In this paper, we shall give some new results about the oscillatory behavior of nonlinear fractional order integro-differential equations with forcing term $v(t)$ of form \[ D_a^\alpha x(t)=v(t)-\int\limits_a^t K(t,s) F(s,x(s))ds, \,\, 0<\alpha <1,\,\, \lim\limits_{t\to a^+} J_a^{1-\alpha} x(t)=b_1, \]where $v$, $K$ and $F$ are continuous functions, $b_1\in\mathbb{R}$, and $D_a^\alpha$ and $J_a^{1-\alpha}$ denote the Riemann-Liouville fractional order differential and integral operators respectively.
📌 3 Key Takeaways
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Objective & Hypothesis:In this paper, we shall give some new results about the oscillatory behavior of nonlinear fractional order integro-differential equations with forcing term $v(t)$ of form \[ D_a^\alpha x(t)=v(t)-\int\limits_a^t K(t,s) F(s,x(s))ds, \,\, 0<\alpha <1,\,\, \lim\limits_{t\to a^+} J_a^{1-\alpha} x(t)=b_1, \]where $v$, $K$ and $F$ are continuous functions, $b_1\in\mathbb{R}$, and $D_a^\alpha$ and $J_a^{1-\alpha}$ denote the Riemann-Liouville fractional order differential and integral operators respectively.
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Key Findings & Data:In this paper, we shall give some new results about the oscillatory behavior of nonlinear fractional order integro-differential equations with forcing term $v(t)$ of form \[ D_a^\alpha x(t)=v(t)-\int\limits_a^t K(t,s) F(s,x(s))ds, \,\, 0<\alpha <1,\,\, \lim\limits_{t\to a^+} J_a^{1-\alpha} x(t)=b_1, \]where $v$, $K$ and $F$ are continuous functions, $b_1\in\mathbb{R}$, and $D_a^\alpha$ and $J_a^{1-\alpha}$ denote the Riemann-Liouville fractional order differential and integral operators respectively.
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Conclusion & Impact:In this paper, we shall give some new results about the oscillatory behavior of nonlinear fractional order integro-differential equations with forcing term $v(t)$ of form \[ D_a^\alpha x(t)=v(t)-\int\limits_a^t K(t,s) F(s,x(s))ds, \,\, 0<\alpha <1,\,\, \lim\limits_{t\to a^+} J_a^{1-\alpha} x(t)=b_1, \]where $v$, $K$ and $F$ are continuous functions, $b_1\in\mathbb{R}$, and $D_a^\alpha$ and $J_a^{1-\alpha}$ denote the Riemann-Liouville fractional order differential and integral operators respectively.
In this paper, we shall give some new results about the oscillatory behavior of nonlinear fractional order integro-differential equations with forcing term $v(t)$ of form \[ D_a^\alpha x(t)=v(t)-\int\limits_a^t K(t,s) F(s,x(s))ds, \,\, 0<\alpha <1,\,\, \lim\limits_{t\to a^+} J_a^{1-\alpha} x(t)=b_1, \]where $v$, $K$ and $F$ are continuous functions, $b_1\in\mathbb{R}$, and $D_a^\alpha$ and $J_a^{1-\alpha}$ denote the Riemann-Liouville fractional order differential and integral operators respectively.
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