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Full-Text Articles in Mathematics
A New Approach To The Method Of Nonlinear Variation Of Parameters, A. Richard Mitchell, M. E. Lord
A New Approach To The Method Of Nonlinear Variation Of Parameters, A. Richard Mitchell, M. E. Lord
Mathematics Technical Papers - Archive
As is well-known [3] the method of variation of parameters is a very useful tool in studying the properties of solutions of perturbed differential equations. Extending the classical variation of constants formula for linear systems, Alekseev [1] obtained a variation of constants formula for nonlinear systems. In [2], Lakshmikantham developed a nonlinear variation of constants formula for a scalar differential equation whose unperturbed term is of variable separable type, under rather mild assumptions. In this paper, we wish to study the general problem under weaker assumptions. Our approach is parallel to the classical one and yields two different formulas, one …
On Massera Type Converse Theorem In Terms Of Two Different Measures (0), V. Lakshmikantham
On Massera Type Converse Theorem In Terms Of Two Different Measures (0), V. Lakshmikantham
Mathematics Technical Papers - Archive
The importance of uniform asymptotic stability in studying the stability properties of perturbed motions needs no emphasis. The converse theorem of Massera which results from uniform asymptotic stability has been widely used in perturbation theory (See [1,2,0]).Very recently [31 this powerful theorem has been fruitfully employed to investigate bifurcation problems and seems to have much future in that direction. The advantage of studying stability properties by means of two different measures, as given by Movehan [41 , is well-known (See also [2,51 ). However, there does not exist a converse theorem of Massera type when uniform asymptotic stability holds in …
Matrix Fields Over The Integers Modulo M, Robert M. Mcconnel, Jacob T. B. Beard Jr.
Matrix Fields Over The Integers Modulo M, Robert M. Mcconnel, Jacob T. B. Beard Jr.
Mathematics Technical Papers - Archive
Let Zm denote the ring of integers modulo m and let [see pdf for notation] denote the complete ring of all [see pdf for notation] matrices over Zm under the usual matrix addition and multiplication. The primary purposes of this paper are to characterize and count all subfields of the ring [see pdf for notation] . Related results are given on field extensions in [see pdf for notation] of subfields of [see pdf for notation] . Partial results on subfields of [see pdf for notation] when R is an arbitrary finite commutative ring with identity are also given. Having basic …
A Polynomial Dual Of Partitions, Jacob T. B. Beard, Ann D. Dorris
A Polynomial Dual Of Partitions, Jacob T. B. Beard, Ann D. Dorris
Mathematics Technical Papers - Archive
Let [see pdf for notation] be a non-negative integral polynomial. The polynomial P(x) is m-graphical, and a multi-graph G a realization of P(x), provided there exists a multi-graph G containing exactly P(1) points where [see pdf for notation] of these points have degree i for [see pdf for notation]. For multi-graphs G,H having polynomials P(x), Q(x) and number-theoretic partitions (degree sequences) [see pdf for notation], the usual product P(x)Q(x) is shown to be the polynomial of the Cartesian product [see pdf for notation], thus inducing a natural product [see pdf for notation] which extends that of juxtaposing integral multiple copies …
On The Method Of Vector Lyapunov Functions, V. Lakshmikantham
On The Method Of Vector Lyapunov Functions, V. Lakshmikantham
Mathematics Technical Papers - Archive
The method of vector Lyapunov functions consists of employing several Lyapunov-like functions and the theory of vectorial differential inequalities. This general comparison technic leads to a more flexible mechanism to study the qualitative properties of nonlinear differential systems. This method is also effective in discussing interconnected dynamical systems where a construction of a single Lyapunov function is difficult. However, an unpleasant fact in this approach is the requirement of quasimonotone property of the comparison system. An idea to get around this will be indicated which might be useful in applications. Also, an extension of this technic to abstract cones will …
On The Redundancy Of Monotony Assumption, V. Lakshmikantham
On The Redundancy Of Monotony Assumption, V. Lakshmikantham
Mathematics Technical Papers - Archive
It is well known that all the results in integral inequalities of Bellman-Gronwall-Reid type demand an assumption of monotony on the functions involved. Since the corresponding theory of differential inequalities does not require the monotonic assumption, it is believed that this extra condition is due to the technics employed rather than the necessity. It is also well known that in proving the convergence of successive approximations, it becomes crucial to suppose an additional restriction of monotony on the functions satisfying uniqueness criteria. The question whether this additional assumption is really needed has been open for many years. This problem was …
Maximal And Minimal Solutions And Comparison Results For Differential Equations In Abstract Cones, V. Lakshmikantham, Roger W. Mitchell, A. Richard Mitchell
Maximal And Minimal Solutions And Comparison Results For Differential Equations In Abstract Cones, V. Lakshmikantham, Roger W. Mitchell, A. Richard Mitchell
Mathematics Technical Papers - Archive
As is well known, an important technique in the theory of differential equations is concerned with estimating a function satisfying a differential inequality by means of the extremal solutions of the corresponding differential equation. This comparison principle has been widely employed in studying the qualitative theory of differential equations (see [3]). If we desire to develop a similar comparison result in abstract spaces we must consider cones. These results could be of great value in applications to the theory of differential equations in abstract spaces. First, we must consider existence results for maximal and minimal solutions in cones which can …
Lyapunov-Like Vector Functions Using Pointwise Degenerate Systems As Comparison Functions, Bernard A. Asner, V. Lakshmikantham
Lyapunov-Like Vector Functions Using Pointwise Degenerate Systems As Comparison Functions, Bernard A. Asner, V. Lakshmikantham
Mathematics Technical Papers - Archive
The use of Lyapunov-like vector functions is recognized as an important tool for estimating the behavior of a dynamical system. In applications, one needs to determine a suitable comparison function which contains information or properties that can be used to obtain some information about the behavior of the dynamical system. In recent years, the remarkable property of pointwise degeneracy has been discovered for linear, autonomous, delay-differential equations. For this class of equations one knows that all solutions will reach a subspace in finite time and remain on the subspace thereafter. As this class of equations is particularly simple, it becomes …
On The Existence Of Solutions Of Differential Equations In A Banach Space, Jerome Eisenfeld, V. Lakshmikantham
On The Existence Of Solutions Of Differential Equations In A Banach Space, Jerome Eisenfeld, V. Lakshmikantham
Mathematics Technical Papers - Archive
The study of Cauchy problem for differential equations in a Banach space has taken two different directions. One approach is to find compactness type conditions [1,2,4,9,14] to guarantee existence of solutions only and the corresponding results are extensions of the classical Peano's Theorem. The other approach is to employ accretive type conditions [9,10,11,12,15] which assure existence as well as uniqueness of solutions. In fact, this latter technic shows that uniqueness conditions imply existence of solutions [16]. In this paper we follow the first direction. Employing Lyapunov-like functions and the notion of the measure of noncompactness, we prove a local existence …
Asymptotic Equilibrium Of Ordinary Differential Systems In A Banach Space, Roger W. Mitchell, A. Richard Mitchell
Asymptotic Equilibrium Of Ordinary Differential Systems In A Banach Space, Roger W. Mitchell, A. Richard Mitchell
Mathematics Technical Papers - Archive
A differential system [see pdf for notation] where [see pdf for notation] has asymptotic equilibrium if 1) for any initial condition [see pdf for notation] the system has a solution [see pdf for notation] existing on and such that [see pdf for notation] exists and is finite, and 2) for any v e B there exists [see pdf for notation] and a solution x(t) of (1)-(2) with [see pdf for notation] Several papers have appeared dealing with asymptotic equilibrium of (1)-(2) when [see pdf for notation], and f is majorized by a scalar function g(t,u) which is monotone in u …
Fixed Point Theorems Through Abstract Cones, Jerome Eisenfeld, V. Lakshmikantham
Fixed Point Theorems Through Abstract Cones, Jerome Eisenfeld, V. Lakshmikantham
Mathematics Technical Papers - Archive
A well-known theorem of Banach states that if T is a mapping on a complete metric space [see pdf for notation] such that for some number [see pdf for notation], the inequality (1.1) [see pdf for notation] holds, then T has a unique fixed point (i.e., a point u such that Tu = u). Extensions of this theorem [1,2] continue to require that T is a contraction i.e., (1.2) [see pdf for notation] This condition is essential if p is a metric but if p takes values in a partially ordered set k, then the condition (1.2) is avoidable. In …
On Perturbing Lyapunov Functions, S. Leela, V. Lakshmikantham
On Perturbing Lyapunov Functions, S. Leela, V. Lakshmikantham
Mathematics Technical Papers - Archive
It is known [2,3] that in proving uniform boundedness of a differential system by means of Lyapunov functions, it is sufficient to impose conditions in the complement of a compact set in [see pdf for notation], whereas, in the case of equiboundedness, the proofs demand that the assumptions hold everywhere in [see pdf for notation]. We wish to present, in this paper, a new idea which permits us to discuss nonuniform properties of solutions of differential equations under weaker assumptions. Our results will show that the equiboundedness can be proved without assuming conditions everywhere in [see pdf for notation] (as …
Truncated Circular Normal Distribution With Applications In Ballistics And Meteorology, Danny D. Dyer
Truncated Circular Normal Distribution With Applications In Ballistics And Meteorology, Danny D. Dyer
Mathematics Technical Papers - Archive
The use of the circular normal distribution (CND) to describe the behavior of random phenomena of a geophysical nature has been discussed by Crutcher [7, p. 9]. Based on the Mauchly [17] - Hsu [12] test, the harmonic dial points (the Fourier coefficients obtained from harmonic analysis of observations of periodic phenomena) representing i.) lunar semidiurnal atmospheric tides (Chapman and Lindzen [5, p. 66]), ii.) the westerly component of wind in the study of tidal oscillations in the upper atmosphere (Haurwitz [10]), and iii.) terrestrial-magnetic activity relative to solar activity (Bartels [2]) may be treated as observations from a CND. …
Existence Md Estimates For Solutions Of Nonlinear Equations Near A Branch Point, Jerome Eisenfeld, V. Lakshmikantham
Existence Md Estimates For Solutions Of Nonlinear Equations Near A Branch Point, Jerome Eisenfeld, V. Lakshmikantham
Mathematics Technical Papers - Archive
Consider the equation (1.1) [see pdf for notation] on a Hilbert space H. Here n is a scalar and [see pdf for notation] is a linear Fredholm operator. That is: (a) L is closed; (b) The domain, D(L) is dense in H; (c) The range, R(L) is closed in H; (d) The dimension of the null space, dim n(L) <= (e) The dimension of the null space of the adjoint dim n(L*) <= The operator N, which may be nonlinear, is defined for sufficiently small and appropriately restricted [see pdf for notation], and [see pdf for notation] Using the method of Lyapunov-Schmidt (cf., e.g. [4] or [5]) we express w in the form (1.2) [see pdf for notation] where [see pdf for notation] denotes the orthogonal complement of n(L) in H. Suppose (u,v) satisfy the simultaneous equations (1.3) [see pdf for notation] (1.4) [see pdf for notation] where P is the orthogonal projection operator of H onto R(L), I is the identity operator on H, and J is a right inverse of L on R(L), i.e.
On A Boundary Value Problem For A Class Of Differential Equations With A Deviating Argument, V. Lakshmikantham, Jerome Eisenfeld
On A Boundary Value Problem For A Class Of Differential Equations With A Deviating Argument, V. Lakshmikantham, Jerome Eisenfeld
Mathematics Technical Papers - Archive
Recently, J. Chandra [1] obtained comparison estimates for differential equations with deviating argument (1) [see pdf for notation] on the interval I: to [see pdf for notation] , where [see pdf for notation] is a given continuous function defined on a suitable interval (containing I ), together with the boundary conditions (2) [see pdf for notation] for [see pdf for notation] The class of BVP (1) - (2) is incorporated in a larger class discussed in [3]. The introduction in [1] of a maximal solution has furnished comparison results and an iterative procedure for obtaining solutions. The use of the …
Comparison Principle And Non Linear Contract In Abstract Spaces, V. Lakshmikantham, Jerome Eisenfeld
Comparison Principle And Non Linear Contract In Abstract Spaces, V. Lakshmikantham, Jerome Eisenfeld
Mathematics Technical Papers - Archive
The theory of differential and integral equations exploits comparison and iterative techniques which do not fall under the Contractive Mapping Principle. For they make use of partial orderings and maximal solutions; concepts which have no significance in a metric space. These methods take their proper place in the theory of cones [1], [2]. In this paper, Banach's Contraction Mapping Principle and comparison and iterative methods are brought together under a single roof which houses various results from the theory of differential equations [3], in addition to an interesting generalization of Banach's Theorem [4]. One 'It thus led, in a natural …