In this article, we establish some common fixed point theorems for a hybrid pair {g, T} of single valued and multi-valued maps satisfying a generalized contractive condition defined on G-metric spaces. Our results unify, generalize and complement various known comparable results from the current literature. 2000 MSC: 54H25; 47H10; 54E50.

This work provides exact and analytical approximate solutions for a non-linear time-fractional generalized biology population model (FGBPM) with suitable initial data under the time-Caputo fractional derivative, in view of a novel effective and applicable scheme, based upon elegant amalgamation between the Laplace transform operator and the generalized power series method. The solution form obtained by the proposed algorithm of considered FGBPM is an infinite multivariable convergent series toward the exact solutions for the integer fractional order. Some applications of the posed model are tested to confirm the theoretical aspects and highlight the superiority of the proposed scheme in predicting the analytical approximate solutions in closed forms compared to other existing analytical methods. Associated figure representations and the results are displayed in different dimensional graphs. Numerical analyses are performed, and discussions regarding the errors and the convergence of the scheme are presented. The simulations and results report that the proposed modern scheme is, indeed, direct, applicable, and effective to deal with a wide range of non-linear time multivariable fractional models.

In this paper, we compile the fractional power series method and the Laplace transform to design a new algorithm for solving the fractional Volterra integro-differential equation. For that, we assume the Laplace power series (LPS) solution in terms of power $q=\frac{1}{m},m\in Z^{+}$, where the fractional derivative of order $\alpha =q\gamma$, for which $\gamma \in Z^{+}$. This assumption will help us to write the integral, the kernel, and the nonhomogeneous terms as a LPS with the same power. The recurrence relations for finding the series coefficients can be constructed using this form. To demonstrate the algorithm’s accuracy, the residual error is defined and calculated for several values of the fractional derivative. Two strongly nonlinear examples are discussed to provide the efficiency of the algorithm. The algorithm gains powerful results for this kind of fractional problem. Under Caputo meaning of the symmetry order, the obtained results are illustrated numerically and graphically. Geometrically, the behavior of the obtained solutions declares that the changing of the fractional derivative parameter values in their domain alters the style of these solutions in a symmetric meaning, as well as indicates harmony and symmetry, which leads them to fully coincide at the value of the ordinary derivative. From these simulations, the results report that the recommended novel algorithm is a straightforward, accurate, and superb tool to generate analytic-approximate solutions for integral and integro-differential equations of fractional order.

Multi proxy signature schemes (MPSS) allow n proxy signers to generate a valid signature on behalf of the original signer. A new efficient self-certified MPSS with message recovery based on the Elliptic Curve Discrete Logarithm Problem (i.e., ECDLP) is proposed in this paper. A rationale for proposing a new scheme is to improve the verification process performance by combining the verification process with message recovery in one single stage (i.e., simultaneously). The security and performance analysis shows, the new scheme has many advantages over the traditional MPS schemes. These benefits include decreasing costs (i.e., certificate sizes, consumption time, and bandwidth requirements) and increasing speed.

Public key cryptography has received great attention in the field of information exchange through insecure channels. In this paper, we combine the Dependent-RSA (DRSA) and chaotic maps (CM) to get a new secure cryptosystem, which depends on both integer factorization and chaotic maps discrete logarithm (CMDL). Using this new system, the scammer has to go through two levels of reverse engineering, concurrently, so as to perform the recovery of original text from the cipher-text has been received. Thus, this new system is supposed to be more sophisticated and more secure than other systems. We prove that our new cryptosystem does not increase the overhead in performing the encryption process or the decryption process considering that it requires minimum operations in both. We show that this new cryptosystem is more efficient in terms of performance compared with other encryption systems, which makes it more suitable for nodes with limited computational ability.

The profound characteristics of factoring problems and chaotic maps are utilized in this article to produce a new identity-based cryptographic scheme. These characteristics include random behavior and high performance (i.e., encryption and decryption) in terms of so many parameters such as computational delay, rate of security, and speed. To illustrate the performance of the proposed cryptographic scheme, some cryptographic attacks are proposed. The mathematical analysis have confirmed that the combined factoring problems and chaotic maps scheme has a high rate of security which satisfy the cryptographic requirements. To the best of our knowledge, this is the first time an ID-based cryptography based on chaotic maps and factoring problems has been proposed.