A Stieltjes–Volterra Integral Equation Theory

Canadian Journal of Mathematics ◽

10.4153/cjm-1966-035-3 ◽

1966 ◽

Vol 18 ◽

pp. 314-331 ◽

Cited By ~ 13

Author(s):

D. B. Hinton

Keyword(s):

Integral Equation ◽

Volterra Integral Equation ◽

Integral Equation Theory ◽

Algebraic Ring ◽

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Suppose S = [a, b] is a number interval and F is a function from S X S to a normed algebraic ring N with multiplicative identity I. We consider the problem of finding, for appropriate conditions on F, a function M from S X S to N such that for all t and x,where the integral is a Cauchy-left integral.

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Uniqueness of solutions for a class of non-linear Volterra integral equations with convolution kernel

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100068262 ◽

1989 ◽

Vol 106 (3) ◽

pp. 547-552 ◽

Cited By ~ 29

Author(s):

P. J. Bushell ◽

W. Okrasinski

Keyword(s):

Integral Equation ◽

Integral Equations ◽

Volterra Integral Equation ◽

Convolution Kernel ◽

Qualitative Behaviour ◽

Uniqueness Of Solutions ◽

Linear Diffusion ◽

Non Linear ◽

Linear Volterra Integral Equations ◽

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The non-linear Volterra integral equationhas been studied recently in connection with non-linear diffusion and percolation problems [4, 6, 10]. The existence, uniqueness and qualitative behaviour of non-negative, non-trivial solutions are the questions of physical interest.

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EXISTENCE OF RESOLVENT FOR VOLTERRA INTEGRAL EQUATIONS ON TIME SCALES

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972709001166 ◽

2010 ◽

Vol 82 (1) ◽

pp. 139-155 ◽

Cited By ~ 26

Author(s):

MURAT ADıVAR ◽

YOUSSEF N. RAFFOUL

Keyword(s):

Integral Equation ◽

Qualitative Analysis ◽

Integral Equations ◽

Time Scales ◽

Volterra Integral Equation ◽

Sufficient Conditions ◽

Volterra Integral Equations ◽

Future Research ◽

Shift Operators ◽

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AbstractWe introduce the concept of ‘shift operators’ in order to establish sufficient conditions for the existence of the resolvent for the Volterra integral equation on time scales. The paper will serve as the foundation for future research on the qualitative analysis of solutions of Volterra integral equations on time scales, using the notion of the resolvent.

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Convolution, fixed point, and approximation in Stieltjes-Volterra integral equations

Journal of the Australian Mathematical Society ◽

10.1017/s144678870001003x ◽

1972 ◽

Vol 14 (2) ◽

pp. 182-199 ◽

Cited By ~ 2

Author(s):

Carl W. Bitzer

Keyword(s):

Integral Equation ◽

Fixed Point ◽

Fixed Point Theorem ◽

Integral Equations ◽

Point Theorem ◽

Volterra Integral Equation ◽

The Other ◽

Integral Equation Theory ◽

Identity Function ◽

Convolution Integrals

This paper focuses primarily on two aspects of Stieltjes-Volterra integral equation theory. One is a theory for convolution integrals — that is, integrals of the form — and the other is a fixed point theorem for a mapping which is induced by an integral equation. Throughout the paper I will denote the identity function whose range of definition should be clear from the context and all integrals will be left integrals, written , whose simplest approximating sum is [f(b) – f(a)]·g(a) and whose value is the limit of approximating sums with respect to successive refinements of the interval. Also, N will denote the set of elements of a complete normed ring with unity 1 and S will denote a set linearly ordered by ≦.

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