Equilibria in Altruistic Economic Growth Models

In this paper, we consider a stochastic economic growth model in the form of an intergenerational dynamic game. Both paternalistic and non-paternalistic components are present in the model. Under very general assumptions allowing for unbounded utility functions and weakly continuous transitions, we establish the existence of Markov perfect equilibria that consist of a consumption strategy and an indirect utility function. In the pure paternalistic case, we obtain new results on equilibria. An important point of our contribution is that we make no separability assumptions on the utility functions of generations.


Introduction
The literature concerning various aspects of altruism in economic models is pretty large. For a comprehensive survey with historical notes, the reader is referred to [9,21,33,35] and their references. This paper is devoted to study some mathematical issues related to existence of equilibria in a large class of economic growth models with altruism between generations. Intergenerational altruism has usually been modeled in two ways. In the paternalistic model, the utility of the current generation depends on its own consumption and the consumptions of other generations. In other words, the generation cares about what all or some successors will consume, but it does not take into account the utilities the successors derive from the act of consumption. In the non-paternalistic model, each generation derives utility from its own consumption and the utilities of future generations.
First strong results on the existence of Markov perfect equilibria in paternalistic economic growth models with deterministic transitions were established by Bernheim and Ray [11] and Leininger [28]. They assume that each generation cares only about consumption of its immediate successor. From the mathematical point of view, their proofs are rather complicated. A simpler and more direct method was used by Balbus et al. [3]. Certain extensions of the works of [11,28] to models with specific stochastic production functions are surveyed in [8,9,25]. However, more general results the reader may find in [4,5], where the transition probability function obeys a natural weak continuity condition. Such transition probabilities are extensively used in economics, since economic dynamics are typically described by some difference equations with additive or multiplicative shocks. The most general paternalistic model was considered by Balbus et al. [5]. However, their results concern non-atomic transition probabilities. The model studied in [4] allows to cover both deterministic production functions and the stochastic transitions that satisfy a stochastic dominance assumption. The drawback of this approach lies in the compactness of the state space and in the separability of the utility function which takes into account only an immediate descendant for each generation.
In contrast to the paternalistic case, there have been very few rigorous studies of models assuming the non-paternalistic altruism. As suggested by Ray [35], "this framework appears to be of somewhat greater interest in the context of applications." Examples include the works of [10,29]. For instance, Ray [35] described a general model with deterministic transitions involving non-paternalistic altruism and formulated an equilibrium concept, but its existence remains an open problem. Balbus et al. [6] showed that an equilibrium in a stochastic version of Ray's framework exists provided that the transition probabilities are non-atomic. The other group of models were considered by Barro [10] or Loury [29], who dealt with only one descendant for each generation. However, the existence of an equilibrium consisting of an indirect utility and an optimal consumption (or saving) strategy in the aforementioned models can be studied by dynamic programming methods using contraction mapping theorems. It is worth mentioning that these methods were also applied to the wide class of various decision processes with recursive utilities, see for instance [17,18].
In this paper, we study a version of so-called mixed models with both paternalistic and non-paternalistic components. A need for studying mixed models is expressed on page 113 in [35]. An approach to mixed models (two-sided altruism) is given by Hori [23]. He considers a rather specific model with pretty strong assumptions on the utility and deterministic transition functions. An equilibrium is shown to exist in this model by the Schauder fixed-point theorem.
Although the approach of [23] concerns some specific model, it is inspiring for us.
In our model, we assume that every generation considers only its immediate successor. The equilibrium problem studied in this paper is a double fixed-point problem. One fixed point is obtained for an indirect utility function via a contraction mapping with a nonlinear discount function. This is a sort of recursive utility extensively discussed in economics, see Becker and Boyd [15]. The second one (in an appropriate strategy space) corresponds to Nash equilibrium in an intergenerational game. Our basic tool is the Schauder-Tychonoff fixed-point theorem (see e.g., Dugundji and Granas [16]). We would like to emphasize that the indirect utility that we consider depends on both consumption and endowment of the generation. This approach is generalized in the model of [35]. A similar mixed model is studied in the recent paper of [2]. However, he considers risk-sensitive generations (uses different utility functions), makes much stronger assumptions on the stochastic transition function and proves existence of equilibria in the class of Borel measurable randomized strategies. Therefore, his approach is essentially different from the one applied in this paper. Our proofs are based on techniques used in [3,4] with some necessary modifications. We prove a general existence theorem for the mixed model with unbounded utility functions by applying the weighted norm approach, which was well developed in dynamic programming (see Wessels [40]; Jaśkiewicz et al. [27]). However, instead of standard exponential discounting, we use a nonlinear discount function. As a by-product, we considerably extend our earlier result for paternalistic models given in [4]. The most important point is that no separability condition on the utility function is imposed.
The rest of the paper is organized as follows. Section 2 contains preliminaries. Section 3 derives the utility function which incorporates the paternalistic and non-paternalistic altruism and formulates an equilibrium. Section 4 presents basic assumptions and existence theorems. Examples satisfying our conditions are given in Sect. 5. Finally, the proofs are collected in Sects. 6 and 7.

Preliminaries
In this section, we introduce some notation and state a few auxiliary results. As usual, R stands for the set of all real numbers and N is the set of all positive integers. Let S = R + = [0, ∞), Let X be the vector space of all continuous from the left functions φ : S → R such that φ(0) = 0 and that the restriction of φ to any bounded interval [0, m] (m ∈ N) has a bounded variation. We assume that X is endowed with the topology of weak convergence. Recall that a sequence (φ n ) converges weakly to φ ∈ X if and only if φ n (s) → φ(s) as n → ∞ at any continuity point s ∈ S of φ. Here, we point out that s = 0 is considered as a continuity Observe that I ⊂ X . Moreover, s = 0 is the continuity point of every function in F or I . Lemma 1 X is a locally convex topological vector space. The sets I and F are convex and sequentially compact in X .
For a more detailed discussion, consult Lemma 1 and Appendix in [7]. In the sequel, we shall use a generalized version of the contraction mapping principle due to [31], see also Theorem 5.2 in [16]. Proposition 1 Let (Y , ρ) be a complete metric space and δ : S → S be a continuous function such that δ(0) = 0 and δ(r ) < r for all r ∈ S + . Assume that T : Y → Y is a mapping such that ρ(T x, T y) ≤ δ(ρ(x, y)) for all x, y ∈ Y , and define T 1 x := T x, T m+1 x := T T m x, x ∈ Y , m ∈ N. Then, T has a unique fixed point x * ∈ Y and lim n→∞ ρ(T n x 0 , x * ) = 0 for any x 0 ∈ Y .
Let d > 0 and η : [d, ∞) → R be a fixed function. Following Milgrom and Shannon [32], we say that η has the strict single crossing property on [d, ∞), when the following holds: if there exists some x ≥ d such that η(x) ≥ 0, then for each x > x, we have η(x ) > 0. It is worth to note that η need not be increasing, see Example 3 in [3].
Let u : S × S → R + be a function of the form u(a, w) = g(u o (a, w)). We make the following assumptions.

Remark 1
The form of u(a, w) = g(u o (a, w)) is very convenient for verification of our assumptions. From them, it follows that u is nonnegative. Using this representation, one can immediately see that (U1)-(U3) hold for u(a, w) such as ln( The function u plays the role of "aggregator" in our definitions of the utility functions in altruistic growth models. Note that the aggregator of [20] The study of unbounded from below utilities u requires new methods and seem to be difficult to handle in this setup. [11] in their study of an altruistic growth model did not assume that u is the composition of u o and g. They directly imposed conditions on u: u is strictly concave in its first argument and u satisfies the increasing difference property (ID). (ID) says that for each w 1 > w 2 in S, the function x → u(x, w 1 ) − u(x, w 2 ) is non-decreasing. The function u that satisfies (ID) is called supermodular. Supermodular functions turn out to be useful in operations research and game theory, see Topkis [38,39]. Balbus et al. [3], on the other hand, assumed that u meets the strict single crossing property. On page 517 in [3], it is also shown that if u satisfies (ID) and is strictly concave in the first argument, then u possesses the strict single crossing property. Observe that the functions ln (

Remark 2 Bernheim and Ray
. However, they are compositions of an increasing function g with u o satisfying (ID). Below, we give an example of u = g • u o where g is increasing and u o satisfies condition (U3) and does not have the (ID) property. This example also illustrates how (U3) can be checked directly using the definition of u o . Example 1 Let u(a, w) := (w + ln(a + w)) σ where σ ∈ (0, 1). Clearly, g(z) = z σ is increasing and continuous and u o (a, w) = w + ln(a + w). The function u o does not satisfy (ID), since ∂ 2 u o ∂a∂w < 0. For any w 2 > w 1 in S, l > 0 we have If

Markov Perfect Equilibria in Altruistic Growth Economies
Consider an infinite sequence of generations labeled by t ∈ T = N. There is one commodity, which may be consumed or invested. Every generation lives one period and in the paternalistic case derives utility from its own consumption and consumption of its immediate descendant.
In the non-paternalistic case generation, t ∈ T takes into account a utility for consumption of generation t + 1. In this paper, we are interested in mixed model where both paternalistic and non-paternalistic components are present. Generation t ∈ T receives the endowment s t ∈ S and chooses consumption level The investment i t := s t − a t determines the endowment of its successor according to some transition probability q from S to S, which depends on The set of all strategies for each generation is denoted by Π.
Let v : S → R + be a continuous increasing function such that v(0) = 0. Assume that generation t ∈ T consumes a ∈ A(s t ) in state s t = s and the following generation is going to use a strategy c t+1 = c ∈ Π. Then, the term is a generation t's evaluation of consumption policy c of generation t + 1 under investment i = s − a. Let U (c )(s ) denote the (Borel measurable in s ) utility for generation t + 1 resulting from its consumption policy c in state s ∈ S. This utility can also be evaluated by generation t under investment i = s − a by computing the expected value with respect to the probability measure q(·|i). More formally, generation t can consider Assume that E i v(c ) and E i U (c ) are aggregated with the aid of the function W : is calculated. Then, the aggregated utility for generation t is obtained by aggregating a ∈ A(s) and w by the function u discussed in Preliminaries. More precisely, the utility of generation t under investment i = s − a is defined as Similarly as in [35] or [29], we can call U an indirect utility for generation t ∈ T . However, one should note that indirect utilities in their approaches are functions depending on endowments only.
This clearly shows that the utility of generation t depends on its own consumption in state s t , the expectation of its own evaluation v of consumption of generation t + 1 (paternalistic altruism component) and the utility U of consumption of generation t + 1 (non-paternalistic altruism component). In the sequel, we impose additional assumptions on functions v and W and the transition probability q to cover an unbounded case.
P(a, c * , U * (c * ))(s) for every s ∈ S. (4) Note that in (4), we deal with a double fixed-point problem. The strategy c * is the best response for every generation t, if its immediate successor is going to use c * , and each generation evaluates its consumption strategy c * using the same function U * . Following Ray [35], one can say that it is assumed in Definition 1 that "there exist an indirect utility function and a consumption strategy (policy), both depending on current endowment, such that each generation finds it optimal to adopt that consumption strategy, provided its immediate descendant uses the same policy and exhibit the given indirect utility. Moreover, the indirect utility function generated by the generations maximization problem is also the same as that announced by its descendant." In the pure paternalistic case, the utility for generation t is of simpler form The analogous form to (2) isP where a ∈ A(s) is a consumption of generation t, i = s − a is its investment in state s ∈ S and c is a consumption strategy of generation t + 1. The definition of equilibrium is similar to that given in [11,28,34] or [3,4]. (2) and (5), we can rewrite Eqs. (4) and (6) in a more convenient forms for our proofs. Namely, observe that

Remark 4
If W (b, r ) = r , then the utility in (3) reduces to a pure non-paternalistic case, which is of the form This model was studied by Loury [29] with a deterministic transition function and by Ray [35], who considered countably many descendants for each generation. An equilibrium existence in the model of [35] is still an open problem. In the stochastic case with non-atomic transition, the existence of such equilibrium was established by Balbus et al. [6]. Moreover, we would like to point out that in this pure non-paternalistic case with utility (7), the fixed point U * is independent of c ∈ F. This follows from the fact that the fixed point belongs to the subspace of functions considered here. The consumption strategy c * ∈ F in Definition 1, on the other hand, can be found by dynamic programming technique, see Loury [29], Durán [17,18]. Our results within this special framework can be considered as an extension of the results of [18] to models with nonlinear discount function.
In the general case, c * from Definitions 1 and 2 is a solution to a non-cooperative game problem. This is a symmetric Nash equilibrium in a game played by generations. Further comments on this issue, the reader may find in [25].

Basic Assumptions and Main Results
Let Pr(S) be the set of all probability measures on the state space S. We recall that a sequence (μ n ) of probability measures on S converges weakly to some μ 0 ∈ Pr(S) (μ n ⇒ μ 0 in short) if, for any bounded continuous function h : S → R, it holds that We already made three assumptions (U1)-(U3) on the aggregator u. Below, we provide additional conditions on the primitive data that will be imposed in our two main results. To include unbounded from above utilities, we shall apply a weighted norm approach inspired by the papers in dynamic programming (see Wessels [40]; Hernández-Lerma and Lasserre [22]; Jaśkiewicz and Nowak [24]) or recursive utility theory (see Boyd [14]; Durán [17,18]).
Let ω : S → [1, ∞) be a continuous non-decreasing function. Further, ω will be called a weight function. We now make some basic assumptions on the transition probability.
(Q1) Assume that λ j : S → [0, 1], j ∈ J := {1, . . . , N }, are continuous functions such that N j=1 λ j (i) = 1 for all i ∈ S. In addition, suppose that there exist transition probabilities q j from S to S, j ∈ J , such that for each i ∈ S, we have Moreover, for every j ∈ J , q j ({0}|0) = 1, and the transition probability q j (·|i) has the Feller property, i.e., if i n → i 0 in S as n → ∞, then q(·|i n ) ⇒ q(·|i 0 ). (Q2) Every transition probability q j (·|y) in (8)  As in preliminaries, we make the following assumption on the discount function δ.
We can now continue our assumptions on the utility function. Here, v ω is defined as sup x∈S v(x)/ω(x) and is assumed to be finite.
For any f : F × S → R we define Let C(F × S) be the Banach space of all continuous functions f : We can now state our main results.

Remark 5
It should be noted that U * in Theorem 1 is unique. However, in a general case the uniqueness of c * is not guaranteed. This issue for similar models is also considered in [8,30].
In the paternalistic case, we can drop the assumptions involving the discount function δ and weight function ω. (c) Assumption (D) was used in dynamic programming models studied in [27].

Remark 7
The classes of strategies F and I were used to study bequest equilibria by Bernheim and Ray [11] and Leininger [28]. Their proofs are rather complicated. A simpler method was proposed in [3,4]. Moreover, this class of semicontinuous strategies is also useful in dynamic game models of resource extraction, see Sundaram [37], Dutta and Sundaram [19], Jaśkiewicz and Nowak [26].
Remark 8 Theorem 1 is new, even in the pure non-paternalistic case (see Remark 4). Theorem 2, on the other hand, considerably extends the work of [4] (in the risk-neutral case), where the state space S is a compact interval and thus, the utility function u is bounded. More importantly, Balbus et al. [4] assume thatP(a, c)(s) =û(a) + S v(c(s ))q(ds |s − a), i.e., the utility is separable. Hence, these results generalize ones of [4] in the two aforementioned directions. We wish to emphasize that certain techniques and ideas used in [3,4] are useful in our setup, but after a suitable adaptation.

Examples
In this section, we give four examples. The first example is inspired by a two-sided altruism model of [23]. The remaining three examples provide some specific functions in the altruistic growth economies for which the imposed conditions are satisfied. Examples 3 and 4 refer to the assumptions used in Theorem 1, whereas Example 5 illustrates conditions used in Theorem 2.
Example 2 Assume that each generation t ∈ T consists of two populations: old and young members. The young population in generation t becomes old in generation t + 1. The utility functions for consumptions are: u 1 for the young population and u 2 for the old population.
Supposing that the members of generation t cooperate, we can maximize u 1 ( where a t ∈ A(s t ) is the total consumption of generation t in state s t ∈ S. Assuming that the functions u 1 and u 2 are increasing and strictly concave, one can show that there exist continuous increasing functions ψ 1 and ψ 2 on S such that By Lemma 1 in [23],û is strictly concave and increasing. Let c t+1 be a consumption strategy of generation t + 1 and U (c t+1 )(s t+1 ) be a chosen (announced) utility function by generation t + 1 depending on both c t+1 and s t+1 . Then, the aggregated utility for generation t iŝ where β ∈ (0, 1). Thus, the utility of generation t is the sum of the utility resulting from cooperation of populations in period t and the weighted sum of the utility from consumption of old in period t + 1 (who were young in period t) and the expected utility U announced by generation t + 1 calculated for the strategy c t+1 and endowment s t+1 .

Example 3 Let the transition probability be of the following form
Hence, we have κ 0 = 1. Furthermore, we define the function δ satisfying (D) as ∈ (0, 1), z ∈ S, and the function in (U4) as v(s) = 2 √ s, s ∈ S. Clearly, v ω = 1. Finally, we put W (b, r ) = δ(b + r ), (b, r ) ∈ S × S and u(a, w) = ln(1 + √ a + w), (a, w) ∈ S × S (see Remark 1). Then, we have u (a, W (b, r ) Condition (U8) follows from the mean value theorem, because The last inequality is due to this fact It remains to check condition (U7) and find the constant κ 1 . For every s ∈ S we obtain Thus, it suffices to take κ 1 = 3/2.

Example 5
In this example, we assume that the next state evolves according to the following recursive equation where > 0 is a fixed interest rate and (ζ t ) is a sequence of i.i.d. random shocks with values in [0, ∞) and with a distribution π such that S ζ π(dζ ) < ∞. Thus, Note that (Q1)-(Q3) are satisfied, if 0 is not an atom of π. In the paternalistic model, we may define u(a, w) = (1/4 √ a + 3/4 √ w) 2 for (a, w) ∈ S × S and v(s) = √ s for s ∈ S. Note that the function is continuous. Hence, all assumptions in Theorem 2 are satisfied.

Basic Monotonicity Result
In this section, we provide a useful result that may have applications to various models in optimization. Let ξ : S → S be an upper semicontinuous function. For any s ∈ S, define Proof Observe that since g is continuous and increasing, we conclude from (10) that Therefore, a (a) First note that ϕ(0) = 0 ≤ ϕ(s) for each s ∈ S. Assume that there exist s 1 , s 2 ∈ S such that 0 < s 1 < s 2 and ϕ(s 1 ) > ϕ(s 2 ). Let S 1 = [s 1 , ∞) and A 1 = [0, ϕ(s 1 )]. Note that ϕ(s 1 ) > 0. Choose any i 1 , i 2 ∈ A 1 such that i 1 > i 2 and define If ξ(i 1 ) ≤ ξ(i 2 ), then under our monotonicity assumptions on u o , we have η(s) < 0 for all s ∈ S 1 , and η has the strict single crossing property. If ξ(i 1 ) > ξ(i 2 ), then we write By (U3), this function has a strict single crossing property. Let ϕ be any function defined on S 1 such that ϕ (s) ∈ arg max i∈A 1 u o (s − i, ξ(i)) for all s ∈ S 1 and ϕ (s 1 ) = ϕ(s 1 ) and ϕ (s 2 ) = ϕ(s 2 ). By Theorem 4' in [32], ϕ is non-decreasing. Hence, it follows that ϕ(s 1 ) ≤ ϕ(s 2 ). We have come to a contradiction, which finishes the proof of part (a).
(b) Let s 0 > 0. Assume that s n ↑ s 0 as n → ∞. Choose any i ∈ [0, s 0 ). Then, we get for all but finitely many n ∈ N. Since φ is non-decreasing, the limit i 0 := lim n→∞ φ(s n ) exists and i 0 ∈ A(s 0 ). By assumption, the function ξ is continuous from the left. Thus, we have Since ξ is continuous from the left, (11) also holds for i = s 0 . Hence, , then there exists some n ∈ N such that s n < s 0 and φ(s n ) > φ(s 0 ). This inequality contradicts the fact that φ is non-decreasing.
The following auxiliary result is a simple modification of Lemma 3 in [4].

Proofs of Theorems 1 and 2
In this section, we assume that assumptions used in Theorems 1 and 2 are satisfied, although they are not explicitly recalled.  [12] yields that the functions (c, i) → S f (c, s )q(ds |i) and (c, i) → S f ω (c, s )q(ds |i) are lower semicontinuous. Since i → S ω(s )q(ds |i) is continuous, it follows that (c, i) → S f (c, s )q(ds |i) is simultaneously lower and upper semicontinuous. Lemma 4 For any c ∈ F, the function i → S v(c(s ))q(ds |i) is upper semicontinuous.

Proof
Since v is continuous and increasing and c ∈ F, the function v(c(·)) is upper semicontinuous. Hence, ω(s) v ω −v(c(s)) is nonnegative and lower semicontinuous. By Proposition 7.31 in [12], the function i → S (ω(s ) v ω − v(c(s )))q(ds |i) is lower semicontinuous. Since → S ω(s )q(ds |i) is continuous, the assertion follows.

Lemma 5 For any c ∈ F, the function i → S v(c(s ))q(ds |i) is continuous from the left.
Proof Consider any q j from the representation (Q1) of q. Then, recall that q j has the Feller property and the stochastic dominance property. Using the Skorohod representation theorem for weak convergence of probability measures (see Billingsley [13]), continuity from the left of v(c(·)) and the dominated convergence theorem as in the proof of Lemma 6 in [4], one can conclude that the function i → S v(c(s ))q j (ds |i) is continuous from the left. Since all functions λ j in the representation of q in (Q1) are continuous, it follows the assertion of our lemma.
Let  v(c n (z)) ≤ v(c(s )). (14) Proof Let (z n ) be arbitrary sequence in S converging to s . By Lemma 4 in [4], we obtain Thus, (14) follows by taking supremum over all sequences (z n ) converging to s on the left-hand side of (15).
Let V be the closed subset of all nonnegative functions f in the Banach space

Proof
Step 1. First note that from q({0}|0) = 1, u(0, 0) = 0 and W (0, 0) = 0, it follows that (T f )(c, 0) = 0. Assume that c n * → c 0 in F and s n → s 0 in S as n → ∞. For each n ∈ N, choose any i n ∈ A o (c n )(s n ). Without loss of generality, it can be assumed that i n → i 0 ∈ A(s 0 ) as n → ∞. By Lemma 3, E i n f (c n ) → E i 0 f (c 0 ) as n → ∞. From Lemmas 6 and 7, it follows that lim sup Therefore, by our monotonicity and continuity assumptions on u and W , we infer that Observe now that if s 0 = 0, then Let C(c 0 ) be the set of all continuity points of the function c 0 . Clearly, 0 ∈ C(c 0 ) and S\C(c 0 ) is countable. Assume that s 0 > 0 and define D := {i ∈ [0, s 0 ) : q(C(c 0 )|i) = 1}. By (Q3), D is a dense subset of [0, s 0 ). Choose any i ∈ D. Then, i ≤ s n for all but finitely many n ∈ N. Under our conditions on v and ω, the sequence (v(c n (·)) satisfies the assumptions of the dominated convergence theorem and it converges to v(c 0 (·)) q(·|i) − a.e. Therefore, we have that lim n→∞ E i v(c n ) = E i v(c 0 ). By Lemma 3, we also have lim Proof of Theorem 1 If κ 0 ≤ 1, then, by Lemma 9, T is a δ-contraction mapping from V into itself. If κ 0 > 1, then we define δ 0 (r ) = δ(κ 0 r ). By (D) and (25), T is a δ 0 -contraction mapping. By Proposition 1, there exists a unique U * ∈ V such that T U * = U * . → y * 0 as n → ∞. Let C(y * 0 ) be the set of all continuity points of the function y * 0 . Since y * 0 ∈ I , S\C(y * 0 ) is countable, the set C(y * 0 ) is dense in S. Choose any s ∈ C(y * 0 ). From Step 1 of the proof of Lemma 8 (with i n = y * n (s), i 0 = y * 0 (s) and s n = s), we can conclude that y * 0 (s) ∈ A * o (c 0 )(s), for any s ∈ C(y * 0 ). Since y * 0 is continuous at s, by Lemma 2, A * o (c 0 )(s) is a singleton. Therefore, y * (s) = a * o (c 0 )(s) for all s ∈ C(y * 0 ). As both functions y * 0 and c 0 are left continuous on S and C(y * 0 ) is dense in S, we conclude that y * 0 (s) = a * o (c 0 )(s) for all s ∈ S. Thus, we have shown that the mapping a * o : F → I is continuous. Let Ψ (c)(s) := s − a * o (c)(s), s ∈ S, c ∈ F. Obviously, Ψ : F → F is also a continuous mapping. Moreover, we know from Lemma 1 that F is a compact convex subset of a locally convex topological vector space. By the Schauder-Tychonoff fixed-point theorem (see Aliprantis and Border [1] or Dugundji and Granas [16]), there exists some c * ∈ F such that Ψ (c * ) = c * . Clearly, (c * , U * ) is an SM P E.

Remark 9
We have shown that U * is a fixed point of the mapping T defined in (17). By Proposition 1, U * is unique. Moreover, if f 0 ≡ 0 and f 0 ∈ V , then lim n→∞ T n f 0 −U * ω = 0.
In the remaining part of this section, we consider only assumptions of Theorem 2. By replacing the function ω with v in Lemmas 4 and 5 and using condition (U5), one can prove that the mapping Proof of Theorem 2 A simple adaptation of the arguments used in the proof of Theorem 1 yields that the mappingã * o : F → I is continuous in the weak topology on F and I . Therefore, the mappingΨ (c)(s) := s −ã * o (c)(s) from F into F is continuous as well. By the Schauder-Tychonoff fixed-point theorem, there exists some c * ∈ F such thatΨ (c * ) = c * . Clearly, c * is an SM P E.
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