Session S12 - Delay and functional differential equations and applications

Tuesday, July 13, 14:50 ~ 15:25 UTC-3

Solutions for functional Volterra--Stieltjes integral equations

Anna Carolina Lafetá

Universidade de Brasília, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In this work, we study integral functional differential equations of Volterra--Stieltjes type:

$(1)\qquad\left\{\begin{array}{lll}x(t) &= & \phi(0) + \displaystyle \int_{t_0}^t a(t,s)f(x_s, s)\mbox{d}g(s), \qquad t\geq t_0,\\ x_{t_0 } & = & \phi,\end{array} \right.$

where $t_0 \in \mathbb{R}$, $\phi \in G([-r,0],\mathbb{R}^n)$, $f:G([-r,0],\mathbb{R}^n)\times [t_0,+\infty)\to \mathbb{R}^n$, $a:[t_0,+\infty)^2 \to \mathbb{R}$, $g:[t_0,+\infty) \to \mathbb{R}$, $x_s:[-r,0] \to \mathbb{R}^n$ is defined by $x_s(\theta) = x(s + \theta)$ and the integral on the right hand side fo the equality in the sense of Henstock--Kurzweil--Stieltjes.

We present some conditions on the functions $a$ and $g$ and also some conditions with respect the integral $\displaystyle \int^{\tau_2}_{\tau_1}b(t,s) f(x_s,s)\mathrm{d}g(s)$, when $b:[t_0,+\infty)^2 \to \mathbb{R}$ is a regulated function and $t_0 \leq \tau_1 \leq \tau_2 \leq t_0 + \sigma

These conditions will guarantee the existence and uniqueness of local and maximal solutions for equation (1).

Moreover, we present correspondences between (1) and functional Volterra integral equations with impulses and Volterra delta--integral equations on time scales.

References

[1] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001.

[2] M. Federson, R. Grau and J. G. Mesquita, Prolongation of solutions of measure differential equations and dynamic equations on time scales, Mathematische Nachrichten, 292(1), 22-55, 2019.

[3] M. Federson and Š. Schwabik, Generalized ODEs approach to impulsive retarded differential equations. Diff integral equations, 19(11) (2006) 1201-1234.

[4] D. Fraňková, Regulated functions, Mathematica Bohemica. 116 (1) (1991), 20-59.

[5] R. Grau, A. C. Lafetá and J. G. Mesquita, Existence and uniqueness of local and maximal solutions for functional Volterra Stieltjes integral equations and applications, submitted.

[6] G. Gripenberg, S.-O. Londen, O. Steffans, Volterra Integral and Functional Equations, in: Encyclopedia of Mathematics and its Applications, vol. 34, Cambridge University Press, Cambridge, 1990.

[7] R. Henstock, A Riemann-type integral of Lebesgue power. Canad. J. Math. 20, (1968) 79--87.

[8] Š. Schwabik, Generalized Ordinary Differential Equations, World Scientific, Series in Real Anal., vol. 5, 1992.

[9] A. Slavík, Dynamic equations on time scales and generalized ordinary differential equations, J. Math. Anal. Appl. 385 (2012), 534--550.

Joint work with Jaqueline Godoy Mesquita (Universidade de Brasília, Brasília, Brasil) and Rogélio Grau (Universidad del Norte, Barranquilla, Colombia).

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