Integral Volterra Equation on Time Scales

• Integral Volterra equations are functional equations where the unknown function appears within an integral with a variable upper limit, modeling memory-dependent systems.
• The resolvent kernel and reciprocity property enable closed-form solutions through series expansions and provide error-controlled Picard iterative schemes.
• This framework unifies continuous and discrete dynamic equations on arbitrary time scales, offering practical methods for existence, uniqueness, and transformation of linear dynamic equations.

An integral Volterra equation is a functional equation in which the unknown function appears inside an integral whose upper limit is the independent variable. Such equations serve as a prototypical model for hereditary (memory-dependent) phenomena in analysis, mathematical physics, control theory, and dynamic systems. The most general form on a time scale $\mathbb{T}$ is

$x(t) = f(t) + \int_a^t K(t,s)x(s)\,\Delta s,$

where $f$ is a known function, $K$ is the kernel, $\Delta s$ denotes the time scale integral (encompassing both continuous and discrete cases), and the integral is over $s \in [a, t]_{\mathbb{T}}$. The Volterra structure reflects causality: the current state depends only on the values up to $t$.

1. Existence and Uniqueness on Time Scales

A rigorous existence and uniqueness theory for Volterra equations on arbitrary time scales is grounded in a power series expansion. Instead of classical Banach fixed point arguments, which require strong continuity assumptions, the method constructs a formal series solution via Picard iterates: $$\begin{aligned} \varphi_0(t) &= f(t), \qquad \varphi_{n}(t) = \int_a^{t} K(t, \eta) \varphi_{n-1}(\eta)\,\Delta\eta, \quad n \geq 1, \end{aligned}$$ with the full solution $\varphi(t) = \sum_{n=0}^\infty \varphi_n(t)$. This approach leverages bounds via generalized time-scale monomials $h_k(t,a)$, and yields uniform convergence when the norm $|\lambda|$ in the kernel expansion is sufficiently small. Uniqueness is established by applying a Grönwall-type inequality adapted to time scales, which forces the difference of any two solutions to vanish identically.

Notably, the required smoothness of $K$ is relaxed from continuity to rd-continuity (right-dense continuity): $K$ must be continuous at right-dense points and possess finite left limits at left-dense points. This requirement admits kernels with isolated jump discontinuities at scattered points in the time scale—a significant generalization over standard calculus.

2. Resolvent Kernel and Structural Reciprocity

A central analytic device is the resolvent kernel, defined recursively as

$$\begin{aligned} r(\lambda; t, s) &= \sum_{l=0}^\infty \lambda^{l} K_l(t, s),\ K_0(t, s) &:= K(t, s),\ K_n(t, s) &:= \int_{s}^{t} K(t, \eta) K_{n-1}(\eta, s)\,\Delta \eta\quad (n \geq 1). \end{aligned}$$

This expansion is crucial for expressing the solution explicitly: $$q(t) = \int_a^t r(\lambda; t, s) f(s)\, \Delta s + f(t).$$ The resolvent kernel encapsulates the cumulative effect of the kernel iterates and reveals a fundamental reciprocity property: $K$ is itself the resolvent of $r$. That is, the process of forming the resolvent kernel is invertible, yielding a pair of kernels possessing a "resolvent duality".

This dual structural feature has both theoretical and computational implications when analyzing the interplay between a Volterra kernel and its corresponding resolvent.

3. Picard Iteration Schemes and Error Bounds

Picard iterates provide both a proof-of-concept for existence and uniqueness and a practical computational scheme for approximating solutions. The method initializes with an arbitrary $P_0(t)$ and recursively defines

$$P_{n}(t) = f(t) + \int_a^{t} K(t, \eta) P_{n-1}(\eta)\,\Delta \eta.$$

Uniform convergence is established, and explicit error bounds are derived: $|P_{n}(t) - P_{n-1}(t)| < L M h_n(t, a),$ where $L$, $M$ are constants depending on bounds of $K$ and $f$, and $h_n$ is the generalized monomial of order $n$ on the time scale.

This construction is particularly well-suited for problems on arbitrary time scales and supports uniform control of approximation error.

4. Connections to Linear Dynamic Equations

Many initial value problems (IVPs) for linear dynamic equations can be recast as Volterra integral equations. For instance, an $n$th-order dynamic equation

$$y^{\Delta^n}(t) + \sum_{i=0}^{n-1} p_i(t)y^{\Delta^i}(t) = q(t)$$

can be transformed using Taylor's formula and generalized time-scale monomials $h_k(t, s)$ into an integral equation where the kernel reflects the structure of the dynamic coefficients $p_i$.

This correspondence establishes a one-to-one relation between unique solutions of linear dynamic equations and their integral (Volterra-type) counterparts, providing a unified framework and allowing analytic techniques from Volterra theory (e.g., resolvent methods, series solutions) to be applied to dynamic equations on nontrivial time scales.

5. Special Kernel Structures and Transform Methods

The framework accommodates various specialized kernel structures:

• Polynomial-type kernels: For linear dynamic equations with polynomial or monomial terms, the kernel may be represented as $$K(t,s) = \sum_{i=1}^{n-1} p_i(t) h_{n-i-1}(t, \sigma(s))$$, with $p_i$ rd-continuous.
• Convolution-type kernels: On unbounded domains, convolution kernels admit direct application of generalized Laplace transforms on time scales, leading to solutions via inverse transforms. This is particularly effective for problems amenable to spectral techniques or in unbounded temporal settings.

Examples in the literature illustrate direct use of Laplace-transform techniques and convolution integral representations, linking operational calculus directly to Volterra integral equations on generalized time scales.

6. Implications and Extensions

The generalization to rd-continuous kernels and the time-scale setting notably broadens the applicability of Volterra integral equations, encompassing both continuous and discrete (difference) systems, and settings with hybrid or irregular time domains. The recursive existence and uniqueness proof not only circumvents reliance on the Banach fixed point theorem, but also allows treatment of less regular (non-fully continuous) kernels.

The explicit solution and structural properties established—via the resolvent kernel, reciprocity, and error-controlled Picard iteration—form a basis for further developments, including the paper of nonlinear equations, time-varying or state-dependent kernels, systems, and equations with piecewise or otherwise singular structure.

A schematic summary of key concepts:

Concept	Formula / Definition	Context
Picard iterates	$$\displaystyle P_n(t) = f(t) + \int_a^t K(t, \eta) P_{n-1}(\eta)\,\Delta\eta$$	Iterative approximation
Resolvent kernel	$$\displaystyle r(\lambda; t, s) = \sum_{l=0}^\infty \lambda^l K_l(t,s)$$	Closed-form solution representation
Reciprocity property	$K$ is the resolvent of $r$, and vice versa	Kernel–resolvent symmetry
Polynomial-type kernel	$$K(t,s) = \sum_{i=1}^{n-1} p_i(t) h_{n-i-1}(t, \sigma(s))$$	Linear dynamic equations

In summary, the integral Volterra equation on time scales provides a unifying analytic structure for memory systems, supporting existence, uniqueness, closed-form solution constructions, and connections to dynamic equations, all under general assumptions on the kernel and time domain (Karpuz, 2011).