On the numerical solutions for nonlinear Volterra-Fredholm integral equations

In this note, we study a class of multistep collocation methods for the numerical integration of nonlinear Volterra-Fredholm Integral Equations (V-FIEs). The derived method is characterized by a lower triangular or diagonal coefficient matrix of the nonlinear system for the computation of the stages which, as it is known, can beexploited to get an efficient implementation. Convergence analysis and linear stability estimates are investigated. Finally numerical experiments are given, which confirm our theoretical results.

Catalan Transform of k -Balancing Sequences

In this work, the Catalan transformation (CT) of k -balancing sequences, B k , n n ≥ 0 , is introduced. Furthermore, the obtained Catalan transformation C B k , n n ≥ 0 was shown as the product of lower triangular matrices called Catalan matrices and the matrix of k -balancing sequences, B k , n n ≥ 0 , which is an n × 1 matrix. Apart from that, the Hankel transform is applied further to calculate the determinant of the matrices formed from C B k , n n ≥ 0 .

Factorizations of some lower triangular matrices and related combinatorial identities

In this study, we investigate the connection between second order recurrence matrix and several combinatorial matrices such as generalized r-eliminated Pascal matrix, Stirling matrix of the first and of the second kind matrices. We give factorizations and inverse factorizations of these matrices by virtue of the second order recurrence matrix. Moreover, we derive several combinatorial identities which are more general results of some earlier works.

Quasi-nilpotency of Generalized Volterra Operators on Sequence Spaces

We study the quasi-nilpotency of generalized Volterra operators on spaces of power series with Taylor coefficients in weighted $$\ell ^p$$ ℓ p spaces $$1<p<+\infty $$ 1 < p < + ∞ . Our main result is that when an analytic symbol g is a multiplier for a weighted $$\ell ^p$$ ℓ p space, then the corresponding generalized Volterra operator $$T_g$$ T g is bounded on the same space and quasi-nilpotent, i.e. its spectrum is $$\{0\}.$$ { 0 } . This improves a previous result of A. Limani and B. Malman in the case of sequence spaces. Also combined with known results about multipliers of $$\ell ^p$$ ℓ p spaces we give non trivial examples of bounded quasi-nilpotent generalized Volterra operators on $$\ell ^p$$ ℓ p . We approach the problem by introducing what we call Schur multipliers for lower triangular matrices and we construct a family of Schur multipliers for lower triangular matrices on $$\ell ^p, 1<p<\infty $$ ℓ p , 1 < p < ∞ related to summability kernels. To demonstrate the power of our results we also find a new class of Schur multipliers for Hankel operators on $$\ell ^2 $$ ℓ 2 , extending a result of E. Ricard.