This paper develops a fixed point version of the well-known Nehari manifold method from critical point theory. The main result is formulated for systems of operator equations, relying on the fixed point theorems of Schauder and Schaefer. The framework also allows for potential extensions combining our Nehari type approach with other fixed point principles. To demonstrate the applicability of the method, an example involving a system of nonlinear integral equations is provided.

In the paper, results on the existence of critical points in annular subsets of a cone are obtained with the additional goal of obtaining multiplicity results. Compared to other approaches in the literature based on the use of Krasnoselskii’s compression-extension theorem or topological index methods, our approach uses the Nehari manifold technique in a surprising combination with the cone version of Birkhoff-Kellogg’s invariant-direction theorem. This yields a simpler alternative to traditional methods involving deformation arguments or Ekeland’s variational principle. The new method is illustrated on a boundary value problem for p-Laplace equations, and we believe that it will be useful for proving the existence, localization, and multiplicity of solutions for other classes of problems with variational structure.

We analyze a control problem for a general class of coupled systems of stationary Navier-Stokes type equations in the incompressible case, with homogeneous Dirichlet condition on the boundary of a bounded domain in RN, N <= 3, and non-homogeneous terms of reaction type. Such a control problem may describe the flow of a viscous incompressible fluid in multidisperse porous media with a controllability condition imposed on the coefficients of the coupled systems and expressed by means of a continuous functional depending on the velocities and pressures. The controllability conditions are not necessarily given by equalities, but more generally are formulated by inclusions. A lower and upper solution technique is used for the exact and approximate solvability of the control problem, which requires the existence, uniqueness and continuous dependence of the solution on the coefficients.

In this paper, we present a control problem related to a semilinear differential equation with a moving singularity, i.e., the singular point depends on a parameter. The particularity of the controllability condition resides in the fact that it depends on the singular point, which in turn depends on the control variable. We provide sufficient conditions to ensure that the functional determining the control is continuous over the entire domain of the parameter. Lower and upper solutions techniques combined with a bisection algorithm is used to prove the controllability of the equation and to approximate the control. An example is given together with some numerical simulations. The results naturally extend to fractional differential equations.

The aim of this paper is to extend the Nehari manifold method from the variational setting to the nonvariational framework of fixed point equations. This is achieved by constructing a radial energy functional that generalizes the standard one from the variational case. Furthermore, the solutions obtained through our method are localized in conical annular sets, which leads to the existence of multiple solutions. The abstract results are illustrated by two representative applications.

In this paper, we extend the concept of b-metric spaces to the vectorial case, where the distance is vector-valued, and the constant in the triangle inequality axiom is replaced by a matrix. For such spaces, we establish results analogous to those in the b-metric setting: fixed-point theorems, stability results, and a variant of Ekeland's variational principle. As a consequence, we also derive a variant of Caristi's fixed-point theorem

This paper investigates the existence and uniqueness of solutions for Kirchhoff type parabolic equations with reaction terms in bounded domain, subject to Cauchy-Dirichlet boundary conditions. We target two types of nonlocal diffusion coefficient: diffusion coefficient which is nonlocal only in space, most often considered in the literature, and diffusion coefficient which is nonlocal both in space and time, representing a memory term of the model. Under suitable Lipschitz continuity assumptions on the reaction term and on the nonlocal diffusion coefficient, we establish the existence of a unique solution using a fixed point approach based on Banach’s contraction principle. Under weaker conditions, we also prove the existence of solutions using compactness arguments and Darbo’s fixed point theorem. Our analysis relies on specific function spaces and properties of the solution operator associated with the classical parabolic equation. We provide concrete examples of nonlocal diffusion coefficients with physical meaning, highlighting the applicability of the results.

In this paper, we introduce and discuss the concept of a mutual control problem. Our approach relies on a vector fixed-point approach based on the fixed-point theorems of Perov, Schauder, and Avramescu. In our analysis, we employ a novel technique utilizing Bielecki equivalent norms.

We analyze a general class of coupled systems of stationary Navier–Stokes type equations with variable coefficients and non-homogeneous terms of reaction type in the incompressible case. Existence of solutions satisfying the homogeneous Dirichlet condition in a bounded domain in [Formula: see text], [Formula: see text], and localization results for the corresponding kinetic energy and enstrophy are obtained by using a variational approach and the fixed point index theory.

The paper deals with existence, localization and multiplicity of radial positive solutions in the annulus or the ball, for the Neumann problem involving a general ϕ -Laplace operator. Our results apply in particular to the classical Laplacian and to the mean curvature operators in the Euclidean and Minkowski spaces. Numerical experiments with the MATLAB object-oriented package Chebfun are performed to obtain numerical solutions for some concrete equations.

The starting point of this paper is the construction of a general family (L_n)_n≥ 1 of positive linear operators of discrete type. Considering (L_n^k)_k≥ 1 the sequence of iterates of one of such operators, L_n , our goal is to find an expression of the upper edge of the error ‖ L_n^kf-f^*‖ , f∈ C[0,1] , where f^* is the fixed point of L_n. The estimate makes use of the error formula for the sequence of successive approximations in Banach’s fixed point theorem and the error of approximation of the operator L_n. Examples of special operators are inserted. Some extensions to multidimensional approximation operators are also given.

In this paper we examine a mutual control problem for systems of two abstract evolution equations subject to a proportionality final condition. Related observability and semi-observability problems are discussed. The analysis employs a vector fixed-point approach, using matrices rather than constants, and applies the technique of Bielecki equivalent norms.

In this paper, the second-order differential equations and systems of Kolmogorov type are defined. With reference to population dynamics models, unlike the first-order equations which give the expression of the per capita rate, in the case of the second-order equations, the law of change of the per capita rate is given. Several control problems with fixed final time and fixed final state, with additive and multiplicative control, are studied. Their controllability is proved with fixed-point methods, the theorems of Banach, Schauder, Krasnoselskii, Avramescu and Perov.

The paper presents an abstract theory regarding the problems with semilinear operator equations involving iterates of a strongly monotone symmetric linear operator. We obtain existence and localization results of positive solutions for such problems using Krasnosel'skii's technique and abstract Harnack inequality. In particular, we obtain results for problems with semilinear poly-Laplace operators.

The paper deals with fractional optimization problems where the objective function (ratio of two functions) is defined on a Cartesian product of two real normed spaces X and Y. Within this framework, we are interested to determine the so-called partial minimizers, i.e. points in $ X \times Y $ XxY with the property that any of its variables minimizes the objective function, restricted to this variable, with respect to the other one. While any global minimizer is obviously a partial minimizer, the reverse implication holds true only under additional assumptions (e.g. separability properties of the involved functions). By exploiting the particularities of the objective function, we deliver a Dinkelbach type algorithm for computing partial minimizers of fractional optimization problems. Further assumptions on the involved spaces and functions, such as Lipschitz-type continuity, partial Frechet differentiability, and coercivity, enable us to establish the convergence of our algorithm to a partial minimizer.

The paper presents a vector approach to control problems for systems of equations. The method is described in the case of Kolmogorov systems which arise frequently in the dynamics of populations. Three types of problems are discussed: problems with control of both per capita growth rates, problems with control parameters acting on the growth rates, and problems which combine the first two types. The controllability is obtained via a vector approach based on the Perov fixed point theorem and matrices which are convergent to zero. Four concrete illustrative examples are added.

We are concerned with existence, localization and multiplicity of positive radial solutions to Dirichlet problems with ϕ-Laplacians in a ball, in both scalar and system cases. Our approach essentially relies on fixed point index computations and a main feature is that it avoids any Harnack type inequality. Applications to some problems involving operators with Uhlenbeck structure are discussed.

this paper we obtain nonlinear alternatives of Leray-Schauder and Mo center dot nch type for nonself vector-valued operators, under hybrid conditions of Perov contraction and compactness. Thus, we give vector versions of the theorems of Krasnosel'skii, Avramescu, Burton-Kirk and Gao-Li-Zhang. An application is given to a boundary value problem for a system of second order differential equations in which some of the equations are implicit.

Abstract We establish compression-expansion type fixed point theorems for systems of operator inclusions with decomposable multivalued maps. The approach is vectorial allowing to localize individually the components of solutions and to obtain multiple solutions with multiplicity not necessarily concerned with all components of the solution. A general scheme of applicability of the theory is elaborated based on Harnack type inequalities and illustrated on systems of differential inclusions with one-dimentional ϕ-Laplacian.