mathematics Article Applications to Boundary Value Problems and Homotopy Theory via Tripled Fixed Point Techniques in Partially Metric Spaces Hasanen A. Hammad 1 , Praveen Agarwal 2,3,4,5,6,* and Juan L. G. Guirao 7,8 1 Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt; [email protected] 2 Department of Mathematics, Anand International College of Engineering, Jaipur 302012, India 3 Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman AE 346, United Arab Emirates 4 Institute of Mathematical Modeling, Almaty 050010, Kazakhstan 5 International Center for Basic and Applied Sciences, Jaipur 302029, India 6 Harish-Chandra Research Institute (HRI), Allahabad 211019, India 7 Departamento de Matematica Aplicada y Estadistica, Universidad Politecnica de Cartagena, Hospital de Marina, 30203 Murcia, Spain; [email protected] or [email protected] 8 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia * Correspondence: [email protected] or [email protected] Abstract: In this manuscript, some tripled fixed point results were derived under (j, r, `)-contraction in the framework of ordered partially metric spaces. Moreover, we furnish an example which supports our theorem. Furthermore, some results about a homotopy results are obtained. Finally, theoretical results are involved in some applications, such as finding the unique solution to the Citation: Hammad, H.A.; Agarwal, boundary value problems and homotopy theory. P.; Guirao, J.L.G. Applications to Boundary Value Problems and Keywords: tripled fixed point; boundary value problem; homotopy theory; partially ordered Homotopy Theory via Tripled Fixed metric space Point Techniques in Partially Metric Spaces. Mathematics 2021, 9, 2012. MSC: 54H25; 47H10; 34B24 https://doi.org/10.3390/math9162012 Academic Editors: Christopher Goodrich and Hsien-Chung Wu 1. Introduction Fixed point theory is one of the important and indispensable branches of non-linear Received: 9 June 2021 analysis due to the proliferation of its applications in many disciplines such as engineer- Accepted: 14 August 2021 Published: 23 August 2021 ing, computer science, physics, economics, biology, chemistry, etc. In mathematics, this technique is credited with clarifying and studying the behavior of dynamical systems, sta- Publisher’s Note: MDPI stays neutral tistical methods, game theory models, differential equations, and many others. Specifically, with regard to jurisdictional claims in this technique studies the existence and uniqueness of the solution to many integral and published maps and institutional affil- fractional equations, which facilitates the way to find numerical solutions to such problems, iations. see [1–11]. Homotopy theory is a fundamental branch of algebraic topology where topologi- cal objects are studied up to homotopy equivalence. In the last century, strong links have emerged between this theory and many branches of mathematics. For example, this trend plays a prominent role in strengthening ties between homotopy theory and Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. category theory (higher-dimensional), which have received considerable attention in re- This article is an open access article cent years, see [12–15]. Furthermore, it is useful in quantum mechanics for dealing with distributed under the terms and Hamiltonian manifolds. conditions of the Creative Commons The idea of a partially metric space (PMS) was presented by Matthews [16] as a part Attribution (CC BY) license (https:// of the study of denotational semantics of data flow networks. In fact, it is widely shared creativecommons.org/licenses/by/ that PMSs play a prominent role in model building in computational and field theory in 4.0/). computer science, see [17–22]. Mathematics 2021, 9, 2012. https://doi.org/10.3390/math9162012 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 2012 2 of 22 Fixed point results for a single mapping in partially ordered metric space (POMSs) were introduced by Matthews [16,23], Oltra and Valero [24] and Altun et al. [25]. For more papers in common fixed point consequences in abstract spaces, we mention [26–32]. In [33], the notion of a coupled fixed point was presented and some pivotal results from it in the setting of PMSs are studied. Later, coupled fixed and coupled common fixed point theorems are proved by [34–37]. In 2011, Berinde and Borcut [38] generalized the idea of a coupled fixed point to a tripled fixed point (TFP) in the setting of POMSs. Under this space, Borcut [39,40], Karapnar et al. [41], Radenovi´c[42] and Aydi et al. [43] introduced some circular theorems concerning TFP theorems. Moreover, more applications in this line are presented by Mustafa et al. [44] and Hammad and De la Sen [45,46]. The purpose of this manuscript is to prove some TFP consequences via (j, r, `)- contraction in the setting of ordered PMSs. In addition, to support our theoretical results, we give an example. Moreover, as applications, the existence and uniqueness of the solution to an initial value problem (IVP) and a homotopy theory are discussed. 2. Preliminaries This part is devoted to recall the standard definition of a homotopy and some elemen- tary properties for PMSs. A homotopy between two functions is defined as follows: consider two continuous functions from a topological space to another; the two functions are considered homotopic if one can be continuously deformed into the other. This deformity is called a homotopy between the two functions. It can be formulated as follows: Definition 1 ([12]). Assume that U, V : D ! E are continuous mappings defined on topo- logical spaces D and E. Then the continuous function H : D × [0, 1] ! E so that H(J, 0) = UJ and H(J, 1) = VJ is called a homotopy from U to V. Furthermore, U and V are called homotopic mappings. + Definition 2 ([16]). Let k 6= Æ. The function L : k × k ! R is called a partial metric on k if for all J1, J2, J3 2 k, the hypotheses below hold: (L1) J1 = J2 iff L(J1, J1) = L(J1, J2) = L(J2, J2); (L2) L(J1, J1) ≤ L(J1, J2) and L(J2, J2) ≤ L(J1, J2); (L3) L(J1, J2) = L(J2, J1); (L4) L(J1, J2) ≤ L(J1, J3) + L(J3, J2) − L(J3, J3). We say that (k, L) is a PMS. + It should be noted that if L is partial metric on k, then the function WL : k × k ! R , described by WL(J1, J2) ≤ 2L(J1, J2) − L(J1, J1) − L(J2, J2), is a metric on k. Example 1 ([23]). Let k = [0, ¥). The pair (k, L) is a PMS under the distance L(J1, J2) = maxfJ1, J2g. It is obvious that L is not a usual metric and in this case WL(J1, J2) = jJ1 − J2j. Example 2. Assume k = f[J1, J2] : J1, J2 2 R, J1 ≤ J2g and define L([J1, J3], [J2, J4]) = maxfJ3, J4g − minfJ1, J2g. Then (k, L) is a PMS. Every partial metric L on k produces a U0 topology cL on k. The base of a topology cL is a family of open L-balls fOL(z, e), z 2 k, e > 0g, where OL(z, e) = fJ 2 k : L(z, J) < L(z, z) + eg, 8z 2 k and e > 0. Mathematics 2021, 9, 2012 3 of 22 Now, we state some topological properties on PMSs. Definition 3 ([16]). A sequence fJwg in a PMS (k, L) is called: w (1) converges to the limit J iff L(J, J) = limw!¥ L(J, J ). n w (2) a Cauchy sequence if limn,w!¥ L(J , J ) exists and is finite. Definition 4 ([16]). (i) A PMS (k, L) is called complete if every Cauchy sequence fJwg in k n w converges (with respect to cL) to a point J 2 k so that L(J, J) = limn,w!¥ L(J , J ). (ii) A mapping x : k ! k is said to be continuous at J0 2 k, if 8e > 0, 9d > 0 so that 0 0 x OL(J , d) ⊆ OL(xJ , e). The following lemmas are very important in the following. Lemma 1 ([16]). (i) We say that fJwg is a Cauchy sequence in the PMS (k, L) if it is a Cauchy sequence in the metric space (k, WL). (ii) If the metric space (k, WL) is complete, the PMS (k, L) is too. Furthermore, w w u w lim WL(J, J ) = 0 , L(J, J) = lim WL(J, J ) = lim WL(J , J ). w!¥ w!¥ w,u!¥ w w Lemma 2 ([47]). Suppose that (k, L) is a PMS. If fJ g 2 k so that limw!¥ J = r and w L(r, r) = 0. Then limw!¥ L(J , s) = limw!¥ L(r, s) for every r, s 2 k. Lemma 3 ([47]). Assume that (k, L) is a PMS. Then (a) If L(J1, J2) = 0, then J1 = J2; (b) If J1 6= J2, then L(J1, J2) > 0. Remark 1 ([47]). If J1 = J2, then L(J1, J2) may not be 0. Definition 5 ([46]). A mapping x : k3 ! k (where k × k × k = k3) on a partially ordered set (POS) (k, ), has a mixed-monotone property, if for any J, q, w 2 k, J1, J2 2 k, J1 J2 ) x(J1, q, w) x(J2, q, w), q1, q2 2 k, q1 q2 ) x(J, q1, w) x(J, q2, w), w1, w2 2 k, w1 w2 ) x(J, q, w1) x(J, q, w2). ∗ Definition 6 ([43]). A mapping x : k3 ! k on a POS (k, ) has a mixed x −monotone property ∗ where x : k ! k, if for any J, q, w 2 k, ∗ ∗ J1, J2 2 k, x J1 x J2 ) x(J1, q, w) x(J2, q, w), ∗ ∗ q1, q2 2 k, x q1 x q2 ) x(J, q1, w) x(J, q2, w), ∗ ∗ w1, w2 2 k, x w1 x w2 ) x(J, q, w1) x(J, q, w2).