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On discontinuous implicit differential equations in ordered Banach spaces with discontinuous implicit boundary conditions

S. Carl, S. Heikkilä (1999)

Annales Polonici Mathematici

We consider the existence of extremal solutions to second order discontinuous implicit ordinary differential equations with discontinuous implicit boundary conditions in ordered Banach spaces. We also study the dependence of these solutions on the data, and cases when the extremal solutions are obtained as limits of successive approximations. Examples are given to demonstrate the applicability of the method developed in this paper.

On existence of extremal solutions of nonlinear functional integral equations in Banach algebras.

Dhage, B.C. (2004)

Journal of Applied Mathematics and Stochastic Analysis

On fixed point theorems of mixed monotone operators.

Du, Xinsheng, Zhao, Zengqin (2011)

Fixed Point Theory and Applications [electronic only]

On Krasnoselskii's cone fixed point theorem.

Kwong, Man Kam (2008)

Fixed Point Theory and Applications [electronic only]

On monotone minimal cuscos

Karel Pastor, Dušan Bednařík (2001)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

On non-absolute functional Volterra integral equations and impulsive differential equations in ordered Banach spaces.

Heikkilä, Seppo, Seikkala, Seppo (2008)

Electronic Journal of Differential Equations (EJDE) [electronic only]

On Pseudo-monotone Operators and Nonlinear Noncoercive Variational Problems on Unbounded Domains.

Rüdiger Landes, Vesa Mustonen (1980)

Mathematische Annalen

On solvability of mixed monotone operator equations with applications to mixed quasimonotone differential systems involving discontinuities.

Heikkilä, S., Kumpulainen, M., Lakshmikantham, V. (1992)

Journal of Applied Mathematics and Stochastic Analysis

On the Approximative Roots of Maximal Monotone Mappings

L. D. Popov (1996)

The Yugoslav Journal of Operations Research

On the iterative process of solving nonlinear operator equations in PTL-spaces

R. Sen (1978)

Matematički Vesnik

On the monotone convergence of implicit Newton-like methods.

Argyros, Ioannis K. (1996)

Southwest Journal of Pure and Applied Mathematics [electronic only]

On the Rockafellar theorem for $Φ^{γ (\cdot, \cdot)}$ -monotone multifunctions

S. Rolewicz (2006)

Studia Mathematica

Let X be an arbitrary set, and γ: X × X → ℝ any function. Let Φ be a family of real-valued functions defined on X. Let $Γ : X \to 2^{Φ}$ be a cyclic $Φ^{γ (\cdot, \cdot)}$ -monotone multifunction with non-empty values. It is shown that the following generalization of the Rockafellar theorem holds. There is a function f: X → ℝ such that Γ is contained in the $Φ^{γ (\cdot, \cdot)}$ -subdifferential of f, $Γ (x) \subset \partial_{Φ}^{γ (\cdot, \cdot)} {f |}_{x}$ .

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