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Extremal solutions to a coupled system of nonlinear fractional differential equations with ψ–Caputo fractional derivatives

Derbazi Choukri, Baitiche Zidane, Benchohra Mouffak, Graef John R.

Journal of Mathematics and Applications

2021

Vol. 44

19--34

Using the well-known monotone iterative technique together with the method of upper and lower solutions, the authors investigate the existence of extremal solutions to a class of coupled systems of nonlinear fractional differential equations involving the ψ–Caputo derivative with initial conditions. As applications of this work, two illustrative examples are presented.

Impulsive differential equations with initial data difference

Skóra L.

Commentationes Mathematicae

2011

Vol. 51, [Z] 1

55-69

In this paper, we present some results on impulsive differential inequalities and equations with initial and impulsive data difference.

On extremal solutions of differential equations with advanced argument

Augustynowicz A., Jankowski J.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2006

Vol. 46, [Z] 1

17-24

We obtain existence of absolutely continuous extremal solutions of the problem u'(x) = F(x, u(x), u(h(x))), u(0) = u0, and the Darboux problem for u_xy(x, y) = G(x, y, u(x, y), u(H(x, y))), where h and H are arbitrary continuous deviated arguments.

Matrix Quadratic Equations, Column/row Reduced Factorizations and an Inertia Theorem for Matrix Polynomials

Karelin I., Lerer L.

International Journal of Applied Mathematics and Computer Science

2001

Vol. 11, no 6

1285-1310

It is shown that a certain Bezout operator provides a bijective correspondence between the solutions of the matrix quadratic equation and factorizatons of a certain matrix polynomial G(lambda) (which is a specification of a Popov-type function) into a product of row and column reduced polynomials. Special attention is paid to the symmetric case, i.e. to the Algebraic Riccati Equation. In particular, it is shown that extremal solutions of such equations correspond to spectral factorizations of G(lambda). The proof of these results depends heavily on a new inertia theorem for matrix polynomials which is also one of the main results in this paper.