Earthline Journal of Mathematical Sciences

Mixed Variational Inequalities and Nonconvex Analysis

2024-06-30T04:44:46+00:00

In this expository paper, we provide an account of fundamental aspects of mixed variational inequalities with major emphasis on the computational properties, various generalizations, dynamical systems, nonexpansive mappings, sensitivity analysis and their applications. Mixed variational inequalities can be viewed as novel extensions and generalizations of variational principles. A wide class of unrelated problems, which arise in various branches of pure and applied sciences are being investigated in the unified framework of mixed variational inequalities. It is well known that variational inequalities are equivalent to the fixed point problems. This equivalent fixed point formulation has played not only a crucial part in studying the qualitative behavior of complicated problems, but also provide us numerical techniques for finding the approximate solution of these problems. Our main focus is to suggest some new iterative methods for solving mixed variational inequalities and related optimization problems using resolvent methods, resolvent equations, splitting methods, auxiliary principle technique, self-adaptive method and dynamical systems coupled with finite difference technique. Convergence analysis of these methods is investigated under suitable conditions. Sensitivity analysis of the mixed variational inequalities is studied using the resolvent equations method. Iterative methods for solving some new classes of mixed variational inequalities are proposed and investigated. Our methods of discussing the results are simple ones as compared with other methods and techniques. Results proved in this paper can be viewed as significant and innovative refinement of the known results.

Coefficients Estimates for a Subclass of Starlike Functions

2024-06-30T12:16:10+00:00

The paper mainly investigates the initial coefficients for the subclasses of starlike functions defined by using the Cosine function involving $\alpha$ ($0\leq\alpha<1$), we obtain upper bounds for initial order of Hankel determinants and symmetric Toeplitz determinants whose elements are the initial coefficients. Also, we obtain initial coefficient estimation of logarithmic coefficients for the subclass.

Two-Step Hybrid Block Method for Solving Second Order Initial Value Problem of Ordinary Differential Equations

2024-07-04T15:40:19+00:00

A new zero-stable two-step hybrid block method for solving second order initial value problems of ordinary differential equations directly is derived and proposed. In the derivation of the method, the assumed power series solution is interpolated at the initial and the hybrid points while its second ordered derivative is collocated at all the nodal and selected off-step points in the interval of consideration. The relevant properties of the method were examined and the method was found to be zero-stable, consistent and convergent. A comparison of the results by the method with the exact solutions and other results in literature shows that the method is accurate, simple and effective in solving the class of problems considered.

Weakly Reich Type Cyclic Contraction Mapping Principle

2024-07-04T16:19:39+00:00

In this paper we introduce the notion of Reich type cyclic weakly contraction and prove a fixed point theorem. Some Corollaries are consequences of the main result.

Cubic Spline Chebyshev Polynomial Approximation for Solving Boundary Value Problems

2024-07-04T17:06:28+00:00

In this work, a Chebyshev polynomial spline function is derived and used to approximate the solution of the second order two-point boundary value problems of variable coefficients with the associated boundary conditions. In deriving the method, the cubic spline Chebyshev polynomial approximation, $S(x)$ is made to satisfy certain conditions for continuity and smoothness of functions. Numerical examples are presented to illustrate the applications of this method. The solution, $y(x)$ of these examples are obtained at some nodal points in the interval of consideration. The absolute errors in each example are estimated, and the comparison of exact values, and approximate values by the present method and other methods in literature at the nodal points are presented graphically. The comparison shows that the proposed method produces better results than Approaching Spline Techniques and collocation method.

New Class of Multivalent Functions Defined by Generalized (p,q)-Bernard Integral Operator

2024-07-15T15:26:54+00:00

Making use of the generalized $(p, q)$-Bernardi integral operator, we introduce and study a new class $\mathcal{F J}_{p, q}^m(\alpha, \delta, \lambda, \gamma)$ of multivalent analytic functions with negative coefficients in the open unit disk $E$. Several geometric characteristics are obtained, like, coefficient estimate, radii of convexity, close-to-convexity and starlikeness, closure theorems, extreme points, integral means inequalities, neighborhood property and convolution properties for functions belonging to the class $\mathcal{F} \mathcal{J}_{p, q}^m(\alpha, \delta, \lambda, \gamma)$.

Upper Bounds for Certain Families of m-Fold Symmetric Bi-Univalent Functions Associating Bazilevic Functions with λ-Pseudo Functions

2024-07-23T17:03:32+00:00

In this paper, we introduce and study a new families $W_{\Sigma_m}(\lambda, \gamma, \delta ; \alpha), W_{\Sigma_m}^*(\lambda, \gamma, \delta ; \beta)$, $M_{\Sigma_m}(\lambda, \gamma, \delta ; \alpha)$ and $M_{\Sigma_m}^*(\lambda, \gamma, \delta ; \beta)$ of holomorphic and $m$-fold symmetric bi-univalent functions associating the Bazilevic functions with $\lambda$-pseudo functions defined in the open unit disk $U$. We find upper bounds for the first two Taylor-Maclaurin $\left|a_{m+1}\right|$ and $\left|a_{2 m+1}\right|$ for functions in these families. Further, we point out several special cases for our results.

A New Class of Univalent Functions Defined by Differential Operator

2024-07-24T17:33:42+00:00

In the present work, we submit and study a new class $\mathcal{S}_{\lambda, \alpha, b}^{z, m, t}(\beta, \gamma, \omega, \mu)$ containing analytic univalent functions defined by new differential operator $D_{\lambda, \alpha}^{z, m, t}$ in the open unit disk $E=\{s \in \mathbb{C}:|s|<1\}$. We get some geometric properties, such as, coefficient estimate, growth and distortion theorems, convex set, radii of convexity and starlikeness, weighted mean, arithmetic mean and partial sums for functions belonging to the class $\mathcal{S}_{\lambda, \alpha, b}^{z, m, t}(\beta, \gamma, \omega, \mu)$.