Random nonlinear Hammerstein equations

We prove existence theorems for random differential equations defined in a separable reflexive Banach space. These theorems are proved through the use of theory of random analysis established in [X. Z. Yuan, Random nonlinear mappings of monotone type, J. Math. Anal. Appl. 19] which differs from the other means, for example in [R. Kannan and H. Salehi, Random nonlinear equations and monotonic nonlinearities, J. Math. Anal. Appl. 57 (1977), 234–256; D. Kravvaritis, Existence theorems for nonlinear random equations and inequalities, J. Math. Anal. Appl. 86 (1982), 61–73; D. A. Kandilakis and N. S. Papageorgious, On the existence of solutions for random differential inclusions in a Banach space, J. Math. Anal. Appl. 126 (1987), 11–23].     

On the existence of solutions of differential inclusions

We consider differential inclusions corresponding to accretive operators in a Banach space X for the case in which X is the one-dimensional Euclidean space. We prove existence theorems and an asymptotic stability theorem. We also introduce the notion of a generalizednonincreasing (nondecreasing) multivalued function and establish a relationship between nondecreasing multivalued functions and accretive operators in the one-dimensional Euclidean space.     

Fixed points for mixed monotone multivalued operators in Banach spaces with applications

In this paper, the existence and iterative approximation of fixed points for a class of systems of mixed monotone multivalued operator are discussed. We present some new fixed point theorems of mixed monotone operators and increasing operators which need not be continuous or satisfy a compactness condition. We also give some applications to differential inclusions with discontinuous right hand side in Banach spaces and to Hammerstein integral inclusions on RN.     

Existence and relaxation of solutions to differential inclusions with unbounded right-hand side in a Banach space

In a separable Banach space we consider a differential inclusion whose values are nonconvex, closed, but not necessarily bounded sets. Along with the original inclusion, we consider the inclusion with convexified right-hand side. We prove existence theorems and establish relations between solutions to the original and convexified differential inclusions. In contrast to assuming that the right-hand side of the inclusion is Lipschitz with respect to the phase variable in the Hausdorff metric, which is traditional in studying this type of questions, we use the (ρ–H) Lipschitz property. Some example is given.     

二阶微分包含的边值问题及其应用

在Banach中讨论了一类二阶微分包含的边值问题和一类反馈控制问题.结合不动点定理,对解的存在性给出了两个充分性条件,最后应用前面的结果建立了反馈控制问题解的存在性定理.     

A variant of Newton's method and its application

This paper details an existence and uniqueness theorem for solving an operator equation of the form F(x)=0, where F is a Gateaux differentiable operator defined on an open convex subset of a Banach space proved. From the main theorem, an earlier theorem of Argyros follows as a consequence. Other corollaries constitute the semilocal versions of the theorems due to Ozban and Weerakoon and Fernando in a general Banach space. Our main theorem leads to the existence of solutions for a class of nonlinear Urysohn-type integral equations in the n-dimensional Euclidean space.     

Dual variational methods in critical point theory and applications

This paper contains some general existence theorems for critical points of a continuously differentiable functional I on a real Banach space. The strongest results are for the case in which I is even. Applications are given to partial differential and integral equations.     

Banach空间中的两点边值问题*

本文在可分Banach空间中研究带线性分离边值条件的二阶微分包含,在较弱的条件下证明了解的存在性,由此可以得到二阶微分方程的边值问题若干解的存在性定理的推广形式.     

Iterative approximation of a solution of multi-valued variational-like inclusion in Banach spaces: A P-η-proximal-point mapping approach

In this paper, we introduce a class of P-η-accretive mappings, an extension of η-m-accretive mappings [C.E. Chidume, K.R. Kazmi, H. Zegeye, Iterative approximation of a solution of a general variational-like inclusion in Banach spaces, Int. J. Math. Math. Sci. 22 (2004) 1159-1168] and P-accretive mappings [Y.-P. Fang, N.-J. Huang, H-accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces, Appl. Math. Lett. 17 (2004) 647-653], in real Banach spaces. We prove some properties of P-η-accretive mappings and give the notion of proximal-point mapping, termed as P-η-proximal-point mapping, associated with P-η-accretive mapping. Further, using P-η-proximal-point mapping technique, we prove the existence of solution and discuss the convergence analysis of iterative algorithm, for multi-valued variational-like inclusions in real Banach space. The theorems presented in this paper extend and improve many known results in the literature.     

Differential inclusions with measurable-pseudo-Lipschitz right-hand side

We obtain existence theorems and Filippov-Wa?ewski type relaxation theorems for differential inclusions in Banach spaces with measurable-pseudo-Lipschitz right-hand side. For the solution sets of these differential inclusions, we also describe some properties that extend classical theorems on continuous dependence and on differentiation of solutions with respect to initial data.     

Controllability Results for Semilinear Evolution Inclusions with Nonlocal Conditions

In this paper, we prove controllabillity results for mild solutions defined on a compact real interval for first-order semilinear differential and integrodifferential evolution inclusions in Banach spaces with nonlocal conditions. By using suitable fixed-point theorems, we study the cases when the multivalued map has convex values as well as nonconvex values.     

Real and complex interpolation methods for finite and infinite families of Banach spaces

A comparative study is made of the various interpolation spaces generated with respect to n-tuples or infinite families of compatible Banach spaces by real and complex interpolation methods due to Sparr, Favini-Lions, Coifman-Cwikel-Rochberg-Sagher-Weiss, and Fernandez. Certain inclusions are established between these spaces and examples are given showing that in general they do not coincide. It is also shown that, in contrast to the case of couples of spaces, the spaces generated by the above methods may depend on the structure of the containing space in which the Banach spaces of the n-tuple (nϵ 3) or infinite family are embedded. Finally a construction is given which enables the spaces of Sparr and Favini-Lions, hitherto defined only with respect to n-tuples, to also be defined with respect to infinite families of Banach spaces.     

Banach空间中随机微分包含的弱解存在性定理

本文对Banach空间中的微分包含及随机微分包含引入了弱解的概念,并给出了它们的存在性定理。     

Divergence theorems in path space

We obtain divergence theorems on the solution space of an elliptic stochastic differential equation defined on a smooth compact finite-dimensional manifold M. The resulting divergences are expressed in terms of the Ricci curvature of M with respect to a natural metric on M induced by the stochastic differential equation. The proofs of the main theorems are based on the lifting method of Malliavin together with a fundamental idea of Driver.     

Einige Existenz- und Einschließungssätze für Fixpunkte expandierender Integraloperatoren

Some existence theorems for fixed points of nonlinear integral operators expanding a cone in a Banach space in the sense of Krasnoselskii [5] are given. Using special cones we find pointwise inclusions for the fixed points. The question of finding the best bounds for the fixed points leads to a nonlinear optimization problem with infinitely many restrictions.

     

Existence Results for Functional Differential Inclusions with Infinite Delay

The aim of the present paper is to investigate the existence of solutions to functional differential inclusions with infinite delay in Banach spaces. A relevant set of phase space axioms is proposed. The main tools used in this paper are certain fixed point theorems based on the setcontraction theory.     

Weyl-Type Theorems on Banach Spaces Under Compact Perturbations

In this paper, we study Browder-type and Weyl-type theorems for operators \(T+K\) defined on a Banach space X, where K is (a non necessarily commuting) compact operator on X. In the last part, the theory is exemplified in the case of isometries, analytic Toeplitz operators, semi-shift operators, and weighted right shifts.     

On simultaneous triangularization of collections of compact operators

In this paper we consider collections of compact (resp. C_p class) operators on arbitrary Banach (resp. Hilbert) spaces. For a subring R of reals, it is proved that an R-algebra of compact operators with spectra in R on an arbitrary Banach space is triangularizable if and only if every member of the algebra is triangularizable. It is proved that every triangularizability result on certain collections, e.g., semigroups, of compact operators on a complex Banach (resp. Hilbert) space gives rise to its counterpart on a real Banach (resp. Hilbert) space. We use our main results to present new proofs as well as extensions of certain classical theorems (e.g., those due to Kolchin, McCoy, and others) on arbitrary Banach (resp. Hilbert) spaces.     

Existence and convergence theorems for generalized hybrid mappings in uniformly convex metric spaces

In this paper, we show a relationship between strictly convexity of type (I) and (II) defined by Takahashi and Talman, and we prove that any uniformly convex metric space is strictly convex of type (II). Continuity of the convex structure is also shown on a compact domain. Then, we prove the existence of a minimum point of a convex, lower semicontinuous and d-coercive function defined on a nonempty closed convex subset of a complete uniformly convex metric space. By using this property, we prove fixed point theorems for (α, β)-generalized hybrid mappings in uniformly convex metric spaces. Using this result, we also obtain a common fixed point theorem for a countable commutative family of (α, β)-generalized hybrid mappings in uniformly convex metric spaces. Finally, we establish strong convergence of a Mann type iteration to a fixed point of (α, β)-generalized hybrid mapping in a uniformly convex metric space without assuming continuity of convex structure. Our results can be applied to obtain the existence and convergence theorems for (α, β)-generalized hybrid mappings in Hilbert spaces, uniformly convex Banach spaces and CAT(0) spaces.     

Dilation theorems for positive definite operator kernels having majorants

Dilation theorems for Banach space valued stochastic processes and operator valued positive definite kernels are considered. It is shown, e.g., that a Banach space valued stochastic process X can be dilated to another process Y, if and only if the covariance kernel of Y is a majorant of the covariance kernel of X. Positive definite operator kernels having majorants of certain special type are characterized.