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On a fixed point theorem for weakly sequentially continuous mapping

Ireneusz Kubiaczyk (1995)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Let E be a metrizable locally convex topological vector space x ∈ E, and let D be a closed convex subset of E such that x ∈ D. In this paper we prove that the weakly sequentially continuous mapping F: D ∪ D which satisfies V̅ = c̅o̅n̅v̅({x} ∪ F(V))⇒ V is relatively weakly compact, has a fixed point. Employing the above results we prove the existence theorem for the Cauchy problem x'(t) = f(t,x(t)), x(0) = x₀. As compared with the previous...

On a Theorem of Mierczyński

Gerd Herzog (1998)

Colloquium Mathematicae

We prove that the initial value problem x’(t) = f(t,x(t)), x ( 0 ) = x 1 is uniquely solvable in certain ordered Banach spaces if f is quasimonotone increasing with respect to x and f satisfies a one-sided Lipschitz condition with respect to a certain convex functional.

On an elasto-dynamic evolution equation with non dead load and friction

Oanh Chau (2006)

Applications of Mathematics

In this paper, we are interested in the dynamic evolution of an elastic body, acted by resistance forces depending also on the displacements. We put the mechanical problem into an abstract functional framework, involving a second order nonlinear evolution equation with initial conditions. After specifying convenient hypotheses on the data, we prove an existence and uniqueness result. The proof is based on Faedo-Galerkin method.

On application of Rothe's fixed point theorem to study the controllability of fractional semilinear systems with delays

Beata Sikora (2019)

Kybernetika

The paper presents finite-dimensional dynamical control systems described by semilinear fractional-order state equations with multiple delays in the control and nonlinear function f . The relative controllability of the presented semilinear system is discussed. Rothe’s fixed point theorem is applied to study the controllability of the fractional-order semilinear system. A control that steers the semilinear system from an initial complete state to a final state at time t > 0 is presented. A numerical...

On BVPs in l∞(A).

Gerd Herzog, Roland Lemmert (2005)

Extracta Mathematicae

We prove the existence of extremal solutions of Dirichlet boundary value problems for u''a + fa(t,u,u'a) = 0 in l∞(A) between a generalized pair of upper and lower functions with respect to the coordinatewise ordering, and for f quasimonotone increasing in its second variable.

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 theorems for semilinear equations and applications

Fang Zhang, Feng Wang (2013)

Annales Polonici Mathematici

Existence results for semilinear operator equations without the assumption of normal cones are obtained by the properties of a fixed point index for A-proper semilinear operators established by Cremins. As an application, the existence of positive solutions for a second order m-point boundary value problem at resonance is considered.

On fourth-order boundary-value problems

Myelkebir Aitalioubrahim (2010)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

We show the existence of solutions to a boundary-value problem for fourth-order differential inclusions in a Banach space, under Lipschitz’s contractive conditions, Carathéodory conditions and lower semicontinuity conditions.

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