Remarks on existence of positive solutions of some integral equations

Jan Ligęza

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2005)

  • Volume: 44, Issue: 1, page 71-82
  • ISSN: 0231-9721

Abstract

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We study the existence of positive solutions of the integral equation x ( t ) = μ ∫ 0 1 k ( t , s ) f ( s , x ( s ) , x ' ( s ) , ... , x ( n - 1 ) ( s ) ) d s , n ≥ 2 in both C n - 1 [ 0 , 1 ] and W n - 1 , p [ 0 , 1 ] spaces, where p ≥ 1 and μ > 0 . Throughout this paper k is nonnegative but the nonlinearity f may take negative values. The Krasnosielski fixed point theorem on cone is used.

How to cite

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Ligęza, Jan. "Remarks on existence of positive solutions of some integral equations." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 44.1 (2005): 71-82. <http://eudml.org/doc/32452>.

@article{Ligęza2005,
abstract = {We study the existence of positive solutions of the integral equation \[ x(t) = \mu \int \_0^1 k(t, s) f(s, x(s), x^\{\prime \}(s), \ldots , x^\{(n-1)\} (s))\, ds, \quad n \ge 2 \] in both $ C^\{n-1\} [0, 1] $ and $ W^\{n-1, p\} [0, 1] $ spaces, where $ p \ge 1 $ and $ \mu > 0 $. Throughout this paper $k$ is nonnegative but the nonlinearity $f$ may take negative values. The Krasnosielski fixed point theorem on cone is used.},
author = {Ligęza, Jan},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {Positive solutions; Fredholm integral equations; cone; boundary value problems; fixed point theorem; positive solution; nonlinear integral equation},
language = {eng},
number = {1},
pages = {71-82},
publisher = {Palacký University Olomouc},
title = {Remarks on existence of positive solutions of some integral equations},
url = {http://eudml.org/doc/32452},
volume = {44},
year = {2005},
}

TY - JOUR
AU - Ligęza, Jan
TI - Remarks on existence of positive solutions of some integral equations
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2005
PB - Palacký University Olomouc
VL - 44
IS - 1
SP - 71
EP - 82
AB - We study the existence of positive solutions of the integral equation \[ x(t) = \mu \int _0^1 k(t, s) f(s, x(s), x^{\prime }(s), \ldots , x^{(n-1)} (s))\, ds, \quad n \ge 2 \] in both $ C^{n-1} [0, 1] $ and $ W^{n-1, p} [0, 1] $ spaces, where $ p \ge 1 $ and $ \mu > 0 $. Throughout this paper $k$ is nonnegative but the nonlinearity $f$ may take negative values. The Krasnosielski fixed point theorem on cone is used.
LA - eng
KW - Positive solutions; Fredholm integral equations; cone; boundary value problems; fixed point theorem; positive solution; nonlinear integral equation
UR - http://eudml.org/doc/32452
ER -

References

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  1. Agarwal R. P., Grace S. R., O’Regan D., Existence of positive solutions of semipositone Fredholm integral equation, Funkciałaj Equaciaj 45 (2002), 223–235. MR1948600
  2. Agarwal R. P., O’Regan D.: Infinite Interval Problems For Differential, Difference, Integral Equations., Kluwer Acad. Publishers, Dordrecht, Boston, London, , 2001. MR1845855
  3. Agarwal R. P., O’Regan D., Wang J. Y.: Positive Solutions of Differential, Difference, Integral Equations., Kluwer Academic Publishers, Dordrecht, Boston, London, , 1999. (1999) MR1680024
  4. Deimling K.: Nonlinear Functional Analysis., Springer, New York, , 1985. (1985) MR0787404
  5. Guo D., Lakshmikannthan V.: Nonlinear Problems in Abstract Cones., Academic Press, San Diego, , 1988. (1988) MR0959889
  6. Galewski A., On a certain generalization of the Krasnosielskii theorem, J. Appl. Anal. 1 (2003), 139–147. 

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