On the behaviour of solutions to a kind of third order neutral stochastic differential equation with delay

This article demonstrates the behaviour of solutions to a kind of nonlinear third order neutral stochastic differential equations. Setting x′(t)=y(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x^{\prime }(t)=y(t)$\end{document}, y′(t)=z(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$y^{\prime }(t) =z(t)$\end{document} the third order differential equation is ablated to a system of first order differential equations together with its equivalent quadratic function to derive a suitable downright Lyapunov functional. This functional is utilised to obtain criteria which guarantee stochastic stability of the trivial solution and stochastic boundedness of the nontrivial solutions of the discussed equations. Furthermore, special cases are provided to verify the effectiveness and reliability of our hypotheses. The results of this paper complement the existing decisions on system of nonlinear neutral stochastic differential equations with delay and extend many results on third order neutral and stochastic differential equations with and without delay in the literature.


Introduction
To analyse or describe numbers of urbane dynamical systems in sciences, social sciences, engineering and health sciences, neutral and stochastic differential equations, with or without delay or randomness, cannot be disregarded or unnoticed. In general, applications of functional differential equations are found in viscoelasticity, pre-predator and control problems, aeroautoelasticity, Brownian particles found in a limitless environment (or medium), motion of a rigid body under control, stretching of a polymer filament, dynamics of oscillator in a vacuum tube, energy source and their interaction in physics, motion of auto-generators with delay, general theory of relativity [10,13,14,[22][23][24]26]. These amazing practical utilisations of functional differential equations in solving reallife phenomena have recently geared up or accelerated research in these directions, see for example the survey books of Arnold [10], Burton [13,14], Driver [19], Hale [22][23][24], Kolmanovskii and Myshkis [26], Yoshizawa [50], to mention but a few, where theories and applications of functional differential equations are discussed.
Abou-El-Ela et al. [1], by employing Lyapunov direct method, addressed the problem of stochastic asymptotic stability and the uniform stochastic boundedness of nonzero solutions for the third order differential equation where a 1 , b 1 , c 1 and σ 1 are positive constants ρ(t) ∈ R is the standard Brownian motion defined on the probability space. Ademola [4], using the second method of Lyapunov, discussed the problem of stability, boundedness, existence and uniqueness of solution of the third order nonlinear stochastic differential equation with delay, namely where a 2 > 0, b 2 > 0, σ 2 > 0 are constants, h, e 2 are nonlinear continuous functions depending on the displayed arguments, h(0) = 0, τ > 0 is a constant delay and ρ(t) ∈ R is defined above.
By introducing more nonlinear functions into the existing equations, Mahmoud and Tunç [32] constructed a suitable Lyapunov functional and applied it to give criteria for the asymptotic stability of the zero solution to nonlinear third order stochastic differential equations with variable and constant delays defined as where a 3 > 0, σ 3 > 0, h > 0 are constants, r(t) is a continuously differentiable function with 0 ≤ r(t) ≤ γ 1 , γ 1 > 0 is a constant, φ, ψ are continuously differentiable functions defined on R such that φ(0) = 0 = ψ(0), and ρ(t) ∈ R m is defined above.
Many papers have been published on the stability and boundedness of solutions of neutral differential equations, Oudjedi et al. [42] established conditions for integrability, boundedness and convergence of solutions to the third order neutral delay differential equations where β and τ are constants with 0 ≤ β ≤ 1 and τ ≥ 0, e 3 (t) and f (w) continuous functions depending only on the arguments shown and f (w) exists and is continuous for all w. By replacing the linear differentiable function w (t) with a nonlinear delay differentiable function, Ademola et al. [5] itemized criteria for uniform asymptotic stability and boundedness of solutions to the nonlinear third order neutral functional differential equation with delay defined as where τ > 0 is a constant delay, φ is a constant satisfying 0 ≤ φ ≤ 1, the functions ϕ(t), χ(t), ψ(t), g(y), h(w) are continuous in their respective arguments on R + , R + , R + , R, R respectively. Besides, it is supposed that the derivatives g (y) and h (w) exist and are continuous for all w, y and h(0) = 0.
The objective of this paper is to obtain sufficient conditions for the stability and boundedness of solutions of the following neutral stochastic differential equation with delay of third order: where p(·) = p(t, x(t), x(tτ (t)), x (t))), φ is a constant satisfying 0 ≤ φ ≤ 1 2 , the continuous functions ψ(t), h(x) and p(·) depending only on the arguments shown and h (x) exist and are continuous for all x; the constants σ , a, b and β are positive with 0 ≤ τ (t) ≤ β, which will be determined later, ω(t) ∈ R is the standard Brownian motion.
Setting x (t) = y(t), x (t) = z(t) and Y (t) = x (t) + φx (tτ (t)), then (1.1) is equivalent to the system of first order differential equations By a solution of (1.1) or (1.2), we have a continuous function x : Then from (1.2) we get We observed that the stochastic differential equations discussed in [1-4, 6-8, 32] exempt neutral term similar to [5,11,12,15,16,[42][43][44]47] where neutral differential equations are considered and the stochastic term is exempted. Equation (1.1) is therefore an extension of these results and the references listed therein as both terms (neutral and stochastic which formed the major contribution of this paper) are included in equation (1.1). It is noteworthy to mention at this junction that the inclusion of both neutral and stochastic terms to equation (1.1) make the authentication or confirmation of Lyapunov functional more difficult to obtain than before. Thus the Lyapunov functional employed in this study includes and generalises the existing functionals employed in [1-4, 6-8, 32] and [5,11,12,15,16,[42][43][44]47] where qualitative behaviour of solution of stochastic differential equations and neutral functional differential equations are respectively considered. In addition, equation (1.1) is a special case of the systems of neutral stochastic differential equations discussed in [9, 10, 20-22, 34-39, 45, 46].
Consider a non-autonomous n-dimensional stochastic delay differential equation Here f : R + ×R 2n → R n and g : R + × R 2n → R n×m are measurable functions and satisfy the local Lipschitz condition. Let B(t) = (B 1 (t), B 2 (t), . . . , B m (t)) T be an m-dimensional Brownian motion defined on the probability space. Hence, the stochastic delay differential equation admits trivial solution x(t, 0) ≡ 0 for any given initial value x 0 ∈ C([-r, 0]; R n ).

Definition 1.2
The trivial solution of the stochastic differential equation (1.4) is said to be stochastically asymptotically stable if it is stochastically stable and, in addition, if for every ε ∈ (0, 1) and κ > 0 there exists δ = δ(ε) > 0 such that where C : R + × R n → R + is a constant function depending on t 0 and x 0 , E x 0 denotes the expectation operator with respect to the probability low associated with x 0 .

Definition 1.4
The solution x(t 0 ; x 0 ) of the stochastic differential equation (1.4) is said to be uniformly stochastically bounded if C in (1.5) is independent of t 0 .
Section 2 considers the stability of the trivial solution, ultimate boundedness of solution is discussed in Sect. 3, and finally illustrative examples are presented in the last section.

Stability of the trivial solution
Now, we shall state here the stability result of (1.1) with p(·) ≡ 0. (1.1), suppose that there are positive constants ψ 0 , h 0 , h 1 and α such that the following conditions are satisfied:

Theorem 2.1 In addition to the assumptions imposed on the functions that appeared in
and Then the trivial solution of (1.1) is uniformly stochastically asymptotically stable, provided that , .

1)
and assumptions (H 1 ) to (H 7 ) of Theorem 2.1 reduce to Routh Hurwitz criteria a > 0, b > 0, c > 0, ab > c for asymptotic stability of the trivial solution of equation is cut down to that discussed in [4]. The assumptions of Theorem 2.1 include and extend the stability results in [4] Theorems 3.3 and 3.4; (iii) Suppose that φ = 0 and ψ(t) = c 1 , then equation (1.1) is weakened to that discussed in [1] and some of our assumptions are similar. Thus the uniform stability result obtained in Theorem 2.1 include and extend the stochastic stability result (Theorem 2.3) discussed in [1]; (iv) If τ (t) = τ > 0 is a constant delay and σ = 0, then equation (1.1) specialises to that considered in [5] and [42], our assumptions in Theorem 2.1 include Theorem 2.1, Corollary 2.2 in [5] and the asymptotic stability Theorem 2.1 in [42] provided that a(t) = b(t) = constant; (v) To crown it all, Theorem 2.1 includes and extends the stochastic stability results considered in [1,4,5,42] and the references cited therein.
Proof of Theorem 2.1.Let (x t , y t , z t ) be any solution of (1.1) or (1.2) with p(·) ≡ 0, we define a Lyapunov continuously differentiable functional V = V (x t , y t , z t , t) employed in this work as follows: with λ 1 , λ 2 , η 1 and η 2 being positive constants which will be specified later. From conditions (H 1 ) and (H 2 ), we have Furthermore, from the definition of V 1 , we get In the same way, it follows that Then From this inequality and (H 4 ), we can deduce a positive constant K 0 such that Since where Using the fact 2|uv| ≤ u 2 + v 2 , we obtain Then there exists a positive constant K 2 such that Therefore, from (2.3) and (2.5), we note that the Lyapunov functional V satisfies the inequalities By using Itô's formula, the derivative of the Lyapunov functional V is given by

From system (1.2) and (1.3), with conditions (H 1 ) -(H 3 ), it follows that
Applying the estimate |uv| ≤ 1 2 (u 2 + v 2 ), we obtain If we let It follows that From conditions (H 6 ) and (H7), the last inequality becomes Therefore, there exists a positive constant K 3 such that , Thus, from (2.7) the inequality is satisfied, then the trivial solution of (1.1) with p(·) ≡ 0 is uniformly stochastically asymptotically stable. This completes the proof of Theorem 2.1.

Ultimate boundedness of solutions
Our main theorem in this section with respect to (1.1) is as follows. , .
Proof of Theorem 3.1.Consider the Lyapunov functional U(x t , y t , z t , t) as follows: where V is defined as (2.2) and W is defined as follows: Now, we shall prove that is satisfied for (1.1) where p 1 and p 2 are positive constants, p 1 ≥ 1. It suffices to show it for W , since it was already proved for V in Sect. 2. We shall use the same techniques, which have already been demonstrated in the proof of Theorem 2.1. Thus from (3.2) we get Therefore, from (H 2 ) and (H 8 ), we obtain W ≥ L x 2 + y 2 + Z 2 for some L > 0.
where D 1 = min{K 1 , L}. Now, by using conditions (H 1 ) and (H 2 ) of Theorem 2.1, we can rewrite (3.2) as the following form: Since |uv| ≤ 1 2 (u 2 + v 2 ), then we get Combining the foregoing inequalities (2.4), (3.1) and (3.5), we have Then we can find a positive constant D 2 such that the last inequality gives Now from the results (3.4) and (3.6), we can find the Lyapunov functional U which satisfies the inequalities Also we can check that is satisfied since p 1 = p 2 = 2 and = 0. From (3.2), (1.2), (1.3) and the definitions of Y (t) and Z(t), we get First, we show that W 1 is a negative definite function, we can rewrite W 1 as the following form: From the assumptions ψ (t) ≤ 0 and h (x) ≤ a, we get W 1 ≤ 0.
Then one can conclude for some positive constants K and ω that LU ≤ -ω x 2 + y 2 + z 2 + Kω |x| + |y| + |z| Then we find Thus, satisfying the inequality t t 0 for some positive constant M. Now, we have the following: It follows that Hence, all solutions of (1.1) are uniformly stochastically bounded. Therefore, the proof of Proof See (2.4)-(2.6) on page 202 in [38].
Remark 3.2 It is noteworthy to mention here that some of our assumptions, and the result of Corollary 3.1 in particular, complement some existing results on the system of neutral stochastic differential equations with delay in literature.