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On the mixed problem for quasilinear partial functional differential equations with unbounded delay

Tomasz Człapiński — 1999

Annales Polonici Mathematici

We consider the mixed problem for the quasilinear partial functional differential equation with unbounded delay $D_{t} z (t, x) = \sum_{i = 1}^{n} f_{i} (t, x, z_{(t, x)}) D_{x_{i}} z (t, x) + h (t, x, z_{(t, x)})$ , where $z_{(t, x)} \in X ̶_{0}$ is defined by $z_{(t, x)} (τ, s) = z (t + τ, x + s)$ , $(τ, s) \in (- \infty, 0] \times [0, r]$ , and the phase space $X ̶_{0}$ satisfies suitable axioms. Using the method of bicharacteristics and the fixed-point method we prove a theorem on the local existence and uniqueness of Carathéodory solutions of the mixed problem.

On the local Cauchy problem for nonlinear hyperbolic functional differential equations

Tomasz Człapiński — 1997

Annales Polonici Mathematici

We consider the local initial value problem for the hyperbolic partial functional differential equation of the first order (1) $D ₓ z (x, y) = f (x, y, z (x, y), (W z) (x, y), D_{y} z (x, y))$ on E, (2) z(x,y) = ϕ(x,y) on [-τ₀,0]×[-b,b], where E is the Haar pyramid and τ₀ ∈ ℝ₊, b = (b₁,...,bₙ) ∈ ℝⁿ₊. Using the method of bicharacteristics and the method of successive approximations for a certain functional integral system we prove, under suitable assumptions, a theorem on the local existence of weak solutions of the problem (1),(2).

On existence and uniqueness of solutions of nonlocal problems for hyperbolic differential-functional equations in two independent variables

Tomasz Człapiński — 1997

Annales Polonici Mathematici

We seek for classical solutions to hyperbolic nonlinear partial differential-functional equations of the second order. We give two theorems on existence and uniqueness for problems with nonlocal conditions in bounded and unbounded domains.

Difference methods for the Darboux problem for functional partial differential equations

Tomasz Człapiński — 1999

Annales Polonici Mathematici

Generalized solutions to boundary value problems for quasilinear hyperbolic systems of partial differential-functional equations

Tomasz Człapiński — 1992

Annales Polonici Mathematici

Generalized solutions to quasilinear hyperbolic systems in the second canonical form are investigated. A theorem on existence, uniqueness and continuous dependence upon the boundary data is given. The proof is based on the methods due to L. Cesari and P. Bassanini for systems which are not functional.

On the existence of generalized solutions of nonlinear first order partial differential-functional equations in two independent variables

Tomasz Człapiński — 1991

Czechoslovak Mathematical Journal

On the mixed problem for hyperbolic partial differential-functional equations of the first order

Tomasz Człapiński — 1999

Czechoslovak Mathematical Journal

We consider the mixed problem for the hyperbolic partial differential-functional equation of the first order $D_{x} z (x, y) = f (x, y, z_{(x, y)}, D_{y} z (x, y)),$ where $z_{(x, y)} [- τ, 0] \times [0, h] \to ℝ$ is a function defined by $z_{(x, y)} (t, s) = z (x + t, y + s)$ , $(t, s) \in [- τ, 0] \times [0, h]$ . Using the method of bicharacteristics and the method of successive approximations for a certain integral-functional system we prove, under suitable assumptions, a theorem of the local existence of generalized solutions of this problem.

Second order evolution differential functional equations with infinite delay

Tomasz Człapiński; Adrian Karpowicz — 2018

Commentationes Mathematicae

We consider a second order semilinear functional evolution equation with infinite delay in a Banach space. We prove the existence of mild solutions for this equation using the measure of noncompactness technique and the Schauder fixed point theorem.

Iterative methods for the Darboux problem for partial functional differential equations.

Człapiński, Tomasz — 1999

Journal of Inequalities and Applications [electronic only]

The mixed problem for an infinite system of first order functional differential equations.

Człapiński, Tomasz — 2001

Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Universitatis Iagellonicae Acta Mathematica

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Currently displaying 1 – 10 of 10

On the mixed problem for quasilinear partial functional differential equations with unbounded delay

On the local Cauchy problem for nonlinear hyperbolic functional differential equations

On existence and uniqueness of solutions of nonlocal problems for hyperbolic differential-functional equations in two independent variables

Difference methods for the Darboux problem for functional partial differential equations

Generalized solutions to boundary value problems for quasilinear hyperbolic systems of partial differential-functional equations

On the existence of generalized solutions of nonlinear first order partial differential-functional equations in two independent variables

On the mixed problem for hyperbolic partial differential-functional equations of the first order

Second order evolution differential functional equations with infinite delay

Iterative methods for the Darboux problem for partial functional differential equations.

The mixed problem for an infinite system of first order functional differential equations.