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Positive solutions for a class of non-autonomous second order difference equations via a new functional fixed point theorem

Lydia Bouchal a, Karima Mebarki a, Svetlin Georgiev Georgiev b (2022)

Archivum Mathematicum

In this paper, by using recent results on fixed point index, we develop a new fixed point theorem of functional type for the sum of two operators T + S where I - T is Lipschitz invertible and S a k -set contraction. This fixed point theorem is then used to establish a new result on the existence of positive solutions to a non-autonomous second order difference equation.

Positive solutions for one-dimensional singular p-Laplacian boundary value problems

Huijuan Song, Jingxue Yin, Rui Huang (2012)

Annales Polonici Mathematici

We consider the existence of positive solutions of the equation 1 / λ ( t ) ( λ ( t ) φ p ( x ' ( t ) ) ) ' + μ f ( t , x ( t ) , x ' ( t ) ) = 0 , where φ p ( s ) = | s | p - 2 s , p > 1, subject to some singular Sturm-Liouville boundary conditions. Using the Krasnosel’skiĭ fixed point theorem for operators on cones, we prove the existence of positive solutions under some structure conditions.

Positive solutions for systems of generalized three-point nonlinear boundary value problems

Johnny Henderson, Sotiris K. Ntouyas, Ioannis K. Purnaras (2008)

Commentationes Mathematicae Universitatis Carolinae

Values of λ are determined for which there exist positive solutions of the system of three-point boundary value problems, u ' ' + λ a ( t ) f ( v ) = 0 , v ' ' + λ b ( t ) g ( u ) = 0 , for 0 < t < 1 , and satisfying, u ( 0 ) = β u ( η ) , u ( 1 ) = α u ( η ) , v ( 0 ) = β v ( η ) , v ( 1 ) = α v ( η ) . A Guo-Krasnosel’skii fixed point theorem is applied.

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