LYAPUNOV STABILITY SOLUTIONS OF FRACTIONAL INTEGRODIFFERENTIAL EQUATIONS

Lyapunov stability and asymptotic stability conditions for the solutions of the fractional integrodiffrential equations x(α)(t) = f (t, x(t)) + t t0 K(t, s, x(s))ds ,0 <α ≤ 1, with the initial condition x (α−1) (t0) = x0, have been investigated. Our methods are applications of Gronwall’s lemma and Schwartz inequality.


Introduction. Consider the fractional integrodiffrential equations of the type
x (α) with the initial condition where R is the set of real numbers, J = [t 0 ,t 0 + a], f ∈ C[J × R n , R n ], and where R n denotes the real n-dimensional Euclidean space, and x 0 is a real constant.
The existence and uniqueness of solution of fractional differential equations, when the integral part in (1.1) is identically zero, has been investigated by some authors, see [1,3,5,6].
In recent papers [7,8], we used Schauder's fixed-point theorem to obtain local existence, and Tychonov's fixed-point theorem to obtain global existence of solution of the fractional integrodifferential equations (1.1) and (1.2).The existence of extremal (maximal and minimal) solutions of the fractional integrodiffrential equations (1.1) and (1.2) using comparison principle and Ascoli lemma has been investigated in [9].
In this paper, we are concerned with the stability and asymptotic stability, in the sense of Lyapunov, for the solution of the fractional integrodiffrential equations (1.1) and (1.2).We will assume that f (t,0) ≡ 0 and K(t, s(t), 0) ≡ 0 for all t ∈ J, so that x = 0 is a solution of (1.1).
The zero solution is said to be stable (in the sense of Lyapunov) if, given ε > 0, there exists δ > 0 such that any solution x(t) of (1.1) satisfying |x(t 0 )| < δ for t = t 0 also satisfies |x(t)| < ε for all t ≥ t 0 .The zero solution is said to be asymptotically stable if, in addition to being stable, |x(t)| → 0 as t → ∞.
Our result is a generalization of Hadid and Alshamani [4], in which it was shown that under certain conditions on f the zero solution of the initial value problem (IVP): with the initial condition x (α−1) (t 0 ) = x 0 is stable and hence it is asymptotically stable.
Next we set forth definitions and lemmas to be used in this paper.For proofs and details see [1,2,3].
provided that this integral (Lebesgue) exists, where Γ is the Gamma function.
Lemma 1.2.The IVP (1.1) and (1.2) is equivalent to the nonlinear integral equation where 0 < t 0 < t ≤ t 0 + a.In other words, every solution of the integral (1.5)

is also a solution of the original IVP (1.1) and (1.2), and vice versa.
Proof.It can be proved easily by applying the integral operator (1.4) with a = t 0 and b = t to both sides of (1.1), as we did in [4], and using some classical results from fractional calculus in [2] to get (1.5).
Lemma 1.3 (Gronwall's lemma).Let u(t) and v(t) be nonnegative continuous functions on some interval t 0 ≤ t ≤ t 0 + a.Also, let the function f (t) be positive, continuous, and monotonically nondecreasing on t 0 ≤ t ≤ t 0 + a and satisfy the inequality then, there exists Proof.For the proof of Lemma 1.3, see [10].

Stability conditions.
In this section, we will prove our main results, and discuss the stability and asymptotic stability of the solution of (1.1) satisfying (1.2).
where C 0 is a positive constant, and hence the solution of (1.1) and (1.2) is asymptotically stable.
Corollary 2.3.It can easily be shown from (2.10) that (2.11) Next, we will prove another important stability result.The result is in connection with α; the method we will use is an application of Schwartz inequality.
Proof.For 0 ≤ t 0 < s < t ≤ t 0 + a, it follows from (1.5) that (2.12) By applying the absolute value, we get ( By Schwartz inequality, we obtain (2.14) Now, using (i) and (ii) in the statement of the theorem and integrating, we obtain where C 1 and C 2 are positive constants (we can calculate them easily).By (2.15), and for 0 ≤ t 0 < s < t ≤ t 0 + a, we have This implies that the zero solution of (1.1) and (1.2) is asymptotically stable.Hence the theorem is proved.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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