A surjectivity result for quasibounded operators

AbstractUsing a degree theory for countably 1-contractive operators, we show a surjectivity theorem for such quasibounded operators. Moreover, the existence of an eigenvalue for these operators is presented

Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems

Let X be a real locally uniformly convex reflexive separable Banach space with locally uniformly convex dual space X∗. Let T:X⊇D(T)→2X∗ be maximal monotone and S:X⊇D(S)→X∗ quasibounded generalized pseudomonotone such that there exists a real reflexive separable Banach space W⊂D(S), dense and continuously embedded in X. Assume, further, that there exists d≥0 such that 〈v∗+Sx,x〉≥-dx2 for all x∈D(T)∩D(S) and v∗∈Tx. New surjectivity results are given for noncoercive, not everywhere defined, and possibly unbounded operators of the type T+S. A partial positive answer for Nirenberg's problem on surjectivity of expansive mapping is provided. Leray-Schauder degree is applied employing the method of elliptic superregularization. A new characterization of linear maximal monotone operator L:X⊇D(L)→X∗ is given as a result of surjectivity of L+S, where S is of type (M) with respect to L. These results improve the corresponding theory for noncoercive and not everywhere defined operators of pseudomonotone type. In the last section, an example is provided addressing existence of weak solution in X=Lp(0,T;W01,p(Ω)) of a nonlinear parabolic problem of the type ut-∑i=1n(∂/∂xi)ai(x,t,u,∇u)=f(x,t),  (x,t)∈Q; u(x,t)=0,  (x,t)∈∂Ω×(0,T); u(x,0)=0,  x∈Ω, where p>1, Ω is a nonempty, bounded, and open subset of RN, ai:Ω×(0,T)×R×RN→R  (i=1,2,…,n) satisfies certain growth conditions, and f∈Lp′(Q), Q=Ω×(0,T), and p′ is the conjugate exponent of p

A New Spectrum for Nonlinear Operators in Banach Spaces

Given any continuous self-map f of a Banach space E over K (where K is R or C) and given any point p of E, we define a subset sigma(f,p) of K, called spectrum of f at p, which coincides with the usual spectrum sigma(f) of f in the linear case. More generally, we show that sigma(f,p) is always closed and, when f is C^1, coincides with the spectrum sigma(f'(p)) of the Frechet derivative of f at p. Some applications to bifurcation theory are given and some peculiar examples of spectra are provided.Comment: 23 pages, 3 figure

Maximality Theorems on the Sum of Two Maximal Monotone Operators and Application to Variational Inequality Problems

Let X be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space X⁎. Let T:X⊇D(T)→2X⁎ and A:X⊇D(A)→2X⁎ be maximal monotone operators. The maximality of the sum of two maximal monotone operators has been an open problem for many years. In this paper, new maximality theorems are proved for T+A under weaker sufficient conditions. These theorems improved the well-known maximality results of Rockafellar who used condition D(T)∘∩D(A)≠∅ and Browder and Hess who used the quasiboundedness of T and condition 0∈D(T)∩D(A). In particular, the maximality of T+∂ϕ is proved provided that D(T)∘∩D(ϕ)≠∅, where ϕ:X→(-∞,∞] is a proper, convex, and lower semicontinuous function. Consequently, an existence theorem is proved addressing solvability of evolution type variational inequality problem for pseudomonotone perturbation of maximal monotone operator

Generalized iterative methods and nonlinear functional equations.

Perturbations of nonlinear operators are also investigated. If F'(x)(B(0; 1)) (R-HOOK) B(0; c((VBAR)(VBAR)x(VBAR)(VBAR))) and if F is perturbed by a nonlinear operator G satisfying a boundedness condition, then F + G is an open mapping from X onto Y. The case where both F and G are Gateaux differentiable operators satisfying various coercive conditions again yields surjectivity results for the sum F + G. These proofs rely on the existence of contractor inequalities derived from the hypotheses. Finally, if G is a compact operator and I - F is compact, then F + G is surjective; the proof uses methods of algebraic topology.Let X and Y be Banach spaces, P be a Gateaux differentiable mapping from X to Y and c : {0, (INFIN)) (--->) (0, (INFIN)) be a continuous nonincreasing function for which (INT)('(INFIN)) c(u)du = (INFIN). If P'(x)(B(0; 1)) contains B(0; c((VBAR)(VBAR)x(VBAR)(VBAR))) for each x (epsilon) X, then P is an open mapping of X onto Y. If the differentiability assumption on P is removed and instead P is both open and locally expansive, then P(X) = Y. If A is a continuous mapping from X to X satisfying for each x (epsilon) X, (GREATERTHEQ) c(max{(VBAR)(VBAR)x(VBAR)(VBAR), (VBAR)(VBAR)y(VBAR)(VBAR)}) (VBAR)(VBAR)x - y(VBAR)(VBAR)('2) for some j (epsilon) J(x -y), then A is a homeomorphism of X onto X. The main technique used in establishing these results is a new fixed point theorem which includes Ekland's Theorem as a special case

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, coincidence-degree theory for semilinear operators, L-k-set contractions, A-proper 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 Sturm-Liouville 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 m-point 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 A-proper 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]