Let T be an asymptotically nonexpansive self-mapping of a closed bounded and convex subset of a uniformly convex Banach space which satisfies Opial's condition. It is shown that, under certain assumptions, the sequence given by xn+1 = αnTn(xn) + (1 - αn)xn converges weakly to some fixed point of T. In arbitrary uniformly convex Banach spaces similar results are obtained concerning the strong convergence of (xn) to a fixed point of T, provided T possesses a compact iterate or satisfies a Frum-Ketkov condition of the fourth kind.

In this paper we have established some fixed point theorems in complete and compact metric spaces.

If a certain weakening of convexity holds for the objective and all constraint functions in a nonconvex constrained minimization problem, Hanson showed that the Kuhn-Tucker necessary conditions are sufficient for a minimum. This property is now generalized to a property, called K-invex, of a vector function in relation to a convex cone K. Necessary conditions and sufficient conditions are obtained for a function f to be K-invex. This leads to a new second order sufficient condition for a constrained minimum.

We prove the convergence of an iterative method to a fixed point of a nonexpansive mapping in a uniformly smooth Banach space. We are able to relax one of the control conditions of P.L. Lions (1977).

A family of third-order iterative processes (that includes Chebyshev and Halley's methods) is studied in Banach spaces. Results on convergence and uniqueness of solution are given, as well as error estimates. This study allows us to compare the most famous third-order iterative processes.

After a review of the work of Lawvere and Tierney, it is shown that every topos may be exactly embedded in a product of topoi each with 1 as a generator, and near-exactly embedded in a power of the category of sets. Several metatheorems are then derived. Natural numbers objects are shown to be characterized by exactness properties, which yield the fact that some topoi can not be exactly embedded in powers of the category of sets, indeed that the “arithmetic” arising from a topos dominates the exactness theory. Finally, several, necessarily non-elementary, conditions are shown to imply exact embedding in powers of the category of sets.

This paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the p-Kirchhoff type and where Ω is a bounded smooth domain of ℝN, 1 < p < N, s ≥ p* = (pN)/(N – p) and M and f are continuous functions.

Let R denote a near-ring such that for each x ∈ R, there exists an integer n(x) > 1 for which xn(x) = x. We show that the additive group of R is commutative if 0.x; = 0 for all x ∈ R and every non-trivial homomorphic image R¯ of R contains a non-zero idempotent e commuting multiplicatively with all elements of R¯. As the major consequence, we obtain the result that if R is distributively-generated, then R is a ring – a generalization of a recent theorem of Ligh on boolean near-rings.

In this paper, we first prove a fixed point theorem for a family of multivalued maps defined on product spaces. We then apply our result to prove an equilibrium existence theorem for an abstract economy. We also consider a system of variational inequalities and prove the existence of its solutions by using our fixed point theorem.

A slant immersion is defined as an isometric immersion from a Riemannnian manifold into an almost Hermitian manifold with constant Wirtinger angle. In this article we give some fundamental results concerning slant immersions. Several results on slant surfaces in ℂ2 are also proved.

Suppose that R is a commutative ring with 1 ≠ 0. In this paper, we introduce the concept of 2-absorbing ideal which is a generalisation of prime ideal. A nonzero proper ideal I of R is called a 2-absorbing ideal of R if whenever a, b, c ∈ R and abc ∈ I, then ab ∈ I or ac ∈ I or bc ∈ I. It is shown that a nonzero proper ideal I of R is a 2-absorbing ideal if and only if whenever I1I2I3 ⊆ I for some ideals I1,I2,I3 of R, then I1I2 ⊆ I or I2I3 ⊆ I or I1I3 ⊆ I. It is shown that if I is a 2-absorbing ideal of R, then either Rad(I) is a prime ideal of R or Rad(I) = P1 ⋂ P2 where P1,P2 are the only distinct prime ideals of R that are minimal over I. Rings with the property that every nonzero proper ideal is a 2-absorbing ideal are characterised. All 2-absorbing ideals of valuation domains and Prüfer domains are completely described. It is shown that a Noetherian domain R is a Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M2 for some maximal ideal M of R or M1M2 where M1,M2 are some maximal ideals of R. If RM is Noetherian for each maximal ideal M of R, then it is shown that an integral domain R is an almost Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M2 for some maximal ideal M of R or M1M2 where M1,M2 are some maximal ideals of R.

A new technique, using the contraction mapping theorem, for solving quadratic equations in Banach space is introduced. The results are then applied to solve Chandrasekhar's integral equation and related equations without the usual positivity assumptions.

Many problems lead to the consideration of “algebras”, given by an object A of a category A together with “actions” TkA → A on A of one or more endofunctors of A, subjected to equational axioms. Such problems include those of free monads and free monoids, of cocompleteness in categories of monads and of monoids, of orthogonal subcategories (= generalized sheaf-categories), of categories of continuous functors, and so on; apart from problems involving the algebras for their own sake.

Desirable properties of the category of algebras - existence of free ones, cocompleteness, existence of adjoints to algebraic functors - all follow if this category can be proved reflective in some well-behaved category: for which we choose a certain comma-category T/A

We show that the reflexion exists and is given as the colimit of a simple transfinite sequence, if A is cocomplete and the Tk preserve either colimits or unions of suitably-long chains of subobjects.

The article draws heavily on the work of earlier authors, unifies and simplifies this, and extends it to new problems. Moreover the reflectivity in T/A is stronger than any earlier result, and will be applied in forthcoming articles, in an enriched version, to the study of categories with structure.

It is shown that all solutions of ẍ + 2x3 = p(t) are bounded, the notation indicating that p is periodic. It is not necessary to have a small parameter multiplying p.

The essential step is to show by appeal to Moser's theorem that, under the mapping (of the initial-value plane) which corresponds to the equation, there are invariant simple closed curves. This implies also that there is an uncountable infinity of almost-periodic solutions and, for each positive integer m, an infinity of periodic solutions of least period 2mπ (2π being taken as the least period of p ).

It is suggested that for a large class of equations the same attack would show all solutions of ẍ + g(x) = p(t) bounded. However, in order to show the method clearly, no generalisation is attempted here.

A polyhedral decomposition of a finite trivalent graph G is defined as a set of circuits = {C1, C2, … Cm} with the property that every edge of G occurs exactly twice as an edge of some Ck. The decomposition is called even if every Ck is a simple circuit of even length. If G has a Tait colouring by three colours a, b, c then the (a, b), (b, c) and (c, a) circuits obviously form an even polyhedral decomposition. It is shown that the converse is also true: if G has an even polyhedral decomposition then it also has a Tait colouring. This permits an equivalent formulation of the four colour conjecture (and a much stronger conjecture of Branko Grünbaum) in terms of polyhedral decompositions alone.

A new graph product is introduced, and the characteristic polynomial of a graph so–formed is given as a function of the characteristic polynomials of the factor graphs. A class of trees produced using this product is shown to be characterized by spectral properties.

Matrix quadratic equations have found the most diverse applications. The present article gives a connected account of their theory, and contains some new results and new proofs of known results.

In this paper, we obtain a version of the sliding plane method of Gidas, Ni and Nirenberg which applies to domains with no smoothness condition on the boundary. The method obtains results on the symmetry of positive solutions of boundary value problems for nonlinear elliptic equations. We also show how our techniques apply to some problems on half spaces.

A class of functions, called pre-invex, is defined. These functions are more general than convex functions and when differentiable are invex. Optimality conditions and duality theorems are given for both scalar-valued and vector-valued programs involving pre-invex functions.

For a constrained minimization problem with cone constraints, lagrangean necessary conditions for a minimum are well known, but are subject to certain hypotheses concerning cones. These hypotheses are now substantially weakened, but a counter example shows that they cannot be omitted altogether. The theorem extends to minimization in a partially ordered vector space, and to a weaker kind of critical point (a quasimin) than a local minimum. Such critical points are related to Kuhn-Tucker conditions, assuming a constraint qualification; in certain circumstances, relevant to optimal control, such a critical point must be a minimum. Using these generalized critical points, a theorem analogous to duality is proved, but neither assuming convexity, nor implying weak duality.