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Articles 541 - 550 of 550
Full-Text Articles in Mathematics
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 …