an equilibrium existence theorem bewley 14 theorem l proved an infinite dimensional equilibrium existence theorem which is a significant extension of the classical finite dimensional theorem of arrow and debreu 11 the assumptions on technology and preferences are natural and applicable in a wide variety of cases the proof is based on a limit argument that makes direct use of the existence of equilibrium in the finite dimensional case this paper establishes the existence of equilibrium under assumptions which are essentially the same as those given by bewley with the additional assumption that the preference orderings of consumers are representable by real valued utility functions this approach is related to the welfare approach of negishi 9 and arrow and hahn 2 chap 51 in the finite dimensional case and simplifies the approach originally adopted by bewley 5 in addition to making clear the role played by each of the assumptions in establishing the existence of equilibrium this approach has the merit of constructing directly a certain real valued function that is maximised at an equilibrium a result that provides a powerful tool in the analysis of qualitative properties of an equilibrium a model of resource allocation in continuous time over an infinite horizon that may be viewed as an application of the model that follows is given in 181

the thermodynamic limit on bethe lattices the author emphasises the fact that to obtain the exact solution of hamiltonian models on bethe lattices one can apply explicitly the formal method of rigorous statistical mechanics i e the thermodynamic limit of probability measures this approach solves the well known dichotomy between clashing alternative solutions and provides a very simple analytic solution for a large class of hamiltonian models

equilibria in exchange economies with a countable number of agents abstract the existence of equilibria is established in an overlapping generations exchange economy where each generation lives for two periods and the commodity space is the positive cone of an infinite dimensional riesz space in particular we establish the existence of equilibria in the stochastic overlapping generations model i e we establish the existence of equilibria when the commodity space in each period is l equipped with the mackey topology l l 1

some ab percolation problems in the antiferromagnetic potts model abstract the ab correlated site random bond percolation problem in a q state antiferromagnetic potts model on bethe lattices is solved we find the analytic expression of the ab percolation characteristic functions in terms of the temperature the external field and the active bond concentration p b the ab threshold and the phase boundary of the system coincide at zero temperature and at most in two other points for every constant p b 1 the properties of the bethe lattice allow us to find the temperature dependent p b which defines the ab droplets i e those special ab clusters which diverge with thermal exponents along the phase boundary

phase diagrams of the q state potts model on bethe lattices abstract we use a rigorous recently developed method of solution of spin models on bethe lattices cayley trees it allows us to detect all the phases of the ferromagnetic and antiferromagnetic q state potts models in a field fpm and apm respectively the phase diagrams of the fpm are characterized by three critical lines of first order transitions in the apm the ordered phases which appear always for h 0 and under certain conditions for h 0 and or h

random walk on a disordered directed bethe lattice the random walk of a particle on a directed bethe lattice of constant coordinancez is examined in the case of random hopping rates as a result the higher the coordinance the narrower the regions of anomalous drift and diffusion the annealed and quenched mean square dispersions are calculated in all dynamical phases in opposition to the one dimensional z 2 case the annealed and quenched mean quadratic dispersions are shown to be identical in all phases

on the oscillatory behavior of solutions of nonlinear fractional differential equations the study of oscillation theory for fractional differential equations has been initiated by grace et al 14 in this paper we establish some new criteria for the oscillation of fractional differential equations with the caputo derivative of the form c d a x t e t f t x t a 1 1 2 we also present the conditions under which all solutions of this equation are asymptotic to at b as t for some real numbers a b we shall employ a different technique rather than that in 14

a new approach to an analysis of henry type integral inequalities and their bihari type versions abstract in this paper we propose a new method to solve integral inequalities of henry gronwall type and their bihari nonlinear version nonlinear integral inequalities with weakly singular kernels and with multiple integrals as well as a modification of the ou iang pachpatte inequality are also treated

on solution of nonlinear abel volterra integral equation abstract this paper is devoted to studying the nonlinear abel volterra integral equation of the form m x a x x 0 t x t 1 dt f x 0 x d with 0 andm r m 0 1 2 the uniqueness of the solution is proved and the asymptotic behavior of x asx 0 is obtained provided thata x andf x have special power asymptotics near zero the solution x in closed form is given in some cases

qualitative analysis for solutions of a certain more generalized two dimensional fractional differential system with hadamard derivative based on an elementary inequality and two gronwall bellman type integral inequality systems this paper generalizes the results for the constructions of explicit bounds and the qualitative properties for the solutions of certain fractional systems with hadamard derivative established in a recent paper by the authors the major generalizations come from linearization method and de singular approach after overcoming some difficulties in the hadamard type singular kernel

probability measures and hamiltonian models on bethe lattices i properties and construction of mrt probability measures the properties of one step markov rotationally and m step m 1 or 2 translationally invariant mrt probability measures on q state site qss bethe lattices are studied a theorem is proven which completely defines such measures in terms of m q2 q fundamental probabilities these are explicitly calculated for any mrt qss hamiltonian model as a consequence of our approach the dychotomy between alternative solutions of hamiltonian models on bethe lattices is solved

phase transitions for quantum markov chains associated with ising type models on a cayley tree the main aim of the present paper is to prove the existence of a phase transition in quantum markov chain qmc scheme for the ising type models on a cayley tree note that this kind of models do not have one dimensional analogous i e the considered model persists only on trees in this paper we provide a more general construction of forward qmc in that construction a qmc is defined as a weak limit of finite volume states with boundary conditions i e qmc depends on the boundary conditions our main result states the existence of a phase transition for the ising model with competing interactions on a cayley tree of order two by the phase transition we mean the existence of two distinct qmc which are not quasi equivalent and their supports do not overlap we also study some algebraic property of the disordered phase of the model which is a new phenomena even in a classical setting

long time behavior for a nonlinear fractional model the asymptotic behavior for solutions of a weighted cauchy type nonlinear fractional problem is investigated we find bounds for solutions on infinite time intervals and also provide sufficient conditions assuring decay to zero this work improves earlier results by the same authors

topological vector spaces admissible in economic equilibrium theory in models of economic equilibrium in markets with infinitely many commodities the commodity space is an ordered topological vector space endowed with additional structure in the present paper we consider ordered topological vector spaces which are admissible for equilibrium analysis in the sense that every economy which is reasonably well behaved posesses an equilibrium it turns out that this condition may be characterized in terms of topology and order this characterization implies that the commodity space has the structure of a kakutani space

first order transitions in percolation models abstract we describe the analytic solution of a potts correlated site random bond percolation model which exhibits a first order percolative transition at the potts critical point the subject is discussed in connection with the droplet theory in lattice spin models

on some qualitative properties for solutions of a certain two dimensional fractional differential systems abstract by means of the modification of medve s de singular method and a result of two dimensional linear integral inequalities components wise not on some norm upper bounds are obtained for solutions of a class of nonlinear two dimensional systems of fractional differential equations the uniqueness and continuous dependence of the solutions are also discussed here

equilibria with discontinuous preferences new fixed point theorems abstract we prove new equilibrium existence results for games and economies with discontinuous and non ordered preferences to do so we introduce the notion of continuous inclusion property and prove new fixed point theorems which extend and generalize the results of fan 17 glicksberg 20 browder 7 and gale and mas colell 19 our results also extend the previous work of yannelis 47 and wu and shen 46

on a market equilibrium theorem with an infinite number of commodities abstract a direct proof is given of the market equilibrium theorem of gale nikaido and debreu for an infinite dimensional commodity space the theorem is closely related to a recent result of aliprantis and brown but allows for excess demand correspondences rather than excess demand functions

power type estimates for a nonlinear fractional differential equation abstract in this work we are concerned with a family of nonlinear ordinary differential equations of fractional order it is proved that solutions of these equations with weighted initial data exist globally and decay as a power function

on the asymptotic integration of a class of sublinear fractional differential equations we estimate the growth in time of the solutions to a class of nonlinear fractional differential equations d0 x x0 f t x which includes d0 x x0 h t x with 0 1 for the case of slowly decaying coefficients h the proof is based on the triple interpolation inequality on the real line and the growth estimate reads as x t o ta when t for 1 1 a 0 our result can be thought of as a noninteger counterpart of the classical bihari asymptotic integration result for nonlinear ordinary differential equations by a carefully designed example we show that in some circumstances such an estimate is optimal

clusters and droplets in the q state potts model a potts correlated polychromatic percolation is studied the clusters are made of sites corresponding to a given value of the q state potts variables connected by bonds being active with probability pb to treat this problem an s state potts hamiltonian diluted with q state potts variables instead of lattice gas variables is introduced to which the the migdal kadanoff renormalisation group is applied it is found for a particular choice of pb 1 e k where k is the potts coupling constant divided by the boltzmann factor that these clusters called droplets diverge at the potts critical point with potts exponents

probability measures and hamiltonian models on bethe lattices ii the solution of thermal and configurational problems in a previous paper we introduced a method for the construction of rotationally and translationally invariant probability measures generated by one step markov hamiltonian models on q state site bethe lattices here the corresponding thermal problems are solved by finding the relative free energy which gives complete information on the properties of the models under study configurational problems also can be solved with the present tools as an example the solution of polychromatic correlated site random bond percolation models is found