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In a recent paper ?Filomat, 35:5 (2021), 1555-1564? the authors proved some best proximity point theorems for classes of contractive pair of mappings as well as cyclic ?-contractions under some sufficient conditions. In this article, we present some counterexamples of the existence results of contractive pair of mappings and present the corrected version of these results. We also show that the best proximity point theorem for cyclic ?-contractions is a straightforward consequence of Boyd-Wong fixed point theorem by dropping an additional assumption.

In this short paper, we will show that the space of real valued uniformly continuous functions defined on a metric space (X,d) is a ring if and only if every subset A ? X has one of the following properties: ? A is Bourbaki-bounded, i.e., every uniformly continuous function on X is bounded on A. ? A contains an infinite uniformly isolated subset, i.e., there exist ? > 0 and an infinite subset F ? A such that d(a,x) ? ? for every a ? F, x ? X n \{a}.

This short paper is motivated by the paper of Bu et al. [C. Bu, C. Feng, P. Dong, A note on computational formulas for the Drazin inverse of certain block matrices, J. Appl. Math. Comput.(38) (2012) 631-640], where the authors gave additive formula for Drazin inverse for matrices under new conditions, and two representations under some specific conditions. Here is shown that the additive formula is not valid for all matrices which satisfy given conditions. Also, here is proved that the representations which were given in mentioned paper do not extend the results given by Hartwig et al. [R. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of a 2 _ 2 block matrix, SIAM J. Matrix. Anal. Appl. (27)(2006) 757-771 ], in fact they are equivalent.

In this short paper, generally, we define a family of functions fk depends on the Eisenstein series with weight 2k, for k ( N. More detail, by considering the function fk, we define a derivative formula for generating weakly modular forms with weight 12. As a result for this, we claim that this formula gives an advantage to find the special solutions of some differential equations.

In this paper, we propose a necessary and sufficient condition for the solvability to a system of matrix equations over the commutative quaternion ring, and establish an expression of its general solution when it is solvable. We also present an algorithm for finding an approximate solution to the system when it is inconsistent. Finally, we give an example to illustrate the main results of this paper.

In this paper, it is our aim in this paper to introduce a new iterative algorithmfor approximation of a solution of an equation of Hammerstein-type. The proposed scheme does not involve computation of inverse of operators under study; it does not involve passing through computation of a certain set that must contain a solution of the equation of Hammerstein-type before convergence takes place. The proposed scheme requires only one parameter satisfying verifiable mild conditions. Moreover, the mappings involved are neither defined on compact subset of the space under study, nor assumed to be angle bounded. Our theorems complement several results that have been obtained in this direction.

In this paper, we consider a linear operator of the Korteweg-de Vries type Lu = ?u ?y + R2(y)?3u ?x3 + R1(y)?u ?x + R0(y)u initially defined on C? 0,?(??), where ?? = {(x, y) : ?? ? x ? ?,?? < y < ?}. C? 0,?(??) is a set of infinitely differentiable compactly supported function with respect to a variable y and satisfying the conditions: u(i) x (??, y) = u(i) x (?, y), i = 0, 1, 2. With respect to the coefficients of the operator L , we assume that these are continuous functions in R(??,+?) and strongly growing functions at infinity. In this paper, we proved that there exists a bounded inverse operator and found a condition that ensures the compactness of the resolvent under some restrictions on the coefficients in addition to the above conditions. Also, two-sided estimates of singular numbers (s-numbers) are obtained and an example is given of how these estimates allow finding estimates of the eigenvalues of the considered operator.

The paper aims at developing the most simplified axiom for a pseudo (k0, k1)-covering space. To make this a success, we need to strongly investigate some properties of a weakly local (WL-, for short) (k0, k1)-isomorphism. More precisely, we initially prove that a digital-topological imbedding w.r.t. a (k0, k1)- isomorphism implies a WL-(k0, k1)-isomorphism. Besides, while a WL-(k0, k1)-isomorphism is proved to be a (k0, k1)-continuous map, it need not be a surjection. However, the converse does not hold. Taking this approach, we prove that aWL-(k0, k1)-isomorphic surjection is equivalent to a pseudo-(k0, k1)-covering map, which simplifies the earlier axiom for a pseudo (k0, k1)-covering space by using one condition. Finally, we further explore some properties of a pseudo (k0, k1)-covering space regarding lifting properties. The present paper only deals with k-connected digital images.

The multiplicative order convergence was studied and investigated on Riesz algebras. This paper deals with Riesz algebras and different topologies on them. In this paper, we investigate Riesz algebras on which we define various kinds of continuities. We give the relation between them under certain specific conditions. We show some relations among locally full, locally convex and locally solid Riesz algebras. Also, we introduce the notions of order and topological continuity of algebraic multiplications on topological Riesz algebras. Also, we extend the multiplication to quotient spaces of Riesz algebras. © 2022, University of Nis. All rights reserved.

In this paper, we devote our research to the essential spectra of linear relations defined on a Banach space. We extend the main results of paper [1] to linear relations.

In this paper, we introduce a new class of submanifolds which are called pointwise quasi bi-slant submanifolds in almost Hermitian manifolds which extends quasi bi-slant, bi-slant, hemi-slant, semi-slant and slant submanifolds in a very natural way. Several basic results in this respect are proved in this paper. Moreover, we obtain some conditions of the distributions which is involved in the definition of the new submanifolds. We also get some results for totally geodesic and mixed totally geodesic conditions for pointwise quasi bi-slant submanifolds. Finally, we illustrate some examples in order to guarantee the new kind of submanifolds.

In this article, we discuss interpolating sesqui-harmonic slant curves in generalized Sasakian space form and find the necessary and sufficient conditions for slant curves to be interpolating sesquiharmonic. Next, we study sesqui minimal slant curves in generalized Sasakian space form. In particular we give an example of interpolating sesqui-harmonic slant curve in Sasakian space form. Our paper generalizes the results of the papers [8, 16, 20].

In this paper, Lupa? Bernstein-Kantorovich operators have been studied using Jackson and Riemann type (p,q)-integrals. It has been shown that (p, q)-integrals as well as Riemann type (p, q)-integrals are not well defined for 0 < q < p < 1 and thus further analysis is needed. Throughout the paper, the case 1 ? q < p < ? has been used. Advantages of using Riemann type (p, q)-integrals are discussed over general (p, q)-integrals. Lupa? Bernstein-Kantorovich operators constructed via Jackson integral need not be positive for every f ? 0. So to make these operators based on general (p, q)-integral positive, one need to consider strictly monotonically increasing functions, and to handle this situation Lupa? Bernstein-Kantorovich operators are constructed using Riemann type (p, q)-integrals. However Lupa? (p, q)-Bernstein-Kantorovich operators based on Riemann type (p, q)-integrals are always positive linear operators. Approximation properties for these operators based on Korovkin?s type approximation theorem are investigated. The rate of convergence via modulus of continuity and function f belonging to the Lipschitz class is computed.

In this paper, we study additive properties of the generalized Drazin inverse in a Banach algebra. We first show that a + b ? Ad under the condition that a, b ? Ad, aba? = ?a?bab?a?, and then give some explicit expressions for the generalized Drazin inverse of the sum a+b under some weaker conditions than those used in the previous papers. Some known results are extended.

In this paper, we investigate the recent paper of Shakoor et al. [A. Shakoor, I. Ali, S. Wali, A. Rehman, Some formulas on the Drazin inverse for the sum of two matrices and block matrices, Bull. Iran. Math. Soc. 48 (2022) 351-366]. Here we prove that the main, additive result from the mentioned paper is actually a corollary of one known result. Furthermore, we give new representations for the Drazin inverse of anti-triangular block matrix, which generalize some representations from current literature on the topic.

Compound random sums are extensions of random sums when the random number of summands is a partial sum of independent and identically distributed positive integer-valued random variables, which assumed independent of summands. In the paper, upper bounds for the large O?rates of convergence in weak limit theorems for compound random sums of arrays of row-wise independent random variables, in term of Trotter distance are studied. The main results are approximation theorems which give the Trotter distance between normalized compound random sums of the given independent random variables and the compound ??decomposable random variables. By these results the converging rates in central limit theorem, weak law of large numbers and stable limit theorem for compound random sums are then established. The obtained results in this paper are closely related to the classical ones.

In this paper, we introduce a new concept of a generalized analytic Feynman integral combining the bounded linear operators on abstract Wiener space. We then obtain some Feynman integration formulas involving the generalized first variation. These formulas are more generalized forms rather than the formulas studied in previous papers. Finally, we establish a generalized Cameron-Storvick theorem, and give some examples to illustrate the usefulness of our results and formulas.

This paper is concerned with a boundary value problem (BVP) for discrete Sturm-Liouville equation with point interaction and boundary conditions depending on a hyperbolic eigenvalue parameter. This paper presents some spectral and scattering properties of this BVP in terms of Jost solution, scattering solutions, scattering function, continuous and discrete spectrum. In addition, the resolvent operator of the BVP is obtained to get the properties of eigenvalues. Furthermore, an example is considered as a special case of the main problem to demonstrate the effectiveness of our results.

In this paper, we study the symmetric and the generating functions for odd and even terms of the second-order linear recurrence sequences. we introduce a operator in order to derive a new family of generating functions of odd and even terms of Mersenne numbers, Mersenne Lucas numbers, (p,q)- Fibonacci-like numbers, k-Pell polynomials and k-Pell Lucas polynomials. By making use of the operator defined in this paper, we give some new generating functions of the products of (p,q)-Fibonacci-like numbers with odd and even terms of certain numbers and polynomials.

In this paper, we establish some new quantum Hermite-Hadamard type inequalities for differentiable convex functions by using the q?2-quantum integral. The results presented in this paper extend the results of Bermudo et al. (On q-Hermite-Hadamard inequalities for general convex functions, Acta Mathematica Hungarica, 2020, 162, 363-374). Finally, we give some examples to show validation of new results of this paper.

Let Hmxn be the set of all m x n matrices over the real quaternion algebra. We call that A ? Hnxn is ?-Hermitian if A = A?* where A?* = -?A*?,? ? {i,j,k},i,j,k are the quaternion units. In this paper, we derive some solvability conditions and the general solution to a system of real quaternion matrix equations. As an application, we present some necessary and sufficient conditions for the existence of an ?-Hermitian solution to some systems of real quaternion matrix equations. We also give the expressions of the general ?-Hermitian solutions to these systems when they are solvable. Some numerical examples are given to illustrate the results of this paper.

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk D is due to the dissertation of Luzin. The paper [11] was devoted to the Dirichlet problem with continuous boundary data for quasilinear Poisson equations in smooth (C1) domains. The present paper is devoted to the Dirichlet problem with arbitrary measurable (over natural parameter) boundary data for the quasilinear Poisson equations in any Jordan domains with rectifiable boundaries. For this purpose, it is constructed completely continuous operators generating nonclassical solutions of the Dirichlet boundary-value problem with arbitrary measurable data for the Poisson equations ?U = G with the sources G ? Lp, p > 1. The latter makes it possible to apply the Leray-Schauder approach to the proof of theorems on the existence of regular nonclassical solutions of the measurable Dirichlet problem for quasilinear Poisson equations of the form ?U(z) = H(z)?Q(U(z)) for multipliers H ? Lp with p > 1 and continuous functions Q : R ? R with Q(t)/t ? 0 as t ? ?. Here the boundary values are interpreted in the sense of angular (along nontangential paths) limits that are a traditional tool of the geometric function theory in comparison with variational interpretations in PDE. As consequences, we give applications to some concrete semi-linear equations of mathematical physics, arising under modelling various physical processes such as diffusion with absorption, plasma states, stationary burning etc.

In this paper, the nature of the singularity of a meromorphic functions of the form f (z) = 1 h(z) + a for a ? C and h is an entire function having a Baker wandering domain, lying over the Baker omitted value is discussed. Various dynamical issues relating to the singular values of f have been studied. Also following are shown in this paper. If a be the Baker omitted value of f then f has a Quasi-nested wandering domain U if and only if there exists {nk}k>0 such that each Unk surrounds a and Unk ? a as k ? ?. If f is a function having Quasi-nested wandering domain then all the Fatou components of f are bounded. In particular, f has no Baker domain. Also existence of Quasi-nested wandering domain ensures that the Julia component containing ? i.e., J? is a singleton buried component. At the end of the paper a result about the non existence of Quasi-nested wandering domain is given.

The purpose of this paper is to continue the study initiated in [7] concerning the notion of pseudo-generalized inverse. In this work, we are interested in investigating the relationship between compactness, finite rank and pseudo-generalized invertibility for bounded operators. The preserving properties of products of pseudo-generalized invertible operators are also treated and related results are given. At the end of this paper, we discuss in some special cases the relationship between the sets of n-left pseudo-generalized inverses of a left pseudo-generalized invertible operator T.

The purpose of this paper is concerned with the existence of extremal mild solutions for Sobolev type Hilfer fractional evolution equations with nonlocal conditions in an ordered Banach spaces E. By using monotone iterative technique coupled with the method of lower and upper solutions, with the help of the theory of propagation family as well as the theory of the measure of noncompactness and Sadovskii?s fixed point theorem, we obtain some existence results of extremal mild solutions for Hilfer fractional evolution equations. Finally, an example is provided to show the feasibility of the theory discussed in this paper

In this paper, we establishes certain new weighted Young and P?lya-Szeg?-type inequalities involving Marichev-Saigo-Maeda fractional integral operators. Meanwhile, corresponding weighted Cauchy-Schwarz type inequalities, Shisha-Mond type inequalities and Diaz-Metcalf type inequalities for Marichev-Saigo-Maeda fractional integral operators are also obtained. As applications, some estimates for weighted Chebyshev-type inequalities with two unknown functions for Marichev-Saigo-Maeda fractional integral operators are presented. The main results of this paper are more general and include a great number of existing classical inequalities.

In this paper, the complete moment convergence of weighted sums for widely orthant dependent (WOD, in short) random variables are established. The results obtained in the paper generalize and improve some known ones. As an application of the main results, we present a result on complete consistency for the weighted estimator in a nonparametric regression model based on WOD errors.

Let (un:n=1,2,...) be a sequence of real or complex numbers. We aim in this paper to determine necessary and/or sufficient conditions under which convergence of a sequence (un) or its certain subsequences follows from summability by deferred Ces?ro means. We also investigate the limiting behavior of deferred moving averages of (un). The conditions in our theorems are one-sided if (un) is a sequence of real numbers, and two-sided if (un) is a sequence of complex numbers. The theory developed in this paper should be useful for developing more interesting and useful results in connection with other sophisticated summability means as well as to extend to other spaces like ordered linear spaces.

In the present paper, the solvability of the stochastic Helmholtz problem is investigated in the class of stochastic differential equations equivalent in distribution. Earlier, by additional variables method the Helmholtz problem was investigated in the class of stochastic differential equations equivalent almost surely (a.s.). The study of the stochastic Helmholtz problem in the class of equations equivalent in distribution allows us to significantly expand the region of its solvability. This is due to the possibility of using well-known methods of the theory of stochastic processes, such as the method of the phase space transformation, the method of absolutely continuous change of measure, and the method of random change of time. In that paper stochastic equations of the Lagrangian structure equivalent in distribution are constructed by the given second order Ito stochastic equations using the methods of phase space transformation, absolutely continuous measure transformation and randomtime substitution. The obtained results are illustrated by specific examples.

The main contributions of this paper is twofold. First, our primary concern is to suggest a new iterative algorithm using the P-?-proximal-point mapping technique and Nadler?s technique for finding the approximate solutions of a system of generalized multi-valued nonlinear variational-like inclusions. Under some appropriate conditions imposed on the parameters and mappings involved in the system of generalized multi-valued nonlinear variational-like inclusions, the strong convergence of the sequences generated by our proposed iterative algorithm to a solution of the aforesaid system is proved. Second, the H(.,.)-?-cocoercive mapping considered in [R. Ahmad, M. Dilshad, M. Akram, Resolvent operator technique for solving a system of generalized variational-like inclusions in Banach sapces, Filomat 26(5)(2012) 897- 908] is investigated and analyzed, and the fact that under the assumptions imposed on H(., .)-?-cocoercive mapping, every H(.,.)-?-cocoercive mapping is P-?-accretive and is not a new one is pointed out. At the same time, some important comments on H(.,.)-?-cocoercive mapping and the results given in the above-mentioned paper are stated.

The purpose of this paper is to give and prove the fundamental theorem of conformable fractional calculus which is given only in previous associated papers for differentiable functions, and then we scrutinize the existence and uniqueness of solutions to some semilinear Cauchy problems for nonlocal conformable fractional integrodifferential or differential equations which their nonlinear terms include fractional derivative or fractional integral. Two examples are investigated to elucidate the main results.

There are three weighted decompositions of tensors proposed in this paper, and the corresponding definitions of the weighted generalized tensor functions are given. The Cauchy integral formula of the weighted Moore-Penrose inverse is developed for solving the tensor equations. Besides above, we give the weighted projection tensors to discuss the representations of the weighted generalized power of tensors. Finally, some special tensors are studied which can preserve the structural invariance under the tensor functions defined in this paper.

In this paper, our interest is devoted to study the convex combinations of the form (1 ? ?) f + ?1, where ? ? (0, 1), of biholomorphic mappings on the Euclidean unit ball Bn in the case of several complex variables. Starting from a result proved by S. Trimble [26] and then extended by P.N. Chichra and R. Singh [3, Theorem 2] which says that if f is starlike such that Re[f ?(z)] > 0, then (1??)z+?f (z) is also starlike, we are interested to extend this result to higher dimensions. In the first part of the paper, we construct starlike convex combinations using the identity mapping on Bn and some particular starlike mappings on Bn. In the second part of the paper, we define the class L* ?(Bn) and prove results involving convex combinations of normalized locally biholomorphic mappings and Loewner chains. Finally, we propose a conjecture that generalize the result proved by Chichra and Singh.

In this paper, two collocation methods based on the shifted Legendre polynomials are proposed for solving system of nonlinear Fredholm-Volterra integro-differential equations. The equation considered in this paper involves the derivative of unknown functions in the integral term, which makes its numerical solution more complicated. We first introduce a single-step Legendre collocation method on the interval [0, 1]. Next, a multi-step version of the proposed method is derived on the arbitrary interval [0, T] that is based on the domain decomposition strategy and specially suited for large domain calculations. The first scheme converts the problem to a system of algebraic equations whereas the later solves the problem step by step in subintervals and produces a sequence of systems of algebraic equations. Some error estimates for the proposed methods are investigated. Numerical examples are given and comparisons with other methods available in the literature are done to demonstrate the high accuracy and efficiency of the proposed methods.

© 2022, University of Nis. All rights reserved.In this paper the authors gave an iteration identity for the generalized Laplace transform L2n and the generalized Glasser transform G2n. Using this identity a Parseval-Goldstein type theorem for the L2n-transform and the G2n-transform is given. By making use of these results a number of new Parseval-Goldstein type identities are obtained for these and many other well-known integral transforms.The identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions. Some examples are also given.

The authors discuss mainly that the Riemannian manifold Mn admitting a unit preserving circle field ? in the present paper. A sufficient and necessary condition is given that Riemannian manifold Mn is an Einstein manifold by imposing some conditions on W2 curvature tensor. Further, this paper obtains the algebra representation of curvature tensors of a W2-recurrent Riemannian manifold Mn given by R???? = 1/d2 [d?d?R?? ? d?d?R?? + d?d?R?? ? d?d?R??].

In this paper, we propose a self-adaptive algorithm for solving the split common null point problem and the fixed point problem for multivalued Bregman quasi-nonexpansive mappings in Banach spaces. We prove that the sequence generated by our iterative scheme converges strongly to a common solution of the above-mentioned problems under some suitable conditions. We also apply our main result to split feasibility problems in Banach spaces. Finally, numerical examples are given to support our main theorem. The results presented in this paper improve and extend many recent results in the literature.

In this paper, the integral identity for differentiable functions with the wide range of applications is investigated. As an effect of this result, the paper provides some new Hermite-Hadamard type inequalities for differentiable functions that are in absolute value h-preinvex. Several special cases are also discussed. At the end, some applications for special means are given as well.

In this paper we propose a stochastic SIR epidemic model to evaluate effect of the randomness on treatment and nonlinear incidence rate. More precisely, we perturb both nonlinear incidence and treatment rates in deterministic SIR model with Gaussian white noise and obtain two diffusion stochastic model. For the model, we theoretically prove that it?s solution is positive and global, and then, we obtain the conditions under which we can claim the existence of the stationary distribution. Also, by constructing suitable Lyapunov functions, weestablish sufficient conditions for p-th moment and almost sure exponential stability of disease-free equilibrium. Conditions for disease extinction are obtained, as well. We close the paper by presenting numerical simulations to verify our theoretical results. For that purpose we use real-life data for spread of cholera in the Department of Artibonite in Haiti, as well as for influenza A H1N1 in Guangdong Province, China.

Throughout the present paper, K denotes a complete, non-trivially valued, ultrametric (or nonarchimedean) field. Sequences, infinite series and infinite matrices have their entries in K. The sequence spaces m?, c?, c?0 were introduced in K earlier by the author in [8-10] and some studies were made. The purpose of the present paper is to characterize the matrix classes (c?0, c? 0), (c?0,m?), (c?0, c?) and (c?, c?0).

In domain theory, by a poset model of a T1 topological space X we usually mean a poset P such that the subspace Max(P) of the Scott space of P consisting of all maximal points is homeomorphic to X. The poset models of T1 spaces have been extensively studied by many authors. In this paper we investigate another type of poset models: lower topology models. The lower topology ?(P) on a poset P is one of the fundamental intrinsic topologies on the poset, which is generated by the sets of the form P\?x, x ? P. A lower topology poset model (poset LT-model) of a topological space X is a poset P such that the space Max?(P) of maximal points of P equipped with the relative lower topology is homeomorphic to X. The studies of such new models reveal more links between general T1 spaces and order structures. The main results proved in this paper include (i) a T1 space is compact if and only if it has a bounded complete algebraic dcpo LT-model; (ii) a T1 space is second-countable if and only if it has an ?-algebraic poset LT-model; (iii) every T1 space has an algebraic dcpo LT-model; (iv) the category of all T1 space is equivalent to a category of bounded complete posets. We will also prove some new results on the lower topology of different types of posets.

In this paper, we present the necessary and sufficient conditions for a poset to be a poset of the union of join and meet irreducible elements of the slim lattice. Slim lattices are special finite lattices that are intensively investigated recently. The problem that we solved in this paper is a generalization of the problem proposed very recently by Cz?dli.

Throughout this paper, K denotes a complete, non-trivially valued, ultrametric (or nonarchimedean) field. Sequences, infinite series and infinite matrices have entries in K. In this paper, we record some interesting properties about the matrix classes (c0,c0) and (c0,c0;P).

In this paper, a new form of separation axioms called r-fuzzy soft Ti;(i = 0,1,2,3,4), r-fuzzy soft regular and r-fuzzy soft normal axioms are introduced in a fuzzy soft topological space based on the paper Ayg?no?lu et al. [7]. Also, the relations of these axioms with each other are investigated with the help of examples. Furthermore, some fuzzy soft invariance properties, namely fuzzy soft topological property and hereditary property are specified.

In this paper, we point out that the proof of Theorem 2.4(5), Proposition 2.6(1) and Proposition 2.8(1) in the paper titled ?On (L,M)-fuzzy convex structures? (Filomat 33(13): 4151-4163, 2019) are not true in general. Then, we give three correct proofs of these results

This paper presents the regularity of a split-quaternionic function and a corresponding split- Cauchy-Riemann system of a split quaternion. The properties of an inverse and an implicit mapping theory for a split-quaternionic map are investigated. In addition, the paper proposes a definition and expression for a split biregular mapping in an open set in C2. The obtained results are illustrated with some examples.

The main goal of this paper is to establish the Lq-convergence of the truncated Euler-Maruyama method for neutral stochastic differential equations with time-dependent delay under the Khasminskii-type condition. Whole consideration is influenced by the presence of the neutral term and delay function. The main theoretical result is illustrated by an example. Since the main result is related to the Lq-convergence of the Euler-Maruyama method under the global Lipschitz condition on the coefficients of the equation under consideration, for completeness of the paper, the appropriate results are given in Appendix.

The subject of this paper is an analytic approximate method for a class of stochastic differential equations with coefficients that do not necessarily satisfy the Lipschitz and linear growth conditions but behave like a polynomials. More precisely, equations from the observed class have unique solutions with bounded moments and their coefficients satisfy polynomial condition. Approximate equations are defined on partitions of a time interval, and their coefficients are Taylor approximations of the coefficients of the initial equation. The rate of Lp convergence increases when degrees in Taylor approximations of coefficients increase. At the end of the paper, an example is provided to support the main theoretical result.

In this paper, we define and introduce some new concepts of the higher order strongly generalized preinvex functions and higher order strongly monotone operators involving the arbitrary bifunction and function. Some new relationships among various concepts of higher order strongly general preinvex functions have been established. It is shown that the optimality conditions of the higher order strongly general preinvex functions are characterized by a class of variational inequalities, which is called the higher order strongly generalized variational-like inequality. Auxiliary principle technique is used to suggest an implicit method for solving higher order strongly generalized variational-like inequalities. Convergence analysis of the proposed method is investigated using the pseudo-monotonicity of the operator. It is shown that the new parallelogram laws for Banach spaces can be obtained as applications of higher order strongly affine generalized preinvex functions, which is itself a novel application. Some special cases also discussed. Results obtained in this paper can be viewed as refinement and improvement of previously known results.

Karapinar introduced the notion of interpolative Ciric-Reich-Rus type contractions in the setting of complete metric space. Taking his approach forward, H. Aydi, initiated the concept of w-admissibility and proved some fixed point results on the same. This approach has been applied to partial-metric space as well. But the question is, can the above result be applied in quasi-partial b-metric space as well? Our paper deals with the above raised question. The paper discusses how w-admissibility can be used to obtain fixed point results for Ciric-Reich-Rus type contractions in quasi-partial b-metric space. Few examples are given to justify the result.