Rakhee Basu | Mahindra Ecole Centrale (original) (raw)

Papers by Rakhee Basu

Asian-European Journal of Mathematics, 2021

This paper deals with the oscillatory results of first order nonlinear delay differential equatio... more This paper deals with the oscillatory results of first order nonlinear delay differential equations with several deviating arguments by employing an iterative process. The results presented here has improved the outcomes of [1, 2, 8]. Various examples are solved in MATLAB software to illustrate the relevance of the main results.

Oscillation criteria for odd higher order nonlinear neutral differential equations with positive ... more Oscillation criteria for odd higher order nonlinear neutral differential equations with positive and negative coefficients DIFFERENTIAL EQUATIONS & APPLICATIONS AIMS AND SCOPE Differential Equations and Applications (DEA, Differ. Equ. Appl.) will publish original papers from all branches of ordinary, functional-differential, and partial differential equations. Only papers of the highest quality will be accepted for the publication. The papers which demonstrate novelty or give relations of differential equations with other fields of mathematics and applications are preferred.

The aim of this work is to provide some simple sufficient conditions for topological transitivity... more The aim of this work is to provide some simple sufficient conditions for topological transitivity of piecewise monotone maps on [0, 1]. Here we introduce a steepness condition that will imply that the map is expanding (in the sense that for every interval, the length of its image is greater than the length of that interval, unless the image is the whole space), and then we prove that these expanding maps are transitive. The theorems stated in this paper improve some known recent results. Moreover, they are simpler to state.

Oscillatory and asymptotic behaviour of a class of nonlinear fourth order neutral equation with q... more Oscillatory and asymptotic behaviour of a class of nonlinear fourth order neutral equation with quasi-derivatives of the form L 4 (y(t) + p(t)y(t − τ)) + q(t)G(y(t − α)) − h(t)H(y(t − β)) = 0, and L 4 (y(t) + p(t)y(t − τ)) + q(t)G(y(t − α)) − h(t)H(y(t − β)) = f (t) have been studied under the assumption ∞ 0 t r n (t) dt = ∞, n = 1, 2, 3. for various ranges of p(t), where L n u(t) = r n d dt L n−1 u(t), n = 0, 1, 2, 3.

In this paper, the authors study oscillatory and asymptotic behavior of solutions of a class of n... more In this paper, the authors study oscillatory and asymptotic behavior of solutions of a class of nonlinear third order neutral differential equations with positive and negative coefficients of the form a(t)(b(t)(y(t) + p(t)y(σ(t)))) + q(t)G(y(α(t))) − h(t)H(y(β(t))) = 0 (E) for 0 p(t) p 1 < 1 and −1 < p 2 p(t) 0. The results in this paper generalize the results of [LI, T.-ZHANG, C.-XING, G.: Oscillation of third-order neutral delay differential equations, Abstr. Appl. Anal. 2012 (2012), Article ID 569201] and various results in the literature. We establish new conditions which guarantees that every solutions of (E) either oscillatory or converges to zero. Examples are considered to illustrate the main results.

In this paper, oscillatory and asymptotic behavior of solutions of a class of nonlinear second or... more In this paper, oscillatory and asymptotic behavior of solutions of a class of nonlinear second order neutral differential equations with positive and negative coefficients of the form (r 1 (t)(x(t) + p 1 (t)x(τ (t)))) + r 2 (t)(x(t) + p 2 (t)x(σ(t))) +p(t)G(x(α(t))) − q(t)H(x(β(t))) = 0 and (r 1 (t)(x(t) + p 1 (t)x(τ (t)))) + r 2 (t)(x(t) + p 2 (t)x(σ(t))) +p(t)G(x(α(t))) − q(t)H(x(β(t))) = f (t) are studied for various ranges of p 1 (t), p 2 (t).

In this paper oscillatory and asymptotic behavior of solutions of a class of nonlinear second ord... more In this paper oscillatory and asymptotic behavior of solutions of a class of nonlinear second order neutral differential equations with positive and negative coefficients of the form (r 1 (t)(x(t) + p 1 (t)x(τ (t))) ′) ′ + r 2 (t)(x(t) + p 2 (t)x(σ(t))) ′ +p(t)G(x(α(t))) − q(t)H(x(β(t))) = 0 and (r 1 (t)(x(t) + p 1 (t)x(τ (t))) ′) ′ + r 2 (t)(x(t) + p 2 (t)x(σ(t))) ′ +p(t)G(x(α(t))) − q(t)H(x(β(t))) = f (t) are studied for p 1 (t), p 2 (t) ∈ C([t 0 , ∞), R). Moreover, using Banach fixed point theorem, sufficient conditions are obtained for the existence of bounded positive solutions of the forced equation.

Oscillatory and asymptotic behaviour of bounded solutions of a class of fourth order nonlinear ne... more Oscillatory and asymptotic behaviour of bounded solutions of a class of fourth order nonlinear neutral equation with quasi-derivatives of the form L 4 (y(t) + p(t)y(t − τ)) + q(t)G(y(t − α)) − h(t)H(y(t − β)) = 0, and L 4 (y(t) + p(t)y(t − τ)) + q(t)G(y(t − α)) − h(t)H(y(t − β)) = f (t) have been studied under the assumption ∫ ∞ 0 1 r n (t) dt < ∞, n = 1, 2, 3 for various ranges of p(t), where L n u(t) = r n d dt L n−1 u(t), n = 0, 1, 2, 3.

In this paper, Oscillatory and asymptotic behaviour of solutions of a class of nonlinear fourth o... more In this paper, Oscillatory and asymptotic behaviour of solutions of a class of nonlinear fourth order neutral differential equations with positive and negative coefficients of the form (H) (r(t)(y(t) + p(t)y(t − τ))) + q(t)G(y(t − α)) − h(t)H(y(t − β)) = 0 and (NH) (r(t)(y(t) + p(t)y(t − τ))) + q(t)G(y(t − α)) − h(t)H(y(t − β)) = f (t) are studied under the assumption ∞ 0 t r(t) dt = ∞ for various ranges of p(t). Using Schauder's fixed point theorem, sufficient conditions are obtained for the existence of bounded positive solutions of (NH).

With a geometric idea, we obtain a set of new oscillation criteria for the forced second order ne... more With a geometric idea, we obtain a set of new oscillation criteria for the forced second order neutral delay differential equation with deviating argument of the form (x(t) + p(t)x(σ(t))) ′′ + q(t)f (x(τ (t))) = g(t). This criteria improves the results obtained by Kong and Wong [4].

In this paper, oscillatory and asymptotic behaviour of solutions of a class of nonlinear fourth o... more In this paper, oscillatory and asymptotic behaviour of solutions of a class of nonlinear fourth order neutral differential equations with several delay of the form (r(t)(y(t) + p(t)y(t − τ))) + m i=1 q i (t)G(y(t − α i)) = 0 and (E) (r(t)(y(t) + p(t)y(t − τ))) + m i=1 q i (t)G(y(t − α i)) = f (t) are studied under the assumption ∞ 0

This paper deals with the oscillatory results of first order nonlinear delay differential equatio... more This paper deals with the oscillatory results of first order nonlinear delay differential equations with several deviating arguments by employing an iterative process. The results presented here has improved the outcomes of [2, 6] and [16]. Various examples are solved in MATLAB software to illustrate the relevance of the main results.

Rocky Mountain Journal of Mathematics, 2021

This paper deals with various new formulas involving a quotient of two or more functions using ap... more This paper deals with various new formulas involving a quotient of two or more functions using applicable techniques of the differential transform method (DTM). These newly developed formulas could be helpful for solving delay differential equations and integro-differential equations, which are highly challenging to solve directly. Moreover, a delay differential equation is completely solved using various algorithms in MATLAB incorporating these new formulas.

Differential Equations & Applications, 2015

In this paper, the authors study oscillatory and asymptotic behavior of solutions of a class of n... more In this paper, the authors study oscillatory and asymptotic behavior of solutions of a class of nonlinear higher order neutral differential equations with positive and negative coefficients of the form (a(t)(b(t)(y(t) + p(t)y(σ (t))))) (n−2) + q(t)G(y(α(t))) − h(t)H(y(β (t))) = 0 (E) for n 3 , n is an odd integer, 0 p(t) p 1 < 1 and −1 < p 2 p(t) 0. The results in this paper generalize the results of Panigrahi and Basu [9] and various results in the literature. We establish new conditions which guarantees that every solutions of (E) either oscillatory or converges to zero. Examples are considered to illustrate the main results.

Annals of the Academy of Romanian Scientists: Series on Mathematics and its Applications

In this paper, oscillatory and asymptotic behaviour of solutions of a class of nonlinear fourth o... more In this paper, oscillatory and asymptotic behaviour of solutions of a class of nonlinear fourth order neutral differential equations with several delay of the form (r(t)(y(t) + p(t)y(t − τ))) + m i=1 q i (t)G(y(t − α i)) = 0 and (E) (r(t)(y(t) + p(t)y(t − τ))) + m i=1 q i (t)G(y(t − α i)) = f (t) are studied under the assumption ∞ 0

The College Mathematics Journal, 2019

Asian-European Journal of Mathematics, 2021

This paper deals with the oscillatory results of first order nonlinear delay differential equatio... more This paper deals with the oscillatory results of first order nonlinear delay differential equations with several deviating arguments by employing an iterative process. The results presented here has improved the outcomes of [1, 2, 8]. Various examples are solved in MATLAB software to illustrate the relevance of the main results.

Oscillation criteria for odd higher order nonlinear neutral differential equations with positive ... more Oscillation criteria for odd higher order nonlinear neutral differential equations with positive and negative coefficients DIFFERENTIAL EQUATIONS & APPLICATIONS AIMS AND SCOPE Differential Equations and Applications (DEA, Differ. Equ. Appl.) will publish original papers from all branches of ordinary, functional-differential, and partial differential equations. Only papers of the highest quality will be accepted for the publication. The papers which demonstrate novelty or give relations of differential equations with other fields of mathematics and applications are preferred.

The aim of this work is to provide some simple sufficient conditions for topological transitivity... more The aim of this work is to provide some simple sufficient conditions for topological transitivity of piecewise monotone maps on [0, 1]. Here we introduce a steepness condition that will imply that the map is expanding (in the sense that for every interval, the length of its image is greater than the length of that interval, unless the image is the whole space), and then we prove that these expanding maps are transitive. The theorems stated in this paper improve some known recent results. Moreover, they are simpler to state.

Oscillatory and asymptotic behaviour of a class of nonlinear fourth order neutral equation with q... more Oscillatory and asymptotic behaviour of a class of nonlinear fourth order neutral equation with quasi-derivatives of the form L 4 (y(t) + p(t)y(t − τ)) + q(t)G(y(t − α)) − h(t)H(y(t − β)) = 0, and L 4 (y(t) + p(t)y(t − τ)) + q(t)G(y(t − α)) − h(t)H(y(t − β)) = f (t) have been studied under the assumption ∞ 0 t r n (t) dt = ∞, n = 1, 2, 3. for various ranges of p(t), where L n u(t) = r n d dt L n−1 u(t), n = 0, 1, 2, 3.

In this paper, the authors study oscillatory and asymptotic behavior of solutions of a class of n... more In this paper, the authors study oscillatory and asymptotic behavior of solutions of a class of nonlinear third order neutral differential equations with positive and negative coefficients of the form a(t)(b(t)(y(t) + p(t)y(σ(t)))) + q(t)G(y(α(t))) − h(t)H(y(β(t))) = 0 (E) for 0 p(t) p 1 < 1 and −1 < p 2 p(t) 0. The results in this paper generalize the results of [LI, T.-ZHANG, C.-XING, G.: Oscillation of third-order neutral delay differential equations, Abstr. Appl. Anal. 2012 (2012), Article ID 569201] and various results in the literature. We establish new conditions which guarantees that every solutions of (E) either oscillatory or converges to zero. Examples are considered to illustrate the main results.

In this paper, oscillatory and asymptotic behavior of solutions of a class of nonlinear second or... more In this paper, oscillatory and asymptotic behavior of solutions of a class of nonlinear second order neutral differential equations with positive and negative coefficients of the form (r 1 (t)(x(t) + p 1 (t)x(τ (t)))) + r 2 (t)(x(t) + p 2 (t)x(σ(t))) +p(t)G(x(α(t))) − q(t)H(x(β(t))) = 0 and (r 1 (t)(x(t) + p 1 (t)x(τ (t)))) + r 2 (t)(x(t) + p 2 (t)x(σ(t))) +p(t)G(x(α(t))) − q(t)H(x(β(t))) = f (t) are studied for various ranges of p 1 (t), p 2 (t).

In this paper oscillatory and asymptotic behavior of solutions of a class of nonlinear second ord... more In this paper oscillatory and asymptotic behavior of solutions of a class of nonlinear second order neutral differential equations with positive and negative coefficients of the form (r 1 (t)(x(t) + p 1 (t)x(τ (t))) ′) ′ + r 2 (t)(x(t) + p 2 (t)x(σ(t))) ′ +p(t)G(x(α(t))) − q(t)H(x(β(t))) = 0 and (r 1 (t)(x(t) + p 1 (t)x(τ (t))) ′) ′ + r 2 (t)(x(t) + p 2 (t)x(σ(t))) ′ +p(t)G(x(α(t))) − q(t)H(x(β(t))) = f (t) are studied for p 1 (t), p 2 (t) ∈ C([t 0 , ∞), R). Moreover, using Banach fixed point theorem, sufficient conditions are obtained for the existence of bounded positive solutions of the forced equation.

Oscillatory and asymptotic behaviour of bounded solutions of a class of fourth order nonlinear ne... more Oscillatory and asymptotic behaviour of bounded solutions of a class of fourth order nonlinear neutral equation with quasi-derivatives of the form L 4 (y(t) + p(t)y(t − τ)) + q(t)G(y(t − α)) − h(t)H(y(t − β)) = 0, and L 4 (y(t) + p(t)y(t − τ)) + q(t)G(y(t − α)) − h(t)H(y(t − β)) = f (t) have been studied under the assumption ∫ ∞ 0 1 r n (t) dt < ∞, n = 1, 2, 3 for various ranges of p(t), where L n u(t) = r n d dt L n−1 u(t), n = 0, 1, 2, 3.

In this paper, Oscillatory and asymptotic behaviour of solutions of a class of nonlinear fourth o... more In this paper, Oscillatory and asymptotic behaviour of solutions of a class of nonlinear fourth order neutral differential equations with positive and negative coefficients of the form (H) (r(t)(y(t) + p(t)y(t − τ))) + q(t)G(y(t − α)) − h(t)H(y(t − β)) = 0 and (NH) (r(t)(y(t) + p(t)y(t − τ))) + q(t)G(y(t − α)) − h(t)H(y(t − β)) = f (t) are studied under the assumption ∞ 0 t r(t) dt = ∞ for various ranges of p(t). Using Schauder's fixed point theorem, sufficient conditions are obtained for the existence of bounded positive solutions of (NH).

With a geometric idea, we obtain a set of new oscillation criteria for the forced second order ne... more With a geometric idea, we obtain a set of new oscillation criteria for the forced second order neutral delay differential equation with deviating argument of the form (x(t) + p(t)x(σ(t))) ′′ + q(t)f (x(τ (t))) = g(t). This criteria improves the results obtained by Kong and Wong [4].

In this paper, oscillatory and asymptotic behaviour of solutions of a class of nonlinear fourth o... more In this paper, oscillatory and asymptotic behaviour of solutions of a class of nonlinear fourth order neutral differential equations with several delay of the form (r(t)(y(t) + p(t)y(t − τ))) + m i=1 q i (t)G(y(t − α i)) = 0 and (E) (r(t)(y(t) + p(t)y(t − τ))) + m i=1 q i (t)G(y(t − α i)) = f (t) are studied under the assumption ∞ 0

This paper deals with the oscillatory results of first order nonlinear delay differential equatio... more This paper deals with the oscillatory results of first order nonlinear delay differential equations with several deviating arguments by employing an iterative process. The results presented here has improved the outcomes of [2, 6] and [16]. Various examples are solved in MATLAB software to illustrate the relevance of the main results.

Rocky Mountain Journal of Mathematics, 2021

This paper deals with various new formulas involving a quotient of two or more functions using ap... more This paper deals with various new formulas involving a quotient of two or more functions using applicable techniques of the differential transform method (DTM). These newly developed formulas could be helpful for solving delay differential equations and integro-differential equations, which are highly challenging to solve directly. Moreover, a delay differential equation is completely solved using various algorithms in MATLAB incorporating these new formulas.

Differential Equations & Applications, 2015

In this paper, the authors study oscillatory and asymptotic behavior of solutions of a class of n... more In this paper, the authors study oscillatory and asymptotic behavior of solutions of a class of nonlinear higher order neutral differential equations with positive and negative coefficients of the form (a(t)(b(t)(y(t) + p(t)y(σ (t))))) (n−2) + q(t)G(y(α(t))) − h(t)H(y(β (t))) = 0 (E) for n 3 , n is an odd integer, 0 p(t) p 1 < 1 and −1 < p 2 p(t) 0. The results in this paper generalize the results of Panigrahi and Basu [9] and various results in the literature. We establish new conditions which guarantees that every solutions of (E) either oscillatory or converges to zero. Examples are considered to illustrate the main results.

Annals of the Academy of Romanian Scientists: Series on Mathematics and its Applications

In this paper, oscillatory and asymptotic behaviour of solutions of a class of nonlinear fourth o... more In this paper, oscillatory and asymptotic behaviour of solutions of a class of nonlinear fourth order neutral differential equations with several delay of the form (r(t)(y(t) + p(t)y(t − τ))) + m i=1 q i (t)G(y(t − α i)) = 0 and (E) (r(t)(y(t) + p(t)y(t − τ))) + m i=1 q i (t)G(y(t − α i)) = f (t) are studied under the assumption ∞ 0

The College Mathematics Journal, 2019