Iterative approximation to common fixed points of a countable family of nonexpansive mappings

We proved several strong convergence results by using the conception of a uniformly asymptotically regular sequence {T _n } of nonexpansive mappings in a reflexive Banach space which admits a weakly continuous duality mapping J _?(l ^p (1?p?t)?=?t ^p?1. The results presented develop and complement the corresponding ones by Song, Y. and Chen, R., 2007 [Iterative approximation to common fixed points of nonexpansive mapping sequences in reflexive Banach spaces. Nonlinear Analysis, 66, 591–603], Song, Y., Chen, R. and Zhou, H., 2007 [Viscosity approximation methods for nonexpansive mapping sequences in Banach spaces. Nonlinear Analysis, 66, 1016–1024] and O'Hara, J.G., Pillay, P. and Xu, H.K., 2006 [Iterative approaches to convex feasibility problem in Banach Space. Nonlinear Analysis, 64, 2022–2042], O'Hara, J.G., Pillay, P. and Xu, H.K., 2003 [Iterative approaches to fineding nearest common fixed point of nonexpansive mappings in Hilbert spaces. Nonlinear Analysis, 54, 1417–1426] and Jung, J.S., 2005 [Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications, 302, 509–520] and many other existing literatures.     

Blow-up for the Timoshenko-type equation with variable exponents

The finite time blow-up of solutions to a nonlinear Timoshenko-type equation with variable exponents is studied. More concretely, we prove that the solutions blow up in finite time with positive initial energy. Also, the existence of finite time blow-up solutions with arbitrarily high initial energy is established. Meanwhile, the upper and lower bounds of the blow-up time are derived. These results deepen and generalize the ones obtained in [Nonlinear Anal. Real World Appl., 61: Paper No. 103341, 2021].     

Blow-up solutions of quadratic differential systems

The main aim of this paper is to investigate the existence problem for blow-up solutions of quadratic differential systems, Riccati equations, and Lotka-Volterra systems. For this purpose, we introduce the concept of negative and positive blow-up times of solutions of the above-mentioned systems and provide upper or lower bounds for these times. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 15, Differential and Functional Differential Equations. Part 1, 2006.     

Global minimization algorithms for concave quadratic programming problems

Abstract: In this article, we consider the concave quadratic programming problem which is known to be NP hard. Based on the improved global optimality conditions by [Dür, M., Horst, R. and Locatelli, M., 1998, Necessary and sufficient global optimality conditions for convex maximization revisited, Journal of Mathematical Analysis and Applications, 217, 637–649] and [Hiriart-Urruty, J.B. and Ledyav, J.S., 1996, A note in the characterization of the global maxima of a convex function over a convex set, Journal of Convex Analysis, 3, 55–61], we develop a new approach for solving concave quadratic programming problems. The main idea of the algorithms is to generate a sequence of local minimizers either ending at a global optimal solution or at an approximate global optimal solution within a finite number of iterations. At each iteration of the algorithms we solve a number of linear programming problems with the same constraints of the original problem. We also present the convergence properties of the proposed algorithms under some conditions. The efficiency of the algorithms has been demonstrated with some numerical examples.     

Sur l'existence de solutions d'équations différentielles stochastiques progréssives rétrogrades couplées

Nonlinear BSDEs were first introduced by Pardoux and Peng, 1990, Adapted solutions of backward stochastic differential equations, Systems and Control Letters, 14, 51–61, who proved the existence and uniqueness of a solution under suitable assumptions on the coefficient. Fully coupled forward–backward stochastic differential equations and their connection with PDE have been studied intensively by Pardoux and Tang, 1999, Forward–backward stochastic differential equations and quasilinear parabolic PDE's, Probability Theory and Related Fields, 114, 123–150; Antonelli and Hamadène, 2006, Existence of the solutions of backward–forward SDE's with continuous monotone coefficients, Statistics and Probability Letters, 76, 1559–1569; Hamadème, 1998, Backward–forward SDE's and stochastic differential games, Stochastic Processes and their Applications, 77, 1–15; Delarue, 2002, On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case, Stochastic Processes and Their Applications, 99, 209–286, amongst others.

Unfortunately, most existence or uniqueness results on solutions of forward–backward stochastic differential equations need regularity assumptions. The coefficients are required to be at least continuous which is somehow too strong in some applications. To the best of our knowledge, our work is the first to prove existence of a solution of a forward–backward stochastic differential equation with discontinuous coefficients and degenerate diffusion coefficient where, moreover, the terminal condition is not necessary bounded.

The aim of this work is to find a solution of a certain class of forward–backward stochastic differential equations on an arbitrary finite time interval. To do so, we assume some appropriate monotonicity condition on the generator and drift coefficients of the equation.

The present paper is motivated by the attempt to remove the classical condition on continuity of coefficients, without any assumption as to the non-degeneracy of the diffusion coefficient in the forward equation.

The main idea behind this work is the approximating lemma for increasing coefficients and the comparison theorem. Our approach is inspired by recent work of Boufoussi and Ouknine, 2003, On a SDE driven by a fractional brownian motion and with monotone drift, Electronic Communications in Probability, 8, 122–134; combined with that of Antonelli and Hamadène, 2006, Existence of the solutions of backward–forward SDE's with continuous monotone coefficients, Statistics and Probability Letters, 76, 1559–1569. Pursuing this idea, we adopt a one-dimensional framework for the forward and backward equations and we assume a monotonicity property both for the drift and for the generator coefficient.

At the end of the paper we give some extensions of our result.     

Localization on the Boundary of Blow-up for Reaction–Diffusion Equations with Nonlinear Boundary Conditions

Abstract

In this work we analyze the existence of solutions that blow-up in finite time for a reaction–diffusion equation u _t  ? Δu = f(x, u) in a smooth domain Ω with nonlinear boundary conditions ?u/?n = g(x, u). We show that, if locally around some point of the boundary, we have f(x, u) = ?βu ^p , β ≥ 0, and g(x, u) = u ^q then, blow-up in finite time occurs if 2q > p + 1 or if 2q = p + 1 and β < q. Moreover, if we denote by T _b the blow-up time, we show that a proper continuation of the blowing up solutions are pinned to the value infinity for some time interval [T, τ] with T _b  ≤ T < τ. On the other hand, for the case f(x, u) = ?βu ^p , for all x and u, with β > 0 and p > 1, we show that blow-up occurs only on the boundary.     

Some blow-up problems for a semilinear parabolic equation with a potential

The blow-up rate estimate for the solution to a semilinear parabolic equation ut=Δu+V(x)|u|^p−1u in Ω×(0,T) with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data u(x,0)=Mφ(x) as M goes to infinity, which have been found in [C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, preprint, arXiv: math.AP/0607055, July 2006], is improved under some reasonable and weaker conditions compared with [C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, preprint, arXiv: math.AP/0607055, July 2006].     

Interior hölder estimates for solutions of schrödinger equations and the regularity of nodal sets

Abstract

Motivated by the study of selfdual vortices in gauge field theory, we consider a class of Mean Field equations of Liouville-type on compact surfaces involving singular data assigned by Dirac measures supported at finitely many points (the so called vortex points). According to the applications, we need to describe the blow-up behavior of solution-sequences which concentrate exactly at the given vortex points. We provide accurate pointwise estimates for the profile of the bubbling sequences as well as “sup + inf” estimates for solutions. Those results extend previous work of Li [Li, Y. Y. (1999). Harnack type inequality: The method of moving planes. Comm. Math. Phys. 200:421–444] and Brezis et al. [Brezis, H., Li, Y. Shafrir, I. (1993). A sup + inf inequality for some nonlinear elliptic equations involving the exponential nonlinearities. J. Funct. Anal. 115: 344–358] relative to the “regular” case, namely in absence of singular sources.     

On inexact generalized proximal methods with a weakened error tolerance criterion

Two inexact versions of a Bregman-function-based proximal method for finding a zero of a maximal monotone operator, suggested in [J. Eckstein (1998). Approximate iterations in Bregman-function-based proximal algorithms. Math. Programming, 83, 113–123; P. da Silva, J. Eckstein and C. Humes (2001). Rescaling and stepsize selection in proximal methods using separable generalized distances. SIAM J. Optim., 12, 238–261], are considered. For a wide class of Bregman functions, including the standard entropy kernel and all strongly convex Bregman functions, convergence of these methods is proved under an essentially weaker accuracy condition on the iterates than in the original papers.

Also the error criterion of a logarithmic–quadratic proximal method, developed in [A. Auslender, M. Teboulle and S. Ben-Tiba (1999). A logarithmic-quadratic proximal method for variational inequalities. Computational Optimization and Applications, 12, 31–40], is relaxed, and convergence results for the inexact version of the proximal method with entropy-like distance functions are described.

For the methods mentioned, like in [R.T. Rockafellar (1976). Monotone operators and the proximal point algorithm. SIAM J. Control Optim., 14, 877–898] for the classical proximal point algorithm, only summability of the sequence of error vector norms is required.     

Blow-up of smooth solutions to the compressible fluid models of Korteweg type

We show the blow-up of smooth solutions to a non-isothermal model of capillary compressible fluids in arbitrary space dimensions with initial density of compact support. This is an extension of Xin’s result [Xin, Z.: Blow-up of smooth solutions to the compressible Navier-Stokes equations with compact density. Comm. Pure Appl. Math., 51, 229–240 (1998)] to the capillary case but we do not need the condition that the entropy is bounded below. Moreover, from the proof of Theorem 1.2, we also obtain the exact relationship between the size of support of the initial density and the life span of the solutions. We also present a sufficient condition on the blow-up of smooth solutions to the compressible fluid models of Korteweg type when the initial density is positive but has a decay at infinity.     

Boundary blow-up solutions for a class of quasilinear elliptic equations

For a given bounded domain Ω in R ^N with smooth boundary ?Ω, we give sufficient conditions on f so that the m-Laplacian equation △ _m u = f(x, u, ?u) admits a boundary blow-up solution u ∈ W ^1,p (Ω). Our main results are new and extend the results in J.V. Concalves and Angelo Roncalli [Boundary blow-up solutions for a class of elliptic equations on a bounded domain, Appl. Math. Comput. 182 (2006), pp. 13–23]. Our approach employs the method of lower–upper solution theorem, fixed point theory and weak comparison principle.     

The blow-up rate for a system of heat equations with neumann boundary conditions

This paper deals with the blow-up properties of solutions to a system of heat equations u _t=Δu, v _t=Δv in B _R×(0, T) with the Neumann boundary conditions εu/εη=e ^v, εv/εη=e ^u on S _R×[0, T). The exact blow-up rates are established. It is also proved that the blow-up will occur only on the boundary. This work is supported by the National Natural Science Foundation of China     

Periodic solutions for predator–prey systems with Beddington–DeAngelis functional response on time scales

This paper deals with the question of existence of periodic solutions of nonautonomous predator–prey dynamical systems with Beddington–DeAngelis functional response. We explore the periodicity of this system on time scales. New sufficient conditions are derived for the existence of periodic solutions. These conditions extend previous results presented in [M. Bohner, M. Fan, J. Zhang, Existence of periodic solutions in predator–prey and competition dynamic systems, Nonlinear. Anal.: Real World Appl. 7 (2006) 1193–1204; M. Fan, Y. Kuang, Dynamics of a nonautonomous predator–prey system with the Beddington–DeAngelies functional response, J. Math. Anal. Appl. 295 (2004) 15–39; J. Zhang, J. Wang, Periodic solutions for discrete predator–prey systems with the Beddington–DeAngelis functional response, Appl. Math. Lett. 19 (2006) 1361–1366].     

On the boundedness of some rational difference equations

We study the boundedness character of solutions of some third order rational difference equations and we confirm some of the conjectures posed in Camouzis et al. [“Progress report on the boundedness character of third-order rational equations”, Journal of Difference Equations and Applications 11 (2005), 1029–1035] and [“On third order rational difference equations, part 6”, Journal of Difference Equations and Applications 11 (2005), 759–777].     

Global existence and nonexistence of the non-Newtonian polytropic filtration equations coupled with nonlinear boundary conditions

In this article, we deal with the global existence and nonexistence of solutions to the non-Newtonian polytropic filtration equations coupled with nonlinear boundary conditions. By constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions for global existence of nonnegative solutions. The critical curve of Fujita type is conjectured with the aid of some new results, which extend the recent results of Zheng, Song, and Jiang [Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, J. Math. Anal. Appl. 298 (2004), pp. 308–324], Zhou and Mu [Critical curve for a non-Newtonian polytropic filtration system coupled via nonlinear boundary flux, Nonlinear Anal. 68 (2008), pp. 1–11], and Zhou and Mu [Algebraic criteria for global existence or blow-up for a boundary coupled system of nonlinear diffusion equations, Appl. Anal. 86 (2007), pp. 1185–1197] to more general equations.     

Equivalences between categories of modules and categories of comodules

We show the close connection between apparently different Galois theories for comodules introduced recently in [J. Gomez-Torrecillas and J. Vercruysse, Comatrix corings and Galois Comodules over firm rings, Algebr. Represent. Theory, 10 （2007）, 271 306] and [Wisbauer, On Galois comodules, Comm. Algebra 34 （2006）, 2683-2711]. Furthermore we study equivalences between categories of comodules over a coring and modules over a firm ring. We show that these equivalences are related to Galois theory for comodules.     

Blow-up properties for a degenerate parabolic system with nonlinear localized sources

In this paper, we investigate the initial-boundary problem of a degenerate parabolic system with nonlinear localized sources. We classify the blow-up solutions into global blow-up cases and single-point blow-up cases according to the values of m,n,p_i,q_i. Furthermore, we obtain the uniform blow-up profiles of solutions for the global blow-up case. Finally, we give some numerical examples to verify the results. These extend and generalize a recent work of one of the authors [L. Du, Blow-up for a degenerate reaction-diffusion systems with nonlinear localized sources, J. Math. Anal. Appl. 324 (2006) 304-320], which only considered uniform blow-up profiles under the special case p₁=p₂=0.     

Semicontinuity of the solution mappings to weak generalized parametric Ky Fan inequality problems with trifunctions

In this article, we discuss the lower semicontinuity of solution maps without the condition of C-strict monotonicity for two classes of weak generalized parametric Ky Fan inequalities under the case that the f-solution set be a general set-valued one. Our results extend the recent ones in the literature (e.g. Cheng, Y.H., Zhu, D.L.: Global stability results for the weak vector variational inequality. J. Glob. Optim. 32, 543–550 (2005); C.R. Chen and S.J. Li, On the solution continuity of parametric generalized systems, Pac. J. Optim. 6 (2010), pp. 141–151; Gong, X.H., Yao, J.C.: Lower semicontinuity of the set of efficient solutions for generalized systems. J. Optim. Theory Appl. 138, 197–205 (2008); Gong, X.H.: Continuity of the solution set to parametric weak vector equilibrium problems. J. Optim. Theory Appl. 139, 35–46 (2008)). Several examples are given for the illustration of our results.     

Blow-up rates of radially symmetric large solutions

This paper adapts a technical device going back to [J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations 224 (2006) 385-439] to ascertain the blow-up rate of the (unique) radially symmetric large solution given through the main theorem of [J. López-Gómez, Uniqueness of radially symmetric large solutions, Discrete Contin. Dyn. Syst., Supplement dedicated to the 6th AIMS Conference, Poitiers, France, 2007, pp. 677-686]. The requested underlying estimates are based upon the main theorem of [S. Cano-Casanova, J. López-Gómez, Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimensional problems on the half-line, J. Differential Equations 244 (2008) 3180-3203]. Precisely, we show that if Ω is a ball, or an annulus, f∈C[0,∞) is positive and non-decreasing, V∈C[0,∞)∩C²(0,∞) satisfies V(0)=0, V^′(u)>0, V^″(u)?0, for every u>0, and V(u)∼Hup−1 as u↑∞, for some H>0 and p>1, then, for each λ?0,

−Δu=λu−f(dist(x,∂Ω))V(u)u     

Global Regularity of Solutions to Systems of Reaction–Diffusion with Sub-Quadratic Growth in Any Dimension

This paper is devoted to the study of the regularity of solutions to some systems of reaction–diffusion equations, with reaction terms having a subquadratic growth. We show the global boundedness and regularity of solutions, without smallness assumptions, in any dimension N. The proof is based on blow-up techniques. The natural entropy of the system plays a crucial role in the analysis. It allows us to use of De Giorgi type methods introduced for elliptic regularity with rough coefficients. In spite these systems are entropy supercritical, it is possible to control the hypothetical blow-ups, in the critical scaling, via a very weak norm. Analogies with the Navier–Stokes equation are briefly discussed in the introduction.