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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 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 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 ] × [ 0 , h ] is a function defined by z ( x , y ) ( t , s ) = z ( x + t , y + s ) , ( t , s ) [ - τ , 0 ] × [ 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.

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 ) = 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.

On the stability of solutions of nonlinear parabolic differential-functional equations

Stanisław Brzychczy (1996)

Annales Polonici Mathematici

We consider a nonlinear differential-functional parabolic boundary initial value problem (1) ⎧A z + f(x,z(t,x),z(t,·)) - ∂z/∂t = 0 for t > 0, x ∈ G, ⎨z(t,x) = h(x)     for t > 0, x ∈ ∂G, ⎩z(0,x) = φ₀(x)     for x ∈ G, and the associated elliptic boundary value problem with Dirichlet condition (2) ⎧Az + f(x,z(x),z(·)) = 0  for x ∈ G, ⎨z(x) = h(x)    for x ∈ ∂G ⎩ where x = ( x , . . . , x m ) G m , G is an open and bounded domain with C 2 + α (0 < α ≤ 1) boundary, the operator     Az := ∑j,k=1m ajk(x) (∂²z/(∂xj ∂xk)) is...

On the vanishing viscosity method for first order differential-functional IBVP

Krzysztof A. Topolski (2008)

Czechoslovak Mathematical Journal

We consider the initial-boundary value problem for first order differential-functional equations. We present the `vanishing viscosity' method in order to obtain viscosity solutions. Our formulation includes problems with a retarded and deviated argument and differential-integral equations.

Oscillation criteria for nonlinear differential equations with p ( t ) -Laplacian

Yutaka Shoukaku (2016)

Mathematica Bohemica

Recently there has been an increasing interest in studying p ( t ) -Laplacian equations, an example of which is given in the following form ( | u ' ( t ) | p ( t ) - 2 u ' ( t ) ) ' + c ( t ) | u ( t ) | q ( t ) - 2 u ( t ) = 0 , t > 0 . In particular, the first study of sufficient conditions for oscillatory solution of p ( t ) -Laplacian equations was made by Zhang (2007), but to our knowledge, there has not been a paper which gives the oscillatory conditions by utilizing Riccati inequality. Therefore, we establish sufficient conditions for oscillatory solution of nonlinear differential equations with...

Oscillation properties for parabolic equations of neutral type

Bao Tong Cui (1992)

Commentationes Mathematicae Universitatis Carolinae

The oscillation of the solutions of linear parabolic differential equations with deviating arguments are studied and sufficient conditions that all solutions of boundary value problems are oscillatory in a cylindrical domain are given.

Oscillations of higher order differential equations of neutral type

N. Parhi (2000)

Czechoslovak Mathematical Journal

In this paper, sufficient conditions have been obtained for oscillation of solutions of a class of n th order linear neutral delay-differential equations. Some of these results have been used to study oscillatory behaviour of solutions of a class of boundary value problems for neutral hyperbolic partial differential equations.

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