Akbar Zada

Hyers-Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems

In this paper, the concepts of Hyers-Ulam stability are generalized for non-autonomous linear differential systems. We prove that the k-periodic linear differential matrix system Z ? ( t ) = A ( t ) Z ( t ) , t ? R is Hyers-Ulam stable if and only if the matrix family L = E ( k , 0 ) has no eigenvalues on the unit circle, i.e. we study the Hyers-Ulam stability in terms of dichotomy of the differential matrix system Z ? ( t ) = A ( t ) Z ( t ) , t ? R . Furthermore, we relate Hyers-Ulam stability of the system Z ? ( t ) = A ( t ) Z ( t ) , t ? R to the boundedness of solution of the following Cauchy problem: { Y ? ( t ) = A ( t ) Y ( t ) + ? ( t ) , t ? 0 Y ( 0 ) = x - x 0 , where A(t) is a square matrix for any t ? R , ?(t) is a bounded function and x , x 0 ? C m .

Uniform Exponential Stability of Discrete Evolution Families on Space of -Periodic Sequences

We prove that the discrete system is uniformly exponentially stable if and only if the unique solution of the Cauchy problem , , is bounded for any real number and any -periodic sequence with . Here, is a sequence of bounded linear operators on Banach space .

On Uniform Exponential Stability and Exact Admissibility of Discrete Semigroups

We prove that a discrete semigroup of bounded linear operators acting on a complex Banach space is uniformly exponentially stable if and only if, for each , the sequence belongs to . Similar results for periodic discrete evolution families are also stated.

Uniform exponential stability of periodic discrete switched linear system

Abstract Let {Πτ(m, n): m ≥ n ≥ 0} be the family of periodic discrete transition matrices generated by bounded valued square matrices Λτ(n), where τ : [ 0 , 1 , 2 , ⋯ ) → Ω is an arbitrary switching signal. We prove that the family {Πτ(m, n): m ≥ n ≥ 0} of bounded linear operator is uniformly exponentially stable if and only if the sequence n ↦ ∑ k = 0 n e i α k Π τ ( n , k ) w ( k ) : Z + → R is bounded.

A fixed point approach to the stability of a nonlinear volterra integrodifferential equation with delay

By using a fixed point method, we prove the Hyers-Ulam-Rassias stability and the Hyers-Ulam stability of a nonlinear Volterra integrodifferential equation with delay. Two examples are presented to support the usability of our results.

Uniform Exponential Stability of Discrete Semigroup and Space of Asymptotically Almost Periodic Sequences

We prove that the discrete semigroup T = {T (n) : n ∈ Z+} is uniformly exponentially stable if and only if for each z(n) ∈ AAP0(Z+,X ) the solution of the Cauchy problem { yn+1 = T (1)yn + z(n + 1), y(0) = 0 belongs to AAP0(Z+,X ). Where T (1) is the algebraic generator of T, Z+ is the set of all non-negative integers and X is a complex Banach space. Our proof uses the approach of discrete evolution semigroups.

Analysis of Implicit Type Nonlinear Dynamical Problem of Impulsive Fractional Differential Equations

We study the existence, uniqueness, and various kinds of Ulam–Hyers stability of the solutions to a nonlinear implicit type dynamical problem of impulsive fractional differential equations with nonlocal boundary conditions involving Caputo derivative. We develop conditions for uniqueness and existence by using the classical fixed point theorems such as Banach fixed point theorem and Krasnoselskii’s fixed point theorem. For stability, we utilized classical functional analysis. Also, an example is given to demonstrate our main theoretical results.

Fixed point theorems in dislocated quasi-metric spaces

In this paper, we discuss the existence and uniqueness of a fixed point in a dislocated quasi-metric space. Several fixed point theorems for distinct type of contractive conditions are presented that generalize, extend, and unify a number of related results reported in the literature. Illustrative examples are provided. c ©2017 All rights reserved.

Criteria for the exponential stability of linear evolution difference equations

Let xV ,f be the solution of the discrete Cauchy problem ζ (n + 1) = B(n)ζ (n) + f (n), ζ(0) = 0, where B(n) is a sequence of bounded linear operators on a Banach space X . We prove that the discrete evolution family V := {V(n, k) : n, k ∈ Z+, n ≥ k} of bounded linear operators acting on a complex Banach space X is uniformly exponentially stable if and only if for each f ∈ CAAP0 (Z+, X ) the solution xV ,f belongs to CAAP0 (Z+, X ), where CAAP0 (Z+, X ) is the space of asymptotically almost periodic sequences. We use the approach of discrete evolution semigroups.

Discrete Characterization of Exponential Stability of Evolution Family Over Hilbert Space

In this article we prove that if U = {U(m,n)}m≥n≥0 is a positive q-periodic discrete evolution family of bounded linear operators acting on a complex Hilbert space H then U is uniformly exponentially stable if for each unit vector x in H the series ∑∞ m=0 φ(|〈U(m, 0)x, x〉|) is bounded, where φ : R+ := [0,∞) → R+ is a non decreasing function such that φ(0) = 0 and φ(t) > 0 for all t ∈ (0,∞). We also prove the converse of the above result by putting an extra condition i.e. if U is uniformly exponentially stable and ∑∞ i=0 φ(xi) = φ( ∑∞ i=0(xi)) for any xi ∈ R+ then the series ∑∞ m=0 φ(|〈U(m, 0)x, x〉|) is bounded. 2000 Mathematics Subject Classification: 35B35.