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The method of quasilinearization for system of hyperbolic functional differential equations

100%

Karpowicz A.

Commentationes Mathematicae

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2008

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tom 48

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nr 2

EN

We deal with monotone iterative method for the Darboux problem for the system of hyperbolic partial functional-differential equations \[ \begin{cases} \frac{\partial^2 u}{\partial x\partial y} (x,y) = f(x,y,u_{(x,y)}, u_{(x,y)}, \text{a.e. in}\ [0,1]\times [0,b]\\ u(x,y) = \psi (x,y), \text{on}\ [-a_0,a]\times [-b_0,b] \setminus (0,a] \times (0,b], \end{cases} \] where the function $u_{(x,y)}\colon [-a_0,0]\times [-b_0,0] \to \mathbb{R}^k$ is defined by $u_{(x,y)} (s, t) = u(s + x, t + y)$ for $(s, t)\in [-a_0 , 0] \times [-b_0 , 0]$.

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The existence of Carathéodory solutions of hyperbolic functional differential equations

100%

Karpowicz A.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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2010

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tom 30

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nr 1

121-140

EN

We consider the following Darboux problem for the functional differential equation $∂²u/∂x∂y(x,y) = f(x,y,u_{(x,y)},∂u/∂x(x,y),∂u/∂y(x,y))$ a.e. in [0,a]×[0,b], u(x,y) = ψ(x,y) on [-a₀,a]×[-b₀,b]\(0,a]×(0,b], where the function $u_{(x,y)}:[-a₀,0]×[-b₀,0] → ℝ^{k}$ is defined by $u_{(x,y)}(s,t) = u(s+x,t+y)$ for (s,t) ∈ [-a₀,0]×[-b₀,0]. We prove a theorem on existence of the Carathéodory solutions of the above problem.

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Carathéodory solutions of hyperbolic functional differential inequalities with first order derivatives

100%

Karpowicz A.

Annales Polonici Mathematici

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2008

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tom 94

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nr 1

53-78

EN

We consider the Darboux problem for a functional differential equation: $(∂²u)/(∂x∂y) (x,y) = f(x,y,u_{(x,y)},u(x,y),∂u/∂x (x,y),∂u/∂y (x,y))$ a.e. in [0,a]×[0,b], u(x,y) = ψ(x,y) on [-a₀,a]×[-b₀,b]∖(0,a]×(0,b], where the function $u_{(x,y)}:[-a₀,0]×[-b₀,0] → ℝ^{k}$ is defined by $u_{(x,y)}(s,t) = u({s+x},{t+y})$ for (s,t) ∈ [-a₀,0]×[-b₀,0]. We give a few theorems about weak and strong inequalities for this problem. We also discuss the case where the right-hand side of the differential equation is linear.

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Second order evolution differential functional equations with infinite delay

64%

Człapiński T., Karpowicz A.

Commentationes Mathematicae

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2018

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tom 58

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nr 1-2

EN

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.

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On the Cauchy problem for hyperbolic functional-differential equations

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Karpowicz A., Leszczyński H.

Annales Polonici Mathematici

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2015

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tom 115

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nr 1

53-74

EN

We consider the Cauchy problem for a nonlocal wave equation in one dimension. We study the existence of solutions by means of bicharacteristics. The existence and uniqueness is obtained in $W^{1,∞}_{loc}$ topology. The existence theorem is proved in a subset generated by certain continuity conditions for the derivatives.

Ograniczanie wyników

2 Annales Polonici Mathematici

2 Commentationes Mathematicae

1 Discussiones Mathematicae, Differential Inclusions, Control and Optimization

5 Karpowicz A.

1 Człapiński T.

1 Leszczyński H.

1 2018

1 2015

1 2010

2 2008

The method of quasilinearization for system of hyperbolic functional differential equations

The existence of Carathéodory solutions of hyperbolic functional differential equations

Carathéodory solutions of hyperbolic functional differential inequalities with first order derivatives

Second order evolution differential functional equations with infinite delay

On the Cauchy problem for hyperbolic functional-differential equations