In this research we establish necessary and sufficient conditions for the stability of the zero solution of scalar Volterra integro-dynamic equation on general time scales. Our approach is based on the construction of suitable Lyapunov functionals. We will compare our findings with known results and provides application to quantum calculus.
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...
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.
This paper presents an approximate method of solving the fractional (in the time variable) equation which describes the processes lying between heat and wave behavior. The approximation consists in the application of a finite subspace of an infinite basis in the time variable (Galerkin method) and discretization in space variables. In the final step, a large-scale system of linear equations with a non-symmetric matrix is solved with the use of the iterative GMRES method.