Conditions for the existence of bounded solutions of nonlinear differential and functional differential equations

Let E be a finite-dimensional Banach space, let C⁰(R; E) be a Banach space of functions continuous and bounded on R and taking values in E; let K:C ⁰(R ,E) → C ⁰(R, E) be a c-continuous bounded mapping, let A: E → E be a linear continuous mapping, and let h ∈ C ⁰(R, E). We establish conditions for the existence of bounded solutions of the nonlinear equation

Conditions for the existence of bounded solutions of one class of nonlinear differential equations

For systems of nonlinear differential equations (dx/dt) = A(x)x + f(t) in a Banach space, we establish sufficient conditions for the existence of their solutions bounded on the entire real axis R. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 3, pp. 317–325, March, 2006.     

On the existence and uniqueness of solutions continuous and bounded on the real axis for nonlinear functional equations

For one class of nonlinear functional equations, we establish conditions for the existence and uniqueness of solutions continuous and bounded on the real axis.     

Numerical existence and uniqueness proof for solutions of nonlinear hyperbolic equations

We consider a numerical method to verify the existence and uniqueness of the solutions of nonlinear hyperbolic problems with guaranteed error bounds. Using a C¹ finite element solution and an inequality constituting a bound on the norm of the inverse operator of the linearized operator, we numerically construct a set of functions which satisfy the hypothesis of Banach's fixed point theorem for a continuous map on L^p-space in a computer. We present detailed verification procedures and give some numerical examples.     

On the existence of solutions for strongly nonlinear differential equations

The objectives of this paper are twofold. Firstly, to prove the existence of an approximate solution in the mean for some nonlinear differential equations, we also investigate the behavior of the class of solutions which may be associated with the differential equation. Secondly, we aim to implement the homotopy perturbation method (HPM) to find analytic solutions for strongly nonlinear differential equations.     

The existence and uniqueness of fuzzy solutions for hyperbolic partial differential equations

Fuzzy hyperbolic partial differential equation, one kind of uncertain differential equations, is a very important field of study not only in theory but also in application. This paper provides a theoretical foundation of numerical solution methods for fuzzy hyperbolic equations by considering sufficient conditions to ensure the existence and uniqueness of fuzzy solution. New weighted metrics are introduced to investigate the solvability for boundary valued problems of fuzzy hyperbolic equations and an extended result for more general classes of hyperbolic equations is initiated. Moreover, the continuity of the Zadeh’s extension principle is used in some illustrative examples with some numerical simulations for \(\alpha \) -cuts of fuzzy solutions.     

非线性一阶常微分方程解的存在惟一性

讨论了一类非线性一阶常微分方程边值问题解的存在惟一性.得到了当参数在一定的范围取值时解存在惟一的充分条件,并包含了一些已知结果.主要结果基于Leray-Schauder非线性抉择理论和Banach不动点定理.     

Existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations

In this paper, we investigate existence and uniqueness of solutions for nonlinear impulsive fractional differential equations. By utilizing the well-known fixed point theorems, we obtain some sufficient conditions for the uniqueness and existence of the solutions. At the end of the paper, an example is given to illustrate our main results.     

On bounded nonoscillatory solutions of third-order nonlinear differential equations

This paper is concerned with the asymptotic behavior of solutions of nonlinear differential equations of the third-order with quasiderivatives. We give the necessary and sufficient conditions guaranteeing the existence of bounded nonoscillatory solutions. Sufficient conditions are proved via a topological approach based on the Banach fixed point theorem.     

Global existence and nonexistence of solutions for second-order nonlinear differential equations

This paper deals with the global existence and nonexistence of solutions of the second-order nonlinear differential equation (φ(x^′))^′+λφ(x)=0${(\phi (x^{\prime }))}^{\prime }+\lambda \phi (x)=0$ satisfying x(0)=_x0$x(0)=x_0$ and x^′(0)=_x1$x^{\prime }(0)=x_1$, where λ   is a positive parameter and φ:(−ρ,ρ)→(−σ,σ)$\phi :(-\rho ,\rho )\to (-\sigma ,\sigma )$ with 0<ρ?∞$0<\rho ?\infty$ and 0<σ?∞$0<\sigma ?\infty$ is strictly increasing odd bijective and continuous on (−ρ,ρ)$(-\rho ,\rho )$. Necessary and sufficient conditions are obtained for the initial value problem to have a unique global solution which is oscillatory and periodic. Examples are given to illustrate our main result. Finally, a nonexistence result for the equation with a damping term is discussed as an application to our result.     

非线性方程极限环的存在惟一性及惟二性

考虑非线性方程 x=(y) - h(y) F(x) ,y=- g(x)的极限环问题 .通过考察方程轨线的走向及比较沿闭轨线的发散量积分 ,给出了极限环的存在惟一性及惟二性的若干组充分条件 ,推广了已有文献的相应结果     

On existence and uniqueness of solutions to uncertain backward stochastic differential equations简

This paper is concerned with a class of uncertain backward stochastic differential equations （UBSDEs） driven by both an m-dimensional Brownian motion and a d-dimensional canonical process with uniform Lipschitzian coefficients. Such equations can be useful in mod- elling hybrid systems, where the phenomena are simultaneously subjected to two kinds of un- certainties： randomness and uncertainty. The solutions of UBSDEs are the uncertain stochastic processes. Thus, the existence and uniqueness of solutions to UBSDEs with Lipschitzian coeffi- cients are proved.     

A criterion for the existence of bounded nonoscillatory solutions for nonlinear retarded differential equatory

Summary For the equation (⋆) below we give a condition which guarantees the existence of a bounded nonoscillatory solution x with Entrata in Redazione il 19 luglio 1977.