An existence theory for functional-differential equations and functional-differential systems

Ifantis, E. K.

doi:10.1016/0022-0396(78)90042-6

Cited by 35 publications

(20 citation statements)

References 10 publications

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“…Consider now the linear manifold of all f (z) ∈ H 2 (D) which satisfy the condition Using representation (2.1) the abstract forms of the terms appearing in the ODEs that we will study were determined in [3,12,13,26]. Indeed: Proposition 2.3.…”

Section: The Methodsmentioning

confidence: 98%

“…However, it was in [3], that this technique was developed in detail for the study of linear functional-differential equations and systems in H 2 (D). This technique was also used in [4][5][6][7][8][9][10][11], for the investigation of analytic and entire solutions of linear ODEs and in [12,13], it was extended to the study of non-linear ODEs, for non-linearities which were powers of the unknown function.…”

Section: Introductionmentioning

confidence: 99%

“…The main idea of this functional-analytic technique used in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] is the transformation of the ODE under consideration, into an equivalent operator equation in an abstract Hilbert (H ) or Banach (H 1 ) space, using an isomorphism between H 2 (D) and H or H 1 (D) and H 1 .…”

Section: Introductionmentioning

confidence: 99%

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A “discretization” technique for the solution of ODEs

Petropoulou

Siafarikas

Tzirtzilakis

2007

Journal of Mathematical Analysis and Applications

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A functional-analytic technique was developed in the past for the establishment of unique solutions of ODEs in H 2 (D) and H 1 (D) and of difference equations in 2 and 1 . This technique is based on two isomorphisms between the involved spaces. In this paper, the two isomorphisms are combined in order to find discrete equivalent counterparts of ODEs, so as to obtain eventually the solution of the ODEs under consideration. As an application, the Duffing equation and the Lorenz system are studied. The results are compared with numerical ones obtained using the 4th order Runge-Kutta method. The advantages of the present method are that, it is accurate, the only errors involved are the round-off errors, it does not depend on the grid used and the obtained solution is proved to be unique.

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Section: The Methodsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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A “discretization” technique for the solution of ODEs

Petropoulou

Siafarikas

Tzirtzilakis

2007

Journal of Mathematical Analysis and Applications

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“…It is clear that (2) includes equation (1) as a special case. In the linear case, such equations with proportional delays also have been studied to some extent by many authors [4][5][6]. By means of the method of majorant series, we will discuss the existence of analytic solutions of equation (2) in a neighborhood of the origin.…”

Section: G(f( Gi Z)) + P2g(f(q 2 Z)) + + P M G(f(q M Z))mentioning

confidence: 99%

Analytic Solutions of a Nonlinear Functional Differential Equation With Proportional Delays

Si¹

2000

Demonstratio Mathematica

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Abstract. This paper is concerned with a nonlinear functional differential equation of the form m ¿=i By means of the method of majorant series conditions are given for existence of analytic solutions of the above equation.It is well-known that the third Jabotinsky equationplays an important role in iteration theory. This equation has been studied by many authors [1][2][3]. In this paper, we will be concerned with a more general class of equation of the formwhere pj and qj are complex numbers. It is clear that (2) includes equation (1) as a special case. In the linear case, such equations with proportional delays also have been studied to some extent by many authors [4][5][6]. By means of the method of majorant series, we will discuss the existence of analytic solutions of equation (2) in a neighborhood of the origin. Some ideas in the proof of our main results are similar to that used in [3,7,8].We now state and prove our main results in this note.1991 Mathematics Subject Classification: 34K25.

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“…Also it is known [1] (see for details in [2] and [3]). Every bounded operator on//2(4)(//) is defined on//I(A)(H1) and maps, in general, elements of//I(A)(//1) into //2(4)(//).…”

mentioning

confidence: 99%