The main aim of this work is to investigate the existence and approximate controllability of mild solutions of impulsive fractional nonlinear control system with a nonsingular kernel in infinite dimensional space. Firstly, we set sufficient conditions to demonstrate the existence and uniqueness of the mild solution of the control system using the Banach fixed point theorem. Further, we prove the approximate controllability of the control system using the sequence method.
Let a continuous maps f:X → X, g:X → X be defined on a compact metric space X. We showed that f has bi-shadowing with g is equivalent to f has backwards bi-shadowing with g and f has two-sided bi-shadowing with g when the maps f, g are onto. We used this result to go on prove that, for expansive surjective maps f, g the properties f has bi-shadowing with g, f has two-sided bi-shadowing with g, f has s- limit bi-shadowing with g and f has two-sided s-limit bi-shadowing with g are equivalent. Finally we concluded that if a continuous maps f, g are surjective. Then f has two-sided s-limit bi-shadowing with g if and only if it has L-bi-shadowing with g.
The inverse shadowing property is concentrated, it has important properties and applications in maths. In this paper, some general properties of this concept are proved. Let ( be a metric space: ( ? ( be maps have the inverse shadowing property. We show the maps ? , have the inverse shadowing property.
If and :( , ????) ?( ,????) are mapped on a metric space ( ,????) have the inverse shadowing property, We show the maps + and . have the inverse shadowing property.
Let (M, d) be a compact metric @-space, Φ : M → M be a continuous map. This paper aims to study the idea of the sequence-asymptotic average shadowing property ( {si
}-AASP ) for a continuous map on-space, ( {si:i≥1} be a given positive integers sequence, where s
0 = 0 ) and achieves the relative of the {si
}-AASP with the sequence AASP ( {si
}-AASP ). We prove that if (M, d) are metric 1-spaces, (X, d) then metric 2-spaces and ƒ : 1 χ → Μ, ψ : 2 x Χ → Χ are continuous maps, then ƒ has the 1 {si
}-AASP and ψ has the 2{si
}-AASP if and only if the product ƒ x ψ has the 1 x 2{si
}-AASP. Also, we show that if Φ has the {si
}-AASP then Φ is-chain transitive.
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