Existence and Uniqueness of Strong Solutions for a Class of Quasi‐Linear Hyperbolic Equations with Order Degeneration

Let k(y) > 0, 𝓁(y) > 0 for y > 0, k(0) = 𝓁(0) = 0 and lim_{y → 0}k(y)/𝓁(y) exists; then the equation L(u) ≔ k(y)u_xx – ∂_y(𝓁(y)u_y) + a(x, y)u_x = f(x, y, u) is strictly hyperbolic for y > 0 and its order degenerates on the line y = 0. Consider the boundary value problem Lu = f(x, y, u) in G, u|_AC = 0, where G is a simply connected domain in ℝ² with piecewise smooth boundary ∂G = AB∪AC∪BC; AB = {(x, 0) : 0 ≤ x ≤ 1}, AC : x = F(y) = ∫^y₀(k(t)/𝓁(t))^1/2dt and BC : x = 1 – F(y) are characteristic curves. Existence of generalized solution is obtained by a finite element method, provided f(x, y, u) satisfies Carathéodory condition and |f(x, y, u)| ≤ Q(x, y) + b|u| with Q ∈ L²(G), b = const > 0. It is shown also that each generalized solution is a strong solution, and that fact is used to prove uniqueness under the additional assumption |f(x, y, u₁) – f(x, y, u₂| ≤ C|u₁ – u₂|, where C = const > 0.     

Weak solution of (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0))u′(0) = ku(0), r(L, u(L))u′ (L) = hu(L) and k, h are suitable elements of [0, ∞]

We consider weak solutions to the nonlinear boundary value problem (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0)) u′(0) = ku(0), r(L, u(L)) u′(L) = hu(L) and k, h are suitable elements of [0, ∞]. In addition to studying some new boundary conditions, we also relax the constraints on r(x, u) and (Fu)(x). r(x, u) > 0 may have a countable set of jump discontinuities in u and r(x, u)^?1?L_q((0, L) × (0, p)). F is an operator from a suitable set of functions to a subset of L_p(0, L) which have nonnegative values. F includes, among others, examples of the form (Fu)(x) = (1 ? H(x ? x₀)) u(x₀), (Fu)(x) = ∫_x^Lf(y, u(y)) dy where f(y, u) may have a countable set of jump discontinuities in u or F may be chosen so that (Fu)′(x) = ? g(x, u(x)) u′(x) ? q(x) u(x) ? f(x, u(x)) where q is a distributional derivative of an L₂(0, L) function.     

Double reduction of a nonlinear (2+1) wave equation via conservation laws

Conservation laws of a nonlinear (2+1) wave equation u_tt = (f(u)u_x)_x +  (g(u)u_y)_y involving arbitrary functions of the dependent variable are obtained, by writing the equation in the partial Euler-Lagrange form. Noether-type operators associated with the partial Lagrangian are obtained for all possible cases of the arbitrary functions. If either of f(u) or g(u) is an arbitrary nonconstant function, we show that there are an infinite number of conservation laws. If both f(u) and g(u) are arbitrary nonconstant functions, it is shown that there exist infinite number of conservation laws when f′(u) and g′(u) are linearly dependent, otherwise there are eight conservation laws. Finally, we apply the generalized double reduction theorem to a nonlinear (2+1) wave equation when f′(u) and g′(u) are linearly independent.     

The surpluses immediately before and at ruin,and the amount of the claim causing ruin

In the classical compound Poisson model of the collective risk theory we consider X, the surplus before the claim that causes ruin, and Y, the deficit at the time of ruin. We denote by f(u; x, y) their joint density (u initial surplus) which is a defective probability density (since X and Y are only defined, if ruin takes place). For an arbitrary claim amount distribution we find that f(0; x, y) = ap(x + y), where p(z) is the probability density function of a claim amount and a is the ratio of the Poisson parameter and the rate of premium income. In the more realistic case, where u is positive, f(u; x, y) can be calculated explicitly, if the claim amount distribution is exponential or, more generally, a combination of exponential distributions. We are also interested in X + Y, the amount of the claim that causes ruin. Its density h(u; z) can be obtained from f(u; x, y). One finds, for example, that h(0; z) = azp(z).     

The Instability of Nonmonotonic Wave Solutions of Parabolic Equations

We consider the class of equations u_t=f(u_xx, u_x, u) under the restriction that for all a,b,c. We first consider this equation over the unbounded domain ? ∞ < x < + ∞, and we show that very nearly every bounded nonmonotonic solution of the form u(t, x)=?(x?ct) is unstable to all nonnegative and all nonpositive perturbations. We then extend these results to nonmonotonic plane wave solutions u(t, x, y)=?(x?ct) of u_t = F(u_xx, u_xy, u_x, u_y, u). Finally, we consider the class of equations u_t=f(u_xx, u_x, u) over the bounded domain 0 < x < 1 with the boundary conditions u(t, x)=A at x=0 and u(t, x)=B at x=1, and we find the stability of all steady solutions u(t, x)=?(x).     

A Generalized Cubic Functional Equation

In this paper, we determine the general solution of the functional equation f1 （2x ＋ y） ＋ f2（2x - y） = f3（x ＋ y） ＋ f4（x - y） ＋ f5（x） without assuming any regularity condition on the unknown functions f1,f2,f3, f4, f5 ： R→R. The general solution of this equation is obtained by finding the general solution of the functional equations f（2x ＋ y） ＋ f（2x - y） = g（x ＋ y） ＋ g（x - y） ＋ h（x） and f（2x ＋ y） - f（2x - y） = g（x ＋ y） - g（x - y）. The method used for solving these functional equations is elementary but exploits an important result due to Hosszfi. The solution of this functional equation can also be determined in certain type of groups using two important results due to Szekelyhidi.     

Generalized functional equation and its applications in information theory

The functional equationf(x,y)+g(x)h(y)F(u/1?x,ν/1?y)=f(u,ν)+g(u)h(ν)F(x/1?u,y/1?ν) ... (1) forx, y, u, ν ∈ [0, 1) andx+u,y+ν ∈ [0,1) whereg andh satisfy the functional equationφ (x+y?xy)=φ(x)φ(y)... (2) has been solved for some non-constant solution of (2) in [0, 1] withφ (0)=1,φ(1)=0 and the solution is used in characterising some measures of information.     

A stability analysis for a semilinear parabolic partial differential equation

We consider a parabolic partial differential equation u_t = u_xx + f(u), where ? ∞ < x < + ∞ and 0 < t < + ∞. Under suitable hypotheses pertaining to f, we exhibit a class of initial data φ(x), ? ∞ < x < + ∞, for which the corresponding solutions u(x, t) approach zero as t → + ∞. This convergence is uniform with respect to x on any compact subinterval of the real axis.     

A general class of fuzzy operators,the demorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators

In this paper we examine operators which can be derived from the general solution of functional equations on associativity. We define the characteristics of those functions f(x) which are necessary for the production of operators. We shall show, that with the help of the negation operator for every such function f(x) a function g(x) can be given, from which a disjunctive operator can be derived, and for the three operators the DeMorgan identity is fulfilled. For the fulfillment of the DeMorgan identity the necessary and sufficient conditions are given.We shall also show that an f_λ(x) can be constructed for every f(x), so that for the derived k_λ(x,y) and d_λ(x,y) lim_λ→∞k_λ(x,y) and lim_λ→∞d_λ(x,y) = max(x,y).As Yager's operator is not reducible, for every λ there exists an α, for which, in case x < α and y<α, k_λ(x,y) = 0.We shall give an f(x) which has the characteristics of Yager's operator, and which is strictly monotone.Finally we shall show, that with the help of all those f(x), which are necessary when constructing a k(x,y), an F(x) can be constructed which has the properties of the measures of fuzziness introduced by A. De Luca and S. Termini. Some classical fuzziness measures are obtained as special cases of our system.     

Distributional analog of a functional equation

In this paper, we shall apply an operator method for casting and solving the distributional analog of functional equations. In particular, the method will be employed to solve f₁(x + y) + f₂ (x - y) + f₃(xy) = 0     

Inverse functions of polynomials and its applications to initialize the search of solutions of polynomials and polynomial systems

In this paper we present a new algorithm for solving polynomial equations based on the Taylor series of the inverse function of a polynomial, f _P (y). The foundations of the computing of such series have been previously developed by the authors in some recent papers, proceeding as follows: given a polynomial function \(y=P(x)=a_0+a_1x+\cdots+a_mx^m\), with \(a_i \in \mathcal{R}, 0 \leq i \leq m\), and a real number u so that P′(u)?≠?0, we have got an analytic function f _P (y) that satisfies x?=?f _P (P(x)) around x?=?u. Besides, we also introduce a new proof (completely different) of the theorems involves in the construction of f _P (y), which provide a better radius of convergence of its Taylor series, and a more general perspective that could allow its application to other kinds of equations, not only polynomials. Finally, we illustrate with some examples how f _P (y) could be used for solving polynomial systems. This question has been already treated by the authors in preceding works in a very complex and hard way, that we want to overcome by using the introduced algorithm in this paper.     

Sixth-order superstable two-step methods for second-order initial-value problems

For the numerical integration of general second-order initial-value problems y″ = f(x, y, y′), y(x₀) = y₀, y′(x₀) = y′₀, we report a family of two-step sixth-order methods which are superstable for the test equation y″ + 2αy′ + β²y = 0, α, β ⩾ 0, α + β\s>0, in the sense of Chawla [1].     

Stability and bifurcation of equilibrium solutions of reaction-diffusion equations on an infinite space interval

Under the condition that f(x, y, z, α) and its partial derivatives decay sufficiently fast as $\text{¦x¦ → ∞}$ we will study the (linear) stability and bifurcation of equilibrium solutions of the scalar problem u_t = u_xx + f(x, u, u_x, α), u_x(?∞, t) = u_x(∞, t) = 0 (1) where α is a real bifurcation parameter. After introducing appropriate function spaces X and Y the problem (1) can be rewritten $\text{d}\text{dt}\text{u = G(u, α)}$, (7) where G:X×R → Y is given by G(u, α)(x) = u″(x) + f(x, u(x), u′(x), α). It will be shown, for each (u, α)?X × R, that the Fréchet derivative G_u(u,a): X → Y is not a Fredholm operator. This difficulty is due to the fact that the domain of the space variable x, is infinite and cannot be eliminated by making another choice of X and Y. Since G_u(u, α) is not Fredholm, the hypotheses of most of the general stability and bifurcation results are not satisfied. If (u₀, α₀?S = {(u, α): G(u, α) = 0}, (i.e., (u₀,α₀) is an equilibrium solution of (7)), a necessary condition on the spectrum of G_u(u₀, α₀) for a change in the stability of points in S to occur at G_u(u₀, α₀) will be given. When this condition is met, the principle of exchange of stability which means, in a neighborhood of (u₀, α₀), that adjacent equilibrium solutions for the same α have opposite stability properties in a weakened sense will be established. Also, when G_u or its first order partial derivatives, evaluated at (u₀, α₀), are not too degenerate, the shape of S in a neighborhood of (u₀, α₀) will be described and a strenghtened form of the principle of exchange of stability will be obtained.     

On an optimization problem of Bellman

This paper considers a problem proposed by Bellman in 1970: given a continuous kernel K(x, y) defined on I × I, find a pair of continuous functions f and g such that f(x) + g(y) ? K(x, y) on I × I and ∝_I (f + g) is minimum. The notion of basic decomposition of K is defined, and it is shown that whenever K(x, y) or K(x, a + b ? y), I = [a, b], admits a basic decomposition, Bellman's problem has a unique differentiable solution, provided K is differentiable. Explicit formulas for such solutions are given. More generally, there are kernels which admit basic decompositions on subintervals which can be “pasted together” to define a unique piecewise differentiable solution.     

A nonlinear evolution system with two subdifferentials and monotone differential games

It is shown that the first order multivalued equation for V = V(t, x, y, z) involving the sum of two subdifferentials composed with the partials of V (V_t +f(t, x, y, z) · ▽_xV + β(V_y) + γ(V_z) + h(t, x, y, z) ? 0 a.e.) has a Lipschitz solution. This solution is shown to be the value of a differential game in which the players are restricted to choosing monotone nondecreasing functions of time. Accordingly, the multivalued equation is interpreted as the corresponding Hamilton-Jacobi equation of the game.     

Nonlinear parabolic equations with p-growth and unbounded data

The purpose of this paper is to prove the existence of a solution for a nonlinear parabolic equation in the form u_t - div(a(t, x, u, Du)) = H(t, x, u, Du) - div(g(t, x)) in Q_T =]0,T[×Ω, Ω ⊂ R^N, with an initial condition u(0) = u₀, where u₀ is not bounded, |H(t,x, u, ξ)⩽ β|ξ|^p + f(t,x) + βe^λ₁^|u|f, |g|^p/(p-1) ∈ L^r(Q_T) for some r = r{N) ⩾ 1, and - div(a(t,x,u, Du)) is the usual Leray-Lions operator.     

Nonlinear equations with mixed derivatives

The nonlinear hyperbolic equation ∂²u(x, y)/∂x ∂y + g(x, y)f(u(x, y)) = 0 with u(x, 0) = φ(x) and u(0, y) = Ψ(y), considered by [1.], 31–45) under appropriate smoothness conditions, is solvable by the author's decomposition method (“Stochastic Systems,” Academic Press, 1983 and “Nonlinear Stochastic Operator Equations,” Academic Press, 1986).     

Skew Derivations of Prime Rings

Given a prime ring R, a skew g-derivation for g : R → R is an additive map f : R → R such that f(xy) = f(x)g(y) + xf(y) = f(x)y + g(x)f(y) and f(g(x)) = g(f(x)) for all x, y ∈ R. We generalize some properties of prime rings with derivations to the class of prime rings with skew derivations.     

Periodic solutions to one-dimensional wave equation with x-dependent coefficients

In this paper, we study the nonlinear one-dimensional periodic wave equation with x-dependent coefficients u(x)ytt−(ux(x)yx)+g(x,t,y)=f(x,t) on (0,π)×R under the boundary conditions a₁y(0,t)+b₁yx(0,t)=0, a₂y(π,t)+b₂yx(π,t)=0 ( for i=1,2) and the periodic conditions y(x,t+T)=y(x,t), yt(x,t+T)=yt(x,t). Such a model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. A main concept is the notion “weak solution” to be given in Section 2. For T is the rational multiple of π, we prove some important properties of the weak solution operator. Based on these properties, the existence and regularity of weak solutions are obtained.