Research Article | Open Access
V. Pragadeeswarar, R. Gopi, M. De la Sen, "Best Proximity Point for the Sum of Two Non-Self-Operators", Journal of Mathematics, vol. 2020, Article ID 9310750, 7 pages, 2020. https://doi.org/10.1155/2020/9310750
Best Proximity Point for the Sum of Two Non-Self-Operators
In the present paper, we focus our attention on the existence of the fixed point for the sum of the cyclic contraction and the noncyclic accretive operator. Also, we study the best proximity point for the sum of two non-self-mappings. Moreover, we provide the existence of the best proximity point for the cyclic contraction through the notion of the nonlinear -set contraction. Finally, we give the existence of the best proximity point for the sum of the nonlinear -set contraction mapping and partially completely continuous mapping in the setting of the partially ordered complete normed linear space.
Fixed point theory plays an important role in the area of nonlinear functional analysis, and it has many applications in the study of nonlinear differential and integral equations. The study of nonlinear equations of the form , where are mappings on the Banach space , helps to solve many physical nonlinear real-life problems. For example, Dhage and Otrocol  gave the existence and approximation of solutions to the following hybrid differential equation:for all and are continuous functions. Also, Banaś and Amar  obtained the existence of the solution to the nonlinear integral equation of the formfor and . So, the researchers involved in finding the solution of the equation , which is clearly the problem of finding the sufficient condition for the existence of fixed point for the sum of two mappings.
In the sequel, in 1955, Krasnoselskii gave an existence of the solution for the equation in the Banach space setting, where is the contraction and is the compact operator. Later, many researchers extended Krasnoselskii’s theorem in different directions (see [3–7] and the references therein). Vijayaraju  proved the theorems pointing the existence of the fixed point for a sum of nonexpansive and continuous mappings and also a sum of asymptotically nonexpansive and continuous mappings in the setting of locally convex spaces. O’Regan  established the fixed point theorem for the sum of two operators if is compact and is nonexpansive. Moreover, the results were used to prove the existence of the solution for second-order boundary value problem. O’Regan and Taoudi  proved the fixed point theorems for the sum of two weakly sequentially continuous mappings in the Banach space. Dhage  proved the fixed point result by combining two fixed point theorems of Krasnoselskii and Dhage and also derived the existence result for the product of two operators in Banach algebra. Dhage  found the local version of fixed point theorems of Krasnoselskii and Nashed et al. By using this result, he provided the application to nonlinear functional integral equations.
Agarwal et al.  obtained the existence of the fixed point for two mappings, compact mapping and nonexpansive mapping, in the setting of both the weak and the strong topology of a Banach space. Ben Amar and Garcia-Falset  proved the existence of fixed point theorems for different kinds of contractions such as nonlinear weakly condensing, 1-set weakly contractive, and pseudo-contractive and nonexpansive operators defined on unbounded domains and provided application to generalized Hammerstein integral equations. Arunchai and Plubtieng  improved the Krasnoselskii theorem on fixed points for the sum of operators , where is the weakly-strongly continuous mapping and is the asymptotically nonexpansive mapping. Banaś and Amar  proved some new types of fixed point theorems for the sum of on an unbounded closed convex subset of a Hausdorff topological vector space, which are used to provide the solution of integral equations in the Lebesgue space. Wang  derived fixed point results for the sum of two operators and if is contractive with respect to the measure of weak noncompactness and is the -nonlinear contraction in the setting of the Banach space. By having this result, he obtained the existence of solutions to a nonlinear Hammerstein integral equation in the space. Ben Amar et al.  improved Krasnoselskii-type fixed point results for the equation , where the operator is -compact and is -compact and asymptotically -nonexpansive operator on an unbounded closed convex subset of a Banach space.
Later, the people were interested in finding the fixed point in partially ordered metric spaces which are more general than metric spaces. First, Ran and Reurings  initiated the fixed point theorems for the contraction mappings in the partially ordered metric space, which is further improved by Nieto and Rodríguez-López  and by Petruşel and Rus , and used to give the existence of the solution for boundary value problems of nonlinear first-order ordinary differential equations.
Recently, the researchers show interest in finding the optimum solution for real-time modeling problems. In this direction, instead of studying about the fixed point (whenever the fixed point does not exist), the researchers are working on approximating the fixed point in some sense, known as the best proximity point. The existence of best proximity point theorems helps to obtain the optimum solution for different types of modelings. In the literature, there are more number of articles about the existence of the best proximity point for single operators. For example, Basha  derived best proximity point theorems for proximal contractions of the first and second kind in the setting of the metric space. Basha  proved the existence of the best proximity point for principal cyclic contractive mappings, proximal cyclic contractive mappings, and proximal contractive mappings which are defined on the metric space. In 2000, Eldred and Veeramani  proved the best proximity point result for contractive type mappings in the metric space. Al-Thagafi and Shahzad  obtained convergence and existence results of the best proximity points for cyclic -contraction maps in the metric space. For more existence of the best proximity point results, we refer the reader to [20–25].
In the light of the above literature survey, we want to find the approximate solution for the fixed point equation of the form , where are non-self-mappings. So, in this work, we initiate to study the best proximity point for the sum of two non-self-operators, and we provide the existence of the best proximity point for the sum of two operators using best proximity point theorems for the single operator. Additionally, we prove an existence result of the fixed point for the sum of cyclic and noncyclic operators, which involves the concept of accretive operators. Finally, we discuss some notions of the ordered normed linear space, and we find sufficient conditions for the existence of the best proximity point in this space.
First, we collect some notions from . Throughout the paper, we denote as the Banach space and metric space, respectively, and letbe its dual. For each , we associate the setwhere denotes . The multivalued operator is called the duality mapping of . Suppose is an operator from to . Then, the operator is defined by .
Here, we give the definition for weak accretive via the accretive operator in .
Definition 1. An operator is said to be weak accretive if for , there exists such that , where is as in (4).
The following lemma is helpful for our one of the results.
Lemma 1. If is weak accretive on , then , for all , and .
Proof. Let with . Since is weak accretive, for , there exists such that . Now, we have
Theorem 1 (see ). Let be nonempty subsets of which are complete. Suppose satisfies(1) and .(2) for ,where . Then, there exists such that .
3. Main Results
Theorem 2. Let be subsets of such that is the subspace. Let , where and are cyclic and noncyclic mappings which satisfy the following:(1) is the weak accretive operator, and is onto(2) is invertible on , where is the identity operator(3) and (4), for and Then, there exists such that .
Proof. First, we note that assumption (3) is valid only if is the subspace. Since is onto, . Since is invertible on , we can defineNow, we show that is the cyclic mapping. Let ; then, . Since is invertible on , there exists unique such that . Then, the equation has a unique solution . Then, by (3), we obtain . Then, . Similarly, we can prove . Next, we prove that is nonexpansive on . Since is invertible on , it implies that . Let ; then, there exist such that . Using Lemma 1, we getTherefore, for , we obtainThen, the operator agrees with the hypothesis of Theorem 1, and then there is such that , which implies .
4. Best Proximity Point Theorems
The following notions are used in this article: let be nonempty subsets of .
The pair is said to have -property if for and ,
Definition 2. A function is called Lipschitzian if there exists a real constant such that, for all ,
Definition 3 (see ). Let be nonempty subsets of . A map is said to be a weakly contractive mapping iffor all , where is a nondecreasing and continuous function such that is positive on and .
Theorem 3 (see ). Let be complete and be nonempty, closed sets such that is nonempty. Let be a weakly contractive mapping such that . Assume the pair has the -property. Then, there exists unique in such that .
The following theorem tells that the sum of two non-self-operators has the best proximity point in Banach space settings. The notion , where is the subset of , is used in the following theorem.
Theorem 4. Let be two nonempty convex, closed subsets of . Assume is nonempty. Let be two mappings which satisfy(1)If , then (2) is -Lipschitzian, where (3), for all , where is as in Definition 3 and (4) and If the pair has the -property, then there exists unique in such that .
Proof. Because of (1), we define by . First, we show that satisfies . For , we have and . Therefore, there exist such that . Then, . Since is convex, . This shows . Now we prove the mapping is weakly contractive. For , we haveThis gives is a weakly contractive mapping from to . Clearly, the pair has the -property, and . Then, the map satisfies the requirements of Theorem 3, and there exists such that .
5. Best Proximity Point Theorems in the Ordered Normed Linear Space
In this section, first we extract some notions from  to obtain best proximity point results. Let be the real vector space. The pair is called the partially ordered linear space, where is the partial order. Two elements are called comparable if either or holds. A nonempty subset of is said to be a chain or totally ordered if any two elements of are comparable. The space is called the partially ordered normed linear space, where is the norm on .
Definition 4 (see ). A mapping is called monotone nondecreasing if implies for all . Similarly, is called monotone nonincreasing if implies for all .
Definition 5 (see ). A subset of is called partially bounded if every chain in is bounded.
We denote by the family of all bounded chains and relatively compact chains of , respectively.
Definition 6 (see ). A mapping is called a partial measure of noncompactness in if it satisfies(1).(2).(3) is nondecreasing, i.e., if .(4)If sequence of closed chains in with and if , then the set is nonempty. The notion in (1) is known as the kernel of , that is, Clearly, . And so, . Because of for all , we obtain . Then, . The measure is called sublinear if it satisfies(5) for all .(6) for . And has the maximum property if(7). Finally, is said to be full if(8).
Definition 7 (see ). A mapping is called a -function if it is monotone nondecreasing and upper semicontinuous with .
Definition 8 (see ). A nondecreasing mapping is said to be partially nonlinear -set-Lipschitz if there is a -function such thatfor any bounded chain in . If for , then is called a partially nonlinear -set-contraction.
Lemma 2 (see ). If is a -function with for , then for all and vice-versa.
Definition 9 (see ). Let be nonempty subsets of . A map is a partially cyclic contraction map if it satisfies(1) and (2), for some and for , with and are comparableFrom (2), one can easily verify that holds.
Lemma 3 (see ). Let be nonempty subsets of . Suppose is a cyclic contraction map. Then, for any in , we have , where .
First, we show the result for a single operator.
Theorem 5. Let be two nonempty closed and partially bounded subsets of . Let be nondecreasing, partially cyclic contraction, and partially nonlinear -set contraction in . If there exists an element such that , then has a best proximity point.
Proof. Since is cyclic, for , then . Define . Now, again , and define . In the same way, we construct such that and , for . Since and is nondecreasing, it implies that . Again using is nondecreasing, we obtain . Denotefor . Here, each is a bounded chain in , and . From the construction of , we obtainConsequently,Now,Letting limit superior as and using Lemma 2, we obtainTherefore, by condition (4) of , we getSince , then for , there exists such that , for . This implies that and are compact chains in . Hence, has a convergent subsequence, that is, . Now,As and by Lemma 3, we obtainTherefore, . Since is a partially cyclic contraction, we haveAs , we getwhich implies that .
Using the above theorem, here, we provide the result on the best proximity point for the sum of two operators.
Theorem 6. Let be a nonempty, closed, and partially bounded subset of . Let be nondecreasing cyclic mappings which satisfy(1) and .(2) is partially completely continuous, and where and with and are comparable.(3) is a nonlinear -set contraction on , and where and with and are comparable.(4)There exists an element such that .
Then, there exists such that .
Proof. Define by . Clearly, because of (1), is a cyclic mapping. And one can easily prove is nondecreasing on . Since implies , now, we show that is a partial contraction. For and with and are comparable, we havewhich show that is a partial contraction. Next, we prove that is a partially nonlinear -set contraction. Let be a bounded chain in . Therefore, . By conditions sublinearity and full of , we getTherefore, the mapping follows Theorem 5, and there exists such that .
In the nonlinear functional analysis, many mathematical problems can be solved by the existence result of fixed points. The fixed point theorems provide sufficient conditions to ensure the fixed point equation , where is the self-mapping, has a solution. In case of nonlinear problems, it is written as , where are self-mappings, and then the fixed point theorems for the sum of two mappings help to obtain the solution for such an equation. Suppose the mappings are non-self-cases, then the fixed point equation does not possess a solution. In the literature, there are many research papers which deal with the existence of the best proximity point for the equation of the form , where is the non-self-mapping. However, there is no single research work which gives the existence result of the best proximity point for the sum of mappings. So, we want to obtain an approximate solution via finding the best proximity point for such an equation in some sense. So, in this research article, we study the existence of the best proximity point for the sum of two non-self-mappings using best proximity point theorems for a single operator. Moreover, using the notion accretive operators, we prove an existence result of the fixed point for the sum of cyclic and noncyclic operators. Also, we study some notions of the ordered normed linear space, and we provide sufficient conditions for the existence of the best proximity point.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This work was partially funded by Basque Government through Grant IT1207-19.
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