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Existence, uniqueness and blow-up of solutions for generalized auto-convolution Volterra integral equations

Author

Listed:
  • Mostafazadeh, Mahdi
  • Shahmorad, Sedaghat
  • Erdoğan, Fevzi

Abstract

In this paper, our intention is to investigate the blow-up theory for generalized auto-convolution Volterra integral equations (AVIEs). To accomplish this, we will consider certain conditions on the main equation. This will establish a framework for our analysis, ensuring that the solution of the equation exists uniquely and is positive. Firstly, we analyze the existence and uniqueness of a local solution for a more general class of AVIEs (including the proposed equation in this paper) under certain hypotheses. Subsequently, we demonstrate the conditions under which this local solution blows up at a finite time. In other words, the solution becomes unbounded at that time. Furthermore, we establish that this blow-up solution can be extended to an arbitrary interval on the non-negative real line, thus referred to as a global solution. These results are also discussed for a special case of generalized AVIEs in which the kernel functions are taken as positive constants.

Suggested Citation

  • Mostafazadeh, Mahdi & Shahmorad, Sedaghat & Erdoğan, Fevzi, 2024. "Existence, uniqueness and blow-up of solutions for generalized auto-convolution Volterra integral equations," Applied Mathematics and Computation, Elsevier, vol. 471(C).
    • Handle: RePEc:eee:apmaco:v:471:y:2024:i:c:s0096300324000808
    DOI: 10.1016/j.amc.2024.128608
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