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This research paper proposes a derivative-free method for solving systems of nonlinearequations with closed and convex constraints, where the functions under consideration are continuousand monotone. Given an initial iterate, the process first generates a specific direction and then employsa line search strategy along the direction to calculate a new iterate. If the new iterate solves theproblem, the process will stop. Otherwise, the projection of the new iterate onto the closed convex set(constraint set) determines the next iterate. In addition, the direction satisfies the sufficient descentcondition and the global convergence of the method is established under suitable assumptions.Finally, some numerical experiments were presented to show the performance of the proposedmethod in solving nonlinear equations and its application in image recovery problems.

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One of the fastest growing and efficient methods for solving the unconstrained minimization problem is the conjugate gradient method (CG). Recently, considerable efforts have been made to extend the CG method for solving monotone nonlinear equations. In this research article, we present a modification of the Fletcher–Reeves (FR) conjugate gradient projection method for constrained monotone nonlinear equations. The method possesses sufficient descent property and its global convergence was proved using some appropriate assumptions. Two sets of numerical experiments were carried out to show the good performance of the proposed method compared with some existing ones. The first experiment was for solving monotone constrained nonlinear equations using some benchmark test problem while the second experiment was applying the method in signal and image recovery problems arising from compressive sensing.

In this work, we describe the model of a new 5-D hyperchaotic system with three positive Lyapunov exponents. Since the maximum positive Lyapunov exponent of the proposed hyperchaotic system is larger than twelve, the new hyperchaotic system is highly hyperchaotic. We also show that the new 5-D hyperchaotic system exhibits multistability with coexisting attractors. Using Multisim, we design an electronic circuit for the new 5-D hyperchaotic system. The hardware implementation of the new 5-D hyperchaotic system is done by applying two numerical methods. From the experimental results of the FPGA-based implementation, we show that the attractors observed in a Lecroy oscilloscope are in good agreement with numerical simulations. To prove the reliability of the proposed system for cybersecurity purposes, we presented a new image cryptosystem using our hyperchaotic system. Experimental outcomes show the efficiency and the reliability of our cryptosystem based on the proposed hyperchaotic system.INDEX TERMS Hyperchaos, bifurcations, multi-stability, attractors, Lyapunov exponents, circuit design, FPGA, numerical methods, data security; image cryptosystem.

In this paper, we propose a modified conjugate descent (CD) projection algorithm for solving system of nonlinear monotone equations with convex constraints. The search direction in this algorithm use a convex combination of the steepest descent algorithm and the well-known CD method. The algorithm proves to be quite efficient for solving large scale monotone nonlinear equations, as it has low storage requirement and does not need the computation of Jacobian matrix. We prove the convergence of the algorithm using some conditions, and perform numerical experiments on some test problems. In order to show the efficiency of our proposed algorithm, the numerical performance is compared with some existing algorithms. Finally, by reformulating 1 regularized problem into monotone equation, we successfully apply the algorithm to restore some blurred images. The numerical results obtained prove that the algorithm can be used as an efficient and qualitative solver for image restoration problems.

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