Zobrazeno 1 - 10
of 120
pro vyhledávání: '"Theodore Burton"'
Autor:
Theodore Burton, Ioannis Purnaras
Publikováno v:
Electronic Journal of Qualitative Theory of Differential Equations, Vol 2019, Iss 94, Pp 1-24 (2019)
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
Externí odkaz:
https://doaj.org/article/ea07f4cd38054366a84a449ba8202fb5
Autor:
Theodore Burton, Ioannis Purnaras
Publikováno v:
Electronic Journal of Qualitative Theory of Differential Equations, Vol 2017, Iss 49, Pp 1-6 (2017)
In the theory of progressive contractions an equation such as \[ x(t) = L(t)+\int^t_0 A(t-s)[ f(s,x(s)) + g(s,x(s-r(s))]ds, \] with initial function $\omega$ with $\omega (0) =L(0)$ defined by $ t\leq 0 \implies x(t) =\omega (t)$ is studied on an int
Externí odkaz:
https://doaj.org/article/5056a9398eab4a938eb0c0fc45b278d3
Autor:
Theodore Burton
Publikováno v:
Electronic Journal of Qualitative Theory of Differential Equations, Vol 2016, Iss 66, Pp 1-13 (2016)
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 dif
Externí odkaz:
https://doaj.org/article/983d7b8224734d2c83c0579285cb97fe
Autor:
Theodore Burton, Ioannis Purnaras
Publikováno v:
Electronic Journal of Qualitative Theory of Differential Equations, Vol 2018, Iss 44, Pp 1-21 (2016)
There is substantial literature going back to 1965 showing boundedness properties of solutions of the integro-differential equation \[ x'(t) = -\int^t_0 A(t-s) h(s,x(s))ds \] when $A$ is a positive kernel and $h$ is a continuous function using \[ \in
Externí odkaz:
https://doaj.org/article/33c43f6be5d74c97832ce26329973b87
Publikováno v:
Electronic Journal of Qualitative Theory of Differential Equations, Vol 2015, Iss 12, Pp 1-24 (2015)
It is shown that a continuous, absolutely integrable function satisfies the initial value problem \[ D^{q}x(t) = f(t,x(t)), \qquad \lim_{t \to 0^+} t^{1-q}x(t) = x^{0} \qquad (0 < q < 1) \] on an interval $(0, T]$ if and only if it satisfies the Vol
Externí odkaz:
https://doaj.org/article/8307aeb97c204364a5fcef3c91a77eef
Autor:
Theodore Burton
Publikováno v:
Electronic Journal of Qualitative Theory of Differential Equations, Vol 2010, Iss 10, Pp 1-10 (2010)
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
Externí odkaz:
https://doaj.org/article/2987f87f1a984c3e9a4193fe36416f68
Autor:
Theodore Burton
Publikováno v:
Electronic Journal of Qualitative Theory of Differential Equations, Vol 2008, Iss 2, Pp 1-22 (2008)
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
Externí odkaz:
https://doaj.org/article/01c9676a22954071a58533d466840278
Autor:
Theodore Burton
Publikováno v:
Electronic Journal of Qualitative Theory of Differential Equations, Vol 2006, Iss 2, Pp 1-17 (2006)
This paper concerns several variants of an integral equation $$ x(t)=a(t)-\int^t_0 C(t,s) x(s)ds,$$ a resolvent $$ R(t,s),$$ and a variation-of-parameters formula $$ x(t)=a(t)-\int^t_0 R(t,s) a(s)ds $$ with special accent on the case in which $a(t)$
Externí odkaz:
https://doaj.org/article/c1a2ab3328e74f0daaacd3e62d5c8c55
Autor:
Theodore Burton
Publikováno v:
Electronic Journal of Qualitative Theory of Differential Equations, Vol 2004, Iss 11, Pp 1-31 (2004)
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
Externí odkaz:
https://doaj.org/article/f2c93d388c7a4d76a6b7bc8565fedb98
Autor:
Theodore Burton, Géza Makay
Publikováno v:
Electronic Journal of Qualitative Theory of Differential Equations, Vol 2002, Iss 14, Pp 1-13 (2002)
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
Externí odkaz:
https://doaj.org/article/6f32174f5070489e8e00f40d86e70e8c