Functional and impulsive differential equations of by Ivanka Stamova, Gani Stamov

By Ivanka Stamova, Gani Stamov

The booklet provides qualitative effects for various periods of fractional equations, together with fractional sensible differential equations, fractional impulsive differential equations, and fractional impulsive practical differential equations, that have now not been coated through different books. It manifests varied positive equipment via demonstrating how those suggestions might be utilized to enquire qualitative houses of the options of fractional structures. given that many purposes were incorporated, the confirmed thoughts and types can be utilized in education scholars in mathematical modeling and within the examine and improvement of fractional-order models.

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Examples will be presented to illustrate our theoretical findings. 8 is devoted to the existence and stability of integral manifolds for fractional functional differential systems and impulsive fractional functional differential systems. 1). Throughout the rest of the book we shall use the class K = {w ∈ C[R+ , R+ ] : w(u) is strictly increasing and w(0) = 0} of functions w, which are called wedges. 1 V (t, 0) = 0, t ≥ t0 . 2) is valid whenever V (t + θ , φ (θ )) ≤ V (t, φ (0)) for −r ≤ θ ≤ 0, where t ∈ [t0 , ∞), φ ∈ C[[−r, 0], Ω ], β ∈ (0, 1).

6). 14 Given a function V ∈ V0 . For (t, φ ) ∈ R × PC[[−r, 0], Ω ], t ̸= tk , k = ±1, ±2, ... 6) is defined by c β D+V (t, φ (0)) = lim sup χ →0+ 1 [V (t, φ (0)) −V (t − χ , φ (0) − χ β f (t, φ ))]. χβ In the next chapters we shall also use the following classes of piecewise continuous Lyapunov-like functions { V1 = V : R × Ω × Ω → R+ , V is continuous in (tk−1 ,tk ] × Ω × Ω , } V (tk , x, y) = V (tk− , x, y) and lim V (t, x, y) = V (tk+ , x, y), x, y ∈ Ω . e. for x1 , x2 ∈ Ω , y1 , y2 ∈ Ω and for t ∈ R it follows ( ) |V (t, x1 , y1 ) −V (t, x2 , y2 )| ≤ LV ||x1 − x2 || + ||y1 − y2 || , t ̸= tk , k = ±1, ±2, ....

Bk ∈ C[R+ , R+ ] and ψ˜ k (u) = u + Bk (u) ≥ 0, k = ±1, ±2, ... are non-decreasing with respect to u. Introduction 27 3. 15) exists on [t0 , ∞). 4. The function V ∈ V0 is such that V (t + , x + Ik (x)) ≤ ψ˜k (V (t, x)), x ∈ Ω , t = tk , tk > t0 , c β D+V (t, x) ≤ g(t,V (t, x)), (t, x) ∈ G, t ∈ [t0 , ∞). Then V (t0+ , x0 ) ≤ u0 implies V (t, x(t;t0 , x0 )) ≤ u+ (t;t0 , u0 ), t ∈ [t0 , ∞). 5. 3 Assume that the function V ∈ V0 is such that: V (t + , x + Ik (x)) ≤ V (t, x), x ∈ Ω , t = tk , tk > t0 , c β D+V (t, x) ≤ qV (t, x), (t, x) ∈ G, t ∈ [t0 , ∞).

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