Nonlinear Differential Equations with “Maxima”: Parametric Stability in Terms of two Measures

Böhner, Martin; Georgieva, Atanaska; Hristova, Snezhana

doi:10.12785/amis/070105

Cited by 9 publications

(6 citation statements)

References 11 publications

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“…(1) The functions 0 , 0 : Z[ − ℎ + 1, ] → R form a couple of quasi lower and upper solutions of (1)- (2) such that 0 ( ) ≤ 0 ( ) for ∈ Z[ − ℎ + 1, ].…”

Section: Monotone-iterative Methodsmentioning

confidence: 99%

“…The presence of the maximum function in equations mostly requires new techniques for investigation of the qualitative behavior of their solutions. Note also differential equations with maxima are not well studied and still not widely applied as models (see, e.g., [1][2][3][4][5], the monograph [6], and cited therein references). The character of the maximum function leads to a variety of different types of difference equations.…”

Section: Introductionmentioning

confidence: 99%

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Monotone-Iterative Method for Solving Antiperiodic Nonlinear Boundary Value Problems for Generalized Delay Difference Equations with Maxima

Golev

Hristova

Nenov

2013

Abstract and Applied Analysis

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A nonlinear generalized difference equation with both delays and the maximum value of the unknown function over a discrete past time interval are studied. A nonlinear boundary value problem of antiperiodic type for the given difference equation is set up. One of the main characteristics of the considered difference equation is the presence of the unknown function in both sides of the equation. It leads to impossibility for using the step method for explicit solving of the nonlinear difference equation. In this paper, an approximate method, namely, the monotone iterative technique, is applied to solve the problem. An important feature of the given algorithm is that each successive approximation of the unknown solution is equal to the unique solution of an appropriately constructed initial value problem for a linear difference equation with “maxima,” and an algorithm for its explicit solving is given. Also, each approximation is a lower/upper solution of the given nonlinear boundary value problem. The suggested scheme for approximate solving is computer realized, and it is applied to a particular example, which is a generalization of a model in population dynamics.

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“…(1) The functions 0 , 0 : Z[ − ℎ + 1, ] → R form a couple of quasi lower and upper solutions of (1)- (2) such that 0 ( ) ≤ 0 ( ) for ∈ Z[ − ℎ + 1, ].…”

Section: Monotone-iterative Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Monotone-Iterative Method for Solving Antiperiodic Nonlinear Boundary Value Problems for Generalized Delay Difference Equations with Maxima

Golev

Hristova

Nenov

2013

Abstract and Applied Analysis

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“…Hence, the fact that 1 − α > 0, together with the continuity of I 1−α U resulting from Equation (8), imply that I 1−α U (0) = 0. Consequently, Equation (9) becomes…”

Section: Definition 2 the Caputo Fractional Derivative Of A Function Umentioning

confidence: 97%

“…Such problems are often modeled by differential equations that contain the maximum values of the unknown function (see [3][4][5]). Recently, ordinary differential equations with "maxima" have received wide attention and have been investigated in diverse directions (see, for example, [4,[6][7][8][9][10][11] and the references therein). As far as we know, in the fractional case, these equations are not yet sufficiently discussed in the existing literature, and thus form a natural subject for further investigation.…”

Section: Introductionmentioning

confidence: 99%

An Iterative Method for Solving a Class of Fractional Functional Differential Equations with “Maxima”

Nisse

2017

Mathematics

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Abstract:In the present work, we deal with nonlinear fractional differential equations with "maxima" and deviating arguments. The nonlinear part of the problem under consideration depends on the maximum values of the unknown function taken in time-dependent intervals. Proceeding by an iterative approach, we obtain the existence and uniqueness of the solution, in a context that does not fit within the framework of fixed point theory methods for the self-mappings, frequently used in the study of such problems. An example illustrating our main result is also given.

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“…Recently, the research interest in differential equations with maxima has increased exponentially. Some stability results for such equations can be found in the monographs [1,2], the papers [3][4][5][6][7][8][9], and references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Practical Stability and Integral Stability for Singular Differential Systems with Maxima

Bao

Wang

2020

Mathematical Problems in Engineering

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Nonlinear Differential Equations with “Maxima”: Parametric Stability in Terms of two Measures

Cited by 9 publications

References 11 publications

Monotone-Iterative Method for Solving Antiperiodic Nonlinear Boundary Value Problems for Generalized Delay Difference Equations with Maxima

Monotone-Iterative Method for Solving Antiperiodic Nonlinear Boundary Value Problems for Generalized Delay Difference Equations with Maxima

An Iterative Method for Solving a Class of Fractional Functional Differential Equations with “Maxima”

Practical Stability and Integral Stability for Singular Differential Systems with Maxima

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