The existence and uniqueness of solutions to a functional equation arising in psychological learning theory

1 Preliminary discussion and introduction

The study of mathematical modeling of perceptual, intellectual, and cognitive processes is known as mathematical psychology. Alternatively, learning in animals and humans can be viewed as a sequence of decisions among several feedback possibilities. Preference sequences are typically surprising, indicating that reaction decisions are determined randomly even in basic recurrent research under well-controlled settings. As a result, it is instructive to include systematic changes in a succession of options to signify differences in response likelihood over trials. From this perspective, most of the learning study is devoted to elucidating the probability of the events that constitute a stochastic method over individual practices.

Recent mathematical learning research has proven that the most superficial learning experiment’s behavior can be modeled using stochastic processes. So, it is a partially original thought (for further details, see [1,2]). Nevertheless, after 1950, two significant features became apparent, according to the research conducted primarily by Bush, Estes, and Mosteller. First, all recommended models have an inclusive learning process, which is crucial. Second, the statistical features of such models cannot be hidden from the analysis [3,4].

In 1976, Istraţéscu [5] investigated the possible participation of predator animals eating two different types of food by the following equation:

where H : Y → R with H ( 0 ) = 0 , H ( 1 ) = 1 , and 0 < ϖ 1 ≤ ϖ 2 < 1 . This behavior is theoretically defined by a Markov process with the state space Y and the probability of transition from state x to state ϖ 1 + ( 1 − ϖ 1 ) x and from state x to state ( 1 − ϖ 2 ) x , which is provided by

respectively. In the aforementioned functional equation (1.1), the solution H represents the ultimate probability of an occurrence when the predator is focused on a specific type of prey, considering that the starting probability for this class to be selected is equal to x .

Turab and Sintunavarat [6] addressed Bush and Mosteller’s experimental studies [7] in 2019. They examined a paradise fish’s behavior using their associated probabilities and presented the subsequent model

where H : Y → R such that H ( 0 ) = 0 , H ( 1 ) = 1 , and 0 < ϖ 1 ≤ ϖ 2 < 1 .

In 2020, the authors employed the aforesaid approach and proposed the following functional equation (see [8]):

where H : Y → R , 0 < ϖ 1 ≤ ϖ 2 < 1 , and η 1 , η 2 ∈ Y . The outlined functional equation was used in this research to investigate a particular form of psychological barrier shown by dogs when confined in a tiny box.

Berinde and Khan, in [9], extended the concepts described above and discussed the existence of a unique solution to the following equation:

where H : Y → R and S 1 , S 2 : Y → Y are given contraction mappings. In addition, functional equations of this kind are utilized to explain the connection between predator animals and their two prey options. On the other hand, a few researchers discussed the functional equation’s solution [10], stability [11], and convergence results [12].

For an arbitrary interval [ ξ 1 , ξ 2 ] , Turab and Sintunavarat [13], in 2020, proposed a functional equation that is an extension of the idea presented in [9]

where H : [ ξ 1 , ξ 2 ] → R and S 1 , S 2 : [ ξ 1 , ξ 2 ] → [ ξ 1 , ξ 2 ] are given contraction mappings.

Different conclusions have been drawn from various human and animal behavior investigations in probability-learning contexts (for further details, see [14–18]) using the transition operators given in Table 1.

In 2022, by depending on the reward and selected side discussed by Bush and Wilson [7] (Table 1), Turab et al. [19] extended the aforementioned work by introducing the following functional equation:

for all x ∈ Y , where κ : Y → Y stands for the proportional of the event probability occurring with the constant fraction of occurrences falling inside 0 ≤ ζ ≤ 1 , H : Y → R is an unknown, and S 1 , S 2 , S 3 , S 4 : Y → Y are given mappings.

By following the work of Bush and Wilson [7], recently, Turab extended the above model (1.7) to the arbitrary interval [ ξ 1 , ξ 2 ] by proposing the following functional equation (for the detail, see [20]):

for all x ∈ Y ˜ = [ ξ 1 , ξ 2 ] , where 0 ≤ ν ≤ 1 , H : Y ˜ → R is an unknown, and S 1 , S 2 , S 3 , S 4 : Y ˜ → Y ˜ are given mappings.

Various studies have examined how different animals respond to learning new probabilistic patterns, and their findings have changed considerably (for further details, see [21–24]). In these types of investigations, a typical issue that arises is

To what extent would the structure be affected if an animal refused to shift its position and remained in the middle?

A proper answer to this question requires a consideration of the innovative studies that Neimark has undertaken on human behavior in two-choice perspective scenarios (see [25]). During trials, the subjects were required to choose which of two lights would be illuminated, while others were given the option of doing nothing. Therefore, the “blank trials,” as she introduced them, are the third kind of event.

In light of the notable study mentioned above, we offer a generic functional equation as follows:

for all x ∈ Y , where H : Y → R , υ 1 , υ 2 , υ 3 , υ 4 , υ 5 : Y → Y are corresponding probabilities of the events with υ 5 ( x ) = ( 1 − υ 1 ( x ) − υ 2 ( x ) − υ 3 ( x ) − υ 4 ( x ) ) , and S 1 , S 2 , S 3 , S 4 , S 5 : Y → Y are given mappings.

In this case, we shall employ the Banach fixed point theorem to demonstrate that the proposed equation (1.9) has a unique solution and that it exists. The stability of the recommended functional equation’s solution shall be tested under the results proposed by Hyers-Ulam and Hyers-Ulam-Rassias. Two examples then show the significance of our research in this area.

Moreover, the philosophy of fixed point theory is engaged with the criteria that guarantee the existence of solutions to the fitted system. It includes methods that can be employed to address issues in several branches of mathematics. We recommend [26–33] and the references within it for the most current study done in this area.

For the progression to continue, the next outcome is required.

2 Existence of a unique solution

Throughout this work, we let Y = [ 0 , 1 ] and D is a category of all real-valued continuous functions H : Y → R with sup μ ≠ ν ∣ H ( μ ) − H ( ν ) ∣ ∣ μ − ν ∣ < ∞ and H ( 0 ) = 0 . There is no room for doubt in the fact that ( D , ∥ ⋅ ∥ ) is a Banach space (see [35]) with

In this subsequent development, we shall use the following suppositions:

1. There exists C ≠ ∅ ⊂ T ≔ { H ∈ D ∣ H ( 1 ) ≤ 1 } such that ( C , ‖ ⋅ ‖ ) be a Banach space, where ‖ ⋅ ‖ is defined in (2.1).

2. The mappings υ k : Y → Y with υ 1 ( 0 ) = 0 = υ 2 ( 0 ) , for k = 1 , 2 , 3 , 4 , 5 , satisfy the following properties:

   (2.2) d ( υ k μ , υ k ν ) ≤ d ( μ , ν ) , ∀ μ , ν ∈ Y ( non-expansive ) ,

   and

   (2.3) ∣ υ k ( x ) ∣ ≤ ω k , ∀ x ∈ Y and ω k ≥ 0 ( boundedness ) .

3. The mappings S k : Y → Y with S 3 ( 0 ) = S 4 ( 0 ) = S 5 ( 0 ) = 0 , for k = 1 , 2 , 3 , 4 , 5 , satisfy the following properties:

   (2.4) d ( S k μ , S k ν ) ≤ α k d ( μ , ν ) , ∀ μ , ν ∈ Y and α k < 1 ( contraction ) ,

   and

   (2.5) ∣ S k ( x ) ∣ ≤ τ k , ∀ x ∈ Y and τ k ≥ 0 ( boundedness ) .

4. (Hyers-Ulam-Rassias stability [36]) For a function δ : C → [ 0 , ∞ ) and for H ∈ C with d ( P H , H ) ≤ δ ( H ) , there exists a unique H ⋆ ∈ C such that P H ⋆ = H ⋆ and d ( H , H ⋆ ) ≤ γ δ ( H ) with γ > 0 .

5. (Hyers-Ulam stability [37]) For ϑ > 0 and for H ∈ C with d ( P H , H ) ≤ ϑ , there exists a unique H ⋆ ∈ C such that P H ⋆ = H ⋆ and d ( H , H ⋆ ) ≤ γ ϑ with γ > 0 .

In contrast to the initial conditions ( λ 2 ) and ( λ 3 ) , here, we present the following relaxed conditions:

1. The mappings υ k : Y → Y with υ 1 ( 0 ) = 0 = υ 2 ( 0 ) , for k = 1 , 2 , 3 , 4 , 5 , satisfy the following properties:

   (2.8) d ( υ k μ , υ k ν ) ≤ β k d ( μ , ν ) , ∀ μ , ν ∈ Y and β k < 1 ( contraction ) ,

   and

   (2.9) ∣ υ k ( x ) ∣ ≤ ω ˜ , ∀ x ∈ Y and ω ˜ ≥ 0 ( boundedness ) .

2. The mappings S k : Y → Y with S 3 ( 0 ) = S 4 ( 0 ) = S 5 ( 0 ) = 0 , for k = 1 , 2 , 3 , 4 , 5 , satisfy the following properties:

   (2.10) d ( S k μ , S k ν ) ≤ α k d ( μ , ν ) , ∀ μ , ν ∈ Y and α k < 1 ( contraction ) ,

   with α 1 ≤ α 2 ≤ α 3 ≤ α 4 ≤ α 5 and

   (2.11) ∣ S k ( x ) ∣ ≤ τ ˜ , ∀ x ∈ Y and τ ˜ ≥ 0 ( boundedness ) .

Based on the above results, we can deduce the following conclusion.

Next, we shall discuss a weaker criteria compared to ( λ ˜ 2 ) , which can be utilized to establish the existence of a unique solution to equation (1.9).

1. The mappings υ k : Y → Y with υ 1 ( 0 ) = 0 = υ 2 ( 0 ) , for k = 1 , 2 , 3 , 4 , 5 , satisfy the following properties:

   (2.16) d ( υ k μ , υ k ν ) ≤ β k d ( μ , ν ) , ∀ μ , ν ∈ Y and β k < 1 ( contraction ) ,

   with β 1 ≤ β 2 ≤ β 3 ≤ β 4 ≤ β 5 , and

   (2.17) ∣ υ k ( x ) ∣ ≤ ω ˜ , ∀ x ∈ Y and ω ˜ ≥ 0 ( boundedness ) .