove convergence of an iterative method based on Uzawa′s algorithm for implementation of our regularization procedure.

In this paper, by applying $\left(p,k\right)$-epi mapping theory, we introduce a new definition of spectrum for nonlinear operators which contains all eigenvalues, as in the linear case. Properties of this spectrum are given and comparison is made with the other definitions of spectra. We also give applications of the new theory.