Title:

Semilinear problems and spectral theory

The subject of this thesis is that part of nonlinear functional analysis which deals with the solvability of semilinear differential equations and the study of spectral theory for nonlinear operators. Chapter one is an introduction to the concepts used through the thesis, including measures of non compactness, (p, A:)epi mappings and related properties, Fredholm operators of index zero, coincidencedegree theory for semilinear operators, Lkset contractions, Aproper operators and so on. The work in chapter two is based on the study of [16]. In [16], a spectrum for nonlinear operators was introduced by Furi, Martelli and Vignoli. Their spectrum need not contain the eigenvalues [9]. We establish a new spectral theory for nonlinear operators which contains all eigenvalues as in the linear case. We compare the new spectrum with that of [16] and the one of [48] and prove that all three spectra may be empty, which answers one of the open questions in [48]. Some applications of the new theory, including the generalization of three well known theorems, the study of the solvability of a Cauchy problem and a Hammerstein integral equation, are obtained in the last section of this chapter. In chapter three, by generalizing the concept of (0, k)epi mappings to that of (0, L, k) epi mappings, we introduce the definition of spectrum for semilinear operators (L,N), where L is a Fredholm operator of index zero, N is a. nonlinear operator. When L is the identity map, this spectrum reduces to the spectrum defined in Chapter 2. We prove that it has similar properties with the spectrum of nonlinear operators. Also in the last section, by using this theory, we discuss the solvability of semilinear operator equations and extend some existence results. In chapter four, we obtain some surjectivity results on the mapping lambdaT  S, where T is a homeomorphism and S is a nonlinear map. We generalize one of the results of [12] in finite dimensional space to infinite dimensional space, which solves the open question of [12]. We also apply our theorems to the study of a nonlinear SturmLiouville problem on the half line following the work by Toland [66] and to prove the existence of a solution for a second order differential equations which was studied in [29]. Much of the work in this Chapter is joint work with J.R.L. Webb and has been published in [21]. Chapter five is related to some recent work by Gupta, Ntouyas, Tsamatos and Lakshmikantham [24][30]. They proved existence results for mpoint boundary value problems for second order ordinary differential equations under nonresonance assumptions and they also assume that the nonlinear part has a linear growth. We obtain results for these boundary value problems in the resonance case. Moreover, our assumptions allow the nonlinear part to have nonlinear growth. Some examples show that there exist equations to which our theorems can be used but the previous results do not apply. Much of the work in this Chapter is joint work with J.R.L. Webb and part of this chapter will be published in [18], [22], [23]. In chapter six, we study second order ordinary differential equations subject to Dirich let, Neumann, periodic and antiperiodic boundary conditions. We make use of an abstract continuation type theorem [56], [57] for semilinear equations involving Aproper mappings to obtain approximation solvability results for these boundary value problems. The results in this chapter generalize the results of [60], [61]. Also we give examples to show that our theorems permit the treatment of equations to which the results of [4], [32], [57] can not be used. Part of this chapter has been submitted for publication, [19].
