Differential Equations Of Neutral Type Research Articles

Convergence of the Solutions for a Neutral Difference Equation with Negative Coefficients

Abstract In this paper, we investigate the asymptotic behavior of the solutions of a neutral type difference equation of the form where τj (n), j = 1, . . . , w are general retarded arguments, σ(n) is a general deviated argument, [###] is a sequence of positive real numbers such that p(n) ≥ p, p ∈ R+, and Δ denotes the forward difference operator Δx(n) = x(n + 1) − x(n).

Asymptotically convergent solutions of a system of nonlinear functional differential equations of neutral type with iterated deviating arguments

Some conditions that guarantee the existence of a family of solutions converging to zero of a system of nonlinear functional differential equations with iterated deviating arguments are given.

Approximation scheme of higher accuracy of differential equations of neutral type

Approximation scheme of higher accuracy of differential equations of neutral type

An infinite dimensional bifurcation problem with application to a class of functional differential equations of neutral type

In this paper we consider an infinite dimensional bifurcationequation depending on a parameter $ \varepsilon>0 . $ By means ofthe theory of condensing operators, we prove the existence of abranch of solutions, parametrized by $ \varepsilon , $ bifurcatingfrom a curve of solutions of the bifurcation equation obtained for$\varepsilon =0 . $ We apply this result to a specific problem,namely to the existence of periodic solutions bifurcating from thelimit cycle of an autonomous functional differential equation ofneutral type when it is periodically perturbed by a nonlinearperturbation term of small amplitude.

Existence of Some Semilinear Nonlocal Functional Differential Equations of Neutral Type

This paper is concerned with the existence of mild and strong solutions on the interval <svg style="vertical-align:-2.22495pt;width:38.037498px;" id="M1" height="14.85" version="1.1" viewBox="0 0 38.037498 14.85" width="38.037498" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.012)"><path id="x5B" d="M290 -163h-170v866h170v-28q-79 -7 -94 -19.5t-15 -72.5v-627q0 -59 14.5 -71.5t94.5 -19.5v-28z" /></g><g transform="matrix(.017,-0,0,-.017,5.927,12.012)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105&#xA;q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" /></g><g transform="matrix(.017,-0,0,-.017,14.086,12.012)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,20.784,12.012)"><path id="x1D447" d="M649 676l-22 -187l-33 -2q3 56 -12 94q-8 20 -31.5 26.5t-86.5 6.5h-74l-90 -491q-4 -23 -6 -36.5t1 -25t7 -16.5t18 -9t27 -5t41 -3l-6 -28h-286l4 28q68 5 84 18.5t28 76.5l94 491h-55q-74 0 -100.5 -6t-41.5 -23q-24 -29 -54 -98l-32 1q32 98 53 188h22q7 -18 15 -22&#xA;t37 -4h417q23 0 33.5 5t25.5 21h23z" /></g><g transform="matrix(.017,-0,0,-.017,32.105,12.012)"><path id="x5D" d="M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z" /></g> </svg> for some neutral partial differential equations with nonlocal conditions. The linear part of the equations is assumed to generate a compact analytic semigroup of bounded linear operators, whereas the nonlinear part satisfies the Carath&#xeb;odory condition and is bounded by some suitable functions. We first employ the Schauder fixed-point theorem to prove the existence of solution on the interval <svg style="vertical-align:-2.22495pt;width:38.674999px;" id="M2" height="15" version="1.1" viewBox="0 0 38.674999 15" width="38.674999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x5B"/></g><g transform="matrix(.017,-0,0,-.017,5.927,12.162)"><path id="x1D6FF" d="M494 514l-10 -15q-56 76 -106.5 117t-110.5 41q-45 0 -45 -30q0 -28 109 -152q59 -68 86 -118.5t27 -107.5q0 -62 -29.5 -119.5t-86.5 -96.5q-64 -45 -138 -45q-76 0 -121.5 53t-45.5 136q0 74 40.5 135t98.5 95q43 26 87 42q-40 59 -57.5 92t-17.5 62q0 46 39 77.5&#xA;t90 31.5q54 0 97.5 -35.5t65 -78.5t28.5 -84zM359 234q0 84 -90 191q-47 -17 -74 -47q-81 -89 -81 -200q0 -71 32.5 -109.5t72.5 -38.5q47 0 80 33t46.5 77.5t13.5 93.5z" /></g><g transform="matrix(.017,-0,0,-.017,14.715,12.162)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,21.413,12.162)"><use xlink:href="#x1D447"/></g><g transform="matrix(.017,-0,0,-.017,32.734,12.162)"><use xlink:href="#x5D"/></g> </svg> for <svg style="vertical-align:-0.1638pt;width:36.5px;" id="M3" height="12.4375" version="1.1" viewBox="0 0 36.5 12.4375" width="36.5" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x1D6FF"/></g><g transform="matrix(.017,-0,0,-.017,13.56,12.162)"><path id="x3E" d="M512 230l-437 -233v58l378 199v2l-378 200v58l437 -233v-51z" /></g><g transform="matrix(.017,-0,0,-.017,28.263,12.162)"><use xlink:href="#x30"/></g> </svg> that is small enough, and, then, by letting <svg style="vertical-align:-0.1638pt;width:48.400002px;" id="M4" height="12.4375" version="1.1" viewBox="0 0 48.400002 12.4375" width="48.400002" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x1D6FF"/></g><g transform="matrix(.017,-0,0,-.017,16.398,12.162)"><path id="x2192" d="M901 255q-71 -62 -185 -187l-22 15l102 147h-727v50h727l-102 147l22 15q114 -125 185 -187z" /></g><g transform="matrix(.017,-0,0,-.017,40.163,12.162)"><use xlink:href="#x30"/></g> </svg> and using a diagonal argument, we have the existence results on the interval <svg style="vertical-align:-2.22495pt;width:38.037498px;" id="M5" height="14.85" version="1.1" viewBox="0 0 38.037498 14.85" width="38.037498" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.012)"><use xlink:href="#x5B"/></g><g transform="matrix(.017,-0,0,-.017,5.927,12.012)"><use xlink:href="#x30"/></g><g transform="matrix(.017,-0,0,-.017,14.086,12.012)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,20.784,12.012)"><use xlink:href="#x1D447"/></g><g transform="matrix(.017,-0,0,-.017,32.105,12.012)"><use xlink:href="#x5D"/></g> </svg>. This approach allows one to drop the compactness assumption on a nonlocal condition, which generalizes recent conclusions on this topic. The obtained results will be applied to a class of functional partial differential equations with nonlocal conditions.

Some oscillation results for second order neutral type difference equations

This paper is concerned with the oscillatory behavior of second order neutral differ- ence equations. Four oscillation theorems for such equations are established and examples are given to illustrate the results. 1. Inroduction In this paper we are concernedwith the oscillation problemof secondorderneutral

On solutions of a class of systems of nonlinear functional differential equations of neutral type with complicated deviations of an argument

Sufficient conditions for the existence of a unique bounded solution together with its first derivative on R+ of a system of nonlinear functional differential equations with an iterated deviating argument dependent of unknown function, and satisfying the condition limt→+∞|x(t+1)-x(t)|=0, are given.

Stability of solutions to differential equations of neutral type

We consider systems of quasilinear delay differential equations of neutral type and establish the condition of stability of the zero solution, as well as uniform estimates for the solution on the half-axis. In the case of asymptotic stability, these estimates provides the decay rate of the solution at infinity. Bibliography: 17 titles.

Periodic solutions in the α-norm for some neutral partial functional differential equations with finite delay

The aim of this work is to study the existence of periodic solutions in the α-norm for some partial differential equations of neutral type with finite delay. We assume that the linear part is densely defined and is the generator of an analytic semigroup. The delayed part is assumed to be periodic with respect to the first argument. In nonhomogeneous linear case, we show that the existence of a bounded solution in \({\mathbb{R}^+}\) implies the existence of periodic solution. In nonlinear case, we use two approaches, the first one is based on the ultimate boundedness of the solutions and the second one is based on the multivalued fixed point theory.

Continuous Dependence of Fixed Points on Parameters and Initial Conditions

The main purpose of the paper is to establish conditions for a continuous dependence of fixed points and an application to non-linear functional differential equation of neutral type have been made.DOI: http://dx.doi.org/10.3329/dujs.v60i1.10330 Dhaka Univ. J. Sci. 60(1): 21-24, 2012 (January)

Oscillation of nonlinear impulsive parabolic differential equations of neutral type

Oscillation of nonlinear impulsive parabolic differential equations of neutral type

A regularization for discontinuous differential equations with application to state-dependent delay differential equations of neutral type

We consider a regularization for a class of discontinuous differential equations arising in the study of neutral delay differential equations with state dependent delays. For such equations the possible discontinuity in the derivative of the solution at the initial point may propagate along the integration interval giving rise to so-called “breaking points”, where the solution derivative is again discontinuous. Consequently, the problem of continuing the solution in a right neighborhood of a breaking point is equivalent to a Cauchy problem for an ode with a discontinuous right-hand side (see e.g. Bellen et al., 2009 [4]). Therefore a classical solution may cease to exist.The regularization is based on the replacement of the vector-field with its time average over an interval of length ε>0. The regularized solution converges as ε→0+ to the classical Filippov solution (Filippov, 1964, 1988 [13,14]). Several properties of the solutions corresponding to small ε>0 are presented.

An existence result for fractional differential equations of neutral type with infinite delay

In this paper, the existence of mild solutions for the fractional differential equations of neutral type with infinite delay is obtained under the conditions in respect of the Kuratowski’s measure of noncompactness. As an application, the existence of mild solution for some integrodifferential equation is obtained.

On the oscillation of third order half-linear neutral type difference equations

In this paper, the authors study the oscillatory properties of third order quasilinear neutral difference equation of the form

Convergence and Divergence of the Solutions of a Neutral Difference Equation

We investigate the asymptotic behavior of the solutions of a neutral type difference equation of the form Δ[x(n) + cx(τ(n))] + p(n)x(σ(n)) = 0, where τ(n) is a general retarded argument, σ(n) is a general deviated argument (retarded or advanced), c ∈ ℝ, (p(n)) n≥0 is a sequence of positive real numbers such that p(n) ≥ p, p ∈ ℝ+, and Δ denotes the forward difference operator Δx(n) = x(n + 1) − x(n). Also, we examine the asymptotic behavior of the solutions in case they are continuous and differentiable with respect to c.

Smooth solutions of an initial-value problem for some differential difference equations

The subject matter of this paper is an initial-value problem with an initial function for a linear differential difference equation of neutral type. The problem is to find an initial function such that the solution generated by this function has some given smoothness at the points multiple of the delay. The problem is solved using a method of polynomial quasisolutions, which is based on a representation of the unknown function in the form of a polynomial of some degree. Substituting this into the initial problem yields some incorrectness in the sense of degree of polynomials, which is compensated for by introducing some residual into the equation. For this residual, an exact analytical formula as a measure of disturbance of the initial-value problem is obtained. It is shown that if a polynomial quasisolution of degree N is chosen as an initial function for the initial-value problem in question, the solution generated will have smoothness not lower than N at the abutment points.

Existence Results for Semilinear Perturbed Functional Differential Equations of Neutral Type with Infinite Delay

In this paper, by using semigroup theory and a nonlinear alternative for the sum of a completely continuous and a contraction operator, we establish sufficient conditions for the existence of solutions for perturbed semilinear neutral functional differential equations with infinite delay.

PERIODIC SOLUTIONS FOR SOME NONLINEAR PARTIAL NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS

In this work, we study the existence of periodic solutions for some nonlinear partial functional differential equation of neutral type. We assume that the linear part is nondensely defined and satisfies the Hille–Yosida condition. The delayed part is assumed to be ω-periodic with respect to the first argument. Using a fixed point theorem for multivalued mapping, some sufficient conditions are given to prove the existence of periodic solutions.

On existence of solutions of a neutral differential equation with deviating argument

We establish theorems on the existence and asymptotic characterization of solutions of a differential equation of neutral type with deviated argument on neutral type. The mentioned differential equation admits both delayed and advanced arguments. In our considerations we use technique linking measures of noncompactness with the Tikhonov fixed point principle in suitable Frechet space. This approach admits us to improve and extend some results.

A Note on a Semilinear Fractional Differential Equation of Neutral Type with Infinite Delay

We deal in this paper with the mild solution for the semilinear fractional differential equation of neutral type with infinite delay: , , , , with and . We prove the existence (and uniqueness) of solutions, assuming that is a linear closed operator which generates an analytic semigroup on a Banach space by means of the Banach's fixed point theorem. This generalizes some recent results.