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New conditions for the existence of non trivial solutions to some Volterra equations.

W. Okrasinski (1990)

Extracta Mathematicae

We consider the following Volterra equation:(1)       u(x) = ∫0x k(x-s) g(u(s)) ds,   where,k: [0, δ0] → R is an increasing absolutely continuous function such thatk(0) = 0g: [0,+ ∞) → [0,+ ∞) is an increasing absolutely continuous function such that g(0) = 0 and g(u)/u → ∞ as u → 0+ (see [3]).Let us note that (1) has always the trivial solution u = 0.Some necessary and sufficient conditions for the existence of nontrivial solutions to (1) with k(x) = xα - 1 (α>0) are given in [1], [2] and...

Non-autonomous implicit integral equations with discontinuous right-hand side

Giovanni Anello, Paolo Cubiotti (2004)

Commentationes Mathematicae Universitatis Carolinae

We deal with the implicit integral equation h ( u ( t ) ) = f ( t , ∫ I g ( t , z ) u ( z ) d z ) for a.a. t ∈ I , where I : = [ 0 , 1 ] and where f : I × [ 0 , λ ] → ℝ , g : I × I → [ 0 , + ∞ [ and h : ] 0 , + ∞ [ → ℝ . We prove an existence theorem for solutions u ∈ L s ( I ) where the contituity of f with respect to the second variable is not assumed.

Non-autonomous vector integral equations with discontinuous right-hand side

Paolo Cubiotti (2001)

Commentationes Mathematicae Universitatis Carolinae

We deal with the integral equation u ( t ) = f ( t , ∫ I g ( t , z ) u ( z ) d z ) , with t ∈ I : = [ 0 , 1 ] , f : I × ℝ n → ℝ n and g : I × I → [ 0 , + ∞ [ . We prove an existence theorem for solutions u ∈ L s ( I , ℝ n ) , s ∈ ] 1 , + ∞ ] , where f is not assumed to be continuous in the second variable. Our result extends a result recently obtained for the special case where f does not depend explicitly on the first variable t ∈ I .

Nonlinear mixed Volterra-Fredholm integrodifferential equation with nonlocal condition

Haribhau L. Tidke, Machindra B. Dhakne (2012)

Applications of Mathematics

The aim of the present paper is to investigate the global existence of mild solutions of nonlinear mixed Volterra-Fredholm integrodifferential equations, with nonlocal condition. Our analysis is based on an application of the Leray-Schauder alternative and rely on a priori bounds of solutions.

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