Nonlinear random operator equations and inequalities in Banach spaces

Nonlinear random equations involving operators of monotone type

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.

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.

We study an initial value problem for a class of integro-differential equations of Volterra type in a real Banach space. Using method of upper and lower solutions and M¨ onch and Von Harten theorem, we obtain an existence theorem of coupled quasi-solutions, which is an extension of those established by Y. Chen and W. Zhuang in [1].

The solutions of initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces

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].

Nonlinear inequalities in Banach spaces

This paper studies fractional differential equations (FDEs) with mixed fractional derivatives. Existence, uniqueness, stability, and asymptotic results are derived.

Under certain conditions a comparison is made between solutions of a pair of initial value problems in a Banach space. This comparison includes and unifies several recent results on differential inequalities in Banach spaces.

A unique solution of initial value problems for first order impulsive integro-differential equations of mixed type in Banach spaces

Recurrence relations for a Newton-like method in Banach spaces

We consider an extension of the notion of well‐posedness by perturbations, introduced by Zolezzi (1995, 1996) for a minimization problem, to a class of generalized mixed variational inequalities in Banach spaces, which includes as a special case the class of mixed variational inequalities. We establish some metric characterizations of the well‐posedness by perturbations. On the other hand, it is also proven that, under suitable conditions, the well‐posedness by perturbations of a generalized mixed variational inequality is equivalent to the well‐posedness by perturbations of the corresponding inclusion problem and corresponding fixed point problem. Furthermore, we derive some conditions under which the well‐posedness by perturbations of a generalized mixed variational inequality is equivalent to the existence and uniqueness of its solution.

The BCQ and the Abadie CQ for infinite systems of convex inequalities in Banach spaces are characterized in terms of the upper semi–continuity of the convex cones generated by the subdifferentials of active convex functions. Some relationships with other constraint qualifications such as the CPLV and the Slate condition are also studied. Applications in best approximation theory are provided.

Motivated and inspired by Korpelevich′s and Noor′s extragradient methods, we suggest an extragradient method by using the sunny nonexpansive retraction which has strong convergence for solving the generalized variational inequalities in Banach spaces.

This paper aims to establish the Tikhonov regularization method for generalized mixed variational inequalities in Banach spaces. For this purpose, we firstly prove a very general existence result for generalized mixed variational inequalities, provided that the mapping involved has the so-called mixed variational inequality property and satisfies a rather weak coercivity condition. Finally, we establish the Tikhonov regularization method for generalized mixed variational inequalities. Our findings extended the results for the generalized variational inequality problem (for short, GVIP(F, K)) in \(R^n\) spaces (He in Abstr Appl Anal, 2012) to the generalized mixed variational inequality problem (for short, GMVIP\((F,\phi , K)\)) in reflexive Banach spaces. On the other hand, we generalized the corresponding results for the generalized mixed variational inequality problem (for short, GMVIP\((F,\phi ,K)\)) in \(R^n\) spaces (Fu and He in J Sichuan Norm Univ (Nat Sci) 37:12–17, 2014) to reflexive Banach spaces.