Afif Ben Amar

Some Fixed Point Theorems and Application to Biological Model

In this paper, we establish some results regarding the existence of solution on L 1 spaces to a nonlinear boundary value problem originally proposed by Rotenberg (J. Theo. Biol. 1983; 103:181–199) to model the growth of cell population. Our strategy consists in establishing new variants of fixed point theorems in general Banach spaces. These topological results can be used to resolve some open problems posed by Latrach and Jeribi (Nonlinear Anal. T.M.A. 1999; 36:843–862).

Fixed point theorems for block operator matrix and an application to a structured problem under boundary conditions of Rotenberg’s model type

In this manuscript, we introduce and study the existence of solutions for a coupled system of differential equations under abstract boundary conditions of Rotenberg’s model type, this last arises in growing cell populations. The entries of block operator matrix associated to this system are nonlinear and act on the Banach space Xp:= Lp([0, 1] × [a, b]; dµ dv), where 0 ≤ a < b < ∞; 1 < p < ∞.

Nonlinear Leray-Schauder alternatives and application to nonlinear problem arising in the theory of growing cell population

Motivated by a mathematical model of an age structured proliferating cell population, we state some new variants of Leray-Schauder type fixed point theorems for (ws)-compact operators. Further, we apply our results to establish some new existence and locality principles for nonlinear boundary value problem arising in the theory of growing cell population in L1-setting. Besides, a topological structure of the set of solutions is provided.

Nonlinear Leray-Schauder Alternatives for Decomposable Operators in Dunford-Pettis Spaces and Application to Nonlinear Eigenvalue Problems

We present some new variants of Leray–Schauder type fixed point theorems and eigenvalue results for decomposable single-valued nonlinear weakly compact operators in Dunford-Pettis spaces.

Fixed-Point Theory on a Frechet Topological Vector Space

We establish some versions of fixed-point theorem in a Frechet topological vector space . The main result is that every map (where is a continuous map and is a continuous linear weakly compact operator) from a closed convex subset of a Frechet topological vector space having the Dunford-Pettis property into itself has fixed-point. Based on this result, we present two versions of the Krasnoselskii fixed-point theorem. Our first result extend the well-known Krasnoselskiis fixed-point theorem for U-contractions and weakly compact mappings, while the second one, by assuming that the family where and a compact is nonlinear equicontractive, we give a fixed-point theorem for the operator of the form .

Fixed point theorems for the sum of two operators on unbounded convex sets and an application

In this paper, we establish new fixed point results for the sum of two operators A and B, where the operator A is assumed to be weakly compact and (ws)-compact, while B is a weakly condensing and expansive operator defined on unbounded domains under different boundary conditions as well as other additional assumptions. In addition, we get new generalized forms of the Krasnosel’skii fixed point theorem in a Banach space by using the concept of measure of weak noncompactness of De Blasi. Later on, we give an application to solve a nonlinear Hammerstein integral equation in L1-space.

Approximate Fixed Point Theorems in Banach Spaces

Let \(\Omega \) be a nonempty convex subset of a topological vector space X. An approximate fixed point sequence for a map \(F: \Omega \longrightarrow \overline{\Omega }\) is a sequence \(\{x_{n}\}_{n} \in \Omega \) so that \(x_{n} - F(x_{n})\longrightarrow \theta\). Similarly, we can define approximate fixed point nets for F. Let us mention that F has an approximate fixed point net if and only if

Nonlinear Eigenvalue Problems in Dunford–Pettis Spaces

\displaystyle{\theta \in \overline{\{x - F(x): x \in \Omega \}}.}