Muhammad Aslam Noor

COMSATS University Islamabad | CUI · Department of Mathematics

In this paper, we use the variational iteration technique to suggest some iterative methods for solving the nonlinear equations involving an auxiliary function. For appropriate and suitable choice of the auxiliary function,one can obtain a wide class of iterative methods for solving the nonlinear equations, which is a novel aspect of this technique...

In this paper, we introduce the notion of coordinated harmonically convex functions. We derive some new integral inequalities of Hermite–Hadamard type for coordinated Harmonically convex functions. The interested readers are encouraged to find the applications of harmonically convex functions in pure and applied sciences.

We prove that the set of second degree cumulative frontier gaps, via fractional integrals of positive orders, of twice differentiable functions having generalized convexity at the level of the second derivative is upper bounded. A sharp Hermite-Hadamard type inequality via fractional integrals leads to an evaluation of this upper bound, which does...

It is known that the general variational inclusions involving difference of two (or more) monotone operators are equivalent to the resolvent equations. We use this equivalent form to develop the sensitivity analysis for general variational inclusions. This technique does not require the differentiability of the given data. Some special cases are di...

In this paper, we consider a new class of quasi variational inequalities involv- ing three operators, which is called the extended general quasi variational inequality. It is shown that the extended general quasi variational inequalities are equivalent to the fixed problems and the extended general implicit Wiener-Hopf equations. These alternative...

In this paper, we introduce a new class of convex functions, which is called nonconvex functions. We show that this class unifies several previously known and new classes of convex functions. We derive several new inequalities of Hermite-Hadamard type for nonconvex functions. Some special cases are also discussed. Results proved in this paper conti...

In this paper, we introduce some new classes of s-Godunova-Levin functions, which are called (s,m)-Godunova-Levin functions of first and second kind. We show that these classes contain previously known classes of convex functions. Finally, we establish some new Ostrowski inequalities for (s,m)- Godunova-Levin functions via fractional integrals, Som...

In this paper, we define and study a new class of analytic functions by using the concept of generalized close-to-convexity. Coefficient results, Hankel determinant problem and some other interesting properties of this class are investigated. Results proved in this paper may stimulate further research in this area.

A new Hermite-Hadamard inequality for p-convex(nonconvex) functions is obtained , which can be viewed as a refinement of known results. We derive some new inequalities of functions whose derivatives in absolute value are nonconvex. Results obtained in this paper continue to hold for special cases. Techniques and ideas of this paper may stimulate fu...

In this paper, we discuss the existence of a solution of extended general quasi variational inequalities using projection technique. Using the technique of Noor, we suggest and analyze a new class of three-step iterative schemes for solving extended general quasi variational inequalities. Under some certain conditions on operators, we also discuss...

In this paper, we derive a new refinement of Hermite-Hadamard inequality for the class of h-convex functions. We also discuss some new and known special cases which can be obtained from our main results. Results obtained in this paper may be viewed as significant improvement of the known results.

In this article, we derive a new refinement of Hermite-Hadamard inequality for the class of h-convex functions. We also discus some new and known special cases, which can be obtained from our min results. Results obtained in this paper may be viewed as significant improvement of the known results.

. A new Hermite-Hadamard inequality for p -convex(nonconvex) functions is ob- tained, which can be viewed as a refinement of known results. We derive some new inequali- ties for functions whose derivatives in absolute value are nonconvex.Results obtained in this paper continue to hold for special cases. Techniques and ideas of this paper may stimul...

In this paper, we introduce the notion of coordinated harmonically convex functions. We derive some new integral inequalities of Hermite–Hadamard type for coordinated Harmonically convex functions. The interested readers are encouraged to find the applications of harmonically convex functions in pure and applied sciences.

In this paper, we use the modified decomposition method for solving the system of third-order nonlinear boundary value problems associated with obstacle problems. Some examples of system of third-order nonlinear boundary value problems are given. The comparison of the results obtained by modified decomposition method with modified variation of para...

In this chapter, we investigate some unified iterative methods for solving the general equilibrium problems using the auxiliary principle technique. The convergence of the proposed methods is analyzed under some suitable conditions. As special cases, we obtain a number of known and new classes of equilibrium and variational inequality problems. Res...

In this paper, some new classes of Godunova-Levin functions are introduced and investigated. Several new fractional Hermite-Hadamard inequalities are derived for these new classes of Godunova-Levin functions. The ideas and techniques of this paper may stimulate and inspire further research.

a b s t r a c t In this paper, we consider a new system of extended general quasi variational inequalities involving six nonlinear operators. Using projection operator technique, we show that the system of extended general quasi variational inequalities is equivalent to a system of fixed point problems. Using this alternative equivalent formulation...

In this paper, we use a new decomposition technique to suggest and consider some new iterative methods for solving system of linear equations. We prove that these iterative methods are similar to the iterative methods derived by using homotopy perturbation method and Adomian decomposition method. We consider the elliptic partial differential equati...

In this paper, we suggest and analyze some new derivative free iterative methods for solving nonlinear equation f (x) = 0 using a suitable transformation. We also give several examples to illustrate the efficiency of these methods. Comparison with other similar method is also given. These new methods can be considered as alternative to the develope...

In this paper we define and study a class of analytic functions which map the open unit disk onto same conic regions and are related to Bazilevic and higher order close-to-convex functions. We investigate rate of growth of coefficients, Hankel determinant problem, inclusion result and establish univalence criterion. Some other interesting propertie...

In this paper, we suggest and analyze a new family of iterative methods for finding zeros of multiplicity of nonlinear equations by using the variational iteration technique. These new methods include the Halley method and its variants forms as special cases. We also give several examples to illustrate the efficiency of these methods. Comparison wi...

Inspired by the very recent work by Noor and Noor [9] and given a closed convex set-valued mapping C, we propose a split algorithm for solving the problem of finding an element x * which is a zero of a given maximal monotone operator T such that its image, Ax * , under a linear operator, A, is in a closed convex set C(x *). Then, we present two str...

In this paper, we suggest and analyze a new recurrence relation which generates the iterative methods of higher order for solving nonlinear equation f (x) = 0. This general recurrence relation is obtained by using variational iteration technique. We purpose some new iterative methods for solving nonlinear equations. We also test different examples...

In this paper, we consider extended general variational inequalities involving nonlinear operators. Using projection operator technique, we show that extended gen-eral variational inequality is equivalent to a fixed point problem. Using this alternative equivalent formulation, we propose and analyze some new algorithms for solving extended general...

In this paper, we consider the class of geometrically-arithmetically s-convex functions. A generalized integral identity for differentiable functions is obtained. Then using this new integral identity we establish our main results which are Hermite-Hadamard inequalities for geometrically-arithmetically s-convex functions. Some special cases are als...

In this paper, we apply a relatively new technique which is called the exp-function method to construct generalized solitary and periodic solutions of Calogero-Degasperis-Fokas (CDF) equation which plays a very important role in mathematical physics, applied and engineering sciences. The suggested algorithm is quite efficient and is practically wel...

We apply a relatively new technique which is called the homotopy perturbation method (HPM) for solving linear and nonlinear partial differential equations. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discretization, linearization or...

In this paper, we apply a modified version of the variational iteration method (MVIM) for solving Burgers’ and coupled Burgers’ equations. The proposed modification is made by introducing He’s polynomials in the correction functional of the variational iteration method (VIM). The use of Lagrange multiplier coupled with He’s polynomials are the clea...

In this paper, we introduce and consider a new system of extended general variational inequalities in Banach spaces. We establish the equivalence between the extended general variational inequalities and the fixed point problems. We use this equivalent formulation to suggest some iterative methods for solving this new system.We prove the convergenc...

In this paper, we derive some new fractional Ostrowski type in-equalities for s-Godunova-Levin functions introduced by Dragomir [3, 4]. Some special cases are also discussed.

We use Ruscheweyh derivative to define certain new classes of analytic functions with bounded radius rotation and related to conic domains. Some interesting and significant results such as inclusion results, growth rate of coefficients and radius problems for these new classes of k-uniformly functions. Several special cases are discussed. Results o...

In this paper, some new concepts of geometrically relative convex sets and relative convex functions are defined. These new classes of geometrically relative convex functions unify several known and new classes of relative convex functions such as exponential convex functions. New Hermite-Hadamard type integral inequalities are derived for these ne...

In this paper, we introduce and consider a new class of general variational inclusions involving the difference of operators in a Hilbert space. We establish the equivalence between the general variational inclusions and the fixed point problems as well as with a new class of resolvent equations using the resolvent operator technique. We use this a...

In this paper, we consider the parametric nonconvex variational inequalities and parametric nonconvex Wiener–Hopf equations. Using the projection technique, we establish the equivalence between the parametric nonconvex variational inequalities and parametric nonconvex Wiener–Hopf equations. We use this alternative equivalence formulation to study t...

In this paper, a multivalued variational inequality problem on uniformly prox-regular set is studied. The existence theorems for such aforementioned problem are presented and, consequently, some algorithms for finding those solutions are also constructed. The results in this paper can be viewed as an improvement of the significant result that prese...

In this paper, we prove some Hermite-Hadamard inequalities for the class of relative semi-convex functions. Several special cases are also discussed. Thus it is worth mentioning that our results can be viewed as a generalization of previous results. Some applications to special means are also presented. Ideas and techniques of this paper may inspir...

In this paper, we derive some new fractional Ostrowski type in-equalities for s-Godunova-Levin functions introduced by Dragomir [3, 4]. Some special cases are also discussed.

It is well known that the general variational inequalities plays a fundamental and significant role in the study of the nonsymmetric and odd-order problems. In this paper, some implicit methods are suggested and investigated for solving the general variational inequalities using the technique of auxiliary principle technique. These methods include...

In this paper, we consider a class of geometrically convex function which is called geometrically-arithmetically h-convex function. Some inequalities of Hermite-Hadamard type for geometrically-arithmetically h-convex functions are derived. Several special cases are discussed.

We use Ruscheweyh derivative to define certain new classes of analytic functions with bounded radius rotation and related to conic domains. Some interesting and significant results such as inclusion results, growth rate of coefficients and radius problems for these new classes of k-uniformly functions. Several special cases are discussed. Results o...

In this paper, we introduce some new classes of s-Godunova-Levin functions, which are called s-Godunova-Levin preinvex functions. Some new Hermite-Hadamard inequalities for these new classes of s-Godunova-Levin functions are obtained.

In this paper, we introduce and consider a new class of convex function which is called harmonically log-convex function. Several inequalities of Hermite-Hadamard type for harmonically log-convex functions are derived. It is shown that harmonically log-convex function implies harmonically convex functions which implies harmonically quasi-convex fun...

In this paper, some new classes of harmonically convex functions are in-troduced and investigated. We derive several Hermite-Hadamard inequalities for these new classes of harmonically convex functions. The ideas and techniques of this paper may be extended for other classes of convex functions.

In this paper, we consider the class of modified h-convex functions, which was introduced by Toader [14]. We derive Hermite-Hadamard type inequalities for the modified h-convex functions. Some special cases are also discussed. We try to show that this class enjoys some nice properties which the convex functions have.

In this paper, we use the concept of bounded Mocanu variation to introduce a new class of analytic functions, defined in the open unit disc, which unifies a number of classes previously studied such as those of functions with bounded radius rotation and bounded Mocanu variation. It also generalizes the concept of β-spiral likeness in some sense. So...

The objective of this paper is to obtain some Hermite-Hadamard type inequalities for h-preinvex functions. Firstly a new kind of generalized h-convex functions, termed h-preinvex functions, is introduced through relaxing the concept of h-convexity introduced by Varosanec. Some Hermite-Hadamard type inequalities for h-preinvex functions are establis...

It is well known that the general variational inequalities plays a fundamental and significant role in the study of the nonsymmetric and odd-order problems. In this paper, some implicit method are suggested and investigated for solving the general variational inequalities using the technique of auxiliary principle technique. It is shown that the co...

In this paper, we show that the parametric general nonconvex variational inequalities are equivalent to the parametric Wiener-Hopf equations. We use this alternative equivalent formulation to study the sensitivity analysis for the nonconvex variational inequalities without assuming the differentiability of the given data. Our results can be conside...

In the paper, the authors introduce the notion “log-h-convex functions” and establish several Hermite-Hadamard type integral inequalities for this kind of functions.

In this paper, we suggest and analyze an iterative scheme for finding the approximate element of the common set of solutions of a generalized equilibrium problem, a variational inequality problem and a hierarchical fixed point problem in a real Hilbert space. We also consider the strong convergence of the proposed method under some conditions. Resu...

In this paper, we introduce a new system of quasi variational inequalities. The projection technique is used to establish the equivalence between this new system of quasi variational inequalities and the fixed point problem. The fixed point formulation enables us to suggest some parallel projection iterative methods for solving the system of quasi...

It is well known that variational inequalities are equivalent to fixed point problems. We use this equivalent formulation to suggest and analyze a new unified implicit method of solving variational inequalities. This new method includes several known implicit and extragradient methods as special cases. We prove that the convergence of this new meth...

We study a new class of convex functions that are called relative semi-convex functions. Some Hermite-Hadamard inequalities for the relative semi-convex function and its variant forms are derived. Several special cases are also discussed.

In thispaper, we introduce and consider a new problem of finding u ∈ K(u) such that Au ∈ C, where K : u → K(u) is a closed convex-valued set in the real Hilbert space H 1 , C is closed convex set in the real Hilbert space H 2 respectively and A is linear bounded self-adjoint operator from H 1 and H 2 . This problem is called the quasi split feasibi...

In this paper, we used Global Error Minimization (GEM)method for nonlinear oscillators. This method convert the nonlinear oscillators into an equivalent optimization problem to obtain an analytical solution of the problem. Approximate solution obtained by GEM method is compared with the solution of He's variational approach. We observe from the res...

In thispaper, we introduce and consider a new problem of finding u ∈ K(u) such that Au ∈ C, where K : u → K(u) is a closed convex-valued set in the real Hilbert space H 1 , C is closed convex set in the real Hilbert space H 2 respectively and A is linear bounded self-adjoint operator from H 1 and H 2 . This problem is called the quasi split feasibi...

In this paper, we use the variational iteration technique to suggest and analyze some new iterative methods for solving a system of nonlinear equations. We prove that the new method has fourth-order convergence. Several numerical examples are given to illustrate the efficiency and performance of the new iterative methods. Our results can be viewed...

In this paper, we consider the system of general mixed variational inequalities. We suggest and analyze a new iterative algorithm for finding a solution of the aforementioned system by using the resolvent operator technique. We also prove the convergence analysis of the proposed algorithm under some suitable mild conditions.

The objective of this paper is to obtain some important properties about convex fuzzy mappings based on a linear ordering of fuzzy numbers proposed by Goetschel and Voxman. Firstly, a new kind of fuzzy mapping, termed semistrictly convex fuzzy mapping, is defined through this linear ordering. Note that semistrict convexity does not imply convexity....

In this paper, we consider a new class of quasi variational inequalities involving three operators, which is called the extended general quasi variational inequality. It is shown that the extended general quasi variational inequalities are equivalent to the fixed problems. This equivalence is used to suggest and analyze some iterative methods for s...

In this paper, we suggest and analyze some new higher-order iterative methods by using Householder's method free from second derivative for solving nonlinear equations. Here we use new and different technique for implementation of higher-order derivative of the function and derive new higher-order predictor-corrector iterative methods free from sec...

In this paper, we suggest and analyze a symmetric accelerated over relaxation (SAOR) method for absolute complementarity problems of finding x ∈ R n , such that x ≥ 0, Ax − |x| − b ≥ 0, 〈x, Ax − |x| − b〉 = 0, where A ∈ R n×n and b ∈ R n . We discuss the convergence of SAOR method when the system matrix A is an L-matrix. Several examples are given t...

In this paper, we use Adomian decomposition method for suggesting an iterative methods for solving system of linear equations. We provide several numerical examples to verify the theoretical results. Comparsion with other methods show that the results obtained in this paper are better. Our results can be viewed as an improvement and extension of th...

In this paper, we introduce a new class of nonconvex functions which is called hϕ-convex function. We prove several Hermite-Hadamard integral inequalities for hϕ-convex function and differentiable hϕ-convex function. Some special cases are also discussed. Thus our results can be viewed as generalization of previous results [3, 6, 17].

In this paper, we suggest and analyze a new resolvent algorithm for finding the common solutions for a generalized system of relaxed cocoercive mixed variational inequality problems and fixed point of a nonexpansive mapping in Hilbert spaces. We also prove the convergence analysis of the proposed algorithm under some suitable mild conditions. In th...

In this paper, we consider the classes of s-convex function of first and second kinds respectively. We derive several new Hermite-Hadamard type of inequalities for these classes via fractional integrals.

In this paper, we introduce and consider a new system of extended general vari-ational inclusions involving eight different operators. Using the resolvent operator techniques, we show that the new system of extended general variational inclusions is equivalent to the fixed point problem. We prove the strong convergence of some new explicit iterativ...

In this paper, we develop a unified and general framework of the sensitivity analysis for a class of quasi variational inequalities involving three operators using the Wiener-Hopf equations technique. Our analysis does not involve the differentiability of the given data. Since the extended general quasi variational inequalities include classi-cal v...

In this paper, we consider the system of general mixed variational inequalities. We suggest and analyze a new iterative algorithm for finding a solution of the aforementioned system by using the resolvent operator technique. We also prove the convergence analysis of the proposed algorithm under some suitable mild conditions.

In this paper, we used Global Error Minimization (GEM)method for nonlinear oscillators. This method convert the nonlinear oscillators into an equivalent optimization problem to obtain an analytical solution of the problem. Approximate solution obtained by GEM method is compared with the solution of He's variational approach. We observe from the res...

Let K and C be nonempty, closed and convex sets in Rn and Rm respectively and A be an m × n real matrix. The split feasibility problem is to find u ε K with Au ε C: Many problems arising in the image reconstruction can be formulated inthis form. In this paper, we use the auxiliary principle technique to suggest and analyze some new iterative algori...

In this paper, we use Adomian decomposition method for suggesting an iterative methods for solving system of linear equations. We provide several numerical examples to verify the theoretical results. Comparsion with other methods show that the results obtained in this paper are better. Our results can be viewed as an improvement and extension of th...

In this paper, we suggest and analyze a new class of iterative methods for solving nonlinear equations by using the homotopy perturbation method. Convergence of their method is also considered. Here we also discuss the efficiency index and computational order of con-vergence of new methods. Several numerical examples are given to illustrate the eff...

In this paper, we introduce and consider a new class of equilibrium variational inequalities , called the mixed general equilibrium bifunction variational inequalities. We suggest and analyze some proximal methods for solving mixed general equilibrium bifunction vari-ational inequalities using the auxiliary principle technique. Convergence of these...

The Editors would like to express their deepest gratitude to the authors for their fascinating and interesting contributions as well as to the staff and the editorial office of the journal for the great and invaluable support. The Editors would like also to express their greatest appreciation to more than 200 reviewers for their important time and...

This paper is to illustrate that the main result of the paper [R.U. Verma, Generalized over-relaxed proximal algorithm based on AA-maximal monotonicity framework and applications to inclusion problems, Mathematical and Computer Modelling 49 (2009) 1587–1594] is incorrect. The convergence rate of the over-relaxed proximal point algorithm should be g...

Let Ω and C be nonempty, closed and convex sets in R n and R m respectively and A be an \({m \times n}\) real matrix. The split feasibility problem is to find \({u \in \Omega}\) with \({Au \in C.}\) Many problems arising in the image reconstruction can be formulated in this form. In this paper, we propose a descent-projection method for solving the...

In this paper, we introduce and study a new class of quasi-variational inequalities, which are called the extended general nonconvex quasi-variational inequalities. Using the projection technique, we establish the equivalence between the extended general nonconvex quasi-variational inequalities and the fixed point problem. We use this alternative e...

In this paper, we introduce and consider a new class of variational inequalities, which is called the nonconvex bifunction variational inequality. We suggest and analyze some iterative methods for solving nonconvex bifunction variational inequalities using the auxiliary principle technique. We prove that the convergence of implicit method requires...

The authors would like to express their deepest gratitude to the reviewers, whose professional comments and valuable suggestions guaranteed the high quality of these selected papers. The editors would like to express their gratitude to the authors for their interesting and novel contributions. They would also like to thank the editorial boards memb...

In this paper, we suggest and analyze an iterative method for solving the equilibrium problems on Hadamard manifolds using the auxiliary principle technique. We also consider the convergence analysis of the proposed method under suitable conditions. Some special cases are considered. Results and ideas of this paper may stimulate further research in...

In this paper, we construct a new projection method and prove the strong convergence for finding a common element of a set of fixed points of strict pseudo-contractions and a set of solutions of a variational inequality with inverse strongly monotone mappings in the setting of real Hilbert spaces. Our results improve and extend the recent results b...

In this paper, we propose two new methods for solving mixed quasi-variational inequalities using the resolvent operator technique. Under certain conditions, the global convergence of both methods is proved. The skew-symmetry of the nonlinear bifunction plays a crucial part in the convergence analysis of these new iterative methods. The comparison o...

We consider and study a new class of variational inequality, which is called the extended general mixed quas-variational inequality. We use the auxiliary principle technique to study the existence of a solution of the extended general mixed quasi-variational inequality. Several special cases are also discussed. Results proved in this paper may stim...