Showing papers in "International Journal of Mathematics and Mathematical Sciences in 1986"
TL;DR: In this paper, a common fixed point theorem of S.L. and S.P. Singh is generalized by weakening commutativity hypotheses and by increasing the number of functions involved.
Abstract: A common fixed point theorem of S.L. and S.P. Singh is generalized by weakening commutativity hypotheses and by increasing the number of functions involved.
1,249 citations
Journal Article•
206 citations
TL;DR: In this paper, it is shown that if Lipschitz conditions of a certain order are imposed on a function f(x), then these conditions affect considerably the absolute convergence of the Fourier series and Fourier transforms of f.
Abstract: It is well known that if Lipschitz conditions of a certain order are imposed on a function f(x), then these conditions affect considerably the absolute convergence of the Fourier series and Fourier transforms of f. In general, if f(x) belongs to a certain function class, then the Lipschitz conditions have bearing as to the dual space to which the Fourier coefficients and transforms of f(x) belong. In the present work we do study the same phenomena for the wider Dini-Lipschitz class as well as for some other allied classes of functions.
56 citations
TL;DR: In this article, the authors consider two selfmaps T and I of a closed convex subset C of a Banach space X which are weakly commuting in X, i.e.
Abstract: We consider two selfmaps T and I of a closed convex subset C of a Banach space X which are weakly commuting in X, i.e.‖TIx−ITx‖≤‖Ix−Tx‖ for any x in X,and satisfy the inequality‖Tx−Ty‖≤a‖Ix−Iy‖
46 citations
TL;DR: In this article, a large part of present knowledge about the behavior of Fourier series of functions of generalized bounded variation is unified, and connections between various concepts are discussed, particular attention being paid to those due to Waterman and Chanturiya.
Abstract: The aim of this article is to unify a large part of present knowledge about the behaviour of Fourier series of functions of generalized bounded variation. The connections between various concepts are discussed, particular attention being payed to those due to Waterman and Chanturiya. Exploiting the existing interactions and utilizing the power of one or another approach to some typical questions of the Fourier theory, a number of previously unnoticed results are obtained in the course of this exposition.
32 citations
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TL;DR: In this article, it was shown that all even integers are representable in infinitely many ways as a proper nonsquare difference, i.e., proper difference of two powerful numbers neither of which is a perfect square.
Abstract: A powerful number is a positive integer n satisfying the property that p2 divides n whenever the prime p divides n; i.e., in the canonical prime decomposition of n, no prime appears with exponent 1. In [1], S.W. Golomb introduced and studied such numbers. In particular, he asked whether (25,27) is the only pair of consecutive odd powerful numbers. This question was settled in [2] by W.A. Sentance who gave necessary and sufficient conditions for the existence of such pairs. The first result of this paper is to provide a generalization of Sentance's result by giving necessary and sufficient conditions for the existence of pairs of powerful numbers spaced evenly apart. This result leads us naturally to consider integers which are representable as a proper difference of two powerful numbers, i.e. n=p1−p2 where p1 and p2 are powerful numbers with g.c.d. (p1,p2)=1. Golomb (op.cit.) conjectured that 6 is not a proper difference of two powerful numbers, and that there are infinitely many numbers which cannot be represented as a proper difference of two powerful numbers. The antithesis of this conjecture was proved by W.L. McDaniel [3] who verified that every non-zero integer is in fact a proper difference of two powerful numbers in infinitely many ways. McDaniel's proof is essentially an existence proof. The second result of this paper is a simpler proof of McDaniel's result as well as an effective algorithm (in the proof) for explicitly determining infinitely many such representations. However, in both our proof and McDaniel's proof one of the powerful numbers is almost always a perfect square (namely one is always a perfect square when n≢2(mod4)). We provide in §2 a proof that all even integers are representable in infinitely many ways as a proper nonsquare difference; i.e., proper difference of two powerful numbers neither of which is a perfect square. This, in conjunction with the odd case in [4], shows that every integer is representable in infinitely many ways as a proper nonsquare difference. Moreover, in §2 we present some miscellaneous results and conclude with a discussion of some open questions.
29 citations
TL;DR: In this paper, the uniqueness of the common fixed point of two set-valued mappings and of two single-value mappings defined on a complete metric space was established under a contractive condition and a weak commutativity concept.
Abstract: Our main theorem establishes the uniqueness of the common fixed point of two set-valued mappings and of two single-valued mappings defined on a complete metric space,
under a contractive condition and a weak commutativity concept. This improves a theorem
of the second author.
26 citations
TL;DR: In this paper, a number of generalizations of the classical Heisenberg-Weyl uncertainty inequality are given, and the n-dimensional Hirschman entropy in-equality (Theorem 2.1) is proved from the optimal form of the Hausdorff-Young theorem and deduce a higher dimensional uncertainty inequality.
Abstract: In this paper a number of generalizations of the classical Heisenberg-Weyl uncertainty inequality are given. We prove the n-dimensional Hirschman entropy in- equality (Theorem 2.1) from the optimal form of the Hausdorff-Young theorem and deduce a higher dimensional uncertainty inequality (Theorem 2.2). From a general weighted f.orm of the Hausdorff-Young theorem, a one-dimensional weighted entropy inequality is proved and some weighted forms of the Heisenberg-Weyl inequalities are given.
22 citations
TL;DR: In this article, the authors studied the topological properties of two kinds of fine topologies on the space C ( X, Y ) of all continuous functions from X into Y.
Abstract: This paper studies the topological properties of two kinds of fine topologies on the space C ( X , Y ) of all continuous functions from X into Y .
22 citations
TL;DR: In this article, certain subclasses B(α,β) and B1(β, β) of Bazilevic functions of type α are introduced and a lot of interesting properties of these classes are derived.
Abstract: Certain subclasses B(α,β) and B1(α,β) of Bazilevic functions of type α are introduced. The object of the present paper is to derive a lot of interesting properties of the classes B(α,β) and B1(α,β).
16 citations
TL;DR: In this article, the reader is encouraged to consider whether there is only one natural category of graphs and how theories of directed graphs and undirected graphs are related, from the view of the portion of category theory which has become common knowledge.
Abstract: This paper presents some graph-theoretic questions from the viewioint of the portion of category theory which has become common knowledge. In particular, the reader is encouraged to consider whether there is only one natural category of graphs and how theories of directed graphs and undirected graphs are related.
TL;DR: In this paper, a topology on the state set of an automaton is considered and it is shown that under this topology, genetically closed subsets and primaries, in the sense of Bavel [1] turn out to be precisely the regular closed subset and minimal regular closed subsset respectively.
Abstract: A topology on the state set of an automaton is considered and it is shown that under this topology, genetically closed subsets and primaries, in the sense of Bavel [1] turn out to be precisely the regular closed subsets and minimal regular closed subsets respectively. The concept of a compact automaton is introduced and it is indicated that it can be viewed as a generalization of a finite automaton. Included also is an observation showing that our topological considerations can help recover some of the results of Dorfler [2].
TL;DR: In this paper, the decomposition method is applied to solve the Van der Pol oscillators without customary restrictive assumptions (I-4) and without resort to perturbation methods, without resorting to the usual I-4.
Abstract: The decomposition method is applied to solve the Duffing and Van der Pol oscillators without customary restrictive assumptions (I-4) and without resort to perturbation methods.
TL;DR: In this article, the class of functions f(z)=zp is defined in terms of functions in the complex number Vkλ(α,b,p) (k≥2, b≠0 is any complex number, 0≤α
Abstract: Let Vkλ(α,b,p) (k≥2, b≠0 is any complex number, 0≤α
TL;DR: In this article, the stability properties of linear multistep methods for delay differential equations with respect to the test equation y'(t) ay (t) + by(t), t O, 0 < % i, are investigated.
Abstract: Stability properties of linear multistep methods for delay differential equations with respect to the test equation y'(t) ay(t) + by(t), t O, 0 < % i, are investigated. It is known that the solution of this equation is bounded if and only if lal -b and we examine whether this property is inherited by multistep methods with Lagrange interpolation and by parametrized Adams methods.
TL;DR: In this paper, it was shown that r23(Pi,Pj,Pk)=[(4k) √ √ 4k √ n) is the smallest number p such that, if the edges of the complete graph Kp are coloured in any fashion with c colors, then for some i, the subgraph whose edges are colored in the ith subset of colors contains a Gi.
Abstract: Chung and Liu have defined the d-chromatic Ramsey number as follows. Let 1≤d≤c and let t=(cd). Let 1,2,…,t be the ordered subsets of d colors chosen from c distinct colors. Let G1,G2,…,Gt be graphs. The d-chromatic Ramsey number denoted by rdc(G1,G2,…,Gt) is defined as the least number p such that, if the edges of the complete graph Kp are colored in any fashion with c colors, then for some i, the subgraph whose edges are colored in the ith subset of colors contains a Gi. In this paper it is shown that r23(Pi,Pj,Pk)=[(4k
TL;DR: In this article, a finite difference method of order 2 and 4 for computing eigenvalues of a two point boundary value problem associated with a fourth order linear differential equation y(4) (p(x) q(x))y 0.
Abstract: This brief report describes some new finite difference methods of order 2 and 4 for computing eigenvalues of a two point boundary value problem associated with a fourth order linear differential equation y(4) (p(x) q(x))y 0. These methods are derived from the formula
TL;DR: Here, the general solution of the functional equation (2) on an open domain, without using 0-probability and 1- Probability, is found.
Abstract: In this series, this paper is devoted to the study of a functional equation connected with the characterization of weighted entropy and weighted entropy of degree β. Here, we find the general solution of the functional equation (2) on an open domain, without using 0-probability and 1-probability.
TL;DR: The best known upper bound for the number of vertices of an (11,6)-cage is given in this article, where it is shown that f(11, 6) and #8804;240.
Abstract: Let f(11,6) be the number of vertices of an (11,6)-cage. By giving a regular graph of girth 6 and valency 11, we show that f(11,6)≤240. This is the best known upper bound for f(11,6).
TL;DR: In this paper, the authors studied the asymptotic behavior in time of the solutions of nonlinear Klein-Gordon equations in the L∞(ℝ3) norm.
Abstract: We study the asymptotic behavior in time of the solutions of a system of nonlinear Klein-Gordon equations. We have two basic results: First, in the L∞(ℝ3) norm, solutions decay like 0(t−3/2) as t→
TL;DR: In this paper, explicit formulae for the cyclotomic numbers of order 15 over GF(p2) were obtained for primes p≡4, 11(mod15) and p ≥ 4.
Abstract: For primes p≡4, 11(mod15) explicit formulae are obtained for the cyclotomic numbers of order 15 over GF(p2).
TL;DR: In this paper, the authors established new discrete inequalities of the Gronwall type in N-independent variables for finite difference equations involving several independent variables and for numerical analysis, and their consequence for the case of N = 3, generalizes all of the known theorems obtained by Pachpatte and Singare in [1].
Abstract: In this paper we shall establish some new discrete inequalities of the Gronwall type in N-independent variables. They will have many applications for finite difference equations involving several independent variables and for numerical analysis. Their consequence for the case of N=3, generalizes all of the known theorems obtained by Pachpatte and Singare in [1]. An example, to which those results established in [1] are inapplicable, is given here to convey the usefulness of the results obtained.
TL;DR: The maximum number of pairwise edge disjoint forests of order five in the complete graph K n and the minimum number of forests whose union is K n, are determined in this article.
Abstract: The maximum number of pairwise edge disjoint forests of order five in the complete graph K n, and the minimum number of forests of order five whose union is K n, are determined.
TL;DR: In this paper, the authors prove existence theorems for differential inclusions with a nonconvex right hand side, which is lower semicontinuous or continuous in the state variable, measurable in the time variable and takes values in a finite or infinite dimensional separable Banach space.
Abstract: In this paper we prove some new existence theorems for differential inclusions with a nonconvex right hand side, which is lower semicontinuous or continuous in the state variable, measurable in the time variable and takes values in a finite or infinite dimensional separable Banach space.
TL;DR: In this paper, the authors proposed an operational procedure based on exploiting the properties of the Mellin transforms, and readily reduced the dual integral equations to a single equation, which then can be inverted by the Hankel inversion to give us an equation of Fredholm type, involving the unknown function.
Abstract: In this paper we deal with dual integral equations with an arbitrary weight function and Hankel kernels of distinct and general order. We propose an operational procedure, which depends on exploiting the properties of the Mellin transforms, and readily reduces the dual equations to a single equation. This then can be inverted by the Hankel inversion to give us an equation of Fredholm type, involving the unknown function. Most of the known results are then derived as special cases of our general result.
TL;DR: In this paper, a formula of inversion was established for an integral transform whose kernel is the Bessel function J (kr) where r varies over the finite interval (0,a) and the u order u is taken to be the eigenvalue parameter.
Abstract: A formula of inversion is established for an integral transform whose kernel is the Bessel function J (kr) where r varies over the finite interval (0,a) and the u order u is taken to be the eigenvalue parameter. qnen this parameter is large the u Bessel function behaves for varying r like the power function r and by relating the 5essel functions to their corresponding power functions the proof of the inversion iormula can be reduced to one depending on the Mellin inversion theorem.
TL;DR: A satisfactory theory of the Global MagnetoFluidoStatic (GMYS) Fields, where symmetric and non-symmetric configurations can be dealt with on the same foot- ing, has not yet been developed as mentioned in this paper.
Abstract: A satisfactory theory of the Global MagnetoFluidoStatic (GMYS) Fields, where symmetric and non-symmetric configurations can be dealt with on the same foot- ing, has not yet been developed. However the formulation of the Nowhere-Force-Free, Local-Global MFS problem about a given smooth isobaric toroidal surface o (ac- tually, a degenerate initial-value problem) can be weakened so as to include certain generallzed solutions as formal power series in a "natural" transverse coordinate. lt is reasonable to conjecture that these series converge, for sufficiently smooth data on . in the same function space which their coefficients belong to (in es- O' sence, a complete linear space over the 2-torus).
TL;DR: In this paper, velocity potentials describing the irrotational infinitesimal motion of two superposed inviscid and incompressible fluids under gravity with a horizontal plane of mean surface of separation, are derived due to a vertical line source present in either of the fluids, whose strength, besides being harmonic in time, varies sinusiodal along its length.
Abstract: Velocity potentials describing the irrotational infinitesimal motion of two superposed inviscid and incompressible fluids under gravity with a horizontal plane of mean surface of separation, are derived due to a vertical line source present in either of the fluids, whose strength, besides being harmonic in time, varies sinusiodal(y along its length. The technique of deriving the potentials here is an extension of the technique used for the case of only time harmonic vertical line source. The present case is concerned with the two-dimensional modified Helmholtz's equation while the previous is concerned with the two-dimenslonal Laplace's equation.
TL;DR: In this article, the Hadamard product of f(z) and z/(1−z) is defined as the Ruscheweyh derivative defined by using the Hadami product.
Abstract: Let Dαf(z) be the Ruscheweyh derivative defined by using the Hadamard product of f(z) and z/(1−z)1
TL;DR: In this article, the existence and cardinality of solutions of multilinear differential equations were studied, and upper bounds on the number of solutions were derived for each of them.
Abstract: We study the existnece and cardinality of solutions of multilinear differential equations giving upper bounds on the number of solutions.