Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications

1 Introduction and preliminaries

The Banach contraction principle for multivalued mappings was given by Nadler [1].

Subsequently, Ćirić [2], Feng and Liu [3], Minak et al. [4], Nicolae [5], and Rasham et al. [6] extended this work. Bakhtin [7] and Czerwik [8,9,10] studied Banach’s theorem in metric and b -metric spaces. Mlaiki et al. [11] discussed Banach’s theorem in controlled metric-type space, while Aiadi et al. [12] extended Banach’s theorem to controlled J-metric spaces. Haque et al. [13] studied Fredholm-type integral equation in controlled rectangular metric-like spaces. Also, Shatanawi et al. [14] investigated Banach’s theorem in tvs-cone metric space by applying Mizoguchi-Takahashi-type theorems. Wardowski [15] presented a seminal extension of Banach’s contraction named it F -contraction and established a fixed point results. Consequently, Sgroi and Vetro [16] showed the presence of specific outcomes for multivalued F -contractive mappings and demonstrated the use of their principal hypothesis for integral and functional equations. Nicolae [5] introduced some new fixed point problems obeying Feng-Liu-type contractive conditions that carry out definite roles.

Padcharoen et al. [17] showed periodic fixed point results for advanced-type F -contractive mappings in modular-like-metric-spaces. Aydi et al. [18] discussed F -contraction via admissible mappings in complete metric space and found the solution of the integral equation using their main theorem. Nashine et al. [19] proved some new fixed point results in orbital b-metric spaces and applied their results to prove the existence of a solution to an integral equation under a set of conditions.

Rasham et al. [20] discussed new theorems in modular-like metric spaces linked with fuzzy-dominated F -contractive mappings and showed applications to obtain the solution of fractional and integral equations. Moreover, Rasham et al. [21] achieved some new common fixed points of two families of multivalued dominated mappings obeying advanced rational type α * − ψ contractive conditions on a closed set in a complete dislocated b -metric space. They used a specific weaker class of strictly increasing function A in place of the class of function F used by Wardowski [15]. Recently, Nashine et al. [19] achieved fixed point results for multivalued mappings fulfilling the Wardowski-Feng-Liu-type contraction on orbitally complete b -metric space; they showed new definitions and examples to clarify their results and proved an application for fractal integral equations. However, Qawaqneh et al. [22] obtained some results for multivalued contractions in b -metric spaces.

In this research work, we prove the existence of fixed point results for a coupled families of dominated set-valued mappings fulfilling Nashine, Wardowski, Feng-Liu, and Ćirić-type contraction for orbitally lower several semi-continuous functions in a complete orbitally b -metric space shown by Nashine et al. [19]. Moreover, some new multi-common fixed point results are proved for two families of dominated set-valued mappings in the setting of ordered complete orbitally b -metric space. Some new definitions and illustrative examples are given to validate our new results. To show the novelty of our results, applications are given to obtain the solution of nonlinear integral equations and fractional differential equations.

The pair ( Γ , d b ) is called a b -metric space (or simply d b -metric).

Let R ( Γ ) represent the set that consists of all compact subsets of Γ .

where H d b is called as Hausdorff b -metric on R ( Γ ) .

there exists δ > 0 so that (for all α , β ∈ Z )

D ( T ( α ) , T ( β ) ) > 0 ⇒ δ + F ( D ( T ( α ) , T ( β ) ) ) ≤ F ( D ( α , β ) ) .

The mapping F : R + → R holds the given restrictions:

( F 1 ) F is a function of strictly-increasing mapping;

( F 2 ) lim j → + ∞ ρ j = 0 iff lim j → + ∞ F ( ρ j ) = − ∞ , for each non-negative sequence { ρ j } j = 1 ∞ ;

( F 3 ) for all θ ∈ ( 0,1 ) , lim j → ∞ ρ j θ F ( ρ j ) = 0 ;

( F 4 ) there is a δ > 0 so that for each non-negative sequence { ρ j } ,

δ + F ( b ρ j ) ≤ F ( ρ j − 1 ) for all j ∈ N , then δ + F ( b j ρ j ) ≤ ( b j − 1 ρ j − 1 ) for all j ∈ N .

( F 5 ) F ( inf β ) = inf F ( β ) for all β ⊂ ( 0 , ∞ ) with inf β > 0 .

Obviously Ω χ κ * is not an empty set included with F ( a ) = ln a or F ( a ) = a + ln a .

If for each convergent sequence { Φ j } in H with lim j → + ∞ Φ j = Φ ∈ H , then a mapping g : H → R is said a lower semi-continuous, whenever satisfying g ( Φ ) ≤ lim j → + ∞ inf g ( Φ j ) .

Θ * ( La , Lb ) = inf { Θ ( t , z ) : t ϵ La , z ϵ Lb } ≥ 1 , whenever Θ ( t , z ) ≥ 1 , for all t , z ∈ A

is called an Θ * − admissible.

A mapping L : A → R ( A ) satisfying Θ * ( a , La ) = inf { Θ ( a , h ) : h ϵ La } ≥ 1 is called a Θ * − dominated on E .

and mappings G , R : A → R ( A ) defined by

G s = [ − 4 + s , − 3 + s ] and Rm = [ − 2 + m , − 1 + m ] respectively . Then, both G and R are not Θ * − admissible, but they are Θ * − dominated.

2 Main results

Let ( Γ , d b ) be a complete b -metric space and ℮ 0 ∈ Γ , and let { P μ : μ ∈ H } and { R ϑ : ϑ ∈ } be two families of set-valued mappings from Γ to R ( Γ ) . Let c ∈ H and ℮ 1 ∈ P c ( ℮ 0 ) , then d b ( ℮ 0 , P c ( ℮ 0 ) ) = d b ( ℮ 0 , ℮ 1 ) . Let ℮ 2 ∈ R g ( ℮ 1 ) be such that d b ( ℮ 1 , R g ( ℮ 1 ) ) = d b ( ℮ 1 , ℮ 2 ) , where g ∈ . Proceeding this method, we achieve a sequence { R ϑ P μ ( ℮ j ) } in Γ , where ℮ 2 j + 1 ∈ P z ( ℮ 2 j ) , ℮ 2 j + 2 ∈ R l ( ℮ 2 j + 1 ) , where z ∈ H , l ∈ and j ∈ N ∪ { 0 } . Also, d b ( ℮ 2 j , P z ( ℮ 2 j ) ) = d b ( ℮ 2 j , ℮ 2 j + 1 ) , d b ( ℮ 2 j + 1 , R l ( ℮ 2 j + 1 ) ) = d b ( ℮ 2 j + 1 , ℮ 2 j + 2 ) , then { R ϑ P μ ( ℮ j ) } is the sequence in Γ generated by an initial guess point ℮ 0 . If { P μ : μ ∈ H } = { R ϑ : ϑ ∈ }, then we write { Γ P μ ( ℮ j ) } alternative of { R ϑ P μ ( ℮ j ) } .

Let P μ , R ϑ : Γ → Γ , and for any ϖ 0 ∈ Γ , O ( ϖ 0 ) = { ϖ 0 , P μ ( ϖ 0 ) , R ϑ ( ϖ 1 ) , . … . . } denote the orbit of ϖ 0 . A function g : Γ → R is said to be a ( P μ , R ϑ ) -orbitally lower semi-continuous if g ( ϖ ) < lim n → + ∞ inf g ( ϖ 0 ) for all sequences { P μ R ϑ ( ϖ n ) } ⊂ O ( ϖ 0 ) with lim n → + ∞ { P μ R ϑ ( ϖ n ) } = ϖ ∈ X .

It is remarked that orbitally complete b -metric space may be not complete. Now, we start our main results.

1. for all ℮ , ϖ ∈ { P μ R ϑ ( ℮ j ) } with α ( ℮ , ϖ ) ≥ 1 or α ( ϖ , ℮ ) ≥ 1 and max{ d b ( ℮ , P μ ( ℮ ) ) , d b ( ϖ , R ϑ ( ϖ ) ) } > 0 , there exists ϖ ∈ E L ℮ satisfying

If (2.1) exists, then P μ and R ϑ admit a multi-common fixed point q in Γ .

This implies that

If, max { d b ( ℮ 2 j , ℮ 2 j + 1 ) , d b ( ℮ 2 j + 1 , ℮ 2 j + 2 ) } = d b ( ℮ 2 j , ℮ 2 j + 1 ) , then from (2.2), we have

which is a contradiction. According to ( F 1 ) , the function F is strictly increasing. So, we obtain

for all j ∈ N ∩ { 0 } , we deduce that

for each j ∈ N ∩ { 0 } , also from inequality (2.3) and using ( F 4 ) we obtain

Since, ℮ j + 1 ∈ E L ϖ , then by the definition of E L ϖ , we have

which implies that

From (2.4) and (2.5), we have

Similarly, for each j ∈ N ∩ { 0 } , we have

Substituting (2.7) into (2.5), we have

Now, put 2 j + 1 = a and also d b ( ℮ 2 j , ℮ 2 j + 1 ) = σ a for each a ∈ N ∩ { 0 } , σ a > 0 and also from the inequality (2.8), it is observed that sequence { σ a } is non-increasing. Furthermore, if there lies a ε > 0 so that lim a → + ∞ σ a = ε . As ε > 0 , let ϒ ( t ) = lim inf t → s + τ ( t ) − lim inf t → s + η ( t ) ≥ 0 . Then, using (2.6), we obtain

Take l a , which is the largest number from the { 0 , 1 , 2 , 3 , … , a − 1 } so that

For each a ∈ N , so { σ a } is the sequence of non-decreasing. From inequality (2.9), we obtain

Similarly, from (2.6), we can obtain

Now, we consider two different cases about the sequence { ϒ ( σ l a ) } .

Hence,

for every k . Consequently, lim k → ∞ F ( b a k σ a k ) = − ∞ , and by ( F 2 ) lim k → + ∞ b a k σ a k = 0 , which is not true that lim k → + ∞ σ a k > 0 , for any b > 1 .

Hence, lim e → + ∞ F ( σ e ) = − ∞ , and by ( F 2 ) , lim e → + ∞ σ e = 0 , which is wrong due to the lim e → + ∞ σ e > 0 . Hence, lim e → + ∞ σ e = 0 . From ( R 3 ) , there is k ∈ ( 0 , 1 ) so that lim a → + ∞ ( b a σ a ) k F ( b a σ a ) = − ∞ , and from ( 2.10 ) , the upcoming conditions satisfies for all a ∈ N :

Taking a → + ∞ in (2.11), we obtain

Since ζ = lim a → + ∞ ϒ > 0 there exists a 0 ∈ N , such that ϒ ( σ l a ) > ζ 2 for all a ≠ a 0 . Thus,

for each a > a 0 . Letting a → + ∞ in ( 2.12 ) , we have

0 ≤ lim a → + ∞ F ( b a σ a ) k ζ 2 < lim a → + ∞ F ( b a σ a ) k ϒ ( σ l a ) = 0 .