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We analyze the implicit Euler discretization for a class of convex linear-quadratic optimal control problems with control appearing linearly. Constraints are defined by lower and upper bounds for the controls, and the cost functional may depend on a regularization parameterźź. Without any structural assumption on the optimal control we prove convergence of orderź1 w.r.t. the mesh size for the discrete optimal values. Under the additional assumption that the optimal control is of bang-bang type and the switching function satisfies a growth condition around their zeros we show that the solutions are calm functions of perturbation and regularization parameters. By applying this result to the implicit Euler discretization we improve existing error estimates for discretizations based on the explicit Euler method. Numerical experiments confirm the theoretical findings and demonstrate the usefulness of implicit methods and regularization in case of bang-bang controls.

In this paper, we consider the semilocal convergence on a class of improved Chebyshev-Halley type methods for solving F ( x ) = 0 , where F: ? ? X ? Y is a nonlinear operator, X and Y are two Banach spaces, ? is a non-empty open convex subset in X. To solve the problems that F?'''(x) is unbounded in ? and it can not satisfy the whole Lipschitz or H o ? lder continuity, ?F?'''(x)? ? N is replaced by ? F ' ' ' ( x 0 ) ? ? N ? , for all x ? ?, where N , N ? ? 0 , x0 is an initial point. Moreover, F?'''(x) is assumed to be local H o ? lder continuous. So the convergence conditions are relaxed. We prove an existence-uniqueness theorem for the solution, which shows that the R-order of these methods is at least 5 + q , where q ? (0, 1. Especially, when F?'''(x) is local Lipschitz continuous, the R-order will become six.

This paper is concerned with impulsive fractional differential inclusions with sectorial operators, nonlocal conditions and infinite delay in Banach spaces. Using tools involving the measure of noncompactness and multi-valued fixed point theory, we derive some existence results of PC-mild solutions when the multivalued map F is convex as well as nonconvex. Furthermore, the compactness of the set of solutions and continuous dependence results are also obtained. As an application, an impulsive neutral partial functional differential equation is also investigated at last.

This paper proposes a class of one parameter conjugate gradient methods, which can be regarded as some kinds of convex combinations of some modified form of PRP and HS methods. The scalar βk has the form of ? k ? k - 1 µ k . The convergence of the given methods is analyzed by some unified tools which show the global convergence of the proposed methods. Numerical experiments with the CUTE collections show that the proposed methods are promising.

This paper investigates existence of PC-mild solutions of impulsive fractional differential inclusions with nonlocal conditions when the linear part is a fractional sectorial operators like in Bajlekova (2001) 1 on Banach spaces. We derive two existence results of PC-mild solutions when the values of the semilinear term F is convex as well as another existence result when its values are nonconvex. Further, the compactness of the set of solutions is characterized.

We present a semi-local convergence analysis of the Gauss-Newton method for solving convex composite optimization problems in Riemannian manifolds using the notion of quasi-regularity for an initial point. Using a combination the L-average Lipszhitz condition and the center L 0 -average Lipschitz condition we introduce majorizing sequences for the Gauss-Newton method that are more precise than in earlier studies. Consequently, our semi-local convergence analysis for the Gauss-Newton method has the following advantages under the same computational cost: weaker sufficient convergence conditions; more precise estimates on the distances involved and an at least as precise information on the location of the solution.

Zhao (2001) introduced a logarithmic barrier interior point method for solving two-stage stochastic linear programs with recourse using Benders' decomposition. This decomposition algorithm and its analysis have been extended by Alzalg and Ariyawansa (2014) for solving two-stage stochastic programs with recourse over all symmetric cones. In this paper, we utilize their work to derive a logarithmic barrier decomposition-based interior point algorithm for two-stage stochastic convex quadratic second-order cone programs with recourse.

Univariate symmetrization technique has many good properties. In this paper, we adopt the high-dimensional viewpoint, and propose a new symmetrization procedure in arbitrary (convex) polytopes of R^d with central symmetry. Moreover, the obtained results are used to extend to the arbitrary centrally symmetric polytopes the well-known Hermite-Hadamard inequality for convex functions.

In this paper, we establish convergence theorems for a general algorithm of an asymptotically nonexpansive map in a uniformly convex hyperbolic space. Our results generalize simultaneously the approximation results of Rhoades (1994) [18], Suantai (2005) [20] and Xu and Noor (2002) [26] on a nonlinear domain. Our results are refinements and generalizations of the corresponding ones in uniformly convex Banach spaces and CAT ( 0 ) spaces.

In this paper, we study harmonic mappings by using the shear construction, introduced by Clunie and Sheil-Small in 1984. We consider two classes of conformal mappings, each of which maps the unit disk D univalently onto a domain which is convex in the horizontal direction, and shear these mappings with suitable dilatations \omega. Mappings of the first class map the unit disk D onto four-slit domains and mappings of the second class take D onto regular n-gons. In addition, we discuss the minimal surfaces associated with such harmonic mappings. Furthermore, illustrations of mappings and associated minimal surfaces are given by using Mathematica., Comment: 17 pages, 6 figures, 2 tables

In this paper we perform the analysis of the following planar Newtonian restricted (n þ 1)body problem: the first n particles have equal mass and are known as primaries. They are subject to their mutual gravitational attraction and are on the vertices of a regular polygon that rotates with an uniform angular velocity x. The remaining particle has infinitesimal mass and moves between the primaries. We limit our inquiry to energy levels that allow us to break down the motion of the infinitesimal in an appropriate family of Poincare maps. Due to the special features of the problem we can deal with the possible collisions and their numerical complexities working in Hill regions, doing several symplectic changes of coordinates and using regularization techniques similar to Birkhoff’s. The method involves several symplectic changes of coordinates to get a workable form. We show several numerical simulations of the Poincare maps to display the dynamics of the problem. This paper deals with the motion in the plane of a infinitesimal particle subject to the gravitational attraction of n particles, called the primaries, of mass m ¼ 1. The primaries are disposed in the vertices of a regular polygon, it rotates rigidly around their center of mass with a constant angular velocity x. Let us observe that the energy of the infinitesimal particle is conserved and we will focus in certain negative energy levels which, among other things, imply the boundedness of the motion. The existence of singularities due to binary collisions of the infinitesimal mass with one of the primaries is a main difficulty and will play a key role in our study. In the other hand it is enough to analyze the motion of the infinitesimal particle in a region between two chosen primaries due to the 2p=n-symmetry of the problem. Our procedure starts with the election of two primaries. After a series of Hamiltonian transformations we obtain a Hamiltonian where it is possible to use the Birkhoff simultaneous regularization of the singularities to get a region without singularities. Now we choose several convenient transversal sections. The associated Poincare maps carry all the information of our dynamical system. A related problem to the (n þ 1)-body problem is the n-center problem, that is, the study of the motion of a infinitesimal particle moving under the gravitational effect of n fixed bodies, called the centers of the problem. If the centers are at the

In this paper, we propose a splitting positive definite mixed finite element method (MFEM) for the approximation of convex optimal control problem governed by elliptic equations with control constraints. By selecting the variation functional properly, the presented procedure can be split into two independent, symmetric and positive definite weak formula for the unknown state variable y and for the unknown flux variable @s. It then follows from the first order necessary and sufficient optimality condition, we deduce another two corresponding adjoint state equations for z and w, which are also independent, symmetric and positive definite. Also, a variational inequality for the control variable u is involved. Convergence analysis shows that the method yields the approximate solutions with optimal accuracy in L^2(@W"U)-norm for the control u,L^2(@W)-norm for the original state y and adjoint state z,H(div;@W)-norm for the flux state @s and adjoint state w without requiring the LBB consistency condition. Finally, some numerical examples are presented to confirm our theoretical results.

In this paper, we study the combined effect of concave and convex nonlinearities on the number of solutions for an indefinite semilinear elliptic system ( E λ , μ ) involving critical exponents and sign-changing weight functions. Using Nehari manifold, the system is proved to have at least two nontrivial nonnegative solutions when the pair of the parameters ( λ , μ ) belongs to a certain subset of R 2 .

In the present paper, by using the Srivastava–Attiya operator, we define two new subclasses of k -fold symmetric analytic functions. Some interesting properties of these subclasses such as integral representations, extreme points, close convex hulls and subordinations are obtained, which generalize and refine some previous results.

Several inequalities for differentiable convex, wright-convex and quasi-convex mapping are obtained respectively that are connected with the celebrated Hermite–Hadamard integral inequality. Also, some error estimates for weighted Trapezoid formula and higher moments of random variables are given.

Bounded solutions for a functional equation were investigated by Ng and Zhang [C. T. Ng, W. Zhang, Invariant curves for planar mapping, J. Difference Eqn. Appl. 3 (1997) 147–168]. In this paper, we consider its solutions which may be unbounded. We give not only existence and uniqueness of its solutions but also the convex and concave solutions. Moreover, some examples are given.

In the paper we consider the class Γ of analytic and univalent functions f in the unit disk Δ , normalized by f (0) = f ′(0) − 1 = 0, having real coefficients and such that f ( Δ ) is convex in the direction of the real axis. We are especially interested in some subclasses of Γ . The most important of them is Γ ( c ) consisting of those functions which have the second coefficients of the Taylor expansion fixed and equal to c . We obtain the Koebe set for this class as well as for the classes Γ + ( c ) and Γ − ( c ) of functions which are in some sense convex in the direction of positive and negative axes respectively.

In this paper we study hybrid iterative scheme for finding a common element of set of solutions of generalized mixed equilibrium problem, set of common fixed points of finite family of weak relatively nonexpansive mapping and null spaces of finite family of γ-inverse strongly monotone mappings in a 2-uniformly convex and uniformly smooth real Banach space. Our results extend, improve and generalize the results of several authors which are announced recently. Application of our theorem to solution of equations of Hammerstein-type is of independent interest.

This work is concerned with the analysis of convergence of generalized iterative methods for solving some variational inequalities with pseudomonotone operators and convex nondifferentiable functionals in Banach spaces. Such inequalities occur, in particular, in descriptions of steady-state filtration processes and equilibrium problems for soft shells. The results obtained in this paper include and extend the results of B. Badriev et al. [I.B. Badriev, O.A. Zadvornov, A.M. Saddeek, Convergence analysis of iterative methods for some variational inequalities with pseudomonotone operators, Diff. Equ. 37 (7) (2001) 934–942].

In this paper, we introduce a new iterative method for finding a common element of the set of fixed points of a finite family of relatively nonexpansive mappings and the set of solutions of an equilibrium problem in uniformly convex and uniformly smooth Banach spaces. Then we prove a strong convergence theorem by using the generalized projection.

In this paper, we will illustrate and visualize the chaotic behavior for solving the sign-changing solutions of concave–convex nonlinearities. We propose contractor iteration algorithm and bifurcation method to improve the computation speed and avoid local convergence of Newton–Raphson method. By this method, sign-changing numerical solutions will be computed and visualized in various domains.

In this paper, a new two-step iterative scheme for a finite family of I"i-asymptotically nonexpansive nonself-mappings {T"i}"i"="1^r is constructed in a uniformly convex Banach space. Weak and strong convergence theorems of this iterative scheme to a common fixed point of {T"i}"i"="1^r and {I"i}"i"="1^r are proved in a uniformly convex Banach space. The results of this paper improve and extend the corresponding results of Temir [2].

In this paper we introduce and study some new subclasses of p-valent starlike, convex, close-to-convex and quasi-convex functions defined by Dziok-Srivastava operator. Inclusion relationships are established and integral operator of functions in these subclasses is discussed.

A quasiconformal extension for the class of k-uniformly convex functions, denoted k-UCV([email protected]?[0,~)), and for the class of k-starlike functions, denoted k-ST is provided. Also, estimation of the norm of pre-Schwarzian derivative in k-UCV is given.

In this paper, fixed point theorems of Rothe and Altman types about convex-power condensing operator are given and applied to the differential equation of one order with integral boundary conditions in abstract space.

Numerical methods of finding the roots of a system of non-linear algebraic equations are treated in this paper. This paper attempts to give an answer to the selection of the most efficient method in a complex problem of Celestial Dynamics, the so-called ring problem of (N+1) bodies. We apply Newton and Broyden's method to these problems and we investigate, by means of their use, the planar equilibrium points, the five equilibrium zones, which are symbolized by A"1, A"2, B, C"2, and C"1 (by order of appearance from the center O to the periphery of the imaginary circle on which the primaries lie) [T.J. Kalvouridis, A planar case of the N+1 body problem: the ring problem. Astrophys. Space Sci. 260 (3) (1999) 309-325], and the attracting regions of the system. The efficiency of these methods is studied through a comparative process. The obtained results are demonstrated in figures and are discussed.

By using a convex sequence Bor [H. Bor, A study on local properties of factored Fourier series, Nonlinear Anal. 57 (2004) 191-197] has obtained a result dealing with local properties of factored Fourier series for |[email protected]?,p"n|"k summability. In this paper, we have generalized that result for |[email protected]?,p"n,@q"n|"k summability. Some new results have also been obtained.

Let C be a nonempty closed convex subset of a real strictly convex and reflexive Banach space E which has a uniformly Gâteaux differentiable norm. Let f:C→C be a given contractive mapping and {Tn}n=1∞:C→C be an infinite family of nonexpansive mappings such that the common fixed point sets F:=⋂n=1∞F(Tn)≠∅. Let {αn} and {βn} be two real sequences in [0, 1]. For given x0∈C arbitrarily, let the sequence {xn} be generated iteratively by xn+1=αnf(xn)+βnxn+(1-αn-βn)Wnxn, whereWn is the W-mapping generated by the mappings Tn,Tn-1,…,T1 and ξn,ξn-1,…,ξ1. Suppose the iterative parameters {αn} and {βn} satisfy the following control conditions: (C1) limn→∞αn=0; (C2) ∑n=0∞αn=∞; (B5) limsupn→∞βn

The object of this paper is to introduce two interesting subclasses T n ∗ ( p , q , α , β , γ ) and C n ( p , q , α , β , γ ) of p-valent starlike functions defined by using a differential operator. We obtain coefficient estimates and distortion theorems for functions belonging to these classes. The radii of convexity for functions belonging to these classes are also determined. Finally, several applications involving an integral operator and certain fractional calculus operator are also considered.

Let E be a real uniformly convex Banach space, K a nonempty closed convex subset of E, and T 1 , T 2 : K → E two uniformly L-Lipschitzian, nonself generalized asymptotically quasi-nonexpansive mappings with nonnegative real sequences { k n ( i ) } , { δ n ( i ) } ( i = 1 , 2 ) , respectively satisfying ∑ n = 1 ∞ ( k n ( i ) - 1 ) + ∞ , ∑ n = 0 ∞ δ n ( i ) + ∞ . Suppose F = F ( T 1 ) ⋂ F ( T 2 ) ≠ ∅ . If, for any x i ∈ K ( i = 0 , 1 , 2 , … , q and q ∈ N is a fixed number), { x n } be a sequence in K defined by y n = P ( α ¯ n x n + β ¯ n T 2 ( PT 2 ) n - 1 x n + γ ¯ n v n ) , n = 0 , 1 , 2 , … , x n + 1 = P ( α n x n + β n T 1 ( PT 1 ) n - 1 y n - q + γ n u n ) , n = q , q + 1 , q + 2 , … , where ∑ γ n ∞ and ∑ γ ¯ n ∞ . Then { x n } converges strongly to a common fixed point of T 1 , T 2 under suitable conditions.

Using slope enclosures may provide sharper bounds for the range of a function than using enclosures of the derivative. Hence, slope enclosures may be useful in verifying the assumptions for existence tests or in algorithms for global optimization. Previous papers by Kolev and Ratz show how to compute slope enclosures for convex and concave functions. In this paper, we generalize these formulas and show how to obtain slope enclosures for a function that has exactly one point of inflection or whose derivative has exactly one point of inflection.

Let E be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, K a nonempty closed convex subset of E , T : K → K an asymptotically nonexpansive mapping with sequence { k n } ⊂ [ 1 , ∞ ) , lim n → ∞ k n = 1 . Let { α n } ⊂ ( 0 , 1 ) be such that lim n → ∞ α n = 0 , lim n → ∞ k n - 1 α n = 0 and f be a contraction on K . Under suitable conditions, we show the existence of a sequence { z n } satisfying the relation z n = α n f ( z n ) + ( 1 - α n ) T n z n , and prove that { z n } converges strongly to the fixed point of T , which solves some variational inequality, provided T is asymptotically regular. As an application, we prove that the iterative process defined by x 0 ∈ K , x n + 1 ≔ α n f ( x n ) + β n x n + γ n T n x n , converges strongly to the same fixed point of T .

Let E be a real uniformly convex Banach space which admits a weakly sequentially continuous duality mapping from E to E∗E∗, C a closed convex subset of E which is also a sunny nonexpansive retract of E . Let T:C→ET:C→E be a nonexpansive nonself-mapping with F(T)≠∅F(T)≠∅, and f:C→Cf:C→C be a fixed contractive mapping. The implicit viscosity iterative sequence {zm}{zm} is defined by {tm}⊂(0,1){tm}⊂(0,1), zm=1m+1∑j=0mP(tmf(zm)+(1-tm)(TP)jzm), for m⩾0m⩾0 and two explicit viscosity iterative sequences {xn}{xn} and {yn}{yn} are given by x0∈C,xn+1=αnf(xn)+(1-αn)1n+1∑j=0n(PT)jxn and y0∈C,yn+1=1n+1∑j=0nP(αnf(yn)+(1-αn)(TP)jyn) for n⩾0n⩾0, where {αn}{αn} is a sequence in (0,1)(0,1) and P is a sunny nonexpansive retraction of E onto C . We prove that under appropriate conditions imposed on {tm}{tm} and {αn}{αn}, the sequences {zm}{zm}, {xn}{xn} and {yn}{yn} converge strongly to some fixed point of T which solves some variational inequalities. The results presented extend and improve the corresponding results of Matsushita and Kuroiwa [S. Matsushita, D. Kuroiwa, Strong convergence of averaging iterations of nonexpansive nonself-mappings, J. Math. Anal. Appl. 294 (2004) 206–214], Song and Chen [Y. Song, R. Chen, Viscosity approximation methods to Cesaro means for nonexpansive mappings, Appl. Math. Comput. 186 (2007) 1120–1128] and many authors.

In the present work, a linearized finite-difference scheme is proposed in order to approximate the solution of some non-linear equation characterized by mixed concave and convex non-linearities. It is proved that the scheme is uniquely solvable and convergent. The resulting method is also analyzed for consistency and stability. Some numerical examples illustrating the method described in our work are given.

Let K be a nonempty, closed and convex subset of a real Banach space E. Let T:K->K be a strictly pseudocontractive map in the sense of Browder and Petryshyn. For a fixed x"[email protected]?K, define a sequence {x"n} byx"n"+"1=([email protected]"n)x"[email protected]"nTx"n,where {@a"n} is a real sequence defined in [0,1] satisfying the following conditions (i) @?"n"="1^[email protected]"n=~, (ii) @?"n"="1^[email protected]"n^2 "[email protected]?x"n-Tx"[email protected]?=0. If, in addition, T is demicompact, then {x"n} converges strongly to some fixed point of T.

Let K be a nonempty closed and convex subset of a real Banach space E. Let T : K → E be a nonexpansive weakly inward mapping with F (T) ≠ ∅ and f : K → K be a contraction. Then for t ∈ (0, 1), there exists a sequence {yt} ⊂ K satisfying yt = (1 - t) f (yt) + tT (yt). Furthermore, if E is a strictly convex real reflexive Banach space having a uniformly Gâteaux differentiable norm, then {yt} converges strongly to a fixed point p of T such that p is the unique solution in F (T) to a certain variational inequality. Moreover, if {Ti, i = 1, 2, ..., r} is a family of nonexpansive mappings, then an explicit iteration process which converges strongly to a common fixed point of {Ti, i = 1, 2, ..., r} and to a solution of a certain variational inequality is constructed. Under the above setting, the family Ti, i = 1, 2, ..., r need not satisfy the requirment that {Mathematical expression}.

The main object of the present paper is to investigate α-convex and α-quasiconvex functions with respect to n-symmetric points. Some interesting properties such as inclusion relationships, integral representations, convolution conditions are obtained.

In this paper, weak and strong convergence theorems are established for the modified Noor iterations with errors for asymptotically nonexpansive mappings in the intermediate sense in a uniformly convex Banach space. Mann-type and Ishikawa-type iterations are included by the modified Noor iterations with errors. The results obtained in this paper extend and improve the recent ones announced by Schu [J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 158 (1991) 407-413; J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43 (1991) 153-159], Xu and Noor [B.L. Xu, M.A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267 (2002) 444-453], Cho et al. [Y.J. Cho, H.Y. Zhou, G. Guo, Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings, Comput. Math. Appl. 47 (2004) 707-717], Suantai [S. Suantai, Weak and strong convergence criteria of Noor Iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 311 (2005) 506-517], Nammanee et al. [K. Nammanee, M.A. Noor, S. Suantai, Convergence criteria of modified Noor iterations with errors for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 314 (2006) 320-334], and many others.

Let q 1 and q 2 be univalent in A := {z: |z| < 1} with q 1 (0) = q 2 (0) = 1. We give some applications of first order differential subordination and superordination to obtain sufficient conditions for a normalized analytic functions f(z) with y(0) = f'(0) = 1 to satisfy q1 (z) zf'(z) f(z) q 2 (z).

In this paper, we study the convergence of two type iteration processes as follows: (I) x n + 1 = α n x + ( 1 - α n ) T ( β n x + ( 1 - β n ) Tx n ) , (II) x n + 1 = α n ( β n x + ( 1 - β n ) Tx n ) + ( 1 - α n ) A n x n , where A n = 1 n + 1 ∑ j = 0 n T j : C → C in uniformly convex Banach space X, which possesses a weakly sequentially continuous duality mapping J and in uniformly convex Banach space X with a uniformly Gateaux differentiable norm, respectively. And prove that above sequences converge strongly to Px when the real sequence {αn}, {βn} satisfies appropriate conditions, where P is sunny non-expansive from C onto F(T).

In this paper, we prove the weak and strong convergence of the Ishikawa iterative scheme with errors to a common fixed point for two asymptotically nonexpansive mappings in a uniformly convex Banach space under a condition weaker than compactness. Our theorems improve and generalize recent known results in the literature.

This paper deals with a class of quasilinear elliptic Dirichlet boundary value problem of type −Δ u = λ (1 + u ) q , where the effect of the superlinear term allows us to find some results about the existence and multiplicity of numerical positive solutions for varying λ > 0 and q > 1 by using finite difference method and we compare the obtained solutions with each other.

The aim of this paper is to present sharp results concerning coefficients, distortion theorem and the radius of convexity for P"p^*(@a,@b,@x,@W). Integral operators of functions belonging to the class P"p^*(@a,@b,@x,@W) are studied here. Furthermore it is shown that the class P"p^*(@a,@b,@x,@W) is closed under convex linear combinations. The extreme points of P"p^*(@a,@b,@x,@W) are also determined.

Some inequalities are presented here for differentiable convex mappings, using Hermite-Hadamard's integral inequality holding for convex functions. A parallel development is deduced for differentiable concave functions. Some applications to special means of real numbers are also given. Finally, some error estimates for the midpoint formula are obtained.

In this paper we will extend the concept of convex fuzzy bodies in a Riemannian manifold to a generalized convex fuzzy bodies which are called invex fuzzy bodies. Some geometrical properties of this kind of bodies are discussed.

In this paper the asymptotic and oscillatory properties of solutions to the first order neutral difference inclusion of the [email protected](x"n-r"nx"n"-"@t)@?p"nF(x"n"-"@s"""1,x"n"-"@s"""2,...,x"n"-"@s"""s),n=0,1,2,...are discussed, where F:R^s->2^R^^^1 is a compact convex upper semicontinuous multifunction. Some examples are given to illustrate main theorems.

In this paper we examine controllability problems of evolution inclusions with nonlocal conditions. Using Kakutani's fixed point theorem and Schauder's fixed point theorem, we establish sufficient conditions for the controllability under convex and nonconvex orientor fields respectively.

The main purpose of this paper is to introduce two classes P"@b(@l,b) and R"@b(@l,b) (b, a complex number with Re(b)>0) of functions which are analytic in the unit disc. Coefficient estimates for functions in these classes and radii of close-to convexity, starlikeness and convexity are obtained. Further applications in the fractional calculus are given for functions in P"@b(@l,b) and R"@b(@l,b).

A new concept of exceptional family for nonlinear variational inequalities over polyhedral sets is introduced in this paper. It generalizes the concepts for complementarity problem introduced by Smith and Isac. We applied the new analytical tool to the study of existence problem for variational inequalities. It is shown that our existence condition is weaker than most of the sufficient conditions which have been known.

In this paper, we present a new interior point method—combined homotopy interior point method (CHIP method) for convex nonlinear programming. Without strict convexity of the logarithmic barrier function and boundedness and nonemptiness of the solution set, we prove that for any ϵ > 0, an ϵ-solution of the problem can be obtained by the CHIP method. To our knowledge, strict convexity of the logarithmic barrier function and nonemptiness and boundedness of the solution set are the essential assumptions of the well-known center path-following method. Therefore, the CHIP method essentially reduces the assumptions of the center path-following method and can be applied to more general problems.