Journal of Nonlinear Analysis and Optimization: Theory & Applications (JNAO)

THE NAKAJO-TAKAHASHI TYPE PROJECTION METHOD FOR A NONEXPANSIVE MAPPING ON A HADAMARD SPACE

S. Sudo — 2025-12-31

In this work, we propose a new iterative scheme on a Hadamard space to find a fixed point of a nonexpansive mapping. In this purpose, we deal with a projection method introduced by Nakajo and Takahashi, and we use a tangent space on a Hadamard space to generate an iterative set-sequence.

SOLVING FREDHOLM INTEGRAL EQUATION VIA FIXED POINT THEOREM IN CONTROLLED METRIC TYPE SPACES

V. Sihag — 2025-12-31

In the present article, the concepts of hesitant fuzzy mapping, contraction of hesitant fuzzy mapping, and generalized contraction of hesitant fuzzy mapping in the settings of the controlled metric-type spaces. The newly introduced notions are utilized to establish fixed-point theorems. Finally, the results are utilized to solve a Fredholm-type integral equation.

SHRINKING PROJECTION METHOD WITH ALLOWABLE RANGES FOR ZERO POINT PROBLEMS IN A BANACH SPACE

T. Ibaraki — 2025-12-31

In this paper, we study the shrinking projection method with allowable ranges introduced by Takeuchi (J. Nonlinear Anal. Optim, 10:83-94, 2019) for the zero point problem. We obtain strong convergence theorems for finding a zero point of a maximal monotone operator in a Banach space. Using our results, we discuss the convex minimization problem.

FIXED POINT ASSOCIATED WITH A NEW CLASS OF CONDENSING OPERATORS AND SOLVABILITY OF VOLTERRA INTEGRAL EQUATION HAVING DELAY

V. Nikam — 2025-12-31

This work presents a novel class of condensing operators to explore the possibility of solutions for the Volterra integral equation with a singular kernel and proportional delay. These equations are significant in many domains, including engineering and physics, yet conventional solution techniques face substantial difficulties due to single kernels and delays. To solve this, we provide a more flexible method of handling such equations by creating a class of condensing operators based on pairs of functions that satisfy specific local requirements. We define these operators and also show some fixed point theorems that expand the application of Darbo's fixed point theorem to a broader class of situations. At the end we provide examples to illustrate our theoretical results and show that the suggested approach works well.

GENERALISED α-m MONOTONICITY AND β-WELL POSEDNESS OF SET VARIATIONAL INEQUALITY PROBLEM

M. Chaudhary — 2025-12-31

We define generalised α-m monotonicity and generalised α-m pseudo-monotonicity for set maps. We also present a new concept of well-posedness, namely β-well-posedness for set variational inequality problem (VI). We further study the relationship of these monotone maps along with (VI). Then we demonstrate a gap function for the above (VI). Beneath the assumption of the said pseudo-monotonicity, a result is obtained showing the relation between the solution and gap function of the said (VI) problem. Finally, with the help of this gap function, we formulate said variational inequality problem into a corresponding mathematical programming problem (MP) and establish the relations between the β-well-posedness of both problems.

SOLUTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS INVOLVING MIXED PARTIAL DERIVATIVE BY ELZAKI SUBSTITUTION METHOD

A. S. Salve — 2025-12-31

In this paper we have to find the solution of nonlinear partial differential equation involving mixed partial derivatives by Elzaki Substitution method. Elzaki trans-form is applied and then nonlinear term handled with the help of Adomian polynomial. We get exact solution of nonlinear partial differential equations involving mixed partial derivatives. It is believed that this work will make it easy to study the nonlinear partial differential equations involving mixed partial derivatives arising different areas of research and innovation. Therefore the current method can be extended for the solution of higher order nonlinear problems. We give illustration through three problems.

SOLUTION OF MIXED ORDERDED SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS VIA LAPALCE DECOMPOSITION METHOD

M. Kangale — 2025-12-31

In this work, we solve a system of partial differential equations with mixed fractional-order derivatives, described in the Caputo sense. The Laplace Decomposition Method (LDM) is used for the solution of PDE. The efficacy of the recommended methodology is illustrated by a number of cases. The correctness of the method is confirmed by the solution graphs, which closely match the exact and LDM solutions. The method's dependability is further confirmed by the observation that the solutions to the fractional-order issues converge towards the solutions to the equivalent integer-order problems.

ON THE REALM OF WEAKLY NONEXPANSIVE MAPS ON QUASI-METRIC SPACES

S. Park — 2025-12-31

Let $(X,d)$ be a metric space. There have appeared thousands of works on nonexpansive maps $f: X\to X$ and the way to get their fixed points. Recently, many maps of the type satisfying $d(fx, f^2x) \le d(x, fx)$ on $x\in X$ appeared in the literature and they are called the weakly nonexpansive maps. There are a large number of fixed point theorems on weakly nonexpansive maps on metric spaces. Their proofs are different each other. In the present article, we trace the history of their applications to ordered fixed point theory. Certain new proofs are also given for known theorems.

LEVITIN-POLYAK WELL-POSEDNESS OF MIXED VARIATIONAL INEQUALITIES INVOLVING A BIFUNCTION

M. Mehta — 2025-12-31

In this paper, we investigate the existence and Levitin–Polyak (LP) well-posedness of a mixed variational inequality problem involving a bifunction. Sufficient conditions for the existence of solutions are established. We explore the connection between saddle points of the associated Lagrangian and solutions to both the original variational inequality and its Minty counterpart. The study presents several key results on LP well-posedness and generalized LP well-posedness, formulated in terms of the behaviour of the approximate solution sets. A notable aspect of this work is the well-posedness analysis based on the gap function approach. More specifically, sufficient conditions for LP well-posedness of mixed variational inequality problem are provided in terms of the level boundedness of its gap function. Furthermore, an equivalence between the LP well-posedness of the mixed variational inequality problem and that of a related optimization problem is also established.