Displaying similar documents to “On the mixed problem for hyperbolic partial differential-functional equations of the first order”

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

Tomasz Człapiński (1999)

Annales Polonici Mathematici

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We consider the mixed problem for the quasilinear partial functional differential equation with unbounded delay D t z ( t , x ) = ∑ 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 ) ∈ X ̶ 0 is defined by z ( t , x ) ( τ , s ) = z ( t + τ , x + s ) , ( τ , s ) ∈ ( - ∞ , 0 ] × [ 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.

Semilinear hyperbolic functional equations

László Simon (2014)

Banach Center Publications

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We consider second order semilinear hyperbolic functional differential equations where the lower order terms contain functional dependence on the unknown function. Existence and uniqueness of solutions for t ∈ (0,T), existence for t ∈ (0,∞) and some qualitative properties of the solutions in (0,∞) are shown.

On the local Cauchy problem for nonlinear hyperbolic functional differential equations

Tomasz Człapiński (1997)

Annales Polonici Mathematici

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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).

Generalized Cauchy problems for hyperbolic functional differential systems

Elżbieta Puźniakowska-Gałuch (2014)

Annales Polonici Mathematici

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A generalized Cauchy problem for hyperbolic functional differential systems is considered. The initial problem is transformed into a system of functional integral equations. The existence of solutions of this system is proved by using the method of successive approximations. Differentiability of solutions with respect to initial functions is proved. It is important that functional differential systems considered in this paper do not satisfy the Volterra condition.

Noncharacteristic mixed problems for hyperbolic systems of the first order

Ewa Zadrzyńska

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CONTENTS1. Introduction....................................................................................................................................................52. Notations and preliminaries .........................................................................................................................11 2.1. Function spaces and spaces of distributions............................................................................................11 2.2. Perturbations...