Fixed Point Results for Almost Generalized Cyclic (𝜓,𝜑)-Weak Contractive Type Mappings with Applications

Abstract

We define a class of almost generalized cyclic (𝜓,𝜙)-weak contractive mappings and discuss the existence and uniqueness of fixed points for such mappings. We present some examples to illustrate our results. Moreover, we state some applications of our main results in nonlinear integral equations.

1. Introduction

Fixed point theory is a crucial tool in the analysis of nonlinear problems. Banach contraction mapping principle [1] is the most known result in this direction: A self-mapping 𝑇∶𝑋→𝑋 on a complete metric space (𝑋,𝑑) has a unique fixed point if there exists 𝑘∈[0,1) such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝑘𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋. In this theorem, a self-mapping 𝑇 is necessarily continuous. Due to its importance to fixed point theory in nonlinear analysis, desired Banach fixed point theorem have been heavily investigated by many authors (see, e.g., [2–15]). One of the remarkable generalizations of the banach contraction mapping principle was reported by Kirk et al. [16] via cyclic contraction. A mapping 𝑇∶𝐴∪𝐵→𝐴∪𝐵 is called cyclic if 𝑇(𝐴)⊆𝐵 and 𝑇(𝐵)⊆𝐴, where 𝐴,𝐵 are nonempty subsets of a metric space (𝑋,𝑑). Moreover, 𝑇 is called cyclic contraction if there exists 𝑘∈(0,1) such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝑘𝑑(𝑥,𝑦) for all 𝑥∈𝐴 and 𝑦∈𝐵. Notice that although a contraction is continuous, cyclic contraction need not to be. This is one of the important gains of this theorem. In this paper, the authors also introduced the following notion.

Definition 1.1 (see [16]). Let 𝑋 be a nonempty set, 𝑝 be a positive integer, and 𝑇∶𝑋→𝑋 be a mapping. 𝑋=⋃^𝑝_𝑖=1𝐴_𝑖 is said to be a cyclic representation of 𝑋 with respect to 𝑇 if (i)𝐴_𝑖,𝑖=1,2,…,𝑝 are nonempty closed sets, (ii)𝑇(𝐴₁)⊆𝐴₂,…,𝑇(𝐴_𝑝−1)⊆𝐴_𝑝,𝑇(𝐴_𝑝)⊆𝐴₁.

Following the paper in [16], a number of fixed point theorems on cyclic representation of 𝑋 with respect to a self-mapping 𝑇 have appeared (see, e.g., [17–26]).

The concept of almost contractions were introduced by Berinde [27, 28]. It was shown in [27] that any strict contraction, the Kannan [12] and Zamfirescu [29] mappings, as well as a large class of quasicontractions, are all almost contractions. Almost contractions and its generalizations were further considered in several works like [7, 30–37]. Recently, Ćirić et al. [7] proved some fixed point results in ordered metric spaces using almost generalized contractive condition, which is given in the following definition.

Definition 1.2. Let 𝑇∶𝑋→𝑋 be a self-mapping on a metric space (𝑋,𝑑). It is said to satisfy almost generalized contractive condition if there exists 𝑘∈(0,1) and 𝐿≥0 such that 𝑑𝑑(),𝑑𝑇𝑥,𝑇𝑦≤𝑘max𝑥,𝑦,𝑑𝑥,𝑇𝑥,𝑑𝑦,𝑇𝑦𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥2+𝐿min𝑑(𝑥,𝑇𝑥),𝑑𝑦,𝑇𝑦,𝑑𝑥,𝑇𝑦,𝑑𝑦,𝑇𝑥, (1.1) for all 𝑥,𝑦∈𝑋.

In this paper, we introduce a class of almost generalized cyclic (𝜓,𝜑)-weak contractive mappings and we investigate the existence and uniqueness of fixed points for almost generalized cyclic (𝜓,𝜑)-weak contractive type mappings. Our main result generalizes and improves some well-known theorems in the literature (see, e.g., [16, 18, 19, 24, 26]). We state some examples to illustrate our results. Furthermore, we apply our main result to analyze the existence and uniqueness of solutions for a class of nonlinear integral equations.

2. Main Result

We start this section by defining two classes of real valued functions. Let Φ be the set of functions 𝜑∶[0,∞)→[0,∞) satisfying the following conditions: (Φ₁)𝜑 is lower semicontinuous; (Φ₂)𝜑⁻¹({0})={0}. We denote by Ψ the set of all continuous functions 𝜓∶[0,∞)→[0,∞).

Definition 2.1. Let (𝑋,𝑑) be a metric space. Let 𝑝 be a positive integer and let 𝐴₁,𝐴₂,…,𝐴_𝑝 be nonempty subsets of 𝑋 and 𝑌=⋃^𝑝_𝑖=1𝐴_𝑖. Let 𝑇∶𝑌→𝑌 be a mapping such that (I)𝑌=⋃^𝑝_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑌 with respect to 𝑇,(II) there exist 𝐿≥0, 𝜓∈Ψ and 𝜑∈Φ such that 𝜓𝑑𝑇𝑥,𝑇𝑦≤𝜓𝑀_𝑇𝑥,𝑦−𝜑𝑀_𝑇𝑑(),𝑥,𝑦+𝐿min𝑥,𝑇𝑥,𝑑𝑦,𝑇𝑦,𝑑𝑥,𝑇𝑦,𝑑𝑦,𝑇𝑥 (2.1)for all (𝑥,𝑦)∈𝐴_𝑖×𝐴_𝑖+1, 𝑖=1,2,…,𝑝 (with 𝐴_𝑝+1=𝐴₁), where 𝑀_𝑇𝑑(),𝑑𝑥,𝑦∶=max𝑥,𝑦,𝑑𝑇𝑥,𝑥,𝑑𝑇𝑦,𝑦𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥2. (2.2) Then 𝑇 is called an almost generalized cyclic (𝜓,𝜑)-weak contractive mapping.

Remark 2.2. Taking in the above definition, 𝑝=1, 𝐴₁=𝑋, 𝜓(𝑡)=𝑡, and 𝜑(𝑡)=(1−𝑘)𝑡, where 𝑘∈(0,1) is a constant, we obtain an almost generalized contractive condition. Then any almost generalized contractive mapping is an almost generalized cyclic (𝜓,𝜑)-weak contractive mapping.

Our main result is the following.

Theorem 2.3. Let {𝐴_𝑖}^𝑝_𝑖=1 be nonempty closed subsets of a complete metric space (𝑋,𝑑) and 𝑌=⋃^𝑝_𝑖=1𝐴_𝑖. Let 𝑇∶𝑌→𝑌 be an almost generalized cyclic (𝜓,𝜑)-weak contractive mapping. Then 𝑇 has a unique fixed point that belongs to ⋂^𝑝_𝑖=1𝐴_𝑖.

Proof. Let 𝑥₀∈𝐴₁ (such a point exists since 𝐴₁≠∅). Define the sequence {𝑥_𝑛} in 𝑋 by 𝑥_𝑛+1=𝑇𝑥_𝑛,𝑛=0,1,2,…. (2.3) We will prove that lim_𝑛→∞𝑑𝑥_𝑛,𝑥_𝑛+1=0. (2.4) If for some 𝑘, we have 𝑥_𝑘+1=𝑥_𝑘, then (2.4) follows immediately. So, we can suppose that 𝑑(𝑥_𝑛,𝑥_𝑛+1)>0 for all 𝑛. From the condition (I), we observe that for all 𝑛, there exists 𝑖=𝑖(𝑛)∈{1,2,…,𝑝} such that (𝑥_𝑛,𝑥_𝑛+1)∈𝐴_𝑖×𝐴_𝑖+1. Then, from the condition (II), we have 𝜓𝑑𝑥_𝑛,𝑥_𝑛+1≤𝜓𝑀_𝑇𝑥_𝑛−1,𝑥_𝑛−𝜑𝑀_𝑇𝑥_𝑛−1,𝑥_𝑛,𝑛=1,2,…. (2.5) On the other hand, we have 𝑀_𝑇𝑥_𝑛−1,𝑥_𝑛=max𝑑𝑥_𝑛−1,𝑥_𝑛,𝑑𝑇𝑥_𝑛−1,𝑥_𝑛−1,𝑑𝑇𝑥_𝑛,𝑥_𝑛,𝑑𝑥_𝑛−1,𝑇𝑥_𝑛+𝑑𝑥_𝑛,𝑇𝑥_𝑛−12=max𝑑𝑥_𝑛−1,𝑥_𝑛,𝑑𝑥_𝑛,𝑥_𝑛−1,𝑑𝑥_𝑛+1,𝑥_𝑛,𝑑𝑥_𝑛−1,𝑥_𝑛+12=max𝑑𝑥_𝑛−1,𝑥_𝑛,𝑑𝑥_𝑛+1,𝑥_𝑛,𝑑𝑥_𝑛−1,𝑥_𝑛+12=max𝑑𝑥_𝑛−1,𝑥_𝑛,𝑑𝑥_𝑛+1,𝑥_𝑛. (2.6) Suppose that 𝑀_𝑇(𝑥_𝑛−1,𝑥_𝑛)=𝑑(𝑥_𝑛+1,𝑥_𝑛). Using (2.5), we obtain 𝜓𝑑𝑥_𝑛,𝑥_𝑛+1≤𝜓𝑑𝑥_𝑛,𝑥_𝑛+1−𝜑𝑑𝑥_𝑛,𝑥_𝑛+1, (2.7) which implies that 𝜑(𝑑(𝑥_𝑛,𝑥_𝑛+1))=0. From condition (Φ₂), we get that 𝑑(𝑥_𝑛,𝑥_𝑛+1)=0, a contradiction with our assumption 𝑑(𝑥_𝑛,𝑥_𝑛+1)>0 for all 𝑛. Thus, we have 𝑀_𝑇(𝑥_𝑛−1,𝑥_𝑛)=𝑑(𝑥_𝑛−1,𝑥_𝑛), which implies that {𝑑(𝑥_𝑛,𝑥_𝑛+1)} is a decreasing sequence of positive numbers. Then there exists 𝑟≥0 such that lim_𝑛→∞𝑑𝑥_𝑛,𝑥_𝑛+1=𝑟. (2.8) Letting 𝑛→∞ in (2.5), using (2.8), the continuity of 𝜓 and the lower semicontinuity of 𝜑, we obtain 𝜓(𝑟)≤𝜓(𝑟)−𝜑(𝑟), (2.9) which implies that 𝜑(𝑟)=0, that is, 𝑟=0. Thus, we proved (2.4).
Now, we will prove that {𝑥_𝑛} is a Cauchy sequence in (𝑋,𝑑). Suppose that {𝑥_𝑛} is not a Cauchy sequence. Then there exists 𝜀>0 for which we can find two sequences of positive integers {𝑚(𝑘)} and {𝑛(𝑘)} such that for all positive integers 𝑘, 𝑛(𝑘)>𝑚(𝑘)>𝑘,𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)≥𝜀,𝑑𝑥_𝑚(𝑘),𝑥_{𝑛(𝑘)−1}<𝜀. (2.10) Using (2.10) and the triangular inequality, we get 𝜀≤𝑑𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)≤𝑑𝑥_𝑚(𝑘),𝑥_{𝑛(𝑘)−1}+𝑑𝑥_{𝑛(𝑘)−1},𝑥_𝑛(𝑘)<𝜀+𝑑𝑥_𝑛(𝑘),𝑥_{𝑛(𝑘)−1}. (2.11) Thus, we have 𝜀≤𝑑𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)<𝜀+𝑑𝑥_𝑛(𝑘),𝑥_{𝑛(𝑘)−1}. (2.12) Letting 𝑘→∞ in the above inequality and using (2.4), we obtain lim_𝑘→∞𝑑𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)=𝜀⁺. (2.13) On the other hand, for all 𝑘, there exists 𝑗(𝑘)∈{1,…,𝑝} such that 𝑛(𝑘)−𝑚(𝑘)+𝑗(𝑘)≡1[𝑝]. Then 𝑥_{𝑚(𝑘)−𝑗(𝑘)} (for 𝑘 large enough, 𝑚(𝑘)>𝑗(𝑘)) and 𝑥_𝑛(𝑘) lie in different adjacently labelled sets 𝐴_𝑖 and 𝐴_𝑖+1 for certain 𝑖∈{1,…,𝑝}. Using (II), we obtain 𝜓𝑑𝑇𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑇𝑥_𝑛(𝑘)≤𝜓𝑀_𝑇𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)−𝜑𝑀_𝑇𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)+𝐿min𝑑𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_{𝑚(𝑘)−𝑗(𝑘)+1},𝑑𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑑𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)+1,𝑑𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−𝑗(𝑘)+1} (2.14) for all 𝑘, that is, 𝜓𝑑𝑥_{𝑚(𝑘)−𝑗(𝑘)+1},𝑥_𝑛(𝑘)+1≤𝜓𝑀_𝑇𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)−𝜑𝑀_𝑇𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)+𝐿min𝑑𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_{𝑚(𝑘)−𝑗(𝑘)+1},𝑑𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑑𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)+1,𝑑𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−𝑗(𝑘)+1}, (2.15) for all 𝑘. Now, we have 𝑀_𝑇𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)=max𝑑𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘),𝑑𝑥_{𝑚(𝑘)−𝑗(𝑘)+1},𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑑𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘),𝑑𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)+1+𝑑𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−𝑗(𝑘)+1}2 (2.16) for all 𝑘. Using the triangular inequality, we get 𝑑𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)−𝑑𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)≤𝑑𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑚(𝑘)≤^{𝑗(𝑘)−1}_𝑙=0𝑑𝑥_{𝑚(𝑘)−𝑗(𝑘)+𝑙},𝑥_{𝑚(𝑘)−𝑗(𝑘)+𝑙+1}≤^𝑝−1_𝑙=0𝑑𝑥_{𝑚(𝑘)−𝑗(𝑘)+𝑙},𝑥_{𝑚(𝑘)−𝑗(𝑘)+𝑙+1}⟶0,as𝑘⟶∞(from(2.4)), (2.17) which implies from (2.13) that lim_𝑘→∞𝑑𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)=𝜀. (2.18) Using (2.4), we have lim_𝑘→∞𝑑𝑥_{𝑚(𝑘)−𝑗(𝑘)+1},𝑥_{𝑚(𝑘)−𝑗(𝑘)}=0,lim_𝑘→∞𝑑𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)=0. (2.19)(2.20) Again, using the triangular inequality, we get 𝑑𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)+1−𝑑𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)≤𝑑𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1. (2.21) Letting 𝑘→∞ in the above inequality, using (2.20) and (2.18), we get lim_𝑘→∞𝑑𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)+1=𝜀. (2.22) Similarly, we have 𝑑𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−𝑗(𝑘)+1}−𝑑𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)≤𝑑𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_{𝑚(𝑘)−𝑗(𝑘)+1}. (2.23) Letting 𝑘→∞, using (2.4) and (2.18), we obtain lim_𝑘→∞𝑑𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−𝑗(𝑘)+1}=𝜀. (2.24) Similarly, we have lim_𝑘→∞𝑑𝑥_{𝑚(𝑘)−𝑗(𝑘)+1},𝑥_𝑛(𝑘)+1=𝜀. (2.25) Now, it follows from (2.18)–(2.24) that lim_𝑘→∞𝑀_𝑇𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)=max𝜀,0,0,𝜀+𝜀2=𝜀,lim_𝑘→∞min𝑑𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_{𝑚(𝑘)−𝑗(𝑘)+1},𝑑𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑑𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)+1,𝑑𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−𝑗(𝑘)+1}=min{0,0,𝜀,𝜀}=0. (2.26)
Letting 𝑘→∞ in (2.15), using (2.25), (2.26), the continuity of 𝜓 and the lower semicontinuity of 𝜑, we obtain 𝜓(𝜀)≤𝜓(𝜀)−𝜑(𝜀), (2.27) which implies that 𝜑(𝜀)=0, that is, 𝜀=0, a contradiction with 𝜀>0. Then we deduce that {𝑥_𝑛} is a Cauchy sequence in the metric space (𝑋,𝑑).
Since (𝑋,𝑑) is complete, there exists 𝑥^∗∈𝑋 such that lim_𝑛→∞𝑥_𝑛=𝑥^∗. (2.28) We will prove that 𝑥^∗∈^𝑝_𝑖=1𝐴_𝑖. (2.29) From condition (I), and since 𝑥₀∈𝐴₁, we have {𝑥_𝑛𝑝}_𝑛≥0⊆𝐴₁. Since 𝐴₁ is closed, from (2.28), we get that 𝑥^∗∈𝐴₁. Again, from the condition (I), we have {𝑥_𝑛𝑝+1}_𝑛≥0⊆𝐴₂. Since 𝐴₂ is closed, from (2.28), we get that 𝑥^∗∈𝐴₂. Continuing this process, we obtain (2.29).
Now, we will prove that 𝑥^∗ is a fixed point of 𝑇. Indeed, from (2.29), since for all 𝑛, there exists 𝑖(𝑛)∈{1,2,…,𝑝} such that 𝑥_𝑛∈𝐴_𝑖(𝑛), Applying (II) with 𝑥=𝑥^∗ and 𝑦=𝑥_𝑛, we obtain 𝜓𝑑𝑇𝑥^∗,𝑥_𝑛+1=𝜓𝑑𝑇𝑥^∗,𝑇𝑥_𝑛≤𝜓𝑀_𝑇𝑥^∗,𝑥_𝑛−𝜑𝑀_𝑇𝑥^∗,𝑥_𝑛+𝐿min𝑑𝑥^∗,𝑇𝑥^∗,𝑑𝑥_𝑛,𝑥_𝑛+1,𝑑𝑥^∗,𝑥_𝑛+1,𝑑𝑥_𝑛,𝑇𝑥^∗, (2.30) for all 𝑛. On the other hand, we have 𝑀_𝑇𝑥^∗,𝑥_𝑛=max𝑑𝑥^∗,𝑥_𝑛,𝑑𝑥^∗,𝑇𝑥^∗,𝑑𝑥_𝑛,𝑥_𝑛+1,𝑑𝑥^∗,𝑥_𝑛+1+𝑑𝑥_𝑛,𝑇𝑥^∗2. (2.31) Using (2.28), we obtain that lim_𝑛→∞𝑀_𝑇𝑥^∗,𝑥_𝑛=𝑑𝑥^∗,𝑇𝑥^∗,lim_𝑛→∞min𝑑𝑥^∗,𝑇𝑥^∗,𝑑𝑥_𝑛,𝑥_𝑛+1,𝑑𝑥^∗,𝑥_𝑛+1,𝑑𝑥_𝑛,𝑇𝑥^∗=0. (2.32) Letting 𝑛→∞ in (2.30), using (2.32), the continuity of 𝜓 and the lower semicontinuity of 𝜑, we get 𝜓𝑑𝑥^∗,𝑇𝑥^∗≤𝜓𝑑𝑥^∗,𝑇𝑥^∗−𝜑𝑑𝑥^∗,𝑇𝑥^∗, (2.33) which implies that 𝑑(𝑥^∗,𝑇𝑥^∗)=0, that is, 𝑥^∗ is a fixed point of 𝑇.
Finally, we prove that 𝑥^∗ is the unique fixed point of 𝑇. Assume that 𝑦^∗ is another fixed point of 𝑇, that is, 𝑇𝑦^∗=𝑦^∗. From the condition (I), this implies that 𝑦^∗∈⋂^𝑝_𝑖=1𝐴_𝑖. Then we can apply (II) for 𝑥=𝑥^∗ and 𝑦=𝑦^∗. We obtain 𝜓𝑑𝑥^∗,𝑦^∗≤𝜓𝑀_𝑇𝑥^∗,𝑦^∗−𝜑𝑀_𝑇𝑥^∗,𝑦^∗. (2.34) Since 𝑥^∗ and 𝑦^∗ are fixed points of 𝑇, we can show easily that 𝑀_𝑇(𝑥^∗,𝑦^∗)=𝑑(𝑥^∗,𝑦^∗). Then we get 𝜓𝑑𝑥^∗,𝑦^∗≤𝜓𝑑𝑥^∗,𝑦^∗−𝜑𝑑𝑥^∗,𝑦^∗, (2.35) which implies that 𝑑(𝑥^∗,𝑦^∗)=0, that is, 𝑥^∗=𝑦^∗. Thus, we proved the uniqueness of the fixed point.

3. Consequences

In this section, we derive some fixed point theorems from our main result given by Theorem 2.3.

If we take 𝑝=1 and 𝐴₁=𝑋 in Theorem 2.3, then we get immediately the following fixed point theorem.

Corollary 3.1. Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝑋 satisfies the following condition: there exist 𝐿≥0, 𝜓∈Ψ and 𝜑∈Φ such that 𝜓𝑑𝑇𝑥,𝑇𝑦≤𝜓𝑀_𝑇𝑥,𝑦−𝜑𝑀_𝑇𝑑(),𝑥,𝑦+𝐿min𝑥,𝑇𝑥,𝑑𝑦,𝑇𝑦,𝑑𝑥,𝑇𝑦,𝑑𝑦,𝑇𝑥 (3.1) for all 𝑥,𝑦∈𝑋. Then 𝑇 has a unique fixed point.

An immediate consequence of Corollary 3.1 is the following fixed point theorem (see Remark 2.2).

Corollary 3.2. Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝑋 satisfies the following condition: there exist 𝐿≥0 and 𝑘∈(0,1) such that 𝑑𝑇𝑥,𝑇𝑦≤𝑘𝑀_𝑇𝑥,𝑦+𝐿min𝑑(𝑥,𝑇𝑥),𝑑𝑦,𝑇𝑦,𝑑𝑥,𝑇𝑦,𝑑𝑦,𝑇𝑥, (3.2) for all 𝑥,𝑦∈𝑋. Then 𝑇 has a unique fixed point.

Remark 3.3. Taking 𝐿=0 in Corollary 3.1, we obtain Theorem  2.2 in [8]. Moreover, in Corollary 3.1, it is not supposed that 𝜓 is nondecreasing and 𝜓⁻¹({0})={0}, as in [8].

Taking 𝜓(𝑡)=𝑡, 𝜑(𝑡)=(1−𝑘)𝑡, with 𝑘∈(0,1), in Theorem 2.3, we derive the following result.

Corollary 3.4. Let {𝐴_𝑖}^𝑝_𝑖=1 be nonempty closed subsets of a complete metric space (𝑋,𝑑) and suppose that 𝑇∶⋃^𝑝_𝑖=1𝐴_𝑖→⋃^𝑝_𝑖=1𝐴_𝑖 satisfies the following conditions (where 𝐴_𝑝+1=𝐴₁): (i)𝑇(𝐴_𝑖)⊆𝐴_𝑖+1 for 1≤𝑖≤𝑝; (ii)there exists a constant 𝑘∈(0,1) such that 𝑑𝑑(),𝑑𝑇𝑥,𝑇𝑦≤𝑘max𝑥,𝑦,𝑑𝑇𝑥,𝑥,𝑑𝑇𝑦,𝑦𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥2+𝐿min𝑑(𝑥,𝑇𝑥),𝑑𝑦,𝑇𝑦,𝑑𝑥,𝑇𝑦,𝑑𝑦,𝑇𝑥, (3.3) for all (𝑥,𝑦)∈(𝐴_𝑖,𝐴_𝑖+1), for 1≤𝑖≤𝑝.
Then 𝑇 has a unique fixed point that belongs to ⋂^𝑝_𝑖=1𝐴_𝑖.

The following fixed point theorems established in [16, 25] are immediate consequences of the above result.

Corollary 3.5. Let {𝐴_𝑖}^𝑝_𝑖=1 be nonempty closed subsets of a complete metric space (𝑋,𝑑) and suppose that 𝑇∶⋃^𝑝_𝑖=1𝐴_𝑖→⋃^𝑝_𝑖=1𝐴_𝑖 satisfies the following conditions (where 𝐴_𝑝+1=𝐴₁): (i)𝑇(𝐴_𝑖)⊆𝐴_𝑖+1 for 1≤𝑖≤𝑝; (ii)there exists a constant 𝑘∈(0,1) such that 𝑑𝑇𝑥,𝑇𝑦≤𝑘𝑑𝑥,𝑦, (3.4) for all (𝑥,𝑦)∈(𝐴_𝑖,𝐴_𝑖+1), for 1≤𝑖≤𝑝.
Then 𝑇 has a unique fixed point that belongs to ⋂^𝑝_𝑖=1𝐴_𝑖.

Corollary 3.6. Let {𝐴_𝑖}^𝑝_𝑖=1 be nonempty closed subsets of a complete metric space (𝑋,𝑑) and suppose that 𝑇∶⋃^𝑝_𝑖=1𝐴_𝑖→⋃^𝑝_𝑖=1𝐴_𝑖 satisfies the following conditions (where 𝐴_𝑝+1=𝐴₁): (i)𝑇(𝐴_𝑖)⊆𝐴_𝑖+1 for 1≤𝑖≤𝑝; (ii)there exists a constant 𝑘∈(0,1) such that 𝑑≤𝑘𝑇𝑥,𝑇𝑦2𝑑𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥, (3.5) for all (𝑥,𝑦)∈(𝐴_𝑖,𝐴_𝑖+1), for 1≤𝑖≤𝑝.
Then 𝑇 has a unique fixed point that belongs to ⋂^𝑝_𝑖=1𝐴_𝑖.

Corollary 3.7. Let {𝐴_𝑖}^𝑝_𝑖=1 be nonempty closed subsets of a complete metric space (𝑋,𝑑) and suppose that 𝑇∶⋃^𝑝_𝑖=1𝐴_𝑖→⋃^𝑝_𝑖=1𝐴_𝑖 satisfies the following conditions (where 𝐴_𝑝+1=𝐴₁): (i)𝑇(𝐴_𝑖)⊆𝐴_𝑖+1 for 1≤𝑖≤𝑝; (ii)there exists a constant 𝑘∈(0,1) such that 𝑑𝑇𝑥,𝑇𝑦≤𝑘max𝑑(𝑥,𝑇𝑥),𝑑𝑦,𝑇𝑦, (3.6) for all (𝑥,𝑦)∈(𝐴_𝑖,𝐴_𝑖+1), for 1≤𝑖≤𝑝.
Then 𝑇 has a unique fixed point that belongs to ⋂^𝑝_𝑖=1𝐴_𝑖.

Corollary 3.8. Let {𝐴_𝑖}^𝑝_𝑖=1 be nonempty closed subsets of a complete metric space (𝑋,𝑑) and suppose that 𝑇∶⋃^𝑝_𝑖=1𝐴_𝑖→⋃^𝑝_𝑖=1𝐴_𝑖 satisfies the following conditions (where 𝐴_𝑝+1=𝐴₁): (i)𝑇(𝐴_𝑖)⊆𝐴_𝑖+1 for 1≤𝑖≤𝑝; (ii)there exist 𝑎₁,𝑎₂,𝑎₃,𝑎₄>0 with 𝑎₁+𝑎₂+𝑎₃+𝑎₄<1 such that 𝑑𝑇𝑥,𝑇𝑦≤𝑎₁𝑑𝑥,𝑦+𝑎₂𝑑(𝑇𝑥,𝑥)+𝑎₃𝑑𝑇𝑦,𝑦+𝑎₄𝑑𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥2, (3.7) for all (𝑥,𝑦)∈(𝐴_𝑖,𝐴_𝑖+1), for 1≤𝑖≤𝑝.
Then 𝑇 has a unique fixed point that belongs to ⋂^𝑝_𝑖=1𝐴_𝑖.

Remark 3.9. Taking 𝑝=1 and 𝐴₁=𝑋, we get (i)from Corollary 3.5, the Banach contraction principle [1]; (ii)from Corollary 3.6, Kannan's fixed point theorem [12]; (iii)from Corollary 3.7, Bianchini’s fixed point theorem [2] (see also [3]); (iv)from Corollary 3.8, Hardy and Rogers fixed point theorem [9].

Now, we derive a fixed point result for cyclic mappings satisfying a contractive condition of integral type.

Denote by Λ the set of functions 𝛼∶[0,∞)→[0,∞) satisfying the following hypotheses: (Λ₁)𝛼 is a Lebesgue integrable mapping on each compact subset of [0,∞); (Λ₂) for any 𝜀>0, we have ∫^𝜀₀𝛼(𝑠)𝑑𝑠>0.

We have the following result.

Corollary 3.10. Let {𝐴_𝑖}^𝑝_𝑖=1 be nonempty closed subsets of a complete metric space (𝑋,𝑑) and suppose 𝑇∶⋃^𝑝_𝑖=1𝐴_𝑖→⋃^𝑝_𝑖=1𝐴_𝑖 satisfies the following conditions (where 𝐴_𝑝+1=𝐴₁): (i)𝑇(𝐴_𝑖)⊆𝐴_𝑖+1 for 1≤𝑖≤𝑝; (ii)there exist 𝐿≥0, 𝛼,𝛽∈Λ such that ₀^{𝑑(𝑇𝑥,𝑇𝑦)}𝛼(𝑠)𝑑𝑠≤^𝑀𝑇₀^(𝑥,𝑦)𝛼(𝑠)𝑑𝑠−^𝑀𝑇₀^(𝑥,𝑦)𝛽(𝑠)𝑑(),𝑑𝑠+𝐿min𝑥,𝑇𝑥,𝑑𝑦,𝑇𝑦,𝑑𝑥,𝑇𝑦,𝑑𝑦,𝑇𝑥 (3.8) for all (𝑥,𝑦)∈(𝐴_𝑖,𝐴_𝑖+1), for 1≤𝑖≤𝑝. Then 𝑇 has a unique fixed point that belongs to ⋂^𝑝_𝑖=1𝐴_𝑖.

Proof. It follows immediately from Theorem 2.3 by observing that the functions 𝜓(𝑡)=∫^𝑡₀𝛼(𝑠)𝑑𝑠 and 𝜑(𝑡)=∫^𝑡₀𝛽(𝑠)𝑑𝑠 belong to Ψ.

Taking 𝐿=0 in Corollary 3.10, we obtain the following result.

Corollary 3.11. Let {𝐴_𝑖}^𝑝_𝑖=1 be nonempty closed subsets of a complete metric space (𝑋,𝑑) and suppose 𝑇∶⋃^𝑝_𝑖=1𝐴_𝑖→⋃^𝑝_𝑖=1𝐴_𝑖 satisfies the following conditions (where 𝐴_𝑝+1=𝐴₁): (i)𝑇(𝐴_𝑖)⊆𝐴_𝑖+1 for 1≤𝑖≤𝑝; (ii)there exist 𝛼,𝛽∈Λ such that ₀^{𝑑(𝑇𝑥,𝑇𝑦)}𝛼(𝑠)𝑑𝑠≤^𝑀𝑇₀^(𝑥,𝑦)𝛼(𝑠)𝑑𝑠−^𝑀𝑇₀^(𝑥,𝑦)𝛽(𝑠)𝑑𝑠, (3.9) for all (𝑥,𝑦)∈(𝐴_𝑖,𝐴_𝑖+1), for 1≤𝑖≤𝑝. Then 𝑇 has a unique fixed point that belongs to ⋂^𝑝_𝑖=1𝐴_𝑖.

4. Some Examples

In this section, we give some examples to illustrate our obtained results.

Example 4.1. Let 𝑋=[0,1] be endowed with the standard metric 𝑑(𝑥,𝑦)=|𝑥−𝑦| for all 𝑥,𝑦∈𝑋. Consider the closed subsets 𝐴₁ and 𝐴₂ defined by 𝐴₁=[0,1/4] and 𝐴₂=[1/4,1]. Define the mapping 𝑇∶𝑋→𝑋 by 1𝑇𝑥=4if[),0𝑥∈0,1if𝑥=1. (4.1) Clearly, we have 𝑇(𝐴₁)⊂𝐴₂ and 𝑇(𝐴₂)⊂𝐴₁.
Now, let (𝑥,𝑦)∈𝐴₁×𝐴₂. We distinguish two cases.

Case 1. If 𝑦=1. In this case, we have 𝑑==1𝑇𝑥,𝑇𝑦𝑇𝑥−𝑇𝑦4=1−04=12⋅12≤12𝑑≤1𝑦,𝑇𝑦2𝑀_𝑇𝑥,𝑦=𝑀_𝑇−1𝑥,𝑦2𝑀_𝑇𝑥,𝑦. (4.2) Define the functions 𝜓,𝜑∶[0,∞)→[0,∞) by 𝜓(𝑡)(𝑡)=𝑡=𝑡,𝜑2,∀𝑡≥0. (4.3) Then we have 𝜓𝑑𝑇𝑥,𝑇𝑦≤𝜓𝑀_𝑇𝑥,𝑦−𝜑𝑀_𝑇𝑥,𝑦. (4.4)

Case 2. If 𝑦≠1. In this case, we have 𝑑(𝑇𝑥,𝑇𝑦)=0, so inequality (4.4) is satisfied.
Similarly, if (𝑥,𝑦)∈𝐴₂×𝐴₁, we can show that (4.4) is satisfied.
Thus, we checked that all conditions of Theorem 2.3 are satisfied (with 𝑝=2). We deduce that 𝑇 has a unique fixed point 𝑥^∗∈𝐴₁∩𝐴₂={1/4}.

Example 4.2. Let 𝑋=[−𝜋,𝜋] be endowed with the standard metric 𝑑(𝑥,𝑦)=|𝑥−𝑦| for all 𝑥,𝑦∈𝑋. Consider the closed subsets 𝐴₁ and 𝐴₂ defined by 𝐴₁=[0,𝜋] and 𝐴₂=[−𝜋,0]. Define the mapping 𝑇∶𝑋→𝑋 by −1𝑇𝑥=3𝑥1cos𝑥if[)∪(],0𝑥∈−𝜋,00,𝜋if𝑥=0. (4.5) Clearly, we have 𝑇(𝐴₁)⊂𝐴₂ and 𝑇(𝐴₂)⊂𝐴₁.
Now, let (𝑥,𝑦)∈(𝐴₁×𝐴₂) with 𝑥≠0 and 𝑦≠0. We have 𝑑==−1𝑇𝑥,𝑇𝑦𝑇𝑥−𝑇𝑦3𝑥1cos𝑥+13𝑦1cos𝑦=13𝑥1||cos𝑥+𝑦1cos𝑦≤13|𝑥|+𝑦. (4.6) On the other hand, we have |𝑥|1=𝑥≤𝑥+3𝑥1cos𝑥=1𝑥+3𝑥1cos𝑥(),𝑦1=𝑑𝑥,𝑇𝑥=−𝑦≤−𝑦+31𝑦cos𝑦1=−𝑦−3𝑦1cos𝑦≤1𝑦+3𝑦1cos𝑦=𝑑𝑦,𝑇𝑦. (4.7) Then we have 𝑑≤2𝑇𝑥,𝑇𝑦3𝑑()≤2max𝑥,𝑇𝑥,𝑑𝑦,𝑇𝑦3𝑀_𝑇𝑥,𝑦. (4.8) Consider the functions 𝜓,𝜑∶[0,∞)→[0,∞) defined by 𝜓(𝑡)(𝑡)=𝑡=𝑡,𝜑3,∀𝑡≥0. (4.9) We have 𝜓𝑑𝑇𝑥,𝑇𝑦≤𝜓𝑀_𝑇𝑥,𝑦−𝜑𝑀_𝑇𝑥,𝑦. (4.10) Moreover, we can show that the above inequality holds if 𝑥=0 or 𝑦=0.
Now, all conditions of Theorem 2.3 are satisfied (with 𝑝=2), we deduce that 𝑇 has a unique fixed point 𝑥^∗∈𝐴₁∩𝐴₂={0}.

5. An Application

In this section, we apply the result given by Theorem 2.3 to study the existence and uniqueness of solutions to a class of nonlinear integral equations.

We consider the nonlinear integral equation 𝑢(𝑡)=¹₀𝑘(𝑡,𝑠,𝑢(𝑠))𝑑𝑠,∀𝑡∈[0,1], (5.1) where 𝑘∶[0,1]×[0,1]×ℝ→ℝ is a continuous function.

Let 𝑋=𝐶([0,1]) be the set of real continuous functions on [0,1]. We endow 𝑋 with the following standard metric: 𝑑_∞(𝑢,𝑣)=max_[]𝑡∈0,1|𝑢(𝑡)−𝑣(𝑡)|,∀𝑢,𝑣∈𝑋. (5.2) It is well known that (𝑋,𝑑_∞) is a complete metric space. Let (𝛼,𝛽)∈𝑋², let (𝛼₀,𝛽₀)∈ℝ² such that 𝛼₀≤𝛼≤𝛽≤𝛽₀. (5.3) We suppose that for all 𝑡∈[0,1], we have 𝛼(𝑡)≤¹₀𝑘𝑡,𝑠,𝛽(𝑠)𝑑𝑠,𝛽(𝑡)≥¹₀𝑘(𝑡,𝑠,𝛼(𝑠))𝑑𝑠. (5.4)(5.5) We suppose that for all 𝑡,𝑠∈[0,1], 𝑘(𝑡,𝑠,⋅) is a decreasing function, that is, 𝑥,𝑦∈ℝ,𝑥≥𝑦⟹𝑘(𝑡,𝑠,𝑥)≤𝑘𝑡,𝑠,𝑦. (5.6) Finally, we suppose that for all 𝑡,𝑠∈[0,1], for all 𝑥,𝑦∈ℝ with 𝑥≤𝛽₀ and 𝑦≥𝛼₀ or 𝑥≥𝛼₀ and 𝑦≤𝛽₀, 𝑘(𝑡,𝑠,𝑥)−𝑘𝑡,𝑠,𝑦≤𝜉𝑥−𝑦, (5.7) where 𝜉∶[0,∞)→[0,∞) is continuous nondecreasing and 𝜑∶𝑡↦𝑡−𝜉(𝑡) belongs to Φ. Now, define the set 𝒲=𝑢∈𝐶([0,1])∶𝛼≤𝑢≤𝛽. (5.8)

We have the following result.

Theorem 5.1. Under the assumptions (5.3)–(5.7), Problem (5.1) has one and only one solution 𝑢^∗∈𝒲.

Proof. Define the closed subsets of 𝑋, 𝐴₁, and 𝐴₂ by 𝐴₁=,𝐴𝑢∈𝑋∶𝑢≤𝛽₂={𝑢∈𝑋∶𝑢≥𝛼}. (5.9) Define the mapping 𝑇∶𝑋→𝑋 by 𝑇𝑢(𝑡)=¹₀𝑘(𝑡,𝑠,𝑢(𝑠))𝑑𝑠,∀𝑡∈[0,1]. (5.10) We will prove that 𝑇𝐴₁⊆𝐴₂,𝑇𝐴₂⊆𝐴₁. (5.11) Let 𝑢∈𝐴₁, that is, 𝑢(𝑠)≤𝛽(𝑠),∀𝑠∈[0,1]. (5.12) Using condition (5.6), we obtain that 𝑘(𝑡,𝑠,𝑢(𝑠))≥𝑘𝑡,𝑠,𝛽(𝑠),∀𝑡,𝑠∈[0,1]. (5.13) The above inequality with condition (5.4) imply that ¹₀𝑘(𝑡,𝑠,𝑢(𝑠))𝑑𝑠≥¹₀𝑘𝑡,𝑠,𝛽(𝑠)𝑑𝑠≥𝛼(𝑡), (5.14) for all 𝑡∈[0,1]. Then we have 𝑇𝑢∈𝐴₂.
Similarly, let 𝑢∈𝐴₂, that is, 𝑢(𝑠)≥𝛼(𝑠),∀𝑠∈[0,1]. (5.15) Using condition (5.6), we obtain that 𝑘(𝑡,𝑠,𝑢(𝑠))≤𝑘(𝑡,𝑠,𝛼(𝑠)),∀𝑡,𝑠∈[0,1]. (5.16) The above inequality with condition (5.5) imply that ¹₀𝑘(𝑡,𝑠,𝑢(𝑠))𝑑𝑠≤¹₀𝑘(𝑡,𝑠,𝛼(𝑠))𝑑𝑠≤𝛽(𝑡), (5.17) for all 𝑡∈[0,1]. Then we have 𝑇𝑢∈𝐴₁. Finally, we deduce that (5.11) holds.
Now, let (𝑢,𝑣)∈𝐴₁×𝐴₂, that is, for all 𝑡∈[0,1], 𝑢(𝑡)≤𝛽(𝑡),𝑣(𝑡)≥𝛼(𝑡). (5.18) This implies from condition (5.3) that for all 𝑡∈[0,1], 𝑢(𝑡)≤𝛽₀,𝑣(𝑡)≥𝛼₀. (5.19) Now, using condition (5.7), we can write that for all 𝑡∈[0,1], we have |𝑇𝑢−𝑇𝑣|(𝑡)≤¹₀|𝑘((𝑠((𝑠≤𝑡,𝑠,𝑢))−𝑘𝑡,𝑠,𝑣))|𝑑𝑠₁₀𝜉𝑢(𝑠)(𝑠𝑑(|−𝑣)|)𝑑𝑠≤𝜉_∞(𝑢,𝑣)since𝜉isnondecreasing≤𝜉𝑀_𝑇(𝑢,𝑣). (5.20) This implies that 𝑑_∞(𝑇𝑢,𝑇𝑣)≤𝜉𝑀_𝑇(𝑢,𝑣)=𝑀_𝑇(𝑢,𝑣)−𝜑𝑀_𝑇(𝑢,𝑣). (5.21) Using the same technique, we can show that the above inequality holds also if we take (𝑢,𝑣)∈𝐴₂×𝐴₁.
Now, all the conditions of Theorem 2.3 are satisfied (with 𝜓(𝑡)=𝑡 and 𝐿=0), we deduce that 𝑇 has a unique fixed point 𝑢^∗∈𝐴₁∩𝐴₂=𝒲, that is, 𝑢^∗∈𝒲 is the unique solution to (5.1).

Acknowledgment

M. Jleli and B. Samet are supported by the Research Center, College of Science, King Saud University.