We shall not dwell on the proof of picards theorem. Solutions of nonlinear integral equations and their application to singular perturbation problems. The problem of course is that fy y is not lipschitz. C a, and the maximum slope over this interval is s d. Finogenko 1 differential equations volume 39, pages 1124 11 2003 cite this article. Request pdf boundaryvalue problems for thirdorder lipschitz ordinary differential equations for the thirdorder differential equation y. Boundaryvalue problems for thirdorder lipschitz ordinary. Then, insensitivity of the solution under small changes of data of equation is examined. If, is bounded and the lipschitz condition is satisfied, then the sequence of functions converges to a continuous function this function satisfies the differential equation this is the unique solution to this differential equation with the given initial condition. Spatial besov regularity for stochastic partial di. Existence and uniqueness theorem on uncertain dierential. Maximiliansuniversitat, germany, 2018, available in pdf format at. Almost every textbook on ordinary differential equations contains a version of the wellknown uniqueness theorem published by lipschitz in 1877, see 7 and, for instance, the monographs 4, 14.
Lipschitz condition, uniqueness, and stability of solutions. The key is the fact that if the mth iterate of f, denoted by fm is a contraction, then f itself has a unique. If the lipschitz condition is not satisfied, there is at least 1 other solution citation needed. An educated guess can take you far, on all areas of mathematics. Existenceuniqueness of solutions to quasilipschitz odes. We have considered some special cases in which one can explicitly determine a solution of a differential equation with initial condition, and conclude that the. Ordinary differential equationssuccessive approximations. Lipschitz condition an overview sciencedirect topics. Our equations in particular include constant delays, time variable delay, distributed delay, etc. We combine regularity results by kim 30 on stochastic partial di. This is not surprising since x 0 is a point of the discontinuity of the coe. In this paper, we study lipschitz stability of caputo fractional differential equations with noninstantaneous impulses and state dependent delays. Does lipschitz with respect to x imply uniqueness for the. Jinqiao duan, wei wang, in effective dynamics of stochastic partial differential equations, 2014.
Theorem on uncertain differential equations with local lipschitz condition. Differential equations university of wisconsinmadison. Lipschitz condition, uniqueness, and stability of solutions of differential equations dr. Chapter 5 the initial value problem for ordinary differential. The lipschitz condition implies that g is continuous on a, b. The paper contains a discussion on solutions to symmetric type of fuzzy stochastic differential equations. Lipschitz continuous ordinary differential equations are. For such integral equations the convergence technique bas been. Bipartite fuzzy stochastic differential equations with.
In a recent study of the strong maximum principle for integro differential equation 8, the third. Adapted solutions of backward stochastic differential. A special type of lipschitz continuity, called contraction, is used in the banach fixedpoint theorem. We say that ft,x mapping i into fn is uniformly lipschitz continuous with respect to xif there is a constant lcalled the lipschitz constant for which. So if the lipschitz condition is satisfied, and, and, is bounded, there is a solution and the solution is unique. Clearly f does not satisfy lipschitz condition near origin. Symmetric fuzzy stochastic differential equations with. On the right lipschitz condition for differential equations with piecewise continuous righthand sides i.
The existence and uniqueness theorem are also valid for certain system of rst order equations. Ordinary differential equationsexistence wikibooks, open. More precisely, it is shown in li that, if a periodic function. It has been proved that uncertain differential equation ude has a unique solution, under the conditions that the coefficients are global lipschitz continuous. The theorem guarantees that a unique solutionexists at least. Pdf it has been proved that uncertain differential equation ude has a unique. Example 1 y c ekx is the unique solution to the di. This is the strategy followed in proving both the existence theorems that we are going to discuss. We define a class of polynomialspace computation tableaux with equally weak. For this reason some authors especially in the past use the term lipschitz condition for the weaker inequality \eqrefeq.
Islam university of dayton we consider an initial value problem of a system of nonlinear ordinary. An ordinary differential equation or ode is an equation involving derivatives of an unknown quantity with respect to a single variable. However, given some initial condition, the solution is unique. Pdf existence and uniqueness theorem on uncertain differential. However, if we choose the initial condition as y1 1 the solution is y x2 is well. Differential equations are essential for a mathematical descrip tion of nature. Ordinary differential equationsexistence wikibooks. We claim that such symmetric equations have unique solutions in the case that equations coefficients satisfy a certain generalized lipschitz condition. Existence and uniqueness theorem on uncertain differential. Lipschitz regularity of solutions for mixed integro. Local lipschitz condition an overview sciencedirect topics. The condition was first considered by lipschitz in li in his study of the convergence of the fourier series of a periodic function. In section 4, we use the same techniques solve several other problems, including the ones about volterra integral equations 20. A firstorder initial value problemis a differential equation whose solution must satisfy an initial condition example 2 show that the function is a solution to the firstorder initial value problem solution the equation is a firstorder differential equation with.
Lipschitz condition contraction operator successive approximations banach fixed point theorem 7. The condition 3 is called a lipschitz condition3, and lis called the lipschitz constant for f. Under the lipschitz and boundedness conditions imposed on drifts and diffusions coefficients we prove existence of a unique solution. Lipschitz stability of nonlinear systems of differential. Finally, we mention that all results can be repeated for solutions to bipartite setvalued stochastic differential equations. We talk about uniform continuity of a function with respect to a domain. In the first instance, perhaps one would like to try the local lipschitz condition plus the linear growth condition, as these conditions guarantee the existence and uniqueness of the solution for a forward stochastic differential equation. Finogenko 1 differential equations volume 39, pages 1124 11. In mathematics, an ordinary differential equation ode is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. In the theory of differential equations, lipschitz continuity is the central condition of the picardlindelof theorem which guarantees the existence and uniqueness of the solution to an initial value problem. The theorem concerns the initial value problem \beginequation\labele. The study is based on lyapunov functions and the razumikhin technique.
In answer to kos question raised in 1983, we show that an initial value problem given by a polynomialtime computable, lipschitz continuous function can have a polynomialspace complete solution. Youre trying to find a counter example for something, this doesnt have to come through a deductive reasoning. This solution is usually a trivial solution where k is a constant. Let us assume that there is more than one root and let a and. Lipschitz condition with respect to one if and only if there is one with respect to the other. Lipschitz condition and functions satisfying lipschitz. Lipschitz continuity the study of existence and uniqueness.
That is, the lipschitz condition on functions is somewhere in between being continuous and continuously di. Lastly, the concept of the lipschitz condition is important and is addressed. However yt 0 is also a solution with initial data y0 0, so we have nonuniqueness of solutions for this equation. On the right lipschitz condition for differential equations. This conclusion can be derived, for instance, from the dinilipschitz criterion and the convergence is indeed uniform. Lipschitz continuous at 0, if this slope is bounded in the neighborhood of every point. But this an example that isnt specific to differential equations.
The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable. Let y 1 and y 2 be two solutions and consider zx q y 1x y 2x 2. Cauchylipschitz theorem encyclopedia of mathematics. We consider the case of impulses that start abruptly at some points and their. First eliminate the by term by multiplying throughout by i. It is important to note that the solution becomes unbounded as x 0. R 1, we know that the function satis s the same lipschitz inequality in r 2. Finding conditions to guarantee existence of solutions to ivp. In a recent study of the strong maximum principle for integrodifferential equation 8, the third. Condition b can be replaced by a weaker condition which is known as lipschitz condition. Let us summarize the proof of the existence of solutions as follows. The symmetric equations under study have drift and diffusion terms symmetrically on both sides of equations.