Caratheodorys function

Boundary value problems model many phenomena in applied sciences and engin- eering, that’s why their study is an important research area despite it’s di¢ cult as long as there is no general method to apply, and then to discuss the qualitative and quantitative properties of solutions for a given boundary value problem, di⁄er- ent methods are used such the upper and lower solutions method, Mawhin theory, numerical methods… [25],The objective of this thesis is the study of the existence of solutions of boundary value problems generated by a class of second order nonlinear di⁄erential equations with boundary conditions of integral and multipoint type, by applying the degree of coincidence of Mawhin.The Mawhin theory permits the use of an approach of topological degree type to problems which can be written as an abstract operator equation of the form Lx = N x, where L is a linear noninvertible operator and N is a nonlinear operator acting on a given Banach space.

In 1972, Mawhin has developed a method to solve this equation in his famous paper ”Topological degree and boundary value problems for nonlinear di⁄erential equations” [40], he assumed that L is a Fredholm operator of index zero. Hence he has developed a new theory of topological degree known as the degree of coincidenceA boundary value problem is said to be at resonance if the corresponding linear homogenous problem has nontrivial solution, otherwise it’Many authors studied ordinary boundary value problems at resonance using Mawhin coincidence degree theory, we can cite Feng and Webb [11], Guezane-Lakoud and Frioui [14], Mahin and Ward [42], Infante [26], and the references [4,10,12,15,18- 20,30-34,40,42,46].Let us consider the following second order di⁄erential equation:In [24], Gupta, Ntouyas and Tsamatos used the Leray-Schauder continuation theorem to prove the existence of solutions of problem (1.1),(1.2) in the case 6= 1Later Feng and Webb in [11], considered the problems (1.1),(1.2) and (1.1),(1.3) in the resonance case that is when = 1 for problem (1.1),(1.2) and = 1 for problem (1.1),(1.3). The authors proved by using Mawhin coincidence degree theory, that these problems have at least one solution.

Recently in [14], by the help of Mawhin coincidence degree, Guezane-Lakoud and Frioui, proved the existence of solutions for a third order multipoint boundary value problem at resonance of the formOrdinary di⁄erential equations with multipoint boundary conditions occur natur- ally arise in some applications such in population dynamics model, in semiconductor problems, thermal conduction problems, hydrodynamic problems…, see [7, 8, 27, 50]. For example, if a dynamic system has m degrees of freedom, then we have exactly m states observed at m di⁄erent times, and consequently we obtain an m-point bound- ary value problem. The discretization of some boundary value problems for partial di⁄erential equations on irregular domains with the line method leads to multipoint boundary value problems.

The aim of this thesis is the study of a resonance boundary value problem gener-ated by a second order nonlinear di⁄erential equation with multipoint and integral2: We apply the Mawhin coincidence degree to prove the existence of at least one solution in a Banach space that we will de…ne later.Di⁄erential equations with nonlocal conditions, especially integral conditions, plays an important role in both theory and applications. The study of these problems is motivated by various applications, including thermoelasticity, chemical engineer- ing, plasma physics…..Let us give a brief outline of each chapter of the thesis.The …rst chapter presents a review of some …xed point theorems, notably theBanach contraction principle, Leray-Schauder’s nonlinear alternative, notion of ho- motopy, Fredholm’s operators of zero index, the concept of the topological degree and its properties is discussed; two degrees are de…ned: the degree of Brouwer in …nite dimension then the degree of Leray-Schauder in in…nite dimension. We cite the theorem of Mawhin coincidence degree and its proof.

In the second chapter, we collect interesting results for some classes of second- order boundary value problems at resonance. For this purpose we will summarize the basic results in the literature and present the main ideas of some research on these problems.The third chapter is devoted to the study of a second order boundary value problem at resonance. In fact we propose to establish the existence of the solution of a di⁄erential equation with multipoint conditions of integral type. The proofs are based on Mahwin’s theory of coincidence. The results of this chapter are published:R. Khaldi, M. Kouidri, Solvability of multipoint value problems with integral condition at resonance, International Journal of Analysis and Applications, Vol 16,In the second chapter, we collect interesting results for some classes of second- order boundary value problems at resonance. For this purpose we will summarize the basic results in the literature and present the main ideas of some research on these problems.