The method of finite differences for nonlinear functional differential equations of the first order
-
Elżbieta Puźniakowska-Gałuch
Abstract
Nonlinear functional partial differential equations with initial conditions are considered on the cone. The weak convergence of a sequence of successive approximations is proved. The proof is given by the duality principle.
A Appendix
In this section the proofs of the used lemmas are presented. First we will present the proof of Lemma 3.3.
Proof of Lemma 3.3.
Take
The function w satisfies
Let
such that
where
On the other hand,
where
and δ is defined by (3.1). Hence, it follows that
From the Riesz representation theorem, we have
where
in the following way:
Now we define a function
on
where
in
Since the matrix
for
where the constants are defined by Assumption 3.2.
For the function
The proof of Lemma 3.3 is complete. ∎
Proof of Lemma 4.3.
This lemma is proved by induction.
For
The sequence of successive approximations (4.1) satisfies
From the mean value theorem, there exists an intermediate point
Now we calculate the absolute value of the
where
It is obvious that for every
This completes the proof of Lemma 4.3. ∎
The next proof is also carried out by induction.
Proof of Lemma 4.4.
For
Making some estimations on the given functions we have that
We will need the following remark for Lemma 4.5.
Remark A.1.
The function
Proof of Lemma 4.5.
From Assumption 3.2 on the given functions,
This gives
The assertion follows. ∎
Proof of Lemma 4.7.
From (4.1) we conclude that
for
whence
and for
from (4.3) and (4.2), we obtain
Applying the above estimate to (4.4) and taking into account that
for
References
[1] P. Brandi and R. Ceppitelli, Existence, uniqueness and continuous dependence for a hereditary nonlinear functional partial differential equation of the first order, Ann. Polon. Math. 47 (1986), no. 2, 121–136. 10.4064/ap-47-2-121-136Suche in Google Scholar
[2] P. Brandi, A. Salvadori and Z. Kamont, Existence of generalized solutions of hyperbolic functional differential equations, Nonlinear Anal. 50 (2002), no. 7, 919–940. 10.1016/S0362-546X(01)00793-3Suche in Google Scholar
[3] M. Cinquini Cibrario, Sopra i sistemi di equazioni alle derivate parziali a caratteristiche reali e multiple, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 4 (1948), 682–688. Suche in Google Scholar
[4] M. Cinquini Cibrario, Nuovi teoremi di esistenza e di unicità per sistemi di equazioni a derivate parziali, Ann. Sc. Norm. Super. Pisa (3) 9 (1955), 65–113. Suche in Google Scholar
[5]
M. Cinquini Cibrario and S. Cinquini,
Sopra una forma più ampia del problema di Cauchy per l’equazione
[6] M. Cinquini Cibrario and S. Cinquini, Sopra una nuova estensione di un teorema di esistenza per equazioni a derivate parziali del primo ordine, Ann. Mat. Pura Appl. (4) 43 (1957), 51–81. 10.1007/BF02411904Suche in Google Scholar
[7] T. Człapiński, On the local Cauchy problem for nonlinear hyperbolic functional-differential equations, Ann. Polon. Math. 67 (1997), no. 3, 215–232. 10.4064/ap-67-3-215-232Suche in Google Scholar
[8] M. Gyllenberg and H. J. A. M. Heijmans, An abstract delay-differential equation modelling size dependent cell growth and division, SIAM J. Math. Anal. 18 (1987), no. 1, 74–88. 10.1137/0518006Suche in Google Scholar
[9] Z. Kamont and H. Leszczyński, Uniqueness result for the generalized entropy solutions to the Cauchy problem for first-order partial differential-functional equations, Z. Anal. Anwend. 13 (1994), no. 3, 477–491. 10.4171/ZAA/500Suche in Google Scholar
[10] Z. A. Kamont, Hyperbolic Functional Differential Inequalities and Applications, Math. Appl. 486, Kluwer Academic, Dordrecht, 1999. 10.1007/978-94-011-4635-7Suche in Google Scholar
[11] S. N. Kružkov, The Cauchy problem in the large for non-linear equations and for certain first-order quasilinear systems with several variables (in Russian), Dokl. Akad. Nauk SSSR 155 (1964), 743–746. Suche in Google Scholar
[12] S. N. Kružkov, Generalized solutions of nonlinear equations of the first order with several variables. I (in Russian), Mat. Sb. (N.S.) 70(112) (1966), 394–415. 10.1070/SM1967v001n01ABEH001969Suche in Google Scholar
[13] S. N. Kružkov, The method of finite differences for a nonlinear equation of the first order with several independent variables (in Russian), Zh. Vychisl. Mat. Mat. Fiz. 6 (1966), 884–894. Suche in Google Scholar
[14] L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa (3) 13 (1959), 115–162. 10.1007/978-3-642-10926-3_1Suche in Google Scholar
[15] E. Puźniakowska, Classical solutions of quasilinear functional differential systems on the Haar pyramid, Differ. Equ. Appl. 1 (2009), no. 2, 179–197. 10.7153/dea-01-09Suche in Google Scholar
[16] E. Puźniakowska-Gałuch, Differentiability with respect to initial functions for partial functional differential equations, Univ. Iagell. Acta Math. 48 (2010), 111–131. Suche in Google Scholar
[17] E. Puźniakowska-Gałuch, On the local Cauchy problem for first order partial differential functional equations, Ann. Polon. Math. 98 (2010), no. 1, 39–61. 10.4064/ap98-1-3Suche in Google Scholar
[18] E. Puźniakowska-Gałuch, Initial problems for hyperbolic functional differential systems, Georgian Math. J. 20 (2013), no. 2, 357–376. 10.1515/gmj-2013-0008Suche in Google Scholar
[19] A. Salvadori, The Cauchy problem for a hereditary hyperbolic structure. Existence, uniqueness and continuous dependence (in Italian), Atti Sem. Mat. Fis. Univ. Modena 32 (1983), no. 2, 329–356. Suche in Google Scholar
[20] J. Szarski, Generalized Cauchy problem for differential-functional equations with first order partial derivatives, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), no. 8, 575–580. Suche in Google Scholar
[21] J. Szarski, Cauchy problem for an infinite system of differential-functional equations with first order partial derivatives, Comment. Math. 1 (1978), 293–300; Special issue dedicated to Władysław Orlicz on the occasion of his seventy-fifth birthday. Suche in Google Scholar
[22] M. Tabata and N. Eshima, The Fokker–Planck equation and the master equation in the theory of migration, IMA J. Appl. Math. 69 (2004), no. 6, 585–603. 10.1093/imamat/69.6.585Suche in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Stabilization of 3D Navier–Stokes–Voigt equations
- On weakly 2-absorbing δ-primary ideals of commutative rings
- General Tauberian theorems for the Cesàro integrability of functions
-
- From simplicial homotopy to crossed module homotopy in modified categories of interest
- Riesz potential in the local Morrey–Lorentz spaces and some applications
- Some remarks on Sierpiński–Zygmund functions in the strong sense
- Weighted grand mixed-norm Lebesgue spaces and boundedness criteria for integral operators
- Solution of the Ulam stability problem for Euler–Lagrange k-quintic mappings
- Construction of Green’s functional for a third order ordinary differential equation with general nonlocal conditions and variable principal coefficient
- The method of finite differences for nonlinear functional differential equations of the first order
- Local variation formulas of solutions for nonlinear controlled functional differential equations with constant delays and the discontinuous initial condition
- Solutions to Neumann boundary value problems with a generalized 𝑝-Laplacian
- Off-diagonal extrapolation on mixed variable Lebesgue spaces and its applications to strong fractional maximal operators
- The Arnon bases in the Steenrod algebra
- Inner functions as multipliers and zero sets in weighted Dirichlet spaces
Artikel in diesem Heft
- Frontmatter
- Stabilization of 3D Navier–Stokes–Voigt equations
- On weakly 2-absorbing δ-primary ideals of commutative rings
- General Tauberian theorems for the Cesàro integrability of functions
-
- From simplicial homotopy to crossed module homotopy in modified categories of interest
- Riesz potential in the local Morrey–Lorentz spaces and some applications
- Some remarks on Sierpiński–Zygmund functions in the strong sense
- Weighted grand mixed-norm Lebesgue spaces and boundedness criteria for integral operators
- Solution of the Ulam stability problem for Euler–Lagrange k-quintic mappings
- Construction of Green’s functional for a third order ordinary differential equation with general nonlocal conditions and variable principal coefficient
- The method of finite differences for nonlinear functional differential equations of the first order
- Local variation formulas of solutions for nonlinear controlled functional differential equations with constant delays and the discontinuous initial condition
- Solutions to Neumann boundary value problems with a generalized 𝑝-Laplacian
- Off-diagonal extrapolation on mixed variable Lebesgue spaces and its applications to strong fractional maximal operators
- The Arnon bases in the Steenrod algebra
- Inner functions as multipliers and zero sets in weighted Dirichlet spaces
