On stability of solutions of integral equations in the class of measurable functions

2021 ◽  
pp. 44-54
Author(s):  
Wassim Merchela
Keyword(s):  
Integral Equation ◽  
Uniform Convergence ◽  
Metric Space ◽  
Measurable Function ◽  
Sufficient Conditions ◽  
Stability Of Solutions ◽  
Lebesgue Measurable Function ◽  
Lipschitz Property ◽  
Measurable Functions ◽  
The Stability

Consider the equation G(x)=(y,) ̃ where the mapping G acts from a metric space X into a space Y, on which a distance is defined, y ̃ ∈ Y. The metric in X and the distance in Y can take on the value ∞, the distance satisfies only one property of a metric: the distance between y,z ∈Y is zero if and only if y= z. For mappings X → Y the notions of sets of covering, Lipschitz property, and closedness are defined. In these terms, the assertion is obtained about the stability in the metric space X of solutions of the considered equation to changes of the mapping G and the element y ̃. This assertion is applied to the study of the integral equation f(t,∫_0^1▒K (t,s)x(s)ds,x(t))= y ̃(t),t ∈[0,1], with respect to an unknown Lebesgue measurable function x: [0,1] ∈R. Sufficient conditions are obtained for the stability of solutions (in the space of measurable functions with the topology of uniform convergence) to changes of the functions f,K,(y.) ̃

scholarly journals Uniform Convergence and Transitive Subsets

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Lei Liu ◽  
Shuli Zhao ◽  
Hongliang Liang
Keyword(s):  
Uniform Convergence ◽  
Metric Space ◽  
Sufficient Conditions ◽  
Continuous Maps

Let(X,d)be a metric space and a sequence of continuous mapsfn:X→Xthat converges uniformly to a mapf. We investigate the transitive subsets offnwhether they can be inherited byfor not. We give sufficient conditions such that the limit mapfhas a transitive subset. In particular, we show the transitive subsets offnthat can be inherited byfiffnconverges uniformly strongly tof.

scholarly journals Stability of solutions for generalized fractional differential problems by applying significant inequality estimates

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohammed D. Kassim ◽  
Thabet Abdeljawad ◽  
Saeed M. Ali ◽  
Mohammed S. Abdo
Keyword(s):  
Fractional Derivative ◽  
Power Function ◽  
Sufficient Conditions ◽  
Research Paper ◽  
Stability Of Solutions ◽  
Initial Value ◽  
Regularization Techniques ◽  
Fractional Differential ◽  
The Stability ◽  
Generalized Fractional Derivative

AbstractIn this research paper, we intend to study the stability of solutions of some nonlinear initial value fractional differential problems. These equations are studied within the generalized fractional derivative of various orders. In order to study the solutions’ decay to zero as a power function, we establish sufficient conditions on the nonlinear terms. To this end, some versions of inequalities are combined and generalized via the so-called Bihari inequality. Moreover, we employ some properties of the generalized fractional derivative and appropriate regularization techniques. Finally, the paper involves examples to affirm the validity of the results.

scholarly journals Pointwise measurable functions

2004 ◽  
Vol 95 (2) ◽  
pp. 305
Author(s):  
Herman Render ◽  
Lothar Rogge
Keyword(s):  
Large Class ◽  
Metric Space ◽  
Measurable Function ◽  
Polish Space ◽  
Topological Spaces ◽  
Range Space ◽  
Compact Metric Space ◽  
Delta Set ◽  
Measurable Functions ◽  
Borel Measurable

We introduce the new concept of pointwise measurability. It is shown in this paper that a measurable function is measurable at each point and that for a large class of topological spaces the converse also holds. Moreover it can be seen that a function which is continuous at a point is Borel-measurable at this point too. Furthermore the set of measurability points is considered. If the range space is a $\sigma$-compact metric space, then this set is a $G_{\delta}$-set; if the range space is only a Polish space this is in general not true any longer.

scholarly journals On the stability of solutions of fractional non conformable differential equations

2020 ◽  
Vol 65 (4) ◽  
pp. 495-502
Author(s):  
Paulo M. Guzman ◽  
Luciano M. Lugo Motta Bittencurt ◽  
Juan E. Napoles Valdes
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lyapunov Method ◽  
Equations Of Motion ◽  
Sufficient Conditions ◽  
Direct Lyapunov Method ◽  
Stability Of Solutions ◽  
Fractional Differential ◽  
The Stability

In this note we obtain sufficient conditions under which we can guarantee the stability of solutions of a fractional differential equations of non conformable type and we obtain some fractional analogous theorems of the direct Lyapunov method for a given class of equations of motion.

scholarly journals Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching

2021 ◽  
Vol 19 (1) ◽  
pp. 614-628
Author(s):  
Xiaozhi Zhang ◽  
Zhangsheng Zhu ◽  
Chenggui Yuan
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Markovian Switching ◽  
Stability Of Solutions ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
The Stability ◽  
Delay Differential

Abstract The aim of this work is to study the asymptotic stability of the time-changed stochastic delay differential equations (SDDEs) with Markovian switching. Some sufficient conditions for the asymptotic stability of solutions to the time-changed SDDEs are presented. In contrast to the asymptotic stability in existing articles, we present the new results on the stability of solutions to time-changed SDDEs, which is driven by time-changed Brownian motion. Finally, an example is given to demonstrate the effectiveness of the main results.

scholarly journals Stability of a Functional Differential System with a Finite Number of Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Josef Rebenda ◽  
Zdeněk Šmarda
Keyword(s):  
Asymptotic Stability ◽  
Differential System ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Stability Of Solutions ◽  
Functional Differential ◽  
The Stability ◽  
Complex Valued ◽  
Functional Differential System

The paper is devoted to the study of asymptotic properties of a real two-dimensional differential system with unbounded nonconstant delays. The sufficient conditions for the stability and asymptotic stability of solutions are given. Used methods are based on the transformation of the considered real system to one equation with complex-valued coefficients. Asymptotic properties are studied by means of Lyapunov-Krasovskii functional. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one or more constant delays or one nonconstant delay were studied.

ON THE STABILITY OF SOLUTIONS OF NONLINEAR SYSTEMS WITH IMPULSE STRUCTURE

2018 ◽  
pp. 624-636
Author(s):  
Natalia Igorevna Zhelonkina ◽  
Alexander Nikolaevich Sesekin
Keyword(s):  
Nonlinear Systems ◽  
Asymptotic Stability ◽  
Stability Property ◽  
Sufficient Conditions ◽  
Functions Of Bounded Variation ◽  
Stability Of Solutions ◽  
Smooth Solutions ◽  
Systems Of Differential Equations ◽  
The Stability ◽  
The Right

In this paper we review the results of the authors related to the study of the stability property of solutions for nonlinear systems of differential equations, on the right-hand side of which there are terms containing products of discontinuous functions and distributions. The solutions of such systems are formalized by the closure of the set of smooth solutions in the space of functions of bounded variation. For such systems, sufficient conditions are obtained for the asymptotic stability of unperturbed solutions.

scholarly journals STABILITY OF ZERO SOLUTION OF SYSTEM WITH SWITCHES CONSISTING OF LINEAR SUBSYSTEMS

2020 ◽  
pp. 89-96
Author(s):  
D. Khusainov ◽  
A. Bychkov ◽  
A. Sirenko
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Dynamic Systems ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Switching Systems ◽  
Stability Of Solutions ◽  
Quadratic Lyapunov Function ◽  
Linear Differential ◽  
The Stability

In this paper, discusses the study of the stability of solutions of dynamic systems with switching. Sufficient conditions are obtained for the asymptotic stability of the zero solution of switching systems consisting of linear differential and difference subsystems. It is proved that the existence of a common quadratic Lyapunov function is sufficient for asymptotic stability.

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