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In this paper we present some results on the question of existence of generalized or measure valued solutions for semilinear impulsive evolution equations on Banach spaces with the nonlinear parts being merely measurable and bounded on bounded sets. This is a far reaching generalization of the previous results of the author and others. It may be interesting to consider extension of the results of this paper to cover impulsive differential inclusions and their potential applications to control theory and uncertain dynamics.

A generalization of Mallat's classic multiresolution analysis (MRA), based on the theory of spectral pairs, was considered in two papers by Gabardo and Nashed. In this nonstandard setting, the translation set is no longer a subgroup or a translate of a subgroup of R, but is a spectrum associated with a one-dimensional spectral pair. In this paper, we continue the study based on this nonstandard setting and give the characterization for nonuniform wavelets associated with a nonuniform MRA. These characterizations are consistent with both the known necessary and sufficient conditions for the existence of nonuniform MRA wavelets and the known characterization for standard dyadic wavelets associated with an MRA.

The paper considers a generalization of the optimal redundancy problem. We consider any mixed series and parallel network consisting of N modules. The ith module has a reliability cost function of ri(di) where di is the investment in module i. The reliability investment functions are not required to be concave or continuous in the dynamic programming model. The objective is to maximize the system reliability subject to cost, weight, or volume constraints. The paper differs from others in that we permit modules in logical parallel to be of different designs, and investment in a module does not necessarily imply that redundant components will be used. Many previous models have been restricted to the parallel series situation which is a special case of this model. A dynamic programming model is presented for the problem. The application of the generalized decomposition operator is used to develop a set of recursive relations for any mixed series and parallel system of modules. An example and comments on generalizations are included.

In this paper, we use a probabilistic setting to introduce a double sequence (L〈k〉n) of linear polynomial operators which includes, as particular cases, the classical Bernstein operators, the Kantorovic operators, and the operators recently introduced by Cao. For these operators, we discuss several approximation properties. In particular, we deal with the convergence properties according to the way in which the different parameters vary, and the preservation of global smoothness and classes of functions determined by concave moduli of continuity. A remarkable feature of our approach is that if f is differentiable, the approximation properties of both L〈k〉nf and its derivatives can be discussed simultaneously. Throughout the paper, probabilistic methods play an important role.

In this paper, a generalization and improvement of Noor's theorem for generalized set-valued variational inclusion in real Banach spaces is studied. The results presented in this paper generalize, improve, and unify a number of recent results.

The concern of this paper is to introduce and study multivariate Baskakov–Kantorovich operators, which are non-tensor product generalization of the one variable. We establish a strong direct theorem of L p approximation by using a decomposition technique.

The sequence space l ( p ) was introduced and studied by Maddox [I.J. Maddox, Spaces of strongly summable sequences, Quart. J. Math. Oxford (2) 18 (1967) 345–355]. In the present paper, the sequence spaces l ( u , v ; p ) of non-absolute type which is derived by the generalized weighted mean is defined and proved that the spaces l ( u , v ; p ) and l ( p ) are linearly isomorphic. Besides this, the β- and γ-duals of the space l ( u , v ; p ) are computed and the basis of that space is constructed. Further, it is established that the sequence space l p ( u , v ) has AD property and given f-dual of the space l p ( u , v ) . Finally, the matrix mappings from the sequence spaces l ( u , v ; p ) to the sequence space μ and from the sequence space μ to the sequence space l ( u , v ; p ) are characterized.

In this paper analysis of the concept of associated homogeneous distributions (generalized functions) is given, and some problems related to these distributions are solved. It is proved that (in the one-dimensional case) there exist only associated homogeneous distributions of order k=1. We introduce a definition of quasi associated homogeneous distributions which is a natural generalization of the notion of associated homogeneous distributions and provide a mathematical description of all quasi associated homogeneous distributions and their Fourier transforms. It is proved that the class of quasi associated homogeneous distributions coincides with the class of distributions introduced in [I.M. Gel'fand, G.E. Shilov, Generalized Functions, vol. 1, Properties and Operations, Academic Press, New York, 1964, Chapter I, §4] as the class of associated homogeneous distributions. For the multidimensional case it is proved that f is a quasi associated homogeneous distribution if and only if it satisfies an Euler type system of differential equations. A new type of Γ-functions generated by quasi associated homogeneous distributions is defined.

The extension to infinite dimensional domains of Clarke's generalized Jacobian is the focus of this paper. First, a generalization of a Fabian–Preiss theorem to the infinite dimensional setting is obtained. As a consequence, a new formula relating the Clarke's generalized Jacobians corresponding to finite dimensional spaces K, L with K⊆L is established. Furthermore, in the infinite dimensional case, basic properties pertaining the generalized Jacobian are developed and then an identification of this set-valued map is produced. Applications of these results in the form of chain rules including sum and product rules, and a computational formula for continuous selections are derived.

In case (Xn) is an i.i.d. sequence, and Sn = Xl + ··· + Xn, the Hsu-Robbins-Erdős theorem states that Σxn=1P(|Sn| >nϵ) < ∞, ϵ > 0, iff EX21 < ∞ and EX1 = 0. The purpose of this paper is to generalize this result to the case in which the steps Xn are independent, but their distributions are taken from a finite set.

In this paper we prove a generalization of the stability of approximately additive mappings in the spirit of Hyers, Ulam, and Rassias.

The purpose of this paper is to study a periodic boundary value problem for a nonlinear ordinary differential equation of second order. We generalize the monotone iterative method to cover some new situations and present a new approach considering the monotone iterates as orbits of a (discrete) dynamical system.

In this paper, we establish a generalization of Hadamard's inequality to r-convex functions on Carnot groups.

If P ( z ) = ∑ ν = 0 n c ν z ν is a polynomial of degree n having no zeros in | z | 1 , then for | β | ⩽ 1 , it was proved by Jain [V.K. Jain, Generalization of certain well known inequalities for polynomials, Glas. Mat. 32 (52) (1997) 45–51] that | z P ′ ( z ) + n β 2 P ( z ) | ⩽ n 2 { | 1 + β 2 | + | β 2 | } max | z | = 1 | P ( z ) | , | z | = 1 . In this paper, we shall first obtain a result concerning minimum modulus of polynomials and next we improve upon the above inequality for the polynomials with restricted zeros. Our results refine and generalize certain well-known polynomial inequalities including some results of Bernstein, Lax, Malik and Vong, and Aziz and Dawood.

A formula for the sum of any positive-integral power of the first N positive integers was published by Johann Faulhaber in the 1600s. In this paper, we generalize Faulhaber's formula to non-integral complex powers with real part greater than −1.

In the present paper we define the generalization of the Phillips operators. Direct and converse theorems are established for Phillips operators and its new generalization. The direct theorems are given for the second order Ditzian–Totik modulus of smoothness. The inverse results are derived in the terminology of [J. Anal. Math. 61 (1993) 61–111].

In this paper we introduce the concept of G-H-space which is a generalization of H-space and a G-H-KKM map which unifies H-KKM maps and generalized KKM maps, and obtain intersection and fixed point theorems in G-H-space. We also give applications to the existence of greatest and maximal elements for preference relations in topological spaces.

A remarkable fundamental theorem established by Mehta plays an important role in proving existence of fixed points, maximal elements, and equilibria in abstract economies. In this paper, we extend Himmelbergʼs measure of precompactness to the general setting of l . c . -spaces and obtain a generalization of Mehtaʼs theorem. As an application, we develop some new fixed point theorems involving a kind of condensing mappings.

In this paper, we derive several existence results for the variational and the generalized variational inequality problems where the mappings under consideration need not have any continuity properties. Uniqueness of the solution of the generalized variational inequality problem is also investigated. As a by-product, we obtain several existence and uniqueness results for the generalized complementarity problem. Applications to minimization problems with generalized convex functions are considered.

In this paper we prove a generalization of the stability of the Pexider equation f(x+y)=g(x)+h(y) in the spirit of Hyers, Ulam, Rassias, and Găvruta.

The concept of a multiplicative transformator formalized by Brodskii, Gokhberg, and Krein in a by now classical 1967 paper can play in causality theory a role similar to that played by the more familiar additive transformator concept. Among its advantages, one finds an alternative point of view (a system cascade connection representation in lieu of a parallel connection), a more compact formulation of some proofs and results, a natural tool for a generalization of these results into a nonlinear setting.

Consider a renewal process, and let K ⩾ 0 denote the random duration of a typical renewal cycle. Assume that on any renewal cycle, a rare event called “success” can occur. Such successes lend themselves naturally to approximation by Poisson point processes. If each success occurs after a random delay, however, Poisson convergence can be relatively slow, because each success corresponds to a time interval, not a point. If K is an arithmetic variable, a “finite-size correction” (FSC) is known to speed Poisson convergence by providing a second, subdominant term in the appropriate asymptotic expansion. This paper generalizes the FSC from arithmetic K to general K . Genomics applications require this generalization, because they have already heuristically applied the FSC to p -values involving absolutely continuous distributions. The FSC also sharpens certain results in queuing theory, insurance risk, traffic flow, and reliability theory.

We consider a quasilinear integrodifferential system in non-normal form. Such a system is a generalization of a phase-field model with memory and includes, as a particular case, the system describing the combustion of a material with memory. In this paper, we study both the direct and the inverse problems. Our fundamental tools are: the theory of analytic semigroups, optimal regularity results and fixed point arguments.

In this paper, using the idea of P. Gavruta (1994, J. Math. Anal. Appl.184, 431–436), we prove a generalization of the stability of approximately additive mappings in the spirits of Hyers, Ulam, and Rassias.

In this paper, we consider K-rotundity in a complex normed linear space and prove a theorem which is the generalization of the Taylor-Foguel theorem. We also consider the relations between K-uniform rotundity, fully K-convexity, and K-rotundity.

In this paper, we consider the nonlinear elliptic systems under controllable growth condition. We use a new method introduced by Duzaar and Grotowski, for proving partial regularity for weak solutions, based on a generalization of the technique of harmonic approximation. We extend previous partial regularity results under the natural growth condition to the case of the controllable growth condition, and directly establishing the optimal Hölder exponent for the derivative of a weak solution.

In 1969, Meir and Keeler [29] obtained a remarkable generalization of the Banach contraction principle. Since then, there have appeared a number of generalizations of their result. In 1981, the second author and Bae [33] extended the Meir-Keeler theorem to two commuting maps by adopting Jungck’s method. This influenced many authors, and, consequently, a number of new results in this line followed. Recent works of Sessa and others [46,47] contain common fixed point theorems of four maps satisfying certain contractive type conditions. In the present paper, we give a new result which encompasses most of such generalizations of the Meir-Keeler theorem. Further our result also includes many other generalizations of the Banach contraction principle. Some authors have obtained their results on 2-metric spaces. However, 2-metric versions are easily obtained from metric ones by an obvious

In this paper we gave a generalization of power means which include positive nonlinear functionals. For these means we obtained generalizations of fundamental mean inequality, Holder's and Minkowski's, and their converse inequalities.

In this paper we consider the weak convergence of the weighted multivariate empiricalU-statistic indexed by rectangles under mixing conditions in the Skorohod topology. It is a generalization of a result of F. H. Ruymgaart and M. C. A. van Zuijlen (J. Statist. Plann. Inference32, 1992, 259–269) from the i.i.d. case to the dependent case. M. Harel, C. A. O'Cinneide and M. L. Puri (J. Multivariate Anal.48, 1994, 297–314) gave only a generalization to the univariate case of the weighted empiricalU-statistic indexed by point; a generalization to the multivariate case is obtained by B. Ragbi (J. Statist. Plann. Inference,61, 1997, 1–16). There are a number of interesting applications dealing with the Cramer–Von Mises statistic or the Anderson–Darling statistic given in Section 4; applications to a Markov process as well as to an ARMA process are also provided.

Local fractional derivative (LFD) operators have been introduced in the recent literature (Chaos 6 (1996) 505–513). Being local in nature these derivatives have proven useful in studying fractional differentiability properties of highly irregular and nowhere differentiable functions. In the present paper we prove Leibniz rule, chain rule for LFD operators. Generalization of directional LFD and multivariable fractional Taylor series to higher orders have been presented.