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Multiple positive solutions for nonlinear fractional boundary value problems.

Daliang Zhao1, Yansheng Liu1

  • 1School of Mathematical Sciences, Shandong Normal University, Jinan, Shandong 250014, China.

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Summary
This summary is machine-generated.

This study establishes criteria for multiple positive solutions to fractional boundary value problems using fixed-point theorems. The research provides new existence conditions for these complex mathematical models.

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Area of Science:

  • Mathematics
  • Applied Mathematics
  • Fractional Calculus

Background:

  • Fractional boundary value problems are increasingly studied for their applications.
  • Understanding the existence of multiple positive solutions is crucial for modeling complex phenomena.
  • Existing methods may not fully capture the behavior of certain fractional differential equations.

Purpose of the Study:

  • To investigate the existence of multiple positive solutions for a specific fractional boundary value problem.
  • To establish new existence criteria using advanced fixed-point theorems.
  • To demonstrate the applicability of the developed criteria through examples.

Main Methods:

  • Application of Guo-Krasnoselskii's fixed point theorem.
  • Utilization of the Leggett-Williams fixed point theorem.
  • Employing a novel extension of Krasnoselskii's fixed point theorem.
  • Construction of a specialized cone in the function space.

Main Results:

  • New existence criteria for multiple positive solutions to the fractional boundary value problem are established.
  • A sufficient condition for the existence of multiple positive solutions is derived.
  • The theoretical results are validated with illustrative examples.

Conclusions:

  • The study successfully provides new theoretical tools for analyzing fractional boundary value problems.
  • The findings contribute to the understanding of multiple positive solutions in fractional calculus.
  • The established criteria offer practical methods for identifying the existence of solutions.