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Non-local second-order boundary value problems with derivative-dependent nonlinearity.

J R L Webb1

  • 1School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8SQ, UK.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|January 4, 2021
PubMed
Summary
This summary is machine-generated.

This study proves multiple positive solutions for nonlinear differential equations with derivative dependence. It utilizes a novel Gronwall-type inequality for derivative bounds, demonstrating broad applicability.

Keywords:
a priori boundsfixed-point indexmultiple positive solutions

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Area of Science:

  • Differential Equations
  • Nonlinear Analysis
  • Mathematical Physics

Background:

  • Nonlinear second-order boundary value problems are crucial in modeling various phenomena.
  • Understanding the existence of multiple solutions is key for comprehensive analysis.
  • Derivative dependence in nonlinear terms presents unique analytical challenges.

Purpose of the Study:

  • To establish the existence of multiple positive solutions for nonlinear second-order nonlocal boundary value problems.
  • To analyze problems where the nonlinearity depends on the derivatives, allowing quadratic growth.
  • To extend the applicability of fixed point theories in differential equations.

Main Methods:

  • Utilizing a recently developed Gronwall-type inequality.
  • Deriving a priori bounds for the norms of derivatives.
  • Applying techniques from topological degree and fixed point theories.

Main Results:

  • The existence of at least two positive solutions is proven.
  • The method provides a way to handle quadratic growth in nonlinear terms with respect to derivatives.
  • The derived bounds are essential for the existence proofs.

Conclusions:

  • The study successfully demonstrates the existence of multiple positive solutions for a complex class of differential equations.
  • The employed methods offer a robust framework for analyzing similar nonlinear problems.
  • This work contributes to the theme issue on topological degree and fixed point theories in differential and difference equations.