Remarks on the Existence of Nontrivial Solutions for a Class of Volterra Equations with Smooth Kernels.
The blow-up solutions of integral equations
An initial value problem fora third order differential equation
For an initial value problem u'''(x) = g(u(x)), u(0) = u'(0) = u''(0) = 0, x > 0, some theorems on existence and uniqueness of solutions are established.
A singular initial value problem for the equation
We consider the problem of the existence of positive solutions u to the problem , (g ≥ 0,x > 0, n ≥ 2). It is known that if g is nondecreasing then the Osgood condition is necessary and sufficient for the existence of nontrivial solutions to the above problem. We give a similar condition for other classes of functions g.
The existence of solutions to a Volterra integral equation
We study the equation u = k∗g(u) with k such that ln k is convex or concave and g is monotonic. Some necessary and sufficient conditions for the existence of nontrivial continuous solutions u of this equation are given.
A singular initial value problem for second and third order differential equations
Galerkin methods for multidimensional nonlinear Volterra type equations
Uniqueness of solutions to an Abel type nonlinear integral equation on the half line
We consider a convolution-type integral equation u = k ⋆ g(u) on the half line (−∞; a), a ∈ ℝ, with kernel k(x) = x α−1, 0 < α, and function g(u), continuous and nondecreasing, such that g(0) = 0 and 0 < g(u) for 0 < u. We concentrate on the uniqueness problem for this equation, and we prove that if α ∈ (1, 4), then for any two nontrivial solutions u 1, u 2 there exists a constant c ∈ ℝ such that u 2(x) = u 1(x +c), −∞ < x. The results are obtained by applying Hilbert projective metrics....
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