picard sequence
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2022 ◽  
Vol 27 (1) ◽  
pp. 163-178
Author(s):  
Mudasir Younis ◽  
Aleksandra Stretenović ◽  
Stojan Radenović

In this manuscript, we generalize, improve, and enrich recent results established by Budhia et al. [L. Budhia, H. Aydi, A.H. Ansari, D. Gopal, Some new fixed point results in rectangular metric spaces with application to fractional-order functional differential equations, Nonlinear Anal. Model. Control, 25(4):580–597, 2020]. This paper aims to provide much simpler and shorter proofs of some results in rectangular metric spaces. According to one of our recent lemmas, we show that the given contractive condition yields Cauchyness of the corresponding Picard sequence. The obtained results improve well-known comparable results in the literature. Using our new approach, we prove that a Picard sequence is Cauchy in the framework of rectangular metric spaces. Our obtained results complement and enrich several methods in the existing state-ofart. Endorsing the materiality of the presented results, we also propound an application to dynamic programming associated with the multistage process.


Author(s):  
Anh Tuan Nguyen ◽  
Tomás Caraballo ◽  
Nguyen Huy Tuan

In this study, we investigate the intial value problem (IVP) for a time-fractional fourth-order equation with nonlinear source terms. More specifically, we consider the time-fractional biharmonic with exponential nonlinearity and the time-fractional Cahn–Hilliard equation. By using the Fourier transform concept, the generalized formula for the mild solution as well as the smoothing effects of resolvent operators are proved. For the IVP associated with the first one, by using the Orlicz space with the function $\Xi (z)={\textrm {e}}^{|z|^{p}}-1$ and some embeddings between it and the usual Lebesgue spaces, we prove that the solution is a global-in-time solution or it shall blow up in a finite time if the initial value is regular. In the case of singular initial data, the local-in-time/global-in-time existence and uniqueness are derived. Also, the regularity of the mild solution is investigated. For the IVP associated with the second one, some modifications to the generalized formula are made to deal with the nonlinear term. We also establish some important estimates for the derivatives of resolvent operators, they are the basis for using the Picard sequence to prove the local-in-time existence of the solution.


Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1921
Author(s):  
Slobodanka Mitrović ◽  
Vahid Parvaneh ◽  
Manuel De La Sen ◽  
Jelena Vujaković ◽  
Stojan Radenović

In this article, we generalize, improve, unify and enrich some results for Jaggi-W-contraction-type mappings in the framework of b-metric-like spaces. Our results supplement numerous methods in the existing literature, and we created new approach to prove that a Picard sequence is Cauchy in a b-metric-like space. Among other things, we prove Wardowski’s theorem, but now by using only the property (W1). Our proofs in this article are much shorter than ones in recently published papers.


2021 ◽  
Vol 69 (2) ◽  
pp. 338-354
Author(s):  
Zoran Mitrović ◽  
Mudasir Jounis ◽  
Miloje Rajović

Introduction/purpose: This article establishes several new contractive conditions in the context of so-called F-metric spaces. The main purpose was to generalize, extend, improve, complement, unify and enrich the already published results in the existing literature. We used only the property (F1) of Wardowski as well as one well-known lemma for the proof that Picard sequence is an F-Cauchy in the framework of F-metric space. Methods: Fixed point metric theory methods were used. Results: New results are enunciated concerning the F-contraction of two mappings S and T in the context of F-complete F-metric spaces. Conclusions: The obtained results represent sharp and significant improvements of some recently published ones. At the end of the paper, an example is given, claiming that the results presented in this paper are proper generalizations of recent developments.


2021 ◽  
Vol 69 (1) ◽  
pp. 8-30
Author(s):  
Mudasir Younis ◽  
Nicola Fabiano ◽  
Zaid Fadail ◽  
Zoran Mitrović ◽  
Stojan Radenović

Introduction/purpose: This paper considers, generalizes and improves recent results on fixed points in rectangular metric spaces. The aim of this paper is to provide much simpler and shorter proofs of some new results in rectangular metric spaces. Methods: Some standard methods from the fixed point theory in generalized metric spaces are used. Results: The obtained results improve the well-known results in the literature. The new approach has proved that the Picard sequence is Cauchy in rectangular metric spaces. The obtained results are used to prove the existence of solutions to some nonlinear problems related to chemical sciences. Finally, an open question is given for generalized contractile mappings in rectangular metric spaces. Conclusions: New results are given for fixed points in rectangular metric spaces with application to some problems in chemical sciences.


Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 940
Author(s):  
Ersin Gilić ◽  
Diana Dolićanin-Đekić ◽  
Zoran D. Mitrović ◽  
Dženis Pučić ◽  
Hassen Aydi

The purpose of this manuscript is to provide much simpler and shorter proofs of some recent significant results in the context of generalized F-Suzuki-contraction mappings in b-complete b-metric spaces. By using our new approach for the proof that a Picard sequence is b-Cauchy, our results generalize, complement and improve many known results in the existing literature. Further, some new contractive conditions are provided here to illustrate the usability of the obtained theoretical results.


Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 767 ◽  
Author(s):  
Jelena Vujaković ◽  
Slobodanka Mitrović ◽  
Mirjana Pavlović ◽  
Stojan Radenović

In this manuscript we discuss, consider, generalize, improve and unify several recent results for so-called F-contraction-type mappings in the framework of complete metric spaces. We also introduce ( φ , F ) -weak contraction and establish the corresponding fixed point result. Using our new approach for the proof that a Picard sequence is a Cauchy in metric space, our obtained results complement and enrich several methods in the existing literature. At the end we give one open question for F-contraction of Ćirić-type mapping.


2020 ◽  
Vol 68 (4) ◽  
pp. 697-714 ◽  
Author(s):  
Jelena Vujaković ◽  
Stojan Radenović

Introduction/purpose: This paper establishes some new results of Piri-Kumam-Dung-type mappings in a complete metric space.The goal was to improve the already published results. Methods: Using the property of a strictly increasing function as well as the known Lemma formulated in (Radenović et al, 2017), the authors have proved that a Picard sequence is a Cauchy sequence. Results: New results were obtained concerning the F-contraction mappings of S in a complete metric space. To prove it, the authors used only property (W1). Conclusion:The authors believe that the obtained results represent a significant improvement of many known results in the existing literature.


Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1190 ◽  
Author(s):  
Manuel De la Sen ◽  
Nebojša Nikolić ◽  
Tatjana Došenović ◽  
Mirjana Pavlović ◽  
Stojan Radenović

In this paper we consider ( s − q ) -graphic contraction mapping in b-metric like spaces. By using our new approach for the proof that a Picard sequence is Cauchy in the context of b-metric-like space, our results generalize, improve and complement several approaches in the existing literature. Moreover, some examples are presented here to illustrate the usability of the obtained theoretical results.


2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Marta Demma ◽  
Pasquale Vetro

We obtain some fixed point results for single-valued and multivalued mappings in the setting of ab-metric space. These results are generalizations of the analogous ones recently proved by Khojasteh, Abbas, and Costache.


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