Nonlinear random equations involving operators of monotone type

Let X be a real Banach space, X* its dual space and ω a measurable space. Let D be a subset of X, L: Ω × D → X* a random operator and η:Ω →X* a measurable mapping. The random equation corresponding to the double [L, η] asks for a measurable mapping ξ: Ω → D such thatRandom equations with operators of monotone type have been studied recentely by Kannan and Salehi [7], Itoh [6] and Kravvarits [8].

In this paper we give some new existence theorems for nonlinear random equations and inequalities involving operators of monotone type in Banach spaces. A random Hammerstein integral equation is also studied. In order to obtain random solutions we use some results from the existing deterministic theory as well as from the theory of measurable multifunctions and, in particular, the measurable selection theorems of Kuratowski/Ryll‐Nardzewski and of Saint‐Beuve.

Periodic solutions for nonlinear differential equations with maximal monotone terms

Existence and stability of stationary solutions of nonlinear random difference equations are studied in this note. Firstly, we give the weak conditions that guarantee the continuity of Lypanunov exponents under small random perturbations. Secondly, we find out the conditions under which the ratio of the random norm and the standard Euclidean norm has deterministic bounds. Based on these new results, we provide easy-to-use conditions that guarantee the existence and almost sure stability of a stationary solution. In addition, we also prove that the stationary solution converges with probability one to the fixed point of the corresponding deterministic system as the noise intensity tends to zero.

The present paper is a continuation of [1]. By using more general type of random contractor than that defined by Lee and Padgett, we show several existence and uniqueness theorems for solutions to a system of nonlinear random operator equations with stochastic domain. Next, we obtain a random coincidence theorem and a random implicit function theorem, which improves and generalizes the corresponding results of Ding, Reddy and Subrahmanyam, Altman, Eng, Krasnoselskii and others. Finally, some applications of our results are given for the solutions of a system of nonlinear random integral equations.

Existence theorems for nonlinear noncoercive operator equations and nonlinear elliptic boundary value problems

Let T(w,x) = y( w ) be a nonlinear random operator equation with not necessarily unique solution. For this and similar equations, we prove results about convergence of solutions of suitable approximate problems Tn(w,x) =Yn(w) to solutions of the original equations. We do this for rather general notions of convergence for random variables. Concepts like consistency, stability, and compactness in sets of measurable functions are introduced and used. For all assumptions that are needed in the general theory, sufficient conditions are given with respect to convergence in probability and almost-sure convergence. As a specific method for constructing approximate equations we discuss "discretization schemes ", where the underlying probability space is discretized. Some results might be of interest also in different contexts; these include criteria for almost-sure convergence of measurable multifunctions and results about compactness with respect to convergence in probability and almost-sure convergence.

In this paper we study the numerical homogenization of nonlinear random parabolic equations. This procedure is developed within a finite element framework. A careful choice of multiscale finite element bases and the global formulation of the problem on the coarse grid allow us to prove the convergence of the numerical method to the homogenized solution of the equation. The relation of the proposed numerical homogenization procedure to multiscale finite element methods is discussed. Within our numerical procedure one is able to approximate the gradients of the solutions. To show this feature of our method we develop numerical correctors that contain two scales, the numerical and the physical. Finally, we would like to note that our numerical homogenization procedure can be used for the general type of heterogeneities.

Solution of nonlinear partial differential equations from base equations

In this paper we study nonlinear parabolic equations using the method of upper and lower solutions. Using truncation and penalization techniques and results from the theory of operators of monotone type, we prove the existence of a periodic solution between an upper and a lower solution. Then with some monotonicity conditions we prove the existence of extremal solutions in the order interval defined by an upper and a lower solution. Finally we consider problems with discontinuities and we show that their solution set is a compact Rδ-set in (CT, L2(Z)).

The aim of the present paper is to study a random equation of the general form x(t, ω)=(Ux)(t, ω), t∈R+ and its special case a nonlinear random functional integral equation given by $$x\left( {t,w} \right) = F\left( {t,\int\limits_0^{g_1 \left( t \right)} {f_1 } \left( {t,s,x\left( {s,w} \right),w} \right)ds, \ldots ,\int\limits_0^{g_m \left( t \right)} {f_m } \left( {t,s,x\left( {s,w} \right),w} \right)ds,x\left( {h_1 \left( t \right),w} \right), \ldots ,x\left( {h_p \left( t \right),\omega } \right),\omega } \right).$$ The existence and uniqueness of a random solution, a second-order stochastic process, of the equations is considered.

PurposeThis paper aims to solve linear and non-linear shallow water wave equations using homotopy perturbation method (HPM). HPM is a straightforward method to handle linear and non-linear differential equations. As such here, one-dimensional shallow water wave equations have been considered to solve those by HPM. Interesting results are reported when the solutions of linear and non-linear equations are compared.Design/methodology/approachHPM was used in this study.FindingsSolution of one-dimensional linear and non-linear shallow water wave equations and comparison of linear and non-linear coupled shallow water waves from the results obtained using present method.Originality/valueCoupled non-linear shallow water wave equations are solved.

A method of deletion of dimensions for a mathematical model which gives the number of variables less than that prescribed by the π-theorem was proposed. In a number of cases, it is possible to exclude from consideration all similarity criteria or, in other words, achieve self-similarity in them. In the framework of deletion of dimensions, this is expressed in a transition from criteria to similarity numbers. Thus, information is reduced without its loss. The limit reduction of the number of variables in the mathematical model makes it possible to use analytical approximation dependences as approximate solutions. Such dependences are obtained from the initial solutions by using the coefficients of stretch, which corresponds to the group methods for solving equations. It was proposed to use solutions of the linearized forms of the original nonlinear equations as initial solutions. Such approach makes it possible to take into account physical character of the change in the studied quantities in the approximation dependences when solution of nonlinear equations cannot be realized using standard functions. Efficiency of the method was illustrated by the example of study of the pendulum motion, which is a counterpart of inertial link of the second order in the theory of automatic control. Solutions were obtained for the cases of presence and absence of environmental resistance. The last variant is interesting by the feasibility of comparing the proposed and available analytical solutions in terms of elliptic integrals.

Existence results for time scale boundary value problem

Nonlinear random equations involving operators of type ( M)