A comparison between applications of the Lyapunov’s second (direct) method and fixed point theory

In this article, we will discuss the application of the Lyapunov’s second method and fixed point theories to certain differential equations of first and second order. First, we will introduce some basic information about these subjects, and later, we give their applications concerning some specific attitude of Solutions of Delay Differential Equations. We will also do a comparison between them. Include keywords, mathematical subject classification numbers as needed. c ©2019 All rights reserved.


Some Basic Information
In this section, we first give some basic information about the presence and uniqueness solutions of ordinary differential equations.
Theorem 2.2. If g(t, u) and ∂g ∂u are the continuous of t and u in the region R(a, b) : |t − t 0 | ≤ a, |u − u 0 | ≤ b, then there exists a unique solution u(t) to the initial value problem. u = g(t, u), u(t 0 ) = u 0 , on the some interval |t − t 0 | ≤ h ≤ a ,where h is a sufficiently small positive constant. It should be noted that the linear initial value problem has a unique solution on the whole interval |t − t 0 | ≤ h . In other words, there is no apparent relationship between the region where the function g(t, u) is continuous and the interval of existence of the solution.
where both a and b are positive real numbers, and satisfies the Lipschitz condition in B 0 . Let Then, the initial value problem has unique solution x(t) on [t 0 , t 0 + α] . Let us consider the system of differential equations with initial condition where i = 1, 2, . . . , n. In vector notation, the above equations written as follows . where x= (x 1 , x 2 , . . . , x n ) , f = ( f 1 , f 2 , . . . , f n ) , and x 0 = (x 1o , x 2o , . . . , x no ) are vectors in R n . We shall is called a continuation of the solution u(t) to I 1 ⊃ I if v(t) is defined on the interval I 1 , v(t) ≡ u(t) on I, and v(t) satisfies the initial value problem u = g (t, u) , u(t 0 ) = u 0 on I 1 . Definition 2.5. A metric space (X, d) is complete if every Caochy seqeance in X is convergent to an aliment in X.
In Math, the fixed point of a function is a point in a function field that is assigned to the function alone. Sometimes a set of elements is called a fixed set. That is, c is the fixed point of the function Many different kinds of problems can be solved by means of fixed point theory. Generally, to solve a problem with fixed point theory is to find: (a) a set S consisting of points which would be acceptable solutions; (b) a mapping P : S → S, with the property that a fixed point solves the problem; (c) a fixed point theorem stating that this mapping on this set will have a fixed point. Theorem 2.6. If φ:(S, d) → (S, d) is a contraction mapping and if (S, d) is complete metric sparse, then φ has aunique fixed point; that's, there is a unique s * ∈ S, such that φ(s * ) = s * . Definition 2.7. The solution x(t) of EQ.(4) it said to be stable if for each ε > 0, there exists a δ = δ (t 0 , ε) > 0 such that, for any solution Hence, we have It follows that When we use the above initial conditions, we get Let us apply the definition of the stability. When then it follows that This implies the zero solution of the equation is stable. Definition 2.9. The solution x(t) of EQ.(4) is said to be uniformly stable if for each ε > o, there exists a δ=δ (ε) > o such that, for any solution x (t) = x(t, t 0 , x 0 ) of EQ. (4), the inequalities t 1 ≥ t 0 and <ε for all t ≥ t 1 . Example 2.10. Consider the initial value problem It follows that Let us apply the definition of the stability. When This means that the zero solutions of the equation is uniformly stable. Definition 2.11. The solution x(t) of EQ.(4) is called asymptotically stable if it is stable and if there exists Consider the above first example. When we take into consideration it solution clearly, Then the zero solution is also asymptotically stable. Definition 2.12. The solution x(t) of EQ. (4) is known as unstable if it is not stable. Example 2.13. Consider the initial value problem Hence, we have It follows that When we use the above initial conditions, we get Let us apply the definition of the stability. When then it follows that when t sufficiently bigger. This means that the zero solution of the system is unstable. Definition 2.14.
The solution x(t) of EQ. (4) is called uniformly asymptotically stable if it is uniformly stable and there is a δ 0 > 0 and,

Definition 2.15.(Lyapunov Function) is any differentiable and continuous function
also the time derivative of this function is quasi-specific negative, given negative or positive definite. Consider the general non -autonomous delay differential system: [2]). Assume that a Lyapunov function for EQ.(4) and wedges satisfying; Then the zero solution of EQ.(4) is uniformly stable. Theorem 2.17. (Yoshizawa [37]). Suppose that there exists a continuous Lyapunov function V (t, ϕ) defined for all t ∈ R + and ϕ ∈ S · , which satisfies the following conditions; Then, the solutions of EQ.(4) are uniformly bounded. Theorem 2.18. If there is a Lyapunov function for EQ.(4) and wedges S.T Then x = 0 is uniformly asymptotical stable.

Application of the Lyapunov's second method
3.1. The Lyapunov's first method. By solving an initial value problem, after finding solutions, we apply definition of the stability whether the founded solution is stable or not. This method is called the Lyapunov's first method.

The Lyapunov's second (direct) method.
Without solving any initial value problem, by define a suitable Lyapunuov function, we can determine the stability of solution subject to the Lyapunov's theorems; this method is called Lyapunov's second method. We begin with the scalar equation where a , b, and r are constants, r ≥ 0. Theorem3.3. If a > 0 and |b| < a, then the zero solution for EQ.(5) above is uniformaly asymptoticaly stable for every r ≥ 0. Proof. Define Lyapunov functional The derivative of V along an unknown solution of Eq. (5) satisfies and In view of Theorems (2.16) and (3.3), we can conclude the zero solution of EQ.(5) is stable, uniformly stable, uniformly asymptotical stable, and al solutions are uniformly bounded, too. It should be noted that without solving EQ.(5), we can determine the stability, uniform stability, uniformly roundedness and uniformly asymptotical stability of solutions of equation considered. This shows the advantage of the Lyapunov's second method.
We are now discussing the stability of the same equation with the fixed point theory: Proof. Let ψ: [−r, 0] → R be continuous initial function and let (M, . ) be the complete metric space of continuous functions φ: [−r,∞) → R with φ 0 =ψ and ψ ( t) → 0 as t → ∞. Recall that the notation φ 0 =ψ means that φ ( t) =ψ ( t) for −r ≤ t ≤ 0. Use the variation of parameters formula is and use this to define a mapping P : M → M by φ ∈ M and −r ≤ t ≤ 0 implies that It is clear that φ ∈ M . To prove that P is a contraction, suppose that φ ,η ∈ M. Then This is a contraction since a > |b| and so P had the unique fixed point φ ∈ M. Since the fixed point φ resides in M it follows that φ(t) → 0. Of course from EQ.(6), we see that the fixed point solves EQ. (5). Moreover, to prove stability, we apply norms to both sides of EQ.(6) at the fixed point and obtain From which the stability relation is easily derived. We motivate our next example by considering the following delay differential equation with variable coefficients: where a and b are bounded continuous functions, It should be noted that this delay differential equation includes the preceding delay differential equation.
Using the Lyapunov functional With the triangle inequality, we have It can now be argued that this yields uniformly asymptotically stability of solutions. Remark3.4. Consider the delay differential equation of the form It follows that the assumptions a > 0 and |b| < a guarantee, then the stability of zero solution given EQ.(5). Next, when we apply the fixed point theory it follows that the assumption a > |b| guarantee stability of zero solution of above equation, too. From which it follows that when we apply the Lyapunov second method and the fixed point theory, then the assumptions of theorems are the same for delay differential equation of first order with the constant coefficients. Consider the differential equation of first order linear delay with variable delay of the model Theorem3.5. We assume the following condition satisfied: ) , for some α ∈ (0, 1) . So the zero solution of EQ. (7) is uniformaly asymptoticaly stable. Proof. Define Lyapunov functionalV (t) by x 2 (s) ds.
It's clear V (0) = 0, and V(x) ≥ x 2 . This indicates that V(x) is positive. Then, along with solutions EQ. (7), we have In view of the conditions 0 < α < 1 and 1 − g (t) > 0, it follows that V (x) is negative defined. Through theory 10, we deduce that the zero solution of EQ. (7) is uniformly asymptotically stable. Now we will prove the stability result for EQ.(7) by fixed point theory Theorem3.6. We supposes that the following assumptions contract: a, b, and g are continuous functions such that there is an L > 0, | x | , | y | ≤ L, g (0) = 0 and |g (x) − g (y)| ≤ | x − y |. Then, every solution of EQ. (7) with small continuous initial function tends to o as t → ∞. So the zero solution is stable. Proof. We will take t 0 = 0 . For the α and L, find δ> 0 with δ + αL ≤ L. Let ψ : (−∞, 0] → R be a given continuous function with |ψ(t)| < δ & let where . is the supremum metric. Then (S, . ) is complete metric space.
Clearly, Pφ ∈ C. We now show that (Pφ) (t) → 0 as t → ∞. Let φ ∈ S and ε > 0 be given. Then To see that P is contraction under the supremum metric, if φ, η ∈ S, then With α< 1 by the condition t 0 e − t s a(u)du |b(s)| ds ≤ α, t ≥ 0, Hence, P has a unique fixed point in S which shows that every solutions EQ.(7) tends to zero, and moreover the zero solution of EQ. (7) is stable. Now we study the stability and asymptotically stability of the zero solution of EQ.(7) by funding the Lyapunov's direct method. Remark3.7. Consider the linear delay differential equation of first order with variable delay of the model It follows that, when we apply the Lyapunov's direct method, the following assumptions g (t) is bounded, and |b (t)| ≤ α (1 − g (t)) , for some α ∈ (0, 1) . Are suficient for the uniformly asymptotically stability of zero solution of EQ. (7). Besides, when we apply fixed point theory, instate of the mentioned conditions, the following assumptions guarantee the asymptotically stability of zero solution: and there is an L > 0 so that if | x | , |y| ≤ L then g (0) = 0 and In the application of the Lyapunov method, the differentiability of the function g is needed, however, this condition is not needed for fixed point theories. This is a disadvantage for the Lyapunov method, however, an advantage in the case of applications of fixed point theories. However, the case of application of fixed point theories, we want impose the condition.
This is a stronger condition. However, during the application of Lyapunov method, we don't need this condition. This is advantage of Lyapunov method.
Construction or definition of a Lyapunov functional is needed when we apply this method. This is the disadvantage of this method. We can give more discussion about the application of these methods. But, we would not like to give here more details. We now consider non-autonomous differential equation of second order And also we can write out the EQ. (8) in the sample order as follows x (t) = y(t) g (x (s)) y (s) ds + q (t), (9) Where deviation τ (t) is the median of the variable deviation, f , b, q and g are continuous functions in R 3 , R and [0, ∞) respectively, and are also dependent on arguments candor presented with g(0) = 0. Continoity these functions are suficient requirements to equate the solution of EQ. (8). It is also assumed that the functions f and g meet the Lipchitz state in numbers x , x and x (t − τ(t)). Through this assumption the uniqueness of EQ. (8) solutions is guaranteed. The derivatives ( dg dx = g (x)) is occur and continuous.
By determining the Lyapunov function (se Krasonvskii [23]) sufficient conditions for stabilization and limits of solutions Eq.(8) are obtained. We prove the following theorem by the Lyapunov's second method. Theorem3.8. Further assuming the basic concepts for the functions f, b and g we assume that the continuum has a constant L, hence to keep the following terms: Hence the zeroes solution of EQ.(4) is stable. Proof. Defining the Lyapunov function Where λ > 0, which will be obtained later and e (t) is a continuous function on R + = [0, ∞) and t 0 |e (s)| ds < ∞, then it follows that By the time derived the Lyapunov function V (t) over the system (5), we have By assumptions of Theorem (3.1) and the estimate |mn| ≤ m 2 2 + n 2 2 , it follows that Hence Let λ = L 2 , so that Will not cause any confusion even if we using ψ(s) us supremam on[m (t 0 ) , ∞). It known in [2], for a given continuous function Φ and for the number y 0 , there is a solution for the system (10) at an interval [t 0 , T) if the solution residue selected, then T = ∞). Let A (t) = f (t, x (t) , y (t) . So the system (10) can write as Pi [19] proved the followings theory through fixed points. Theorem3. 10. We suppose that the following assumptions hold: (ii) ∃ l > 0 such that g(x) satisfies the Lipchitz condition on [−l, l]. g(x) is odd and it is increasing on (iii) ∃ α ∈ (0, 1) and a continuous function a (t)  (Pi [19]). We produce the following basic assumptions about the delay function t(t) in EQ. (8): Lemma3.11. Let ψ : [m (t 0 ) , t 0 ] → R be a given continuous function. If x (t) , y(t) is the solution of the system (10) on [t 0 , T 1 ] satisfying x (t) = ψ (t) , t ∈ [m (t 0 ) , t 0 ], and y (t 0 ) = x (t 0 ), then x (t) is solution of integral equation Conversely, if the continuous function is a solution of system (10) on [t 0 , T 2 ].
Then, system (10), can be stated as follows x = y, Multiplying both sides of Eq.(13) by e − t ∞ A(s)ds and integrating, it follows that so that A(s)ds = B(t), then from Eq. (14) we have Hence, it follows so that This completes the proof of Lemma. Theorem3.12. We suppose that the following assumptions hold: (i) ∃ l >0 S.T, the function g Satisfies the condition of Lipschitz on [−l, l] and g is odd and is strictly increasing on [−l, l] and x − g (x) is no decreasing on [−l, l].
(ii) ∃ α ∈ (0, 1), and the continuous function By referring to the assumption g (0) = 0 of the theory, it is clear that g (l) ≤ l. Because g (x) meets the Lipschitz condition in [−l, l] , and g (x) is a continuous function in [−l, l] , there is a constant δ so that δ < l. Thus, from the statement (Pφ)(t), it follows ≤ δ + M t t 0 e − t u A(s)ds |g (x (u − τ (u)))| du We can conclude that |x(t)| + |y(t)| < l 2 + M Q + e −a 0 Q a 0 .
This finishes the proof of theorem.

Conclusion
A Linard equation type with multiple variable delays, EQ.(4), is considered. First, the stability, uniformly stability, bondedness and uniformly bondedness of solutions of this equation (4), have been discussed by the Lyapunov-Krasovskii functional approached. Later, the stability of the solutions of the same equation, when p(t)=0 in equation (4) is studied by fixed point techniques. It is clear that the assumptions of Theorem (3.8) are completely different except the similarity of the assumption of Theorem (2.18) and the assumption of Theorem (3.10).Theoretical assumptions of theorem (3.8) are cleared, gracious and understandable. That is, Theorem (2.16) assumptions had quite simple shapes and can easily prove their applicability. However, the best of our knowledge, it may be difficult to say the same for the assumptions of Theorem (3.8). When we change EQ.(1) into a more complex form, finding an appropriate Lyapunov-Krasovskii functional, which gives meaningful results, may be difficulty. It should be noted construction or definition Lyapunov-Krasovskii functional Residue us the open problem in literature, till now. This fact shows the advantage of the fixed point theory over the Lyapunov's direct method in the special case. Further, in spite of the more effectiveness of the Lyapunov-Krasovskii works in natural and functional differential equations of the highest order, and the application of the fixed point theory of these equations is very difficult, and sometimes it may be impossible. By this fact, we mean that depending on the form and order of the differential equation given, sometimes the Lyapunov-Krasovskii functional approach has an advantage over the fixed point theory, and sometimes it is in the contrast. However, to date, the most effective way to study the qualitative behaviors of nonlinear differential equations in the higher ranks is still the Lyapunov second method. At the end, this method is old but yet active in scientific literature.