We give relationships between some Banach-space geometric properties that guarantee the weak fixed point property. The results extend some known results ofDalby and Xu.

We study a nonlinear problem of pendulum-type for a $p$-Laplacian with nonlinearperiodic-type boundary conditions. Using an extension of Mawhin's continuation theorem for nonlinear operators, we prove the existence of a solution under a Landesman-Lazer type condition. Moreover, using the method of upper and lower solutions, we generalize a celebrated result by Castro for the classical pendulum equation.

The paper presents general existence principles which can be used for a large class of nonlocal boundary value problems of the form ${\left(\phi \left(x^{\prime }\right)\right)}^{\prime }=f_1\left(t,x,x^{\prime }\right)+f_2\left(t,x,x^{\prime }\right)F_{1}x+f_3\left(t,x,x^{\prime }\right)F_{2}x,\alpha \left(x\right)=0$, $\beta \left(x\right)=0$, where $f_j$ satisfy local Carathéodory conditions on some $\left[0,T\right]\times {𝒟}_j\subset {ℝ}^2$, $f_j$ are either regular or have singularities in their phase variables $\left(j=1,2,3\right)$, $F_i:C^1\left[0,T\right]\to C^0\left[0,T\right]\left(i=1,2\right)$, and $\alpha ,\beta :C^1\left[0,T\right]\to ℝ$ are continuous. The proofs are based on the Leray-Schauder degree theory and use regularization and sequential techniques. Applications of general existence principles to singular BVPs are given.

Let $D$ be a bounded domain in ${ℝ}^n\left(n\ge 2\right)$. We consider the following nonlinear elliptic problem: $\Delta u=f\left(\cdot ,u\right)$ in $D$ (in the sense of distributions), $u_{|\partial D}=\varphi$, where $\varphi$ is a nonnegative continuous function on $\partial D$ and $f$ is a nonnegative function satisfying some appropriate conditions related to some Kato class of functions $K\left(D\right)$. Our aim is to prove that the above problem has a continuous positive solution bounded below by a fixed harmonic function, which is continuous on $\overline{D}$. Next, we will be interested in the Dirichlet problem $\Delta u=-\rho \left(\cdot ,u\right)$ in $D$ (in the sense of distributions), $u_{|\partial D}=0$, where $\rho$ is a nonnegative function satisfying some assumptions detailed below. Our approach is based on the Schauder fixed-point theorem.

We consider a class of singularly perturbed parabolic equations for which the degenerate equations obtained by setting the small parameter equal to zero are algebraic equations that have several roots. We study boundary layer type solutions that, as time increases, periodically go through two fairly long lasting stages with extremely fast transitions in between. During one of these stages the solution outside the boundary layer is close to one of the roots of the degenerate (reduced) equation, while during the other stage the solution is close to the other root. Such equations may be used as models for bio-switches where the transitions between various stationary states of biological systems are initiated by comparatively slow changes within the systems.

By means of a variational structure and $Z_2$-group index theory, we obtain multiple periodic solutions to a class of second-order nonlinear neutral delay equations of the form$x^{″}\left(t-\tau \right)+\lambda \left(t\right)f\left(t,x\left(t\right),x\left(t-\tau \right),x\left(t-2\tau \right)\right)=x\left(t\right),\lambda \left(t\right)>0,\tau >0$.

We consider a quasi-linear parabolic system with respect to unknown functions $u$ and $v$on a bounded domain of $n$-dimensional Euclidean space. We assume that the diffusion coefficient of $u$ is a positive smooth function $A\left(u\right)$, and that the diffusion coefficient of $v$ is a positive constant. If $A\left(u\right)$ is a positive constant, the system is referred to as so-called Keller-Segel system. In the case where the domain is a bounded domain of two-dimensional Euclidean space, it is shown that some solutions to Keller-Segel system blow up in finite time. In three and more dimensional cases, it is shown that solutions to so-called Nagai system blow up in finite time. Nagai system is introduced by Nagai. The diffusion coefficients of Nagai system are positive constants. In this paper, we describe that solutions to the quasi-linear parabolic system exist globally in time, if the positive function $A\left(u\right)$ rapidly increases with respect to $u$.

The initial-boundary value problem for the mathematical model of low-concentrated aqueous polymer solutions is considered. For this initial-boundary value problem a concept of a weak solution is introduced and the existence theorem for such solutions is proved.

This work provides the solution of the direct Electroencephalography (EEG) problem for the complete ellipsoidal shell-model of the human head. The model involves four confocal ellipsoids that represent the successive interfaces between the brain tissue, the cerebrospinal fluid, the skull, and the skin characterized by different conductivities. The electric excitation of the brain is due to an equivalent electric dipole, which is located within the inner ellipsoid. The proposed model is considered to be physically complete, since the effect of the substance surrounding the brain is taken into account. The direct EEG problem consists in finding the electric potential inside each conductive space, as well as at the nonconductive exterior space. The solution of this multitransmission problem is given analytically in terms of elliptic integrals and ellipsoidal harmonics, in such way that makes clear the effect that each shell has on the next one and outside of the head. It is remarkable that the dependence on the observation point is not affected by the presence of the conductive shells. Reduction to simpler ellipsoidal models and to the corresponding spherical models is included.

We will be concerned with the existence result of unilateral problem associated to the equations of the form $Au+g\left(x,u,\nabla u\right)=f$, where $A$ is a Leray-Lions operator from its domain $D\left(A\right)\subset W_0^1{L}_M\left(\Omega \right)$ into $W^{-1}{E}_{\overline{M}}\left(\Omega \right)$. On the nonlinear lower order term $g\left(x,u,\nabla u\right)$, we assume that it is a Carathéodory function having natural growth with respect to $\left|\nabla u\right|$, and satisfies the sign condition. The right-hand side $f$ belongs to $W^{-1}{E}_{\overline{M}}\left(\Omega \right)$.

We study the norming points and norming functionals of symmetric operators on $L_p$ spaces for $p=2m$ or $p=2m/\left(2m-1\right)$. We prove some general result relating uniqueness of minimal projection to the set of norming functionals. As a main application, we obtain that the Fourier projection onto span $\left[1,\sin x,\cos x\right]$ is a unique minimal projection in $L_p$.

Generators $f$ for $\sigma$-algebras can be used to view the dynamics of an invertible measurable transformation $T$ in terms of the range values of $f\circ T$. Such generators are the norm rather than the exception. Related measurable and quantitative methods of estimating a function from the behavior of ergodic averages are also discussed.

The system of quasilinear equations for symmetric flows of a viscous heat-conducting gas with a free external boundary is considered. For global in time weak solutions having nonstrictly positive density, the linear in time two-sided bounds for the gas volume growth are established.

We will study the following polyharmonic nonlinear elliptic equation ${\left(-∆\right)}^{m}u+f\left(‧,u\right)=0$ in ${ℝ}^n$, $n>2m$. Under appropriate conditions on the nonlinearity $f\left(x,t\right)$, related to a class of functions called $m$-Green-tight functions, we give some existence results for the above equation.

We establish a $3G$-theorem for the iterated Green function of ${\left(-∆\right)}^{pm}$, ($p\ge 1,m\ge 1$), on the unit ball $B$ of ${ℝ}^n$ $\left(n\ge 1\right)$ with boundary conditions ${\left(\partial /\partial \nu \right)}^j{\left(-∆\right)}^{km}u=0$ on $\partial B$, for $0\le k\le p-1$ and $0\le j\le m-1$. We exploit this result to study a class of potentials ${𝒥}_{m,n}^{\left(p\right)}$. Next, we aim at proving the existence of positive continuous solutions for the following polyharmonic nonlinear problems ${\left(-∆\right)}^{pm}u=h\left(‧,u\right)$, in $D$ (in the sense of distributions), ${\text{lim}}_{\left|x\right|\to 1}\left({\left(-∆\right)}^{km}u\left(x\right)/{\left(1-\left|x\right|\right)}^{m-1}\right)=0$, for $0\le k\le p-1$, where $D=B$ or $B\backslash \left\{0\right\}$ and $h$ is a Borel measurable function on $D\times \left(0,\infty \right)$ satisfying some appropriate conditions related to ${𝒥}_{m,n}^{\left(p\right)}$.

We apply a coincidence degree theorem of Mawhin to show the existence of at least one symmetric solution of the nonlinear second-order multipoint boundary value problem $u^{″}\left(t\right)=f\left(t,u\left(t\right),\left|u^{\prime }\left(t\right)\right|\right),t\in \left(0,1\right)$, $u\left(0\right)={\sum }_{i=1}^n{\mu }_{i}u\left({\xi }_i\right),u\left(1-t\right)=u\left(t\right),t\in \left[0,1\right]$, where $0<{\xi }_1<{\xi }_2<\dots <{\xi }_n\le 1/2$, ${\sum }_{i=1}^n{\mu }_i=1$, $f:\left[0,1\right]\times {ℝ}^2\to ℝ$ with $f\left(t,x,y\right)=f\left(1-t,x,y\right)$, $\left(t,x,y\right)\in \left[0,1\right]\times {ℝ}^2$, satisfying the Carathéodory conditions.

We consider a biharmonic equation under the Navier boundary condition and with a nearly critical exponent ($P_{\epsilon }$): ${∆}^{2}u=u^{9-\epsilon }$, $u>0$ in $\Omega$ and $u=∆u=0$ on $\partial \Omega$, where $\Omega$ is a smooth bounded domain in ${ℝ}^5$, $\epsilon >0$. We study the asymptotic behavior of solutions of ($P_{\epsilon }$) which are minimizing for the Sobolev quotient as $\epsilon$ goes to zero. We show that such solutions concentrate around a point $x_0\in \Omega$ as $\epsilon \to 0$, moreover $x_0$ is a critical point of the Robin's function. Conversely, we show that for any nondegenerate critical point $x_0$ of the Robin's function, there exist solutions of ($P_{\epsilon }$) concentrating around $x_0$ as $\epsilon \to 0$.

The nonlocal boundary value problem for hyperbolic-elliptic equation $d^{2}u\left(t\right)/d{t}^2+Au\left(t\right)=f\left(t\right)$, $\left(0\le t\le 1\right)$, $-d^{2}u\left(t\right)/d{t}^2+Au\left(t\right)=g\left(t\right)$, $\left(-1\le t\le 0\right)$, $u\left(0\right)=\varphi$, $u\left(1\right)=u\left(-1\right)$ in a Hilbert space $H$ is considered. The second order of accuracy difference schemes for approximate solutions of this boundary value problem are presented. The stability estimates for the solution of these difference schemes are established.

We present an integer valued degree theory for locally compact perturbations of Fredholm maps of index zero between (open sets in) Banach spaces (quasi-Fredholm maps, for short). The construction is based on the Brouwer degree theory and on the notion of orientation for nonlinear Fredholm maps given by the authors in some previous papers. The theory includes in a natural way the celebrated Leray-Schauder degree.

Thetwo-point boundary value problem for second-order differential inclusions ofthe form $\left(D/dt\right)\dot{m}\left(t\right)\in F\left(t,m\left(t\right),\dot{m}\left(t\right)\right)$ on complete Riemannian manifolds is investigated for a couple of points, nonconjugate along at least one geodesic ofLevi-Civitá connection, where $D/dt$ is the covariant derivative of Levi-Civitá connection and $F\left(t,m,X\right)$ is a set-valued vector with quadratic or less than quadratic growth in the third argument. Some interrelations between certain geometric characteristics, the distance between points, and the norm of right-hand side are found that guarantee solvability of the above problem for $F$ with quadratic growth in $X$. It is shown that this interrelation holds for all inclusions with $F$ having less than quadratic growth in $X$, and so for them the problem is solvable.

One of our main results is the following convergence theorem for one-parameter nonexpansive semigroups: let $C$ be a bounded closed convex subset of a Hilbert space $E$, and let $\left\{T\left(t\right):t\in {ℝ}_{+}\right\}$ be a strongly continuous semigroup of nonexpansive mappings on $C$. Fix $u\in C$ and $t_1,t_2\in {ℝ}_{+}$ with $t_1<t_2$. Define a sequence $\left\{x_n\right\}$ in $C$ by $x_n=\left(1-{\alpha }_n\right)/\left(t_2-t_1\right){\displaystyle {\int }_{t_1}^{t_2}T\left(s\right)x_{n}ds+{\alpha }_{n}u}$ for $n\in \mathbb{N}$, where $\left\{{\alpha }_n\right\}$ is a sequence in $\left(0,1\right)$ converging to $0$. Then $\left\{x_n\right\}$ converges strongly to a common fixed point of $\left\{T\left(t\right):t\in {ℝ}_{+}\right\}$.

The aim of this paper is twofold. First, several basic mathematical concepts involved in the construction and study of Bregman type iterative algorithms are presented from a unified analytic perspective. Also, some gaps in the current knowledge about those concepts are filled in. Second, we employ existing results on total convexity, sequential consistency, uniform convexity and relative projections in order to define and study the convergence of a new Bregman type iterative method of solving operator equations.

The equation of symmetric equilibrium of an isothermal gas with an unknown boundary in the field of a body force is considered. Conditions for solvability and insolvability of the problem as well as for uniqueness and nonuniqueness of solutions are presented. Examples of finite, countable, or continual sets of solutions are constructed including equipotential ones. Static stability of solutions is analyzed too.

This paper is concerned with pointwise estimates for the gradient of the heat kernel $K_t$, $t>0$, of the Laplace operator on a Riemannian manifold $M$. Under standard assumptions on $M$, we show that $\nabla K_t$ satisfies Gaussian bounds if and only if it satisfies certain uniform estimates or estimates in $L^p$ for some $1\le p\le \infty$. The proof is based on finite speed propagation for the wave equation, and extends to a more general setting. We also prove that Gaussian bounds on $\nabla K_t$ are stable under surjective, submersive mappings between manifolds which preserve the Laplacians. As applications, we obtain gradient estimates on covering manifolds and on homogeneous spaces of Lie groups of polynomial growth and boundedness of Riesz transform operators.

We formulate and study robust control problems for a two-dimensional time-dependent Ginzburg-Landau model with Robin boundary conditions on phase-field parameter, which describes the phase transitions taking place in superconductor films with variable thickness. The objective of such study is to control the motion of vortices in the superconductor films by taking into account the influence of noises in data. Firstly, we introduce the perturbation problem of the nonlinear governing coupled system of equations (the deviation from the desired target). The existence and the uniqueness of the solution of the perturbation are proved as well as stability under mild assumptions. Afterwards, the robust control problems are formulated in the case when the control is in the external magnetic field and in the case when the control is in the initial condition of the vector potential. We show the existence of an optimal solution, and we also find necessary conditions for a saddle point optimality.

The exact controllability of a nonlinear stochastic heat equation with null Dirichlet boundary conditions, nonzero initial and target values, and an interior control is established.

This paper deals with a class of nonlinear elliptic equations in an unbounded domain $D$ of ${ℝ}^n$, $n\ge 3$, with a nonempty compact boundary, where the nonlinear term satisfies some appropriate conditions related to a certain Kato class $K^{\infty }\left(D\right)$. Our purpose is to give some existence results and asymptotic behaviour for positive solutions by using the Green function approach and the Schauder fixed point theorem.

The existence of a nontrivial solution for semilinear elliptic problems with strictly differentiable nonlinearity is proved. A result of homological linking under nonstandard geometrical assumption is also shown. Techniques of Morse theory are employed.

We study the existence of solutions an $H$-system for a revolution surface without boundary for $H$ depending on the radius $f$. Under suitable conditions we prove that the existence of a solution is equivalent to the solvability of a scalar equation $N\left(a\right)=L/\sqrt{2}$, where $N:𝒜\subset {ℝ}^{+}\to ℝ$ is a function depending on $H$. Moreover, using the method of upper and lower solutions we prove existence results for some particular examples. In particular, applying a diagonal argument we prove the existence of unbounded surfaces with prescribed $H$.

The exact controllability for a semilinear stochastic wave equation with a boundary control is established. The target and initial spaces are $L^2\left(G\right)\times H^{-1}\left(G\right)$ with $G$ being a bounded open subset of $R^3$ and the nonlinear terms having at most a linear growth.

We give necessary and sufficient conditions for uniform exponential dichotomy of evolution families in terms of the admissibility of the pair $\left(L^p\left(ℝ,X\right),L^q\left(ℝ,X\right)\right)$. We show that the admissibility of the pair $\left(L^p\left(ℝ,X\right),L^q\left(ℝ,X\right)\right)$ is equivalent to the uniform exponential dichotomy of an evolution family if and only if $p\ge q$. As applications we obtain characterizations for uniform exponential dichotomy of semigroups.

We establish an existence result on positive solution for a class of reaction-diffusion equation with semipositone structure. In particular, our results apply to the diffusive logistic equation with a class of sign changing weight and constant yield harvesting. We establish the result via the method of subsuper solutions.

This paper deals with a class of singular nonlinear polyharmonic equations on the unit ball $B$ in ${ℝ}^n\hspace{0.17em}(n\ge 2)$ where the combined effects of a singular and a sublinear term allow us by using the Schauder fixed point theorem to establish an existence result for the following problem: ${\left(-\Delta \right)}^{m}u=\phi \left(\cdot ,u\right)+\psi \left(\cdot ,u\right)$ in $B$ (in the sense of distributions), $u>0$, ${{\lim }_{\left|x\right|\to 1}u\left(x\right)/\left(1-\left|x\right|\right)}^{m-1}=0$. Our approach is based on estimates for the polyharmonic Green function on $B$ with zero Dirichlet boundary conditions.

We consider eigenvalues of elliptic boundary value problems, written in variational form, when the leading coefficients are perturbed by terms which are small in some integral sense. We obtain asymptotic formulae. The main specific of these formulae is that the leading term is different from that in the corresponding formulae when the perturbation is small in $L^{\infty }$-norm.

Let $X$ be a complete CAT$\left(0\right)$ space with the geodesic extension property and Alexandrov curvature bounded below. It is shown that if $C$ is a closed subset of $X$, then the set of points of $X$ which have a unique nearest point in $C$ is $G_{\delta }$ and of the second Baire category in $X$. If, in addition, $C$ is bounded, then the set of points of $X$ which have a unique farthest point in $C$ is dense in $X$. A proximity result for set-valued mappings is also included.

We consider a reaction-diffusion system modeling chemotaxis, which describes the situation of two species of bacteria competing for the same nutrient. We use Moser-Alikakos iteration to prove the global existence of the solution. We also study the existence of nontrivial steady state solutions and their stability.