Solution of Fractional Differential Equations Utilizing Symmetric Contraction

Abstract

The aim of this paper is to present another family of fractional symmetric --contractions and build up some new results for such contraction in the context of -metric space. The author derives some results for Suzuki-type contractions and orbitally -complete and orbitally continuous mappings in -metric spaces. The inspiration of this paper is to observe the solution of fractional-order differential equation with one of the boundary conditions using fixed-point technique in -metric space.

1. Preliminaries and Scope

Fixed-point theory has been promoted by a few particular works in the most recent decades [1–3]. One of the intriguing methodologies was presented in Karapinar et al.’s work [4] which starts a thought of interpolative kind of contractions and sest up shiny new fixed-point results in partial metric space. Recently, Jleli and Samet [5] introduced a new generalization of metric space and named it as -metric space.

Definition 1. (see [5]). Let be the set function that meets the following conditions:  is nondecreasing; that is, for all , it implies .  For each iteration , we haveThe generalized notion of metric space is as follows.

Definition 2. (see [5]). Let with be a given mapping. Suppose that there exists such that  , .  for all .  Each , , , and for each with , we haveThen, it is said that is an -metric on .
Here, the pair is called an -metric space and it is abbreviated as -MS. A sequence in is -Cauchy, if . Furthermore, is -complete, if every -Cauchy sequence is -convergent in .
The following example is stated in [5].

Example 1. (see [5]). The set of natural numbers is an -MS if we define byfor all , , and . Moreover, does not form a metric but it is an -MS.
Jleli and Samet proposed a simple Banach fixed-point theorem as follows.

Theorem 1. (see [5]). Let be an -MS. Let be a self-mapping. Suppose that the following conditions are met:(i) is -complete.(ii) a constant such thatThen, attains a unique fixed-point .
In 2012, Samet et al. introduced a class of -admissible mappings as follows.

Definition 3. (see [6]). Let and . is said to be -admissible if , and implies that .
Next, Salimi et al. [7] modified the concept of -admissible mapping as follows.

Definition 4. (see [7]). Let and be two functions. is called an -admissible mapping with respect to , if , and implies that .
If , then the above definition reduces to Definition 3. If , then is called an -subadmissible mapping.

Definition 5. (see [8]). Consider a metric space and assume that and are two functions. A mapping is considered as --continuous mapping in whenever is given, and the sequence is as follows:For more details, see, for example, [9, 10].
A mapping is called orbitally continuous in if implies that . mapping is orbitally continuous on if is orbitally continuous .

2. Fractional Symmetric --Contraction of Type-I

In this segment, first we present a new fractional symmetric --contraction of type-I.

Definition 6. Let be an -metric space and two functions. We consider that is a fractional symmetric --contraction of type-I along with constants and such that, whenever , we havewherewhere , for all .

Example 2. Let with grace of -metric defined byand consider and . Define byand byIf , clearly such thatBy taking any value of constants and , clearly, (6) holds for all , . Point out that has two fixed points, which are 0 and 1.
Now, we initiate brand new fixed-point theorems for fractional symmetric --contraction of type-I in the configuration of -complete -MS.

Theorem 2. Let be a complete -metric space and is a fractional symmetric --contraction of type-I satisfying the following statements:(i) is an -admissible mapping concerning (ii)There exists to such an extent that (iii) is --continuousAt that point, possesses a fixed point at .

Proof. Consider in with the goal that . For , we build a chain in such a way that and . Proceeding with this exercise, , for every . Presently, as long as mapping is -admissible with respect to , at that time . Carrying on in this way, we getProvided that for some , then is a fixed point of . So, we assume that accompanied byAs is fractional symmetric --contraction of type-I, a part of , we havewhich implies thatand we deduce thatWe conclude that is a nonincreasing sequence with nonnegative terms. Thus, there is a nonnegative constant such that . Note that . From (16), we haveThis providesConsidering for as much asthere subsist some corresponding toLet be fixed and let be cognate and is satisfied. By , there exists which connotes thatHence, by (21) and , we getwhere with the goal that together with . Therefore, by using and (22), we havewhich implies that by , we haveConsequently, is an -Cauchy sequence. Meanwhile is an -complete metric space and there exists such that is -convergent to ; that is,and is --continuous as well as , each one of at that point as ; in other words, . Now we are going to prove that is a fixed point of . We argue by contradiction by supposing that . By , we haveBy using () and the contractive condition, we havefor all . In other words, by using () and (25), we getwhich gives a contradiction. Therefore ; hence possesses a fixed point of .

Theorem 3. Let be an -complete -metric space and let be a fractional symmetric --contraction of type-I fulfilling the accompanying affirmations:(i) is an -admissible mapping concerning (ii)There exists a to such extent that (iii)An iteration in is such that escorted by at the same time ; after that holds for each Afterwards, possesses a fixed point in .

Proof. On closing lines of the proof of Theorem 2, we acquire for each . Using (), we haveFrom (6) connecting , we haveEmploy (25) with certitude thatand we obtainand making use of , we havewhich is a contradiction. Therefore, ; in other words, possesses a fixed point of .

Example 3. Let with an -metric byaccompanied by together with . Define along withand by(i)Case I. If , clearly . Hence, every condition of Theorem 2 is satisfied.(ii)Case II. If , clearly is an -admissible mapping with respect to , whenever , such that(iii)By taking constant , and , for all .(iv)Case III. If any , then we haveTherefore, whole constraints of Theorem 2 are satisfied. Hence, is fractional symmetric --contraction of type-I.

Definition 7. Consider an -metric space and two functions. Then an -metric space on is said to be --complete if and only if every -Cauchy sequence , along with-converges in .

Remark 1. Theorems 2 and 3 also hold for --complete -metric space instead of -complete -metric space (for details, see [10]).

3. Fractional Symmetric --Contraction of Type-II

In this section, a fractional symmetric --contraction of type-II is introduced and in the structure of -complete -metric space. Using this notion, we shall provide a fixed-point theorem.

Definition 8. Consider a self-map on an -metric space and two functions . We presume that is a fractional symmetric --contraction of type-II provided that there are constants and such that, whenever , we ownwherewhere , for all .
Now we show and demonstrate our next theorem.

Theorem 4. Let be an -complete -metric space and let be a fractional symmetric --contraction of type-II fulfilling the accompanying affirmations:(i) is an -admissible mapping concerning (ii)There exists to such an extent that (iii) is --continuousAfter that, possesses a fixed point in .

Proof. Consider in correspondent to . For , we build an iteration in such a way that and . Proceeding with this exercise, , for all . Now, as long as mapping is -admissible with respect to , at that time . Carrying on in this way, we ownProvided that for some , then is a fixed point of . So, we assume that accompanied byAs is fractional symmetric --contraction of type-II, a part of , we ownwhich implies thatand we deduce thatWe conclude that is a nonincreasing sequence with nonnegative terms. As a result, there is a nonnegative constant such that . We shall indicate that . Indeed, from (46), we derive thatThe rest of the test follows the same lines of Theorem 2.

Theorem 5. Consider an -complete -metric space and let be a fractional symmetric --contraction of type-II meeting the following assertions:(i) is an -admissible mapping with respect to (ii)There exists such that (iii)An iteration in is such that escorted by at the same time ; after that holds for each Afterwards, possesses a fixed point in .

Proof. Similar to the lines of Theorem 3, since, by (iii), holds for every . Using (), we meetFrom (40) and , we haveMaking use of (25), we getand we procureUsing , we havewhich is a logical inconsistency. Along these lines ; that is, possesses a fixed point of .

4. Fractional Symmetric --Contraction of Type-III

In this section, fractional symmetric --contraction of type-III is considered in the environment of -complete -metric space. After stating a fixed-point theorem for such maps, we set up fractional symmetric --contraction of type-III as follows.

Definition 9. Consider an -metric space with a self-map and two functions . We say that is fractional symmetric --contraction of type-III along with constants and such that, whenever , we havewherewhere , for all .
Now we declare and demonstrate our next theorem.

Theorem 6. Let an -complete be an -metric space along with being a fractional symmetric --contraction of type-III which meets the following assertions:(i) is an -admissible mapping concerning (ii)There exists such that (iii) is --continuousAfter that, possesses a fixed point in .

Proof. Consider in with the aim that . Take any ; we erect an recapitulate in such a way that and . Continuing with this practice, , every . As long as mapping is -admissible with respect to , at that time . Carrying on in this way, we findProvided that for some , then is a fixed point of . So, we assume that accompanied byAs is fractional symmetric --contraction of type-III, a part of