A Liapunov functional for a linear integral equation

In this note we consider a scalar integral equation $x(t)= a(t)-\int^t_0 C(t,s)x(s)ds$, together with its resolvent equation, $R(t,s)= C(t,s)-\int^t_s C(t,u) R(u,s)du$, where $C$ is convex. Using a Liapunov functional we show that for fixed $s$ then $|R(t,s) - C(t,s)| \to 0$ as $t \to \infty$ and $\int^{\infty}_s (R(t,s)-C(t,s))^2 dt < \infty$. We then show that the variation of parameters formula $x(t)=a(t)-\int^t_0 R(t,s) a(s)ds$ can be replaced by $X(t)=a(t)-\int^t_0 C(t,s)a(s)ds$ when $a \in L^1[0,\infty)$ and that $|X(t) - x(t)|\to 0$ as $t \to \infty$ and $\int^{\infty}_0 (x(t)-X(t))^2 dt < \infty$. A mild nonlinear extension is given

Integral equations with contrasting kernels

In this paper we study integral equations of the form $x(t)=a(t)-\int^t_0 C(t,s)x(s)ds$ with sharply contrasting kernels typified by $C^*(t,s)=\ln (e+(t-s))$ and $D^*(t,s)=[1+(t-s)]^{-1}$. The kernel assigns a weight to $x(s)$ and these kernels have exactly opposite effects of weighting. Each type is well represented in the literature. Our first project is to show that for $a\in L^2[0,\infty)$, then solutions are largely indistinguishable regardless of which kernel is used. This is a surprise and it leads us to study the essential differences. In fact, those differences become large as the magnitude of $a(t)$ increases. The form of the kernel alone projects necessary conditions concerning the magnitude of $a(t)$ which could result in bounded solutions. Thus, the next project is to determine how close we can come to proving that the necessary conditions are also sufficient. The third project is to show that solutions will be bounded for given conditions on $C$ regardless of whether $a$ is chosen large or small; this is important in real-world problems since we would like to have $a(t)$ as the sum of a bounded, but badly behaved function, and a large well behaved function

Asymptotic stability in differential equations with unbounded delay

In this paper we consider a functional differential equation of the form $$x'=F(t,x,\int_0^t C(at-s) x(s)\,ds)$$ where $a$ is a constant satisfying $0<a<\infty$. Thus, the integral represents the memory of past positions of the solution $x$. We make the assumption that $\int_0^\infty |C(t)|\, dt<\infty$ so that this is a fading memory problem and we are interested in studying the effects of that memory over all those values of $a$. Very different properties of solutions emerge as we vary $a$ and we are interested in developing an approach which handles them in a unified way

Continuity, compactness, fixed points, and integral equations

An integral equation, $x(t)=a(t)-\int^t_{-\infty} D(t,s)g(x(s))ds$ with $a(t)$ bounded, is studied by means of a Liapunov functional. There results an a priori bound on solutions. This gives rise to an interplay between continuity and compactness and leads us to a fixed point theorem of Schaefer type. It is a very flexible fixed point theorem which enables us to show that the solution inherits properties of $a(t)$, including periodic or almost periodic solutions in a Banach space

Marachkov type stability results for functional differential equations

This paper is concerned with systems of functional differential equations with either finite or infinite delay. We give conditions on the system and on a Liapunov function to ensure that the zero solution is asymptotically stable. The main result of this paper is that the assumption on boundedness in Marachkov type stability results may be replaced (in both the finite and the infinite delay case) with the condition that $|f(t,\varphi)|\le F(t)$ such that $\int^{\infty} 1/F(t) dt=\infty$

Addendum to asymptotic stability in differential equations with unbounded delay

This addendum concerns the paper of the above title found in EJQTDE No. 13 (1999). Throughout that paper was the tacit assumption that the coefficient functions $h(t)$, $b(t)$, and $C(t)$ are all continuous on their respective domains. Every result, as well as the existence result stated at the end of the first section, depended on those functions being continuous. A search of the paper indicates that we failed to state this. We regret any inconvenience which this may have caused any reader

Necessary and sufficient conditions for large contractions in fixed point theory

Many problems in integral and differential equations involve an equation in which there is almost a contraction mapping. Through some type of transformation we arrive at an operator of the form $H(x) =x-f(x)$. The paper contains two main parts. First we offer several transformations which yield that operator. We then offer necessary and sufficient conditions to ensure that the operator is a large contraction. These operators yield unique fixed points. A partial answer to a question raised in [D. R. Smart, Fixed point theorems, Cambridge University Press, Cambridge, 1980] is given. The last section contains examples and applications

Complementary equations: a fractional differential equation and a Volterra integral equation

Abstract. It is shown that a continuous, absolutely integrable function satisfies the initial value problem Dqx(t) = f (t, x(t)), lim t→0+ t1−qx(t) = x0 (0 &lt; q &lt; 1) on an interval (0, T] if and only if it satisfies the Volterra integral equation x(t) = x0tq−1 + 1 Γ(q) ∫ t 0 (t − s)q−1 f (s, x(s)) ds on this same interval. In contradistinction to established existence theorems for these equations, no Lipschitz condition is imposed on f (t, x). Examples with closed-form solutions illustrate this result

Integral equations, transformations, and a Krasnoselskii-Schaefer type fixed point theorem

In this paper we extend the work begun in 1998 by the author and Kirk for integral equations in which we combined Krasnoselskii's fixed point theorem on the sum of two operators with Schaefer's fixed point theorem. Schaefer's theorem eliminates a difficult hypothesis in Krasnoselskii's theorem, but requires an a priori bound on solutions. Here, we simplify the work by means of a transformation which often reduces the a priori bound to a triviality. Our work is focused on an integral equation in which the goal is to prove that there is a unique continuous positive solution on $[0,\infty)$. In addition to the transformation, there are two techniques which we would emphasize. A technique is introduced yielding a lower bound on the solutions which enables us to deal with problems threatening non-uniqueness. The technique offers a solution to a classical problem and it seems entirely new. We show that when the equation defines the sum of a contraction and a Lipschitz operator, then we first get existence on arbitrary intervals $[0,E]$ and then introduce a technique which we call a progressive contraction which allows us to prove uniqueness and then parlay the solution to $[0,\infty)$. The technique is well suited to integral equations

Fixed points and differential equations with asymptotically constant or periodic solutions

Cooke and Yorke developed a theory of biological growth and epidemics based on an equation $x'(t)=g(x(t))-g(x(t-L))$ with the fundamental property that $g$ is an arbitrary locally Lipschitz function. They proved that each solution either approaches a constant or $\pm \infty$ on its maximal right-interval of definition. They also raised a number of interesting questions and conjectures concerning the determination of the limit set, periodic solutions, parallel results for more general delays, and stability of solutions. Although their paper motivated many subsequent investigations, the basic questions raised seem to remain unanswered. We study such equations with more general delays by means of two successive applications of contraction mappings. Given the initial function, we explicitly locate the constant to which the solution converges, show that the solution is stable, and show that its limit function is a type of "selective global attractor." In the last section we examine a problem of Minorsky in the guidance of a large ship. Knowledge of that constant to which solutions converge is critical for guidance and control