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<p>In this paper, we derive sufficient conditions for the existence, uniqueness, and Hyers-Ulam stability of solutions to a class of nonlinear integro-dynamic equations on time scales. The analysis focuses on equations that combine discrete and continuous dynamics, characterized by nonlinear integral operators. To establish the existence of solutions, we employ Schauder’s fixed point theorem, while uniqueness is ensured using the Banach contraction principle. Furthermore, we investigate the Hyers-Ulam stability of solutions, demonstrating their continuous dependence on initial conditions. As an application, we model the population dynamics of a species that exhibits continuous growth during specific seasons and undergoes discrete breeding events. The model is formulated on a time scale <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_937_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {T}=(1/2)\mathbb {Z}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">T</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation> to capture the interaction between population growth, survival rates, and seasonal resource availability. We utilize Python programming to perform numerical simulations, providing computational estimates of population sizes over time. These simulations demonstrate the effectiveness of integro-dynamic models in capturing mixed continuous-discrete behaviors in dynamic systems.</p>

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Qualitative results for nonlinear iterative integro-dynamic equation with deviating arguments on time scales

  • 期刊论文

  • 发表日期：

    2025年8月8日

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  • Mahammad Khuddush,
  • S. Kalesha Vali,
  • M. V. Ramakrishna

摘要

In this paper, we derive sufficient conditions for the existence, uniqueness, and Hyers-Ulam stability of solutions to a class of nonlinear integro-dynamic equations on time scales. The analysis focuses on equations that combine discrete and continuous dynamics, characterized by nonlinear integral operators. To establish the existence of solutions, we employ Schauder’s fixed point theorem, while uniqueness is ensured using the Banach contraction principle. Furthermore, we investigate the Hyers-Ulam stability of solutions, demonstrating their continuous dependence on initial conditions. As an application, we model the population dynamics of a species that exhibits continuous growth during specific seasons and undergoes discrete breeding events. The model is formulated on a time scale \(\mathbb {T}=(1/2)\mathbb {Z}\) T = ( 1 / 2 ) Z to capture the interaction between population growth, survival rates, and seasonal resource availability. We utilize Python programming to perform numerical simulations, providing computational estimates of population sizes over time. These simulations demonstrate the effectiveness of integro-dynamic models in capturing mixed continuous-discrete behaviors in dynamic systems.