RGNAA PublishingCommunications in Nonlinear Analysis2371-79209220211001AN APPLICATION OF FIXED POINT THEORY TO A NONLINEAR INTEGRAL EQUATION IN BANACH SPACE124112710ENAustine EfutOfemDepartment of Mathematics, University of Uyo, Uyo, Nigeria.0000-0001-8064-2326Journal Article20200514In this paper we propose a new iterative scheme, called the AF iteration process, for approximating the unique solution of a mixed type Volterra-Fredholm functional nonlinear integral equation. We prove in the sense of Berinde \cite{Berin} that our new iterative scheme converges at a rate faster than some of the leading iterative schemes in the literature which have been employed recently to approximate the unique solution of a mixed type Volterra Fredholm functional nonlinear integral equation. We also prove that our new iterative method converges strongly to the unique solution of a mixed type Volterra Fredholm functional nonlinear integral equation. In addition, we give data dependence result for the solution of the nonlinear integral equation which we are considering with the help of our new iterative scheme. Our results improve and unify some well known results in the existing literature.RGNAA PublishingCommunications in Nonlinear Analysis2371-79209220211001Periodic and fixed points of the Leader-type contractions in quasi-triangular spaces]{Periodic and fixed points of the Leader-type contractions in quasi-triangular spaces116118111ENSaurabhManroANIMESHGuptaDepartment of Mathematics, Sagar Institute of Engineering, Technology and Research, Ratibad Bhopal (M.P.), IndiaTejwantSinghDepartment of Mathematics, Desh Bhagat University, Mandi Gobindgarh, Punjab, IndiaRajvirKaurDesh Bhagat universityJournal Article20191030Let \(C=\{C_{\alpha}\}_{\alpha\in\mathcal{A}}\in[1;\infty)^{\mathcal{A}}\) with index set \(\mathcal{A}\). A quasi-triangular space \((X,\mathcal{P}_{C;\mathcal{A}})\) is a set X with family \(\mathcal{P}_{C;\mathcal{A}}=\{p_{\alpha}:X^{2}\rightarrow[0,\infty),\alpha \in \mathcal{A}\}\) satisfying \(\forall_{\alpha\in\mathcal{A}}\forall _{u,v,w\in X}\{p_{\alpha}(u,w)\leq C_{\alpha}[p_{\alpha }(u,v)+p_{\alpha }(v,w)]\}\). In \((X,\mathcal{P}_{C;\mathcal{A}})\), using the left (right) families \(\mathcal{J}_{C;\mathcal{A}}\) generated by \(\mathcal {P}_{C;\mathcal{A}}\) (\(\mathcal{P}_{C;\mathcal{A}}\) is a particular case of \(\mathcal {J}_{C;\mathcal{A}}\)), we establish theorems concerning left (right) \(\mathcal {P}_{C;\mathcal{A}}\)-convergence, existence, periodic point, fixed point, and (when\((X,\mathcal{P}_{C;\mathcal{A}})\) is separable) uniqueness for \(\mathcal{J}_{C;\mathcal{A}}\)-contractions and weak \(\mathcal {J}_{C;\mathcal{A}}\)-contractions \(T:X\rightarrow X\) satisfying <br />\begin{eqnarray*}\begin{aligned}<br />& \forall_{x,y\in X}\forall _{\alpha\in\mathcal{A}}\forall_{\varepsilon>0}\exists_{\eta >0}\exists _{r\in\mathbb{N}}\forall_{s,l\in\mathbb{N}} \{J_{\alpha }(T^{[s]}(x),T^{[s+r]}(x)) + J_{\alpha }(T^{[l]}(y),T^{[l+r]}(y)) < \eta+\varepsilon \\ & \Rightarrow C_{\alpha }J_{\alpha}(T^{[s+r]}(x),T^{[l+r]}(y))<\varepsilon\}<br />\end{aligned}\end{eqnarray*}<br />and <br />\begin{eqnarray*}\begin{aligned}<br />& \exists _{w^{0}\in X}\forall_{\alpha\in\mathcal{A}}\forall_{\varepsilon >0}\exists_{\eta>0}\exists_{r\in\mathbb{N}} \forall_{s,l\in \mathbb{N}} \{J_{\alpha}(T^{[s+r]}(w^{0}), T^{[s]}(w^{0}))+J_{\alpha}(T^{[l]}(w^{0}), T^{[l+r]}(w^{0}))<\eta+\varepsilon \\ & \Rightarrow C_{\alpha}J_{\alpha }(T^{[s+r]}(w^{0}),T^{[l+r]}(w^{0}))<\varepsilon\},<br />\end{aligned} \end{eqnarray*} <br />respectively. The spaces \((X,\mathcal{P}_{C;\mathcal{A}})\), in particular, generalize metric, ultrametric, quasi-metric, ultra-quasi-metric, b-metric, partial metric, partial b-metric, pseudometric, quasi-pseudometric, ultra-quasi-pseudometric, partial quasi-pseudometric, topological, uniform, quasi-uniform, gauge, ultra gauge, partial gauge, quasi-gauge, ultra-quasi-gauge, and partial quasi-gauge spaces. Results are new in all these spaces. Examples are provided.RGNAA PublishingCommunications in Nonlinear Analysis2371-79209220211001On Hardy-Rogers Type Contraction Mappings in Cone A_{b}-metric spaces116136011ENIsaYildirimAtaturk University0000-0001-6165-716XJournal Article20210522In this manuscript, a generalized fixed point theorem of Hardy-Rogers type contraction is proved in cone A_{b}-metric spaces, which relaxes the contraction condition. Also, some fixed point results for different contraction mappings are given in such spaces.RGNAA PublishingCommunications in Nonlinear Analysis2371-79209220211001Convergence of an M-step Iteration Scheme to the Commo Fixed Points of Finite Family of Mixed-Type Total Asymtotically Nonexpansive Mappings in Uniformly Convex Banach Spaces117136013ENImo KaluAgwuMichael Okpara University of Agriculture, Umudike, Umuahia0000-0003-4535-1974Daonatus IkechiIgbokweDepartment of Mathematics, Michael Okpara University of Agriculture, Umudike, Abia State - NigeriaJournal Article20210531We propose an m-step iteration scheme of hybrid mixed-type for m-total asymptotically nonexpansive self mappings and m-total asymptotically nonexpansive nonself mappings and establish some weak convergence theorems of the scheme to the common xed point of the mappings in the setting of uniformly convex Banach spaces. In additon, we proved demiclosedness principle for total asymptotically nonexpansive nonself mappings in uniformly convex Banach spaces. Our results extend and genralise the results in [23] and other numerous results currently in literature.RGNAA PublishingCommunications in Nonlinear Analysis2371-79209220211001CONVERGENCE THEOREMS FOR MONOTONE GENERALIZED α-NONEXPANSIVE MAPPINGS IN ORDERED BANACH SPACE BY A NEW FOUR-STEP ITERATION PROCESS WITH APPLICATION117138222ENUnwanaUdofiaDEPARTMENT OF MATHEMATICS AND STATISTICS, AKWA IBOM STATE UNIVERSITY, IKOT AKPADEN, MKPAT ENIN, NIGERIA.0000-0002-8640-5804DonatusIgbokweDepartment of Mathematics,
Michael Okpara University of Agriculture,
Umudike, Abia State,
Nigeria.0000-0002-8574-6658Journal Article20210212We introduce a new four-step iterative algorithm and show that the new algorithm converges faster than a number of existing iterative algorithms for contraction mappings. We prove strong and weak convergence results for approximating fixed points of monotone generalized α-nonexpansive mappings. Further, we utilize our proposed algorithm to solve Split Feasibility<br />Problem (SFP). Our result complements, extends and generalizes some existing results in literature.