Oscillation Criteria for Matrix Fractional Differential Equations via Riccati Transformation and Integral Averaging

Abstract

In this work, we use the Riemann–Liouville (R-L) fractional derivative of order $\alpha \in (0,1)$ to study the oscillation criteria for damped matrix fractional differential equations and determine sufficient conditions under which all prepared solutions of the system show oscillatory behaviour. The criteria are novel even for the linear undamped case and extend conventional oscillation results for integer-order matrix differential systems to the fractional setting. The goal of the current effort is to better understand the relationships between solutions and their derivatives. Using the matrix-valued Riccati transformation converts the system into a Riccati-type inequality, and the oscillation conditions are then derived by integrating against a weighted kernel via the operator L. Both results generalise the integer-order oscillation criteria to the fractional matrix setting, extending their applicability to fractional-order control systems, viscoelastic structural models, and anomalous diffusion processes. This work develops new conditions and analytical techniques that deepen insight and provide useful results for analysing oscillatory behaviour and asymptotic stability of of the considered systems. To illustrate the significance of the obtained oscillation results, we give two examples.

1. Introduction

Oscillation theory is an intriguing field of research that has seen a notable surge in popularity and attention in recent years. Many fields have witnessed the growth of oscillatory theory since its evolution. It is one of the important branches of applied theory. The theory focuses on both theoretical and practical applications. In dynamical models, damping and oscillation scenarios are often formulated by means of external sources and/or nonlinear diffusion, perturbing the natural evolution of related systems (see, e.g., [1,2]). Further, the research also enables deeper insights in the theory of oscillation equations in fields like engineering, mathematical modelling, bifurcation theory, optical science, etc. (see [3]). The problem of oscillation and non-oscillation in differential equations (DEs) has been studied by various authors like Agarwal et al. [4], Butler [5], Grace et al. [6], Swanson [7], and Kiguradze et al. [8]. In the literature, an extensive amount of discussion over the oscillation of solutions for several types of second-order matrix DEs and dynamic equations has been been had with regard to damping [9,10,11,12].

A matrix DE is a mathematical equation which contains multiple functions stacked into vector form with a matrix relating the functions to their derivatives. The matrix DE was first initiated by Howard [13] in 1956. In 1968, oscillation of the nonlinear matrix DE was first studied by Tomastik [14]. It is important to present a clear connection between properties of systems or even n-th order DEs and so-called oscillation properties of matrix equations. In a matrix equation, we consider the Wronskian of the fundamental system. Oscillation is defined as vanishing Wronskian at corresponding points. The Wronskian is one of the classical objects in the theory of DEs. For ordinary DEs, the Wronskian of fundamental systems cannot vanish. For fractional equations like delay DEs, Wronskian can be zero at corresponding points. Fractional equations in this sense are similar to delay equations. Note that non-vanishing of the Wronskian at the point a (i.e., $W\left(a\right)$ is not zero) implies the fact of unique solvability of a one-point boundary value problem with all conditions at the point a. Non-oscillation of solutions allows us to achieve the unique solvability for sufficiently large r and oscillation disturbs this property. For second-order equations, non-vanishing Wronskian is equivalent to the Sturm separation theorem, claiming that between two adjacent zeros of every nontrivial solution there is one, and only one, zero of nonproportional to it solutions. This allows us to conclude that “standard” scalar solutions are either all oscillatory or all non-oscillatory. The importance of the non-vanishing Wronskian in functional differential boundary value problems was obtained in the monograph [15], where non-vanishing of the Wronskian is one of the essential conditions in the theorem on sing-constancy of Green’s functions. The Wronskian of second-order delay equations is the central focus of the monograph [16], where the growth of the Wronskian is highlighted as a crucial factor in determining the existence of unbounded solutions. Fractional equations are similar according to their behaviour to delay DEs. Xu et al. [17] developed the oscillation phenomenon for nonlinear matrix DEs by using a specific function $v(r,s,t)$. Many authors investigated the second-order matrix DE in linear and nonlinear cases, which is addressed below using references [18,19]. Yang and Tang [20] established oscillation conditions for differential systems with self adjoint using the generalised integral averaging methodology and the Riccati transformation method. Several authors have shown interest in studying the self adjoint matrix DEs as given in references [21,22,23,24].

The idea of a non-integer order of integration originated with differential calculus itself. In the late 17th century, G.W. Leibnitz, the philosopher and father of modern calculus, made some observations about the possibility and significance of the fractional derivative of order. Further, Liouville conducted a thorough examination and defined the first instance of a fractional integration operator in a series of papers published between 1832 and 1837. Subsequent research and advancements, including work by Riemann, resulted in the creation of the integral-based Riemann–Liouville fractional integral operator, a crucial component of fractional calculus ever since. The most significant extension of standard calculus is the Riemann–Liouville fractional derivative. It is a specific example that almost all definitions of fractional derivatives incorporate. A significant body of literature was produced by several authors whose papers included the Riemann derivative [25,26,27,28].

Numerous applications of matrix DEs can be shown in many disciplines, including computer science, physics, engineering, and economics [29]. The matrix fractional DE studied in this paper (see Equation (1) below) arises naturally in several important physical and engineering contexts, which we now describe.

• Fractional viscoelastic beam: Consider a two-degree-of-freedom Kelvin–Voigt beam under harmonic loading. The governing equation of motion takes the form of (1), where $Q\left(r\right)$ represents the frequency-dependent stiffness matrix, $R\left(r\right)$ encodes the fractional damping arising from the viscous component of the Kelvin–Voigt model, and $P\left(r\right)$ encodes the restoring force coupling between the two degrees of freedom. The Riemann–Liouville derivative of order $\alpha \in (0,1)$ captures the material’s memory-dependent stress–strain response, which cannot be modelled by classical integer-order derivatives [30]. Oscillation of the prepared solution $X\left(r\right)$ in this setting corresponds physically to resonance of the viscoelastic structure under periodic excitation.
• Fractional-order control: In the state-space representation of a multi-input multi-output (MIMO) fractional-order controller, the closed-loop system evolution takes the form of (1), where the nonlinear operators G and F represent actuator and sensor characteristics, respectively, satisfying assumptions ($A_2$)–($A_3$). The matrix $Q\left(r\right)$ plays the role of the system gain matrix, $R\left(r\right)$ captures the fractional feedback damping, and $P\left(r\right)$ encodes the state-coupling structure. Oscillation of the prepared solutions of (1) is directly linked to the proximity of the closed-loop system trajectory to the stability boundary, making the oscillation criteria of Theorems 1 and 2 directly applicable to the stability analysis of such systems.
• Anomalous diffusion: The evolution of a vector-valued concentration field $X\left(r\right)={\left[x_{ij}\left(r\right)\right]}_{n\times n}$ in a heterogeneous medium is governed by a matrix fractional differential system of the form (1), where the convolution kernel ${(r-s)}^{-\alpha }$ in the integral term encodes the sub-diffusive memory of the medium. The order $\alpha \in (0,1)$ characterises the degree of sub-diffusion: as $\alpha \to 1^{-}$, the system recovers classical Fickian diffusion, consistent with Remark 1. The oscillation criteria established in this paper determine the long-time oscillatory behaviour of the concentration profiles across the coupled components of the field.

In recent times, numerous researchers have been working to expand the applications of fractional DEs into the matrix framework of fractional DEs. Matrix fractional DEs in nonlinear and linear forms are important in a variety of fields, including physics, mathematics, statistics, control, engineering, and linear differential system problems. Many motivating problems in the aforementioned fields result in nonhomogeneous linear matrix fractional DEs or a system of nonhomogeneous linear matrix fractional DEs [31,32,33]. Garrappa and Popolizio [34], for instance, have talked about using matrix functions for fractional partial DEs. Kiliçman and Ahmood [35] examined solutions to specific matrix fractional DEs. Juraev et al. [36] recorded the Cauchy problem for matrix factorizations of the Helmholtz equation in a three-dimensional bounded domain. The generalised fractional Taylor’s series is extended to a matrix form by Ahmad El-Ajou [37], who provides results on its convergence and also highlights the usefulness of the method in matrix theory, which is illustrated by applications for solving linear and nonlinear matrix fractional DEs and computing Frobenius norm approximations.

Yang [38] acquired new oscillation conditions for linear second-order matrix differential systems with damping, which can be obtained by utilizing the following sequence of subintervals:

$$\begin{array}{c}\hfill \left(Q\right(r)Y\prime (r\left)\right)\prime +r\left(r\right)Q\left(r\right)Y\prime \left(r\right)+P\left(r\right)Y\left(r\right)=0.\end{array}$$

Parhi and Praharaj [39] studied the nonlinear matrix DEs of the form

$${\left(P\left(r\right)Y^{\prime }\right)}^{\prime }+Q\left(r\right)F\left(Y\right)=0.$$

In addition, Yang et al. [20] established new oscillation criteria for second-order matrix DEs of the form

$${\left(q\left(r\right)P\left(r\right)X^{\prime }\left(r\right)\right)}^{\prime }+p\left(r\right)P\left(r\right)X^{\prime }\left(r\right)+Q\left(r\right)F\left(X^{\prime }\left(r\right)GX\left(r\right)\right)=0.$$

To the best of the authors’ knowledge, no earlier work has established oscillation criteria for the specific class of damped nonlinear matrix fractional DEs of the type (1). Motivated by this observation, we consider the oscillatory behaviour of matrix fractional DEs of the form

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dr}\left(Q\left(r\right)D_{+}^{\alpha }X\left(r\right)\right)+R\left(r\right)D_{+}^{\alpha }X\left(r\right)+P\left(r\right)G\left({\int }_0^r{(r-s)}^{-\alpha }X\left(s\right)ds\right)F\left(D_{+}^{\alpha }X\left(r\right)\right)=0,}\end{array}$$

(1)

where $r\ge r_0>0$ and $D_{+}^{\alpha }(·)$ is the Riemann–Liouville (R-L) derivative of $\alpha \in (0,1)$. Here, X, P, R, Q, G, and F satisfy the following assumptions:

(A₁)

The matrix-valued functions $P\left(r\right)$, $Q\left(r\right)$, and $R\left(r\right)$ are of order $n\times n$ and have real-valued continuous entries on the interval $[r_0,\infty )$. Moreover, $Q\left(r\right)$ is symmetric and positive-definite, while $R\left(r\right)$ is symmetric for all $r\in [r_0,\infty )$. The solution X satisfies $x_{ij}\in A{C}_{\mathrm{loc}}^1\left([r_0,\infty )\right)$, ensuring that $D_{+}^{\alpha }X\left(r\right)$ exists pointwise.

(A₂)

The function G is continuously differentiable on ${\mathbb{R}}^{n^2}$. For any nonzero matrix $X\left(r\right)$, the product $X\left(r\right)G\left(X\right(r\left)\right)$ is positive-definite. In addition, the derivative satisfies $G^{\prime }\left(K\left(r\right)\right)\ge \mu I_n>0$, where $\mu$ is a positive constant and $I_n$ denotes the identity matrix. The inverse $G^{-1}$ exists for $K\ne 0$, and $G{\left(K\left(r\right)\right)}^{-1}$ is positive-definite. Moreover, $G^{\prime }\left(K\left(r\right)\right)$ is symmetric for all $K\ne 0$, so the inequality $G^{\prime }\left(K\left(r\right)\right)\ge \mu I_n$ holds in the Löwner ordering.

(A₃)

Let $F\in C({\mathbb{R}}^{n^2},{\mathbb{R}}^{n^2})$. The matrix $F\left(D_{+}^{\alpha }X\left(r\right)\right)$ is continuous and uniformly positive-definite, satisfying

$$F\left(D_{+}^{\alpha }X\left(r\right)\right)\ge M{I}_n>0,$$

where M is a positive constant and $I_n$ is the identity matrix.

A solution $X\left(r\right)$ of (1) is called nontrivial if there exists at least one $r\in [r_0,\infty )$ such that $detX\left(r\right)\ne 0$. A nontrivial solution $X\left(r\right)$ of (1) is said to be prepared (or self-conjugate) provided that it satisfies

$$\begin{array}{c}\hfill {\left(D_{+}^{\alpha }X\left(r\right)\right)}^{*}\left(Q\left(r\right)D_{+}^{\alpha }X\left(r\right)\right)={\left(Q\left(r\right)D_{+}^{\alpha }X\left(r\right)\right)}^{*}\left(D_{+}^{\alpha }X\left(r\right)\right),\end{array}$$

for all $r\in [r_0,\infty )$. Here, for any matrix Y, the notation $Y^{*}$ denotes its transpose.

A prepared solution $X\left(r\right)$ of (1) is termed oscillatory if $detX\left(r\right)$ has zeros that occur arbitrarily far to the right on $[r_0,\infty )$. Otherwise, the solution is called non-oscillatory.

The article’s structure is further organised as follows. In the preliminary section, we recall several definitions that are used throughout the paper and present a lemma establishing a connection between the Riemann–Liouville fractional derivative and the classical derivative. Next, some new oscillation criteria are established for Equation (1) with the integral average method by employing the matrix Riccati-type transformation. The importance of the new results is shown in the last section with illustrations. In conclusion, the new oscillation criteria for matrix DEs are derived using the new linear integral operator.

2. Preliminaries

In this section, certain necessary definitions and lemmas that are related to the paper are reviewed.

Definition 1 

([30]). Let $0<\alpha <1$. The Riemann–Liouville fractional derivative of order α of a function $Y\left(r\right)$ with respect to r is defined by

$$D_{+}^{\alpha }Y\left(r\right):={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\displaystyle \frac{d}{dr}}{\int }_{r_0}^r{(r-s)}^{-\alpha }Y\left(s\right)ds,$$

(2)

provided that the above expression exists pointwise on the interval $[r_0,\infty )$. Here, $\Gamma (·)$ denotes the Gamma function.

Definition 2 

([30]). The Riemann–Liouville fractional integral of order $\alpha >0$ of a function $Y:[r_0,\infty )\to R^{n^2}$ on the half-axis $[r_0,\infty )$ is given by

$$\begin{array}{c}\hfill I_{+}^{\alpha }Y\left(r\right):={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{r_0}^r{(r-s)}^{\alpha -1}Y\left(s\right)ds\mathit{for}r\ge r_0,\end{array}$$

where $\Gamma \left(\alpha \right)$ denotes the gamma function defined by $\Gamma \left(\alpha \right)=\int e^{-r}{r}^{\alpha -1}dr$, $\alpha >0$.

Definition 3. 

For any function $b:b(r,s)\in C([r_0,\infty )\times [r_0,r)),r\ge r_0\ge 0$, we define the linear integral operator $L_r^{\sigma }$ as

$$\begin{array}{c}\hfill L_r^{\sigma }\left(b\right)={\int }_{r_0}^r\sigma \left(s\right){(r-s)}^{\alpha }\left(b\right)ds,\end{array}$$

where $\alpha \in (0,1)$ is consistent with the standing assumption, and $\sigma \in C^1\left([r_0,\infty )\right)$ with $\sigma >0$. If

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial b}{\partial s}}\in C([r_0,\infty )\times [r_0,r)),\end{array}$$

$$\begin{array}{c}\hfill L_r^{\sigma }\left({\displaystyle \frac{\partial b}{\partial s}}\right)=-\sigma \left(r\right){(r-r_0)}^{\alpha }b(r_0,r)-L_r^{\sigma }\left[\left({\displaystyle \frac{-\alpha }{r-s}}+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right)b\right].\end{array}$$

Proposition 1 

([30]). Let $X\left(r\right)={\left[x_{ij}\right]}_{n\times n}$, $i,j=1,2,\dots ,n$ be the solution for (1) and

$$K\left(r\right)={\int }_{r_0}^r{(r-s)}^{-\alpha }X\left(s\right)ds,$$

where $K\left(r\right)={\left[k_{ij}\right]}_{n\times n}$. Then,

$$K^{\prime }\left(r\right)=\Gamma (1-\alpha )D_{+}^{\alpha }X\left(r\right),$$

for $\alpha \in (0,1)$, $r\ge 0$.

Proof. 

Consider

$$\begin{array}{c}\hfill K\left(r\right)={\int }_{r_0}^r{(r-s)}^{-\alpha }X\left(s\right)ds.\end{array}$$

Taking the $ij$th component,

$$k_{ij}\left(r\right)={\int }_{r_0}^r{(r-s)}^{-\alpha }{x}_{ij}\left(s\right)ds,i,j=1,2,\dots ,n.$$

Taking differentiation on both sides,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d}{dr}}\left(k_{ij}\left(r\right)\right)& =& {\displaystyle \frac{d}{dr}}{\int }_{r_0}^r{(r-s)}^{-\alpha }{x}_{ij}\left(s\right)ds\hfill \\ & =& \Gamma (1-\alpha )\left[{\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\displaystyle \frac{d}{dr}}{\int }_{r_0}^r{(r-s)}^{-\alpha }{x}_{ij}\left(s\right)ds\right]\hfill \\ \hfill k_{ij}^{\prime }\left(r\right)& =& \Gamma (1-\alpha )D_{+}^{\alpha }{x}_{ij}\left(r\right),\hfill \end{array}$$

which implies

$$\begin{array}{cc}\hfill {\left(k_{ij}^{\prime }\left(r\right)\right)}_{n\times n}& =\Gamma (1-\alpha )D_{+}^{\alpha }{\left[x_{ij}\left(r\right)\right]}_{n\times n}.\end{array}$$

Therefore,

$$\begin{array}{c}\hfill K^{\prime }\left(r\right)=\Gamma (1-\alpha )D_{+}^{\alpha }X\left(r\right).\end{array}$$

□

Remark 1. 

The restriction $\alpha \in (0,1)$ ensures that $\Gamma (1-\alpha )$ is finite. As $\alpha \to 1^{-}$, the R-L derivative reduces to the classical derivative $X^{\prime }\left(r\right)$, which is consistent with the integer-order theory.

Remark 2. 

Since $x_{ij}\in C^{\beta }$ with $\beta >\alpha$ and ${(r-s)}^{-\alpha }$ is integrable on $[0,r)$ for $\alpha \in (0,1)$, differentiation of $K\left(r\right)$ under the integral sign is supported by [[30], Theorem 2.2].

3. Main Results

The newly obtained results for the oscillation of the matrix fractional DE based on Riccati transformation are established within this section. This section’s matrix inequalities are all understood via Löwner ordering: $A-B$ is positive semi-definite if $A\ge B$.

Theorem 1. 

Let $f\in C^{\prime }[r_0,\infty )$ such that

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim\; sup}{\displaystyle \frac{1}{r^{\alpha }}}{L}_{r_0}^{\sigma }[\Phi \left(s\right)-{\displaystyle \frac{v\left(s\right)Q\left(s\right)}{4\mu \Gamma (1-\alpha )}}(R\left(s\right)Q^{-1}\left(s\right)+2\mu h\left(s\right)I_n-2\mu \Gamma \left(1-\alpha \right)h\left(s\right)I_n)\\ \hfill -{\left(\left({\displaystyle \frac{-\alpha }{r-s}}\right)+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right)}^2]=\infty ,\end{array}$$

(3)

where

$$\begin{array}{cc}\hfill v\left(s\right)& =exp(-2\mu )\int h\left(s\right)ds,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \Phi \left(s\right)& =v\left(s\right)\left[LP\left(s\right)-h\left(s\right)R\left(s\right)-\mu \Gamma \left(1-\alpha \right)h^2\left(s\right)Q\left(s\right)-{\left(h\left(s\right)Q\left(s\right)\right)}^{\prime }\right]\hfill \end{array}$$

(5)

and $h\in C^1[r_0,\infty )$ is a freely chosen auxiliary function. Then, each solution $X\left(r\right)$ of (1) is oscillatory.

Proof. 

Let’s assume the contradiction that there exists a prepared solution $X\left(r\right)$ of (1) that does not oscillate. If $detX\left(r\right)\ne 0$, by substituting the value of $K\left(r\right)={\int }_0^r{(r-s)}^{-\alpha }X\left(s\right)ds$, Equation (1) becomes

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dr}}\left(Q\left(r\right)D_{+}^{\alpha }X\left(r\right)\right)+R\left(r\right)D_{+}^{\alpha }X\left(r\right)+P\left(r\right)G\left(K\left(r\right)\right)F\left(D_{+}^{\alpha }X\left(r\right)\right)=0.\end{array}$$

Let

$$\begin{array}{cc}\hfill U\left(r\right)=& v\left(r\right)Q\left(r\right)\left[D_{+}^{\alpha }X\left(r\right)G^{-1}\left(K\left(r\right)\right)+h\left(r\right)I_n\right],\hfill \end{array}$$

(6)

where $Q\left(r\right)$ is symmetric by $\left(A_1\right)$ and $X\left(r\right)$ is prepared, and $U\left(r\right)$ is symmetric for all $r\ge r_0$. Now, $U\left(r\right)$ is differentiable almost everywhere, $v,h\in C^1$, $Q\in C^1$ by $\left(A_1\right)$, and $G^{-1}\left(K\left(r\right)\right)$ is of class $C^1$ by $\left(A_2\right)$, we get

$$\begin{array}{cc}\hfill U^{\prime }\left(r\right)=& -2\mu v\left(r\right)h\left(r\right)\left[Q\left(r\right)(D_{+}^{\alpha }X\left(r\right)G^{-1}K\left(r\right)+h\left(r\right)I_n)\right]\hfill \\ \hfill & +v\left(r\right)\left[{\left(Q\left(r\right)D_{+}^{\alpha }X\left(r\right)G^{-1}\left(K\left(r\right)\right)\right)}^{\prime }+{\left(h\left(r\right)Q\left(r\right)\right)}^{\prime }\right].\hfill \end{array}$$

(7)

By using $G^{\prime }\left(K\left(r\right)\right)>\mu I_n>0$, where $\mu$ is a constant and $I_n$ is the $n\times n$ identity matrix, $F\left(D_{+}^{\alpha }X\left(r\right)\right)>M{I}_n>0$.

$$\begin{array}{cc}\hfill U^{\prime }\left(r\right)\le & {\displaystyle \frac{-\mu \Gamma \left(1-\alpha \right)U\left(r\right)Q^{-1}\left(r\right)U\left(r\right)}{v\left(r\right)}}-U\left(r\right)[R\left(r\right)Q^{-1}\left(r\right)\hfill \\ \hfill & +2\mu h\left(r\right)I_n-2\mu \Gamma \left(1-\alpha \right)h\left(r\right)I_n]-v\left(r\right)[LP\left(r\right)-h\left(r\right)R\left(r\right)\hfill \\ \hfill & -\mu \Gamma \left(1-\alpha \right)h^2\left(r\right)Q\left(r\right)-{\left(h\left(r\right)Q\left(r\right)\right)}^{\prime }].\hfill \end{array}$$

(8)

Using $G^{\prime }\left(K\left(r\right)\right)\ge \mu I_n$ and $F\left(D_{+}^{\alpha }X\left(r\right)\right)\ge M{I}_n$ from $\left(A_2\right)$–$\left(A_3\right)$, and substituting $K^{\prime }\left(r\right)=\Gamma (1-\alpha )D_{+}^{\alpha }X\left(r\right)$ from Proposition 1 into (7), we obtain

$$\begin{array}{cc}\hfill U^{\prime }\left(r\right)\le & {\displaystyle \frac{-\mu \Gamma \left(1-\alpha \right)U\left(r\right)Q^{-1}\left(r\right)U\left(r\right)}{v\left(r\right)}}+U\left(r\right)[R\left(r\right)Q^{-1}\left(r\right)\hfill \\ \hfill & +2\mu h\left(r\right)I_n-2\mu \Gamma \left(1-\alpha \right)h\left(r\right)I_n]-\Phi \left(r\right),\hfill \end{array}$$

(9)

for $r\ge r_0$. We introduce $W\left(r\right)$ to complete the square in the Riccati inequality (9),

$$\begin{array}{cc}\hfill W\left(r\right)=& \sqrt{\mu \Gamma \left(1-\alpha \right)}{\left(\sqrt{Q\left(r\right)}\right)}^{-1}U\left(r\right)+{\displaystyle \frac{v\left(r\right)\sqrt{Q\left(r\right)}}{2\sqrt{\mu \Gamma \left(1-\alpha \right)}}}[R\left(r\right)Q^{-1}\left(r\right)+2\mu h\left(r\right)I_n\hfill \\ \hfill & -2\mu \Gamma \left(1-\alpha \right)h\left(r\right)I_n].\hfill \end{array}$$

(10)

Since $Q\left(r\right)$, $U\left(r\right)$, and $W\left(r\right)$ are all symmetric, the completing-the-square step in the subsequent inequality is valid in the matrix setting. Now, applying the operator $L_r^{\sigma }$ to (10) and by replacing r with s, we get

$$\begin{array}{c}\hfill -L_r^{\sigma }\left[\left(({\displaystyle \frac{-\alpha }{r-s}}\right)+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right)U\left(s\right)]+L_r^{\sigma }{\displaystyle \frac{W^2\left(s\right)}{v\left(s\right)}}+L_r^{\sigma }[\Phi \left(s\right)-{\displaystyle \frac{v\left(s\right)Q\left(s\right)}{4\mu \Gamma \left(1-\alpha \right)}}(R\left(s\right)Q^{-1}\left(s\right)\\ \hfill +2\mu h\left(s\right)I_n-2\mu \Gamma \left(1-\alpha \right)h\left(s\right)I_n{)}^2]\le \sigma \left(r_0\right){(r-r_0)}^{\alpha }U\left(r_0\right),\end{array}$$

(11)

for $r\ge r_0$. Since

$$tr\left(Q^{-1}\left(s\right)W^2\left(s\right)\right)=tr\left[{\left(Q^{-1/2}\left(s\right)W\left(s\right)\right)}^T\left(Q^{-1/2}\left(s\right)W\left(s\right)\right)\right]\ge 0,$$

the quadratic term involving $W\left(s\right)$ is non-negative. Hence, interpreting the matrix inequality in the scalar sense through the trace operator and applying the generalised proportional fractional integral operator $L_r^{\alpha }[·]$ to both sides of $\left(11\right)$, we obtain

$$\begin{array}{cc}\hfill & L_r^{\sigma }\left[{\displaystyle \frac{1}{v\left(s\right)}}{\left(W\left(s\right)-{\displaystyle \frac{v\left(s\right)}{2\sqrt{\mu \Gamma \left(1-\alpha \right)}}}\left({\displaystyle \frac{-\alpha }{r-s}}+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right)\sqrt{P\left(s\right)}\right)}^2\right]\hfill \\ \hfill & +L_r^{\sigma }[\Phi \left(s\right)-{\displaystyle \frac{v\left(s\right)Q\left(s\right)}{4\mu \Gamma \left(1-\alpha \right)}}((R\left(s\right)Q^{-1}\left(s\right)+2\mu h\left(s\right)I-2\mu \Gamma \left(1-\alpha \right)h\left(s\right)I_n)\hfill \\ \hfill & -{\left({\displaystyle \frac{-\alpha }{r-s}}+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right)}^2{I}_n]\le \sigma \left(r_0\right){(r-r_0)}^{\alpha }U\left(r_0\right)\mathrm{for}r\ge r_0.\hfill \end{array}$$

(12)

We note that all matrix inequalities in (11)–(12) are understood in the scalar sense via the trace functional. Since $W\left(r\right)$ is symmetric (as verified above), the matrix ${\left(W\left(s\right)-c(r,s)\sqrt{Q\left(s\right)}\right)}^2$ is positive semi-definite for any scalar function $c(r,s)$, and its trace is non-negative. The non-negativity of the first term in (12) therefore follows from $tr\left(S^2\right)\ge 0$ for symmetric S, without requiring commutativity of W and $Q^{1/2}$.

If the 1st term is non-negative, we have

$$\begin{array}{ccc}& & L_r^{\sigma }[\Phi \left(s\right)-{\displaystyle \frac{v\left(s\right)Q\left(s\right)}{4\mu \Gamma \left(1-\alpha \right)}}\left(\right(R\left(s\right)Q^{-1}\left(s\right)+2\mu h\left(s\right)I_n\hfill \\ & & -2\mu \Gamma \left(1-\alpha \right)h\left(s\right)I_n)-{\left({\displaystyle \frac{-\alpha }{r-s}}+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right)}^2{I}_n]\le \sigma \left(r\right){(r-r)}^{\alpha }U\left(r\right).\hfill \end{array}$$

(13)

Applying $r=r_0$, dividing (13) by $r^{\alpha }$, and taking ${\displaystyle \underset{r\to \infty }{lim\; sup}}$ for (13), we get

$$\begin{array}{cc}\hfill & \underset{r\to \infty }{lim\; sup}{\displaystyle \frac{1}{r^{\alpha }}}{L}_{r_0}^{\sigma }[\Phi \left(s\right)-{\displaystyle \frac{v\left(s\right)Q\left(s\right)}{4\mu \Gamma \left(1-\alpha \right)}}(R\left(s\right)Q^{-1}\left(s\right)+2\mu h\left(s\right)I_n-2\mu \Gamma \left(1-\alpha \right)h\left(s\right)I_n)\hfill \\ \hfill & -{\left(\left({\displaystyle \frac{-\alpha }{r-s}}\right)+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right)}^2{I}_n]\le \sigma \left(r_0\right)U\left(r_0\right)<\infty ,\hfill \end{array}$$

which leads to a contradiction with (1). Hence, it is proved. □

Remark 3. 

Theorem 1 gives a single integral-average condition (3) that is both closed-form and computable: one substitutes the system matrices P, Q, and R, and then one easily chooses h and σ to optimise the upper limit. The result reduces the oscillation criterion—which concerns the infinite-dimensional space of admissible solutions to a scalar asymptotic condition on known data. When the damping term drops out $(R=0)$ and $\alpha =1^{-}$, condition (3) reduces to a classical Philos oscillation condition for classical self-adjoint matrix systems.

Corollary 1. 

If

$$\underset{r\to \infty }{lim\; sup}{\displaystyle \frac{1}{r^{\alpha }}}{L}_{r_0}^{\sigma }(\Phi \left(s\right))=\infty$$

is replaced in (3) and

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}sup{\displaystyle \frac{1}{r^{\alpha }}}{L}_{r_0}^{\sigma }[{\displaystyle \frac{v\left(s\right)Q\left(s\right)}{4\mu \Gamma \left(1-\alpha \right)}}\left(R\left(s\right)Q^{-1}\left(s\right)+2\mu h\left(s\right)I_n-2\mu \Gamma \left(1-\alpha \right)h\left(s\right)I_n\right)\\ \hfill -{\left(\left({\displaystyle \frac{-\alpha }{r-s}}\right)+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right)}^2{I}_n]<\infty .\end{array}$$

Then, each solution $X\left(r\right)$ of (1) is oscillatory.

Theorem 2. 

If $f\in C^{\prime }[r_0,\infty )$ such that

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim\; sup}{\displaystyle \frac{1}{r^{\alpha }}}{L}_{r_0}^{\sigma }\left[{\displaystyle \frac{v\left(s\right)}{4\mu \Gamma \left(1-\alpha \right)}}{\left({\displaystyle \frac{-\alpha }{r-s}}+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right)}^{2}Q\left(s\right)\right]<\infty \end{array}$$

(14)

and there exists $A\in C[r_0,\infty )$, then

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim\; sup}{\displaystyle \frac{1}{r^{\alpha }}}{L}_r^{\sigma }[\Phi \left(s\right)-{\displaystyle \frac{v\left(s\right)Q\left(s\right)}{4\mu \Gamma \left(1-\alpha \right)}}\left(R\left(s\right)Q^{-1}\left(s\right)+2\mu h\left(s\right)I_n-2\mu \Gamma \left(1-\alpha \right)h\left(s\right)I_n\right)\\ \hfill -{\left(\left({\displaystyle \frac{-\alpha }{r-s}}\right)+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right)}^2{I}_n]\ge A\left(r\right),r\ge r_0\end{array}$$

(15)

and

$$\begin{array}{cc}\hfill & {\int }_{r_0}^{\infty }{\displaystyle \frac{\sigma \left(s\right)}{v\left(s\right)}}[{\displaystyle \frac{\sqrt{\mu \Gamma \left(1-\alpha \right)}A\left(s\right){\left(\sqrt{Q\left(s\right)}\right)}^{-1}}{\sigma \left(s\right)}}\hfill \\ \hfill & +{\displaystyle \frac{v\left(s\right)\sqrt{Q\left(s\right)}}{2\sqrt{\mu \Gamma (1-\alpha )}}}{\left(R\left(s\right)Q^{-1}\left(s\right)+2\mu h\left(s\right)I_n-2\mu \Gamma (1-\alpha )h\left(s\right)I_n\right]}^{2}ds=\infty ,\hfill \end{array}$$

(16)

where $v\left(s\right)$ and $\Phi \left(s\right)$ are already defined in the previous theorem. As a result, Equation (1) oscillates.

Proof. 

Assume that $detX\left(r\right)\ne 0$ for $r\ge r_0$. If (3.8) holds, ∀$r\ge s\ge r_0$, for $r\ge s\ge r_0$. Thus, we get

$$\begin{array}{ccc}& & \underset{r\to \infty }{lim\; inf}{\displaystyle \frac{1}{r^{\alpha }}}{L}_r^{\sigma }\left[{\displaystyle \frac{1}{v\left(s\right)}}{\left(W\left(s\right)-{\displaystyle \frac{v\left(s\right)}{2\sqrt{\mu \Gamma \left(1-\alpha \right)}}}\left({\displaystyle \frac{\alpha }{r-s}}+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right)\sqrt{Q\left(s\right)}\right)}^2\right]\hfill \\ & & +\underset{r\to \infty }{lim\; sup}{\displaystyle \frac{1}{r^{\alpha }}}{L}_r^{\sigma }[\Phi \left(s\right)-{\displaystyle \frac{v\left(s\right)Q\left(s\right)}{4\mu \Gamma \left(1-\alpha \right)}}(R\left(s\right)Q^{-1}\left(s\right)+2\mu h\left(s\right)I_n-2\mu \Gamma \left(1-\alpha \right)h\left(s\right)I_n)\hfill \\ & & -{\left({\displaystyle \frac{-\alpha }{r-s}}+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right)}^2{I}_n]\le \sigma \left(r\right)U\left(r\right)\forall r\ge r_0.\hfill \end{array}$$

(17)

Hence, by (15),

$$\begin{array}{c}\hfill \sigma \left(r\right)U\left(r\right)\ge A\left(r\right)+\underset{r\to \infty }{lim\; inf}{\displaystyle \frac{1}{r^{\alpha }}}{L}_s^{\sigma }\left[{\displaystyle \frac{1}{v\left(s\right)}}{\left(W\left(s\right)-{\displaystyle \frac{v\left(s\right)}{2\sqrt{\mu \Gamma \left(1-\alpha \right)}}}\left({\displaystyle \frac{\alpha }{r-s}}+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right)\sqrt{Q\left(s\right)}\right)}^2\right],\end{array}$$

for all $r\ge r_0$,

$$\begin{array}{cc}\hfill \sigma \left(r\right)U\left(r\right)& \ge A\left(r\right)\end{array}$$

(18)

and

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}inf{\displaystyle \frac{1}{r^{\alpha }}}{L}_{r_0}^{\sigma }\left[{\displaystyle \frac{1}{v\left(s\right)}}{\left(W\left(s\right)-{\displaystyle \frac{v\left(s\right)}{2\sqrt{\mu \Gamma \left(1-\alpha \right)}}}\times \left({\displaystyle \frac{\alpha }{r-s}}+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right)\sqrt{Q\left(s\right)}\right)}^2\right]<\infty .\end{array}$$

(19)

Define

$$\begin{array}{ccc}\hfill B\left(r\right)& =& {\displaystyle \frac{1}{r^{\alpha }}}{\mathcal{L}}_{r_0}^{\sigma }\left[{\displaystyle \frac{\mathrm{tr}\left(W^2\left(s\right)\right)}{v\left(s\right)}}\right]={\displaystyle \frac{1}{r^{\alpha }}}{\int }_{r_0}^r\sigma \left(s\right){(r-s)}^{\alpha }{\displaystyle \frac{\mathrm{tr}\left(W^2\left(s\right)\right)}{v\left(s\right)}}ds\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill C\left(r\right)& =& {\displaystyle \frac{1}{r^{\alpha }}}{L}_{r_0}^{\sigma }\left(\left({\displaystyle \frac{-\alpha }{r-s}}+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right){\displaystyle \frac{v\left(s\right)tr\left(Q\left(s\right)W\left(s\right)\right)}{2\mu \Gamma \left(1-\alpha \right)}}\right).\hfill \end{array}$$

(21)

Here, $B\left(r\right)$ and $C\left(r\right)$ are interpreted as scalar quantities via the trace functional, making the ratio ${\displaystyle \frac{C\left(r_n\right)}{B\left(r_n\right)}}$ and the subsequent Cauchy–Schwarz inequality valid in the scalar sense. Then,

$$\begin{array}{c}\hfill B\left(r\right)-C\left(r\right)\le {\displaystyle \frac{1}{r^{\alpha }}}{L}_{r_0}^{\sigma }\left[{\displaystyle \frac{1}{v\left(s\right)}}{\left(W\left(s\right)-{\displaystyle \frac{v\left(s\right)}{2\sqrt{\mu \Gamma \left(1-\alpha \right)}}}\left({\displaystyle \frac{\alpha }{r-s}}+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right)\sqrt{Q\left(s\right)}\right)}^2\right].\end{array}$$

(22)

Since $W\left(s\right)$ is symmetric, $S{(r,s)}^2$ is positive semi-definite and $\mathrm{tr}\left(S{(r,s)}^2\right)\ge 0$; the right-hand side of (22) is non-negative, and the completing-the-square step holds under the trace by linearity and $\mathrm{tr}\left(AB\right)=\mathrm{tr}\left(BA\right)$.

From (19) and (22), we get

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim\; inf}\left[B\left(r\right)-C\left(r\right)\right]<\infty .\end{array}$$

(23)

Now, we claim that

$$\begin{array}{c}\hfill {\int }_{r_0}^{\infty }{\displaystyle \frac{\sigma \left(s\right)\mathrm{tr}\left(W^2\left(s\right)\right)}{v\left(s\right)}}ds<\infty .\end{array}$$

(24)

Suppose to the contrary that

$$\begin{array}{c}\hfill {\int }_{r_0}^{\infty }{\displaystyle \frac{\sigma \left(s\right)\mathrm{tr}\left(W^2\left(s\right)\right)}{v\left(s\right)}}ds=\infty .\end{array}$$

(25)

Thus,

$$\begin{array}{cc}\hfill \underset{r\to \infty }{lim}B\left(r\right)& =\infty .\end{array}$$

(26)

Now, suppose ${\left\{r_n\right\}}_{n=1}^{\infty }\subset [r_0,\infty )$, satisfying ${\displaystyle \underset{n\to \infty }{lim}{r}_n=\infty }$ and

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}[B\left(r_n\right)-C\left(r_n\right)]=\underset{r\to \infty }{lim}inf[B\left(r\right)-C\left(r\right)].\end{array}$$

(27)

By (23) and (27), there exists a constant $\eta >0$∋

$$B\left(r_n\right)-C\left(r_n\right)\le \eta ,(n=1,2,\dots ).$$

This together with (26) means that ${lim}_{r\to \infty }C\left(r_n\right)=\infty$. So, for large enough,

$$1-{\displaystyle \frac{C\left(r_n\right)}{B\left(r_n\right)}}<{\displaystyle \frac{1}{2}},$$

i.e., ${\displaystyle \frac{C\left(r_n\right)}{B\left(r_n\right)}}<{\displaystyle \frac{1}{2}}$. Moreover,

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}{\displaystyle \frac{C^2\left(r_n\right)}{B\left(r_n\right)}}=\infty .\end{array}$$

(28)

Applying the scalar Cauchy–Schwarz inequality

$${\left(\int fg\right)}^2\le \left(\int f^2\right)\left(\int g^2\right)$$

with $f\left(s\right)=\sqrt{\sigma \left(s\right)/v\left(s\right)}\mathrm{tr}{\left(W^2\left(s\right)\right)}^{1/2}$ and $g\left(s\right)=\sqrt{\sigma \left(s\right)v\left(s\right)}\left({\displaystyle \frac{-\alpha }{r-s}}+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right){\displaystyle \frac{\mathrm{tr}{\left(Q\left(s\right)\right)}^{1/2}}{\sqrt{\mu \Gamma (1-\alpha )}}}$, where $\mathrm{tr}\left(W^2\left(s\right)\right)={\parallel W\left(s\right)\parallel }_F^2\ge 0$, gives

$$\begin{array}{cc}\hfill C^2\left(r_n\right)& ={\displaystyle \frac{1}{r_n^{2\alpha }}}{\left[L_{r_0}^{\sigma }\left({\displaystyle \frac{-\alpha }{r-s}}+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right){\displaystyle \frac{v\left(s\right)\mathrm{tr}\left(Q\right(s\left)W\right(s\left)\right)}{\mu \Gamma (1-\alpha )}}\right]}^2\hfill \\ \hfill & \le {\displaystyle \frac{1}{r_n^{\alpha }}}{L}_{r_0}^{\sigma }{\left({\displaystyle \frac{-\alpha }{r-s}}+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right)}^2{\displaystyle \frac{v\left(s\right)\mathrm{tr}\left(Q\right(s\left)\right)}{\mu \Gamma (1-\alpha )}}·{\displaystyle \frac{1}{r_n^{\alpha }}}{L}_{r_0}^{\sigma }{\displaystyle \frac{\mathrm{tr}\left(W^2\left(s\right)\right)}{v\left(s\right)}}\hfill \\ \hfill & =B\left(r_n\right)r_n^{-\alpha }{L}_{r_0}^{\sigma }\left[{\left({\displaystyle \frac{-\alpha }{r-s}}+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right)}^2{\displaystyle \frac{v\left(s\right)\mathrm{tr}\left(Q\right(s\left)\right)}{\mu \Gamma (1-\alpha )}}\right]\hfill \end{array}$$

(29)

By (26) and (28), we get

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}{r}_n^{-\alpha }{L}_{r_0}^{\sigma }\left[{\left({\displaystyle \frac{-\alpha }{r-s}}+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right)}^2{\displaystyle \frac{Q\left(s\right)v\left(s\right)}{\mu \Gamma \left(1-\alpha \right)}}\right]=\infty ,\end{array}$$

which contradicts (3). So, we obtain (24). Then, by (18), in view of

$$\begin{array}{cc}\hfill W\left(r\right)& =\sqrt{\mu \Gamma \left(1-\alpha \right)\sqrt{Q^{-1}\left(r\right)})}U\left(r\right)+{\displaystyle \frac{v\left(r\right)\sqrt{Q\left(r\right)}}{2\sqrt{\mu \Gamma \left(1-\alpha \right)}}}(R\left(r\right)Q^{-1}\left(r\right)+2\mu h\left(r\right)I_n\hfill \\ \hfill & -2\mu \Gamma \left(1-\alpha \right)h\left(r\right)I_n),\hfill \end{array}$$

we get

$$\begin{array}{c}\hfill [{\displaystyle \frac{\sqrt{\mu \Gamma \left(1-\alpha \right)}A\left(r\right){\left(\sqrt{Q\left(r\right)}\right)}^{-1}}{\sigma \left(s\right)}}+{\displaystyle \frac{v\left(r\right)\sqrt{Q\left(r\right)}}{2\sqrt{\mu \Gamma \left(1-\alpha \right)}}}(R\left(r\right)Q^{-1}\left(r\right)\\ \hfill +2\mu h\left(r\right)I_n-2\mu \Gamma \left(1-\alpha \right)h\left(r\right)I_n){]}^2\le W^2\left(r\right).\end{array}$$

For all $r\le r_0$, by (24), we get

$$\begin{array}{cc}\hfill {\int }_{r_0}^{\infty }& {\displaystyle \frac{\sigma \left(s\right)}{v\left(s\right)}}[{\displaystyle \frac{\sqrt{\mu \Gamma \left(1-\alpha \right)}A\left(s\right){\left(\sqrt{Q\left(s\right)}\right)}^{-1}}{\sigma \left(s\right)}}+{\displaystyle \frac{v\left(s\right)\sqrt{Q\left(s\right)}}{2\sqrt{\mu \Gamma \left(1-\alpha \right)}}}(R\left(s\right)Q^{-1}\left(s\right)\hfill \\ \hfill & +2\mu h\left(s\right)I_n-2\mu \Gamma \left(1-\alpha \right)h\left(s\right)I_n){]}^{2}ds\le {\int }_{r_0}^{\infty }{\displaystyle \frac{\sigma \left(s\right)}{v\left(s\right)}}{W}^2\left(s\right)ds<\infty ,\hfill \end{array}$$

which ia a contradiction to (16). Hence, it is proved. □

Remark 4. 

As $\alpha \to 1^{-}$, the R-L derivative $D_{+}^{\alpha }X\left(r\right)$ converges to the classical derivative $X^{\prime }\left(r\right)$, consistent with Remark 1 and the standard limit property of the Riemann–Liouville operator [30]. We note that $\alpha =1$ is excluded by the standing assumption $\alpha \in (0,1)$, and $\Gamma (1-\alpha )$ has a pole at $\alpha =1$; the recovery of the classical derivative is therefore understood in the limiting sense $\alpha \to 1^{-}$. Moreover, $\Gamma (1-\alpha )\to 1$ as $\alpha \to 0^{+}$ (since $\Gamma \left(1\right)=1$), not as $\alpha \to 1^{-}$. In this limit, Theorem 3.1 reduces to the oscillation criterion of Yang [20] for integer-order damped matrix systems.

Remark 5. 

If $R\left(r\right)=0$ (no damping) and G is the identity operator, then Theorem 3.1 reduces to an analogue of the Philos-type criterion for self-adjoint matrix systems established by Yang and Tang [20], now extended to the fractional-order setting.

Remark 6. 

The nonlinear operators G and F, constrained by assumptions $\left(A_2\right)$–$\left(A_3\right)$, admit a strictly larger class of equations than the linear systems treated in Yang [20] and Liu–Meng [19]. In particular, G need not be the identity matrix, provided that $G^{\prime }\left(K\right)\ge \mu I_n$.

4. Examples

This section provides two examples that illustrate and verify the theoretical results obtained for system (1).

Example 1. 

Consider the following fractional matrix DE:

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dr}}\left(Q\left(r\right)D_{+}^{\alpha }X\left(r\right)\right)+R\left(r\right)D_{+}^{\alpha }X\left(r\right)+P\left(r\right)G\left({\int }_0^r{(r-s)}^{-\alpha }X\left(s\right)ds\right)F\left(D_{+}^{\alpha }X\left(r\right)\right)=0,\end{array}$$

(30)

for $r\ge r_0\ge 0$. $G\left(K\right(r\left)\right)=K\left(r\right)$, since $F\left(D_{+}^{\alpha }X\right)=I_2+{\left(D_{+}^{\alpha }X\right)}^2\ge I_2>0$, assumption $\left(A_3\right)$ is satisfied with $M=1$.

$$\begin{array}{ccc}\hfill \Phi \left(s\right)& =& v\left(s\right)[LP\left(s\right)-h\left(s\right)R\left(s\right)-\Gamma (1-\alpha )h^2\left(s\right)Q\left(s\right)-{\left(h\left(s\right)Q\left(s\right)\right)}^{\prime }].\hfill \end{array}$$

Here, $v\left(s\right)={\displaystyle \frac{1}{s^2}}$, $h\left(s\right)={\displaystyle \frac{1}{s}},$ $R\left(s\right)=s^3\left[\begin{array}{cc}-4& 0\\ 0& -4\end{array}\right]$, $Q\left(s\right)=s^4\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, $M>1$, $\mu >1$, $\alpha =1/3$, and

$$\begin{array}{c}\hfill P\left(s\right)=\left({\displaystyle \frac{(\frac{\sqrt{3}}{2}{s}^{4}sins+\frac{s^4}{2}coss)\sqrt{3}\Gamma \left(\frac{1}{3}\right)}{\pi (\sqrt{3}coss-sins)(5+2{cos}^{2}s-\sqrt{3}sin2s)}}\right)I_2.\end{array}$$

Assumption $\left(A_1\right)$ is satisfied since $Q\left(s\right)=s^4{I}_2$ is symmetric positive-definite and $R\left(s\right)=-4s^3{I}_2$ is symmetric, both with continuous entries on $[r_0,\infty )$. For $\left(A_2\right)$, $G\left(K\right)=K$ gives $G^{\prime }\left(K\right)=I_2\ge \mu I_2$ with $\mu =1$, and $XG\left(X\right)=X^2>0$ for every nonzero X. Assumption $\left(A_3\right)$ holds since $F\left(D_{+}^{\alpha }X\right)=I_2+{\left(D_{+}^{\alpha }X\right)}^2\ge I_2>0$, giving $M=1$. Finally, $X\left(r\right)=cosr·I_2$ is a scalar multiple of $I_2$ and hence commutes with $Q\left(r\right)$, confirming the prepared solution condition.

Consider

$$\begin{array}{ccc}\hfill \underset{r\to \infty }{lim\; sup}{\displaystyle \frac{1}{r^2}}{\int }_1^r{(r-s)}^2\Phi \left(s\right)ds& =& \underset{r\to \infty }{lim\; sup}{\displaystyle \frac{1}{r^2}}{\int }_1^r{(r-s)}^2\hfill \\ & & \times [{\displaystyle \frac{1}{s^2}}\left({\displaystyle \frac{(\frac{\sqrt{3}}{2}{s}^{4}sins+\frac{s^4}{2}coss)\sqrt{3}\Gamma \left(\frac{1}{3}\right)}{\pi (\sqrt{3}coss-sins)(5+2{cos}^{2}s-\sqrt{3}sin2s)}}\right)I_2\hfill \\ & & 4s^2{I}_2-\Gamma \left({\displaystyle \frac{2}{3}}\right)s^2{I}_2-{\left(s^3{I}_2\right)}^{\prime }]ds\hfill \\ & =& \underset{r\to \infty }{lim\; sup}{\displaystyle \frac{1}{r^2}}{\int }_1^r{(r-s)}^2(4-\Gamma \left({\displaystyle \frac{2}{3}}\right)-{\displaystyle \frac{s^2}{2}})I_{2}ds=\infty .\hfill \end{array}$$

$$\begin{array}{cc}\hfill \underset{r\to \infty }{lim\; sup}& {\displaystyle \frac{1}{r^2}{\int }_1^r{(r-s)}^2\frac{v\left(s\right)Q\left(s\right)}{4\mu \Gamma (1-\alpha )}[R\left(s\right)Q^{-1}\left(s\right)}\hfill \\ \hfill & +2\mu h\left(s\right)I_2-2\mu \Gamma (1-\alpha )h\left(s\right)I_2-\left({\displaystyle \frac{-\alpha }{r-s}}+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right){]}^{2}ds\hfill \\ \hfill & {\displaystyle =\underset{r\to \infty }{lim\; sup}\frac{1}{r^2}{\int }_1^r{(r-s)}^2\frac{s^2{I}_2}{4\Gamma \left(\frac{2}{3}\right)}{\left[-\frac{2}{3}-\frac{2\Gamma \left(\frac{2}{3}\right)}{s}+\frac{2}{r-s}\right]}^2{I}_{2}ds<\infty .}\hfill \end{array}$$

Hence, all conditions of Corollary 1 are satisfied. Hence, $X\left(r\right)=cosr{I}_2$ is an oscillatory solution of (30).

To illustrate the results in a structurally distinct setting, we consider a non-diagonal coefficient matrix $Q\left(r\right)$ with coupling terms, in contrast to the scalar multiple of $I_2$ used in Example 1.

Example 2. 

Consider the following fractional matrix DE:

$${\displaystyle \frac{d}{dr}}\left(Q\left(r\right)D_{+}^{\alpha }X\left(r\right)\right)+R\left(r\right)D_{+}^{\alpha }X\left(r\right)+P\left(r\right)G\left({\int }_0^r{(r-s)}^{-\alpha }X\left(s\right)ds\right)F\left(D_{+}^{\alpha }X\left(r\right)\right)=0$$

(31)

for $r\ge r_0=1$, with $\alpha =\frac{1}{3}$, and

$$\begin{array}{cc}\hfill Q\left(s\right)& =s^4\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],R\left(s\right)=-4s^3\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],\hfill \\ \hfill P\left(s\right)& =\left[\left(\Gamma \left(\frac{2}{3}\right)-1\right)s^2+s^4\left(1+{sin}^{2}s\right)\right]\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],\hfill \\ \hfill G\left(K\right)& =K,F\left(D_{+}^{\alpha }X\right)=2I_2+{\left(D_{+}^{\alpha }X\right)}^2.\hfill \end{array}$$

Take $v\left(s\right)=\frac{1}{s^2}$, $h\left(s\right)=\frac{1}{s}$, $\sigma \left(s\right)=\frac{1}{s^2}$, and $\mu =1$.

Assumptions: The matrix $\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right]$ has eigenvalues 1 and 3, so $Q\left(s\right)$ is symmetric positive-definite and $R\left(s\right)$ is symmetric, both with continuous entries, giving $\left(A_1\right)$. Since $G\left(K\right)=K$, we have $G^{\prime }\left(K\right)=I_2\ge \mu I_2$ with $\mu =1$; hence, assumption $\left(A_2\right)$ is satisfied. Since ${\left(D_{+}^{\alpha }X\right)}^2\ge 0$, we have $F\left(D_{+}^{\alpha }X\right)\ge 2I_2>0$, giving $\left(A_3\right)$ with $M=2$. Since $X\left(r\right)=sin\left(r\right)I_2$, $D_{+}^{\alpha }X\left(r\right)$ is also a scalar multiple of $I_2$. Hence, $D_{+}^{\alpha }X\left(r\right)$ commutes with $Q\left(r\right)$, and the prepared solution condition is satisfied. Since $Q^{-1}\left(s\right)=\frac{1}{3s^4}\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]$, we get $R\left(s\right)Q^{-1}\left(s\right)=-\frac{4}{s}{I}_2$, and so

$$M\left(s\right):=R\left(s\right)Q^{-1}\left(s\right)+2h\left(s\right)I_2-2\Gamma \left(\frac{2}{3}\right)h\left(s\right)I_2=-{\displaystyle \frac{2+2\Gamma \left(\frac{2}{3}\right)}{s}}{I}_2.$$

A direct substitution gives

$$-h\left(s\right)R\left(s\right)-\Gamma \left(\frac{2}{3}\right)h^2\left(s\right)Q\left(s\right)-{\left(h\left(s\right)Q\left(s\right)\right)}^{\prime }=\left(1-\Gamma \left(\frac{2}{3}\right)\right)s^2\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],$$

and with the chosen $P\left(s\right)$, the $\left(\Gamma \left(\frac{2}{3}\right)-1\right)$ terms cancel exactly, yielding

$$\Phi \left(s\right)=s^2\left(1+{sin}^{2}s\right)\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right].$$

Since $1+{sin}^{2}s\ge 1$ and $\mathrm{tr}\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right]=4$,

$$\underset{r\to \infty }{lim\; sup}{\displaystyle \frac{1}{r^{1/3}}}{L}_{r_0}^{\sigma }\left[\Phi \left(s\right)\right]\ge {\displaystyle \frac{4}{r^{1/3}}}{\int }_1^r{(r-s)}^{1/3}ds={\displaystyle \frac{4}{r^{1/3}}}·{\displaystyle \frac{3}{4}}{(r-1)}^{4/3}\to \infty .$$

Since ${\sigma }^{\prime }\left(s\right)/\sigma \left(s\right)=-2/s$, and

$${\displaystyle \frac{v\left(s\right)Q\left(s\right)}{4\mu \Gamma \left(\frac{2}{3}\right)}}M{\left(s\right)}^2={\displaystyle \frac{{\left(2+2\Gamma \left(\frac{2}{3}\right)\right)}^2}{4\Gamma \left(\frac{2}{3}\right)}}\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right]$$

is a constant matrix, its contribution to $\frac{1}{r^{1/3}}{L}_{r_0}^{\sigma }[·]$ satisfies

$${\displaystyle \frac{1}{r^{1/3}}}{\int }_1^r{\displaystyle \frac{{(r-s)}^{1/3}}{s^2}}ds\sim O\left(1\right)\mathit{as}r\to \infty .$$

The remaining term ${\left(\frac{-\alpha }{r-s}+\frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}\right)}^2={\left(\frac{1}{3(r-s)}+\frac{2}{s}\right)}^2$ is subtracted, contributing only a non-positive correction to the integrand. Hence,

$$\begin{array}{cc}\hfill \underset{r\to \infty }{lim\; sup}{\displaystyle \frac{1}{r^{1/3}}}{L}_{r_0}^{\sigma }[& {\displaystyle \frac{v\left(s\right)Q\left(s\right)}{4\mu \Gamma \left(\frac{2}{3}\right)}}{\left(R\left(s\right)Q^{-1}\left(s\right)+2\mu h\left(s\right)I_2-2\mu \Gamma \left(\frac{2}{3}\right)h\left(s\right)I_2\right)}^2\hfill \\ \hfill & -{\left({\displaystyle \frac{-\alpha }{r-s}}+{\displaystyle \frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}}\right)}^2{I}_2]<\infty .\hfill \end{array}$$

Thus, all conditions of Corollary 1 are satisfied. Hence, every prepared solution of (31) is oscillatory.

5. Conclusions

In this study, we have used a generalised Riccati transformation in conjunction with a linear integral operator to examine new oscillation criteria for matrix fractional DEs. This method allows for a more thorough examination of the connection between matrix-valued systems and fractional derivatives, which results in useful adequate conditions for oscillatory behaviour. In addition to validating and supporting previous conclusions from the classical theory, the results of this study also extend and generalise them to the fractional matrix framework, expanding their relevance and scope [12,20]. Furthermore, the suggested criteria are applicable to a broader class of problems since they include appropriate structural conditions on the associated matrices and nonlinear factors. The paper extends oscillation theory to the specific class of damped nonlinear matrix fractional DEs of the form (1), and provides a basis for future work in stability, control, and numerical analysis within this precise framework The results are validated by the examples given.