Existence of unique solution to nonlinear mixed Volterra Fredholm-Hammerstein integral equations in complex-valued fuzzy metric spaces

1 Introduction

Historically, the extension of the set of numbers were developed from natural numbers to complex numbers. Similarly, the concept of set has been extended in a variety of ways. A fuzzy set is one such an extension of conventional sets. From a point of view of the extension of ranges of membership functions, that is, extensions from the discrete set {0,1} to the closed unit interval [0, 1], it looks like the extension from natural numbers to real numbers. Fuzzy sets allow us to model very difficult uncertainties in a very easy way. Fuzzy reasoning is a straightforward formalism for encoding human knowledge or common sense in a numerical framework. The idea of fuzzy sets was introduced by Zadeh in the year 1965 [30]. Since then, significant research activities have been established and lead to the development of the theory of fuzzy systems. Michalek and Kramosil established the notion of fuzzy metric spaces (FMSs) which is the extension of probabilistic metric spaces (PMSs) [18]. A number of beautiful results regarding fixed points are derived by several authors in FMSs [11, 17].

Of interest, the study of fuzzy sets has been extended in different directions. Complex fuzzy sets, which are proposed by Ramot et al. [1], are characterized by complex-valued membership functions. The extension looks like one from real numbers to complex numbers. Complex fuzzy sets are given by a complex degree of membership, represented in polar coordinates, which is a combination of a degree of membership in a fuzzy set along with a crisp phase value that denotes the position within the set. The complex fuzzy set carries more information than a traditional fuzzy set and enables efficient reasoning. Buckley introduced complex numbers and complex analysis for the first time [13–15] and many authors generalized the results already proved in analysis to the setting of complex fuzzy sets, see [20–23].

In the year 2011, Azam et al. initiated the notion of complex-valued metric spaces (CVMSs) as a particular case of the so-called cone metric spaces (see [1, 27] and the references therein). They established common fixed point results for self mappings with rational type contraction and discussed its application to integral equations [12].

Very recently Shukla et al. [10] and others established the innovative concept of complex-valued fuzzy metric spaces (CVFMSs) where they generalized fuzzy set to complex fuzzy set by extending the membership function to unit complex interval. They introduced the concept of Cauchy sequence in the said setting and derived some important generalizations involving fuzzy version of Banach contraction principal.

The integral equations have a significant role in mathematical analysis and have many important applications. Various scientific, engineering, physical problems such as scattering in quantum mechanics, water wave, diffraction problems and conformal mapping can be designated by integral equations, see [7, 28]. Since the said setting deals with the vague situations and uncertainty and we know that fractional differential equations with uncertainty have been attracting more and more attentions, since they are capable of modeling many real world processes or phenomenon. As some parameters which appears in each models may be inherit some vagueness. For this purpose authors can investigate the mentioned problems under uncertainty which are presented by fuzzy concepts. Agarwal et al. [3, 4] are the pioneers who studied fuzzy fractional differential equations. They generalized some fixed point theorems and presented an application to the existence of solutions of fuzzy fractional differential equations [5]. With such a fast development of fractional differential equations in the frame of fractional operators of either singular or nonsingular kernels [2, 9], this study and the previously related literature will be useful in that direction.

Motivated by the above work of Shukla et al., we discussed the fuzzy version of (ψ, ϕ) contraction in the setting of CVFMSs and derived beautiful remark. In application to our main results, we present an analytical method to solve mixed Voltera Fredholm-Hammerstein integral equation of the second kind. The solution of the said integral equations has been a subject of considerable interest. Studies on population dynamics, parabolic boundary value problems, the mathematical modeling of the spatio-temporal development of an epidemic and various physical and biological models lead to nonlinear mixed Volterra Fredholm-Hammerstein integral equations. A discussion of the formulation of such models is given in [8, 26] and the references therein. Also so far, as we know, few manuscripts have yet been attempted for solving mixed Voltera Fredholm-Hammerstein integral equation of the second kind in the complex plane. While there has been much work on developing and analyzing numerical methods for solving one-dimensional integral equation, for instant see [24, 29] and the references therein.

2 Preliminaries

Throughout the paper we have denoted the complex number system over the real numbers by C. Let P = {(x, y) :0 ≤ x < ∞, 0 ≤ y < ∞} ⊂ C . The elements (0, 0), (1, 1) ∈ P are denoted by ϖ and l, respectively.

Define a partial ordering ⪯ on C by ξ₁ ⪯ ξ₂ iff ξ₂ - ξ₁ ∈ P . The relations ξ₁ ⪯ ξ₂ and ξ₁ ≺ ξ₂ indicate that Re (ξ₁) ≤ Re (ξ₂), Im (ξ₁) ≤ Im (ξ₂) and Re (ξ₁) < Re (ξ₂), Im (ξ₁) < Im (ξ₂) respectively. A sequence is monotonic with respect to ⪯ if either ξ _q  ⪯ ξ_q+1 or ξ_q+1 ⪯ ξ _q ∀ q ∈ ℕ . Let the unit closed complex interval be denoted by I = {(z, y) :0 ≤ z ≤ 1, 0 ≤ y ≤ 1}, the open unit complex interval by I₀ = {(z, y) :0 ≤ z < 1, 0 ≤ y < 1} and the set {(z, y) :0 < z < ∞, 0 < y < ∞} is denoted by P _ϖ . Clearly for ξ, z ∈ C, z ⪯ ξ iff z - ξ ∈ P _ϖ  .

Let D ⊂ C . If there exists inf D such that it i the lower bound of D, that is inf D ⪯ c ∀ c ∈ D and v ⪯ inf D for every lower bound v of D, then inf D is called the greatest lower bound of D . In the same way we define sup D, (lub)the least upper bound of D .

Definition 2.1. [10] Let Q ⊆ X be a non-empty set. A complex fuzzy set M is characterized by a mapping μ _Q  (x) with domain Q and the range in the closed unit complex interval I, which assigns each element x ∈ Q, a grade of membership in Q, and are thus of the form μ Q ( x ) = r Q ( x ) l , where r _Q  (x) ∈ [0, 1] . The complex fuzzy set may be written as M = { ( x , μ Q ( x ) | x ∈ X ) } .

Definition 2.2. [10] A binary equation * : I × I → I is said to be complex-valued t-norm if the following conditions hold:

z₁* z₂ = z₂ * z₁;

Example 2.3. Let * _a , *  _b , *  _c  : I × I → I be three binary operations defined, respectively by

o₁ *  _a o₂ = (e₁e₂, c₁c₂), for all o₁ = (e₁, c₁), o₂ = (e₂, c₂)∈ I;

Indeed if I _R  = [0, 1] is the real unit closed interval and * _a , *  _b  : I _R  × I _R  → I _R are two t-norms, then * : I × I → I defined by o 1 * o 2 = ( e 1 * a e 2 , c 1 * b c 2 ) , for all o 1 = ( e 1 , c 1 ) , o 2 = ( e 2 , c 2 ) ∈ I , is a complex valued t-norm.

Definition 2.4. [10] Let Q be a non-empty set and * be a continuous complex-valued t-norm and M : Q × Q × P _ϖ  → I be a complex valued fuzzy metric satisfying conditions:

0 ⪯ M (z, ξ, t),

Example 2.5. Consider any metric space (Q, σ). Define * by o₁ * o₂ = (e₁e₂, c₁, c₂), for all o₁ = (e₁, c₁), o₂ = (e₂, c₂) ∈ I . Let the complex fuzzy set M defined be given by M ( z , ξ , t ) = e 1 + c 1 e 1 + c 1 + σ ( z , ξ ) l for all z, ξ ∈ P _ϖ , t = (e, c) . Then (Q, M, *) is a complex valued fuzzy metric spaces.

Indeed in above example, if g : P _ϖ  → (0, ∞) is continuous and o decreasing function, that is, o₁ ⪯ o₂ implies that g (o₁) ≤ g (o₂), then (Q, M, *) is a complex valued fuzzy metric spaces, where M ( z , ξ , t ) = g ( t ) g ( t ) + σ ( z , ξ ) l . Similarly it is obvious that, for the fuzzy set defined by M ( z , ξ , t ) = [ exp ( σ ( z , ξ ) g ( t ) ) ] - 1 l , (Q, M, *) is a complex valued fuzzy metric spaces

Lemma 2.6. [10] Let (Q, M, *) be a complex valued fuzzy metric space. If t, t′ ∈ P _ϖ and t ⪯ t′, then M (z, ξ, t) ⪯ M (z, ξ, t′) ∀ z, ξ ∈ Q .

Definition 2.7. [10] Let (Q, M, *) be a CVFMS. A sequence {z _n } ∈ Q converges to z ∈ Q, if for each ℓ ∈ I₀ and t ∈ P _ϖ there exists k₀ ∈ N with l - ℓ ≺ M ( z n , z , t ) ∀ n > k 0 .

Lemma 2.8. [10] Let (Q, M, *) be complex valued fuzzy metric space. A sequence {z _q } in Q converges to v ∈ Q iff lim q → ∞ M ( z q , v , t ) = l holds ∀ t ∈ P _ϖ  .

Definition 2.9. [10] Let (Q, M, *) be a complex valued fuzzy metric space. A sequence {x _q } in Q is known as a Cauchy sequence if lim q → ∞ inf ℓ > q M ( x q , x ℓ , t ) = l ∀ t ∈ P ϖ .

The complex valued fuzzy metric space (Q, M, *) is called complete if every Cauchy sequence is convergent in Q.

Remark 2.10. [10] Let z _n  ∈ P ∀ n ∈ N then:

If the sequence {z _n } is monotonic with respect to ⪯ and there exist γ, δ ∈ P with γ ⪯ z _n  ⪯ δ, ∀ n ∈ N, then there exists z ∈ P such that lim n → ∞ z n = z .

Remark 2.11. [10] Let x q , x q ′ , w ∈ P , ∀ q ∈ N, then

If x q ⪯ x q ′ ⪯ l ∀ q ∈ N and lim q → ∞ x q = l , then lim q → ∞ x q ′ = l .

3 Main results

Theorem 3.1. Let (Q, M, *) be a complete CVFMSs and let ϝ : Q → Q be a self-mapping satisfying the inequality

ψ ( l - M ( ϝ z , ϝ ξ , t ) ) ⪯ ψ ( l - M ( z , ξ , t ) ) - ϕ ( l - M ( z , ξ , t ) ) ,(3.1) where ψ, ϕ : P → P are both continuous and monotone nondecreasing functions with ψ (t), ϕ (t) > ϖ and ψ (ϖ) = ϕ (ϖ) = ϖ. Then ϝ has a unique fixed point.

Proof. Let z₀ ∈ Q . Define a sequence z _q in Q by z q = ϝ z q - 1 ∀ q ∈ N . If z _q  = z_q-1 for some q ∈ N then z _q is a fixed point of ϝ . Consequently suppose z _q  ¬ = z_q-1 for t ∈ P _ϖ and q ∈ N by (3.1), we have

ψ ( l - M ( z q , z q + 1 , t ) ) = ψ ( l - M ( ϝ z q - 1 , ϝ z q , t ) ) ⪯ ψ ( l - M ( z q - 1 , z q , t ) ) - ϕ ( l - M ( z q - 1 , z q , t ) ) ≺ ψ ( l - M ( z q - 1 , z q , t ) ) .(3.2) Using monotone property of ψ function

M ( z q , z q + 1 , t ) ≻ M ( z q - 1 , z q , t ) ∀ q ∈ N .(3.3) It follows that M (z _q , z_q+1, t) is an increasing sequence of complex numbers in I .

Let U ( t ) = lim q → ∞ M ( z q , z q + 1 , t ) , we shall show that U (t) = l ∀ t > 0. If not, then there exists t > 0 with U (t) < l, then from (3.2) on taking q → ∞, we have ψ ( l - U ( t ) ) ≺ ψ ( l - U ( t ) ) , which is a contradiction. Therefore, M (z _q , z_q+1, t) → l as q→ ∞.

To prove that z _q is a Cauchy sequence, let define K q = { M ( z q , z ℓ , t ) : ℓ > q } ⊂ I for q ∈ N and fixed t ∈ P ϖ . Since ϖ ≺ M (z _q , z_ℓ, t) ⪯ l ∀ q ∈ N . By Remark 2.10, the infimum inf K _q exists ∀ q ∈ N . Now for each positive integer s, M ( z q , z q + s , t ) ≽ M ( z q , z q + 1 , t s ) * M ( z q + 1 , z q + 2 , t s ) * ⋯ * M ( z q + s - 1 , z q + s , t s ) . It follows that lim q → ∞ inf q + s > q M ( z q , z q + s , t ) ≽ l * l * ⋯ * l = l . Therefore, from Definition 2.4, we have lim q → ∞ inf ℓ > q M ( z q , z ℓ , t ) = l . Hence M (z _q , z_ℓ, t) is a Cauchy sequence. Since Q is complete and by Lemma 2.8, we have v ∈ Q such that

lim q → ∞ M ( z q , v , t ) = l , ∀ t ∈ P ϖ .(3.4) Therefore, for any t ∈ P _ϖ and q ∈ N, it follows from (3.1) that ψ ( l - M ( ϝ z q , ϝ v , t ) ) ⪯ ψ ( l - M ( z q , v , t ) ) - ϕ ( l - M ( z q , v , t ) ) ⪯ ψ ( l - M ( z q , v , t ) ) . Using monotonicity of ψ function, we get

M ( ϝ z q , ϝ v , t ) ≽ ( M ( z q , v , t ) ) .(3.5) For any t ∈ P _ϖ , M ( v , ϝ v , t ) ≽ M ( v , z q + 1 , t 2 ) * M ( z q + 1 , ϝ v , t 2 ) = M ( v , z q + 1 , t 2 ) * M ( ϝ z q , ϝ v , t 2 ) . Taking lim q → ∞ and using (3.4), (3.5) and Remark 2.11, we have M ( v , ϝ v , t ) = l ∀ t ∈ P ϖ , that is ϝv = v .

Let w be any other fixed point of ϝ and there exists t ∈ P _ϖ such that M (v, w, t) ¬ = l, then it follows from (3.1) that ψ ( l - M ( v , w , t ) ) = ψ ( l - M ( ϝ v , ϝ w , t ) ) ⪯ ψ ( l - M ( v , w , t ) ) - ϕ ( l - M ( v , w , t ) ) ≺ ψ ( l - M ( v , w , t ) ) ] l - M ( v , w , t ) ≺ l - M ( v , w , t ) , using monotone property of ψ function which is contradiction. Hence M (v, w, t) = l ∀ t ∈ P _ϖ , i.e v = w .□

The following example demonstrates the applicability of Theorem 3.1.

Example 3.2. Let I _R  = [0, 1] and Q = I _R  ∪ {2}. Define a t-norm by d * e = ( max { d 1 + d 2 - 1 , 0 } , max { e 1 + e 2 - 1 , 0 } ) ∀ d = (d₁, e₁), e = (d₂, e₂) ∈ I. Define ρ : Q × Q → C as follows; ρ ( z , ξ ) = | z - ξ | ( θ , 1 ) , where θ is a nonnegative real number. Then (Q, ρ) is complex valued metric space. Further it is possible to define M ( z , ξ , c ) = l - ρ ( z , ξ ) 1 + de + θ ∀ z , ξ ∈ Q , c = ( d , e ) ∈ P ϖ . Then (Q, M, *) is complex valued fuzzy metric space. Define the mapping ϝ : Q → Q by ϝ ( ξ ) = { ξ 2 if ξ = 0 1 if ξ ¬ = 0 . Also, defined ϕ, ψ : P → P by ψ ( t ) = t 2 , ϕ ( t ) = t 8 . Then, we have

ψ ( l - M ( z , ξ , c ) ) - ϕ ( l - M ( z , ξ , c ) ) = ψ ( | z - ξ | 1 + de + θ ( θ , 1 ) ) - ϕ ( | z - ξ | 1 + de + θ ( θ , 1 ) ) = | z - ξ | 2 ( 1 + de + θ ( θ , 1 ) - | z - ξ | 8 ( 1 + de + θ ( θ , 1 ) ) = 3 | z - ξ | 8 ( 1 + de + θ ) ( θ , 1 ) ) ≽ 2 | z - ξ | 8 ( 1 + de + θ ) ( θ , 1 ) = | ξ 2 - z 2 | 2 ( 1 + de + θ ) ( θ , 1 ) = ψ ( | ξ 2 - z 2 | ( 1 + de + θ ) ( θ , 1 ) )(3.6) = ψ ( l - ( l - | ξ 2 - z 2 | ( 1 + de + θ ) ( θ , 1 ) ) ) = ψ ( l - M ( ϝ ξ , ϝ z , c ) ) . From the last inequality, we conclude that the condition (3.1) is satisfied. i.e. for all z, ξ ∈ P, we note that only (0, 0) satisfies (3.6). Therefore, ϝ has only one fixed point.

Corollary 3.3. Let (Q, M, *) be complete CVFMSs and let the mapping ϝ : Q → Q satisfying the weakly contractive inequality l - M ( ϝ z , ϝ ξ , t ) ⪯ l - M ( z , ξ , t ) - ϕ ( l - M ( z , ξ , t ) ) , where ϕ : P → P is continuous, monotone nondecreasing function with ϕ (t) > ϖ and ϕ (ϖ) = ϖ. Then ϝ has only one fixed point.

Proof. The proof is immediate if ψ (t) = t in Theorem 3.1.□

Now we obtain the existence result of fixed point for mapping satisfying the restricted contraction condition. For this we set B [z₀, ℓ, t] = {z ∈ Q : l - ℓ ⪯ M (z₀, z, ℓ)}, for ℓ ∈ I _θ , t ∈ P _ϖ and z₀ ∈ Q.

Corollary 3.4. Let (Q, M, *) ba complete CVFMSs and ϝ : Q → Q with:

There exists z₀ ∈ Q and ℓ ∈ I _θ with l - ℓ ⪯ M (z₀, ϝ z₀, t) for all t ∈ P _ϖ  .

ψ ( l - M ( ϝ z , ϝ ξ , t ) ) ⪯ ψ ( l - M ( z , ξ , t ) ) - ϕ ( l - M ( z , ξ , t ) ) ,(3.7) where ψ, ϕ : P → P are both continuous, monotone nondecreasing functions with ψ (t), ϕ (t) > ϖ and ψ (ϖ) = ϕ (ϖ) = ϖ. Then ϝ has a unique fixed point in B [z₀, ℓ, t].

Proof. To prove the require result, it is enough to show that B [z₀, ℓ, t] is complete and ϝz ∈ B [z₀, ℓ, t]. Let {z _q } ∈ B [z₀, ℓ, t] be a Cauchy sequence. Since Q is complete and using Lemma 2.8, there exists v ∈ Q with lim q → ∞ M ( z 0 , v , t ) = l for all t ∈ P ϖ . Now for all s, q ∈ N M ( z 0 , v , t + t s ) ≽ M ( z 0 , z q , t ) * M ( z 0 , z q , t s ) . Since z _q  ∈ B [z₀, ℓ, t], for each q ∈ N, and for all t ∈ P _ϖ , lim q → ∞ M ( z 0 , v , t ) = l and by definition of t-norm we have M ( z 0 , v , t + t s ) ≽ ( l - ℓ ) * l , for every s ∈ N . Letting lim s → ∞ and using Remark 2.11, we get M (z₀, v, t) ≽ l - ℓ . Thus v ∈ B [z₀, ℓ, t].

For each z ∈ B [z₀, ℓ, t], it yields from (3.7) ψ ( l - M ( ϝ z 0 , ϝ z , t ) ) ⪯ ψ ( l - M ( z 0 , z , t ) ) - ϕ ( l - M ( z 0 , z , t ) ) , ⪯ ψ ( l - M ( z 0 , z , t ) ) by monotonicity of ψ, ϕ we obtain l - M ( ϝ z 0 , ϝ z , t ) ⪯ l - M ( z 0 , z , t ) , which yields M ( ϝ z 0 , ϝ z , t ) ≽ M ( z 0 , z , t ) . Therefore, for all s ∈ N, we get M ( z 0 , ϝ z , t + t s ) ≽ M ( z 0 , ϝ z q , t s ) * M ( ϝ z 0 , ϝ z , t ) ≽ ( l - ℓ ) * M ( z 0 , z , t ) ≽ ( l - ℓ ) * ( l - ℓ ) ≽ ( l - ℓ ) . Taking lim s → ∞ and using Remark 2.11, we have M (z₀, ϝ z, t) ≽ l - ℓ . Thus ϝz ∈ B [z₀, ℓ, t] .□

Theorem 3.5. Let (S, M, *) be complete CVFMSs such that, for any sequence {t _q } in P _ϖ with lim q → ∞ t q = ∞ , we have lim q → ∞ inf ξ ∈ S M ( z , ξ , t q ) = l , ∀ z ∈ S . If ϝ : S → S satisfies ψ ( M ( ϝ z , ϝ ξ , t ) ) ≽ ψ ( M ( z , ξ , t ) ) - ϕ ( M ( z , ξ , t ) ) ,(3.8) for z, ξ ∈ S, t ∈ P _ϖ and μ ∈ (0, 1), where ψ, ϕ : P → P are both continuous, monotone nondecreasing functions with ψ (ϖ), ϕ (ϖ) > ϖ and ψ (ϖ) = ϕ (ϖ) = ϖ. Then ϝ has a unique fixed point.

Proof. Let z _q  ∈ S be arbitrary fixed. Define a sequence {z _q } in S z q = ϝ z q - 1 ∀ q ∈ N . If z _q  = z_q-1 for some q ∈ N then z _q is a fixed point of ϝ . Let z _q  ¬ = z_q-1 ∀ q ∈ N. Substitute z = z_q-1 and ξ = z _q in (3.8), we have to prove that z _q is a Cauchy sequence. For q ∈ N and fixed t ∈ P _ϖ , define K q = { M ( z q , z ℓ , t ) : ℓ > q } ⊂ I . Since ϖ ≺ M (z _q , z_ℓ, t) ⪯ l ∀ q ∈ N . By Remark 2.10, the infimum, inf K _q  = σ _n , (say) exists ∀ q ∈ N . For t ∈ P _ϖ and q, ℓ ∈ N with ℓ > q, by (3.8) and Lemma 2.6, we have ψ ( M ( z q + 1 , z ℓ + 1 , t ) ) = ψ ( M ( ϝ z q , ϝ z ℓ , t ) ) ≽ ψ ( M ( z q , z ℓ , t ) ) - ϕ ( M ( z q , z ℓ , t ) ) , implies that

ψ ( M ( z q + 1 , z ℓ + 1 , t ) ) ≽ ψ ( M ( z q , z ℓ , t ) ) ≽ ψ ( M ( z q , z ℓ , t μ ) ) .(3.9) Using the monotone property of ψ function, we have M ( z q , z ℓ , t ) ⪯ M ( z q + 1 , z ℓ + 1 , t ) ∀ q , ℓ ∈ N with ℓ > q . Therefore, by definition we have

ϖ ⪯ σ q ⪯ σ q + 1 ⪯ l , ∀ q , ℓ ∈ N .(3.10) Thus {σ _q } is a monotonic sequence in P and, utilizing Remark 2.10 and (3.10), there exists l′ ∈ P such that

lim q → ∞ σ q = l ′ .(3.11) Again, from (3.9) we have for t ∈ P and q ∈ N, σ q + 1 = inf ℓ > q M ( z q + 1 , z ℓ + 1 , t ) ≽ inf ℓ > q M ( z q , z ℓ , t μ ) Similarly for ℓ > q with ℓ, q ∈ N, we get M ( z q + 1 , z ℓ + 1 , t ) ≽ M ( z q , z ℓ , t μ ) = M ( ϝ z q - 1 , ϝ z ℓ - 1 , t μ ) ≽ M ( z q - 1 , z ℓ - 1 , t μ 2 ) = M ( ϝ z q - 2 , ϝ z ℓ - 2 , t μ 2 ) ≽ M ( z q - 2 , z ℓ - 2 , t μ 3 ) ≽ ⋯ ≽ M ( ϝ z 0 , ϝ z ℓ - q , t μ q + 1 ) , hence ∀ q ∈ N and t ∈ P σ q + 1 = inf ℓ > q M ( z q + 1 , z ℓ + 1 , t ) ≽ inf ℓ > q M ( z 0 , z ℓ - q , t μ q + 1 ) ≽ inf ξ ∈ S M ( z 0 , ξ , t μ q + 1 ) . By (3.11) and hypothesis, we have

l ′ ≽ lim q → ∞ inf ℓ > q M ( z 0 , ξ , t μ q + 1 ) = l .(3.12) By (3.11) and (3.12), we get lim q → ∞ σ q = l Thus {z _q } is a Cauchy sequence in S. Since S is complete and by using Lemma 2.8, there exists v ∈ S with

lim q → ∞ M ( z q , v , t ) = l ∀ t ∈ P ϖ .(3.13) For any t ∈ P _ϖ , it yields from (3.8), that M ( v , ϝ v , t ) ≽ M ( v , z q + 1 , t 2 ) * M ( z q + 1 , ϝ v t 2 ) = M ( v , z q + 1 , t 2 ) * M ( ϝ z q , ϝ v t 2 ) ≽ M ( v , z q + 1 , t 2 ) * M ( z q , v t 2 μ ) . Taking the limit as q→ ∞ and using (3.13) and Remark 2.11, we get M (v, ϝ v, t) = l that is ϝv = v . Let w ∈ S be any other fixed point of ϝ and there exists t ∈ P _ϖ such that M (v, w, t) ¬ = l, then it follows from (3.8) that ψ ( M ( v , w , t ) ) = ψ ( M ( ϝ v , ϝ w , t ) ) ≽ ψ ( M ( v , w , t μ ) ) - ϕ ( M ( v , w , t μ ) ) ≽ ψ ( M ( v , w , t μ 2 ) ) - ϕ ( M ( v , w , t μ 2 ) ) ≽ ⋯ ≽ ψ ( M ( v , w , t μ q ) ) - ϕ ( M ( v , w , t μ q ) ) , ∀ q ∈ N. Using M ( v , w , t μ q ) ≽ M ( v , ξ , t μ q ) and lim q → ∞ t μ q = ∞ , it follows that M (v, w, t) ≽ l, which is a contradiction. Thus M (v, w, t) = l ∀ t ∈ P _ϖ , that is v = w and hence uniqueness follows.

□

Remark 3.6.

If ϕ (t) = kt where 0 < k < 1, in Corollary 3.3 then we obtain Theorem 3.1 of [10].

In the following we provide a more general form of statement then of Theorem 3.5 for fixed point, as follows.

Theorem 3.7. Let (S, M, *) be complete CVFMSs such that, for any sequence {t _q } in P _ϖ with lim q → ∞ t q = ∞ , we have lim q → ∞ M ( z , ξ , t q ) = l , ∀ z ∈ S . Moreover, assume that there is an z₀ ∈ S such that for any sequence {t _q } ∈ P _ϖ with lim q → ∞ t q = ∞ , we have lim q → ∞ inf ξ ∈ A z 0 M ( z , ξ , t q ) = l , where A _{z 0} represents the set of ϝ-iterates of z₀, that is, A _{z 0}  = {ϝ  ⁿ  (z₀) : q ∈ N}.

If ϝ : S → S satisfies ψ ( M ( ϝ z , ϝ ξ , t ) ) ≽ ψ ( M ( z , ξ , t ) ) - ϕ ( M ( z , ξ , t ) ) ,(3.14) ∀ z, ξ ∈ S, t ∈ P _ϖ and μ ∈ (0, 1), where ψ, ϕ : P → P are both continuous, monotone nondecreasing functions with ψ (ϖ), ϕ (ϖ) > ϖ and ψ (ϖ) = ϕ (ϖ) = ϖ. Then ϝ has a unique fixed point.

Proof. Define a sequence {z _q } in S by z q = ϝ z q - 1 ∀ q ∈ N . Since, for t ∈ P _ϖ , σ q + 1 = inf ℓ > q M ( z q + 1 , z ℓ + 1 ) ≽ inf ℓ > q M ( z 0 , z ℓ - q , t μ q + 1 ) ≽ inf ℓ > q M ( z 0 , ϝ ℓ - q ( ξ ) , t μ q + 1 ) ≽ inf ξ ∈ A z 0 M ( z 0 , ξ , t μ q + 1 ) .

Thus lim q → ∞ σ q + 1 = lim q → ∞ inf ℓ > q M ( z q + 1 , z ℓ + 1 ) ≽ inf ξ ∈ A z 0 M ( z 0 , ξ , t μ q + 1 ) = l . The proof is completed as the proof of Theorem 3.5. □

Application to the fixed point theorems obtained in this manuscript, we study the existence of the solution to a nonlinear integral equation mentioned in abstract part of this manuscript.

Theorem 4.1. Consider the integral equation 0.1. Assume that the following conditions are satisfied,

Let ψ, ϕ be continuous, monotonic nondecreasing functions with ψ (t), ϕ (t) >0,

Proof. Let S = (C [0, 1], R) . It is easy to check that (S, M, *) is a complex valued fuzzy metric space with the complex valued fuzzy metric defined by M ( z , ξ , t ) = t t + ∥ z - ξ ∥ ∞ l , where ∥ z - ξ ∥ ∞ = max t ∈ P | z ( t ) - ξ ( t ) | ∀ z, ξ ∈ S, t ∈ P, where t₁ * t₂ = min {t₁, t₂} ∀ t₁ = (a₁, b₁) and t₂ = (a₂, b₂) ∈ I. Now define a mapping N : ( S , ∥ . ∥ ∞ ) → ( S , ∥ . ∥ ∞ ) by

N ( ξ ( t ) ) = g ( t ) + ρ ∫ 0 t K 1 ( t , ℘ ) ϝ 1 ( ℘ , ξ ( ℘ ) ) d ℘ + ϱ ∫ 0 1 K 2 ( t , ℘ ) ϝ 2 ( ℘ , ξ ( ℘ ) ) d ℘ ,(4.1) from (4.1), we obtain | N ( ξ ( t ) ) - N ( z ( t ) ) | = | ρ ( ∫ 0 t K 11 * ( t , ℘ ) ϝ 11 * ( ℘ , ξ ( ℘ ) ) d ℘ - K 11 * ( t , ℘ ) ϝ 11 * ( ℘ , z ( ℘ ) ) d ℘ + ι ̇ ∫ 0 t K 12 * ( t , ℘ ) ϝ 12 * ( ℘ , ξ ( ℘ ) ) d ℘ - K 11 * ( t , ℘ ) ϝ 11 * ( ℘ , z ( ℘ ) ) d ℘ ) + ϱ ( ∫ 0 1 K 12 * ( t , ℘ ) ϝ 12 * ( ℘ , ξ ( ℘ ) ) - K 12 * ( t , ℘ ) ϝ 12 * ( ℘ , z ( ℘ ) ) d ℘ + ι ̇ ∫ 0 1 K 22 * ( t , ℘ ) ϝ 22 * ( ℘ , ξ ( ℘ ) ) - K 22 * ( t , ℘ ) ϝ 22 * ( ℘ , z ( ℘ ) ) d ℘ ) | ≤ | ρ | ( A 11 ∫ 0 t | ϝ 11 * ( ℘ , ξ ( ℘ ) ) d ℘ - ϝ 11 * ( ℘ , z ( ℘ ) ) d ℘ | + A 12 ∫ 0 t | ϝ 12 * ( ℘ , ξ ( ℘ ) ) d ℘ - ϝ 12 * ( ℘ , z ( ℘ ) ) d ℘ | ) + | ϱ | ( A 21 ∫ 0 1 | ϝ 21 * ( ℘ , ξ ( ℘ ) ) d ℘ - ϝ 21 * ( ℘ , z ( ℘ ) ) d ℘ | + A 22 ∫ 0 1 | ϝ 22 * ( ℘ , ξ ( ℘ ) ) d ℘ - ϝ 22 * ( ℘ , z ( ℘ ) ) d ℘ | ) ⪯ | ρ | ( A 11 B 11 + A 12 B 12 ) ∫ 0 t | ξ - z | d ℘ + | ϱ | ( A 21 B 21 + A 22 B 22 ) ∫ 0 1 | ξ - z | d ℘ ⪯ | ρ | ( A 11 B 11 + A 12 B 12 ) + | ϱ | ( A 21 B 21 + A 22 B 22 ) ∥ z - ξ ∥ ∞ ⪯ q ∥ ξ - z ∥ ∞ ⪯ ∥ ξ - z ∥ ∞ . So, ∥ N ( ξ ) - N ( z ) ∥ ∞ ⪯ ∥ ξ - z ∥ ∞ . Now for all ξ, z, γ, y ∈ I, with ξ ⪯ z ⪯ γ ⪯ y, it follows that ψ ( M ( N ( ξ ) , N ( z ) , t ) ) = ψ ( min { M ( N ( ξ ) , N ( z ) , t ) , M ( N ( γ ) , N ( y ) , t ) } ) = ψ ( min { t t + ∥ N ( ξ ) - N ( z ) ∥ ∞ l , t t + ∥ N ( γ ) - N ( y ) ∥ ∞ l } ) ≽ ψ ( min { t t + ∥ ξ - z ∥ ∞ l , t t + ∥ γ - y ∥ ∞ l } ) ≽ ψ ( min { t t + ∥ ξ - z ∥ ∞ l , t t + ∥ γ - y ∥ ∞ l } ) - ϕ ( min { t t + ∥ ξ - z ∥ ∞ l , t t + ∥ γ - y ∥ ∞ l } ) = ψ ( M ( ξ , z , t ) * M ( γ , y , t ) ) - ϕ ( M ( ξ , z , t ) * M ( γ , y ) , t ) ) = ψ ( M ( ξ , z , t ) - ϕ ( M ( ξ , z , t ) .

Thus by the application of Theorem 3.5 and defined operator N , problem (0.1) has a unique common solution in S.□

We have developed a novel approach for studying the existence of solutions for Volterra Fredholm-Hammerstein integral equation. The main tool is (ψ, ϕ) contraction for fuzzy valued operator in complete complex valued fuzzy metric space. The obtained theorems extend and improve the corresponding results which given in the literature. Furthermore, the results show that the proposed approach is a promising tool for such type of equations.

Hopefully, the proposed approach can be used for other kinds of systems of integral equations and also it would be possible to extend the mentioned approach for solving systems of integro-differential equations.